Subgroups — #14670235

Order 2816.

462 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₁₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#13066694	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₁₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₁₁	C₂²	1	2	yes	—
9	C₄×C₁₁×C₂²⋊C₄	C₄	1	1	yes	characteristic
10	C₂²×C₁₁⋊C₁₆	—	2	2	no	—
11	C₂²×C₁₁⋊C₁₆	—	2	4	no	—
12	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁⋊C₁₆	C₂²	1	2	yes	—
15	C₁₁×#336	C₄	1	2	yes	—
16	C₂²⋊(C₁₁⋊C₁₆)	—	2	2	no	—
17	C₄⋊(C₁₁⋊C₁₆)	—	2	4	no	—
18	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₁	—	2	2	no	—
20	C₂²×C₈×C₁₁	—	2	2	no	—
21	C₂²×C₈×C₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₁	D₄	1	2	yes	—
24	C₂²×C₈×C₁₁	Q₈	1	2	yes	—
25	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₁₁×C₂²⋊C₄	C₈	1	1	yes	characteristic
27	C₂×C₁₁×C₂²⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₄²×C₁₁	C₂×C₄	1	2	yes	—
29	C₁₁×C₂²⋊C₈	—	2	2	no	—
30	C₄×C₈×C₁₁	—	2	2	no	—
31	C₄×C₈×C₁₁	—	2	2	no	—
32	C₁₁×#95	C₈	1	2	yes	—
33	C₂×C₁₁⋊C₁₆	—	2	8	no	—
34	C₂×C₁₁⋊C₁₆	—	2	2	no	—
35	C₂×C₁₁⋊C₁₆	—	2	2	no	—
36	C₂×C₁₁⋊C₁₆	—	2	2	no	—
37	C₂×C₁₁⋊C₁₆	—	2	1	no	—
38	C₂×C₁₁⋊C₁₆	—	2	1	no	—
39	C₂×C₁₁⋊C₁₆	—	2	2	no	—
40	C₂×C₁₁⋊C₁₆	D₄	1	4	yes	—
41	#12912	—	11	1	no	maximal, Sylow
42	C₂⁴×C₁₁	C₂×C₈	1	1	yes	characteristic
43	C₂²×C₄×C₁₁	—	2	2	no	—
44	C₂²×C₄×C₁₁	—	2	2	no	—
45	C₂²×C₄×C₁₁	C₄⋊C₄	1	2	yes	—
46	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₁	M₁₆	1	2	yes	—
50	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₁	—	2	2	no	—
52	C₂×C₈×C₁₁	—	2	4	no	—
53	C₂×C₈×C₁₁	—	2	2	no	—
54	C₂×C₈×C₁₁	—	2	2	no	—
55	C₂×C₈×C₁₁	—	2	2	no	—
56	C₂×C₈×C₁₁	—	2	2	no	—
57	C₂×C₈×C₁₁	—	2	2	no	—
58	C₂×C₈×C₁₁	C₄⋊C₄	1	2	yes	—
59	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₈×C₁₁	#41	1	2	yes	—
63	C₁₁×C₂²⋊C₄	—	2	2	no	—
64	C₁₁×C₂²⋊C₄	C₁₆	1	4	yes	complemented
65	C₄²×C₁₁	—	2	2	no	—
66	C₄²×C₁₁	—	2	2	no	—
67	C₁₁⋊C₁₆	—	2	4	no	—
68	C₁₁⋊C₁₆	—	4	1	no	—
69	C₁₁⋊C₁₆	—	4	1	no	—
70	C₂³×C₁₆	—	11	1	no	—
71	C₂×C₄⋊C₁₆	—	11	2	no	—
72	C₂×C₂²⋊C₁₆	—	11	1	no	—
73	C₈×C₂²⋊C₄	D₁₁	1	1	yes	characteristic
74	#1374	—	11	2	no	—
75	C₂³×C₁₁	—	2	2	no	—
76	C₂³×C₁₁	—	2	2	no	—
77	C₂³×C₁₁	C₄⋊C₈	1	2	yes	—
78	C₂³×C₁₁	C₂×C₁₆	1	2	yes	—
79	C₂³×C₁₁	M₃₂	1	2	yes	—
80	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₁	—	2	2	no	—
82	C₂×C₄×C₁₁	—	2	2	no	—
83	C₂×C₄×C₁₁	—	2	2	no	—
84	C₂×C₄×C₁₁	—	2	2	no	—
85	C₂×C₄×C₁₁	—	2	2	no	—
86	C₂×C₄×C₁₁	—	2	4	no	—
87	C₂×C₄×C₁₁	—	2	2	no	—
