Subgroups — C₆₇×#60241 #30739492

Order 17152.

2130 conjugacy classes of subgroups.

The subgroup lattice has 858 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2981	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2979	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2978	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2976	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2975	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2974	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2933	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2930	C₂	1	2	yes	maximal, complemented
13	C₂×#436×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2925	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₇×#2880	C₂	1	2	yes	maximal, complemented
17	C₆₇×#2879	C₂	1	2	yes	maximal, complemented
18	C₂²×#41×C₆₇	C₂²	1	2	yes	—
19	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
21	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
22	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
23	#550×C₆₇	C₄	1	1	yes	characteristic
24	D₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
25	#548×C₆₇	C₄	1	2	yes	—
26	#547×C₆₇	C₄	1	2	yes	—
27	#547×C₆₇	C₄	1	1	yes	characteristic
28	#546×C₆₇	C₄	1	4	yes	—
29	#546×C₆₇	C₄	1	2	yes	—
30	D₄²×C₆₇	C₄	1	1	yes	characteristic
31	D₄²×C₆₇	C₄	1	2	yes	—
32	#532×C₆₇	C₂²	1	1	yes	characteristic
33	#532×C₆₇	C₂²	1	2	yes	—
34	#525×C₆₇	C₂²	1	2	yes	—
35	#525×C₆₇	C₂²	1	4	yes	—
36	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
39	#436×C₆₇	C₂²	1	1	yes	characteristic
40	#436×C₆₇	C₂²	1	1	yes	characteristic
41	#436×C₆₇	C₂²	1	2	yes	—
42	#436×C₆₇	C₂²	1	2	yes	complemented
43	#436×C₆₇	C₂²	1	2	yes	complemented
44	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
45	#436×C₆₇	C₂²	1	1	yes	characteristic, complemented
46	#436×C₆₇	C₂²	1	4	yes	complemented
47	#436×C₆₇	C₂²	1	4	yes	complemented
48	#436×C₆₇	C₂²	1	2	yes	complemented
49	#436×C₆₇	C₂²	1	2	yes	complemented
50	#436×C₆₇	C₂²	1	4	yes	complemented
51	#436×C₆₇	C₂²	1	2	yes	complemented
52	#435×C₆₇	C₂²	1	4	yes	complemented
53	#435×C₆₇	C₂²	1	4	yes	complemented
54	#435×C₆₇	C₂²	1	4	yes	complemented
55	#435×C₆₇	C₂²	1	4	yes	complemented
56	#435×C₆₇	C₂²	1	4	yes	complemented
57	#435×C₆₇	C₂²	1	4	yes	complemented
58	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
59	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
60	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
61	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
62	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
63	#433×C₆₇	C₂²	1	2	yes	complemented
64	#433×C₆₇	C₂²	1	2	yes	complemented
65	#433×C₆₇	C₂²	1	4	yes	complemented
66	#432×C₆₇	C₂²	1	2	yes	complemented
67	#431×C₆₇	C₂²	1	2	yes	complemented
68	#431×C₆₇	C₂²	1	2	yes	complemented
69	#431×C₆₇	C₂²	1	2	yes	complemented
70	#424×C₆₇	C₂²	1	1	yes	characteristic, complemented
71	#424×C₆₇	C₂²	1	1	yes	characteristic, complemented
72	#423×C₆₇	C₂²	1	1	yes	characteristic, complemented
73	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
74	#408×C₆₇	C₂²	1	1	yes	characteristic
75	#408×C₆₇	C₂²	1	2	yes	—
76	#408×C₆₇	C₂²	1	1	yes	characteristic
77	#408×C₆₇	C₂²	1	2	yes	complemented
78	#408×C₆₇	C₂²	1	2	yes	complemented
79	#408×C₆₇	C₂²	1	4	yes	complemented
80	#408×C₆₇	C₂²	1	4	yes	complemented
81	#408×C₆₇	C₂²	1	4	yes	complemented
82	#408×C₆₇	C₂²	1	4	yes	complemented
83	#407×C₆₇	C₂²	1	4	yes	complemented
84	#407×C₆₇	C₂²	1	4	yes	complemented
85	#407×C₆₇	C₂²	1	2	yes	complemented
86	#407×C₆₇	C₂²	1	4	yes	complemented
87	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
88	C₂×C₂²⋊C₈×C₆₇	C₂²	1	4	yes	complemented
89	#405×C₆₇	C₂²	1	2	yes	complemented
90	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
91	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
92	C₂×#41×C₆₇	—	4	4	no	—
93	C₂×#41×C₆₇	—	4	4	no	—
94	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
95	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
96	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
97	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
98	C₂×#41×C₆₇	C₂³	1	2	yes	—
99	C₂×#41×C₆₇	C₂³	1	4	yes	—
100	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
101	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
103	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
104	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
105	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
106	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
107	#131×C₆₇	—	4	4	no	—
108	#131×C₆₇	—	4	4	no	—
109	C₂×M₁₆×C₆₇	—	4	4	no	—
110	C₂×M₁₆×C₆₇	—	4	4	no	—
111	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
113	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
114	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
115	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
116	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
117	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
118	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
119	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
120	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
