Subgroups — C₆₇×#60219 #30739470

Order 17152.

1906 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3480	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2970	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2969	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2968	C₂	1	4	yes	maximal, complemented
5	C₆₇×#2967	C₂	1	4	yes	maximal, complemented
6	C₆₇×#2966	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2965	C₂	1	2	yes	maximal, complemented
8	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2935	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#2935	C₂	1	2	yes	maximal, complemented
12	C₆₇×#2932	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2925	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2925	C₂	1	2	yes	maximal, complemented
16	C₆₇×#2875	C₂	1	2	yes	maximal, complemented
17	C₄×#131×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#131×C₆₇	C₂²	1	2	yes	—
21	C₂×#131×C₆₇	C₂²	1	2	yes	—
22	#559×C₆₇	C₄	1	1	yes	characteristic
23	#559×C₆₇	C₄	1	2	yes	—
24	#557×C₆₇	C₄	1	1	yes	characteristic
25	#556×C₆₇	C₄	1	1	yes	characteristic
26	#556×C₆₇	C₄	1	2	yes	—
27	#553×C₆₇	C₄	1	2	yes	—
28	#553×C₆₇	C₄	1	4	yes	—
29	#550×C₆₇	C₄	1	1	yes	characteristic
30	#548×C₆₇	C₄	1	2	yes	—
31	#532×C₆₇	C₂²	1	1	yes	characteristic
32	#532×C₆₇	C₂²	1	2	yes	—
33	#529×C₆₇	C₂²	1	2	yes	—
34	#529×C₆₇	C₂²	1	4	yes	—
35	C₄×#41×C₆₇	C₂²	1	2	yes	—
36	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×#41×C₆₇	C₂²	1	2	yes	—
38	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	#446×C₆₇	C₂²	1	1	yes	characteristic
40	#446×C₆₇	C₂²	1	1	yes	characteristic
41	#446×C₆₇	C₂²	1	2	yes	—
42	#446×C₆₇	C₂²	1	2	yes	—
43	#446×C₆₇	C₂²	1	1	yes	characteristic
44	#446×C₆₇	C₂²	1	1	yes	characteristic
45	#446×C₆₇	C₂²	1	2	yes	—
46	#446×C₆₇	C₂²	1	4	yes	—
47	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
48	#436×C₆₇	C₂²	1	2	yes	—
49	#436×C₆₇	C₂²	1	2	yes	—
50	#436×C₆₇	C₂²	1	2	yes	—
51	#436×C₆₇	C₂²	1	4	yes	—
52	#435×C₆₇	C₂²	1	4	yes	—
53	#435×C₆₇	C₂²	1	4	yes	—
54	#435×C₆₇	C₂²	1	4	yes	—
55	#435×C₆₇	C₂²	1	4	yes	—
56	#435×C₆₇	C₂²	1	4	yes	—
57	#435×C₆₇	C₂²	1	4	yes	—
58	#435×C₆₇	C₂²	1	4	yes	—
59	C₈×D₄×C₆₇	C₂²	1	2	yes	—
60	C₈×D₄×C₆₇	C₂²	1	2	yes	—
61	C₈×D₄×C₆₇	C₂²	1	4	yes	—
62	C₈×D₄×C₆₇	C₂²	1	2	yes	—
63	#433×C₆₇	C₂²	1	4	yes	—
64	#433×C₆₇	C₂²	1	4	yes	—
65	#433×C₆₇	C₂²	1	4	yes	—
66	#432×C₆₇	C₂²	1	2	yes	—
67	#432×C₆₇	C₂²	1	4	yes	—
68	#431×C₆₇	C₂²	1	4	yes	—
69	#431×C₆₇	C₂²	1	2	yes	—
70	#423×C₆₇	C₂²	1	1	yes	characteristic
71	#423×C₆₇	C₂²	1	2	yes	—
72	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
73	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
74	#408×C₆₇	C₂²	1	2	yes	—
75	#408×C₆₇	C₂²	1	2	yes	—
76	#408×C₆₇	C₂²	1	2	yes	—
77	#405×C₆₇	C₂²	1	1	yes	characteristic
78	#405×C₆₇	C₂²	1	1	yes	characteristic
79	#405×C₆₇	C₂²	1	2	yes	—
80	#405×C₆₇	C₂²	1	2	yes	—
81	#405×C₆₇	C₂²	1	2	yes	—
82	#405×C₆₇	C₂²	1	4	yes	—
83	#405×C₆₇	C₂²	1	4	yes	—
84	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
85	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
86	C₄×M₁₆×C₆₇	C₂²	1	2	yes	—
87	C₄×M₁₆×C₆₇	C₂²	1	4	yes	—
88	C₂×#97×C₆₇	C₂²	1	2	yes	—
89	C₂×#97×C₆₇	C₂²	1	4	yes	—
90	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
91	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
92	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
93	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
94	C₂×#41×C₆₇	C₂³	1	2	yes	—
95	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
96	#131×C₆₇	—	2	4	no	—
97	#131×C₆₇	—	2	8	no	—
98	#131×C₆₇	—	2	4	no	—
99	#131×C₆₇	—	2	8	no	—
100	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
101	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
102	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
103	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
104	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
105	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
106	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
107	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
108	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
109	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
110	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
111	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
