Subgroups — C₆₇×#60268 #30739519

Order 17152.

1908 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2994	C₂	1	4	yes	maximal, complemented
3	C₆₇×#2993	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2991	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2990	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2989	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2988	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2960	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2933	C₂	1	4	yes	maximal, complemented
10	C₆₇×#2931	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2925	C₂	1	2	yes	maximal, complemented
12	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2906	C₂	1	2	yes	maximal, complemented
14	C₂×#423×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#131×C₆₇	C₂²	1	4	yes	—
16	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
17	#533×C₆₇	C₄	1	1	yes	characteristic
18	#533×C₆₇	C₄	1	1	yes	characteristic
19	#533×C₆₇	C₂²	1	1	yes	characteristic
20	#532×C₆₇	C₄	1	1	yes	characteristic
21	#532×C₆₇	C₄	1	1	yes	characteristic
22	#532×C₆₇	C₂²	1	1	yes	characteristic
23	#529×C₆₇	C₄	1	2	yes	—
24	#529×C₆₇	C₄	1	2	yes	—
25	#529×C₆₇	C₄	1	2	yes	—
26	#529×C₆₇	C₄	1	2	yes	—
27	#529×C₆₇	C₂²	1	2	yes	—
28	#529×C₆₇	C₂²	1	2	yes	—
29	#525×C₆₇	C₄	1	2	yes	—
30	#525×C₆₇	C₄	1	2	yes	—
31	#525×C₆₇	C₂²	1	2	yes	—
32	C₄×#41×C₆₇	C₂²	1	2	yes	—
33	C₄×#41×C₆₇	C₂²	1	2	yes	—
34	C₄×#41×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
36	#446×C₆₇	C₂²	1	4	yes	complemented
37	#446×C₆₇	C₂²	1	4	yes	complemented
38	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
39	C₈×Q₈×C₆₇	C₂²	1	4	yes	complemented
40	#436×C₆₇	C₂²	1	8	yes	—
41	#436×C₆₇	C₂²	1	4	yes	complemented
42	#436×C₆₇	C₂²	1	4	yes	complemented
43	#435×C₆₇	C₂²	1	8	yes	—
44	#435×C₆₇	C₂²	1	8	yes	—
45	C₈×D₄×C₆₇	C₂²	1	8	yes	—
46	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
47	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
48	#433×C₆₇	C₂²	1	8	yes	complemented
49	#432×C₆₇	C₂²	1	4	yes	complemented
50	#431×C₆₇	C₂²	1	1	yes	characteristic
51	#431×C₆₇	C₂²	1	1	yes	characteristic
52	#431×C₆₇	C₂²	1	1	yes	characteristic
53	#431×C₆₇	C₂²	1	1	yes	characteristic
54	#431×C₆₇	C₂²	1	2	yes	—
55	#431×C₆₇	C₂²	1	2	yes	—
56	#431×C₆₇	C₂²	1	4	yes	complemented
57	#424×C₆₇	C₂²	1	4	yes	—
58	#424×C₆₇	C₂²	1	8	yes	complemented
59	#423×C₆₇	C₂²	1	2	yes	—
60	#423×C₆₇	C₂²	1	2	yes	—
61	#423×C₆₇	C₂²	1	2	yes	—
62	#423×C₆₇	C₂²	1	2	yes	—
63	#423×C₆₇	C₂²	1	4	yes	complemented
64	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
65	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
66	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
67	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
68	C₂×C₄⋊C₈×C₆₇	C₂²	1	4	yes	complemented
69	#408×C₆₇	C₂²	1	4	yes	—
70	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
71	#405×C₆₇	C₂²	1	4	yes	—
72	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
73	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
74	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×#41×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
77	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
78	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
79	#131×C₆₇	—	4	8	no	—
80	C₂×M₁₆×C₆₇	—	4	4	no	—
81	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
82	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
83	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
84	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
85	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
86	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
87	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
88	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
89	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
90	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
91	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
92	#126×C₆₇	C₂×C₄	1	4	yes	—
93	#126×C₆₇	C₂×C₄	1	4	yes	—
94	#125×C₆₇	C₂×C₄	1	2	yes	—
95	#125×C₆₇	C₂×C₄	1	2	yes	—
96	#125×C₆₇	C₂×C₄	1	2	yes	—
97	#124×C₆₇	C₂×C₄	1	2	yes	—
98	#124×C₆₇	C₂×C₄	1	2	yes	—
99	#124×C₆₇	C₂×C₄	1	2	yes	—
100	#123×C₆₇	C₂×C₄	1	4	yes	—
101	#123×C₆₇	C₂×C₄	1	4	yes	—
102	#123×C₆₇	C₂×C₄	1	4	yes	—
103	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
104	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
105	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
106	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
107	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
108	C₂²⋊Q₈×C₆₇	C₂×C₄	1	2	yes	—
109	#121×C₆₇	C₂×C₄	1	2	yes	—
110	#121×C₆₇	C₂×C₄	1	2	yes	—
111	#121×C₆₇	C₂×C₄	1	2	yes	—
112	#121×C₆₇	C₂×C₄	1	2	yes	—
113	#121×C₆₇	C₂×C₄	1	2	yes	—
