Subgroups — #14909326

Order 2816.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079186	C₂	1	2	yes	maximal, complemented
2	#13070895	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070877	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070848	C₂	1	2	yes	maximal, complemented
5	#13070712	C₂	1	2	yes	maximal, complemented
6	#13070707	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070702	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₈₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#66864	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623010	—	2	2	no	—
13	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	#10622729	C₂²	1	2	yes	—
17	#10622728	C₂²	1	2	yes	—
18	#10622727	—	2	2	no	—
19	C₂×Dic₈₈	C₂²	1	1	yes	characteristic
20	C₂×#66864	C₂²	1	2	yes	—
21	C₂×D₁₇₆	C₂²	1	1	yes	characteristic
22	C₄×Dic₄₄	C₂²	1	2	yes	—
23	#10622310	—	2	2	no	—
24	#10622309	C₂²	1	2	yes	—
25	C₄×D₈₈	C₂²	1	2	yes	—
26	C₄×#4314	—	2	2	no	—
27	#10622296	C₂²	1	2	yes	—
28	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
31	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
32	#10622147	C₂²	1	2	yes	—
33	#10622147	C₂²	1	2	yes	—
34	#10622143	C₂²	1	2	yes	—
35	#10622142	C₂²	1	1	yes	characteristic
36	#10622142	C₂²	1	1	yes	characteristic
37	#10622141	C₂²	1	2	yes	—
38	#10622141	C₂²	1	2	yes	—
39	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
42	#66983	—	2	2	no	—
43	#66982	—	4	2	no	—
44	C₄×C₂²⋊D₁₁	—	4	2	no	—
45	#66977	—	2	2	no	—
46	C₂×Dic₄₄	C₂³	1	2	yes	—
47	C₂×D₈₈	C₂³	1	2	yes	—
48	C₂×#4314	—	2	2	no	—
49	#66929	—	4	2	no	—
50	#66928	—	2	2	no	—
51	C₄×D₄₄	—	2	2	no	—
52	#66925	—	4	2	no	—
53	#66923	—	2	2	no	—
54	C₄×Dic₂₂	—	2	2	no	—
55	C₂×C₁₁×C₁₆	—	2	2	no	—
56	C₂×C₁₁×C₁₆	—	2	2	no	—
57	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
65	#66885	—	2	2	no	—
66	#66885	—	2	2	no	—
67	#66882	C₂³	1	2	yes	—
68	#66882	C₂³	1	2	yes	—
69	#66881	—	2	2	no	—
70	#66880	—	2	2	no	—
71	#66880	—	2	2	no	—
72	Dic₈₈	—	2	2	no	—
73	#66864	—	2	4	no	—
74	D₁₇₆	—	2	2	no	—
75	#33735	—	11	1	no	maximal, Sylow
76	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₁₁	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₂²⋊D₁₁	—	4	2	no	—
80	C₂×D₄₄	—	2	2	no	—
81	C₂×C₄×D₁₁	—	4	2	no	—
82	C₂×Dic₂₂	—	2	2	no	—
83	C₂×C₈×C₁₁	—	2	2	no	—
84	C₂×C₈×C₁₁	—	2	1	no	—
85	C₂×C₈×C₁₁	—	2	1	no	—
86	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
90	C₄²×C₁₁	D₈	1	1	yes	characteristic
91	C₄²×C₁₁	D₈	1	1	yes	characteristic
92	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₂²⋊Dic₁₁	—	4	2	no	—
95	(C₂×D₁₁)⋊C₄	—	4	2	no	—
96	(C₂×D₁₁)⋊C₄	—	4	2	no	—
97	(C₂×D₁₁)⋊C₄	—	4	2	no	—
98	(C₂×D₁₁)⋊C₄	—	4	2	no	—
99	C₄⋊Dic₁₁	—	2	2	no	—
100	C₄⋊Dic₁₁	—	2	2	no	—
101	Dic₁₁⋊C₄	—	4	2	no	—
102	Dic₁₁⋊C₄	—	4	2	no	—
103	Dic₁₁⋊C₄	—	4	2	no	—
104	Dic₁₁⋊C₄	—	4	2	no	—
105	C₄×Dic₁₁	—	4	2	no	—
106	Dic₄₄	—	2	2	no	—
107	Dic₄₄	—	2	2	no	—
108	D₈₈	—	2	2	no	—
109	D₈₈	—	2	2	no	—
110	#4314	—	4	4	no	—
111	C₁₁×C₁₆	—	2	1	no	—
112	C₁₁×C₁₆	—	2	1	no	—
113	C₁₁×C₁₆	#41	1	2	yes	—
114	C₁₁×C₁₆	#41	1	2	yes	—
115	#3168	—	11	2	no	—
116	#2253	—	11	1	no	—
117	#2249	—	11	1	no	—