112	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
113	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
115	C₄⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
116	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
117	#127×C₆₇	C₂×C₄	1	2	yes	—
118	#126×C₆₇	C₂×C₄	1	4	yes	—
119	#126×C₆₇	C₂×C₄	1	4	yes	—
120	#126×C₆₇	C₂×C₄	1	4	yes	—
121	#125×C₆₇	C₂×C₄	1	2	yes	—
122	#125×C₆₇	C₂×C₄	1	4	yes	—
123	#124×C₆₇	C₂×C₄	1	2	yes	—
124	#124×C₆₇	C₂×C₄	1	4	yes	—
125	#124×C₆₇	C₂×C₄	1	2	yes	—
126	#124×C₆₇	C₂×C₄	1	2	yes	—
127	#124×C₆₇	C₂×C₄	1	2	yes	—
128	#123×C₆₇	C₂×C₄	1	4	yes	—
129	#123×C₆₇	C₂×C₄	1	4	yes	—
130	#123×C₆₇	C₂×C₄	1	4	yes	—
131	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
132	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
133	#121×C₆₇	C₂×C₄	1	2	yes	—
134	#121×C₆₇	C₂×C₄	1	4	yes	—
135	#121×C₆₇	C₂×C₄	1	2	yes	—
136	#121×C₆₇	C₂×C₄	1	4	yes	—
137	#121×C₆₇	C₂×C₄	1	2	yes	—
138	#121×C₆₇	C₂×C₄	1	4	yes	—
139	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
140	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
141	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
142	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
143	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
144	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
145	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
146	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
147	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
148	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
149	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
150	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
151	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
152	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
153	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
154	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
155	C₄×D₄×C₆₇	C₂×C₄	1	4	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	C₂³	1	2	yes	—
161	C₄×D₄×C₆₇	C₂³	1	4	yes	—
162	C₄×D₄×C₆₇	C₂³	1	4	yes	—
163	C₄×D₄×C₆₇	C₂³	1	2	yes	—
164	#117×C₆₇	C₂×C₄	1	2	yes	—
165	#117×C₆₇	C₂×C₄	1	4	yes	—
166	#117×C₆₇	C₂³	1	1	yes	characteristic
167	#117×C₆₇	C₂³	1	2	yes	—
168	#117×C₆₇	C₂³	1	4	yes	—
169	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
170	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
171	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
172	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
173	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
174	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
175	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
176	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
177	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
178	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
179	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
180	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
181	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
182	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
183	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
184	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
185	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
186	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
187	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
188	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
189	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
190	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
191	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
192	#97×C₆₇	C₂³	1	2	yes	—
193	#97×C₆₇	C₂³	1	2	yes	—
194	#97×C₆₇	C₂³	1	2	yes	—
195	#97×C₆₇	C₂³	1	4	yes	—
196	#97×C₆₇	C₂³	1	4	yes	—
197	#97×C₆₇	C₂³	1	2	yes	—
198	#97×C₆₇	C₂³	1	2	yes	—
199	#97×C₆₇	C₂³	1	2	yes	—
200	#97×C₆₇	C₂³	1	2	yes	—
201	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
202	C₄×C₈×C₆₇	C₂³	1	2	yes	—
203	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
204	C₄×C₈×C₆₇	C₂³	1	2	yes	—
205	C₄×C₈×C₆₇	C₂³	1	2	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	4	yes	—
209	C₄×C₈×C₆₇	C₂³	1	4	yes	—
210	#41×C₆₇	—	2	2	no	—
211	#41×C₆₇	—	2	4	no	—
212	#41×C₆₇	—	2	2	no	—
213	#41×C₆₇	—	2	4	no	—
