Subgroups — #14907201

Order 2816.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622517	C₂	1	1	yes	maximal, characteristic
2	#13077534	C₂	1	2	yes	maximal, complemented
3	#13070503	C₂	1	4	yes	maximal, complemented
4	#13070498	C₂	1	4	yes	maximal, complemented
5	C₂×#10622144	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070345	C₂	1	2	yes	maximal, complemented
7	C₁₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	#10623276	—	2	2	no	—
9	C₂×#66957	C₂²	1	2	yes	—
10	C₂×C₁₁×M₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
12	C₁₁×#405	C₂²	1	1	yes	characteristic
13	C₂×#66883	C₂²	1	2	yes	—
14	#10622618	—	2	2	no	—
15	C₂×Dic₈₈	C₄	1	1	yes	characteristic, complemented
16	#10622517	C₄	1	2	yes	complemented
17	#10622517	C₄	1	2	yes	complemented
18	#10622517	C₂²	1	2	yes	—
19	#10622517	C₂²	1	2	yes	—
20	C₂×#66864	C₄	1	2	yes	complemented
21	C₂×D₁₇₆	C₄	1	1	yes	characteristic, complemented
22	#10622330	—	2	4	no	—
23	#10622313	C₂²	1	8	yes	complemented
24	C₁₁×#346	C₂²	1	2	yes	complemented
25	C₄×C₁₁×C₁₆	C₂²	1	2	yes	complemented
26	#10622148	C₂²	1	8	yes	complemented
27	#10622144	C₂²	1	2	yes	complemented
28	#10622144	C₂²	1	2	yes	complemented
29	C₂×#4339	—	2	2	no	—
30	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₁₁×#117	D₄	1	1	yes	characteristic
33	C₂×#4318	—	2	2	no	—
34	#66960	—	4	4	no	—
35	M₁₆×D₁₁	—	4	4	no	—
36	C₂×Dic₄₄	C₂×C₄	1	2	yes	—
37	#66957	—	2	2	no	—
38	#66957	—	2	2	no	—
39	#66957	C₂×C₄	1	4	yes	—
40	#66957	C₂³	1	4	yes	—
41	C₂×D₈₈	C₂×C₄	1	2	yes	—
42	C₂×#4314	—	2	2	no	—
43	#66954	—	2	8	no	—
44	C₁₁×M₃₂	—	2	1	no	—
45	C₁₁×M₃₂	—	2	1	no	—
46	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₁×C₁₆	C₂³	1	2	yes	—
51	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
52	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
53	C₁₁×#97	D₄	1	2	yes	—
54	C₄×C₈×C₁₁	C₂³	1	2	yes	—
55	#66883	—	2	2	no	—
56	#66883	C₂³	1	4	yes	—
57	#66869	—	2	8	no	—
58	Dic₈₈	C₂×C₄	1	2	yes	—
59	Dic₈₈	C₂×C₄	1	2	yes	—
60	#66864	C₂×C₄	1	4	yes	—
61	#66864	C₂×C₄	1	4	yes	—
62	D₁₇₆	C₂×C₄	1	2	yes	—
63	D₁₇₆	C₂×C₄	1	2	yes	—
64	#33465	—	11	1	no	maximal, Sylow
65	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₁	—	4	2	no	—
67	#4339	—	2	4	no	—
68	#4339	—	2	4	no	—
69	#4339	—	4	2	no	—
70	#4339	—	4	2	no	—
71	C₂×D₄₄	—	2	2	no	—
72	C₂×C₄×D₁₁	—	4	2	no	—
73	C₂×Dic₂₂	—	2	2	no	—
74	C₁₁×M₁₆	D₈	1	2	yes	—
75	C₁₁×M₁₆	D₈	1	2	yes	—
76	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
83	C₁₁×C₄⋊C₄	—	2	1	no	—
84	C₁₁×C₂²⋊C₄	—	2	1	no	—
85	C₄²×C₁₁	C₂×D₄	1	2	yes	—
86	#4318	—	2	4	no	—
87	#4318	—	4	2	no	—
88	C₂×C₁₁⋊C₈	—	4	2	no	—
89	Dic₄₄	—	2	2	no	—
90	Dic₄₄	C₂²×C₄	1	4	yes	—
91	D₈₈	—	2	2	no	—
92	D₈₈	C₂²×C₄	1	4	yes	—
93	#4314	—	2	4	no	—
94	#4314	—	2	4	no	—
95	#4313	—	4	8	no	—
96	C₈×D₁₁	—	4	8	no	—
97	C₁₁×C₁₆	—	2	2	no	—
98	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
99	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
100	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
101	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
102	C₂×#508	—	11	1	no	—
103	#2957	—	11	2	no	—
104	#2179	—	11	4	no	—
105	#2178	—	11	4	no	—
106	C₂×#368	—	11	1	no	—
