Subgroups — C₄⋊C₄×D₈×C₆₇ #30701931

Order 17152.

1134 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	D₄×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1942	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1862	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1802	C₂	1	2	yes	maximal, complemented
8	C₈×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₆₇	—	2	2	no	—
13	C₄×D₈×C₆₇	—	2	8	no	—
14	C₄×D₈×C₆₇	C₄	1	8	yes	complemented
15	C₂×#107×C₆₇	C₂²	1	2	yes	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#102×C₆₇	C₂²	1	4	yes	—
20	C₂×#102×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#389×C₆₇	C₂²	1	4	yes	—
24	#385×C₆₇	—	2	4	no	—
25	#384×C₆₇	C₂²	1	2	yes	—
26	#380×C₆₇	—	2	2	no	—
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
28	#340×C₆₇	C₂²	1	4	yes	—
29	#336×C₆₇	C₂²	1	2	yes	—
30	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
31	C₂³×C₄×C₆₇	—	2	2	no	—
32	C₂³×C₄×C₆₇	—	2	4	no	—
33	C₂×D₈×C₆₇	C₂×C₄	1	4	yes	—
34	C₂×D₈×C₆₇	D₄	1	4	yes	—
35	C₂×D₈×C₆₇	Q₈	1	4	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₄×D₄×C₆₇	—	2	8	no	—
40	C₄×D₄×C₆₇	—	2	4	no	—
41	C₄×D₄×C₆₇	—	4	4	no	—
42	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	4	8	no	—
47	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
53	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#107×C₆₇	—	2	4	no	—
56	#107×C₆₇	—	4	2	no	—
57	#107×C₆₇	C₂×C₄	1	4	yes	complemented
58	C₄⋊C₈×C₆₇	—	2	2	no	—
59	#102×C₆₇	—	2	8	no	—
60	#102×C₆₇	C₂×C₄	1	8	yes	complemented
61	C₄×C₈×C₆₇	—	2	4	no	—
62	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
63	#95×C₆₇	—	2	4	no	—
64	#95×C₆₇	D₄	1	2	yes	—
65	C₂⁴×C₆₇	—	2	2	no	—
66	C₂×D₄×C₆₇	—	2	2	no	—
67	C₂×D₄×C₆₇	—	2	2	no	—
68	C₂×D₄×C₆₇	—	2	2	no	—
69	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
70	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
71	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₆₇	—	2	2	no	—
73	C₂²×C₄×C₆₇	—	2	8	no	—
74	C₂²×C₄×C₆₇	—	2	4	no	—
75	C₂²×C₄×C₆₇	—	4	8	no	—
76	C₂²×C₄×C₆₇	—	4	4	no	—
77	C₂²×C₄×C₆₇	—	4	4	no	—
78	C₂²×C₄×C₆₇	—	4	8	no	—
79	C₂²×C₄×C₆₇	—	8	4	no	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
85	D₈×C₆₇	C₄⋊C₄	1	16	yes	direct product factor
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	2	1	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	—	2	2	no	—
91	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	4	no	—
98	C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₄⋊C₄×C₆₇	—	4	8	no	—
100	C₄⋊C₄×C₆₇	—	4	4	no	—
101	C₄⋊C₄×C₆₇	—	8	2	no	—
102	C₄⋊C₄×C₆₇	D₈	1	4	yes	direct product factor
103	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
104	C₂²⋊C₄×C₆₇	—	2	8	no	—
105	C₂²⋊C₄×C₆₇	—	4	8	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	2	no	—
108	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₆₇	#41	1	2	yes	—
110	C₂³×C₆₇	—	2	4	no	—
111	C₂³×C₆₇	—	2	4	no	—
112	C₂³×C₆₇	—	2	4	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	4	2	no	—
115	C₂³×C₆₇	—	4	2	no	—
116	C₂³×C₆₇	—	4	2	no	—
117	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
118	D₄×C₆₇	—	2	4	no	—
119	D₄×C₆₇	—	2	4	no	—
120	D₄×C₆₇	—	2	4	no	—
121	D₄×C₆₇	C₂×C₄⋊C₄	1	8	yes	complemented
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	4	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	16	no	—
133	C₂×C₄×C₆₇	—	4	4	no	—
134	C₂×C₄×C₆₇	—	4	4	no	—
135	C₂×C₄×C₆₇	—	4	4	no	—
136	C₂×C₄×C₆₇	—	8	16	no	—
137	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
138	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
140	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₆₇	#135	1	2	yes	—
142	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₆₇	—	2	2	no	—
149	C₈×C₆₇	—	4	2	no	—
150	C₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	complemented
151	C₂²×C₆₇	—	2	8	no	—
152	C₂²×C₆₇	—	4	4	no	—
153	C₂²×C₆₇	—	4	4	no	—
154	C₂²×C₆₇	—	4	4	no	—
155	C₂²×C₆₇	—	4	4	no	—
156	C₂²×C₆₇	—	4	4	no	—
157	C₂²×C₆₇	—	4	4	no	—
