Subgroups — #14909409

Order 2816.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079266	C₂	1	1	yes	maximal, characteristic
2	#13070892	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070873	C₂	1	1	yes	maximal, characteristic, complemented
4	#13070861	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070859	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070731	C₂	1	2	yes	maximal, complemented
7	#13070714	C₂	1	2	yes	maximal, complemented
8	#13070478	C₂	1	2	yes	maximal, complemented
9	C₄×#66892	C₂	1	2	yes	maximal, complemented
10	#13070213	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₁×#3177	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₁×#541	—	2	1	no	—
13	#10623066	—	2	1	no	—
14	C₁₁×#499	C₂²	1	1	yes	characteristic
15	C₁₁×#494	C₂²	1	1	yes	characteristic
16	C₁₁×#491	—	2	1	no	—
17	C₁₁×#470	—	2	1	no	—
18	C₁₁×#467	C₂²	1	2	yes	—
19	C₁₁×#441	—	2	1	no	—
20	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
21	C₁₁×#405	C₂²	1	1	yes	characteristic
22	C₂×#66892	C₂²	1	2	yes	—
23	C₂×#66892	C₂²	1	2	yes	—
24	#10622748	—	2	1	no	—
25	#10622728	C₂²	1	2	yes	—
26	C₂×#66876	C₂²	1	1	yes	characteristic
27	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
28	#10622326	—	2	1	no	—
29	#10622324	—	2	1	no	—
30	#10622308	C₂²	1	1	yes	characteristic
31	C₄×D₈₈	C₂²	1	2	yes	—
32	#10622296	C₂²	1	1	yes	characteristic
33	#10622202	C₂²	1	2	yes	—
34	#10622202	C₂²	1	2	yes	—
35	#10622129	C₂²	1	2	yes	—
36	#10622129	C₂²	1	2	yes	—
37	#10622128	C₂²	1	2	yes	—
38	#10622127	C₂²	1	2	yes	—
39	#10622099	C₂²	1	1	yes	characteristic
40	#10622099	C₂²	1	1	yes	characteristic
41	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
43	C₂×C₁₁×Q₁₆	—	2	1	no	—
44	C₂×C₁₁×Q₁₆	C₂³	1	2	yes	—
45	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
47	C₁₁×C₄⋊Q₈	—	2	1	no	—
48	C₁₁×#126	—	4	2	no	—
49	C₁₁×#125	—	2	1	no	—
50	C₁₁×#125	—	4	2	no	—
51	C₁₁×C₂²⋊Q₈	—	2	2	no	—
52	C₄×Q₈×C₁₁	—	2	2	no	—
53	C₁₁×#117	D₄	1	1	yes	characteristic
54	#66983	—	2	2	no	—
55	C₂×D₈₈	—	2	1	no	—
56	C₂×D₈₈	C₂³	1	2	yes	—
57	#66947	—	4	2	no	—
58	#66937	—	4	2	no	—
59	#66927	—	2	1	no	—
60	C₄×D₄₄	—	2	2	no	—
61	#66923	—	2	1	no	—
62	C₁₁×#107	C₂³	1	2	yes	—
63	C₁₁×#106	—	2	1	no	—
64	C₁₁×#103	—	2	2	no	—
65	C₁₁×#103	—	2	2	no	—
66	C₁₁×#97	D₄	1	1	yes	characteristic
67	C₁₁×#97	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
70	#66892	—	2	4	no	—
71	#66892	—	2	4	no	—
72	#66885	—	2	2	no	—
73	#66885	—	2	2	no	—
74	#66882	C₂³	1	2	yes	—
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	1	yes	characteristic
80	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
81	C₂×C₁₁⋊C₁₆	C₂³	1	1	yes	characteristic
82	#33743	—	11	1	no	maximal, Sylow
83	C₂×Q₈×C₁₁	—	2	2	no	—
84	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₁₁	—	4	2	no	—
86	C₂×D₄₄	—	2	2	no	—
87	C₂×D₄₄	—	4	2	no	—
88	C₂×C₄×D₁₁	—	4	2	no	—
89	C₁₁×Q₁₆	—	2	2	no	—
90	C₁₁×Q₁₆	—	2	2	no	—
91	C₁₁×Q₁₆	—	4	2	no	—
92	C₁₁×M₁₆	—	2	2	no	—
93	C₂×C₈×C₁₁	—	2	1	no	—
94	C₂×C₈×C₁₁	—	2	1	no	—
95	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
99	C₁₁×C₄⋊C₄	—	2	2	no	—
100	C₁₁×C₄⋊C₄	—	4	2	no	—
101	C₁₁×C₄⋊C₄	—	4	2	no	—
102	C₁₁×C₄⋊C₄	—	4	2	no	—
