Subgroups — #14907282

Order 2816.

498 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070525	C₂	1	2	yes	maximal, complemented
2	#13070525	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
4	#13070375	C₂	1	2	yes	maximal
5	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
6	#13070213	C₂	1	2	yes	maximal
7	C₄×#66876	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#2960	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#533	C₄	1	1	yes	characteristic
10	C₁₁×#446	C₄	1	2	yes	—
11	C₁₁×#446	C₄	1	2	yes	—
12	C₈×Q₈×C₁₁	C₂²	1	2	yes	—
13	C₈×Q₈×C₁₁	C₂²	1	2	yes	—
14	C₁₁×#424	C₄	1	2	yes	—
15	C₁₁×#423	C₄	1	1	yes	characteristic
16	C₁₁×#405	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	C₂×#66876	C₂²	1	1	yes	characteristic
19	C₂×#66876	C₂²	1	1	yes	characteristic
20	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
21	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
23	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
25	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
26	#10622099	C₂²	1	2	yes	—
27	#10622099	C₂²	1	4	yes	—
28	C₄×C₁₁⋊C₁₆	C₂²	1	2	yes	—
29	C₄×C₁₁⋊C₁₆	C₂²	1	4	yes	—
30	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
33	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
34	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
35	C₁₁×#125	C₈	1	2	yes	—
36	C₁₁×C₂²⋊Q₈	C₈	1	2	yes	—
37	C₁₁×C₂²⋊Q₈	C₈	1	2	yes	—
38	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
40	C₁₁×#117	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
43	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
44	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
45	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
46	C₁₁×#97	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
50	C₄×C₈×C₁₁	C₂³	1	4	yes	—
51	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
52	#66876	—	2	2	no	—
53	#66876	Q₈	1	4	yes	complemented
54	C₂×C₁₁⋊C₁₆	—	2	4	no	—
55	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
56	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
58	C₂×C₁₁⋊C₁₆	C₂³	1	2	yes	—
59	#33476	—	11	1	no	maximal, Sylow
60	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
61	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
65	C₁₁×M₁₆	—	2	4	no	—
66	C₂×C₈×C₁₁	—	2	2	no	—
67	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
70	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
72	C₂×C₈×C₁₁	#41	1	2	yes	—
73	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
74	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
75	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
76	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
77	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
78	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
79	C₁₁×C₄⋊C₄	C₂×C₈	1	2	yes	—
80	C₁₁×C₂²⋊C₄	C₂×C₈	1	4	yes	—
81	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₁	C₂²×C₄	1	4	yes	—
86	C₁₁⋊C₁₆	—	2	2	no	—
87	C₁₁⋊C₁₆	—	2	2	no	—
88	C₁₁⋊C₁₆	—	2	2	no	—
89	C₁₁⋊C₁₆	C₂×Q₈	1	4	yes	—
90	#2960	D₁₁	1	1	yes	characteristic
91	#2183	—	11	2	no	—
92	#2183	—	11	4	no	—
93	Q₈×C₁₆	—	11	2	no	—
94	#2151	—	11	2	no	—
95	C₂×C₄⋊C₁₆	—	11	1	no	—
96	#2108	—	11	2	no	—
97	C₄×M₃₂	—	11	1	no	—
98	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
99	Q₈×C₁₁	—	2	4	no	—
100	Q₈×C₁₁	—	4	2	no	—
101	C₂×C₄×C₁₁	—	2	2	no	—
102	C₂×C₄×C₁₁	—	4	1	no	—
103	C₂×C₄×C₁₁	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
108	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
109	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₁	#131	1	2	yes	—
112	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₁₁	—	2	1	no	—
114	C₈×C₁₁	—	2	1	no	—
115	C₈×C₁₁	—	2	4	no	—
116	C₈×C₁₁	C₄×Q₈	1	2	yes	—
117	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
118	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
119	#533	Dic₁₁	1	1	yes	characteristic
120	C₂×M₃₂	—	11	1	no	—
121	C₂×M₃₂	—	11	1	no	—
122	C₂²×C₁₆	—	11	2	no	—
123	#446	Dic₁₁	1	2	yes	—
124	#446	Dic₁₁	1	2	yes	—
125	C₈×Q₈	C₂×D₁₁	1	2	yes	—
126	C₈×Q₈	C₂×D₁₁	1	2	yes	—
127	#424	Dic₁₁	1	2	yes	—
128	#423	Dic₁₁	1	1	yes	characteristic
129	#405	C₂×D₁₁	1	2	yes	—
130	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
131	C₄⋊C₁₆	—	11	1	no	—
132	C₄⋊C₁₆	—	11	1	no	—
133	C₄⋊C₁₆	—	11	2	no	—
134	C₄⋊C₁₆	—	11	4	no	—
135	C₄⋊C₁₆	—	11	4	no	—
136	#346	—	11	2	no	—
137	#346	—	11	4	no	—
138	C₄×C₁₆	—	11	2	no	—
139	C₄×C₁₆	—	11	4	no	—
140	C₂²×C₁₁	—	2	1	no	—
141	C₂²×C₁₁	—	2	1	no	—
142	C₂²×C₁₁	C₈×Q₈	1	2	yes	—
143	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₁₁	—	2	1	no	—
145	C₄×C₁₁	—	2	4	no	—
146	C₄×C₁₁	—	2	1	no	—
147	C₄×C₁₁	—	2	1	no	—
148	C₄×C₁₁	—	2	4	no	—
