Subgroups — #14908456

Order 2816.

556 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077469	C₂	1	1	yes	maximal, characteristic
2	#13070770	C₂	1	2	yes	maximal, complemented
3	#13070770	C₂	1	2	yes	maximal, complemented
4	#13070760	C₂	1	2	yes	maximal, complemented
5	#13070750	C₂	1	2	yes	maximal, complemented
6	#13070402	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070398	C₂	1	1	yes	maximal, characteristic, complemented
8	#13070375	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070331	C₂	1	1	yes	maximal, characteristic
10	C₂×#10622129	C₂	1	1	yes	maximal, characteristic
11	C₁₁×#3164	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₁×#544	—	2	1	no	—
13	C₂²×D₈₈	C₂²	1	1	yes	characteristic
14	#10623052	—	2	1	no	—
15	C₁₁×#501	—	2	2	no	—
16	C₁₁×#500	C₂²	1	2	yes	complemented
17	C₁₁×#499	C₂²	1	2	yes	complemented
18	C₁₁×#428	—	2	1	no	—
19	C₁₁×#427	C₂²	1	1	yes	characteristic
20	C₂×C₁₁×#107	C₂²	1	1	yes	characteristic
21	C₁₁×#405	C₂²	1	1	yes	characteristic
22	#10622744	—	2	1	no	—
23	#10622719	C₂²	1	1	yes	characteristic
24	C₂×#66876	C₂²	1	1	yes	characteristic
25	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
26	#10622324	—	2	2	no	—
27	C₄×D₈₈	C₂²	1	2	yes	—
28	C₄×D₈₈	C₂²	1	2	yes	—
29	#10622129	C₂²	1	2	yes	—
30	#10622129	C₂²	1	2	yes	—
31	#10622129	C₂²	1	2	yes	—
32	#10622129	C₂²	1	2	yes	—
33	#10622128	C₂²	1	1	yes	characteristic, complemented
34	#10622128	C₂²	1	1	yes	characteristic, complemented
35	#10622128	C₂²	1	2	yes	complemented
36	#10622127	C₂²	1	1	yes	characteristic, complemented
37	#10622127	C₂²	1	1	yes	characteristic, complemented
38	#10622127	C₂²	1	2	yes	complemented
39	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	complemented
40	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	complemented
41	C₂²×D₄₄	—	2	1	no	—
42	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
44	C₁₁×C₄⋊Q₈	—	2	2	no	—
45	C₁₁×C₄⋊Q₈	—	4	1	no	—
46	C₁₁×#125	—	2	2	no	—
47	C₁₁×#125	—	4	1	no	—
48	C₁₁×C₂²⋊Q₈	—	4	2	no	—
49	C₁₁×#117	—	2	1	no	—
50	C₁₁×#117	D₄	1	1	yes	characteristic
51	C₂×C₁₁×C₄⋊C₄	—	2	1	no	—
52	#66979	—	2	1	no	—
53	C₂×D₈₈	—	2	1	no	—
54	C₂×D₈₈	—	2	2	no	—
55	C₂×D₈₈	Q₈	1	4	yes	complemented
56	C₂×D₈₈	C₂³	1	2	yes	—
57	#66946	—	4	1	no	—
58	#66946	—	4	1	no	—
59	C₂²⋊C₄×D₁₁	—	4	2	no	—
60	C₄×D₄₄	—	2	2	no	—
61	C₄×D₄₄	—	2	2	no	—
62	C₁₁×#107	—	2	2	no	—
63	C₁₁×#107	—	2	1	no	—
64	C₁₁×#107	C₂³	1	1	yes	characteristic
65	C₁₁×#107	C₂³	1	2	yes	—
66	C₁₁×#107	C₂³	1	1	yes	characteristic
67	C₁₁×#106	—	2	2	no	—
68	C₁₁×#106	—	2	1	no	—
69	C₁₁×#97	D₄	1	2	yes	—
70	C₄×C₈×C₁₁	C₂³	1	2	yes	—
71	#66885	—	2	2	no	—
72	#66885	—	2	2	no	—
73	#66882	C₂³	1	1	yes	characteristic
74	#66882	C₂³	1	1	yes	characteristic
75	#66881	—	2	1	no	—
76	#66876	—	2	2	no	—
77	C₂×C₁₁⋊C₁₆	—	2	2	no	—
78	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
79	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
80	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
81	#33631	—	11	1	no	maximal, Sylow
82	C₂³×D₁₁	—	4	1	no	—
83	C₂×Q₈×C₁₁	—	4	2	no	—
84	C₂²×C₄×C₁₁	—	4	1	no	—
85	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
86	C₂×D₄₄	—	2	4	no	—
87	C₂×D₄₄	—	2	2	no	—
88	C₂×D₄₄	—	4	1	no	—
89	C₂×D₄₄	—	4	2	no	—
90	C₂×C₄×D₁₁	—	4	2	no	—
91	C₂×C₄×D₁₁	—	4	2	no	—
92	C₁₁×M₁₆	—	2	2	no	—
93	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