114	#121×C₆₇	C₂×C₄	1	2	yes	—
115	#120×C₆₇	C₂×C₄	1	4	yes	—
116	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
117	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
118	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
119	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
120	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
121	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
122	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
123	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
124	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
125	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
126	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
127	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
128	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
129	C₄×D₄×C₆₇	C₂×C₄	1	2	yes	—
130	C₄×D₄×C₆₇	C₂³	1	4	yes	—
131	C₄×D₄×C₆₇	C₂³	1	4	yes	—
132	C₄×D₄×C₆₇	C₂³	1	2	yes	—
133	C₄×D₄×C₆₇	C₂³	1	2	yes	—
134	#117×C₆₇	C₂×C₄	1	2	yes	—
135	#117×C₆₇	C₂×C₄	1	2	yes	—
136	#117×C₆₇	C₂×C₄	1	4	yes	—
137	#117×C₆₇	C₂×C₄	1	4	yes	—
138	#117×C₆₇	C₂³	1	2	yes	—
139	#117×C₆₇	C₂³	1	4	yes	—
140	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
141	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
142	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
143	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
144	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
145	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
146	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
147	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
148	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
149	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
152	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
153	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
154	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
155	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
156	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
157	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
158	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
159	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
160	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
161	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
162	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
163	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
164	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
165	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
166	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
167	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
168	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
169	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
170	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
171	#97×C₆₇	C₂³	1	8	yes	—
172	C₄×C₈×C₆₇	C₂³	1	2	yes	—
173	C₄×C₈×C₆₇	C₂³	1	2	yes	—
174	C₄×C₈×C₆₇	C₂³	1	2	yes	—
175	C₄×C₈×C₆₇	C₂³	1	2	yes	—
176	C₄×C₈×C₆₇	C₂³	1	8	yes	—
177	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
178	#41×C₆₇	—	2	8	no	—
179	#41×C₆₇	—	2	4	no	—
180	#41×C₆₇	—	2	4	no	—
181	#41×C₆₇	—	4	4	no	—
182	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
183	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
184	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
185	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
186	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
187	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
188	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
189	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
190	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
191	C₂²×C₄×C₆₇	—	2	4	no	—
192	C₂²×C₄×C₆₇	—	2	2	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	4	2	no	—
195	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
198	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
199	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
200	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
206	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
207	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
209	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
211	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
214	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
215	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
216	M₁₆×C₆₇	—	2	8	no	—
217	M₁₆×C₆₇	—	4	4	no	—
218	M₁₆×C₆₇	—	4	8	no	—
219	C₂×C₈×C₆₇	—	2	4	no	—
220	C₂×C₈×C₆₇	—	2	4	no	—
221	C₂×C₈×C₆₇	—	4	4	no	—
222	C₂×C₈×C₆₇	—	4	8	no	—
223	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
224	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