88	C₂×C₄×C₁₁	—	2	2	no	—
89	C₂×C₄×C₁₁	—	2	2	no	—
90	C₂×C₄×C₁₁	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₁	C₂×C₁₆	1	4	yes	complemented
92	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
94	C₂×C₄×C₁₁	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₁₁	#131	1	2	yes	—
96	C₈×C₁₁	—	2	2	no	—
97	C₈×C₁₁	—	4	2	no	—
98	C₈×C₁₁	#120	1	1	yes	characteristic
99	C₈×C₁₁	C₂²⋊Q₈	1	2	yes	—
100	C₈×C₁₁	#123	1	1	yes	characteristic
101	C₂³×C₈	C₂×D₁₁	1	1	yes	characteristic
102	C₂²×C₁₆	—	11	2	no	—
103	C₂²×C₁₆	—	11	1	no	—
104	C₂²×C₁₆	—	11	1	no	—
105	C₂²×C₁₆	—	22	4	no	—
106	C₂²×C₁₆	—	22	2	no	—
107	C₂×C₂²⋊C₈	Dic₁₁	1	1	yes	characteristic
108	C₂×C₄×C₈	C₂×D₁₁	1	2	yes	—
109	C₄×C₂²⋊C₄	Dic₁₁	1	1	yes	characteristic
110	C₄⋊C₁₆	—	22	4	no	—
111	C₂²⋊C₁₆	—	22	2	no	—
112	#336	Dic₁₁	1	2	yes	—
113	C₂²×C₁₁	—	2	2	no	—
114	C₂²×C₁₁	—	2	2	no	—
115	C₂²×C₁₁	—	2	4	no	—
116	C₂²×C₁₁	—	2	4	no	—
117	C₂²×C₁₁	C₄⋊C₁₆	1	4	yes	complemented
118	C₂²×C₁₁	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂²×C₁₁	C₈×D₄	1	2	yes	—
120	C₂²×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
121	C₂²×C₁₁	C₂×M₃₂	1	1	yes	characteristic
122	C₂²×C₁₁	#504	1	2	yes	—
123	C₄×C₁₁	—	2	2	no	—
124	C₄×C₁₁	—	2	4	no	—
125	C₄×C₁₁	—	4	2	no	—
126	C₄×C₁₁	#385	1	1	yes	characteristic
127	C₄×C₁₁	#424	1	1	yes	characteristic
128	C₄×C₁₁	#435	1	2	yes	—
129	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	Dic₂₂	1	2	yes	—
133	C₂²×C₈	D₄₄	1	2	yes	—
134	C₂²×C₈	C₂×Dic₁₁	1	2	yes	—
135	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
136	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	C₁₁⋊C₈	1	1	yes	characteristic
138	C₂×C₂²⋊C₄	C₁₁⋊C₈	1	1	yes	characteristic
139	C₂×C₄²	C₂×Dic₁₁	1	2	yes	—
140	C₂×C₁₆	—	11	4	no	—
141	C₂×C₁₆	—	22	8	no	—
142	C₂×C₁₆	—	22	2	no	—
143	C₂×C₁₆	—	22	2	no	—
144	C₂×C₁₆	—	22	1	no	—
145	C₂×C₁₆	—	22	1	no	—
146	C₂×C₁₆	—	22	2	no	—
147	C₂×C₁₆	—	22	2	no	—
148	C₂²⋊C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	—	2	2	no	—
151	#95	C₁₁⋊C₈	1	2	yes	—
152	C₂×C₁₁	—	2	4	no	—
153	C₂×C₁₁	#1850	1	1	yes	characteristic
154	C₂×C₁₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
155	C₂×C₁₁	#2151	1	1	yes	characteristic
156	C₂×C₁₁	D₄×C₁₆	1	2	yes	—
157	C₂×C₁₁	#2168	1	2	yes	—
158	C₂⁴	C₂×C₁₁⋊C₈	1	1	yes	characteristic
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	2	no	—
161	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
162	C₂²×C₄	C₂×C₁₁⋊C₈	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