103	C₁₁×C₄⋊C₄	—	4	2	no	—
104	C₁₁×C₄⋊C₄	D₈	1	2	yes	—
105	C₁₁×C₂²⋊C₄	—	4	2	no	—
106	C₁₁×C₂²⋊C₄	D₈	1	2	yes	—
107	C₄²×C₁₁	—	4	2	no	—
108	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
110	(C₂×D₁₁)⋊C₄	—	4	2	no	—
111	(C₂×D₁₁)⋊C₄	—	4	2	no	—
112	C₄⋊Dic₁₁	—	2	2	no	—
113	Dic₁₁⋊C₄	—	4	2	no	—
114	D₈₈	—	2	2	no	—
115	D₈₈	—	2	2	no	—
116	D₈₈	—	4	2	no	—
117	C₁₁⋊C₁₆	—	2	1	no	—
118	C₁₁⋊C₁₆	—	2	1	no	—
119	C₁₁⋊C₁₆	#41	1	2	yes	—
120	C₁₁⋊C₁₆	#41	1	2	yes	—
121	#3177	D₁₁	1	1	yes	characteristic, complemented
122	#3176	—	11	1	no	—
123	#2253	—	11	1	no	—
124	#2248	—	11	1	no	—
125	#2245	—	11	1	no	—
126	#2244	—	11	1	no	—
127	#2220	—	11	2	no	—
128	#2217	—	11	2	no	—
129	#2175	—	11	2	no	—
130	C₄×SD₃₂	—	11	2	no	—
131	#2108	—	11	1	no	—
132	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
133	C₂²×D₁₁	—	4	2	no	—
134	Q₈×C₁₁	—	4	2	no	—
135	Q₈×C₁₁	—	4	2	no	—
136	C₂×C₄×C₁₁	—	2	1	no	—
137	C₂×C₄×C₁₁	—	2	1	no	—
138	C₂×C₄×C₁₁	—	4	2	no	—
139	C₂×C₄×C₁₁	—	4	2	no	—
140	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
141	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₁₁	—	16	2	no	—
145	C₂×Dic₁₁	—	4	2	no	—
146	D₄₄	—	4	2	no	—
147	D₄₄	—	4	2	no	—
148	D₄₄	—	16	2	no	—
149	C₄×D₁₁	—	16	2	no	—
150	C₈×C₁₁	—	2	1	no	—
151	C₈×C₁₁	—	2	1	no	—
152	C₈×C₁₁	#135	1	2	yes	—
153	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
155	#541	—	2	1	no	—
156	#540	—	22	1	no	—
157	C₂×SD₃₂	—	11	2	no	—
158	C₂×SD₃₂	—	11	2	no	—
159	C₂×M₃₂	—	11	1	no	—
160	C₂²×C₁₆	—	11	1	no	—
161	#499	—	11	1	no	—
162	#499	C₂×D₁₁	1	1	yes	characteristic
163	#494	C₂×D₁₁	1	1	yes	characteristic
164	#493	—	11	1	no	—
165	#491	—	2	1	no	—
166	#490	—	22	1	no	—
167	#470	—	2	1	no	—
168	#469	—	22	1	no	—
169	#467	C₂×D₁₁	1	2	yes	—
170	#466	—	11	2	no	—
171	#442	—	22	1	no	—
172	#441	—	2	1	no	—
173	C₄×Q₁₆	C₂×D₁₁	1	2	yes	—
174	C₄×D₈	—	11	2	no	—
175	#405	C₂×D₁₁	1	1	yes	characteristic
176	#367	—	11	2	no	—
177	#366	—	11	2	no	—
178	#358	—	11	2	no	—
179	#358	—	11	2	no	—
180	#357	—	11	2	no	—
181	#357	—	11	2	no	—
182	#346	—	11	1	no	—
183	#346	—	11	1	no	—
184	C₄×C₁₆	—	11	1	no	—
185	C₄×C₁₆	—	11	1	no	—
186	C₂²×C₁₁	—	2	2	no	—
187	C₂²×C₁₁	—	2	1	no	—
188	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
189	C₂×D₁₁	—	8	2	no	—
190	C₂×D₁₁	—	8	2	no	—
191	C₂×D₁₁	—	8	2	no	—
192	C₄×C₁₁	—	2	2	no	—
193	C₄×C₁₁	—	2	2	no	—
194	C₄×C₁₁	—	2	1	no	—
195	C₄×C₁₁	—	8	2	no	—
196	C₄×C₁₁	—	8	2	no	—
197	C₄×C₁₁	#532	1	1	yes	characteristic
198	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
199	Dic₁₁	—	8	2	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	C₂²×D₁₁	1	2	yes	—
202	C₂×D₈	—	11	2	no	—
203	C₂×D₈	—	22	1	no	—
204	C₂×M₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
205	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	1	no	—
207	#127	—	22	1	no	—
208	#126	—	4	2	no	—
209	#125	—	2	1	no	—
210	#125	—	4	2	no	—
211	#125	—	22	1	no	—
212	#123	—	44	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	#121	—	22	2	no	—
215	#121	—	44	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×D₄	—	22	2	no	—
218	#117	C₂²⋊D₁₁	1	1	yes	characteristic
219	SD₃₂	—	22	4	no	—
220	SD₃₂	—	22	4	no	—
221	M₃₂	—	22	2	no	—
222	C₂×C₁₆	—	11	1	no	—
223	C₂×C₁₆	—	11	1	no	—
