Subgroups — #30754758

Order 17152.

510 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	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
2	#22558600	C₂	1	2	yes	maximal, complemented
3	#22558530	C₂	1	2	yes	maximal, complemented
4	#22558492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	#16863449	C₂²	1	1	yes	characteristic
8	#16863449	C₂²	1	1	yes	characteristic
9	#16863149	—	2	1	no	—
10	#16863147	—	2	1	no	—
11	D₄⋊(C₆₇⋊C₈)	—	2	4	no	—
12	#16863120	—	2	4	no	—
13	#16863117	—	2	1	no	—
14	C₈×C₆₇⋊C₈	—	2	1	no	—
15	C₂²×D₈×C₆₇	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
17	C₄×D₈×C₆₇	—	2	1	no	—
18	C₄×D₈×C₆₇	—	2	1	no	—
19	C₄×D₈×C₆₇	—	2	1	no	—
20	C₄×D₈×C₆₇	—	2	1	no	—
21	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
22	C₂×#102×C₆₇	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
25	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
26	C₄⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
27	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
29	C₂²⋊(C₆₇⋊C₈)	D₄	1	2	yes	—
30	C₄×C₆₇⋊C₈	—	2	1	no	—
31	C₄×C₆₇⋊C₈	—	2	1	no	—
32	C₄×C₆₇⋊C₈	—	2	2	no	—
33	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	—	2	2	no	—
35	C₂×D₈×C₆₇	—	2	1	no	—
36	C₂×D₈×C₆₇	—	2	1	no	—
37	C₂×D₈×C₆₇	—	2	2	no	—
38	C₂×D₈×C₆₇	C₈	1	4	yes	complemented
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₄×D₄×C₆₇	—	2	2	no	—
42	C₄×D₄×C₆₇	—	2	2	no	—
43	C₄×D₄×C₆₇	—	2	2	no	—
44	C₄×D₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
48	#107×C₆₇	—	2	1	no	—
49	#107×C₆₇	—	2	1	no	—
50	#102×C₆₇	—	2	2	no	—
51	#102×C₆₇	—	2	2	no	—
52	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
55	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	2	2	no	—
58	C₂×C₆₇⋊C₈	—	4	2	no	—
59	C₂⁴×C₆₇	—	2	2	no	—
60	C₂×D₄×C₆₇	—	2	2	no	—
61	C₂×D₄×C₆₇	—	2	2	no	—
62	C₂×D₄×C₆₇	—	2	2	no	—
63	C₂×D₄×C₆₇	—	2	2	no	—
64	C₂×D₄×C₆₇	—	2	2	no	—
65	C₂×D₄×C₆₇	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₆₇	—	2	2	no	—
67	C₂²×C₄×C₆₇	—	4	2	no	—
68	C₂²×C₄×C₆₇	—	4	2	no	—
69	C₂²×C₄×C₆₇	—	8	4	no	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
73	D₈×C₆₇	—	2	4	no	—
74	D₈×C₆₇	—	2	4	no	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	1	no	—
77	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂×C₈	1	2	yes	—
82	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
83	C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₂²⋊C₄×C₆₇	—	4	2	no	—
86	C₂²⋊C₄×C₆₇	—	4	2	no	—
87	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₆₇	#41	1	1	yes	characteristic
91	C₂³×C₆₇	—	2	4	no	—
92	C₂³×C₆₇	—	4	2	no	—
93	C₂³×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	—	8	2	no	—
95	C₂³×C₆₇	—	8	2	no	—
96	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
97	D₄×C₆₇	—	2	4	no	—
98	D₄×C₆₇	—	2	4	no	—
99	D₄×C₆₇	—	4	2	no	—
100	D₄×C₆₇	—	4	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	4	1	no	—
103	C₂×C₄×C₆₇	—	4	2	no	—
104	C₂×C₄×C₆₇	—	8	4	no	—
105	C₂×C₄×C₆₇	—	8	4	no	—
106	C₂×C₄×C₆₇	—	8	2	no	—
107	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
112	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#131	1	1	yes	characteristic
117	C₈×C₆₇	—	2	1	no	—
118	C₈×C₆₇	—	2	1	no	—
119	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
120	C₈×C₆₇	C₂²⋊C₈	1	2	yes	—
121	C₆₇⋊C₈	—	4	2	no	—
122	C₆₇⋊C₈	—	4	2	no	—
123	C₆₇⋊C₈	—	8	2	no	—
124	C₂²×C₆₇	—	2	1	no	—
125	C₂²×C₆₇	—	2	1	no	—
126	C₂²×C₆₇	—	4	4	no	—
127	C₂²×C₆₇	—	8	4	no	—
128	C₂²×C₆₇	—	8	4	no	—
129	C₂²×C₆₇	—	8	2	no	—
130	C₂²×C₆₇	—	8	2	no	—
131	C₂²×C₆₇	#390	1	1	yes	characteristic
132	C₂²×C₆₇	C₈×D₄	1	1	yes	characteristic
133	C₂²×C₆₇	#436	1	1	yes	characteristic
134	C₄×C₆₇	—	2	1	no	—
135	C₄×C₆₇	—	2	1	no	—
136	C₄×C₆₇	—	2	1	no	—
137	C₄×C₆₇	—	8	2	no	—
138	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
139	C₄×C₆₇	#408	1	1	yes	characteristic
140	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
141	C₄×C₆₇	#436	1	1	yes	characteristic
142	C₄×C₆₇	#472	1	2	yes	—
143	#9668	—	67	1	no	maximal, Sylow
144	C₂×C₆₇	—	2	1	no	—
145	C₂×C₆₇	—	2	1	no	—
146	C₂×C₆₇	—	8	4	no	—
147	C₂×C₆₇	#1954	1	1	yes	characteristic
148	C₂×C₆₇	#1976	1	1	yes	characteristic
149	C₂×C₆₇	#1978	1	1	yes	characteristic
150	C₂×C₄×D₈	D₆₇	1	1	yes	characteristic
151	#1564	—	67	1	no	—
152	#1478	—	67	2	no	—
153	#1460	—	67	2	no	—