214	#41×C₆₇	—	4	4	no	—
215	C₂×Q₈×C₆₇	—	4	2	no	—
216	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
217	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
218	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
219	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
220	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
221	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
222	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
223	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
224	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
225	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
226	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
227	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
228	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
229	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
230	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
231	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
232	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
233	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
234	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
235	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
236	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
237	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
238	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
239	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
240	M₁₆×C₆₇	—	2	2	no	—
241	M₁₆×C₆₇	—	2	4	no	—
242	M₁₆×C₆₇	—	2	4	no	—
243	M₁₆×C₆₇	—	2	4	no	—
244	M₁₆×C₆₇	—	2	8	no	—
245	M₁₆×C₆₇	—	2	8	no	—
246	M₁₆×C₆₇	#41	1	4	yes	complemented
247	C₂×C₈×C₆₇	—	2	2	no	—
248	C₂×C₈×C₆₇	—	2	4	no	—
249	C₂×C₈×C₆₇	—	2	4	no	—
250	C₂×C₈×C₆₇	—	2	8	no	—
251	C₂×C₈×C₆₇	—	2	8	no	—
252	C₂×C₈×C₆₇	—	2	4	no	—
253	C₂×C₈×C₆₇	#41	1	4	yes	complemented
254	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
255	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
256	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
257	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
258	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
259	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
260	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
261	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
262	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
263	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
264	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
265	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
266	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
267	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
268	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
269	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
270	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
271	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
272	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
273	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
274	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
275	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
276	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
277	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
278	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
279	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
280	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
281	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
282	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
283	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
284	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
285	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
286	C₂²⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
287	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
288	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
289	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
290	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
291	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
292	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
293	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