158	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₆₇	D₄²	1	1	yes	characteristic
160	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
161	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
162	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
163	C₂²×C₆₇	#576	1	1	yes	characteristic
164	C₂²×C₆₇	#578	1	1	yes	characteristic
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	4	4	no	—
168	C₄×C₆₇	—	4	2	no	—
169	C₄×C₆₇	—	8	4	no	—
170	C₄×C₆₇	C₄×D₈	1	4	yes	complemented
171	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
172	C₄×C₆₇	#520	1	1	yes	characteristic
173	C₄×C₆₇	#547	1	1	yes	characteristic
174	C₄×C₆₇	#554	1	1	yes	characteristic
175	C₄⋊C₄×D₈	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	4	8	no	—
177	C₂×C₆₇	D₄×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₆₇	C₂×C₄×D₈	1	1	yes	characteristic
179	C₂×C₆₇	#2951	1	1	yes	characteristic
180	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
181	C₂×C₆₇	#3283	1	1	yes	characteristic
182	C₂×C₆₇	Q₈×D₈	1	1	yes	characteristic
183	C₂×C₆₇	#3380	1	1	yes	characteristic
184	C₂×C₄×D₈	C₂×C₆₇	1	2	yes	—
185	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
186	D₄×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
187	#1942	C₂×C₆₇	1	2	yes	complemented
188	#1862	C₂×C₆₇	1	4	yes	complemented
189	#1845	C₂×C₆₇	1	1	yes	characteristic, complemented
190	#1802	C₂×C₆₇	1	2	yes	complemented
191	C₈×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
192	C₆₇	C₄⋊C₄×D₈	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
194	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	—
195	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
196	C₂²×C₄⋊C₄	—	2	2	no	—
197	C₄×D₈	—	2	8	no	—
198	C₄×D₈	C₄×C₆₇	1	8	yes	complemented
199	C₂×#107	C₂²×C₆₇	1	2	yes	—
200	C₂×#107	C₂²×C₆₇	1	2	yes	—
201	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
202	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
203	C₂×#102	C₂²×C₆₇	1	4	yes	—
204	C₂×#102	C₂²×C₆₇	1	2	yes	—
205	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
206	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
207	#389	C₂²×C₆₇	1	4	yes	—
208	#385	—	2	4	no	—
209	#384	C₂²×C₆₇	1	2	yes	—
210	#380	—	2	2	no	—
211	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
212	#340	C₂²×C₆₇	1	4	yes	—
213	#336	C₂²×C₆₇	1	2	yes	—
214	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
215	C₂³×C₄	—	2	2	no	—
216	C₂³×C₄	—	2	4	no	—
217	C₂×D₈	C₂×C₄×C₆₇	1	4	yes	—
218	C₂×D₈	D₄×C₆₇	1	4	yes	—
219	C₂×D₈	Q₈×C₆₇	1	4	yes	—
220	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
221	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
223	C₄×D₄	—	2	8	no	—
224	C₄×D₄	—	2	4	no	—
225	C₄×D₄	—	4	4	no	—
226	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	complemented
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	4	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	4	8	no	—
231	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
232	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
234	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	4	no	—
237	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
238	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
239	#107	—	2	4	no	—
240	#107	—	4	2	no	—
241	#107	C₂×C₄×C₆₇	1	4	yes	complemented
242	C₄⋊C₈	—	2	2	no	—
243	#102	—	2	8	no	—
244	#102	C₂×C₄×C₆₇	1	8	yes	complemented
245	C₄×C₈	—	2	4	no	—
246	C₄×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
247	#95	—	2	4	no	—
248	#95	D₄×C₆₇	1	2	yes	—
249	C₂⁴	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
254	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
255	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	8	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	8	no	—
260	C₂²×C₄	—	4	4	no	—
261	C₂²×C₄	—	4	4	no	—
262	C₂²×C₄	—	4	8	no	—
263	C₂²×C₄	—	8	4	no	—
264	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
265	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
