Subgroups — #30675798

Order 16768.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553939	C₂	1	1	yes	maximal, characteristic
2	#22553788	C₂	1	1	yes	maximal, characteristic
3	#22553621	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553620	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553619	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553618	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553606	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553595	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553594	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553353	C₂	1	1	yes	maximal, characteristic
11	#22553349	C₂	1	1	yes	maximal, characteristic
12	#22553337	C₂	1	1	yes	maximal, characteristic
13	#22553319	C₂	1	1	yes	maximal, characteristic
14	#103×D₁₃₁	C₂	1	1	yes	maximal, characteristic
15	#477×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×#14523496	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
19	#16862245	C₂²	1	1	yes	characteristic
20	#16862244	—	2	1	no	—
21	#16862238	C₂²	1	1	yes	characteristic
22	#16862237	C₂²	1	1	yes	characteristic
23	#16862236	—	2	1	no	—
24	#16862224	—	2	1	no	—
25	#16862222	—	2	1	no	—
26	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₆₂	—	2	1	no	—
28	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
29	#16862211	—	2	1	no	—
30	#16862211	C₂²	1	1	yes	characteristic
31	#16862204	—	2	1	no	—
32	#16862204	C₂²	1	1	yes	characteristic
33	#16862204	C₂²	1	1	yes	characteristic
34	#16862203	—	2	1	no	—
35	#16862203	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
37	#16862180	C₂²	1	1	yes	characteristic
38	#16862179	C₂²	1	1	yes	characteristic
39	#16862178	C₂²	1	1	yes	characteristic
40	#16862177	C₂²	1	1	yes	characteristic
41	#16862177	C₂²	1	1	yes	characteristic
42	#16862177	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
46	#16862171	C₂²	1	1	yes	characteristic
47	#16862171	C₂²	1	1	yes	characteristic
48	#16862170	C₂²	1	1	yes	characteristic
49	#16862170	C₂²	1	1	yes	characteristic
50	#16862169	C₂²	1	1	yes	characteristic
51	#16862169	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
54	#107×C₁₃₁	C₂²	1	1	yes	characteristic
55	#107×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	#103×C₁₃₁	C₂²	1	1	yes	characteristic
58	#103×C₁₃₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	—	2	1	no	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	Q₈×D₁₃₁	—	2	2	no	—
63	Q₈×D₁₃₁	Q₈	1	4	yes	complemented
64	C₂×C₄×D₁₃₁	—	2	1	no	—
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
67	#14523496	—	2	2	no	—
68	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₁₃₁	—	2	2	no	—
71	C₄⋊Dic₁₃₁	—	2	1	no	—
72	C₄⋊Dic₁₃₁	—	2	1	no	—
73	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
76	Dic₁₃₁⋊C₄	—	2	1	no	—
77	Dic₁₃₁⋊C₄	—	2	1	no	—
78	Dic₁₃₁⋊C₄	—	2	1	no	—
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	—	2	1	no	—
82	Dic₁₃₁⋊C₄	—	2	1	no	—
83	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
85	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₁₃₁)⋊C₄	D₄	1	1	yes	characteristic
89	C₄×Dic₁₃₁	—	2	1	no	—
90	C₄×Dic₁₃₁	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	—	2	1	no	—
93	C₄×Dic₁₃₁	—	2	1	no	—
