Subgroups — #13070911

Order 1408.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10622514	C₂	1	1	yes	maximal, characteristic
2	C₁₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622085	C₂	1	2	yes	maximal, complemented
5	D₁₁×C₃₂	C₂	1	2	yes	maximal, complemented
6	#66954	C₄	1	1	yes	characteristic
7	C₂×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
8	#66876	C₄	1	1	yes	characteristic
9	#66862	C₄	1	2	yes	—
10	C₁₆×D₁₁	C₂²	1	2	yes	—
11	C₁₁×C₃₂	C₂²	1	2	yes	—
12	C₁₁⋊C₃₂	C₂²	1	1	yes	characteristic
13	C₁₁⋊C₃₂	C₂²	1	1	yes	characteristic
14	#4339	C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
16	#4318	C₈	1	1	yes	characteristic
17	#4313	C₈	1	2	yes	—
18	C₈×D₁₁	C₂×C₄	1	2	yes	—
19	C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₁₁×C₁₆	C₂³	1	1	yes	characteristic
21	C₁₁⋊C₁₆	C₂×C₄	1	2	yes	—
22	#2258	—	11	1	no	maximal, Sylow
23	C₂×C₄×C₁₁	C₂×C₈	1	1	yes	characteristic
24	C₂²⋊D₁₁	C₁₆	1	2	yes	—
25	D₄₄	C₁₆	1	1	yes	characteristic
26	C₄×D₁₁	C₂×C₈	1	2	yes	—
27	Dic₂₂	C₁₆	1	1	yes	characteristic
28	C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
29	C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
30	C₁₁⋊C₈	C₂×C₈	1	2	yes	—
31	#504	—	11	1	no	—
32	M₆₄	—	11	2	no	—
33	M₆₄	D₁₁	1	1	yes	characteristic, complemented
34	C₂×C₃₂	—	11	1	no	—
35	C₂×C₃₂	—	11	2	no	—
36	C₂²×C₁₁	C₂×C₁₆	1	1	yes	characteristic
37	C₂×D₁₁	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₁	C₂×C₁₆	1	1	yes	characteristic
39	C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
40	Dic₁₁	C₂×C₁₆	1	2	yes	—
41	#131	—	11	1	no	—
42	M₃₂	—	11	1	no	—
43	M₃₂	—	11	2	no	—
44	C₂×C₁₆	—	11	2	no	—
45	C₂×C₁₆	C₂×D₁₁	1	1	yes	characteristic
46	C₃₂	—	11	1	no	—
47	C₃₂	—	11	1	no	—
48	C₃₂	C₂×D₁₁	1	2	yes	—
49	C₂×C₁₁	—	2	1	no	—
50	C₂×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
51	D₁₁	—	2	2	no	—
52	#41	—	11	1	no	—
53	M₁₆	—	11	1	no	—
54	M₁₆	—	11	2	no	—
55	C₂×C₈	—	11	2	no	—
56	C₂×C₈	C₄×D₁₁	1	1	yes	characteristic
57	C₁₆	—	11	2	no	—
58	C₁₆	C₄×D₁₁	1	1	yes	characteristic
59	C₁₆	C₂²×D₁₁	1	1	yes	characteristic
60	C₁₁	#2258	1	1	yes	characteristic, complemented, Sylow
61	Q₈	—	11	1	no	—
62	D₄	—	11	2	no	—
63	D₄	—	11	1	no	—
64	C₂×C₄	—	11	2	no	—
65	C₂×C₄	C₈×D₁₁	1	1	yes	characteristic
66	C₈	—	11	2	no	—
67	C₈	C₈×D₁₁	1	1	yes	characteristic
68	C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
69	C₂²	—	11	2	no	—
70	C₂²	C₁₆×D₁₁	1	1	yes	characteristic
71	C₄	—	11	2	no	—
72	C₄	C₁₆×D₁₁	1	1	yes	characteristic
73	C₄	C₂×C₈×D₁₁	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	22	2	no	—
76	C₂	C₂×C₁₆×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₁
commutator, socle	C₂×C₁₁	C₂²×C₁₆
Fitting	C₁₁×M₆₄	C₂
prime radical	#4339	C₈
cosocle	C₁₁×C₁₆	C₂³
hypercenter	M₆₄	D₁₁
hypocenter	C₁₁	#2258
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₁₁×M₆₄	C₂	semidirect
3. C₂×C₁₁⋊C₃₂	C₂	semidirect
4. #10622085	C₂	semidirect
5. D₁₁×C₃₂	C₂	semidirect
33. M₆₄	D₁₁	semidirect
60. C₁₁	#2258	semidirect

Maximal subgroups

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

subgroup	index
#10622514	2
C₁₁×M₆₄	2
C₂×C₁₁⋊C₃₂	2
#10622085	2
D₁₁×C₃₂	2
#2258	11