Subgroups — C₆₇×#60920 #30740171

Order 17152.

1886 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#540×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3260	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3259	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3257	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3080	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3077	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3055	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3038	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3036	C₂	1	2	yes	maximal, complemented
11	C₆₇×#2905	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
14	#542×C₆₇	C₂²	1	4	yes	complemented
15	#541×C₆₇	C₂²	1	2	yes	complemented
16	#540×C₆₇	—	2	4	no	—
17	#540×C₆₇	C₂²	1	4	yes	—
18	#540×C₆₇	C₂²	1	4	yes	—
19	#540×C₆₇	C₂²	1	2	yes	complemented
20	#539×C₆₇	—	2	2	no	—
21	#538×C₆₇	—	2	2	no	—
22	#533×C₆₇	C₂²	1	1	yes	characteristic
23	#532×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
25	#529×C₆₇	—	2	4	no	—
26	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
27	#525×C₆₇	C₂²	1	2	yes	—
28	C₂×#123×C₆₇	—	2	2	no	—
29	C₂×#121×C₆₇	—	2	2	no	—
30	C₂×#121×C₆₇	C₂²	1	2	yes	—
31	C₄×#41×C₆₇	C₂²	1	2	yes	—
32	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
33	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
34	#479×C₆₇	C₂²	1	8	yes	complemented
35	#478×C₆₇	C₂²	1	8	yes	complemented
36	#478×C₆₇	C₂²	1	8	yes	complemented
37	#476×C₆₇	C₂²	1	4	yes	complemented
38	#474×C₆₇	C₂²	1	4	yes	complemented
39	#469×C₆₇	C₂²	1	8	yes	complemented
40	#468×C₆₇	C₂²	1	8	yes	complemented
41	#466×C₆₇	C₂²	1	8	yes	complemented
42	#465×C₆₇	C₂²	1	8	yes	complemented
43	#463×C₆₇	C₂²	1	8	yes	complemented
44	#461×C₆₇	C₂²	1	4	yes	complemented
45	#460×C₆₇	C₂²	1	8	yes	complemented
46	#459×C₆₇	C₂²	1	4	yes	complemented
47	#459×C₆₇	C₂²	1	8	yes	complemented
48	#428×C₆₇	C₂²	1	4	yes	—
49	#424×C₆₇	C₂²	1	4	yes	complemented
50	#424×C₆₇	C₂²	1	4	yes	complemented
51	#423×C₆₇	C₂²	1	2	yes	—
52	#423×C₆₇	C₂²	1	2	yes	—
53	#418×C₆₇	C₂²	1	4	yes	—
54	#417×C₆₇	C₂²	1	4	yes	—
55	C₂×#41×C₆₇	C₂³	1	2	yes	—
56	C₂²×D₄×C₆₇	—	2	2	no	—
57	C₂²×D₄×C₆₇	—	2	2	no	—
58	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	—	2	2	no	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	AGL(1,ℤ/8)×C₆₇	—	4	8	no	—
62	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
63	C₂×D₈×C₆₇	C₂³	1	8	yes	—
64	C₂×M₁₆×C₆₇	—	4	4	no	—
65	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
66	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
67	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
68	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
70	#127×C₆₇	—	2	2	no	—
71	#127×C₆₇	C₂³	1	1	yes	characteristic
72	#127×C₆₇	C₂³	1	4	yes	—
73	#126×C₆₇	—	2	4	no	—
74	#126×C₆₇	—	2	4	no	—
75	#125×C₆₇	—	2	4	no	—
76	#125×C₆₇	—	2	2	no	—
77	#125×C₆₇	C₂³	1	2	yes	—
78	#125×C₆₇	C₂³	1	4	yes	—
79	#124×C₆₇	—	2	4	no	—
80	#124×C₆₇	C₂³	1	2	yes	—
81	#123×C₆₇	—	2	2	no	—