121	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
122	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
123	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
124	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
125	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
126	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
127	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
128	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
129	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
130	#127×C₆₇	C₂×C₄	1	2	yes	—
131	#124×C₆₇	C₂×C₄	1	2	yes	—
132	#124×C₆₇	C₂×C₄	1	4	yes	—
133	#123×C₆₇	C₂×C₄	1	4	yes	—
134	#123×C₆₇	C₂×C₄	1	4	yes	—
135	#123×C₆₇	C₂×C₄	1	4	yes	—
136	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
137	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
138	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
139	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
140	#121×C₆₇	C₂×C₄	1	2	yes	—
141	#121×C₆₇	C₂×C₄	1	4	yes	—
142	#121×C₆₇	C₂×C₄	1	2	yes	—
143	#121×C₆₇	C₂×C₄	1	4	yes	—
144	#121×C₆₇	C₂×C₄	1	2	yes	—
145	#121×C₆₇	C₂×C₄	1	2	yes	—
146	#121×C₆₇	C₂×C₄	1	2	yes	—
147	#121×C₆₇	C₂×C₄	1	2	yes	—
148	#121×C₆₇	C₂×C₄	1	4	yes	—
149	#120×C₆₇	C₂×C₄	1	4	yes	—
150	#120×C₆₇	C₂×C₄	1	4	yes	—
151	#120×C₆₇	C₂×C₄	1	4	yes	—
152	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
153	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
154	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
155	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
156	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
157	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
158	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
159	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
160	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
161	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
162	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
163	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
164	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
165	C₄×D₄×C₆₇	C₂³	1	2	yes	—
166	C₄×D₄×C₆₇	C₂³	1	4	yes	—
167	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
168	C₄×D₄×C₆₇	C₂³	1	2	yes	—
169	C₄×D₄×C₆₇	C₂³	1	4	yes	—
170	C₄×D₄×C₆₇	C₂³	1	2	yes	—
171	#117×C₆₇	C₂³	1	1	yes	characteristic
172	#117×C₆₇	C₂³	1	2	yes	—
173	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
174	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
175	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
176	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
177	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
178	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
179	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
180	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
181	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
182	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
183	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
184	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
185	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
186	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
187	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
188	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
189	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
190	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
191	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
192	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
193	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
194	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
195	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
196	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
197	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
198	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
199	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
200	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
201	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
202	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
203	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
204	#97×C₆₇	C₂³	1	2	yes	—
205	#97×C₆₇	C₂³	1	4	yes	—
206	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
207	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
208	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
209	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
210	C₄×C₈×C₆₇	C₂³	1	2	yes	—
211	C₄×C₈×C₆₇	C₂³	1	4	yes	—
212	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
213	C₂⁴×C₆₇	C₂²×C₄	1	4	yes	—
214	#41×C₆₇	—	2	4	no	—
215	#41×C₆₇	—	2	4	no	—
216	#41×C₆₇	—	2	2	no	—
217	#41×C₆₇	—	2	2	no	—
218	#41×C₆₇	—	4	4	no	—
219	#41×C₆₇	—	4	4	no	—
220	#41×C₆₇	—	4	8	no	—
221	#41×C₆₇	—	4	4	no	—
222	#41×C₆₇	—	4	4	no	—
223	C₂×Q₈×C₆₇	—	4	4	no	—
224	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
225	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
226	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
227	C₂×D₄×C₆₇	—	4	2	no	—
228	C₂×D₄×C₆₇	—	4	4	no	—
229	C₂×D₄×C₆₇	—	4	4	no	—
230	C₂×D₄×C₆₇	—	4	8	no	—