118	#2242	—	11	2	no	—
119	#2217	—	11	2	no	—
120	#2216	—	11	1	no	—
121	#2215	—	11	1	no	—
122	C₄×Q₃₂	—	11	1	no	—
123	C₄×SD₃₂	—	11	2	no	—
124	C₄×D₁₆	—	11	1	no	—
125	C₂×C₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
126	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
127	C₂²×D₁₁	—	4	2	no	—
128	C₂×C₄×C₁₁	—	2	2	no	—
129	C₂×C₄×C₁₁	—	2	1	no	—
130	C₂×C₄×C₁₁	—	2	1	no	—
131	C₂×C₄×C₁₁	—	2	2	no	—
132	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
138	C₂²⋊D₁₁	—	8	4	no	—
139	C₂×Dic₁₁	—	4	2	no	—
140	C₂×Dic₁₁	—	4	2	no	—
141	C₂×Dic₁₁	—	4	2	no	—
142	D₄₄	—	4	2	no	—
143	D₄₄	—	4	2	no	—
144	C₄×D₁₁	—	8	4	no	—
145	Dic₂₂	—	4	2	no	—
146	Dic₂₂	—	4	2	no	—
147	C₈×C₁₁	—	2	1	no	—
148	C₈×C₁₁	—	2	1	no	—
149	C₈×C₁₁	#135	1	2	yes	—
150	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
151	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
152	#529	—	22	2	no	—
153	C₂×Q₃₂	—	11	1	no	—
154	C₂×SD₃₂	—	11	2	no	—
155	C₂×D₁₆	—	11	1	no	—
156	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
157	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
158	#499	—	11	2	no	—
159	#495	—	11	2	no	—
160	#488	—	22	2	no	—
161	#467	—	11	2	no	—
162	#466	—	11	2	no	—
163	#465	—	22	2	no	—
164	C₄×Q₁₆	—	11	2	no	—
165	C₄×SD₁₆	—	22	2	no	—
166	C₄×D₈	—	11	2	no	—
167	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
168	#367	—	11	2	no	—
169	#366	—	11	1	no	—
170	#366	—	11	1	no	—
171	#358	—	11	2	no	—
172	#358	—	11	2	no	—
173	#357	—	11	2	no	—
174	#357	—	11	2	no	—
175	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
176	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
177	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
178	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	#508	1	2	yes	—
182	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₁₁	—	8	2	no	—
184	C₂×D₁₁	—	8	2	no	—
185	C₂×D₁₁	—	8	2	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	#508	1	2	yes	—
191	C₄×C₁₁	#532	1	1	yes	characteristic
192	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
193	Dic₁₁	—	8	2	no	—
194	Dic₁₁	—	8	2	no	—
195	Dic₁₁	—	8	2	no	—
196	C₂×Q₁₆	—	11	2	no	—
197	C₂×SD₁₆	—	22	2	no	—
198	C₂×D₈	—	11	2	no	—
199	C₂²×C₈	D₄₄	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
201	#126	—	44	2	no	—
202	#125	—	22	2	no	—
203	#124	—	22	2	no	—
204	#123	—	44	2	no	—
205	C₂²⋊Q₈	—	22	2	no	—
206	#121	—	22	2	no	—
207	C₄×Q₈	—	22	2	no	—
208	C₄×D₄	—	22	2	no	—
209	C₄×D₄	—	44	2	no	—
210	#117	—	44	2	no	—
211	C₂×C₄²	D₄₄	1	1	yes	characteristic
212	Q₃₂	—	22	2	no	—
213	SD₃₂	—	22	4	no	—
214	D₁₆	—	22	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
218	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
219	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
220	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
221	#107	—	11	2	no	—
222	#107	—	11	2	no	—
223	#106	—	22	2	no	—
224	#103	—	22	2	no	—
225	#103	—	22	2	no	—
226	#102	—	22	2	no	—
227	#102	—	22	2	no	—
228	C₄×C₈	D₄₄	1	1	yes	characteristic
229	C₄×C₈	D₄₄	1	1	yes	characteristic
230	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
232	C₂×C₁₁	—	2	2	no	—
233	C₂×C₁₁	#3169	1	1	yes	characteristic