82	#123×C₆₇	—	2	8	no	—
83	#123×C₆₇	—	2	8	no	—
84	#123×C₆₇	—	2	4	no	—
85	#123×C₆₇	—	2	4	no	—
86	C₂²⋊Q₈×C₆₇	—	2	2	no	—
87	C₂²⋊Q₈×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
89	#121×C₆₇	—	2	4	no	—
90	#121×C₆₇	—	2	2	no	—
91	#121×C₆₇	—	2	8	no	—
92	#121×C₆₇	—	2	8	no	—
93	#121×C₆₇	—	2	4	no	—
94	#121×C₆₇	C₂³	1	8	yes	—
95	#121×C₆₇	C₂³	1	4	yes	—
96	#120×C₆₇	—	2	2	no	—
97	C₄×Q₈×C₆₇	—	2	2	no	—
98	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
99	C₄×D₄×C₆₇	—	2	4	no	—
100	C₄×D₄×C₆₇	—	2	4	no	—
101	C₄×D₄×C₆₇	—	2	2	no	—
102	C₄×D₄×C₆₇	—	2	2	no	—
103	C₄×D₄×C₆₇	—	2	2	no	—
104	C₄×D₄×C₆₇	C₂³	1	4	yes	—
105	C₄×D₄×C₆₇	C₂³	1	4	yes	—
106	C₄×D₄×C₆₇	C₂³	1	4	yes	—
107	#117×C₆₇	—	2	2	no	—
108	#117×C₆₇	D₄	1	4	yes	—
109	#117×C₆₇	C₂³	1	2	yes	—
110	#117×C₆₇	C₂³	1	2	yes	—
111	#117×C₆₇	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
113	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
114	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
118	C₂×C₄²×C₆₇	—	2	2	no	—
119	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
121	#107×C₆₇	C₂³	1	8	yes	—
122	#106×C₆₇	C₂³	1	8	yes	—
123	C₄⋊C₈×C₆₇	C₂³	1	8	yes	—
124	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
125	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
126	#103×C₆₇	C₂³	1	8	yes	—
127	#102×C₆₇	C₂³	1	8	yes	—
128	#102×C₆₇	C₂³	1	8	yes	—
129	#102×C₆₇	C₂³	1	8	yes	—
130	C₂⁴×C₆₇	—	2	2	no	—
131	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
132	#41×C₆₇	—	2	4	no	—
133	#41×C₆₇	—	2	4	no	—
134	C₂×Q₈×C₆₇	—	2	2	no	—
135	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
136	C₂×D₄×C₆₇	—	2	2	no	—
137	C₂×D₄×C₆₇	—	2	4	no	—
138	C₂×D₄×C₆₇	—	2	8	no	—
139	C₂×D₄×C₆₇	—	2	2	no	—
140	C₂×D₄×C₆₇	—	2	2	no	—
141	C₂×D₄×C₆₇	—	2	8	no	—
142	C₂×D₄×C₆₇	—	2	4	no	—
143	C₂×D₄×C₆₇	—	2	4	no	—
144	C₂×D₄×C₆₇	—	8	4	no	—
145	C₂×D₄×C₆₇	—	8	4	no	—
146	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	—	2	2	no	—
149	C₂²×C₄×C₆₇	—	2	2	no	—
150	C₂²×C₄×C₆₇	—	2	2	no	—
151	C₂²×C₄×C₆₇	—	2	2	no	—
152	C₂²×C₄×C₆₇	—	2	4	no	—
153	C₂²×C₄×C₆₇	—	2	4	no	—
154	C₂²×C₄×C₆₇	—	2	4	no	—
155	C₂²×C₄×C₆₇	—	8	4	no	—
156	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
159	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
162	SD₁₆×C₆₇	—	4	8	no	—
163	D₈×C₆₇	—	4	8	no	—
164	M₁₆×C₆₇	—	2	8	no	—
165	M₁₆×C₆₇	—	4	4	no	—
166	C₂×C₈×C₆₇	—	4	4	no	—
167	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
168	C₄⋊C₄×C₆₇	—	2	8	no	—
169	C₄⋊C₄×C₆₇	—	2	2	no	—
170	C₄⋊C₄×C₆₇	—	2	4	no	—
171	C₄⋊C₄×C₆₇	—	2	2	no	—
172	C₄⋊C₄×C₆₇	—	2	4	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
176	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
177	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
178	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂²⋊C₄×C₆₇	—	2	2	no	—
180	C₂²⋊C₄×C₆₇	—	2	4	no	—
181	C₂²⋊C₄×C₆₇	—	2	4	no	—
182	C₂²⋊C₄×C₆₇	—	2	4	no	—
183	C₂²⋊C₄×C₆₇	—	2	8	no	—
184	C₂²⋊C₄×C₆₇	—	2	4	no	—
185	C₂²⋊C₄×C₆₇	—	2	4	no	—
186	C₂²⋊C₄×C₆₇	—	2	4	no	—
187	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