154	#1450	—	67	1	no	—
155	C₆₇	#9668	1	1	yes	characteristic, complemented, Sylow
156	C₂²×D₈	Dic₆₇	1	1	yes	characteristic
157	C₂×C₄×D₄	C₂×D₆₇	1	2	yes	—
158	C₄×D₈	—	2	1	no	—
159	C₄×D₈	—	2	1	no	—
160	C₄×D₈	—	2	1	no	—
161	C₄×D₈	—	2	1	no	—
162	#431	—	67	1	no	—
163	#431	—	67	1	no	—
164	#431	—	67	2	no	—
165	C₂×#107	Dic₆₇	1	1	yes	characteristic
166	C₂×#102	Dic₆₇	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
168	#335	—	134	1	no	—
169	#334	—	134	1	no	—
170	D₄⋊C₈	—	134	4	no	—
171	#323	—	134	4	no	—
172	#322	—	134	1	no	—
173	C₈²	—	134	1	no	—
174	C₂²×D₄	C₂×Dic₆₇	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂×D₈	—	2	1	no	—
177	C₂×D₈	—	2	1	no	—
178	C₂×D₈	—	2	2	no	—
179	C₂×D₈	C₆₇⋊C₈	1	4	yes	complemented
180	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	—	2	2	no	—
184	C₄×D₄	—	2	2	no	—
185	C₄×D₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	#107	—	2	1	no	—
190	#107	—	2	1	no	—
191	C₄⋊C₈	—	67	2	no	—
192	C₄⋊C₈	—	134	2	no	—
193	C₄⋊C₈	—	134	1	no	—
194	#102	—	2	2	no	—
195	#102	—	2	2	no	—
196	C₂²⋊C₈	—	67	2	no	—
197	C₂²⋊C₈	—	134	2	no	—
198	C₂²⋊C₈	—	134	1	no	—
199	C₄×C₈	—	134	2	no	—
200	C₄×C₈	—	134	1	no	—
201	C₄×C₈	—	134	1	no	—
202	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
203	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
204	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
205	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
206	C₂⁴	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	C₂×C₆₇⋊C₈	1	4	yes	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	8	4	no	—
217	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
219	C₂²×C₄	#11761627	1	1	yes	characteristic
220	D₈	—	2	4	no	—
221	D₈	—	2	4	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	268	2	no	—
225	C₂×C₈	—	134	2	no	—
226	C₂×C₈	—	134	2	no	—
227	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
228	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
231	C₂×C₈	C₂×C₆₇⋊C₈	1	2	yes	—
232	C₂×C₈	#11761612	1	2	yes	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₄²	D₄×D₆₇	1	1	yes	characteristic
238	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
239	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
240	C₄²	#11761627	1	1	yes	characteristic
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	8	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
247	D₄	—	2	4	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	8	4	no	—
256	C₂×C₄	—	8	2	no	—
257	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
259	C₂×C₄	#14524014	1	1	yes	characteristic
260	C₂×C₄	#14524014	1	1	yes	characteristic
261	C₂×C₄	#14524030	1	1	yes	characteristic
262	C₂×C₄	#14524039	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
264	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
265	C₂×C₄	#14524050	1	1	yes	characteristic
266	C₂×C₄	#14524050	1	1	yes	characteristic
267	C₈	—	2	1	no	—
268	C₈	—	2	1	no	—
269	C₈	—	268	2	no	—
270	C₈	—	268	2	no	—
271	C₈	—	536	2	no	—
272	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
273	C₈	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	4	no	—
278	C₂²	—	8	4	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	#16863299	1	1	yes	characteristic
282	C₂²	D₄×C₆₇⋊C₈	1	1	yes	characteristic
283	C₂²	#16863474	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	8	2	no	—
288	C₄	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
289	C₄	#16863363	1	1	yes	characteristic
290	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
291	C₄	#16863474	1	1	yes	characteristic
292	C₄	#16863663	1	2	yes	—
293	C₂	—	2	1	no	—
294	C₂	—	2	1	no	—
295	C₂	—	8	4	no	—
296	C₂	#22561004	1	1	yes	characteristic
297	C₂	#22561108	1	1	yes	characteristic
298	C₂	#22561120	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×C₄×D₈×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂²×D₈×C₆₇	C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₄×D₈	D₆₇
hypocenter	C₆₇	#9668
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #22558957	C₂	semidirect
2. #22558600	C₂	semidirect
3. #22558530	C₂	semidirect
4. #22558492	C₂	semidirect
38. C₂×D₈×C₆₇	C₈	semidirect
155. C₆₇	#9668	semidirect
179. C₂×D₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22558957	2
#22558600	2
#22558530	2
#22558492	2
C₂×C₄×D₈×C₆₇	2
#9668	67