231	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
232	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
233	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
234	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
235	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
236	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
237	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
238	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
239	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
240	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
241	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
242	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
243	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
244	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
245	C₂²×C₄×C₆₇	—	2	4	no	—
246	C₂²×C₄×C₆₇	—	2	4	no	—
247	C₂²×C₄×C₆₇	—	4	4	no	—
248	C₂²×C₄×C₆₇	—	4	4	no	—
249	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
250	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
251	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
252	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
253	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
254	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
255	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
256	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
257	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
258	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
259	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
260	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
261	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
262	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
263	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
264	M₁₆×C₆₇	—	2	2	no	—
265	M₁₆×C₆₇	—	2	4	no	—
266	M₁₆×C₆₇	—	2	2	no	—
267	M₁₆×C₆₇	—	4	2	no	—
268	M₁₆×C₆₇	—	4	2	no	—
269	M₁₆×C₆₇	—	4	2	no	—
270	M₁₆×C₆₇	—	4	2	no	—
271	M₁₆×C₆₇	—	4	4	no	—
272	M₁₆×C₆₇	—	4	4	no	—
273	M₁₆×C₆₇	—	4	4	no	—
274	C₂×C₈×C₆₇	—	2	2	no	—
275	C₂×C₈×C₆₇	—	2	4	no	—
276	C₂×C₈×C₆₇	—	2	2	no	—
277	C₂×C₈×C₆₇	—	4	4	no	—
278	C₂×C₈×C₆₇	—	4	4	no	—
279	C₂×C₈×C₆₇	—	4	4	no	—
280	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
281	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
282	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
283	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
284	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
285	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
286	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
287	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
288	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
289	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
290	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
291	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
292	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
293	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
294	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
295	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
296	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
297	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
298	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
299	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
300	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
301	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
302	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
303	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
304	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
305	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
306	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
307	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
308	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
309	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
310	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
311	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
312	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
313	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
314	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
315	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
316	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
317	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
318	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
319	C₄²×C₆₇	C₂⁴	1	2	yes	—
320	C₂³×C₆₇	—	2	4	no	—
321	C₂³×C₆₇	—	4	4	no	—
322	C₂³×C₆₇	—	4	2	no	—
323	C₂³×C₆₇	—	4	4	no	—
324	C₂³×C₆₇	—	4	2	no	—
325	C₂³×C₆₇	—	4	4	no	—
326	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
327	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
328	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
329	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
330	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
331	Q₈×C₆₇	—	2	2	no	—
332	Q₈×C₆₇	—	2	4	no	—
333	Q₈×C₆₇	—	4	4	no	—
334	D₄×C₆₇	—	2	4	no	—
335	D₄×C₆₇	—	2	8	no	—
336	D₄×C₆₇	—	2	2	no	—