188	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	4	no	—
191	C₄²×C₆₇	C₂×D₄	1	2	yes	—
192	C₄²×C₆₇	C₂×D₄	1	2	yes	—
193	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
194	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
195	C₄²×C₆₇	C₂⁴	1	2	yes	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	4	no	—
199	C₂³×C₆₇	—	4	2	no	—
200	C₂³×C₆₇	—	4	4	no	—
201	C₂³×C₆₇	—	4	4	no	—
202	C₂³×C₆₇	—	4	4	no	—
203	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
204	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₆₇	—	2	2	no	—
206	Q₈×C₆₇	—	2	2	no	—
207	Q₈×C₆₇	—	8	2	no	—
208	D₄×C₆₇	—	2	2	no	—
209	D₄×C₆₇	—	2	4	no	—
210	D₄×C₆₇	—	2	4	no	—
211	D₄×C₆₇	—	2	2	no	—
212	D₄×C₆₇	—	4	4	no	—
213	D₄×C₆₇	—	4	4	no	—
214	D₄×C₆₇	—	8	2	no	—
215	D₄×C₆₇	—	8	8	no	—
216	D₄×C₆₇	—	8	4	no	—
217	D₄×C₆₇	—	8	8	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	4	no	—
225	C₂×C₄×C₆₇	—	4	2	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	4	no	—
228	C₂×C₄×C₆₇	—	4	4	no	—
229	C₂×C₄×C₆₇	—	8	4	no	—
230	C₂×C₄×C₆₇	—	8	4	no	—
231	C₂×C₄×C₆₇	—	8	4	no	—
232	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
234	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
235	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
238	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
239	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
240	C₈×C₆₇	—	2	8	no	—
241	C₂²×C₆₇	—	2	2	no	—
242	C₂²×C₆₇	—	2	4	no	—
243	C₂²×C₆₇	—	2	2	no	—
244	C₂²×C₆₇	—	4	4	no	—
245	C₂²×C₆₇	—	4	4	no	—
246	C₂²×C₆₇	—	4	4	no	—
247	C₂²×C₆₇	—	4	2	no	—
248	C₂²×C₆₇	—	4	4	no	—
249	C₂²×C₆₇	—	8	2	no	—
250	C₂²×C₆₇	—	8	8	no	—
251	C₂²×C₆₇	#537	1	2	yes	—
252	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
253	C₄×C₆₇	—	2	2	no	—
254	C₄×C₆₇	—	2	2	no	—
255	C₄×C₆₇	—	2	4	no	—
256	C₄×C₆₇	—	2	2	no	—
257	C₄×C₆₇	—	4	4	no	—
258	C₄×C₆₇	—	4	4	no	—
259	C₄×C₆₇	—	4	2	no	—
260	C₄×C₆₇	#537	1	2	yes	—
261	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
262	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
263	#60920	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂×C₆₇	—	2	2	no	—
265	C₂×C₆₇	—	4	2	no	—
266	C₂×C₆₇	—	8	2	no	—
267	C₂×C₆₇	—	8	4	no	—
268	C₂×C₆₇	C₂×#537	1	1	yes	characteristic
269	C₂×C₆₇	#3585	1	2	yes	—
270	#3455	C₂×C₆₇	1	1	yes	characteristic, complemented
271	C₂×#540	C₂×C₆₇	1	1	yes	characteristic, complemented
272	#3260	C₂×C₆₇	1	4	yes	complemented
273	#3259	C₂×C₆₇	1	4	yes	complemented
274	#3257	C₂×C₆₇	1	8	yes	complemented
275	#3080	C₂×C₆₇	1	2	yes	complemented
276	#3077	C₂×C₆₇	1	2	yes	complemented
277	#3055	C₂×C₆₇	1	4	yes	complemented
278	#3038	C₂×C₆₇	1	2	yes	complemented
279	#3036	C₂×C₆₇	1	2	yes	complemented
280	#2905	C₂×C₆₇	1	1	yes	characteristic, complemented
281	C₆₇	#60920	1	1	yes	characteristic, direct product factor, Sylow
282	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	—
283	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	—
284	#542	C₂²×C₆₇	1	4	yes	complemented
285	#541	C₂²×C₆₇	1	2	yes	complemented