224	C₂×C₁₆	—	11	1	no	—
225	C₂×C₁₆	—	11	1	no	—
226	C₂×C₁₆	—	22	2	no	—
227	#107	—	11	2	no	—
228	#107	C₂²×D₁₁	1	2	yes	—
229	#106	—	2	1	no	—
230	#106	—	22	1	no	—
231	#103	—	2	2	no	—
232	#103	—	2	2	no	—
233	#102	—	22	2	no	—
234	#102	—	22	2	no	—
235	#97	C₂²⋊D₁₁	1	1	yes	characteristic
236	#97	C₂²⋊D₁₁	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
238	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
239	C₂×C₁₁	—	4	1	no	—
240	C₂×C₁₁	#3169	1	1	yes	characteristic
241	C₂×C₁₁	#3416	1	2	yes	—
242	D₁₁	—	16	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×D₄	—	22	2	no	—
245	C₂×D₄	—	44	2	no	—
246	C₂×D₄	—	44	2	no	—
247	C₂²×C₄	—	44	2	no	—
248	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	4	2	no	—
252	D₈	—	22	2	no	—
253	D₈	—	22	2	no	—
254	D₈	—	44	2	no	—
255	M₁₆	—	2	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
260	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
261	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	22	2	no	—
268	C₄⋊C₄	—	44	2	no	—
269	C₄⋊C₄	#4323	1	2	yes	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	44	2	no	—
272	C₂²⋊C₄	—	44	2	no	—
273	C₂²⋊C₄	#4323	1	2	yes	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
276	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
277	C₁₆	—	11	2	no	—
278	C₁₆	—	11	2	no	—
279	C₁₆	—	22	1	no	—
280	C₁₆	—	22	1	no	—
281	C₁₁	#33743	1	1	yes	characteristic, complemented, Sylow
282	C₂³	—	44	2	no	—
283	C₂³	C₂×#4323	1	1	yes	characteristic
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	44	2	no	—
287	D₄	—	44	2	no	—
288	D₄	—	176	2	no	—
289	D₄	—	176	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	44	2	no	—
295	C₂×C₄	—	176	2	no	—
296	C₂×C₄	C₂×#4323	1	2	yes	—
297	C₂×C₄	C₂×#4323	1	2	yes	—
298	C₂×C₄	C₂×#4323	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	#67003	1	2	yes	—
303	C₈	#41×D₁₁	1	1	yes	characteristic
304	C₈	#41×D₁₁	1	1	yes	characteristic
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	88	2	no	—
308	C₂²	—	88	2	no	—
309	C₂²	—	88	2	no	—
310	C₂²	C₂²×#4323	1	1	yes	characteristic
311	C₄	—	2	1	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	88	2	no	—
317	C₄	C₂×#67003	1	1	yes	characteristic
318	C₄	#10623363	1	1	yes	characteristic
319	C₂	—	4	1	no	—
320	C₂	—	176	2	no	—
321	C₂	#13079191	1	1	yes	characteristic
322	C₂	#13081760	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#4323
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#3177	C₂
socle	C₂²×C₁₁	C₂²×D₈
prime radical	#13079266	C₂
hypercenter	#3177	D₁₁
hypocenter	C₁₁	#33743
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #13070892	C₂	semidirect
3. #13070873	C₂	semidirect
4. #13070861	C₂	semidirect
5. #13070859	C₂	semidirect
6. #13070731	C₂	semidirect
7. #13070714	C₂	semidirect
8. #13070478	C₂	semidirect
9. C₄×#66892	C₂	semidirect
10. #13070213	C₂	semidirect
11. C₁₁×#3177	C₂	semidirect
121. #3177	D₁₁	semidirect
281. C₁₁	#33743	semidirect

Maximal subgroups

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

subgroup	index
#13079266	2
#13070892	2
#13070873	2
#13070861	2
#13070859	2
#13070731	2
#13070714	2
#13070478	2
C₄×#66892	2
#13070213	2
C₁₁×#3177	2
#33743	11