337	D₄×C₆₇	—	2	4	no	—
338	D₄×C₆₇	—	2	2	no	—
339	D₄×C₆₇	—	2	2	no	—
340	D₄×C₆₇	—	2	4	no	—
341	D₄×C₆₇	—	4	2	no	—
342	D₄×C₆₇	—	4	4	no	—
343	D₄×C₆₇	—	4	4	no	—
344	D₄×C₆₇	—	4	4	no	—
345	D₄×C₆₇	—	4	4	no	—
346	D₄×C₆₇	—	4	2	no	—
347	D₄×C₆₇	—	4	4	no	—
348	C₂×C₄×C₆₇	—	2	4	no	—
349	C₂×C₄×C₆₇	—	2	4	no	—
350	C₂×C₄×C₆₇	—	2	4	no	—
351	C₂×C₄×C₆₇	—	2	2	no	—
352	C₂×C₄×C₆₇	—	2	4	no	—
353	C₂×C₄×C₆₇	—	2	2	no	—
354	C₂×C₄×C₆₇	—	2	4	no	—
355	C₂×C₄×C₆₇	—	2	4	no	—
356	C₂×C₄×C₆₇	—	2	2	no	—
357	C₂×C₄×C₆₇	—	2	2	no	—
358	C₂×C₄×C₆₇	—	2	2	no	—
359	C₂×C₄×C₆₇	—	2	2	no	—
360	C₂×C₄×C₆₇	—	2	2	no	—
361	C₂×C₄×C₆₇	—	4	4	no	—
362	C₂×C₄×C₆₇	—	4	4	no	—
363	C₂×C₄×C₆₇	—	4	4	no	—
364	C₂×C₄×C₆₇	—	4	1	no	—
365	C₂×C₄×C₆₇	—	4	2	no	—
366	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
367	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
368	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
369	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
370	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
371	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
372	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
373	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
374	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
375	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
376	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
377	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
378	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
379	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
380	C₈×C₆₇	—	2	1	no	—
381	C₈×C₆₇	—	2	2	no	—
382	C₈×C₆₇	—	2	1	no	—
383	C₈×C₆₇	—	2	1	no	—
384	C₈×C₆₇	—	2	2	no	—
385	C₈×C₆₇	—	2	1	no	—
386	C₈×C₆₇	—	2	2	no	—
387	C₈×C₆₇	—	2	4	no	—
388	C₈×C₆₇	—	2	2	no	—
389	C₂²×C₆₇	—	2	1	no	—
390	C₂²×C₆₇	—	2	1	no	—
391	C₂²×C₆₇	—	2	2	no	—
392	C₂²×C₆₇	—	2	2	no	—
393	C₂²×C₆₇	—	2	2	no	—
394	C₂²×C₆₇	—	2	4	no	—
395	C₂²×C₆₇	—	2	2	no	—
396	C₂²×C₆₇	—	2	2	no	—
397	C₂²×C₆₇	—	2	2	no	—
398	C₂²×C₆₇	—	2	4	no	—
399	C₂²×C₆₇	—	2	2	no	—
400	C₂²×C₆₇	—	2	2	no	—
401	C₂²×C₆₇	—	2	4	no	—
402	C₂²×C₆₇	—	2	2	no	—
403	C₂²×C₆₇	—	8	4	no	—
404	C₂²×C₆₇	—	8	8	no	—
405	C₂²×C₆₇	#518	1	2	yes	—
406	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
407	C₄×C₆₇	—	2	2	no	—
408	C₄×C₆₇	—	2	2	no	—
409	C₄×C₆₇	—	2	2	no	—
410	C₄×C₆₇	—	2	4	no	—
411	C₄×C₆₇	—	2	1	no	—
412	C₄×C₆₇	—	2	2	no	—
413	C₄×C₆₇	—	2	1	no	—
414	C₄×C₆₇	—	2	1	no	—
415	C₄×C₆₇	—	2	2	no	—
416	C₄×C₆₇	#518	1	2	yes	—
417	C₄×C₆₇	#568	1	2	yes	—
418	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
419	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
420	#60241	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
421	C₂×C₆₇	—	2	2	no	—
422	C₂×C₆₇	—	4	2	no	—
423	C₂×C₆₇	—	4	4	no	—
424	C₂×C₆₇	—	4	2	no	—
425	C₂×C₆₇	—	4	2	no	—
426	C₂×C₆₇	C₂×#518	1	1	yes	characteristic
427	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
428	C₂×C₆₇	#3573	1	1	yes	characteristic
429	D₄×#41	C₂×C₆₇	1	1	yes	characteristic
430	#2981	C₂×C₆₇	1	2	yes	complemented
431	#2979	C₂×C₆₇	1	2	yes	complemented
432	#2978	C₂×C₆₇	1	4	yes	complemented
433	#2976	C₂×C₆₇	1	4	yes	complemented
434	#2975	C₂×C₆₇	1	2	yes	complemented
435	#2974	C₂×C₆₇	1	2	yes	complemented
436	#2935	C₂×C₆₇	1	1	yes	characteristic, complemented
437	D₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
438	#2933	C₂×C₆₇	1	2	yes	complemented
439	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
440	#2930	C₂×C₆₇	1	2	yes	complemented
441	C₂×#436	C₂×C₆₇	1	1	yes	characteristic, complemented
442	#2925	C₂×C₆₇	1	1	yes	characteristic, complemented
443	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
444	#2880	C₂×C₆₇	1	2	yes	complemented
445	#2879	C₂×C₆₇	1	2	yes	complemented
446	C₆₇	#60241	1	1	yes	characteristic, direct product factor, Sylow
447	C₂²×#41	C₂²×C₆₇	1	2	yes	—
448	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
449	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
450	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
451	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
452	#550	C₄×C₆₇	1	1	yes	characteristic
453	D₄×Q₈	C₄×C₆₇	1	1	yes	characteristic
454	#548	C₄×C₆₇	1	2	yes	—
455	#547	C₄×C₆₇	1	2	yes	—
456	#547	C₄×C₆₇	1	1	yes	characteristic
457	#546	C₄×C₆₇	1	4	yes	—
458	#546	C₄×C₆₇	1	2	yes	—
459	D₄²	C₄×C₆₇	1	1	yes	characteristic
460	D₄²	C₄×C₆₇	1	2	yes	—
461	#532	C₂²×C₆₇	1	1	yes	characteristic
462	#532	C₂²×C₆₇	1	2	yes	—
463	#525	C₂²×C₆₇	1	2	yes	—
464	#525	C₂²×C₆₇	1	4	yes	—
465	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
466	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
467	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
468	#436	C₂²×C₆₇	1	1	yes	characteristic
469	#436	C₂²×C₆₇	1	1	yes	characteristic
470	#436	C₂²×C₆₇	1	2	yes	—
471	#436	C₂²×C₆₇	1	2	yes	complemented
472	#436	C₂²×C₆₇	1	2	yes	complemented
473	#436	C₂²×C₆₇	1	1	yes	characteristic, complemented
474	#436	C₂²×C₆₇	1	1	yes	characteristic, complemented
475	#436	C₂²×C₆₇	1	4	yes	complemented
476	#436	C₂²×C₆₇	1	4	yes	complemented
477	#436	C₂²×C₆₇	1	2	yes	complemented
478	#436	C₂²×C₆₇	1	2	yes	complemented
479	#436	C₂²×C₆₇	1	4	yes	complemented
480	#436	C₂²×C₆₇	1	2	yes	complemented