294	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
295	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
296	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
297	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
298	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
299	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
300	C₄²×C₆₇	C₂⁴	1	2	yes	—
301	C₄²×C₆₇	C₂⁴	1	2	yes	—
302	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
303	C₄²×C₆₇	C₂⁴	1	2	yes	—
304	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
305	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
306	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
307	Q₈×C₆₇	—	2	2	no	—
308	Q₈×C₆₇	—	2	4	no	—
309	Q₈×C₆₇	—	2	2	no	—
310	Q₈×C₆₇	—	2	2	no	—
311	Q₈×C₆₇	—	4	2	no	—
312	D₄×C₆₇	—	2	4	no	—
313	D₄×C₆₇	—	2	8	no	—
314	D₄×C₆₇	—	2	2	no	—
315	D₄×C₆₇	—	2	4	no	—
316	D₄×C₆₇	—	2	4	no	—
317	D₄×C₆₇	—	4	4	no	—
318	D₄×C₆₇	—	4	2	no	—
319	D₄×C₆₇	—	4	4	no	—
320	C₂×C₄×C₆₇	—	2	2	no	—
321	C₂×C₄×C₆₇	—	2	4	no	—
322	C₂×C₄×C₆₇	—	2	2	no	—
323	C₂×C₄×C₆₇	—	2	4	no	—
324	C₂×C₄×C₆₇	—	2	4	no	—
325	C₂×C₄×C₆₇	—	2	2	no	—
326	C₂×C₄×C₆₇	—	4	2	no	—
327	C₂×C₄×C₆₇	—	4	4	no	—
328	C₂×C₄×C₆₇	—	4	1	no	—
329	C₂×C₄×C₆₇	—	4	2	no	—
330	C₂×C₄×C₆₇	—	4	2	no	—
331	C₂×C₄×C₆₇	—	4	4	no	—
332	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
333	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
334	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
335	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
336	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
337	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
338	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
339	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
340	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
341	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
342	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
343	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
344	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
345	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
346	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
347	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
348	C₈×C₆₇	—	2	1	no	—
349	C₈×C₆₇	—	2	2	no	—
350	C₈×C₆₇	—	2	1	no	—
351	C₈×C₆₇	—	2	2	no	—
352	C₈×C₆₇	—	2	4	no	—
353	C₈×C₆₇	—	2	2	no	—
354	C₈×C₆₇	C₂×#41	1	2	yes	—
355	C₈×C₆₇	C₂×#41	1	4	yes	—
356	C₈×C₆₇	C₂×#41	1	2	yes	—
357	C₂²×C₆₇	—	2	1	no	—
358	C₂²×C₆₇	—	2	1	no	—
359	C₂²×C₆₇	—	2	2	no	—
360	C₂²×C₆₇	—	2	4	no	—
361	C₂²×C₆₇	—	2	2	no	—
362	C₂²×C₆₇	—	2	4	no	—
363	C₂²×C₆₇	—	2	2	no	—
364	C₂²×C₆₇	—	2	4	no	—
365	C₂²×C₆₇	C₄×#41	1	2	yes	—
366	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
367	C₄×C₆₇	—	2	2	no	—
368	C₄×C₆₇	—	2	4	no	—
369	C₄×C₆₇	—	2	2	no	—
370	C₄×C₆₇	—	2	1	no	—
371	C₄×C₆₇	—	2	2	no	—
372	C₄×C₆₇	—	2	2	no	—
373	C₄×C₆₇	—	2	2	no	—
374	C₄×C₆₇	—	2	1	no	—
375	C₄×C₆₇	—	2	2	no	—
376	C₄×C₆₇	—	2	1	no	—
377	C₄×C₆₇	—	2	2	no	—
378	C₄×C₆₇	C₄×#41	1	2	yes	—
379	C₄×C₆₇	#568	1	2	yes	—
380	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
381	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
382	#60219	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
383	C₂×C₆₇	—	2	2	no	—
384	C₂×C₆₇	—	4	2	no	—
385	C₂×C₆₇	—	4	4	no	—
386	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
387	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
388	C₂×C₆₇	#3573	1	1	yes	characteristic
389	#3480	C₂×C₆₇	1	1	yes	characteristic
390	#2970	C₂×C₆₇	1	2	yes	complemented
391	#2969	C₂×C₆₇	1	2	yes	complemented
392	#2968	C₂×C₆₇	1	4	yes	complemented
393	#2967	C₂×C₆₇	1	4	yes	complemented
394	#2966	C₂×C₆₇	1	2	yes	complemented
395	#2965	C₂×C₆₇	1	2	yes	complemented
396	Q₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
397	C₂×#446	C₂×C₆₇	1	1	yes	characteristic, complemented
398	#2935	C₂×C₆₇	1	1	yes	characteristic, complemented
399	#2935	C₂×C₆₇	1	2	yes	complemented
400	#2932	C₂×C₆₇	1	2	yes	complemented
401	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
402	#2925	C₂×C₆₇	1	1	yes	characteristic, complemented
403	#2925	C₂×C₆₇	1	2	yes	complemented
404	#2875	C₂×C₆₇	1	2	yes	complemented
405	C₄×#131	C₂×C₆₇	1	1	yes	characteristic, complemented
406	C₆₇	#60219	1	1	yes	characteristic, direct product factor, Sylow