286	#540	—	2	4	no	—
287	#540	C₂²×C₆₇	1	4	yes	—
288	#540	C₂²×C₆₇	1	4	yes	—
289	#540	C₂²×C₆₇	1	2	yes	complemented
290	#539	—	2	2	no	—
291	#538	—	2	2	no	—
292	#533	C₂²×C₆₇	1	1	yes	characteristic
293	#532	C₂²×C₆₇	1	1	yes	characteristic
294	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
295	#529	—	2	4	no	—
296	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
297	#525	C₂²×C₆₇	1	2	yes	—
298	C₂×#123	—	2	2	no	—
299	C₂×#121	—	2	2	no	—
300	C₂×#121	C₂²×C₆₇	1	2	yes	—
301	C₄×#41	C₂²×C₆₇	1	2	yes	—
302	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
303	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
304	#479	C₂²×C₆₇	1	8	yes	complemented
305	#478	C₂²×C₆₇	1	8	yes	complemented
306	#478	C₂²×C₆₇	1	8	yes	complemented
307	#476	C₂²×C₆₇	1	4	yes	complemented
308	#474	C₂²×C₆₇	1	4	yes	complemented
309	#469	C₂²×C₆₇	1	8	yes	complemented
310	#468	C₂²×C₆₇	1	8	yes	complemented
311	#466	C₂²×C₆₇	1	8	yes	complemented
312	#465	C₂²×C₆₇	1	8	yes	complemented
313	#463	C₂²×C₆₇	1	8	yes	complemented
314	#461	C₂²×C₆₇	1	4	yes	complemented
315	#460	C₂²×C₆₇	1	8	yes	complemented
316	#459	C₂²×C₆₇	1	8	yes	complemented
317	#459	C₂²×C₆₇	1	4	yes	complemented
318	#428	C₂²×C₆₇	1	4	yes	—
319	#424	C₂²×C₆₇	1	4	yes	complemented
320	#424	C₂²×C₆₇	1	4	yes	complemented
321	#423	C₂²×C₆₇	1	2	yes	—
322	#423	C₂²×C₆₇	1	2	yes	—
323	#418	C₂²×C₆₇	1	4	yes	—
324	#417	C₂²×C₆₇	1	4	yes	—
325	C₂×#41	C₂³×C₆₇	1	2	yes	—
326	C₂²×D₄	—	2	2	no	—
327	C₂²×D₄	—	2	2	no	—
328	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
329	C₂³×C₄	—	2	2	no	—
330	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
331	AGL(1,ℤ/8)	—	4	8	no	—
332	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
333	C₂×D₈	C₂³×C₆₇	1	8	yes	—
334	C₂×M₁₆	—	4	4	no	—
335	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
336	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
337	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
338	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
340	#127	—	2	2	no	—
341	#127	C₂³×C₆₇	1	4	yes	—
342	#127	C₂³×C₆₇	1	1	yes	characteristic
343	#126	—	2	4	no	—
344	#126	—	2	4	no	—
345	#125	—	2	4	no	—
346	#125	—	2	2	no	—
347	#125	C₂³×C₆₇	1	4	yes	—
348	#125	C₂³×C₆₇	1	2	yes	—
349	#124	—	2	4	no	—
350	#124	C₂³×C₆₇	1	2	yes	—
351	#123	—	2	2	no	—
352	#123	—	2	8	no	—
353	#123	—	2	8	no	—
354	#123	—	2	4	no	—
355	#123	—	2	4	no	—
356	C₂²⋊Q₈	—	2	2	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
359	#121	—	2	4	no	—
360	#121	—	2	2	no	—
361	#121	—	2	8	no	—
362	#121	—	2	8	no	—
363	#121	—	2	4	no	—
364	#121	C₂³×C₆₇	1	8	yes	—
365	#121	C₂³×C₆₇	1	4	yes	—
366	#120	—	2	2	no	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	—	2	4	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	C₂³×C₆₇	1	4	yes	—
375	C₄×D₄	C₂³×C₆₇	1	4	yes	—
376	C₄×D₄	C₂³×C₆₇	1	4	yes	—
377	#117	—	2	2	no	—
378	#117	D₄×C₆₇	1	4	yes	—
379	#117	C₂³×C₆₇	1	2	yes	—
380	#117	C₂³×C₆₇	1	2	yes	—
381	#117	C₂³×C₆₇	1	2	yes	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
388	C₂×C₄²	—	2	2	no	—