481	#435	C₂²×C₆₇	1	4	yes	complemented
482	#435	C₂²×C₆₇	1	4	yes	complemented
483	#435	C₂²×C₆₇	1	4	yes	complemented
484	#435	C₂²×C₆₇	1	4	yes	complemented
485	#435	C₂²×C₆₇	1	4	yes	complemented
486	#435	C₂²×C₆₇	1	4	yes	complemented
487	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
488	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
489	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
490	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
491	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
492	#433	C₂²×C₆₇	1	2	yes	complemented
493	#433	C₂²×C₆₇	1	2	yes	complemented
494	#433	C₂²×C₆₇	1	4	yes	complemented
495	#432	C₂²×C₆₇	1	2	yes	complemented
496	#431	C₂²×C₆₇	1	2	yes	complemented
497	#431	C₂²×C₆₇	1	2	yes	complemented
498	#431	C₂²×C₆₇	1	2	yes	complemented
499	#424	C₂²×C₆₇	1	1	yes	characteristic, complemented
500	#424	C₂²×C₆₇	1	1	yes	characteristic, complemented
501	#423	C₂²×C₆₇	1	1	yes	characteristic, complemented
502	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
503	#408	C₂²×C₆₇	1	1	yes	characteristic
504	#408	C₂²×C₆₇	1	2	yes	—
505	#408	C₂²×C₆₇	1	1	yes	characteristic
506	#408	C₂²×C₆₇	1	2	yes	complemented
507	#408	C₂²×C₆₇	1	2	yes	complemented
508	#408	C₂²×C₆₇	1	4	yes	complemented
509	#408	C₂²×C₆₇	1	4	yes	complemented
510	#408	C₂²×C₆₇	1	4	yes	complemented
511	#408	C₂²×C₆₇	1	4	yes	complemented
512	#407	C₂²×C₆₇	1	4	yes	complemented
513	#407	C₂²×C₆₇	1	4	yes	complemented
514	#407	C₂²×C₆₇	1	2	yes	complemented
515	#407	C₂²×C₆₇	1	4	yes	complemented
516	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
517	C₂×C₂²⋊C₈	C₂²×C₆₇	1	4	yes	complemented
518	#405	C₂²×C₆₇	1	2	yes	complemented
519	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
520	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
521	C₂×#41	—	4	4	no	—
522	C₂×#41	—	4	4	no	—
523	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
524	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
525	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
526	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
527	C₂×#41	C₂³×C₆₇	1	2	yes	—
528	C₂×#41	C₂³×C₆₇	1	4	yes	—
529	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
530	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
531	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
532	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
533	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
534	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
535	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
536	#131	—	4	4	no	—
537	#131	—	4	4	no	—
538	C₂×M₁₆	—	4	4	no	—
539	C₂×M₁₆	—	4	4	no	—
540	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
541	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
542	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
543	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
544	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
545	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
546	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
547	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
548	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
549	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
550	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
551	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
552	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
553	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
554	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
555	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
556	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
557	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
558	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
559	#127	C₂×C₄×C₆₇	1	2	yes	—
560	#124	C₂×C₄×C₆₇	1	2	yes	—
561	#124	C₂×C₄×C₆₇	1	4	yes	—
562	#123	C₂×C₄×C₆₇	1	4	yes	—
563	#123	C₂×C₄×C₆₇	1	4	yes	—
564	#123	C₂×C₄×C₆₇	1	4	yes	—
565	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
566	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
567	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
568	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
569	#121	C₂×C₄×C₆₇	1	2	yes	—
570	#121	C₂×C₄×C₆₇	1	4	yes	—
571	#121	C₂×C₄×C₆₇	1	2	yes	—
572	#121	C₂×C₄×C₆₇	1	4	yes	—
573	#121	C₂×C₄×C₆₇	1	2	yes	—
574	#121	C₂×C₄×C₆₇	1	2	yes	—
575	#121	C₂×C₄×C₆₇	1	2	yes	—
576	#121	C₂×C₄×C₆₇	1	2	yes	—
577	#121	C₂×C₄×C₆₇	1	4	yes	—
578	#120	C₂×C₄×C₆₇	1	4	yes	—
579	#120	C₂×C₄×C₆₇	1	4	yes	—
580	#120	C₂×C₄×C₆₇	1	4	yes	—
581	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
582	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
583	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
584	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
585	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
586	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
587	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
588	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
589	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
590	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
