Subgroups — #30675778

Order 16768.

384 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14523512	C₂	1	1	yes	maximal, characteristic
3	#22553622	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553620	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553618	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553616	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553614	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553612	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553610	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553353	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553351	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553334	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553310	C₂	1	1	yes	maximal, characteristic
14	#22553310	C₂	1	1	yes	maximal, characteristic
15	#478×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523512	C₂²	1	1	yes	characteristic
17	C₂×#14523496	C₂²	1	1	yes	characteristic
18	C₂×#14523496	C₂²	1	1	yes	characteristic
19	#16862238	C₂²	1	1	yes	characteristic
20	#16862237	C₂²	1	1	yes	characteristic
21	#16862237	C₂²	1	1	yes	characteristic
22	#16862236	—	2	1	no	—
23	#16862236	—	2	1	no	—
24	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
25	#16862211	C₂²	1	1	yes	characteristic
26	D₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
28	(C₂²⋊D₁₃₁)⋊C₄	—	2	1	no	—
29	#16862204	C₂²	1	1	yes	characteristic
30	#16862202	—	2	1	no	—
31	C₄⋊C₄×D₁₃₁	—	2	1	no	—
32	C₄⋊C₄×D₁₃₁	—	2	1	no	—
33	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₃₁	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₁₃₁	—	2	1	no	—
36	C₄²×D₁₃₁	C₂²	1	1	yes	characteristic
37	#16862180	C₂²	1	1	yes	characteristic
38	#16862179	C₂²	1	1	yes	characteristic
39	#16862179	C₂²	1	1	yes	characteristic
40	#16862178	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	#16862169	C₂²	1	1	yes	characteristic
49	#16862169	C₂²	1	1	yes	characteristic
50	#16862167	C₂²	1	1	yes	characteristic
51	#16862167	C₂²	1	1	yes	characteristic
52	#125×C₁₃₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
54	#107×C₁₃₁	C₂²	1	1	yes	characteristic
55	#106×C₁₃₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
57	#102×C₁₃₁	C₂²	1	1	yes	characteristic
58	#102×C₁₃₁	C₂²	1	1	yes	characteristic
59	#14523512	—	2	2	no	—
60	#14523512	Q₈	1	4	yes	—
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
63	C₂²×Dic₁₃₁	—	2	1	no	—
64	C₂²×Dic₁₃₁	—	2	1	no	—
65	C₂×C₄×D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
70	#14523496	—	2	2	no	—
71	#14523496	—	2	2	no	—
72	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₁₃₁	—	2	1	no	—
75	C₄⋊Dic₁₃₁	—	2	1	no	—
76	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
84	Dic₁₃₁⋊C₄	—	2	1	no	—
85	Dic₁₃₁⋊C₄	—	2	1	no	—
86	Dic₁₃₁⋊C₄	—	2	1	no	—
87	C₂²⋊Dic₁₃₁	—	2	1	no	—
88	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
89	C₄×Dic₁₃₁	—	2	1	no	—
90	C₄×Dic₁₃₁	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
93	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
94	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₁₃₁	—	2	1	no	—
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	C₂²⋊C₄×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	C₂×Q₈	1	2	yes	—