234	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
235	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
236	D₁₁	—	16	2	no	—
237	C₂×Q₈	—	22	2	no	—
238	C₂×D₄	—	22	2	no	—
239	C₂×D₄	—	44	2	no	—
240	C₂²×C₄	—	44	2	no	—
241	C₂²×C₄	D₈₈	1	1	yes	characteristic
242	C₂²×C₄	D₈₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
244	Q₁₆	—	22	2	no	—
245	Q₁₆	—	22	2	no	—
246	SD₁₆	—	44	4	no	—
247	D₈	—	22	2	no	—
248	D₈	—	22	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
254	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
255	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
256	C₄⋊C₄	—	22	2	no	—
257	C₄⋊C₄	—	22	2	no	—
258	C₄⋊C₄	—	44	2	no	—
259	C₄⋊C₄	—	44	2	no	—
260	C₄⋊C₄	—	44	2	no	—
261	C₄⋊C₄	—	44	2	no	—
262	C₂²⋊C₄	—	44	2	no	—
263	C₂²⋊C₄	—	44	2	no	—
264	C₂²⋊C₄	—	44	2	no	—
265	C₂²⋊C₄	—	44	2	no	—
266	C₂²⋊C₄	—	44	2	no	—
267	C₄²	—	44	2	no	—
268	C₄²	D₈₈	1	1	yes	characteristic
269	C₄²	D₈₈	1	1	yes	characteristic
270	C₄²	C₂×D₄₄	1	1	yes	characteristic
271	C₄²	C₂×D₄₄	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	#4339	1	2	yes	—
275	C₁₆	#4339	1	2	yes	—
276	C₁₁	#33735	1	1	yes	characteristic, complemented, Sylow
277	C₂³	—	44	2	no	—
278	C₂³	C₂×D₈₈	1	1	yes	characteristic
279	Q₈	—	44	2	no	—
280	Q₈	—	44	2	no	—
281	D₄	—	44	2	no	—
282	D₄	—	44	2	no	—
283	D₄	—	88	4	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	44	2	no	—
289	C₂×C₄	—	44	2	no	—
290	C₂×C₄	—	44	2	no	—
291	C₂×C₄	—	88	4	no	—
292	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
295	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
296	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	#66957	1	2	yes	—
301	C₈	C₂×#4339	1	1	yes	characteristic
302	C₈	C₂×#4339	1	1	yes	characteristic
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	88	2	no	—
306	C₂²	—	88	2	no	—
307	C₂²	—	88	2	no	—
308	C₂²	#10622517	1	2	yes	—
309	C₂²	C₂²×D₈₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	88	2	no	—
315	C₄	—	88	2	no	—
316	C₄	—	88	2	no	—
317	C₄	#10622517	1	2	yes	—
318	C₄	#10623008	1	1	yes	characteristic
319	C₄	C₂×#66957	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	176	2	no	—
322	C₂	#13079195	1	1	yes	characteristic
323	C₂	C₂×#10622517	1	1	yes	characteristic
324	C₂	C₂×#10622517	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₈₈
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₄×C₁₁×C₁₆	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	#13070702	C₂
hypercenter	C₂×C₄×C₁₆	D₁₁
hypocenter	C₁₁	#33735
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #13079186	C₂	semidirect
2. #13070895	C₂	semidirect
3. #13070877	C₂	semidirect
4. #13070848	C₂	semidirect
5. #13070712	C₂	semidirect
6. #13070707	C₂	semidirect
8. C₄×Dic₈₈	C₂	semidirect
9. C₄×#66864	C₂	semidirect
10. C₄×D₁₇₆	C₂	semidirect
11. C₂×C₄×C₁₁×C₁₆	C₂	semidirect
125. C₂×C₄×C₁₆	D₁₁	semidirect
276. C₁₁	#33735	semidirect

Maximal subgroups

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

subgroup	index
#13079186	2
#13070895	2
#13070877	2
#13070848	2
#13070712	2
#13070707	2
#13070702	2
C₄×Dic₈₈	2
C₄×#66864	2
C₄×D₁₇₆	2
C₂×C₄×C₁₁×C₁₆	2
#33735	11