Subgroups — #30765772

Order 17152.

342 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic
2	#22562007	C₂	1	1	yes	maximal, characteristic
3	#22562006	C₂	1	1	yes	maximal, characteristic
4	#22558189	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊(C₆₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#22558187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×Q₈⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Q₈×D₆₇	C₄	1	1	yes	characteristic
9	Q₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
10	C₈×Dic₁₃₄	—	2	1	no	—
11	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
12	#16863520	—	2	1	no	—
13	#16863520	C₄	1	1	yes	characteristic
14	C₄⋊C₈×D₆₇	—	2	1	no	—
15	C₄⋊C₈×D₆₇	C₄	1	1	yes	characteristic
16	C₄×C₈×D₆₇	—	2	1	no	—
17	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
18	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	1	yes	characteristic
19	Dic₆₇⋊C₁₆	—	2	1	no	—
20	(C₂×D₆₇)⋊C₁₆	—	2	1	no	—
21	(C₂×D₆₇)⋊C₁₆	—	2	1	no	—
22	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₁₆×Dic₆₇	—	2	1	no	—
24	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
26	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₆₇	C₈	1	1	yes	characteristic
28	C₂×C₈×D₆₇	—	2	1	no	—
29	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
30	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
31	Q₈×Dic₆₇	C₈	1	1	yes	characteristic
32	C₄×Dic₁₃₄	—	2	1	no	—
33	C₄×Dic₁₃₄	C₂×C₄	1	1	yes	characteristic
34	#14524008	—	2	1	no	—
35	#14524008	C₈	1	1	yes	characteristic
36	C₄⋊C₄×D₆₇	—	2	1	no	—
37	C₄⋊C₄×D₆₇	C₈	1	1	yes	characteristic
38	C₄²×D₆₇	—	2	1	no	—
39	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₆₇⋊C₁₆	—	2	1	no	—
41	C₂×C₆₇⋊C₁₆	—	2	1	no	—
42	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
43	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	1	yes	characteristic
44	Dic₆₇⋊C₈	—	2	1	no	—
45	Dic₆₇⋊C₈	—	2	1	no	—
46	Dic₆₇⋊C₈	—	2	1	no	—
47	C₄×C₆₇⋊C₈	—	2	1	no	—
48	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
49	C₈×Dic₆₇	—	2	1	no	—
50	C₈×Dic₆₇	—	2	1	no	—
51	C₈×Dic₆₇	D₄	1	1	yes	characteristic
52	C₈×Dic₆₇	D₄	1	1	yes	characteristic
53	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₆₇	—	2	1	no	—
55	C₂×C₁₆×C₆₇	—	2	1	no	—
56	C₄⋊C₈×C₆₇	—	2	1	no	—
57	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×Dic₁₃₄	—	2	1	no	—
61	C₂×Dic₁₃₄	C₂×C₈	1	1	yes	characteristic
62	Q₈×D₆₇	—	2	2	no	—
63	Q₈×D₆₇	C₁₆	1	4	yes	complemented
64	C₂×C₄×D₆₇	—	2	1	no	—
65	C₂×C₄×D₆₇	—	2	1	no	—
66	C₂×C₄×D₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×D₆₇	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₆₇	M₁₆	1	1	yes	characteristic
69	C₂×C₆₇⋊C₈	—	2	1	no	—
70	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₆₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₈×D₆₇	—	2	1	no	—
73	C₈×D₆₇	—	2	1	no	—
74	C₈×D₆₇	—	4	1	no	—
75	C₈×D₆₇	—	4	2	no	—
76	C₄⋊Dic₆₇	—	2	1	no	—
77	C₄⋊Dic₆₇	C₂×C₈	1	1	yes	characteristic
78	Dic₆₇⋊C₄	—	2	1	no	—
79	Dic₆₇⋊C₄	—	2	1	no	—
80	Dic₆₇⋊C₄	—	2	1	no	—
81	C₄×Dic₆₇	—	2	1	no	—
82	C₄×Dic₆₇	—	2	1	no	—
83	C₄×Dic₆₇	—	2	1	no	—
84	C₄×Dic₆₇	C₂²⋊C₄	1	1	yes	characteristic
85	C₄×Dic₆₇	C₂×C₈	1	1	yes	characteristic
86	C₄×Dic₆₇	M₁₆	1	1	yes	characteristic
87	C₆₇⋊C₁₆	—	4	1	no	—
88	C₆₇⋊C₁₆	—	4	1	no	—
89	C₂×Q₈×C₆₇	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₆₇	—	2	1	no	—