107	#2143	—	11	2	no	—
108	#2108	D₁₁	1	1	yes	characteristic, complemented
109	C₂³×C₁₁	#135	1	1	yes	characteristic
110	C₂²×D₁₁	—	4	2	no	—
111	C₂×C₄×C₁₁	—	2	2	no	—
112	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
116	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
118	C₂²⋊D₁₁	—	4	4	no	—
119	C₂²⋊D₁₁	—	4	4	no	—
120	C₂×Dic₁₁	—	4	2	no	—
121	D₄₄	—	2	4	no	—
122	D₄₄	—	4	2	no	—
123	C₄×D₁₁	—	4	4	no	—
124	C₄×D₁₁	—	4	4	no	—
125	Dic₂₂	—	2	4	no	—
126	Dic₂₂	—	4	2	no	—
127	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
128	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
129	C₈×C₁₁	C₂×D₈	1	4	yes	—
130	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
131	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
132	C₁₁⋊C₈	—	4	4	no	—
133	C₂×#135	—	11	2	no	—
134	#568	—	22	2	no	—
135	#508	—	11	2	no	—
136	#508	—	11	2	no	—
137	#508	—	11	2	no	—
138	#508	—	11	2	no	—
139	C₂×Q₃₂	—	11	1	no	—
140	C₂×SD₃₂	—	11	2	no	—
141	C₂×D₁₆	—	11	1	no	—
142	C₂×M₃₂	C₂×D₁₁	1	1	yes	characteristic
143	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
144	#444	—	22	4	no	—
145	#443	—	11	8	no	—
146	C₂×#108	—	11	2	no	—
147	#421	—	22	2	no	—
148	#405	C₂×D₁₁	1	1	yes	characteristic
149	#368	—	11	2	no	—
150	#368	—	11	2	no	—
151	#359	—	11	8	no	—
152	#346	C₂×D₁₁	1	2	yes	complemented
153	C₄×C₁₆	C₂×D₁₁	1	2	yes	complemented
154	C₂²×C₁₁	—	4	1	no	—
155	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
156	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
157	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
158	C₂×D₁₁	—	4	4	no	—
159	C₂×D₁₁	—	16	2	no	—
160	C₄×C₁₁	—	4	2	no	—
161	C₄×C₁₁	C₄×D₈	1	1	yes	characteristic
162	C₄×C₁₁	C₄×D₈	1	1	yes	characteristic
163	C₄×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
164	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
165	Dic₁₁	—	4	4	no	—
166	C₂×#41	—	22	2	no	—
167	#135	—	11	4	no	—
168	#135	—	11	4	no	—
169	#135	—	22	2	no	—
170	#135	—	22	2	no	—
171	C₂×Q₁₆	—	11	2	no	—
172	C₂×SD₁₆	—	22	2	no	—
173	C₂×D₈	—	11	2	no	—
174	#131	—	22	8	no	—
175	#131	—	44	4	no	—
176	C₂×M₁₆	—	22	2	no	—
177	C₂×M₁₆	—	44	4	no	—
178	C₂×M₁₆	D₄₄	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
180	#117	D₄₄	1	1	yes	characteristic
181	Q₃₂	—	11	2	no	—
182	Q₃₂	—	11	2	no	—
183	SD₃₂	—	11	4	no	—
184	SD₃₂	—	11	4	no	—
185	D₁₆	—	11	2	no	—
186	D₁₆	—	11	2	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
190	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
191	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
192	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
193	C₂×C₁₆	C₂²×D₁₁	1	2	yes	—
194	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
195	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
196	#108	—	11	4	no	—
197	#108	—	22	2	no	—
198	#104	—	22	8	no	—
199	#97	D₄₄	1	2	yes	—
200	C₄×C₈	C₂²×D₁₁	1	2	yes	—
201	C₂×C₁₁	—	2	1	no	—
202	C₂×C₁₁	—	2	1	no	—
203	C₂×C₁₁	—	2	1	no	—
204	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
205	D₁₁	—	8	4	no	—
206	#41	—	22	4	no	—
207	#41	—	22	4	no	—
208	#41	—	44	2	no	—
209	#41	—	44	2	no	—
210	C₂×Q₈	—	22	2	no	—
211	C₂×D₄	—	22	2	no	—
212	C₂×D₄	—	44	2	no	—