96	C₂×C₈×C₁₁	#41	1	2	yes	—
97	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
98	C₁₁×C₄⋊C₄	—	2	1	no	—
99	C₁₁×C₄⋊C₄	—	2	2	no	—
100	C₁₁×C₄⋊C₄	—	2	1	no	—
101	C₁₁×C₄⋊C₄	—	4	1	no	—
102	C₁₁×C₄⋊C₄	—	4	2	no	—
103	C₁₁×C₄⋊C₄	—	4	2	no	—
104	C₁₁×C₄⋊C₄	—	4	2	no	—
105	C₁₁×C₄⋊C₄	D₈	1	1	yes	characteristic
106	C₁₁×C₄⋊C₄	D₈	1	1	yes	characteristic
107	C₁₁×C₂²⋊C₄	—	4	1	no	—
108	C₁₁×C₂²⋊C₄	D₈	1	2	yes	—
109	C₄²×C₁₁	—	4	1	no	—
110	C₄²×C₁₁	C₂×D₄	1	2	yes	—
111	C₂²⋊Dic₁₁	—	4	1	no	—
112	(C₂×D₁₁)⋊C₄	—	4	2	no	—
113	(C₂×D₁₁)⋊C₄	—	4	2	no	—
114	C₄⋊Dic₁₁	—	2	1	no	—
115	C₄⋊Dic₁₁	—	2	1	no	—
116	C₄×Dic₁₁	—	4	1	no	—
117	D₈₈	—	2	4	no	—
118	D₈₈	—	2	2	no	—
119	D₈₈	C₂×Q₈	1	4	yes	—
120	C₁₁⋊C₁₆	—	2	1	no	—
121	C₁₁⋊C₁₆	—	2	1	no	—
122	C₁₁⋊C₁₆	—	2	2	no	—
123	#3164	D₁₁	1	1	yes	characteristic, complemented
124	#2948	—	11	1	no	—
125	#2228	—	11	2	no	—
126	#2228	—	11	2	no	—
127	#2226	—	11	2	no	—
128	#2224	—	11	2	no	—
129	#2159	—	11	1	no	—
130	#2158	—	11	1	no	—
131	#2151	—	11	1	no	—
132	#2140	—	11	1	no	—
133	C₂×#357	—	11	1	no	—
134	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₁₁	—	4	2	no	—
136	C₂²×D₁₁	—	4	4	no	—
137	C₂²×D₁₁	—	16	1	no	—
138	C₂²×D₁₁	—	16	1	no	—
139	Q₈×C₁₁	—	16	2	no	—
140	C₂×C₄×C₁₁	—	4	1	no	—
141	C₂×C₄×C₁₁	—	4	1	no	—
142	C₂×C₄×C₁₁	—	4	2	no	—
143	C₂×C₄×C₁₁	—	16	1	no	—
144	C₂×C₄×C₁₁	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₁	#137	1	2	yes	—
149	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×Dic₁₁	—	4	1	no	—
151	C₂×Dic₁₁	—	4	1	no	—
152	D₄₄	—	2	4	no	—
153	D₄₄	—	4	2	no	—
154	D₄₄	—	4	4	no	—
155	C₄×D₁₁	—	16	2	no	—
156	C₄×D₁₁	—	16	2	no	—
157	C₈×C₁₁	—	2	2	no	—
158	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
159	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
160	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
161	C₂²×D₈	—	11	1	no	—
162	#544	—	2	1	no	—
163	#518	—	22	1	no	—
164	C₂×M₃₂	—	11	1	no	—
165	C₂²×C₁₆	—	11	1	no	—
166	#501	—	2	2	no	—
167	#500	C₂×D₁₁	1	2	yes	complemented
168	#499	C₂×D₁₁	1	2	yes	complemented
169	#442	—	22	2	no	—
170	C₄×D₈	—	11	2	no	—
171	C₄×D₈	—	11	2	no	—
172	#428	—	2	1	no	—
173	#427	—	11	1	no	—
174	#427	C₂×D₁₁	1	1	yes	characteristic
175	C₂×#107	C₂×D₁₁	1	1	yes	characteristic
176	#418	—	22	1	no	—
177	#405	C₂×D₁₁	1	1	yes	characteristic
178	#367	—	11	1	no	—
179	#367	—	11	1	no	—
180	#367	—	11	2	no	—
181	#366	—	11	1	no	—
182	#366	—	11	1	no	—
183	#366	—	11	2	no	—
184	C₄⋊C₁₆	—	11	2	no	—
185	C₄⋊C₁₆	—	11	2	no	—
186	#357	—	11	2	no	—
187	#357	—	11	2	no	—
188	#357	—	11	2	no	—
189	#357	—	11	2	no	—
190	C₂²×C₁₁	—	2	1	no	—
191	C₂²×C₁₁	—	2	1	no	—
192	C₂²×C₁₁	#474	1	2	yes	—
193	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
194	C₂×D₁₁	—	4	4	no	—
195	C₂×D₁₁	—	8	2	no	—
196	C₂×D₁₁	—	8	2	no	—
197	C₂×D₁₁	—	16	4	no	—
198	C₄×C₁₁	—	2	2	no	—
199	C₄×C₁₁	—	2	2	no	—
200	C₄×C₁₁	—	8	2	no	—
201	C₄×C₁₁	—	8	1	no	—
202	C₄×C₁₁	—	8	1	no	—
203	C₄×C₁₁	#474	1	2	yes	—
204	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
205	C₄×C₁₁	C₂×#137	1	1	yes	characteristic
206	Dic₁₁	—	8	1	no	—
207	Dic₁₁	—	8	1	no	—
208	C₂²×D₄	—	22	1	no	—
209	C₂×D₈	—	11	4	no	—
210	C₂×D₈	—	11	2	no	—
211	C₂×D₈	—	22	2	no	—
212	C₂×D₈	—	22	1	no	—
213	C₂×M₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	C₄×D₄	—	22	2	no	—
221	C₄×D₄	—	22	2	no	—
222	C₄×D₄	—	44	1	no	—
223	C₄×D₄	—	44	1	no	—
224	#117	—	2	1	no	—