109	C₂²⋊D₁₃₁	—	2	4	no	—
110	C₂×Dic₁₃₁	—	2	1	no	—
111	C₂×Dic₁₃₁	—	2	4	no	—
112	C₂×Dic₁₃₁	—	2	1	no	—
113	C₂×Dic₁₃₁	—	2	1	no	—
114	C₂×Dic₁₃₁	—	2	1	no	—
115	C₂×Dic₁₃₁	—	4	2	no	—
116	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
119	C₄×D₁₃₁	—	4	1	no	—
120	C₄×D₁₃₁	—	4	1	no	—
121	C₄×D₁₃₁	—	4	2	no	—
122	C₄×D₁₃₁	—	4	2	no	—
123	C₄×D₁₃₁	C₂×Q₈	1	2	yes	—
124	C₄×D₁₃₁	#41	1	2	yes	—
125	C₁₃₁⋊C₈	—	2	1	no	—
126	C₁₃₁⋊C₈	—	2	1	no	—
127	C₂³×C₁₃₁	—	2	1	no	—
128	D₄×C₁₃₁	—	2	1	no	—
129	D₄×C₁₃₁	C₂×Q₈	1	2	yes	—
130	C₂×C₄×C₁₃₁	—	2	1	no	—
131	C₂×C₄×C₁₃₁	—	2	1	no	—
132	C₂×C₄×C₁₃₁	—	2	1	no	—
133	C₂×C₄×C₁₃₁	—	4	2	no	—
134	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
137	C₈×C₁₃₁	—	2	1	no	—
138	C₈×C₁₃₁	—	2	1	no	—
139	C₂²×C₁₃₁	—	2	2	no	—
140	C₂²×C₁₃₁	—	4	1	no	—
141	C₂²×C₁₃₁	—	4	1	no	—
142	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₁₃₁	—	2	1	no	—
144	C₂×D₁₃₁	—	2	1	no	—
145	C₂×D₁₃₁	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₁₃₁	—	2	1	no	—
147	C₄×C₁₃₁	—	2	1	no	—
148	C₄×C₁₃₁	—	4	1	no	—
149	C₄×C₁₃₁	—	4	1	no	—
150	C₄×C₁₃₁	—	4	1	no	—
151	C₄×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
153	Dic₁₃₁	—	2	2	no	—
154	Dic₁₃₁	—	2	1	no	—
155	Dic₁₃₁	—	2	1	no	—
156	Dic₁₃₁	—	4	1	no	—
157	Dic₁₃₁	—	4	1	no	—
158	Dic₁₃₁	—	4	1	no	—
159	Dic₁₃₁	C₂²⋊Q₈	1	2	yes	—
160	C₂×C₁₃₁	—	4	2	no	—
161	C₂×C₁₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₁₃₁	#575	1	1	yes	characteristic
163	C₂×C₁₃₁	#575	1	1	yes	characteristic
164	D₁₃₁	—	2	2	no	—
165	C₁₃₁	#3080	1	1	yes	characteristic, complemented, Sylow
166	#3080	—	131	1	no	maximal, Sylow
167	C₂×#125	—	131	1	no	—
168	C₄×#41	—	131	1	no	—
169	#479	—	131	1	no	—
170	#479	—	131	1	no	—
171	#479	—	131	1	no	—
172	#479	—	131	1	no	—
173	#478	—	131	1	no	—
174	#478	—	131	1	no	—
175	#478	—	131	1	no	—
176	#478	D₁₃₁	1	1	yes	characteristic, complemented
177	#428	—	131	1	no	—
178	#428	—	131	1	no	—
179	#423	—	131	1	no	—
180	#417	—	131	1	no	—
181	#417	—	131	1	no	—
182	C₂×#41	—	131	1	no	—
183	C₂×M₁₆	—	131	1	no	—
184	C₂×M₁₆	—	131	1	no	—
185	#125	—	262	1	no	—
186	#125	—	262	1	no	—
187	#125	—	131	1	no	—
188	#125	—	131	1	no	—
189	#125	—	131	1	no	—
190	#125	C₂×D₁₃₁	1	1	yes	characteristic
191	C₄×Q₈	—	131	1	no	—
192	C₄×Q₈	—	131	1	no	—
193	C₄×D₄	—	262	1	no	—
194	C₄×D₄	—	262	1	no	—
195	C₄×D₄	—	131	1	no	—
196	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
197	#117	—	262	1	no	—
198	#117	—	131	1	no	—
199	C₂×C₄⋊C₄	—	262	1	no	—
200	C₂×C₄⋊C₄	—	262	1	no	—
201	C₂×C₄⋊C₄	—	131	1	no	—
202	C₂×C₄⋊C₄	—	131	1	no	—
203	C₂×C₄²	—	262	1	no	—
204	C₂×C₄²	—	131	1	no	—
205	#107	—	131	1	no	—
206	#107	—	131	1	no	—
207	#107	—	131	1	no	—
208	#107	C₂×D₁₃₁	1	1	yes	characteristic
209	#106	—	131	1	no	—
210	#106	—	131	1	no	—
211	#106	—	131	1	no	—
212	#106	C₂×D₁₃₁	1	1	yes	characteristic
213	C₄⋊C₈	—	131	1	no	—
214	C₄⋊C₈	—	131	1	no	—
215	C₄⋊C₈	—	131	1	no	—
216	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
217	#103	—	131	1	no	—
218	#103	—	131	1	no	—
219	#103	—	131	1	no	—
220	#103	—	131	1	no	—
221	#102	—	131	1	no	—
222	#102	—	131	1	no	—
223	#102	C₂×D₁₃₁	1	1	yes	characteristic
224	#102	C₂×D₁₃₁	1	1	yes	characteristic
