Subgroups — #34078913

Order 19328.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752768	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#23752520	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752519	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752516	C₂	1	1	yes	maximal, characteristic, complemented
6	#23752506	C₂	1	1	yes	maximal, characteristic, complemented
7	#23752505	C₂	1	1	yes	maximal, characteristic, complemented
8	#23752496	C₂	1	1	yes	maximal, characteristic, complemented
9	#23752495	C₂	1	1	yes	maximal, characteristic, complemented
10	#23752253	C₂	1	1	yes	maximal, characteristic
11	#23752251	C₂	1	1	yes	maximal, characteristic
12	#23752234	C₂	1	1	yes	maximal, characteristic
13	#23752219	C₂	1	1	yes	maximal, characteristic
14	#23752219	C₂	1	1	yes	maximal, characteristic
15	#479×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₅₁	C₂²	1	1	yes	characteristic
17	C₂×#14558987	C₂²	1	1	yes	characteristic
18	C₂×#14558987	C₂²	1	1	yes	characteristic
19	#18012738	C₂²	1	1	yes	characteristic
20	#18012738	C₂²	1	1	yes	characteristic
21	#18012731	C₂²	1	1	yes	characteristic
22	#18012717	—	2	1	no	—
23	#18012717	—	2	1	no	—
24	Q₈×Dic₁₅₁	C₂²	1	1	yes	characteristic
25	C₄×Dic₃₀₂	—	2	1	no	—
26	C₄×Dic₃₀₂	C₂²	1	1	yes	characteristic
27	#18012704	—	2	1	no	—
28	#18012704	—	2	1	no	—
29	#18012704	—	2	1	no	—
30	#18012704	C₂²	1	1	yes	characteristic
31	#18012697	C₂²	1	1	yes	characteristic
32	#18012697	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₅₁	—	2	1	no	—
34	C₄⋊C₄×D₁₅₁	C₂²	1	1	yes	characteristic
35	C₄²×D₁₅₁	—	2	1	no	—
36	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
37	#18012673	C₂²	1	1	yes	characteristic
38	#18012672	C₂²	1	1	yes	characteristic
39	#18012672	C₂²	1	1	yes	characteristic
40	#18012671	C₂²	1	1	yes	characteristic
41	#18012670	C₂²	1	1	yes	characteristic
42	#18012670	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₅₁⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
46	#18012664	C₂²	1	1	yes	characteristic
47	#18012664	C₂²	1	1	yes	characteristic
48	#18012663	C₂²	1	1	yes	characteristic
49	#18012663	C₂²	1	1	yes	characteristic
50	#18012662	C₂²	1	1	yes	characteristic
51	#18012662	C₂²	1	1	yes	characteristic
52	#125×C₁₅₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
54	#107×C₁₅₁	C₂²	1	1	yes	characteristic
55	#106×C₁₅₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₅₁	C₂²	1	1	yes	characteristic
57	#103×C₁₅₁	C₂²	1	1	yes	characteristic