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₄⋊C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₆₇	—	2	1	no	—
96	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₁₆×C₆₇	—	4	1	no	—
98	C₁₆×C₆₇	—	4	1	no	—
99	C₂²×D₆₇	C₂²⋊C₈	1	1	yes	characteristic
100	Q₈×C₆₇	—	2	1	no	—
101	Q₈×C₆₇	C₂×C₁₆	1	2	yes	—
102	C₂×C₄×C₆₇	—	2	1	no	—
103	C₂×C₄×C₆₇	—	2	1	no	—
104	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×Dic₆₇	—	2	1	no	—
108	C₂×Dic₆₇	—	2	1	no	—
109	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
110	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
111	C₂×Dic₆₇	C₂²⋊C₈	1	1	yes	characteristic
112	C₄×D₆₇	—	2	4	no	—
113	C₄×D₆₇	—	2	1	no	—
114	C₄×D₆₇	—	2	1	no	—
115	C₄×D₆₇	—	4	1	no	—
116	C₄×D₆₇	—	4	2	no	—
117	C₄×D₆₇	C₂×C₁₆	1	2	yes	—
118	C₄×D₆₇	M₃₂	1	2	yes	—
119	Dic₁₃₄	—	2	1	no	—
120	Dic₁₃₄	—	2	4	no	—
121	Dic₁₃₄	C₂×C₁₆	1	2	yes	—
122	C₈×C₆₇	—	2	1	no	—
123	C₈×C₆₇	—	4	1	no	—
124	C₈×C₆₇	#137	1	1	yes	characteristic
125	C₈×C₆₇	#137	1	1	yes	characteristic
126	C₆₇⋊C₈	—	2	1	no	—
127	C₆₇⋊C₈	—	2	1	no	—
128	C₆₇⋊C₈	—	4	1	no	—
129	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
130	C₂×D₆₇	—	2	1	no	—
131	C₂×D₆₇	—	2	1	no	—
132	C₂×D₆₇	C₂²⋊C₁₆	1	2	yes	—
133	C₄×C₆₇	—	2	2	no	—
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	4	1	no	—
136	C₄×C₆₇	#419	1	1	yes	characteristic
137	C₄×C₆₇	#421	1	1	yes	characteristic
138	C₄×C₆₇	C₂²×C₁₆	1	1	yes	characteristic
139	C₄×C₆₇	C₂×M₃₂	1	1	yes	characteristic
140	Dic₆₇	—	2	2	no	—
141	Dic₆₇	—	2	1	no	—
142	Dic₆₇	—	2	1	no	—
143	Dic₆₇	—	4	1	no	—
144	Dic₆₇	C₂²⋊C₁₆	1	2	yes	—
145	#11637	—	67	1	no	maximal, Sylow
146	C₂×C₆₇	#1479	1	1	yes	characteristic
147	C₂×C₆₇	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
148	C₂×C₆₇	#2117	1	1	yes	characteristic
149	D₆₇	—	2	2	no	—
150	C₂×C₈×Q₈	—	67	1	no	—
151	#2162	—	67	1	no	—
152	#2162	—	67	1	no	—
153	Q₈⋊C₁₆	—	67	1	no	—
154	Q₈⋊C₁₆	—	67	1	no	—
155	Q₈⋊C₁₆	—	67	1	no	—
156	Q₈⋊C₁₆	D₆₇	1	1	yes	characteristic, complemented
157	C₆₇	#11637	1	1	yes	characteristic, complemented, Sylow
158	C₂×C₄×Q₈	—	67	1	no	—
159	C₈×Q₈	—	67	1	no	—
160	C₈×Q₈	—	67	1	no	—
161	C₈×Q₈	—	67	1	no	—
162	C₈×Q₈	—	134	1	no	—
163	C₈×Q₈	—	134	1	no	—
164	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
165	C₂×C₄⋊C₈	—	67	1	no	—
166	C₂×C₄⋊C₈	—	134	1	no	—
167	C₂×C₄×C₈	—	67	1	no	—
168	C₂×C₄×C₈	—	134	1	no	—
169	C₄⋊C₁₆	—	67	1	no	—
170	C₄⋊C₁₆	—	134	1	no	—
171	C₄⋊C₁₆	C₂×D₆₇	1	1	yes	characteristic
172	C₂²⋊C₁₆	—	134	1	no	—
173	C₂²⋊C₁₆	—	134	1	no	—
174	C₄×C₁₆	—	67	1	no	—
175	C₄×C₁₆	—	134	1	no	—
176	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
177	C₂²×Q₈	—	67	1	no	—
178	C₂²×C₈	—	67	1	no	—
179	C₂²×C₈	—	67	1	no	—
180	C₂²×C₈	—	134	1	no	—
181	C₄×Q₈	—	67	1	no	—
182	C₄×Q₈	—	67	1	no	—
183	C₄×Q₈	—	67	1	no	—
184	C₄×Q₈	—	134	1	no	—
185	C₄×Q₈	—	134	1	no	—
186	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
187	C₂×C₄⋊C₄	—	67	1	no	—
188	C₂×C₄⋊C₄	—	134	1	no	—
189	C₂×C₄²	—	67	1	no	—
190	C₂×C₄²	—	134	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	134	1	no	—
194	C₂×C₁₆	—	134	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	67	1	no	—
197	C₄⋊C₈	—	134	1	no	—
198	C₄⋊C₈	—	134	1	no	—
199	C₄⋊C₈	—	134	1	no	—
200	C₄⋊C₈	—	134	1	no	—
201	C₄⋊C₈	C₄×D₆₇	1	1	yes	characteristic