225	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
226	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
227	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
228	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
229	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
230	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
231	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
232	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
233	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
234	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
235	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
236	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
237	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
238	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
239	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
240	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
241	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
242	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
243	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
244	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
245	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
246	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
247	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
248	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
249	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
250	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
251	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
252	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
253	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
254	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
255	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
256	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
257	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
258	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
259	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
260	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
261	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
262	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
263	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
264	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
265	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
266	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
267	C₄²×C₆₇	C₂⁴	1	4	yes	—
268	C₂³×C₆₇	—	2	2	no	—
269	C₂³×C₆₇	—	2	2	no	—
270	C₂³×C₆₇	—	4	1	no	—
271	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
272	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
273	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
274	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
275	Q₈×C₆₇	—	2	4	no	—
276	Q₈×C₆₇	—	2	4	no	—
277	Q₈×C₆₇	—	2	4	no	—
278	D₄×C₆₇	—	2	8	no	—
279	D₄×C₆₇	—	2	8	no	—
280	D₄×C₆₇	—	2	4	no	—
281	D₄×C₆₇	—	2	4	no	—
282	D₄×C₆₇	—	2	4	no	—
283	D₄×C₆₇	—	4	4	no	—
284	C₂×C₄×C₆₇	—	2	2	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	2	no	—
287	C₂×C₄×C₆₇	—	2	2	no	—
288	C₂×C₄×C₆₇	—	2	2	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	2	2	no	—
291	C₂×C₄×C₆₇	—	2	2	no	—
292	C₂×C₄×C₆₇	—	2	8	no	—
293	C₂×C₄×C₆₇	—	2	2	no	—
294	C₂×C₄×C₆₇	—	2	2	no	—
295	C₂×C₄×C₆₇	—	2	2	no	—
296	C₂×C₄×C₆₇	—	2	1	no	—
297	C₂×C₄×C₆₇	—	2	4	no	—
298	C₂×C₄×C₆₇	—	2	1	no	—
299	C₂×C₄×C₆₇	—	2	1	no	—
300	C₂×C₄×C₆₇	—	2	1	no	—
301	C₂×C₄×C₆₇	—	2	4	no	—
302	C₂×C₄×C₆₇	—	4	1	no	—
303	C₂×C₄×C₆₇	—	4	1	no	—
304	C₂×C₄×C₆₇	—	4	1	no	—
305	C₂×C₄×C₆₇	—	4	4	no	—
306	C₂×C₄×C₆₇	#131	1	4	yes	complemented
307	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
308	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
309	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
310	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
311	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
312	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
313	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
314	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
315	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
316	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
317	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
318	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
319	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
320	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
321	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
322	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
323	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
324	C₈×C₆₇	—	2	2	no	—
325	C₈×C₆₇	—	2	2	no	—
326	C₈×C₆₇	—	2	8	no	—
327	C₈×C₆₇	—	2	2	no	—
328	C₈×C₆₇	—	2	2	no	—