58	#103×C₁₅₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₃₀₂	—	2	1	no	—
60	C₂×Dic₃₀₂	—	2	1	no	—
61	C₂×Dic₃₀₂	—	2	1	no	—
62	C₂×Dic₃₀₂	C₂³	1	1	yes	characteristic
63	Q₈×D₁₅₁	—	2	2	no	—
64	Q₈×D₁₅₁	Q₈	1	4	yes	complemented
65	C₂×C₄×D₁₅₁	—	2	1	no	—
66	C₂×C₄×D₁₅₁	—	2	1	no	—
67	C₂×C₄×D₁₅₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₅₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
70	#14558987	—	2	2	no	—
71	#14558987	—	2	2	no	—
72	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₅₁	—	2	1	no	—
75	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₅₁	C₂³	1	1	yes	characteristic
78	Dic₁₅₁⋊C₄	—	2	1	no	—
79	Dic₁₅₁⋊C₄	—	2	1	no	—
80	Dic₁₅₁⋊C₄	—	2	1	no	—
81	Dic₁₅₁⋊C₄	—	2	1	no	—
82	Dic₁₅₁⋊C₄	—	2	1	no	—
83	Dic₁₅₁⋊C₄	—	2	1	no	—
84	Dic₁₅₁⋊C₄	—	2	1	no	—
85	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₅₁)⋊C₄	—	2	1	no	—
87	C₄×Dic₁₅₁	—	2	1	no	—
88	C₄×Dic₁₅₁	—	2	1	no	—
89	C₄×Dic₁₅₁	—	2	1	no	—
90	C₄×Dic₁₅₁	—	2	1	no	—
91	C₄×Dic₁₅₁	—	2	1	no	—
92	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
93	C₄×Dic₁₅₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₅₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₅₁	—	2	1	no	—
99	C₄⋊C₄×C₁₅₁	—	2	1	no	—
100	C₄⋊C₄×C₁₅₁	—	2	1	no	—
101	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₅₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₅₁	—	2	1	no	—
105	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₅₁	C₂×D₄	1	1	yes	characteristic
107	Dic₃₀₂	—	2	1	no	—
108	Dic₃₀₂	—	2	4	no	—
109	Dic₃₀₂	—	4	2	no	—
110	Dic₃₀₂	—	4	2	no	—
111	Dic₃₀₂	C₂×Q₈	1	2	yes	—
112	C₂×Dic₁₅₁	—	2	1	no	—
113	C₂×Dic₁₅₁	—	2	1	no	—
114	C₂×Dic₁₅₁	—	2	1	no	—
115	C₂×Dic₁₅₁	—	2	1	no	—
116	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
119	C₄×D₁₅₁	—	2	4	no	—
120	C₄×D₁₅₁	—	2	2	no	—
121	C₄×D₁₅₁	—	4	1	no	—
122	C₄×D₁₅₁	—	4	2	no	—
123	C₄×D₁₅₁	C₂×Q₈	1	2	yes	—
124	C₄×D₁₅₁	#41	1	2	yes	—
125	C₁₅₁⋊C₈	—	2	1	no	—
126	C₁₅₁⋊C₈	—	2	1	no	—
127	Q₈×C₁₅₁	—	2	1	no	—
128	Q₈×C₁₅₁	C₂×Q₈	1	2	yes	—
129	C₂×C₄×C₁₅₁	—	2	1	no	—
130	C₂×C₄×C₁₅₁	—	2	1	no	—
131	C₂×C₄×C₁₅₁	—	2	1	no	—
132	C₂×C₄×C₁₅₁	—	2	1	no	—
133	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
136	C₈×C₁₅₁	—	2	1	no	—
137	C₈×C₁₅₁	—	2	1	no	—
138	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₅₁	—	2	1	no	—
140	C₂×D₁₅₁	—	2	1	no	—
141	C₂×D₁₅₁	C₂²⋊Q₈	1	2	yes	—
142	C₄×C₁₅₁	—	2	2	no	—
143	C₄×C₁₅₁	—	2	1	no	—
144	C₄×C₁₅₁	—	2	1	no	—
145	C₄×C₁₅₁	—	4	1	no	—
146	C₄×C₁₅₁	—	4	1	no	—
147	C₄×C₁₅₁	—	4	1	no	—
148	C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₁₅₁	C₂×#41	1	1	yes	characteristic