591	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
592	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
593	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
594	C₄×D₄	C₂³×C₆₇	1	2	yes	—
595	C₄×D₄	C₂³×C₆₇	1	4	yes	—
596	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
597	C₄×D₄	C₂³×C₆₇	1	2	yes	—
598	C₄×D₄	C₂³×C₆₇	1	4	yes	—
599	C₄×D₄	C₂³×C₆₇	1	2	yes	—
600	#117	C₂³×C₆₇	1	1	yes	characteristic
601	#117	C₂³×C₆₇	1	2	yes	—
602	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
603	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
604	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
605	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	—
606	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
607	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
608	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
609	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
610	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
611	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
612	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
613	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
614	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
615	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
616	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
617	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
618	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
619	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
620	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
621	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
622	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
623	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
624	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
625	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
626	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
627	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
628	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
629	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
630	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
631	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
632	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
633	#97	C₂³×C₆₇	1	2	yes	—
634	#97	C₂³×C₆₇	1	4	yes	—
635	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
636	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
637	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
638	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
639	C₄×C₈	C₂³×C₆₇	1	2	yes	—
640	C₄×C₈	C₂³×C₆₇	1	4	yes	—
641	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
642	C₂⁴	C₂²×C₄×C₆₇	1	4	yes	—
643	#41	—	2	4	no	—
644	#41	—	2	4	no	—
645	#41	—	2	2	no	—
646	#41	—	2	2	no	—
647	#41	—	4	4	no	—
648	#41	—	4	4	no	—
649	#41	—	4	8	no	—
650	#41	—	4	4	no	—
651	#41	—	4	4	no	—
652	C₂×Q₈	—	4	4	no	—
653	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
654	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
655	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
656	C₂×D₄	—	4	4	no	—
657	C₂×D₄	—	4	2	no	—
658	C₂×D₄	—	4	4	no	—
659	C₂×D₄	—	4	8	no	—
660	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
661	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
662	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
663	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
664	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
665	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
666	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
667	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
668	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
669	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
670	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
671	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
672	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
673	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
674	C₂²×C₄	—	2	4	no	—
675	C₂²×C₄	—	2	4	no	—
676	C₂²×C₄	—	4	4	no	—
677	C₂²×C₄	—	4	4	no	—
678	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
679	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
680	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
681	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
682	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
683	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
684	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
685	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
686	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
687	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
688	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
689	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
690	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
691	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
692	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
693	M₁₆	—	2	2	no	—
694	M₁₆	—	2	4	no	—
695	M₁₆	—	2	2	no	—
696	M₁₆	—	4	2	no	—
697	M₁₆	—	4	2	no	—
698	M₁₆	—	4	2	no	—
699	M₁₆	—	4	2	no	—
700	M₁₆	—	4	4	no	—
