Subgroups — #30678624

Order 16768.

362 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554045	C₂	1	1	yes	maximal, characteristic, complemented
2	#22554018	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553669	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553665	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553657	C₂	1	1	yes	maximal, characteristic, complemented
6	#22553611	C₂	1	1	yes	maximal, characteristic, complemented
7	#22553606	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553528	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553526	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553506	C₂	1	1	yes	maximal, characteristic
11	#22553436	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553399	C₂	1	1	yes	maximal, characteristic, complemented
13	#22553394	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#14523498	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14523502	C₂²	1	1	yes	characteristic
17	C₂×D₁₀₄₈	C₂²	1	1	yes	characteristic
18	C₂×#14523498	C₂²	1	1	yes	characteristic
19	#16862245	C₂²	1	1	yes	characteristic
20	#16862242	C₂²	1	1	yes	characteristic
21	#16862237	C₂²	1	1	yes	characteristic
22	#16862234	C₂²	1	1	yes	characteristic
23	#16862233	C₂²	1	1	yes	characteristic
24	#16862232	—	2	1	no	—
25	#16862229	—	2	1	no	—
26	#16862228	—	2	1	no	—
27	#16862226	—	2	1	no	—
28	#16862224	—	2	1	no	—
29	#16862224	—	2	1	no	—
30	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic
31	#16862211	C₂²	1	1	yes	characteristic
32	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
33	#16862208	C₂²	1	1	yes	characteristic
34	#16862208	C₂²	1	1	yes	characteristic
35	#16862208	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₁₃₁	—	2	1	no	—
37	C₄⋊C₄×D₁₃₁	—	2	1	no	—
38	#16862179	C₂²	1	1	yes	characteristic
39	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
41	#16862171	C₂²	1	1	yes	characteristic
42	#16862169	C₂²	1	1	yes	characteristic
43	#16862169	C₂²	1	1	yes	characteristic
44	#16862168	C₂²	1	1	yes	characteristic
45	#16862167	C₂²	1	1	yes	characteristic
46	#16862166	C₂²	1	1	yes	characteristic
47	#16862166	C₂²	1	1	yes	characteristic
48	#16862166	C₂²	1	1	yes	characteristic
49	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₃₁	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	C₂×Dic₂₆₂	—	2	1	no	—
60	C₂×Dic₂₆₂	D₄	1	1	yes	characteristic
61	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
62	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
63	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
65	C₂×C₂²⋊D₁₃₁	—	2	1	no	—
66	C₂×C₄×D₁₃₁	—	2	1	no	—
67	C₂×C₄×D₁₃₁	—	2	1	no	—
68	C₂×C₄×D₁₃₁	—	2	1	no	—
69	C₂×C₄×D₁₃₁	—	2	1	no	—
70	#14523502	—	2	2	no	—
71	D₁₀₄₈	—	2	2	no	—
72	#14523498	—	2	2	no	—
73	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₁₃₁	—	2	1	no	—
76	C₄⋊Dic₁₃₁	D₄	1	1	yes	characteristic
77	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
78	Dic₁₃₁⋊C₄	—	2	1	no	—
79	Dic₁₃₁⋊C₄	—	2	1	no	—
80	Dic₁₃₁⋊C₄	—	2	1	no	—
81	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
82	Dic₁₃₁⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₃₁	—	2	1	no	—
84	C₂²⋊Dic₁₃₁	—	2	1	no	—
85	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
87	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
88	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