225	#117	—	22	1	no	—
226	#117	C₂²⋊D₁₁	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₂²⋊C₄	—	44	2	no	—
229	M₃₂	—	22	2	no	—
230	C₂×C₁₆	—	11	1	no	—
231	C₂×C₁₆	—	11	2	no	—
232	C₂×C₁₆	—	11	1	no	—
233	C₂×C₁₆	—	22	2	no	—
234	#107	—	2	1	no	—
235	#107	—	2	2	no	—
236	#107	—	11	1	no	—
237	#107	—	11	1	no	—
238	#107	C₂²×D₁₁	1	2	yes	—
239	#107	C₂²×D₁₁	1	1	yes	characteristic
240	#107	C₂²×D₁₁	1	1	yes	characteristic
241	#106	—	2	1	no	—
242	#106	—	2	2	no	—
243	#106	—	22	1	no	—
244	#102	—	22	2	no	—
245	#102	—	22	2	no	—
246	#97	C₂²⋊D₁₁	1	2	yes	—
247	C₄×C₈	C₂²×D₁₁	1	2	yes	—
248	C₂×C₁₁	—	2	2	no	—
249	C₂×C₁₁	C₂×#474	1	1	yes	characteristic
250	C₂×C₁₁	#3415	1	1	yes	characteristic
251	C₂×C₁₁	#3416	1	1	yes	characteristic
252	D₁₁	—	8	4	no	—
253	C₂⁴	—	44	1	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×D₄	—	22	4	no	—
256	C₂×D₄	—	22	2	no	—
257	C₂×D₄	—	44	1	no	—
258	C₂×D₄	—	44	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	44	2	no	—
261	C₂²×C₄	—	44	2	no	—
262	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
263	D₈	—	11	4	no	—
264	D₈	—	22	4	no	—
265	D₈	—	22	2	no	—
266	M₁₆	—	2	2	no	—
267	C₂×C₈	Q₈×D₁₁	1	2	yes	—
268	C₂×C₈	#4343	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₁₁	1	2	yes	—
270	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
271	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	22	1	no	—
280	C₄⋊C₄	—	22	1	no	—
281	C₄⋊C₄	#4323	1	1	yes	characteristic
282	C₄⋊C₄	#4323	1	1	yes	characteristic
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	44	1	no	—
285	C₂²⋊C₄	—	44	2	no	—
286	C₂²⋊C₄	—	44	2	no	—
287	C₂²⋊C₄	#4323	1	2	yes	—
288	C₄²	—	4	1	no	—
289	C₄²	—	44	1	no	—
290	C₄²	C₂×C₂²⋊D₁₁	1	2	yes	—
291	C₁₆	—	22	1	no	—
292	C₁₆	—	22	1	no	—
293	C₁₆	—	22	2	no	—
294	C₁₁	#33631	1	1	yes	characteristic, complemented, Sylow
295	C₂³	—	44	2	no	—
296	C₂³	—	44	4	no	—
297	C₂³	—	176	1	no	—
298	C₂³	—	176	1	no	—
299	C₂³	C₂×#4323	1	1	yes	characteristic
300	Q₈	—	16	2	no	—
301	D₄	—	22	4	no	—
302	D₄	—	44	2	no	—
303	D₄	—	44	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	44	1	no	—
309	C₂×C₄	—	44	1	no	—
310	C₂×C₄	—	176	2	no	—
311	C₂×C₄	—	176	2	no	—
312	C₂×C₄	C₂×#4323	1	2	yes	—
313	C₂×C₄	C₂×#4323	1	2	yes	—
314	C₂×C₄	C₂×#4323	1	1	yes	characteristic
315	C₂×C₄	#66995	1	2	yes	—
316	C₂×C₄	#66999	1	2	yes	—
317	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
318	C₈	—	2	2	no	—
319	C₈	#66999	1	2	yes	—
320	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
321	C₈	C₂×#4343	1	1	yes	characteristic
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	44	4	no	—
325	C₂²	—	88	2	no	—
326	C₂²	—	88	2	no	—
327	C₂²	—	176	4	no	—
328	C₂²	#10622700	1	2	yes	—
329	C₂²	C₂²×#4323	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	8	2	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	1	no	—
335	C₄	—	88	1	no	—
336	C₄	—	88	1	no	—
337	C₄	#10622700	1	2	yes	—
338	C₄	C₂×#66995	1	1	yes	characteristic
339	C₄	C₂×#66999	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	88	4	no	—
342	C₂	C₂×#10622700	1	1	yes	characteristic
343	C₂	#13081755	1	1	yes	characteristic
344	C₂	#13081760	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#3164	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	C₂²×D₈₈	C₂²
hypercenter	#3164	D₁₁
hypocenter	C₁₁	#33631
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3164