701	M₁₆	—	4	4	no	—
702	M₁₆	—	4	4	no	—
703	C₂×C₈	—	2	2	no	—
704	C₂×C₈	—	2	4	no	—
705	C₂×C₈	—	2	2	no	—
706	C₂×C₈	—	4	4	no	—
707	C₂×C₈	—	4	4	no	—
708	C₂×C₈	—	4	4	no	—
709	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
710	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
711	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
712	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
713	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
714	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
715	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
716	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
717	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
718	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
719	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
720	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
721	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
722	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
723	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
724	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
725	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
726	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
727	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
728	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
729	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
730	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
731	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
732	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
733	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
734	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
735	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
736	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
737	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
738	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
739	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
740	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
741	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
742	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
743	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
744	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
745	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
746	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
747	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
748	C₄²	C₂⁴×C₆₇	1	2	yes	—
749	C₂³	—	2	4	no	—
750	C₂³	—	4	4	no	—
751	C₂³	—	4	2	no	—
752	C₂³	—	4	4	no	—
753	C₂³	—	4	2	no	—
754	C₂³	—	4	4	no	—
755	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
756	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
757	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
758	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
759	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
760	Q₈	—	2	2	no	—
761	Q₈	—	2	4	no	—
762	Q₈	—	4	4	no	—
763	D₄	—	2	4	no	—
764	D₄	—	2	8	no	—
765	D₄	—	2	2	no	—
766	D₄	—	2	4	no	—
767	D₄	—	2	2	no	—
768	D₄	—	2	2	no	—
769	D₄	—	2	4	no	—
770	D₄	—	4	4	no	—
771	D₄	—	4	4	no	—
772	D₄	—	4	4	no	—
773	D₄	—	4	2	no	—
774	D₄	—	4	4	no	—
775	D₄	—	4	2	no	—
776	D₄	—	4	4	no	—
777	C₂×C₄	—	2	4	no	—
778	C₂×C₄	—	2	4	no	—
779	C₂×C₄	—	2	4	no	—
780	C₂×C₄	—	2	2	no	—
781	C₂×C₄	—	2	4	no	—
782	C₂×C₄	—	2	2	no	—
783	C₂×C₄	—	2	4	no	—
784	C₂×C₄	—	2	4	no	—
785	C₂×C₄	—	2	2	no	—
786	C₂×C₄	—	2	2	no	—
787	C₂×C₄	—	2	2	no	—
788	C₂×C₄	—	2	2	no	—
789	C₂×C₄	—	2	2	no	—
790	C₂×C₄	—	4	4	no	—
791	C₂×C₄	—	4	4	no	—
792	C₂×C₄	—	4	4	no	—
793	C₂×C₄	—	4	1	no	—
794	C₂×C₄	—	4	2	no	—
795	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
796	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
797	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
798	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
799	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
800	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
801	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
802	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
803	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
804	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
805	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
806	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
807	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
808	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
809	C₈	—	2	1	no	—
810	C₈	—	2	2	no	—
811	C₈	—	2	1	no	—
812	C₈	—	2	1	no	—
813	C₈	—	2	2	no	—
814	C₈	—	2	1	no	—
815	C₈	—	2	2	no	—
816	C₈	—	2	4	no	—
817	C₈	—	2	2	no	—
818	C₂²	—	2	1	no	—
819	C₂²	—	2	1	no	—
820	C₂²	—	2	2	no	—
821	C₂²	—	2	2	no	—
822	C₂²	—	2	2	no	—
823	C₂²	—	2	4	no	—
824	C₂²	—	2	2	no	—
825	C₂²	—	2	2	no	—
826	C₂²	—	2	4	no	—
827	C₂²	—	2	2	no	—
828	C₂²	—	2	2	no	—
829	C₂²	—	2	2	no	—
830	C₂²	—	2	2	no	—
831	C₂²	—	2	4	no	—
832	C₂²	—	8	4	no	—
833	C₂²	—	8	8	no	—
834	C₂²	#518×C₆₇	1	2	yes	—
835	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
836	C₄	—	2	1	no	—
837	C₄	—	2	2	no	—
838	C₄	—	2	1	no	—
839	C₄	—	2	1	no	—
840	C₄	—	2	2	no	—
841	C₄	—	2	2	no	—