149	C₄×C₁₁	C₈×Q₈	1	2	yes	—
150	C₄×C₁₁	#504	1	2	yes	—
151	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
153	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
154	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
155	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
156	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
157	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
158	#125	C₁₁⋊C₈	1	2	yes	—
159	C₂²⋊Q₈	C₁₁⋊C₈	1	2	yes	—
160	C₂²⋊Q₈	C₁₁⋊C₈	1	2	yes	—
161	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
162	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
163	#117	C₂×Dic₁₁	1	2	yes	—
164	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
165	M₃₂	—	11	4	no	—
166	M₃₂	—	22	2	no	—
167	C₂×C₁₆	—	11	2	no	—
168	C₂×C₁₆	—	11	2	no	—
169	C₂×C₁₆	—	11	2	no	—
170	C₂×C₁₆	—	11	2	no	—
171	C₂×C₁₆	—	22	4	no	—
172	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
173	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
174	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
175	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
176	#97	C₂×Dic₁₁	1	4	yes	—
177	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
178	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
179	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
180	C₄×C₈	C₂²×D₁₁	1	4	yes	—
181	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
182	C₂×C₁₁	—	2	2	no	—
183	C₂×C₁₁	C₂×C₈×Q₈	1	1	yes	characteristic
184	C₂×C₁₁	C₂×#504	1	1	yes	characteristic
185	C₂×C₁₁	#3407	1	1	yes	characteristic
186	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
187	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
188	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
189	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
190	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	Q₈×D₁₁	1	2	yes	—
194	C₂×C₈	#4343	1	2	yes	—
195	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
196	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
197	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
198	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
199	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
200	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
201	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
202	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
203	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
204	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
205	C₄⋊C₄	C₂×C₁₁⋊C₈	1	2	yes	—
206	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
207	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
208	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
209	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
210	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
211	C₄²	C₂²×Dic₁₁	1	4	yes	—
212	C₁₆	—	11	4	no	—
213	C₁₆	—	22	2	no	—
214	C₁₆	—	22	2	no	—
215	C₁₆	—	22	2	no	—
216	C₁₁	#33476	1	1	yes	characteristic, complemented, Sylow
217	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
223	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
224	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
225	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
226	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
227	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
228	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
229	C₂×C₄	Q₈×Dic₁₁	1	2	yes	—
230	C₂×C₄	#67001	1	2	yes	—
231	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	4	no	—
235	C₈	Q₈×Dic₁₁	1	2	yes	—
236	C₈	C₂×Q₈×D₁₁	1	1	yes	characteristic
237	C₈	C₂×#4343	1	1	yes	characteristic
238	C₂²	—	2	1	no	—
239	C₂²	—	2	1	no	—
240	C₂²	Q₈×C₁₁⋊C₈	1	2	yes	—
241	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	4	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	4	no	—
247	C₄	Q₈×C₁₁⋊C₈	1	2	yes	—
248	C₄	#10622753	1	2	yes	—
249	C₄	C₂×Q₈×Dic₁₁	1	1	yes	characteristic
250	C₄	C₂×#67001	1	1	yes	characteristic
251	C₂	—	2	2	no	—
252	C₂	C₂×Q₈×C₁₁⋊C₈	1	1	yes	characteristic
253	C₂	C₂×#10622753	1	1	yes	characteristic
254	C₂	#13081719	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2960	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2960	D₁₁
hypocenter	C₁₁	#33476
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #13070525	C₂	semidirect
2. #13070525	C₂	semidirect
3. Q₈×C₁₁⋊C₁₆	C₂	semidirect
53. #66876	Q₈	semidirect
216. C₁₁	#33476	semidirect

Maximal subgroups

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

subgroup	index
#13070525	2
Q₈×C₁₁⋊C₁₆	2
#13070375	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
#13070213	2
C₄×#66876	2
C₁₁×#2960	2
#33476	11