329	C₂²×C₆₇	—	2	1	no	—
330	C₂²×C₆₇	—	2	1	no	—
331	C₂²×C₆₇	—	2	1	no	—
332	C₂²×C₆₇	—	2	1	no	—
333	C₂²×C₆₇	—	2	1	no	—
334	C₂²×C₆₇	—	2	4	no	—
335	C₂²×C₆₇	—	2	1	no	—
336	C₂²×C₆₇	—	2	4	no	—
337	C₂²×C₆₇	—	2	1	no	—
338	C₂²×C₆₇	—	2	4	no	—
339	C₂²×C₆₇	—	4	4	no	—
340	C₂²×C₆₇	#520	1	2	yes	—
341	C₂²×C₆₇	C₂×#131	1	2	yes	—
342	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
343	C₄×C₆₇	—	2	1	no	—
344	C₄×C₆₇	—	2	1	no	—
345	C₄×C₆₇	—	2	4	no	—
346	C₄×C₆₇	—	2	1	no	—
347	C₄×C₆₇	—	2	1	no	—
348	C₄×C₆₇	—	2	4	no	—
349	C₄×C₆₇	—	2	1	no	—
350	C₄×C₆₇	—	2	1	no	—
351	C₄×C₆₇	—	2	1	no	—
352	C₄×C₆₇	—	2	4	no	—
353	C₄×C₆₇	#520	1	2	yes	—
354	C₄×C₆₇	C₂×#131	1	2	yes	—
355	C₄×C₆₇	C₂×#131	1	2	yes	—
356	C₄×C₆₇	#568	1	2	yes	—
357	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
358	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
359	#60268	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
360	C₂×C₆₇	—	2	2	no	—
361	C₂×C₆₇	—	2	2	no	—
362	C₂×C₆₇	—	4	4	no	—
363	C₂×C₆₇	—	4	1	no	—
364	C₂×C₆₇	C₂×#520	1	1	yes	characteristic
365	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
366	C₂×C₆₇	C₂×#568	1	1	yes	characteristic
367	#3444	C₂×C₆₇	1	1	yes	characteristic
368	#2994	C₂×C₆₇	1	4	yes	complemented
369	#2993	C₂×C₆₇	1	4	yes	complemented
370	#2991	C₂×C₆₇	1	2	yes	complemented
371	#2990	C₂×C₆₇	1	2	yes	complemented
372	#2989	C₂×C₆₇	1	2	yes	complemented
373	#2988	C₂×C₆₇	1	2	yes	complemented
374	#2960	C₂×C₆₇	1	2	yes	complemented
375	#2933	C₂×C₆₇	1	4	yes	complemented
376	#2931	C₂×C₆₇	1	2	yes	complemented
377	#2925	C₂×C₆₇	1	2	yes	complemented
378	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
379	#2906	C₂×C₆₇	1	2	yes	complemented
380	C₂×#423	C₂×C₆₇	1	1	yes	characteristic, complemented
381	C₆₇	#60268	1	1	yes	characteristic, direct product factor, Sylow
382	C₂×#131	C₂²×C₆₇	1	4	yes	—
383	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
384	#533	C₄×C₆₇	1	1	yes	characteristic
385	#533	C₄×C₆₇	1	1	yes	characteristic
386	#533	C₂²×C₆₇	1	1	yes	characteristic
387	#532	C₄×C₆₇	1	1	yes	characteristic
388	#532	C₄×C₆₇	1	1	yes	characteristic
389	#532	C₂²×C₆₇	1	1	yes	characteristic
390	#529	C₄×C₆₇	1	2	yes	—
391	#529	C₄×C₆₇	1	2	yes	—
392	#529	C₄×C₆₇	1	2	yes	—
393	#529	C₄×C₆₇	1	2	yes	—
394	#529	C₂²×C₆₇	1	2	yes	—
395	#529	C₂²×C₆₇	1	2	yes	—
396	#525	C₄×C₆₇	1	2	yes	—
397	#525	C₄×C₆₇	1	2	yes	—
398	#525	C₂²×C₆₇	1	2	yes	—
399	C₄×#41	C₂²×C₆₇	1	2	yes	—
400	C₄×#41	C₂²×C₆₇	1	2	yes	—
401	C₄×#41	C₂²×C₆₇	1	2	yes	—
402	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
403	#446	C₂²×C₆₇	1	4	yes	complemented
404	#446	C₂²×C₆₇	1	4	yes	complemented
405	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
406	C₈×Q₈	C₂²×C₆₇	1	4	yes	complemented
407	#436	C₂²×C₆₇	1	8	yes	—
408	#436	C₂²×C₆₇	1	4	yes	complemented
409	#436	C₂²×C₆₇	1	4	yes	complemented
410	#435	C₂²×C₆₇	1	8	yes	—
411	#435	C₂²×C₆₇	1	8	yes	—
412	C₈×D₄	C₂²×C₆₇	1	8	yes	—
413	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
414	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
415	#433	C₂²×C₆₇	1	8	yes	complemented
416	#432	C₂²×C₆₇	1	4	yes	complemented
417	#431	C₂²×C₆₇	1	1	yes	characteristic
418	#431	C₂²×C₆₇	1	1	yes	characteristic
419	#431	C₂²×C₆₇	1	1	yes	characteristic
420	#431	C₂²×C₆₇	1	1	yes	characteristic
421	#431	C₂²×C₆₇	1	2	yes	—
422	#431	C₂²×C₆₇	1	2	yes	—
423	#431	C₂²×C₆₇	1	4	yes	complemented
424	#424	C₂²×C₆₇	1	4	yes	—
425	#424	C₂²×C₆₇	1	8	yes	complemented
426	#423	C₂²×C₆₇	1	2	yes	—
427	#423	C₂²×C₆₇	1	2	yes	—
428	#423	C₂²×C₆₇	1	2	yes	—
429	#423	C₂²×C₆₇	1	2	yes	—
430	#423	C₂²×C₆₇	1	4	yes	complemented
431	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
432	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
433	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
434	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
435	C₂×C₄⋊C₈	C₂²×C₆₇	1	4	yes	complemented
436	#408	C₂²×C₆₇	1	4	yes	—
437	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
438	#405	C₂²×C₆₇	1	4	yes	—
439	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
440	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
441	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
442	C₂×#41	C₂³×C₆₇	1	2	yes	—
443	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
444	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
445	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
446	#131	—	4	8	no	—
447	C₂×M₁₆	—	4	4	no	—
448	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
449	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
450	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
451	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
452	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
453	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