842	C₄	—	2	2	no	—
843	C₄	—	2	2	no	—
844	C₄	—	2	4	no	—
845	C₄	#518×C₆₇	1	2	yes	—
846	C₄	#568×C₆₇	1	2	yes	—
847	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
848	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
849	C₂	—	2	2	no	—
850	C₂	—	4	2	no	—
851	C₂	—	4	2	no	—
852	C₂	—	4	2	no	—
853	C₂	—	4	4	no	—
854	C₂	C₂×#518×C₆₇	1	1	yes	characteristic
855	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
856	C₂	C₆₇×#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂⁴×C₄
prime radical	D₄×#41×C₆₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₆₇×#2981	C₂	semidirect
3. C₆₇×#2979	C₂	semidirect
4. C₆₇×#2978	C₂	semidirect
5. C₆₇×#2976	C₂	semidirect
6. C₆₇×#2975	C₂	semidirect
7. C₆₇×#2974	C₂	semidirect
8. C₆₇×#2935	C₂	semidirect
9. D₄×M₁₆×C₆₇	C₂	semidirect
10. C₆₇×#2933	C₂	semidirect
11. C₆₇×#2931	C₂	semidirect
12. C₆₇×#2930	C₂	semidirect
13. C₂×#436×C₆₇	C₂	semidirect
14. C₆₇×#2925	C₂	semidirect
15. C₆₇×#2922	C₂	semidirect
16. C₆₇×#2880	C₂	semidirect
17. C₆₇×#2879	C₂	semidirect
19. C₂×#131×C₆₇	C₂²	semidirect
20. C₂×#131×C₆₇	C₂²	semidirect
21. C₂²×M₁₆×C₆₇	C₂²	semidirect
22. C₂²×M₁₆×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
43. #436×C₆₇	C₂²	semidirect
44. #436×C₆₇	C₂²	semidirect
45. #436×C₆₇	C₂²	semidirect
46. #436×C₆₇	C₂²	semidirect
47. #436×C₆₇	C₂²	semidirect
48. #436×C₆₇	C₂²	semidirect
49. #436×C₆₇	C₂²	semidirect
50. #436×C₆₇	C₂²	semidirect
51. #436×C₆₇	C₂²	semidirect
52. #435×C₆₇	C₂²	semidirect
53. #435×C₆₇	C₂²	semidirect
54. #435×C₆₇	C₂²	semidirect
55. #435×C₆₇	C₂²	semidirect
56. #435×C₆₇	C₂²	semidirect
57. #435×C₆₇	C₂²	semidirect
58. C₈×D₄×C₆₇	C₂²	semidirect
59. C₈×D₄×C₆₇	C₂²	semidirect
60. C₈×D₄×C₆₇	C₂²	semidirect
61. C₈×D₄×C₆₇	C₂²	semidirect
62. C₈×D₄×C₆₇	C₂²	semidirect
63. #433×C₆₇	C₂²	semidirect
64. #433×C₆₇	C₂²	semidirect
65. #433×C₆₇	C₂²	semidirect
66. #432×C₆₇	C₂²	semidirect
67. #431×C₆₇	C₂²	semidirect
68. #431×C₆₇	C₂²	semidirect
69. #431×C₆₇	C₂²	semidirect
70. #424×C₆₇	C₂²	semidirect
71. #424×C₆₇	C₂²	semidirect
72. #423×C₆₇	C₂²	semidirect
73. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
77. #408×C₆₇	C₂²	semidirect
78. #408×C₆₇	C₂²	semidirect
79. #408×C₆₇	C₂²	semidirect
80. #408×C₆₇	C₂²	semidirect
81. #408×C₆₇	C₂²	semidirect
82. #408×C₆₇	C₂²	semidirect
83. #407×C₆₇	C₂²	semidirect
84. #407×C₆₇	C₂²	semidirect
85. #407×C₆₇	C₂²	semidirect
86. #407×C₆₇	C₂²	semidirect
87. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
88. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
89. #405×C₆₇	C₂²	semidirect
90. C₄×M₁₆×C₆₇	C₂²	semidirect
91. C₂×C₄×C₈×C₆₇	C₂²	semidirect
420. #60241	C₆₇	direct (N × H)
430. #2981	C₂×C₆₇	semidirect
431. #2979	C₂×C₆₇	semidirect
432. #2978	C₂×C₆₇	semidirect
433. #2976	C₂×C₆₇	semidirect
434. #2975	C₂×C₆₇	semidirect
435. #2974	C₂×C₆₇	semidirect
436. #2935	C₂×C₆₇	semidirect
437. D₄×M₁₆	C₂×C₆₇	semidirect
438. #2933	C₂×C₆₇	semidirect
439. #2931	C₂×C₆₇	semidirect
440. #2930	C₂×C₆₇	semidirect
441. C₂×#436	C₂×C₆₇	semidirect
442. #2925	C₂×C₆₇	semidirect
443. #2922	C₂×C₆₇	semidirect
444. #2880	C₂×C₆₇	semidirect
445. #2879	C₂×C₆₇	semidirect
446. C₆₇	#60241	direct (N × H)
448. C₂×#131	C₂²×C₆₇	semidirect
449. C₂×#131	C₂²×C₆₇	semidirect
450. C₂²×M₁₆	C₂²×C₆₇	semidirect
451. C₂²×M₁₆	C₂²×C₆₇	semidirect
471. #436	C₂²×C₆₇	semidirect
472. #436	C₂²×C₆₇	semidirect
473. #436	C₂²×C₆₇	semidirect
474. #436	C₂²×C₆₇	semidirect
475. #436	C₂²×C₆₇	semidirect
476. #436	C₂²×C₆₇	semidirect
477. #436	C₂²×C₆₇	semidirect
478. #436	C₂²×C₆₇	semidirect
479. #436	C₂²×C₆₇	semidirect
480. #436	C₂²×C₆₇	semidirect
481. #435	C₂²×C₆₇	semidirect
482. #435	C₂²×C₆₇	semidirect
483. #435	C₂²×C₆₇	semidirect
484. #435	C₂²×C₆₇	semidirect
485. #435	C₂²×C₆₇	semidirect
486. #435	C₂²×C₆₇	semidirect
487. C₈×D₄	C₂²×C₆₇	semidirect
488. C₈×D₄	C₂²×C₆₇	semidirect
489. C₈×D₄	C₂²×C₆₇	semidirect
490. C₈×D₄	C₂²×C₆₇	semidirect
491. C₈×D₄	C₂²×C₆₇	semidirect
492. #433	C₂²×C₆₇	semidirect
493. #433	C₂²×C₆₇	semidirect
494. #433	C₂²×C₆₇	semidirect
495. #432	C₂²×C₆₇	semidirect
496. #431	C₂²×C₆₇	semidirect
497. #431	C₂²×C₆₇	semidirect
498. #431	C₂²×C₆₇	semidirect
499. #424	C₂²×C₆₇	semidirect
500. #424	C₂²×C₆₇	semidirect
501. #423	C₂²×C₆₇	semidirect
502. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
506. #408	C₂²×C₆₇	semidirect
507. #408	C₂²×C₆₇	semidirect
508. #408	C₂²×C₆₇	semidirect
509. #408	C₂²×C₆₇	semidirect
510. #408	C₂²×C₆₇	semidirect
511. #408	C₂²×C₆₇	semidirect
512. #407	C₂²×C₆₇	semidirect
513. #407	C₂²×C₆₇	semidirect
514. #407	C₂²×C₆₇	semidirect
515. #407	C₂²×C₆₇	semidirect
516. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
517. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
518. #405	C₂²×C₆₇	semidirect
519. C₄×M₁₆	C₂²×C₆₇	semidirect
520. C₂×C₄×C₈	C₂²×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
D₄×#41×C₆₇	2
C₆₇×#2981	2
C₆₇×#2979	2
C₆₇×#2978	2
C₆₇×#2976	2
C₆₇×#2975	2
C₆₇×#2974	2
C₆₇×#2935	2
D₄×M₁₆×C₆₇	2
C₆₇×#2933	2
C₆₇×#2931	2
C₆₇×#2930	2
C₂×#436×C₆₇	2
C₆₇×#2925	2
C₆₇×#2922	2
C₆₇×#2880	2
C₆₇×#2879	2
#60241	67

Subgroups — C67×#60241 #30739492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60241 #30739492