Subgroups — #14907200

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	#13077533	C₂	1	2	yes	maximal, complemented
2	#13070502	C₂	1	4	yes	maximal, complemented
3	#13070497	C₂	1	4	yes	maximal, complemented
4	C₂×#10622144	C₂	1	1	yes	maximal, characteristic, complemented
5	#13070344	C₂	1	2	yes	maximal, complemented
6	#13070212	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₁×#508	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#135	C₂²	1	2	yes	—
9	#10623350	—	2	2	no	—
10	C₁₁×#508	C₄	1	2	yes	complemented
11	C₁₁×#508	C₄	1	2	yes	complemented
12	C₁₁×#508	C₂²	1	2	yes	—
13	C₁₁×#508	C₂²	1	2	yes	—
14	C₂×C₁₁×Q₃₂	C₄	1	1	yes	characteristic, complemented
15	C₂×C₁₁×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₁₁×D₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
18	#10622849	—	2	2	no	—
19	#10622810	—	2	4	no	—
20	#10622809	C₂²	1	8	yes	complemented
21	C₂×#66883	C₂²	1	2	yes	—
22	#10622714	C₂²	1	1	yes	characteristic
23	C₂×#66876	C₂²	1	1	yes	characteristic
24	#10622205	C₂²	1	8	yes	complemented
25	#10622144	C₂²	1	2	yes	complemented
26	#10622144	C₂²	1	2	yes	complemented
27	#10622140	C₂²	1	2	yes	complemented
28	C₁₆×Dic₁₁	C₂²	1	2	yes	complemented
29	C₂×C₁₁×#41	—	2	2	no	—
30	C₁₁×#135	—	2	2	no	—
31	C₁₁×#135	—	2	2	no	—
32	C₁₁×#135	C₂×C₄	1	4	yes	—
33	C₁₁×#135	C₂³	1	4	yes	—
34	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×C₁₁×SD₁₆	—	2	2	no	—
36	C₂×C₁₁×D₈	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
38	#67001	—	2	8	no	—
39	#67001	—	4	4	no	—
40	#66979	D₄	1	1	yes	characteristic
41	C₂×#4318	—	2	2	no	—
42	C₂×#4318	—	4	4	no	—
43	C₂×#4318	D₄	1	1	yes	characteristic
44	C₁₁×Q₃₂	C₂×C₄	1	2	yes	—
45	C₁₁×Q₃₂	C₂×C₄	1	2	yes	—
46	C₁₁×SD₃₂	C₂×C₄	1	4	yes	—
47	C₁₁×SD₃₂	C₂×C₄	1	4	yes	—
48	C₁₁×D₁₆	C₂×C₄	1	2	yes	—
49	C₁₁×D₁₆	C₂×C₄	1	2	yes	—
50	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
55	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
56	#66901	—	2	8	no	—
57	#66883	—	2	2	no	—
58	#66883	C₂³	1	4	yes	—
59	#66879	D₄	1	2	yes	—
60	C₈×Dic₁₁	C₂³	1	2	yes	—
61	#66876	—	2	1	no	—
62	#66876	—	2	1	no	—
63	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
64	#33465	—	11	1	no	maximal, Sylow
65	C₁₁×#41	—	2	4	no	—
66	C₁₁×#41	—	2	4	no	—
67	C₁₁×#41	—	4	2	no	—
68	C₁₁×#41	—	4	2	no	—
69	C₂×Q₈×C₁₁	—	2	2	no	—
70	C₂×D₄×C₁₁	—	2	2	no	—
71	C₂×D₄×C₁₁	—	4	2	no	—
72	C₂²×C₄×C₁₁	—	4	2	no	—
73	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₁₁×Q₁₆	—	2	2	no	—
75	C₁₁×Q₁₆	C₂²×C₄	1	4	yes	—
76	C₁₁×SD₁₆	—	2	4	no	—
77	C₁₁×SD₁₆	—	2	4	no	—
78	C₁₁×D₈	—	2	2	no	—
79	C₁₁×D₈	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
86	C₂²⋊Dic₁₁	—	2	1	no	—
87	C₄⋊Dic₁₁	—	2	1	no	—
88	C₄×Dic₁₁	C₂×D₄	1	2	yes	—
89	#4318	—	2	4	no	—
90	#4318	—	4	8	no	—
91	#4318	—	4	2	no	—
92	#4318	D₈	1	2	yes	—
93	#4318	D₈	1	2	yes	—
94	C₂×C₁₁⋊C₈	—	4	8	no	—
95	C₂×C₁₁⋊C₈	—	4	2	no	—
96	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
97	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