407	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
408	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
409	C₂×#131	C₂²×C₆₇	1	2	yes	—
410	C₂×#131	C₂²×C₆₇	1	2	yes	—
411	#559	C₄×C₆₇	1	1	yes	characteristic
412	#559	C₄×C₆₇	1	2	yes	—
413	#557	C₄×C₆₇	1	1	yes	characteristic
414	#556	C₄×C₆₇	1	1	yes	characteristic
415	#556	C₄×C₆₇	1	2	yes	—
416	#553	C₄×C₆₇	1	2	yes	—
417	#553	C₄×C₆₇	1	4	yes	—
418	#550	C₄×C₆₇	1	1	yes	characteristic
419	#548	C₄×C₆₇	1	2	yes	—
420	#532	C₂²×C₆₇	1	1	yes	characteristic
421	#532	C₂²×C₆₇	1	2	yes	—
422	#529	C₂²×C₆₇	1	2	yes	—
423	#529	C₂²×C₆₇	1	4	yes	—
424	C₄×#41	C₂²×C₆₇	1	2	yes	—
425	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
426	C₄×#41	C₂²×C₆₇	1	2	yes	—
427	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
428	#446	C₂²×C₆₇	1	1	yes	characteristic
429	#446	C₂²×C₆₇	1	1	yes	characteristic
430	#446	C₂²×C₆₇	1	2	yes	—
431	#446	C₂²×C₆₇	1	2	yes	—
432	#446	C₂²×C₆₇	1	1	yes	characteristic
433	#446	C₂²×C₆₇	1	1	yes	characteristic
434	#446	C₂²×C₆₇	1	2	yes	—
435	#446	C₂²×C₆₇	1	4	yes	—
436	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
437	#436	C₂²×C₆₇	1	2	yes	—
438	#436	C₂²×C₆₇	1	2	yes	—
439	#436	C₂²×C₆₇	1	2	yes	—
440	#436	C₂²×C₆₇	1	4	yes	—
441	#435	C₂²×C₆₇	1	4	yes	—
442	#435	C₂²×C₆₇	1	4	yes	—
443	#435	C₂²×C₆₇	1	4	yes	—
444	#435	C₂²×C₆₇	1	4	yes	—
445	#435	C₂²×C₆₇	1	4	yes	—
446	#435	C₂²×C₆₇	1	4	yes	—
447	#435	C₂²×C₆₇	1	4	yes	—
448	C₈×D₄	C₂²×C₆₇	1	2	yes	—
449	C₈×D₄	C₂²×C₆₇	1	2	yes	—
450	C₈×D₄	C₂²×C₆₇	1	4	yes	—
451	C₈×D₄	C₂²×C₆₇	1	2	yes	—
452	#433	C₂²×C₆₇	1	4	yes	—
453	#433	C₂²×C₆₇	1	4	yes	—
454	#433	C₂²×C₆₇	1	4	yes	—
455	#432	C₂²×C₆₇	1	2	yes	—
456	#432	C₂²×C₆₇	1	4	yes	—
457	#431	C₂²×C₆₇	1	4	yes	—
458	#431	C₂²×C₆₇	1	2	yes	—
459	#423	C₂²×C₆₇	1	1	yes	characteristic
460	#423	C₂²×C₆₇	1	2	yes	—
461	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
462	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
463	#408	C₂²×C₆₇	1	2	yes	—
464	#408	C₂²×C₆₇	1	2	yes	—
465	#408	C₂²×C₆₇	1	2	yes	—
466	#405	C₂²×C₆₇	1	1	yes	characteristic
467	#405	C₂²×C₆₇	1	1	yes	characteristic
468	#405	C₂²×C₆₇	1	2	yes	—
469	#405	C₂²×C₆₇	1	2	yes	—
470	#405	C₂²×C₆₇	1	2	yes	—
471	#405	C₂²×C₆₇	1	4	yes	—
472	#405	C₂²×C₆₇	1	4	yes	—
473	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
474	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
475	C₄×M₁₆	C₂²×C₆₇	1	2	yes	—
476	C₄×M₁₆	C₂²×C₆₇	1	4	yes	—
477	C₂×#97	C₂²×C₆₇	1	2	yes	—
478	C₂×#97	C₂²×C₆₇	1	4	yes	—
479	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
480	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
481	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
482	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
483	C₂×#41	C₂³×C₆₇	1	2	yes	—
484	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
485	#131	—	2	4	no	—
486	#131	—	2	8	no	—
487	#131	—	2	4	no	—
488	#131	—	2	8	no	—
489	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
490	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
491	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
492	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
493	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
494	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
495	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
496	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
497	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
498	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
499	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
500	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
501	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
502	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
503	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
504	C₄⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
505	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
506	#127	C₂×C₄×C₆₇	1	2	yes	—
507	#126	C₂×C₄×C₆₇	1	4	yes	—
508	#126	C₂×C₄×C₆₇	1	4	yes	—
509	#126	C₂×C₄×C₆₇	1	4	yes	—
510	#125	C₂×C₄×C₆₇	1	2	yes	—
511	#125	C₂×C₄×C₆₇	1	4	yes	—
512	#124	C₂×C₄×C₆₇	1	2	yes	—
513	#124	C₂×C₄×C₆₇	1	4	yes	—
514	#124	C₂×C₄×C₆₇	1	2	yes	—
515	#124	C₂×C₄×C₆₇	1	2	yes	—
516	#124	C₂×C₄×C₆₇	1	2	yes	—
517	#123	C₂×C₄×C₆₇	1	4	yes	—
518	#123	C₂×C₄×C₆₇	1	4	yes	—