454	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
455	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
456	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
457	C₄⋊Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
458	#127	C₂×C₄×C₆₇	1	1	yes	characteristic
459	#126	C₂×C₄×C₆₇	1	4	yes	—
460	#126	C₂×C₄×C₆₇	1	4	yes	—
461	#125	C₂×C₄×C₆₇	1	2	yes	—
462	#125	C₂×C₄×C₆₇	1	2	yes	—
463	#125	C₂×C₄×C₆₇	1	2	yes	—
464	#124	C₂×C₄×C₆₇	1	2	yes	—
465	#124	C₂×C₄×C₆₇	1	2	yes	—
466	#124	C₂×C₄×C₆₇	1	2	yes	—
467	#123	C₂×C₄×C₆₇	1	4	yes	—
468	#123	C₂×C₄×C₆₇	1	4	yes	—
469	#123	C₂×C₄×C₆₇	1	4	yes	—
470	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
471	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
472	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
473	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
474	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
475	C₂²⋊Q₈	C₂×C₄×C₆₇	1	2	yes	—
476	#121	C₂×C₄×C₆₇	1	2	yes	—
477	#121	C₂×C₄×C₆₇	1	2	yes	—
478	#121	C₂×C₄×C₆₇	1	2	yes	—
479	#121	C₂×C₄×C₆₇	1	2	yes	—
480	#121	C₂×C₄×C₆₇	1	2	yes	—
481	#121	C₂×C₄×C₆₇	1	2	yes	—
482	#120	C₂×C₄×C₆₇	1	4	yes	—
483	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
484	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
485	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
486	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
487	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
488	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
489	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
490	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
491	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
492	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
493	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
494	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
495	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
496	C₄×D₄	C₂×C₄×C₆₇	1	2	yes	—
497	C₄×D₄	C₂³×C₆₇	1	4	yes	—
498	C₄×D₄	C₂³×C₆₇	1	4	yes	—
499	C₄×D₄	C₂³×C₆₇	1	2	yes	—
500	C₄×D₄	C₂³×C₆₇	1	2	yes	—
501	#117	C₂×C₄×C₆₇	1	2	yes	—
502	#117	C₂×C₄×C₆₇	1	2	yes	—
503	#117	C₂×C₄×C₆₇	1	4	yes	—
504	#117	C₂×C₄×C₆₇	1	4	yes	—
505	#117	C₂³×C₆₇	1	2	yes	—
506	#117	C₂³×C₆₇	1	4	yes	—
507	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
508	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
509	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
510	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
511	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
512	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
513	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
514	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
515	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
516	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
517	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
518	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
519	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
520	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
521	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
522	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
523	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
524	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
525	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
526	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
527	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
528	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
529	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
530	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
531	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
532	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
533	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
534	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
535	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
536	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
537	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
538	#97	C₂³×C₆₇	1	8	yes	—
539	C₄×C₈	C₂³×C₆₇	1	2	yes	—
540	C₄×C₈	C₂³×C₆₇	1	2	yes	—
541	C₄×C₈	C₂³×C₆₇	1	2	yes	—
542	C₄×C₈	C₂³×C₆₇	1	2	yes	—
543	C₄×C₈	C₂³×C₆₇	1	8	yes	—
544	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
545	#41	—	2	8	no	—
546	#41	—	2	4	no	—
547	#41	—	2	4	no	—
548	#41	—	4	4	no	—
549	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
550	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
551	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
552	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
553	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
554	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