89	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
90	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
91	C₄×Dic₁₃₁	—	2	1	no	—
92	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₃₁	—	2	1	no	—
96	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₃₁	—	2	1	no	—
99	C₄⋊C₄×C₁₃₁	—	2	1	no	—
100	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₁₃₁	—	2	1	no	—
104	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
105	C₂²×D₁₃₁	—	2	1	no	—
106	C₂²×D₁₃₁	—	2	1	no	—
107	Dic₂₆₂	—	2	2	no	—
108	Dic₂₆₂	—	2	1	no	—
109	Dic₂₆₂	—	2	1	no	—
110	Dic₂₆₂	—	8	1	no	—
111	D₅₂₄	—	2	1	no	—
112	D₅₂₄	—	2	1	no	—
113	D₅₂₄	—	2	1	no	—
114	D₅₂₄	#41	1	2	yes	—
115	C₂²⋊D₁₃₁	—	4	2	no	—
116	C₂²⋊D₁₃₁	—	8	1	no	—
117	C₂×Dic₁₃₁	—	2	1	no	—
118	C₂×Dic₁₃₁	—	2	1	no	—
119	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₃₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₁₃₁	—	4	2	no	—
124	C₄×D₁₃₁	—	8	1	no	—
125	C₄×D₁₃₁	—	8	1	no	—
126	C₄×D₁₃₁	—	8	1	no	—
127	C₁₃₁⋊C₈	—	2	1	no	—
128	C₁₃₁⋊C₈	—	2	1	no	—
129	C₂³×C₁₃₁	—	2	1	no	—
130	D₄×C₁₃₁	—	2	1	no	—
131	D₄×C₁₃₁	—	2	1	no	—
132	C₂×C₄×C₁₃₁	—	2	1	no	—
133	C₂×C₄×C₁₃₁	—	2	1	no	—
134	C₂×C₄×C₁₃₁	—	2	1	no	—
135	C₂×C₄×C₁₃₁	—	8	1	no	—
136	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
139	C₈×C₁₃₁	—	2	1	no	—
140	C₈×C₁₃₁	—	2	1	no	—
141	C₂²×C₁₃₁	—	4	1	no	—
142	C₂²×C₁₃₁	—	4	1	no	—
143	C₂²×C₁₃₁	—	4	1	no	—
144	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₃₁	—	2	2	no	—
146	C₂×D₁₃₁	—	4	1	no	—
147	C₂×D₁₃₁	—	4	1	no	—
148	C₂×D₁₃₁	—	4	1	no	—
149	C₂×D₁₃₁	—	4	1	no	—
150	C₂×D₁₃₁	—	4	1	no	—
151	C₄×C₁₃₁	—	2	1	no	—
152	C₄×C₁₃₁	—	2	1	no	—
153	C₄×C₁₃₁	—	4	1	no	—
154	C₄×C₁₃₁	—	4	1	no	—
155	C₄×C₁₃₁	—	4	1	no	—
156	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₁₃₁	—	2	1	no	—
159	Dic₁₃₁	—	2	1	no	—
160	Dic₁₃₁	—	2	1	no	—
161	Dic₁₃₁	—	4	1	no	—
162	Dic₁₃₁	—	4	1	no	—
163	Dic₁₃₁	#135	1	2	yes	—
164	C₂×C₁₃₁	—	8	1	no	—
165	C₂×C₁₃₁	#548	1	1	yes	characteristic
166	C₂×C₁₃₁	C₂×#135	1	1	yes	characteristic
167	C₂×C₁₃₁	#576	1	1	yes	characteristic
168	D₁₃₁	—	4	2	no	—
169	D₁₃₁	—	8	1	no	—
170	C₁₃₁	#3361	1	1	yes	characteristic, complemented, Sylow
171	#3361	—	131	1	no	maximal, Sylow
172	#559	—	131	1	no	—
173	#554	—	131	1	no	—
174	#488	—	131	1	no	—
175	#488	—	131	1	no	—
176	#486	—	131	1	no	—
177	#478	—	131	1	no	—
178	#478	D₁₃₁	1	1	yes	characteristic, complemented
179	#477	—	131	1	no	—
180	#463	—	131	1	no	—
181	#463	—	131	1	no	—
182	#459	—	131	1	no	—
183	C₈×Q₈	—	131	1	no	—
184	C₄×SD₁₆	—	131	1	no	—
185	C₄×D₈	—	131	1	no	—
186	C₄×D₈	—	131	1	no	—
187	C₂×SD₁₆	—	131	1	no	—
188	C₂×D₈	—	131	1	no	—
189	C₂×D₈	—	131	1	no	—
190	C₄⋊Q₈	—	131	1	no	—
191	#127	—	131	1	no	—
192	#125	—	131	1	no	—
193	#125	C₂×D₁₃₁	1	1	yes	characteristic
194	#124	—	262	1	no	—
195	#124	—	131	1	no	—
196	#124	—	131	1	no	—
197	#123	—	262	1	no	—
198	#123	—	262	1	no	—
199	C₂²⋊Q₈	—	262	1	no	—
200	C₂²⋊Q₈	—	262	1	no	—
201	C₂²⋊Q₈	—	262	1	no	—
202	C₄×Q₈	—	131	1	no	—
203	C₄×Q₈	—	131	1	no	—
204	C₄×D₄	—	262	1	no	—
205	C₄×D₄	—	131	1	no	—
206	C₄×D₄	—	131	1	no	—
207	C₄×D₄	—	131	1	no	—
208	C₄×D₄	—	131	1	no	—
209	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
210	C₂×C₄⋊C₄	—	262	1	no	—
211	#107	—	131	1	no	—
212	#107	C₂×D₁₃₁	1	1	yes	characteristic
213	#106	C₂×D₁₃₁	1	1	yes	characteristic
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	#102	—	131	1	no	—