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	—	2	2	no	—
102	C₂×#508	D₁₁	1	1	yes	characteristic
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	—	11	1	no	—
109	C₂³×C₁₁	—	4	2	no	—
110	C₂³×C₁₁	#135	1	1	yes	characteristic
111	Q₈×C₁₁	—	2	4	no	—
112	Q₈×C₁₁	—	4	2	no	—
113	D₄×C₁₁	—	2	4	no	—
114	D₄×C₁₁	—	4	4	no	—
115	D₄×C₁₁	—	4	4	no	—
116	D₄×C₁₁	—	4	2	no	—
117	C₂×C₄×C₁₁	—	4	2	no	—
118	C₂×C₄×C₁₁	—	4	4	no	—
119	C₂×C₄×C₁₁	—	4	4	no	—
120	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₁₁	—	2	2	no	—
127	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
128	C₈×C₁₁	C₄×D₄	1	1	yes	characteristic
129	C₈×C₁₁	C₂³×C₄	1	1	yes	characteristic
130	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
131	C₁₁⋊C₈	—	4	4	no	—
132	C₁₁⋊C₈	C₂×D₈	1	4	yes	—
133	C₂×#135	C₂×D₁₁	1	2	yes	—
134	#568	—	22	2	no	—
135	#508	Dic₁₁	1	2	yes	complemented
136	#508	Dic₁₁	1	2	yes	complemented
137	#508	C₂×D₁₁	1	2	yes	—
138	#508	C₂×D₁₁	1	2	yes	—
139	C₂×Q₃₂	Dic₁₁	1	1	yes	characteristic, complemented
140	C₂×SD₃₂	Dic₁₁	1	2	yes	complemented
141	C₂×D₁₆	Dic₁₁	1	1	yes	characteristic, complemented
142	C₂×M₃₂	—	11	1	no	—
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	—	11	1	no	—
149	#368	—	11	2	no	—
150	#368	—	11	2	no	—
151	#359	—	11	8	no	—
152	#346	—	11	2	no	—
153	C₄×C₁₆	—	11	2	no	—
154	C₂²×C₁₁	—	4	1	no	—
155	C₂²×C₁₁	—	4	4	no	—
156	C₂²×C₁₁	—	16	2	no	—
157	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
158	C₂²×C₁₁	C₄×D₈	1	1	yes	characteristic
159	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
160	C₄×C₁₁	—	4	4	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	2	no	—
166	C₂×#41	—	2	2	no	—
167	#135	—	2	2	no	—
168	#135	—	2	2	no	—
169	#135	C₂×Dic₁₁	1	4	yes	—
170	#135	C₂²×D₁₁	1	4	yes	—
171	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
172	C₂×SD₁₆	—	2	2	no	—
173	C₂×D₈	C₂×Dic₁₁	1	2	yes	—
174	#131	—	22	8	no	—
175	#131	—	44	4	no	—
176	C₂×M₁₆	—	11	1	no	—
177	C₂×M₁₆	—	22	2	no	—
178	C₂×M₁₆	—	44	4	no	—
179	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
180	#117	—	11	1	no	—
181	Q₃₂	C₂×Dic₁₁	1	2	yes	—
182	Q₃₂	C₂×Dic₁₁	1	2	yes	—
183	SD₃₂	C₂×Dic₁₁	1	4	yes	—
184	SD₃₂	C₂×Dic₁₁	1	4	yes	—
185	D₁₆	C₂×Dic₁₁	1	2	yes	—
186	D₁₆	C₂×Dic₁₁	1	2	yes	—
187	M₃₂	—	22	1	no	—
188	M₃₂	—	22	1	no	—
189	C₂×C₁₆	—	11	2	no	—
190	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
191	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
192	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
193	C₂×C₁₆	C₂×Dic₁₁	1	1	yes	characteristic
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	—	11	2	no	—
200	C₄×C₈	—	11	2	no	—
201	C₂×C₁₁	—	2	1	no	—
202	C₂×C₁₁	—	2	1	no	—
203	C₂×C₁₁	—	2	1	no	—
204	C₂×C₁₁	—	8	4	no	—
205	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
206	#41	—	2	4	no	—
207	#41	—	2	4	no	—
208	#41	—	4	2	no	—
209	#41	—	4	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×D₄	—	2	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
215	Q₁₆	—	2	2	no	—