519	#123	C₂×C₄×C₆₇	1	4	yes	—
520	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
521	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
522	#121	C₂×C₄×C₆₇	1	2	yes	—
523	#121	C₂×C₄×C₆₇	1	4	yes	—
524	#121	C₂×C₄×C₆₇	1	2	yes	—
525	#121	C₂×C₄×C₆₇	1	4	yes	—
526	#121	C₂×C₄×C₆₇	1	2	yes	—
527	#121	C₂×C₄×C₆₇	1	4	yes	—
528	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
529	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
530	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
531	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
532	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
533	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
534	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
535	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
536	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
537	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
538	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
539	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
540	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
541	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
542	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
543	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
544	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
545	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
546	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
547	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
548	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
549	C₄×D₄	C₂³×C₆₇	1	2	yes	—
550	C₄×D₄	C₂³×C₆₇	1	4	yes	—
551	C₄×D₄	C₂³×C₆₇	1	4	yes	—
552	C₄×D₄	C₂³×C₆₇	1	2	yes	—
553	#117	C₂×C₄×C₆₇	1	2	yes	—
554	#117	C₂×C₄×C₆₇	1	4	yes	—
555	#117	C₂³×C₆₇	1	1	yes	characteristic
556	#117	C₂³×C₆₇	1	2	yes	—
557	#117	C₂³×C₆₇	1	4	yes	—
558	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
559	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
560	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
561	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
562	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
563	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
564	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
565	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
566	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
567	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
568	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
569	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
570	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
571	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
572	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
573	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
574	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
575	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
576	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
577	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
578	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
579	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
580	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
581	#97	C₂³×C₆₇	1	2	yes	—
582	#97	C₂³×C₆₇	1	2	yes	—
583	#97	C₂³×C₆₇	1	2	yes	—
584	#97	C₂³×C₆₇	1	4	yes	—
585	#97	C₂³×C₆₇	1	4	yes	—
586	#97	C₂³×C₆₇	1	2	yes	—
587	#97	C₂³×C₆₇	1	2	yes	—
588	#97	C₂³×C₆₇	1	2	yes	—
589	#97	C₂³×C₆₇	1	2	yes	—
590	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
591	C₄×C₈	C₂³×C₆₇	1	2	yes	—
592	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
593	C₄×C₈	C₂³×C₆₇	1	2	yes	—
594	C₄×C₈	C₂³×C₆₇	1	2	yes	—
595	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
596	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
597	C₄×C₈	C₂³×C₆₇	1	4	yes	—
598	C₄×C₈	C₂³×C₆₇	1	4	yes	—
599	#41	—	2	2	no	—
600	#41	—	2	4	no	—
601	#41	—	2	2	no	—
602	#41	—	2	4	no	—
603	#41	—	4	4	no	—
604	C₂×Q₈	—	4	2	no	—
605	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
606	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
607	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
608	C₂×Q₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
609	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