94	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₃₁	—	2	1	no	—
96	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₁₃₁	—	2	1	no	—
100	C₄⋊C₄×C₁₃₁	—	2	1	no	—
101	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
104	C₄²×C₁₃₁	—	2	1	no	—
105	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
106	C₂²×D₁₃₁	C₂×D₄	1	1	yes	characteristic
107	Dic₂₆₂	—	2	1	no	—
108	Dic₂₆₂	—	2	4	no	—
109	Dic₂₆₂	—	8	1	no	—
110	Dic₂₆₂	C₂×Q₈	1	2	yes	—
111	C₂×Dic₁₃₁	—	2	1	no	—
112	C₂×Dic₁₃₁	—	2	1	no	—
113	C₂×Dic₁₃₁	—	2	1	no	—
114	C₂×Dic₁₃₁	—	2	1	no	—
115	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
118	C₄×D₁₃₁	—	2	4	no	—
119	C₄×D₁₃₁	—	8	1	no	—
120	C₄×D₁₃₁	C₂×Q₈	1	2	yes	—
121	C₄×D₁₃₁	#41	1	2	yes	—
122	C₁₃₁⋊C₈	—	2	1	no	—
123	C₁₃₁⋊C₈	—	2	1	no	—
124	Q₈×C₁₃₁	—	2	1	no	—
125	Q₈×C₁₃₁	—	8	1	no	—
126	Q₈×C₁₃₁	C₂×Q₈	1	2	yes	—
127	C₂×C₄×C₁₃₁	—	2	1	no	—
128	C₂×C₄×C₁₃₁	—	2	1	no	—
129	C₂×C₄×C₁₃₁	—	2	1	no	—
130	C₂×C₄×C₁₃₁	—	2	1	no	—
131	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
134	C₈×C₁₃₁	—	2	1	no	—
135	C₈×C₁₃₁	—	2	1	no	—
136	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₃₁	—	2	1	no	—
138	C₂×D₁₃₁	—	2	1	no	—
139	C₂×D₁₃₁	C₂²⋊Q₈	1	2	yes	—
140	C₄×C₁₃₁	—	2	2	no	—
141	C₄×C₁₃₁	—	2	1	no	—
142	C₄×C₁₃₁	—	2	1	no	—
143	C₄×C₁₃₁	—	4	1	no	—
144	C₄×C₁₃₁	—	4	1	no	—
145	C₄×C₁₃₁	—	4	1	no	—
146	C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
148	Dic₁₃₁	—	2	2	no	—
149	Dic₁₃₁	—	2	1	no	—
150	Dic₁₃₁	—	2	1	no	—
151	Dic₁₃₁	—	4	1	no	—
152	Dic₁₃₁	—	4	1	no	—
153	Dic₁₃₁	—	4	1	no	—
154	Dic₁₃₁	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₁₃₁	#577	1	1	yes	characteristic
157	C₂×C₁₃₁	#578	1	1	yes	characteristic
158	D₁₃₁	—	2	2	no	—
159	C₁₃₁	#3082	1	1	yes	characteristic, complemented, Sylow
160	#3082	—	131	1	no	maximal, Sylow
161	#544	—	131	1	no	—
162	#519	—	131	1	no	—
163	#479	—	131	1	no	—
164	#479	—	131	1	no	—
165	#479	—	131	1	no	—
166	#479	—	131	1	no	—
167	#477	—	131	1	no	—
168	#477	D₁₃₁	1	1	yes	characteristic, complemented
169	#475	—	131	1	no	—
170	#475	—	131	1	no	—
171	#428	—	131	1	no	—
172	#427	—	131	1	no	—
173	#424	—	131	1	no	—
174	#419	—	131	1	no	—
175	C₂×#103	—	131	1	no	—
176	C₂²×Q₈	—	131	1	no	—
177	C₂×M₁₆	—	131	1	no	—
178	C₂²×C₈	—	131	1	no	—
179	C₄⋊Q₈	—	262	1	no	—
180	C₄⋊Q₈	—	131	1	no	—
181	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
182	#125	—	262	1	no	—
183	#125	—	131	1	no	—
184	#125	—	131	1	no	—
185	C₂²⋊Q₈	—	262	1	no	—
186	C₂²⋊Q₈	—	262	1	no	—
187	C₄×Q₈	—	262	1	no	—
188	C₄×Q₈	—	262	1	no	—
189	C₄×Q₈	—	131	1	no	—
190	C₄×Q₈	—	131	1	no	—
191	C₄×Q₈	—	131	1	no	—
192	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
193	#117	—	262	1	no	—
194	#117	—	262	1	no	—
195	#117	—	131	1	no	—
196	#117	—	131	1	no	—
197	#117	—	131	1	no	—
198	C₂×C₄⋊C₄	—	131	1	no	—
199	#107	—	131	1	no	—
200	#107	—	131	1	no	—
201	#107	C₂×D₁₃₁	1	1	yes	characteristic
202	#107	C₂×D₁₃₁	1	1	yes	characteristic
203	#106	—	131	1	no	—
204	#106	—	131	1	no	—
205	#106	—	131	1	no	—
206	#106	—	131	1	no	—
207	C₄⋊C₈	—	131	1	no	—
208	C₄⋊C₈	—	131	1	no	—
209	C₄⋊C₈	—	131	1	no	—
210	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
211	#103	—	131	1	no	—
212	#103	—	131	1	no	—
213	#103	—	131	1	no	—
214	#103	—	131	1	no	—