555	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
556	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
557	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
558	C₂²×C₄	—	2	4	no	—
559	C₂²×C₄	—	2	2	no	—
560	C₂²×C₄	—	2	2	no	—
561	C₂²×C₄	—	4	2	no	—
562	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
563	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
564	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
565	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
566	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
567	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
568	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
569	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
570	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
571	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
572	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
573	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
574	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
575	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
576	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
577	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
578	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
579	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
580	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
581	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
582	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
583	M₁₆	—	2	8	no	—
584	M₁₆	—	4	4	no	—
585	M₁₆	—	4	8	no	—
586	C₂×C₈	—	2	4	no	—
587	C₂×C₈	—	2	4	no	—
588	C₂×C₈	—	4	4	no	—
589	C₂×C₈	—	4	8	no	—
590	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
591	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
592	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
593	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
594	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
595	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
596	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
597	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
598	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
599	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
600	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
601	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
602	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
603	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
604	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
605	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
606	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
607	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
608	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
609	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
610	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
611	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
612	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
613	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
614	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
615	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
616	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
617	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
618	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
619	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
620	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
621	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
622	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
623	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
624	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
625	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
626	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
627	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
628	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
629	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
630	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
631	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
632	C₄²	C₂⁴×C₆₇	1	4	yes	—
633	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
634	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
635	C₂³	—	2	2	no	—
636	C₂³	—	2	2	no	—
637	C₂³	—	4	1	no	—
638	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
639	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
640	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
641	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
642	Q₈	—	2	4	no	—
643	Q₈	—	2	4	no	—
644	Q₈	—	2	4	no	—
645	D₄	—	2	8	no	—
646	D₄	—	2	8	no	—
647	D₄	—	2	4	no	—
648	D₄	—	2	4	no	—
649	D₄	—	2	4	no	—