221	#102	—	131	1	no	—
222	#102	—	131	1	no	—
223	#102	—	131	1	no	—
224	#102	—	131	1	no	—
225	#102	C₂×D₁₃₁	1	1	yes	characteristic
226	#102	C₂×D₁₃₁	1	1	yes	characteristic
227	C₄×C₈	—	131	1	no	—
228	C₄×C₈	—	131	1	no	—
229	C₄×C₈	—	131	1	no	—
230	C₂×Q₈	—	262	1	no	—
231	C₂×Q₈	—	131	1	no	—
232	C₂×Q₈	—	131	1	no	—
233	C₂×D₄	—	262	1	no	—
234	C₂×D₄	—	262	1	no	—
235	C₂×D₄	—	131	1	no	—
236	C₂×D₄	—	131	1	no	—
237	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	262	1	no	—
240	C₂²×C₄	—	262	1	no	—
241	C₂²×C₄	—	262	1	no	—
242	C₂²×C₄	—	262	1	no	—
243	SD₁₆	—	262	2	no	—
244	D₈	—	262	2	no	—
245	D₈	—	262	2	no	—
246	C₂×C₈	—	131	1	no	—
247	C₂×C₈	—	131	1	no	—
248	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	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	—	131	1	no	—
260	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	262	1	no	—
265	C₂²⋊C₄	—	262	1	no	—
266	C₂²⋊C₄	—	262	1	no	—
267	C₂²⋊C₄	—	262	1	no	—
268	C₂²⋊C₄	—	262	1	no	—
269	C₂²⋊C₄	—	262	1	no	—
270	C₂²⋊C₄	—	262	1	no	—
271	C₂²⋊C₄	—	262	1	no	—
272	C₄²	—	262	1	no	—
273	C₄²	—	131	1	no	—
274	C₄²	—	131	1	no	—
275	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
276	C₂³	—	2	1	no	—
277	C₂³	—	262	1	no	—
278	C₂³	—	262	1	no	—
279	Q₈	—	262	2	no	—
280	Q₈	—	262	1	no	—
281	Q₈	—	262	1	no	—
282	Q₈	—	1048	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	262	1	no	—
286	D₄	—	262	1	no	—
287	D₄	—	262	1	no	—
288	D₄	—	524	2	no	—
289	D₄	—	1048	1	no	—
290	D₄	—	131	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	262	1	no	—
295	C₂×C₄	—	262	1	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	524	2	no	—
298	C₂×C₄	—	1048	1	no	—
299	C₂×C₄	—	1048	1	no	—
300	C₂×C₄	—	1048	1	no	—
301	C₂×C₄	—	131	1	no	—
302	C₂×C₄	—	131	1	no	—
303	C₂×C₄	—	131	1	no	—
304	C₂×C₄	—	131	1	no	—
305	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	262	1	no	—
311	C₈	—	262	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	262	2	no	—
316	C₂²	—	524	1	no	—
317	C₂²	—	524	1	no	—
318	C₂²	—	524	1	no	—
319	C₂²	—	524	1	no	—
320	C₂²	—	524	1	no	—
321	C₂²	C₂×D₄×D₁₃₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	262	1	no	—
328	C₄	—	262	1	no	—
329	C₄	—	262	1	no	—
330	C₄	—	524	1	no	—
331	C₄	—	524	1	no	—
332	C₄	—	131	2	no	—
333	C₄	#41×D₁₃₁	1	1	yes	characteristic
334	C₄	#16862294	1	1	yes	characteristic
335	C₂	—	8	1	no	—
336	C₂	—	524	2	no	—
337	C₂	—	1048	1	no	—
338	C₂	#22553972	1	1	yes	characteristic
339	C₂	#135×D₁₃₁	1	1	yes	characteristic
340	C₂	#22554116	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	#22553506	C₂
hypercenter	#478	D₁₃₁
hypocenter	C₁₃₁	#3361
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. #22554045	C₂	semidirect
2. #22554018	C₂	semidirect
3. #22553669	C₂	semidirect
4. #22553665	C₂	semidirect
5. #22553657	C₂	semidirect
6. #22553611	C₂	semidirect
7. #22553606	C₂	semidirect
8. #22553528	C₂	semidirect
9. #22553526	C₂	semidirect
11. #22553436	C₂	semidirect
12. #22553399	C₂	semidirect
13. #22553394	C₂	semidirect
14. C₄×#14523498	C₂	semidirect
15. #478×C₁₃₁	C₂	semidirect
170. C₁₃₁	#3361	semidirect
178. #478	D₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#22554045	2
#22554018	2
#22553669	2
#22553665	2
#22553657	2
#22553611	2
#22553606	2
#22553528	2
#22553526	2
#22553506	2
#22553436	2
#22553399	2
#22553394	2
C₄×#14523498	2
#478×C₁₃₁	2
#3361	131