389	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
391	#107	C₂³×C₆₇	1	8	yes	—
392	#106	C₂³×C₆₇	1	8	yes	—
393	C₄⋊C₈	C₂³×C₆₇	1	8	yes	—
394	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
395	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
396	#103	C₂³×C₆₇	1	8	yes	—
397	#102	C₂³×C₆₇	1	8	yes	—
398	#102	C₂³×C₆₇	1	8	yes	—
399	#102	C₂³×C₆₇	1	8	yes	—
400	C₂⁴	—	2	2	no	—
401	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
402	#41	—	2	4	no	—
403	#41	—	2	4	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	8	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	8	4	no	—
415	C₂×D₄	—	8	4	no	—
416	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
417	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	8	4	no	—
426	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
427	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
428	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
430	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
432	SD₁₆	—	4	8	no	—
433	D₈	—	4	8	no	—
434	M₁₆	—	2	8	no	—
435	M₁₆	—	4	4	no	—
436	C₂×C₈	—	4	4	no	—
437	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	2	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
446	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
447	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
448	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
449	C₂²⋊C₄	—	2	2	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	8	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
458	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
459	C₄²	—	2	2	no	—
460	C₄²	—	2	4	no	—
461	C₄²	C₂×D₄×C₆₇	1	2	yes	—
462	C₄²	C₂×D₄×C₆₇	1	2	yes	—
463	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
464	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
465	C₄²	C₂⁴×C₆₇	1	2	yes	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	4	no	—
469	C₂³	—	4	2	no	—
470	C₂³	—	4	4	no	—
471	C₂³	—	4	4	no	—
472	C₂³	—	4	4	no	—
473	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
474	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
475	Q₈	—	2	2	no	—
476	Q₈	—	2	2	no	—
477	Q₈	—	8	2	no	—
478	D₄	—	2	2	no	—
479	D₄	—	2	4	no	—
480	D₄	—	2	4	no	—
481	D₄	—	2	2	no	—
482	D₄	—	4	4	no	—
483	D₄	—	4	4	no	—
484	D₄	—	8	8	no	—
485	D₄	—	8	2	no	—
486	D₄	—	8	8	no	—
487	D₄	—	8	4	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	4	no	—
490	C₂×C₄	—	2	2	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	4	2	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	4	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	8	4	no	—
500	C₂×C₄	—	8	4	no	—
501	C₂×C₄	—	8	4	no	—
502	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
503	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
505	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
506	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
507	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
508	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
509	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