650	D₄	—	4	4	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	2	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	2	no	—
659	C₂×C₄	—	2	8	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	2	no	—
663	C₂×C₄	—	2	1	no	—
664	C₂×C₄	—	2	4	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	1	no	—
667	C₂×C₄	—	2	1	no	—
668	C₂×C₄	—	2	4	no	—
669	C₂×C₄	—	4	1	no	—
670	C₂×C₄	—	4	1	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	4	no	—
673	C₂×C₄	#131×C₆₇	1	4	yes	complemented
674	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
675	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
676	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
677	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
678	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
679	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
680	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
681	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
682	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
683	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
684	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
685	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
686	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
687	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
688	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
689	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
690	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
691	C₈	—	2	2	no	—
692	C₈	—	2	2	no	—
693	C₈	—	2	8	no	—
694	C₈	—	2	2	no	—
695	C₈	—	2	2	no	—
696	C₂²	—	2	1	no	—
697	C₂²	—	2	1	no	—
698	C₂²	—	2	1	no	—
699	C₂²	—	2	1	no	—
700	C₂²	—	2	1	no	—
701	C₂²	—	2	4	no	—
702	C₂²	—	2	1	no	—
703	C₂²	—	2	4	no	—
704	C₂²	—	2	1	no	—
705	C₂²	—	2	4	no	—
706	C₂²	—	4	4	no	—
707	C₂²	#520×C₆₇	1	2	yes	—
708	C₂²	C₂×#131×C₆₇	1	2	yes	—
709	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
710	C₄	—	2	1	no	—
711	C₄	—	2	1	no	—
712	C₄	—	2	4	no	—
713	C₄	—	2	1	no	—
714	C₄	—	2	1	no	—
715	C₄	—	2	1	no	—
716	C₄	—	2	4	no	—
717	C₄	—	2	1	no	—
718	C₄	—	2	1	no	—
719	C₄	—	2	4	no	—
720	C₄	#520×C₆₇	1	2	yes	—
721	C₄	C₂×#131×C₆₇	1	2	yes	—
722	C₄	C₂×#131×C₆₇	1	2	yes	—
723	C₄	#568×C₆₇	1	2	yes	—
724	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
725	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
726	C₂	—	2	2	no	—
727	C₂	—	2	2	no	—
728	C₂	—	4	1	no	—
729	C₂	—	4	4	no	—
730	C₂	C₂×#520×C₆₇	1	1	yes	characteristic
731	C₂	C₂²×#131×C₆₇	1	1	yes	characteristic
732	C₂	C₂×#568×C₆₇	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×#3444	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×#2994	C₂	semidirect
3. C₆₇×#2993	C₂	semidirect
4. C₆₇×#2991	C₂	semidirect
5. C₆₇×#2990	C₂	semidirect
6. C₆₇×#2989	C₂	semidirect
7. C₆₇×#2988	C₂	semidirect
8. C₆₇×#2960	C₂	semidirect
9. C₆₇×#2933	C₂	semidirect
10. C₆₇×#2931	C₂	semidirect
11. C₆₇×#2925	C₂	semidirect
12. C₂×#431×C₆₇	C₂	semidirect
13. C₆₇×#2906	C₂	semidirect
14. C₂×#423×C₆₇	C₂	semidirect
36. #446×C₆₇	C₂²	semidirect
37. #446×C₆₇	C₂²	semidirect
38. C₈×Q₈×C₆₇	C₂²	semidirect
39. C₈×Q₈×C₆₇	C₂²	semidirect
41. #436×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
46. C₈×D₄×C₆₇	C₂²	semidirect
47. C₈×D₄×C₆₇	C₂²	semidirect
48. #433×C₆₇	C₂²	semidirect
49. #432×C₆₇	C₂²	semidirect
56. #431×C₆₇	C₂²	semidirect
58. #424×C₆₇	C₂²	semidirect
63. #423×C₆₇	C₂²	semidirect
68. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
306. C₂×C₄×C₆₇	#131	semidirect
359. #60268	C₆₇	direct (N × H)
368. #2994	C₂×C₆₇	semidirect
369. #2993	C₂×C₆₇	semidirect
370. #2991	C₂×C₆₇	semidirect
371. #2990	C₂×C₆₇	semidirect
372. #2989	C₂×C₆₇	semidirect
373. #2988	C₂×C₆₇	semidirect
374. #2960	C₂×C₆₇	semidirect
375. #2933	C₂×C₆₇	semidirect
376. #2931	C₂×C₆₇	semidirect
377. #2925	C₂×C₆₇	semidirect
378. C₂×#431	C₂×C₆₇	semidirect
379. #2906	C₂×C₆₇	semidirect
380. C₂×#423	C₂×C₆₇	semidirect
381. C₆₇	#60268	direct (N × H)
403. #446	C₂²×C₆₇	semidirect
404. #446	C₂²×C₆₇	semidirect
405. C₈×Q₈	C₂²×C₆₇	semidirect
406. C₈×Q₈	C₂²×C₆₇	semidirect
408. #436	C₂²×C₆₇	semidirect
409. #436	C₂²×C₆₇	semidirect
413. C₈×D₄	C₂²×C₆₇	semidirect
414. C₈×D₄	C₂²×C₆₇	semidirect
415. #433	C₂²×C₆₇	semidirect
416. #432	C₂²×C₆₇	semidirect
423. #431	C₂²×C₆₇	semidirect
425. #424	C₂²×C₆₇	semidirect
430. #423	C₂²×C₆₇	semidirect
435. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
673. C₂×C₄	#131×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3444	2
C₆₇×#2994	2
C₆₇×#2993	2
C₆₇×#2991	2
C₆₇×#2990	2
C₆₇×#2989	2
C₆₇×#2988	2
C₆₇×#2960	2
C₆₇×#2933	2
C₆₇×#2931	2
C₆₇×#2925	2
C₂×#431×C₆₇	2
C₆₇×#2906	2
C₂×#423×C₆₇	2
#60268	67

Subgroups — C67×#60268 #30739519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60268 #30739519