Subgroups — C₆₇×#61585 #30740836

Order 17152.

2378 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#547×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3330	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3272	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3206	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3193	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3100	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3006	C₂	1	4	yes	maximal, complemented
9	C₆₇×#2975	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
11	C₂²×#41×C₆₇	C₂²	1	2	yes	complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
13	#547×C₆₇	—	2	4	no	—
14	#547×C₆₇	C₂²	1	8	yes	complemented
15	#547×C₆₇	C₂²	1	4	yes	complemented
16	#546×C₆₇	—	2	4	no	—
17	D₄²×C₆₇	C₂²	1	4	yes	complemented
18	#537×C₆₇	—	2	1	no	—
19	#534×C₆₇	—	2	2	no	—
20	#532×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
22	C₂×#123×C₆₇	—	2	2	no	—
23	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#121×C₆₇	—	2	1	no	—
25	C₂×#121×C₆₇	C₂²	1	4	yes	—
26	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
29	#492×C₆₇	C₂²	1	4	yes	complemented
30	#487×C₆₇	C₂²	1	4	yes	complemented
31	#484×C₆₇	C₂²	1	8	yes	complemented
32	#480×C₆₇	C₂²	1	4	yes	complemented
33	#474×C₆₇	C₂²	1	4	yes	complemented
34	#468×C₆₇	C₂²	1	16	yes	complemented
35	#461×C₆₇	C₂²	1	8	yes	complemented
36	#459×C₆₇	C₂²	1	4	yes	complemented
37	#450×C₆₇	C₂²	1	8	yes	complemented
38	#449×C₆₇	C₂²	1	16	yes	complemented
39	#448×C₆₇	C₂²	1	8	yes	complemented
40	#447×C₆₇	C₂²	1	4	yes	complemented
41	C₄×D₈×C₆₇	C₂²	1	8	yes	complemented
42	#436×C₆₇	C₂²	1	8	yes	complemented
43	#431×C₆₇	C₂²	1	2	yes	complemented
44	#417×C₆₇	C₂²	1	8	yes	complemented
45	#407×C₆₇	C₂²	1	4	yes	complemented
46	H(5,𝔽₂)×C₆₇	—	4	8	no	—
47	C₂×#41×C₆₇	—	2	4	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	4	no	—
50	C₂×#41×C₆₇	—	4	8	no	—
51	C₂×#41×C₆₇	C₂³	1	8	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
53	C₂²×D₄×C₆₇	—	2	2	no	—
54	C₂²×D₄×C₆₇	—	2	4	no	—
55	C₂²×D₄×C₆₇	—	2	4	no	—
56	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
57	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
59	C₂³×C₄×C₆₇	—	2	1	no	—
60	C₂³×C₄×C₆₇	—	2	2	no	—
61	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
62	AGL(1,ℤ/8)×C₆₇	—	4	16	no	—
63	C₂×SD₁₆×C₆₇	C₂³	1	16	yes	—
64	C₂×D₈×C₆₇	C₂³	1	8	yes	—
65	C₂×M₁₆×C₆₇	C₂³	1	8	yes	—
66	C₄⋊Q₈×C₆₇	—	2	2	no	—
67	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
68	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
69	#127×C₆₇	C₂³	1	2	yes	—
70	#124×C₆₇	—	2	2	no	—
71	#123×C₆₇	—	2	16	no	—
72	#123×C₆₇	—	2	8	no	—
73	C₂²⋊Q₈×C₆₇	—	2	4	no	—
74	C₂²⋊Q₈×C₆₇	—	2	4	no	—
75	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
76	#121×C₆₇	—	2	8	no	—
77	#121×C₆₇	—	2	4	no	—
78	#121×C₆₇	—	2	4	no	—
79	#121×C₆₇	—	2	2	no	—
80	#121×C₆₇	—	2	4	no	—
81	#121×C₆₇	C₂³	1	4	yes	—
82	#121×C₆₇	C₂³	1	8	yes	—
83	#120×C₆₇	—	2	8	no	—
84	C₄×D₄×C₆₇	—	2	2	no	—
85	C₄×D₄×C₆₇	—	2	2	no	—
86	C₄×D₄×C₆₇	D₄	1	4	yes	—
87	C₄×D₄×C₆₇	C₂³	1	4	yes	—
88	C₄×D₄×C₆₇	C₂³	1	4	yes	—
89	C₄×D₄×C₆₇	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
94	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
97	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
98	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#107×C₆₇	C₂³	1	8	yes	—
103	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
104	#103×C₆₇	C₂³	1	16	yes	—