150	Dic₁₅₁	—	2	2	no	—
151	Dic₁₅₁	—	2	1	no	—
152	Dic₁₅₁	—	4	1	no	—
153	Dic₁₅₁	—	4	1	no	—
154	Dic₁₅₁	—	4	1	no	—
155	Dic₁₅₁	C₂²⋊Q₈	1	2	yes	—
156	Dic₁₅₁	#137	1	2	yes	—
157	C₂×C₁₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
158	C₂×C₁₅₁	C₂×#137	1	1	yes	characteristic
159	C₂×C₁₅₁	#575	1	1	yes	characteristic
160	D₁₅₁	—	2	2	no	—
161	C₁₅₁	#3078	1	1	yes	characteristic, complemented, Sylow
162	#3078	—	151	1	no	maximal, Sylow
163	#533	—	151	1	no	—
164	C₂×C₄×Q₈	—	151	1	no	—
165	#479	—	151	1	no	—
166	#479	—	151	1	no	—
167	#479	—	151	1	no	—
168	#479	D₁₅₁	1	1	yes	characteristic, complemented
169	#477	—	151	1	no	—
170	#477	—	151	1	no	—
171	#475	—	151	1	no	—
172	#475	—	151	1	no	—
173	#428	—	151	1	no	—
174	#428	—	151	1	no	—
175	#423	—	151	1	no	—
176	#419	—	151	1	no	—
177	#419	—	151	1	no	—
178	C₂²×Q₈	—	151	1	no	—
179	C₂×M₁₆	—	151	1	no	—
180	C₂×M₁₆	—	151	1	no	—
181	C₄⋊Q₈	—	151	1	no	—
182	C₄⋊Q₈	—	151	1	no	—
183	#125	—	151	1	no	—
184	#125	C₂×D₁₅₁	1	1	yes	characteristic
185	C₂²⋊Q₈	—	302	1	no	—
186	C₂²⋊Q₈	—	302	1	no	—
187	C₄×Q₈	—	302	1	no	—
188	C₄×Q₈	—	302	1	no	—
189	C₄×Q₈	—	302	1	no	—
190	C₄×Q₈	—	302	1	no	—
191	C₄×Q₈	—	151	1	no	—
192	C₄×Q₈	—	151	1	no	—
193	C₄×Q₈	—	151	1	no	—
194	C₄×Q₈	C₂×D₁₅₁	1	1	yes	characteristic
195	#117	—	151	1	no	—
196	#117	—	151	1	no	—
197	C₂×C₄⋊C₄	—	302	1	no	—
198	C₂×C₄⋊C₄	—	151	1	no	—
199	C₂×C₄²	—	302	1	no	—
200	C₂×C₄²	—	151	1	no	—
201	#107	—	151	1	no	—
202	#107	—	151	1	no	—
203	#107	—	151	1	no	—
204	#107	C₂×D₁₅₁	1	1	yes	characteristic
205	#106	—	151	1	no	—
206	#106	—	151	1	no	—
207	#106	—	151	1	no	—
208	#106	C₂×D₁₅₁	1	1	yes	characteristic
209	C₄⋊C₈	—	151	1	no	—
210	C₄⋊C₈	—	151	1	no	—
211	C₄⋊C₈	—	151	1	no	—
212	C₄⋊C₈	C₂×D₁₅₁	1	1	yes	characteristic
213	#103	—	151	1	no	—
214	#103	—	151	1	no	—
215	#103	—	151	1	no	—
216	#103	—	151	1	no	—
217	#103	—	151	1	no	—
218	#103	—	151	1	no	—
219	#103	C₂×D₁₅₁	1	1	yes	characteristic
220	#103	C₂×D₁₅₁	1	1	yes	characteristic
221	C₂×Q₈	—	302	1	no	—
222	C₂×Q₈	—	302	1	no	—
223	C₂×Q₈	—	302	1	no	—
224	C₂×Q₈	—	302	2	no	—
225	C₂×Q₈	—	151	4	no	—
226	C₂×Q₈	—	151	1	no	—
227	C₂×Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
228	C₂²×C₄	—	302	1	no	—
229	C₂²×C₄	—	302	1	no	—
230	C₂²×C₄	—	151	1	no	—
231	C₂²×C₄	—	151	1	no	—
232	C₂²×C₄	—	151	1	no	—
233	M₁₆	—	302	2	no	—
234	M₁₆	—	302	2	no	—
235	C₂×C₈	—	151	1	no	—
236	C₂×C₈	—	151	1	no	—
237	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₁₅₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	302	1	no	—
243	C₄⋊C₄	—	302	1	no	—