610	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
611	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
612	C₂×D₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
613	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
614	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
615	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
616	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
617	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
618	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
619	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
620	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
621	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
622	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
623	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
624	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
625	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
626	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
627	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
628	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
629	M₁₆	—	2	2	no	—
630	M₁₆	—	2	4	no	—
631	M₁₆	—	2	4	no	—
632	M₁₆	—	2	4	no	—
633	M₁₆	—	2	8	no	—
634	M₁₆	—	2	8	no	—
635	M₁₆	#41×C₆₇	1	4	yes	complemented
636	C₂×C₈	—	2	2	no	—
637	C₂×C₈	—	2	4	no	—
638	C₂×C₈	—	2	4	no	—
639	C₂×C₈	—	2	8	no	—
640	C₂×C₈	—	2	8	no	—
641	C₂×C₈	—	2	4	no	—
642	C₂×C₈	#41×C₆₇	1	4	yes	complemented
643	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
644	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
645	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
646	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
647	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
648	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
649	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
650	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
651	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
652	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
653	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
654	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
655	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
656	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
657	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
658	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
659	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
660	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
661	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
662	C₄⋊C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
663	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
664	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
665	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
666	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
667	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
668	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
669	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
670	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
671	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
672	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
673	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
674	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
675	C₂²⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
676	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
677	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
678	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
679	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
680	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
681	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
682	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
683	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
684	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
685	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
686	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
687	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
688	C₄²	C₂⁴×C₆₇	1	2	yes	—
689	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
690	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