105	#102×C₆₇	C₂³	1	8	yes	—
106	#102×C₆₇	C₂³	1	8	yes	—
107	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
108	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
109	C₄×C₈×C₆₇	C₂³	1	4	yes	—
110	C₂⁴×C₆₇	—	2	1	no	—
111	C₂⁴×C₆₇	—	2	4	no	—
112	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
113	#41×C₆₇	—	2	16	no	—
114	#41×C₆₇	—	4	8	no	—
115	#41×C₆₇	—	4	16	no	—
116	#41×C₆₇	—	4	16	no	—
117	#41×C₆₇	—	4	8	no	—
118	C₂×Q₈×C₆₇	—	2	2	no	—
119	C₂×Q₈×C₆₇	—	2	2	no	—
120	C₂×Q₈×C₆₇	—	4	4	no	—
121	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
122	C₂×D₄×C₆₇	—	2	4	no	—
123	C₂×D₄×C₆₇	—	2	8	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	8	no	—
126	C₂×D₄×C₆₇	—	2	4	no	—
127	C₂×D₄×C₆₇	—	2	1	no	—
128	C₂×D₄×C₆₇	—	2	2	no	—
129	C₂×D₄×C₆₇	—	2	2	no	—
130	C₂×D₄×C₆₇	—	2	4	no	—
131	C₂×D₄×C₆₇	—	4	8	no	—
132	C₂×D₄×C₆₇	—	4	16	no	—
133	C₂×D₄×C₆₇	—	4	8	no	—
134	C₂×D₄×C₆₇	—	4	16	no	—
135	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
136	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
137	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	2	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	4	no	—
142	C₂²×C₄×C₆₇	—	2	8	no	—
143	C₂²×C₄×C₆₇	—	2	4	no	—
144	C₂²×C₄×C₆₇	—	4	4	no	—
145	C₂²×C₄×C₆₇	—	4	8	no	—
146	C₂²×C₄×C₆₇	—	4	4	no	—
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	SD₁₆×C₆₇	—	4	16	no	—
154	D₈×C₆₇	—	2	16	no	—
155	M₁₆×C₆₇	—	4	8	no	—
156	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
157	C₄⋊C₄×C₆₇	—	2	4	no	—
158	C₄⋊C₄×C₆₇	—	2	8	no	—
159	C₄⋊C₄×C₆₇	—	2	4	no	—
160	C₄⋊C₄×C₆₇	—	2	1	no	—
161	C₄⋊C₄×C₆₇	—	2	4	no	—
162	C₄⋊C₄×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
165	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
166	C₂²⋊C₄×C₆₇	—	2	8	no	—
167	C₂²⋊C₄×C₆₇	—	2	4	no	—
168	C₂²⋊C₄×C₆₇	—	2	4	no	—
169	C₂²⋊C₄×C₆₇	—	2	8	no	—
170	C₂²⋊C₄×C₆₇	—	2	4	no	—
171	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
172	C₄²×C₆₇	C₂×D₄	1	2	yes	—
173	C₄²×C₆₇	C₂⁴	1	2	yes	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	2	4	no	—
177	C₂³×C₆₇	—	2	4	no	—
178	C₂³×C₆₇	—	4	2	no	—
179	C₂³×C₆₇	—	4	2	no	—
180	C₂³×C₆₇	—	4	2	no	—
181	C₂³×C₆₇	—	4	8	no	—
182	C₂³×C₆₇	—	4	2	no	—
183	C₂³×C₆₇	—	8	4	no	—
184	C₂³×C₆₇	—	8	4	no	—
185	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
186	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
187	Q₈×C₆₇	—	2	8	no	—
188	Q₈×C₆₇	—	4	4	no	—
189	Q₈×C₆₇	—	4	4	no	—
190	D₄×C₆₇	—	2	4	no	—
191	D₄×C₆₇	—	2	8	no	—
192	D₄×C₆₇	—	2	2	no	—
193	D₄×C₆₇	—	2	8	no	—
194	D₄×C₆₇	—	4	16	no	—
195	D₄×C₆₇	—	4	16	no	—
196	D₄×C₆₇	—	4	8	no	—
197	D₄×C₆₇	—	4	4	no	—
198	D₄×C₆₇	—	8	2	no	—
199	D₄×C₆₇	—	8	4	no	—
200	D₄×C₆₇	H(5,𝔽₂)	1	4	yes	complemented
201	C₂×C₄×C₆₇	—	2	8	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	2	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	4	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	16	no	—
210	C₂×C₄×C₆₇	—	4	4	no	—
211	C₂×C₄×C₆₇	—	4	8	no	—
212	C₂×C₄×C₆₇	—	4	2	no	—
213	C₂×C₄×C₆₇	—	4	8	no	—
214	C₂×C₄×C₆₇	—	4	8	no	—
215	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	complemented
216	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
221	C₈×C₆₇	—	2	8	no	—
222	C₂²×C₆₇	—	2	1	no	—
223	C₂²×C₆₇	—	2	1	no	—
224	C₂²×C₆₇	—	2	4	no	—
225	C₂²×C₆₇	—	2	4	no	—
226	C₂²×C₆₇	—	2	4	no	—
227	C₂²×C₆₇	—	2	4	no	—
228	C₂²×C₆₇	—	4	8	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	—	4	2	no	—
231	C₂²×C₆₇	—	4	4	no	—
232	C₂²×C₆₇	—	4	4	no	—
233	C₂²×C₆₇	—	4	4	no	—