244	C₄⋊C₄	—	302	1	no	—
245	C₄⋊C₄	—	302	1	no	—
246	C₄⋊C₄	—	302	1	no	—
247	C₄⋊C₄	—	302	1	no	—
248	C₄⋊C₄	—	302	1	no	—
249	C₄⋊C₄	—	302	1	no	—
250	C₄⋊C₄	—	151	1	no	—
251	C₄⋊C₄	—	151	1	no	—
252	C₄⋊C₄	—	151	1	no	—
253	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₁₅₁	1	1	yes	characteristic
256	C₂²⋊C₄	—	302	1	no	—
257	C₂²⋊C₄	—	302	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	302	1	no	—
260	C₄²	—	302	1	no	—
261	C₄²	—	302	1	no	—
262	C₄²	—	302	1	no	—
263	C₄²	—	302	1	no	—
264	C₄²	—	151	1	no	—
265	C₄²	—	151	1	no	—
266	C₄²	—	151	1	no	—
267	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
268	C₂³	—	151	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	302	4	no	—
271	Q₈	—	302	1	no	—
272	Q₈	—	604	2	no	—
273	Q₈	—	604	2	no	—
274	Q₈	—	151	2	no	—
275	Q₈	Q₈×D₁₅₁	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	302	4	no	—
281	C₂×C₄	—	302	2	no	—
282	C₂×C₄	—	302	1	no	—
283	C₂×C₄	—	302	1	no	—
284	C₂×C₄	—	302	1	no	—
285	C₂×C₄	—	302	1	no	—
286	C₂×C₄	—	604	1	no	—
287	C₂×C₄	—	604	2	no	—
288	C₂×C₄	—	151	1	no	—
289	C₂×C₄	—	151	1	no	—
290	C₂×C₄	—	151	1	no	—
291	C₂×C₄	—	151	2	no	—
292	C₂×C₄	—	151	2	no	—
293	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
294	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	302	1	no	—
299	C₈	—	302	1	no	—
300	C₂²	—	302	1	no	—
301	C₂²	—	302	1	no	—
302	C₂²	—	151	2	no	—
303	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	302	2	no	—
311	C₄	—	302	1	no	—
312	C₄	—	604	1	no	—
313	C₄	—	604	1	no	—
314	C₄	—	604	1	no	—
315	C₄	—	151	2	no	—
316	C₄	—	151	2	no	—
317	C₄	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
318	C₄	#41×D₁₅₁	1	1	yes	characteristic
319	C₂	—	302	2	no	—
320	C₂	C₂²⋊Q₈×D₁₅₁	1	1	yes	characteristic
321	C₂	#137×D₁₅₁	1	1	yes	characteristic
322	C₂	#23753009	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#479×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂²×D₁₅₁	C₂×D₄
hypercenter	#479	D₁₅₁
hypocenter	C₁₅₁	#3078
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #23752520	C₂	semidirect
4. #23752519	C₂	semidirect
5. #23752516	C₂	semidirect
6. #23752506	C₂	semidirect
7. #23752505	C₂	semidirect
8. #23752496	C₂	semidirect
9. #23752495	C₂	semidirect
15. #479×C₁₅₁	C₂	semidirect
64. Q₈×D₁₅₁	Q₈	semidirect
161. C₁₅₁	#3078	semidirect
168. #479	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752768	2
C₄×Q₈×D₁₅₁	2
#23752520	2
#23752519	2
#23752516	2
#23752506	2
#23752505	2
#23752496	2
#23752495	2
#23752253	2
#23752251	2
#23752234	2
#23752219	2
#479×C₁₅₁	2
#3078	151