213	C₂²×C₄	—	44	2	no	—
214	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
215	Q₁₆	—	11	4	no	—
216	Q₁₆	—	22	2	no	—
217	SD₁₆	—	22	4	no	—
218	SD₁₆	—	22	4	no	—
219	D₈	—	11	4	no	—
220	D₈	—	22	2	no	—
221	M₁₆	—	22	4	no	—
222	M₁₆	—	44	2	no	—
223	M₁₆	—	44	8	no	—
224	M₁₆	D₈₈	1	2	yes	—
225	M₁₆	D₈₈	1	2	yes	—
226	C₂×C₈	—	44	2	no	—
227	C₂×C₈	—	44	8	no	—
228	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄₄	1	2	yes	—
232	C₂×C₈	#4339	1	1	yes	characteristic
233	C₂×C₈	#4339	1	1	yes	characteristic
234	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₄²	C₂×D₄₄	1	2	yes	—
238	C₁₆	—	2	2	no	—
239	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
240	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
241	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
242	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
243	C₁₁	#33465	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	44	2	no	—
245	C₂³	#66957	1	1	yes	characteristic
246	Q₈	—	22	4	no	—
247	Q₈	—	44	2	no	—
248	D₄	—	22	4	no	—
249	D₄	—	44	4	no	—
250	D₄	—	44	4	no	—
251	D₄	—	44	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	44	2	no	—
254	C₂×C₄	—	44	4	no	—
255	C₂×C₄	—	44	4	no	—
256	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
258	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
260	C₂×C₄	#66957	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
262	C₈	—	44	4	no	—
263	C₈	C₄×D₄₄	1	1	yes	characteristic
264	C₈	C₄×D₄₄	1	1	yes	characteristic
265	C₈	C₂×D₈₈	1	4	yes	—
266	C₈	C₂²×C₄×D₁₁	1	1	yes	characteristic
267	C₈	C₂×#4339	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	44	4	no	—
270	C₂²	—	176	2	no	—
271	C₂²	C₄×D₈₈	1	1	yes	characteristic
272	C₂²	C₄×D₈₈	1	1	yes	characteristic
273	C₂²	C₂×#66957	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	44	4	no	—
276	C₄	C₄×D₈₈	1	1	yes	characteristic
277	C₄	C₄×D₈₈	1	1	yes	characteristic
278	C₄	C₂×C₄×D₄₄	1	1	yes	characteristic
279	C₄	C₂²×D₈₈	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	88	4	no	—
284	C₂	C₂×C₄×D₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₈₈
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₁₁×#2108	C₂
socle	C₂×C₁₁	C₂×C₄×D₈
prime radical	C₂×#10622517	C₂
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2108	D₁₁
hypocenter	C₁₁	#33465
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #405 ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #13077534	C₂	semidirect
3. #13070503	C₂	semidirect
4. #13070498	C₂	semidirect
5. C₂×#10622144	C₂	semidirect
6. #13070345	C₂	semidirect
7. C₁₁×#2108	C₂	semidirect
15. C₂×Dic₈₈	C₄	semidirect
16. #10622517	C₄	semidirect
17. #10622517	C₄	semidirect
20. C₂×#66864	C₄	semidirect
21. C₂×D₁₇₆	C₄	semidirect
23. #10622313	C₂²	semidirect
24. C₁₁×#346	C₂²	semidirect
25. C₄×C₁₁×C₁₆	C₂²	semidirect
26. #10622148	C₂²	semidirect
27. #10622144	C₂²	semidirect
28. #10622144	C₂²	semidirect
108. #2108	D₁₁	semidirect
152. #346	C₂×D₁₁	semidirect
153. C₄×C₁₆	C₂×D₁₁	semidirect
243. C₁₁	#33465	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622517	2
#13077534	2
#13070503	2
#13070498	2
C₂×#10622144	2
#13070345	2
C₁₁×#2108	2
#33465	11