234	C₂²×C₆₇	—	8	4	no	—
235	C₂²×C₆₇	—	16	8	no	—
236	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
237	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
238	C₄×C₆₇	—	2	2	no	—
239	C₄×C₆₇	—	2	4	no	—
240	C₄×C₆₇	—	2	1	no	—
241	C₄×C₆₇	—	4	4	no	—
242	C₄×C₆₇	—	4	2	no	—
243	C₄×C₆₇	—	4	4	no	—
244	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
245	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
246	#61585	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂×C₆₇	—	2	2	no	—
248	C₂×C₆₇	—	4	4	no	—
249	C₂×C₆₇	—	4	4	no	—
250	C₂×C₆₇	—	8	4	no	—
251	C₂×C₆₇	#3484	1	1	yes	characteristic
252	C₂×C₆₇	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₂×C₆₇	#3586	1	1	yes	characteristic
254	#3496	C₂×C₆₇	1	1	yes	characteristic, complemented
255	C₂×#547	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#3330	C₂×C₆₇	1	8	yes	complemented
257	#3272	C₂×C₆₇	1	8	yes	complemented
258	#3206	C₂×C₆₇	1	2	yes	complemented
259	#3193	C₂×C₆₇	1	4	yes	complemented
260	#3100	C₂×C₆₇	1	2	yes	complemented
261	#3006	C₂×C₆₇	1	4	yes	complemented
262	#2975	C₂×C₆₇	1	1	yes	characteristic, complemented
263	C₆₇	#61585	1	1	yes	characteristic, direct product factor, Sylow
264	C₂×H(5,𝔽₂)	—	2	1	no	—
265	C₂²×#41	C₂²×C₆₇	1	2	yes	complemented
266	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
267	#547	—	2	4	no	—
268	#547	C₂²×C₆₇	1	8	yes	complemented
269	#547	C₂²×C₆₇	1	4	yes	complemented
270	#546	—	2	4	no	—
271	D₄²	C₂²×C₆₇	1	4	yes	complemented
272	#537	—	2	1	no	—
273	#534	—	2	2	no	—
274	#532	C₂²×C₆₇	1	2	yes	—
275	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
276	C₂×#123	—	2	2	no	—
277	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
278	C₂×#121	—	2	1	no	—
279	C₂×#121	C₂²×C₆₇	1	4	yes	—
280	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
282	C₂²×C₄⋊C₄	—	2	1	no	—
283	#492	C₂²×C₆₇	1	4	yes	complemented
284	#487	C₂²×C₆₇	1	4	yes	complemented
285	#484	C₂²×C₆₇	1	8	yes	complemented
286	#480	C₂²×C₆₇	1	4	yes	complemented
287	#474	C₂²×C₆₇	1	4	yes	complemented
288	#468	C₂²×C₆₇	1	16	yes	complemented
289	#461	C₂²×C₆₇	1	8	yes	complemented
290	#459	C₂²×C₆₇	1	4	yes	complemented
291	#450	C₂²×C₆₇	1	8	yes	complemented
292	#449	C₂²×C₆₇	1	16	yes	complemented
293	#448	C₂²×C₆₇	1	8	yes	complemented
294	#447	C₂²×C₆₇	1	4	yes	complemented
295	C₄×D₈	C₂²×C₆₇	1	8	yes	complemented
296	#436	C₂²×C₆₇	1	8	yes	complemented
297	#431	C₂²×C₆₇	1	2	yes	complemented
298	#417	C₂²×C₆₇	1	8	yes	complemented
299	#407	C₂²×C₆₇	1	4	yes	complemented
300	H(5,𝔽₂)	—	4	8	no	—
301	C₂×#41	—	2	4	no	—
302	C₂×#41	—	2	2	no	—
303	C₂×#41	—	2	4	no	—
304	C₂×#41	—	4	8	no	—
305	C₂×#41	C₂³×C₆₇	1	8	yes	—
306	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
307	C₂²×D₄	—	2	2	no	—
308	C₂²×D₄	—	2	4	no	—
309	C₂²×D₄	—	2	4	no	—
310	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
311	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
313	C₂³×C₄	—	2	1	no	—
314	C₂³×C₄	—	2	2	no	—
315	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
316	AGL(1,ℤ/8)	—	4	16	no	—
317	C₂×SD₁₆	C₂³×C₆₇	1	16	yes	—
318	C₂×D₈	C₂³×C₆₇	1	8	yes	—
319	C₂×M₁₆	C₂³×C₆₇	1	8	yes	—
320	C₄⋊Q₈	—	2	2	no	—
321	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
322	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
323	#127	C₂³×C₆₇	1	2	yes	—
324	#124	—	2	2	no	—
325	#123	—	2	16	no	—
326	#123	—	2	8	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	4	no	—
329	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