216	Q₁₆	C₂²×Dic₁₁	1	4	yes	—
217	SD₁₆	—	2	4	no	—
218	SD₁₆	—	2	4	no	—
219	D₈	—	2	2	no	—
220	D₈	C₂²×Dic₁₁	1	4	yes	—
221	M₁₆	—	11	2	no	—
222	M₁₆	—	11	2	no	—
223	M₁₆	—	22	4	no	—
224	M₁₆	—	44	8	no	—
225	M₁₆	—	44	2	no	—
226	C₂×C₈	—	11	2	no	—
227	C₂×C₈	—	44	8	no	—
228	C₂×C₈	—	44	2	no	—
229	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
230	C₂×C₈	#4341	1	1	yes	characteristic
231	C₂×C₈	#4341	1	1	yes	characteristic
232	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
234	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
235	C₄⋊C₄	—	22	1	no	—
236	C₂²⋊C₄	—	22	1	no	—
237	C₄²	—	11	2	no	—
238	C₁₆	—	22	2	no	—
239	C₁₆	C₂²×Dic₁₁	1	1	yes	characteristic
240	C₁₆	C₂²×Dic₁₁	1	1	yes	characteristic
241	C₁₆	C₂²×Dic₁₁	1	1	yes	characteristic
242	C₁₆	C₂²×Dic₁₁	1	1	yes	characteristic
243	C₁₁	#33465	1	1	yes	characteristic, complemented, Sylow
244	C₂³	—	4	2	no	—
245	C₂³	#66965	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	22	2	no	—
256	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
257	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
258	C₂×C₄	#66965	1	1	yes	characteristic
259	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
260	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
262	C₈	—	11	4	no	—
263	C₈	—	44	4	no	—
264	C₈	D₄×Dic₁₁	1	1	yes	characteristic
265	C₈	D₄×Dic₁₁	1	1	yes	characteristic
266	C₈	C₂×#4341	1	1	yes	characteristic
267	C₈	C₂³×Dic₁₁	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	D₈×Dic₁₁	1	1	yes	characteristic
272	C₂²	D₈×Dic₁₁	1	1	yes	characteristic
273	C₂²	C₂×#66965	1	1	yes	characteristic
274	C₄	—	4	4	no	—
275	C₄	—	44	2	no	—
276	C₄	D₈×Dic₁₁	1	1	yes	characteristic
277	C₄	D₈×Dic₁₁	1	1	yes	characteristic
278	C₄	C₂×D₈×D₁₁	1	1	yes	characteristic
279	C₄	C₂×D₄×Dic₁₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×D₈×Dic₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₁₁
commutator	C₈×C₁₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₁
Fitting, prime radical	C₂×C₁₁×#508	C₂
socle	C₂×C₁₁	C₂×C₄×D₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	C₂×#508	D₁₁
hypocenter	C₁₁	#33465
LCS	G ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. #13077533	C₂	semidirect
2. #13070502	C₂	semidirect
3. #13070497	C₂	semidirect
4. C₂×#10622144	C₂	semidirect
5. #13070344	C₂	semidirect
6. #13070212	C₂	semidirect
10. C₁₁×#508	C₄	semidirect
11. C₁₁×#508	C₄	semidirect
14. C₂×C₁₁×Q₃₂	C₄	semidirect
15. C₂×C₁₁×SD₃₂	C₄	semidirect
16. C₂×C₁₁×D₁₆	C₄	semidirect
20. #10622809	C₂²	semidirect
24. #10622205	C₂²	semidirect
25. #10622144	C₂²	semidirect
26. #10622144	C₂²	semidirect
27. #10622140	C₂²	semidirect
28. C₁₆×Dic₁₁	C₂²	semidirect
135. #508	Dic₁₁	semidirect
136. #508	Dic₁₁	semidirect
139. C₂×Q₃₂	Dic₁₁	semidirect
140. C₂×SD₃₂	Dic₁₁	semidirect
141. C₂×D₁₆	Dic₁₁	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
#13077533	2
#13070502	2
#13070497	2
C₂×#10622144	2
#13070344	2
#13070212	2
C₂×C₁₁×#508	2
#33465	11