269	D₈	C₄⋊C₄×C₆₇	1	16	yes	direct product factor
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
276	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
277	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
278	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	4	8	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	D₈×C₆₇	1	4	yes	direct product factor
287	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
288	C₂²⋊C₄	—	2	8	no	—
289	C₂²⋊C₄	—	4	8	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
293	C₄²	#41×C₆₇	1	2	yes	—
294	C₂³	—	2	4	no	—
295	C₂³	—	2	4	no	—
296	C₂³	—	2	4	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
302	D₄	—	2	4	no	—
303	D₄	—	2	4	no	—
304	D₄	—	2	4	no	—
305	D₄	C₂×C₄⋊C₄×C₆₇	1	8	yes	complemented
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	16	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	8	16	no	—
321	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
322	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
323	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
324	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
325	C₂×C₄	#135×C₆₇	1	2	yes	—
326	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
328	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
329	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
330	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
331	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	4	2	no	—
334	C₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	complemented
335	C₂²	—	2	8	no	—
336	C₂²	—	4	4	no	—
337	C₂²	—	4	4	no	—
338	C₂²	—	4	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
343	C₂²	D₄²×C₆₇	1	1	yes	characteristic
344	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
345	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
346	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
347	C₂²	#576×C₆₇	1	1	yes	characteristic
348	C₂²	#578×C₆₇	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	4	no	—
353	C₄	—	8	4	no	—
354	C₄	C₄×D₈×C₆₇	1	4	yes	complemented
355	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
356	C₄	#520×C₆₇	1	1	yes	characteristic
357	C₄	#547×C₆₇	1	1	yes	characteristic
358	C₄	#554×C₆₇	1	1	yes	characteristic
359	C₂	—	4	8	no	—
360	C₂	D₄×C₄⋊C₄×C₆₇	1	1	yes	characteristic
361	C₂	C₂×C₄×D₈×C₆₇	1	1	yes	characteristic
362	C₂	C₆₇×#2951	1	1	yes	characteristic
363	C₂	D₄×D₈×C₆₇	1	1	yes	characteristic
364	C₂	C₆₇×#3283	1	1	yes	characteristic
365	C₂	Q₈×D₈×C₆₇	1	1	yes	characteristic
366	C₂	C₆₇×#3380	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₄⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. D₄×C₄⋊C₄×C₆₇	C₂	semidirect
4. C₆₇×#1942	C₂	semidirect
5. C₆₇×#1862	C₂	semidirect
6. C₆₇×#1845	C₂	semidirect
7. C₆₇×#1802	C₂	semidirect
8. C₈×C₄⋊C₄×C₆₇	C₂	semidirect
14. C₄×D₈×C₆₇	C₄	semidirect
42. C₄×D₄×C₆₇	C₂×C₄	semidirect
57. #107×C₆₇	C₂×C₄	semidirect
60. #102×C₆₇	C₂×C₄	semidirect
62. C₄×C₈×C₆₇	C₂×C₄	semidirect
85. D₈×C₆₇	C₄⋊C₄	direct (N × H)
102. C₄⋊C₄×C₆₇	D₈	direct (N × H)
121. D₄×C₆₇	C₂×C₄⋊C₄	semidirect
150. C₈×C₆₇	C₂×C₄⋊C₄	semidirect
170. C₄×C₆₇	C₄×D₈	semidirect
175. C₄⋊C₄×D₈	C₆₇	direct (N × H)
186. D₄×C₄⋊C₄	C₂×C₆₇	semidirect
187. #1942	C₂×C₆₇	semidirect
188. #1862	C₂×C₆₇	semidirect
189. #1845	C₂×C₆₇	semidirect
190. #1802	C₂×C₆₇	semidirect
191. C₈×C₄⋊C₄	C₂×C₆₇	semidirect
192. C₆₇	C₄⋊C₄×D₈	direct (N × H)
198. C₄×D₈	C₄×C₆₇	semidirect
226. C₄×D₄	C₂×C₄×C₆₇	semidirect
241. #107	C₂×C₄×C₆₇	semidirect
244. #102	C₂×C₄×C₆₇	semidirect
246. C₄×C₈	C₂×C₄×C₆₇	semidirect
269. D₈	C₄⋊C₄×C₆₇	direct (N × H)
286. C₄⋊C₄	D₈×C₆₇	direct (N × H)
305. D₄	C₂×C₄⋊C₄×C₆₇	semidirect
334. C₈	C₂×C₄⋊C₄×C₆₇	semidirect
354. C₄	C₄×D₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₆₇	2
D₄×C₄⋊C₄×C₆₇	2
C₆₇×#1942	2
C₆₇×#1862	2
C₆₇×#1845	2
C₆₇×#1802	2
C₈×C₄⋊C₄×C₆₇	2
C₄⋊C₄×D₈	67

Subgroups — C4⋊C4×D8×C67 #30701931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄⋊C₄×D₈×C₆₇ #30701931