330	#121	—	2	8	no	—
331	#121	—	2	4	no	—
332	#121	—	2	4	no	—
333	#121	—	2	2	no	—
334	#121	—	2	4	no	—
335	#121	C₂³×C₆₇	1	4	yes	—
336	#121	C₂³×C₆₇	1	8	yes	—
337	#120	—	2	8	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	D₄×C₆₇	1	4	yes	—
341	C₄×D₄	C₂³×C₆₇	1	4	yes	—
342	C₄×D₄	C₂³×C₆₇	1	4	yes	—
343	C₄×D₄	C₂³×C₆₇	1	4	yes	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	8	no	—
348	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₂²⋊C₄	—	2	4	no	—
354	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
355	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
356	#107	C₂³×C₆₇	1	8	yes	—
357	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
358	#103	C₂³×C₆₇	1	16	yes	—
359	#102	C₂³×C₆₇	1	8	yes	—
360	#102	C₂³×C₆₇	1	8	yes	—
361	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
362	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
363	C₄×C₈	C₂³×C₆₇	1	4	yes	—
364	C₂⁴	—	2	1	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
367	#41	—	2	16	no	—
368	#41	—	4	8	no	—
369	#41	—	4	16	no	—
370	#41	—	4	16	no	—
371	#41	—	4	8	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	8	no	—
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	8	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	1	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	4	8	no	—
386	C₂×D₄	—	4	16	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	16	no	—
389	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
390	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
391	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	4	no	—
395	C₂²×C₄	—	2	4	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	8	no	—
400	C₂²×C₄	—	4	4	no	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	SD₁₆	—	4	16	no	—
408	D₈	—	2	16	no	—
409	M₁₆	—	4	8	no	—
410	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	2	4	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
418	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
419	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	4	no	—
422	C₂²⋊C₄	—	2	4	no	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
426	C₄²	C₂×D₄×C₆₇	1	2	yes	—
427	C₄²	C₂⁴×C₆₇	1	2	yes	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	2	no	—
430	C₂³	—	2	4	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	8	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	8	4	no	—
438	C₂³	—	8	4	no	—
439	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
440	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
441	Q₈	—	2	8	no	—
442	Q₈	—	4	4	no	—
443	Q₈	—	4	4	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	8	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	8	no	—
448	D₄	—	4	16	no	—
449	D₄	—	4	16	no	—
450	D₄	—	4	8	no	—
451	D₄	—	4	4	no	—
452	D₄	—	8	4	no	—
453	D₄	—	8	2	no	—
454	D₄	H(5,𝔽₂)×C₆₇	1	4	yes	complemented
455	C₂×C₄	—	2	8	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	16	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	8	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	8	no	—
468	C₂×C₄	—	4	8	no	—
469	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	complemented
470	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
471	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
472	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
473	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