691	C₄²	C₂⁴×C₆₇	1	2	yes	—
692	C₄²	C₂⁴×C₆₇	1	2	yes	—
693	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
694	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
695	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
696	Q₈	—	2	2	no	—
697	Q₈	—	2	4	no	—
698	Q₈	—	2	2	no	—
699	Q₈	—	2	2	no	—
700	Q₈	—	4	2	no	—
701	D₄	—	2	4	no	—
702	D₄	—	2	8	no	—
703	D₄	—	2	2	no	—
704	D₄	—	2	4	no	—
705	D₄	—	2	4	no	—
706	D₄	—	4	4	no	—
707	D₄	—	4	4	no	—
708	D₄	—	4	2	no	—
709	C₂×C₄	—	2	2	no	—
710	C₂×C₄	—	2	4	no	—
711	C₂×C₄	—	2	2	no	—
712	C₂×C₄	—	2	4	no	—
713	C₂×C₄	—	2	4	no	—
714	C₂×C₄	—	2	2	no	—
715	C₂×C₄	—	4	2	no	—
716	C₂×C₄	—	4	4	no	—
717	C₂×C₄	—	4	2	no	—
718	C₂×C₄	—	4	4	no	—
719	C₂×C₄	—	4	1	no	—
720	C₂×C₄	—	4	2	no	—
721	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
722	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
723	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
724	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
725	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
726	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
727	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
728	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
729	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
730	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
731	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
732	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
733	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
734	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
735	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
736	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
737	C₈	—	2	1	no	—
738	C₈	—	2	2	no	—
739	C₈	—	2	1	no	—
740	C₈	—	2	2	no	—
741	C₈	—	2	4	no	—
742	C₈	—	2	2	no	—
743	C₈	C₂×#41×C₆₇	1	2	yes	—
744	C₈	C₂×#41×C₆₇	1	4	yes	—
745	C₈	C₂×#41×C₆₇	1	2	yes	—
746	C₂²	—	2	1	no	—
747	C₂²	—	2	1	no	—
748	C₂²	—	2	2	no	—
749	C₂²	—	2	4	no	—
750	C₂²	—	2	2	no	—
751	C₂²	—	2	4	no	—
752	C₂²	—	2	2	no	—
753	C₂²	—	2	4	no	—
754	C₂²	C₄×#41×C₆₇	1	2	yes	—
755	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
756	C₄	—	2	2	no	—
757	C₄	—	2	1	no	—
758	C₄	—	2	2	no	—
759	C₄	—	2	2	no	—
760	C₄	—	2	2	no	—
761	C₄	—	2	1	no	—
762	C₄	—	2	2	no	—
763	C₄	—	2	1	no	—
764	C₄	—	2	2	no	—
765	C₄	—	2	2	no	—
766	C₄	—	2	4	no	—
767	C₄	C₄×#41×C₆₇	1	2	yes	—
768	C₄	#568×C₆₇	1	2	yes	—
769	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
770	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
771	C₂	—	2	2	no	—
772	C₂	—	4	2	no	—
773	C₂	—	4	4	no	—
774	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
775	C₂	C₂×#568×C₆₇	1	1	yes	characteristic
776	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	C₆₇×#3480	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×#2970	C₂	semidirect
3. C₆₇×#2969	C₂	semidirect
4. C₆₇×#2968	C₂	semidirect
5. C₆₇×#2967	C₂	semidirect
6. C₆₇×#2966	C₂	semidirect
7. C₆₇×#2965	C₂	semidirect
8. Q₈×M₁₆×C₆₇	C₂	semidirect
9. C₂×#446×C₆₇	C₂	semidirect
10. C₆₇×#2935	C₂	semidirect
11. C₆₇×#2935	C₂	semidirect
12. C₆₇×#2932	C₂	semidirect
13. C₆₇×#2931	C₂	semidirect
14. C₆₇×#2925	C₂	semidirect
15. C₆₇×#2925	C₂	semidirect
16. C₆₇×#2875	C₂	semidirect
17. C₄×#131×C₆₇	C₂	semidirect
246. M₁₆×C₆₇	#41	semidirect
253. C₂×C₈×C₆₇	#41	semidirect
382. #60219	C₆₇	direct (N × H)
390. #2970	C₂×C₆₇	semidirect
391. #2969	C₂×C₆₇	semidirect
392. #2968	C₂×C₆₇	semidirect
393. #2967	C₂×C₆₇	semidirect
394. #2966	C₂×C₆₇	semidirect
395. #2965	C₂×C₆₇	semidirect
396. Q₈×M₁₆	C₂×C₆₇	semidirect
397. C₂×#446	C₂×C₆₇	semidirect
398. #2935	C₂×C₆₇	semidirect
399. #2935	C₂×C₆₇	semidirect
400. #2932	C₂×C₆₇	semidirect
401. #2931	C₂×C₆₇	semidirect
402. #2925	C₂×C₆₇	semidirect
403. #2925	C₂×C₆₇	semidirect
404. #2875	C₂×C₆₇	semidirect
405. C₄×#131	C₂×C₆₇	semidirect
406. C₆₇	#60219	direct (N × H)
635. M₁₆	#41×C₆₇	semidirect
642. C₂×C₈	#41×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3480	2
C₆₇×#2970	2
C₆₇×#2969	2
C₆₇×#2968	2
C₆₇×#2967	2
C₆₇×#2966	2
C₆₇×#2965	2
Q₈×M₁₆×C₆₇	2
C₂×#446×C₆₇	2
C₆₇×#2935	2
C₆₇×#2932	2
C₆₇×#2931	2
C₆₇×#2925	2
C₆₇×#2875	2
C₄×#131×C₆₇	2
#60219	67

Subgroups — C67×#60219 #30739470

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60219 #30739470