510	C₈	—	2	8	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	4	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	4	no	—
517	C₂²	—	4	2	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	8	2	no	—
520	C₂²	—	8	8	no	—
521	C₂²	#537×C₆₇	1	2	yes	—
522	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
523	C₄	—	2	2	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	4	no	—
526	C₄	—	2	2	no	—
527	C₄	—	4	4	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	2	no	—
530	C₄	#537×C₆₇	1	2	yes	—
531	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
532	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
533	C₂	—	2	2	no	—
534	C₂	—	4	2	no	—
535	C₂	—	8	4	no	—
536	C₂	—	8	2	no	—
537	C₂	C₂×#537×C₆₇	1	1	yes	characteristic
538	C₂	C₆₇×#3585	1	2	yes	—

	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×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3455	C₂	semidirect
2. C₂×#540×C₆₇	C₂	semidirect
3. C₆₇×#3260	C₂	semidirect
4. C₆₇×#3259	C₂	semidirect
5. C₆₇×#3257	C₂	semidirect
6. C₆₇×#3080	C₂	semidirect
7. C₆₇×#3077	C₂	semidirect
8. C₆₇×#3055	C₂	semidirect
9. C₆₇×#3038	C₂	semidirect
10. C₆₇×#3036	C₂	semidirect
11. C₆₇×#2905	C₂	semidirect
14. #542×C₆₇	C₂²	semidirect
15. #541×C₆₇	C₂²	semidirect
19. #540×C₆₇	C₂²	semidirect
34. #479×C₆₇	C₂²	semidirect
35. #478×C₆₇	C₂²	semidirect
36. #478×C₆₇	C₂²	semidirect
37. #476×C₆₇	C₂²	semidirect
38. #474×C₆₇	C₂²	semidirect
39. #469×C₆₇	C₂²	semidirect
40. #468×C₆₇	C₂²	semidirect
41. #466×C₆₇	C₂²	semidirect
42. #465×C₆₇	C₂²	semidirect
43. #463×C₆₇	C₂²	semidirect
44. #461×C₆₇	C₂²	semidirect
45. #460×C₆₇	C₂²	semidirect
46. #459×C₆₇	C₂²	semidirect
47. #459×C₆₇	C₂²	semidirect
49. #424×C₆₇	C₂²	semidirect
50. #424×C₆₇	C₂²	semidirect
263. #60920	C₆₇	direct (N × H)
270. #3455	C₂×C₆₇	semidirect
271. C₂×#540	C₂×C₆₇	semidirect
272. #3260	C₂×C₆₇	semidirect
273. #3259	C₂×C₆₇	semidirect
274. #3257	C₂×C₆₇	semidirect
275. #3080	C₂×C₆₇	semidirect
276. #3077	C₂×C₆₇	semidirect
277. #3055	C₂×C₆₇	semidirect
278. #3038	C₂×C₆₇	semidirect
279. #3036	C₂×C₆₇	semidirect
280. #2905	C₂×C₆₇	semidirect
281. C₆₇	#60920	direct (N × H)
284. #542	C₂²×C₆₇	semidirect
285. #541	C₂²×C₆₇	semidirect
289. #540	C₂²×C₆₇	semidirect
304. #479	C₂²×C₆₇	semidirect
305. #478	C₂²×C₆₇	semidirect
306. #478	C₂²×C₆₇	semidirect
307. #476	C₂²×C₆₇	semidirect
308. #474	C₂²×C₆₇	semidirect
309. #469	C₂²×C₆₇	semidirect
310. #468	C₂²×C₆₇	semidirect
311. #466	C₂²×C₆₇	semidirect
312. #465	C₂²×C₆₇	semidirect
313. #463	C₂²×C₆₇	semidirect
314. #461	C₂²×C₆₇	semidirect
315. #460	C₂²×C₆₇	semidirect
316. #459	C₂²×C₆₇	semidirect
317. #459	C₂²×C₆₇	semidirect
319. #424	C₂²×C₆₇	semidirect
320. #424	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3455	2
C₂×#540×C₆₇	2
C₆₇×#3260	2
C₆₇×#3259	2
C₆₇×#3257	2
C₆₇×#3080	2
C₆₇×#3077	2
C₆₇×#3055	2
C₆₇×#3038	2
C₆₇×#3036	2
C₆₇×#2905	2
#60920	67

Subgroups — C67×#60920 #30740171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60920 #30740171