Complements

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

N	H ≅ G/N	type
2. #13070770	C₂	semidirect
3. #13070770	C₂	semidirect
4. #13070760	C₂	semidirect
5. #13070750	C₂	semidirect
6. #13070402	C₂	semidirect
7. #13070398	C₂	semidirect
8. #13070375	C₂	semidirect
11. C₁₁×#3164	C₂	semidirect
16. C₁₁×#500	C₂²	semidirect
17. C₁₁×#499	C₂²	semidirect
33. #10622128	C₂²	semidirect
34. #10622128	C₂²	semidirect
35. #10622128	C₂²	semidirect
36. #10622127	C₂²	semidirect
37. #10622127	C₂²	semidirect
38. #10622127	C₂²	semidirect
39. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
40. C₄⋊(C₁₁⋊C₁₆)	C₂²	semidirect
55. C₂×D₈₈	Q₈	semidirect
123. #3164	D₁₁	semidirect
167. #500	C₂×D₁₁	semidirect
168. #499	C₂×D₁₁	semidirect
294. C₁₁	#33631	semidirect

Maximal subgroups

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

subgroup	index
#13077469	2
#13070770	2
#13070760	2
#13070750	2
#13070402	2
#13070398	2
#13070375	2
#13070331	2
C₂×#10622129	2
C₁₁×#3164	2
#33631	11