164	C₂²×C₄	#4318	1	2	yes	—
165	C₂²×C₄	C₄⋊Dic₁₁	1	2	yes	—
166	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₄⋊Dic₁₁	1	2	yes	—
175	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
177	C₂×C₈	D₄×D₁₁	1	2	yes	—
178	C₂×C₈	#4341	1	2	yes	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	C₁₁⋊C₁₆	1	4	yes	complemented
181	C₄²	—	2	2	no	—
182	C₄²	—	2	2	no	—
183	C₁₆	—	22	4	no	—
184	C₁₆	—	44	1	no	—
185	C₁₆	—	44	1	no	—
186	C₁₁	#12912	1	1	yes	characteristic, complemented, Sylow
187	C₂³	—	2	2	no	—
188	C₂³	—	2	2	no	—
189	C₂³	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
190	C₂³	C₂×C₁₁⋊C₁₆	1	2	yes	—
191	C₂³	#66876	1	2	yes	—
192	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	2	yes	—
203	C₂×C₄	C₂×C₁₁⋊C₁₆	1	4	yes	complemented
204	C₂×C₄	C₂×#4318	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
206	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
207	C₂×C₄	#67001	1	2	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	4	2	no	—
210	C₈	#66931	1	2	yes	—
211	C₈	#66935	1	1	yes	characteristic
212	C₈	#66939	1	1	yes	characteristic
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	C₄⋊(C₁₁⋊C₁₆)	1	4	yes	complemented
218	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
219	C₂²	D₄×C₁₁⋊C₈	1	2	yes	—
220	C₂²	C₂²×C₁₁⋊C₁₆	1	1	yes	characteristic
221	C₂²	C₂×#66876	1	1	yes	characteristic
222	C₂²	#10622753	1	2	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	2	2	no	—
225	C₄	—	4	2	no	—
226	C₄	#10622563	1	1	yes	characteristic
227	C₄	#10622612	1	1	yes	characteristic
228	C₄	#10622624	1	2	yes	—
229	C₂	—	2	4	no	—
230	C₂	#13068796	1	1	yes	characteristic
231	C₂	C₂×C₄⋊(C₁₁⋊C₁₆)	1	1	yes	characteristic
232	C₂	#13070375	1	1	yes	characteristic
233	C₂	D₄×C₁₁⋊C₁₆	1	2	yes	—
234	C₂	#13070442	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₁
commutator	C₂²×C₁₁	C₂²×C₁₆
Fitting	C₈×C₁₁×C₂²⋊C₄	C₂
socle	C₂³×C₁₁	C₂²×C₈
prime radical	C₂⁴×C₁₁	C₂×C₈
cosocle	C₂²×C₈×C₁₁	C₂³
hypercenter	C₈×C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#12912
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₁₁⋊C₁₆)	C₂	semidirect
4. #13066694	C₂	semidirect
64. C₁₁×C₂²⋊C₄	C₁₆	semidirect
91. C₂×C₄×C₁₁	C₂×C₁₆	semidirect
117. C₂²×C₁₁	C₄⋊C₁₆	semidirect
180. C₂²⋊C₄	C₁₁⋊C₁₆	semidirect
186. C₁₁	#12912	semidirect
203. C₂×C₄	C₂×C₁₁⋊C₁₆	semidirect
217. C₂²	C₄⋊(C₁₁⋊C₁₆)	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₁⋊C₁₆	2
C₂×C₄⋊(C₁₁⋊C₁₆)	2
C₂×C₂²⋊(C₁₁⋊C₁₆)	2
#13066694	2
C₈×C₁₁×C₂²⋊C₄	2
#12912	11