475	C₈	—	2	8	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	2	4	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	4	8	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	4	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	8	4	no	—
489	C₂²	—	16	8	no	—
490	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
491	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
492	C₄	—	2	2	no	—
493	C₄	—	2	4	no	—
494	C₄	—	2	1	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	4	no	—
498	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	4	yes	—
499	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
500	C₂	—	2	2	no	—
501	C₂	—	4	4	no	—
502	C₂	—	4	4	no	—
503	C₂	—	8	4	no	—
504	C₂	C₆₇×#3484	1	1	yes	characteristic
505	C₂	C₂²×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
506	C₂	C₆₇×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3496	C₂	semidirect
2. C₂×#547×C₆₇	C₂	semidirect
3. C₆₇×#3330	C₂	semidirect
4. C₆₇×#3272	C₂	semidirect
5. C₆₇×#3206	C₂	semidirect
6. C₆₇×#3193	C₂	semidirect
7. C₆₇×#3100	C₂	semidirect
8. C₆₇×#3006	C₂	semidirect
9. C₆₇×#2975	C₂	semidirect
11. C₂²×#41×C₆₇	C₂²	semidirect
12. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
14. #547×C₆₇	C₂²	semidirect
15. #547×C₆₇	C₂²	semidirect
17. D₄²×C₆₇	C₂²	semidirect
21. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
23. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
27. C₂×C₄×D₄×C₆₇	C₂²	semidirect
29. #492×C₆₇	C₂²	semidirect
30. #487×C₆₇	C₂²	semidirect
31. #484×C₆₇	C₂²	semidirect
32. #480×C₆₇	C₂²	semidirect
33. #474×C₆₇	C₂²	semidirect
34. #468×C₆₇	C₂²	semidirect
35. #461×C₆₇	C₂²	semidirect
36. #459×C₆₇	C₂²	semidirect
37. #450×C₆₇	C₂²	semidirect
38. #449×C₆₇	C₂²	semidirect
39. #448×C₆₇	C₂²	semidirect
40. #447×C₆₇	C₂²	semidirect
41. C₄×D₈×C₆₇	C₂²	semidirect
42. #436×C₆₇	C₂²	semidirect
43. #431×C₆₇	C₂²	semidirect
44. #417×C₆₇	C₂²	semidirect
45. #407×C₆₇	C₂²	semidirect
200. D₄×C₆₇	H(5,𝔽₂)	semidirect
215. C₂×C₄×C₆₇	AGL(1,ℤ/8)	semidirect
246. #61585	C₆₇	direct (N × H)
254. #3496	C₂×C₆₇	semidirect
255. C₂×#547	C₂×C₆₇	semidirect
256. #3330	C₂×C₆₇	semidirect
257. #3272	C₂×C₆₇	semidirect
258. #3206	C₂×C₆₇	semidirect
259. #3193	C₂×C₆₇	semidirect
260. #3100	C₂×C₆₇	semidirect
261. #3006	C₂×C₆₇	semidirect
262. #2975	C₂×C₆₇	semidirect
263. C₆₇	#61585	direct (N × H)
265. C₂²×#41	C₂²×C₆₇	semidirect
266. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
268. #547	C₂²×C₆₇	semidirect
269. #547	C₂²×C₆₇	semidirect
271. D₄²	C₂²×C₆₇	semidirect
275. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
277. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
281. C₂×C₄×D₄	C₂²×C₆₇	semidirect
283. #492	C₂²×C₆₇	semidirect
284. #487	C₂²×C₆₇	semidirect
285. #484	C₂²×C₆₇	semidirect
286. #480	C₂²×C₆₇	semidirect
287. #474	C₂²×C₆₇	semidirect
288. #468	C₂²×C₆₇	semidirect
289. #461	C₂²×C₆₇	semidirect
290. #459	C₂²×C₆₇	semidirect
291. #450	C₂²×C₆₇	semidirect
292. #449	C₂²×C₆₇	semidirect
293. #448	C₂²×C₆₇	semidirect
294. #447	C₂²×C₆₇	semidirect
295. C₄×D₈	C₂²×C₆₇	semidirect
296. #436	C₂²×C₆₇	semidirect
297. #431	C₂²×C₆₇	semidirect
298. #417	C₂²×C₆₇	semidirect
299. #407	C₂²×C₆₇	semidirect
454. D₄	H(5,𝔽₂)×C₆₇	semidirect
469. C₂×C₄	AGL(1,ℤ/8)×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3496	2
C₂×#547×C₆₇	2
C₆₇×#3330	2
C₆₇×#3272	2
C₆₇×#3206	2
C₆₇×#3193	2
C₆₇×#3100	2
C₆₇×#3006	2
C₆₇×#2975	2
#61585	67

Subgroups — C67×#61585 #30740836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61585 #30740836