225	#41	—	262	2	no	—
226	#41	—	131	4	no	—
227	C₂×Q₈	—	131	1	no	—
228	C₂×D₄	—	262	1	no	—
229	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	262	1	no	—
232	C₂²×C₄	—	262	1	no	—
233	C₂²×C₄	—	262	1	no	—
234	C₂²×C₄	—	262	1	no	—
235	C₂²×C₄	—	131	1	no	—
236	C₂²×C₄	—	131	1	no	—
237	C₂²×C₄	—	131	1	no	—
238	M₁₆	—	262	2	no	—
239	M₁₆	—	262	2	no	—
240	C₂×C₈	—	131	1	no	—
241	C₂×C₈	—	131	1	no	—
242	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	262	1	no	—
247	C₄⋊C₄	—	262	1	no	—
248	C₄⋊C₄	—	262	1	no	—
249	C₄⋊C₄	—	262	1	no	—
250	C₄⋊C₄	—	262	1	no	—
251	C₄⋊C₄	—	262	1	no	—
252	C₄⋊C₄	—	262	1	no	—
253	C₄⋊C₄	—	262	1	no	—
254	C₄⋊C₄	—	262	1	no	—
255	C₄⋊C₄	—	262	1	no	—
256	C₄⋊C₄	—	131	1	no	—
257	C₄⋊C₄	—	131	1	no	—
258	C₄⋊C₄	—	131	1	no	—
259	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	262	1	no	—
264	C₂²⋊C₄	—	262	1	no	—
265	C₄²	—	262	1	no	—
266	C₄²	—	262	1	no	—
267	C₄²	—	262	1	no	—
268	C₄²	—	131	1	no	—
269	C₄²	—	131	1	no	—
270	C₄²	—	131	1	no	—
271	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	131	1	no	—
274	Q₈	—	262	1	no	—
275	Q₈	—	131	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	262	4	no	—
278	D₄	Q₈×D₁₃₁	1	2	yes	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	262	1	no	—
284	C₂×C₄	—	262	4	no	—
285	C₂×C₄	—	262	1	no	—
286	C₂×C₄	—	262	1	no	—
287	C₂×C₄	—	262	1	no	—
288	C₂×C₄	—	524	1	no	—
289	C₂×C₄	—	524	1	no	—
290	C₂×C₄	—	524	2	no	—
291	C₂×C₄	—	524	2	no	—
292	C₂×C₄	—	524	2	no	—
293	C₂×C₄	—	131	1	no	—
294	C₂×C₄	—	131	1	no	—
295	C₂×C₄	—	131	1	no	—
296	C₂×C₄	—	131	2	no	—
297	C₂×C₄	—	131	2	no	—
298	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
299	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	262	1	no	—
304	C₈	—	262	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	262	1	no	—
309	C₂²	—	262	1	no	—
310	C₂²	—	131	2	no	—
311	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	262	2	no	—
318	C₄	—	262	1	no	—
319	C₄	—	262	1	no	—
320	C₄	—	524	1	no	—
321	C₄	—	524	1	no	—
322	C₄	—	524	1	no	—
323	C₄	—	131	2	no	—
324	C₄	C₂×Q₈×D₁₃₁	1	1	yes	characteristic
325	C₄	#41×D₁₃₁	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	262	2	no	—
328	C₂	C₂²⋊Q₈×D₁₃₁	1	1	yes	characteristic
329	C₂	#22554108	1	1	yes	characteristic
330	C₂	#22554109	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₁
commutator, cosocle	C₂×C₄×C₁₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₁
Fitting	#478×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×D₄
prime radical	C₂×#14523512	C₂²
hypercenter	#478	D₁₃₁
hypocenter	C₁₃₁	#3080
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #125×D₁₃₁	C₂	semidirect
3. #22553622	C₂	semidirect
4. #22553620	C₂	semidirect
5. #22553618	C₂	semidirect
6. #22553616	C₂	semidirect
7. #22553614	C₂	semidirect
8. #22553612	C₂	semidirect
9. #22553610	C₂	semidirect
10. #22553353	C₂	semidirect
11. #22553351	C₂	semidirect
12. #22553334	C₂	semidirect
15. #478×C₁₃₁	C₂	semidirect
165. C₁₃₁	#3080	semidirect
176. #478	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#125×D₁₃₁	2
C₄×#14523512	2
#22553622	2
#22553620	2
#22553618	2
#22553616	2
#22553614	2
#22553612	2
#22553610	2
#22553353	2
#22553351	2
#22553334	2
#22553310	2
#478×C₁₃₁	2
#3080	131