215	#103	—	131	1	no	—
216	#103	—	131	1	no	—
217	#103	C₂×D₁₃₁	1	1	yes	characteristic
218	#103	C₂×D₁₃₁	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	262	1	no	—
221	C₂×Q₈	—	262	1	no	—
222	C₂×Q₈	—	262	2	no	—
223	C₂×Q₈	—	131	4	no	—
224	C₂×Q₈	—	131	1	no	—
225	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
226	C₂²×C₄	—	262	1	no	—
227	C₂²×C₄	—	262	1	no	—
228	C₂²×C₄	—	131	1	no	—
229	M₁₆	—	262	2	no	—
230	C₂×C₈	—	262	2	no	—
231	C₂×C₈	—	131	1	no	—
232	C₂×C₈	—	131	1	no	—
233	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	262	1	no	—
238	C₄⋊C₄	—	262	1	no	—
239	C₄⋊C₄	—	262	1	no	—
240	C₄⋊C₄	—	262	1	no	—
241	C₄⋊C₄	—	262	1	no	—
242	C₄⋊C₄	—	262	1	no	—
243	C₄⋊C₄	—	262	1	no	—
244	C₄⋊C₄	—	262	1	no	—
245	C₄⋊C₄	—	262	1	no	—
246	C₄⋊C₄	—	131	1	no	—
247	C₄⋊C₄	—	131	1	no	—
248	C₄⋊C₄	—	131	1	no	—
249	C₄⋊C₄	—	131	1	no	—
250	C₄⋊C₄	—	131	1	no	—
251	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
254	C₂²⋊C₄	—	262	1	no	—
255	C₂²⋊C₄	—	262	1	no	—
256	C₂²⋊C₄	—	131	1	no	—
257	C₂²⋊C₄	—	131	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	262	1	no	—
260	C₄²	—	262	1	no	—
261	C₄²	—	262	1	no	—
262	C₄²	—	262	1	no	—
263	C₄²	—	262	1	no	—
264	C₄²	—	131	1	no	—
265	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
266	C₂³	—	131	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	262	4	no	—
269	Q₈	—	262	1	no	—
270	Q₈	—	8	1	no	—
271	Q₈	—	1048	1	no	—
272	Q₈	—	131	2	no	—
273	Q₈	Q₈×D₁₃₁	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	262	4	no	—
279	C₂×C₄	—	262	1	no	—
280	C₂×C₄	—	262	1	no	—
281	C₂×C₄	—	262	1	no	—
282	C₂×C₄	—	262	1	no	—
283	C₂×C₄	—	1048	1	no	—
284	C₂×C₄	—	131	1	no	—
285	C₂×C₄	—	131	1	no	—
286	C₂×C₄	—	131	1	no	—
287	C₂×C₄	—	131	2	no	—
288	C₂×C₄	—	131	2	no	—
289	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
290	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	262	1	no	—
295	C₈	—	262	1	no	—
296	C₂²	—	262	1	no	—
297	C₂²	—	262	1	no	—
298	C₂²	—	131	2	no	—
299	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	262	1	no	—
307	C₄	—	262	1	no	—
308	C₄	—	262	2	no	—
309	C₄	—	524	1	no	—
310	C₄	—	524	1	no	—
311	C₄	—	524	1	no	—
312	C₄	—	131	2	no	—
313	C₄	C₂×Q₈×D₁₃₁	1	1	yes	characteristic
314	C₄	#41×D₁₃₁	1	1	yes	characteristic
315	C₂	—	262	2	no	—
316	C₂	C₂²⋊Q₈×D₁₃₁	1	1	yes	characteristic
317	C₂	#22554122	1	1	yes	characteristic
318	C₂	#22554125	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#477×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂²×D₁₃₁	C₂×D₄
hypercenter	#477	D₁₃₁
hypocenter	C₁₃₁	#3082
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #477

Complements

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

N	H ≅ G/N	type
3. #22553621	C₂	semidirect
4. #22553620	C₂	semidirect
5. #22553619	C₂	semidirect
6. #22553618	C₂	semidirect
7. #22553606	C₂	semidirect
8. #22553595	C₂	semidirect
9. #22553594	C₂	semidirect
15. #477×C₁₃₁	C₂	semidirect
63. Q₈×D₁₃₁	Q₈	semidirect
159. C₁₃₁	#3082	semidirect
168. #477	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22553939	2
#22553788	2
#22553621	2
#22553620	2
#22553619	2
#22553618	2
#22553606	2
#22553595	2
#22553594	2
#22553353	2
#22553349	2
#22553337	2
#22553319	2
#103×D₁₃₁	2
#477×C₁₃₁	2
#3082	131