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	67	1	no	—
204	C₄×C₈	—	67	1	no	—
205	C₄×C₈	—	67	1	no	—
206	C₄×C₈	—	134	1	no	—
207	C₄×C₈	—	134	1	no	—
208	C₄×C₈	—	134	1	no	—
209	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
210	C₂×Q₈	—	67	4	no	—
211	C₂×Q₈	—	67	1	no	—
212	C₂×Q₈	—	134	1	no	—
213	C₂×Q₈	—	134	2	no	—
214	C₂×Q₈	C₈×D₆₇	1	1	yes	characteristic
215	C₂²×C₄	—	67	1	no	—
216	C₂²×C₄	—	67	1	no	—
217	C₂²×C₄	—	67	1	no	—
218	C₂²×C₄	—	134	1	no	—
219	C₂²×C₄	—	134	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	268	1	no	—
222	C₂×C₈	—	268	2	no	—
223	C₂×C₈	—	67	1	no	—
224	C₂×C₈	—	67	1	no	—
225	C₂×C₈	—	134	1	no	—
226	C₂×C₈	—	134	1	no	—
227	C₂×C₈	—	134	1	no	—
228	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
229	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	67	1	no	—
232	C₄⋊C₄	—	134	1	no	—
233	C₄⋊C₄	—	134	1	no	—
234	C₄⋊C₄	—	134	1	no	—
235	C₄⋊C₄	—	134	1	no	—
236	C₄⋊C₄	C₈×D₆₇	1	1	yes	characteristic
237	C₄²	—	2	1	no	—
238	C₄²	—	67	1	no	—
239	C₄²	—	67	1	no	—
240	C₄²	—	67	1	no	—
241	C₄²	—	134	1	no	—
242	C₄²	—	134	1	no	—
243	C₄²	—	134	1	no	—
244	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
245	C₁₆	—	4	1	no	—
246	C₁₆	—	4	1	no	—
247	C₁₆	—	268	1	no	—
248	C₁₆	—	268	1	no	—
249	C₂³	—	67	1	no	—
250	Q₈	—	2	1	no	—
251	Q₈	—	67	2	no	—
252	Q₈	—	134	4	no	—
253	Q₈	—	134	1	no	—
254	Q₈	C₁₆×D₆₇	1	2	yes	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	268	1	no	—
258	C₂×C₄	—	268	2	no	—
259	C₂×C₄	—	67	1	no	—
260	C₂×C₄	—	67	2	no	—
261	C₂×C₄	—	67	2	no	—
262	C₂×C₄	—	67	1	no	—
263	C₂×C₄	—	67	1	no	—
264	C₂×C₄	—	134	4	no	—
265	C₂×C₄	—	134	1	no	—
266	C₂×C₄	—	134	1	no	—
267	C₂×C₄	—	134	1	no	—
268	C₂×C₄	—	134	1	no	—
269	C₂×C₄	C₂²⋊C₄×D₆₇	1	1	yes	characteristic
270	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
271	C₂×C₄	M₁₆×D₆₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	268	1	no	—
275	C₈	—	134	1	no	—
276	C₈	—	134	1	no	—
277	C₈	#14524072	1	1	yes	characteristic
278	C₈	#14524075	1	1	yes	characteristic
279	C₂²	—	67	2	no	—
280	C₂²	—	134	1	no	—
281	C₂²	—	134	1	no	—
282	C₂²	C₂²⋊C₈×D₆₇	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	1	no	—
286	C₄	—	268	1	no	—
287	C₄	—	67	2	no	—
288	C₄	—	134	2	no	—
289	C₄	—	134	1	no	—
290	C₄	—	134	1	no	—
291	C₄	#16863403	1	1	yes	characteristic
292	C₄	#16863411	1	1	yes	characteristic
293	C₄	C₂×C₁₆×D₆₇	1	1	yes	characteristic
294	C₄	M₃₂×D₆₇	1	1	yes	characteristic
295	C₂	—	134	2	no	—
296	C₂	#22558602	1	1	yes	characteristic
297	C₂	C₂²⋊C₁₆×D₆₇	1	1	yes	characteristic
298	C₂	#22561835	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₇
commutator	C₄×C₆₇	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×Q₈⋊C₁₆	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
prime radical	C₂²×D₆₇	C₂²⋊C₈
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	Q₈⋊C₁₆	D₆₇
hypocenter	C₆₇	#11637
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ Q₈⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. #22558189	C₂	semidirect
5. Q₈⋊(C₆₇⋊C₁₆)	C₂	semidirect
6. #22558187	C₂	semidirect
7. C₆₇×Q₈⋊C₁₆	C₂	semidirect
63. Q₈×D₆₇	C₁₆	semidirect
156. Q₈⋊C₁₆	D₆₇	semidirect
157. C₆₇	#11637	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×D₆₇	2
#22562007	2
#22562006	2
#22558189	2
Q₈⋊(C₆₇⋊C₁₆)	2
#22558187	2
C₆₇×Q₈⋊C₁₆	2
#11637	67