Subgroups — #13070925

Order 1408.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10622517	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₁×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#10622087	C₂	1	2	yes	maximal, complemented
5	D₃₅₂	C₂	1	1	yes	maximal, characteristic, complemented
6	#66957	—	2	2	no	—
7	C₂×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
8	Dic₈₈	C₂²	1	2	yes	—
9	#66864	—	2	2	no	—
10	D₁₇₆	C₂²	1	2	yes	—
11	C₁₁×C₃₂	C₂²	1	1	yes	characteristic
12	C₁₁×C₃₂	C₂²	1	1	yes	characteristic
13	#4339	—	4	2	no	—
14	C₂×C₈×C₁₁	D₄	1	1	yes	characteristic
15	Dic₄₄	—	2	2	no	—
16	D₈₈	—	2	2	no	—
17	#4314	—	4	2	no	—
18	C₁₁×C₁₆	D₄	1	1	yes	characteristic
19	C₁₁×C₁₆	C₂³	1	1	yes	characteristic
20	#2262	—	11	1	no	maximal, Sylow
21	C₂×C₄×C₁₁	D₈	1	1	yes	characteristic
22	C₂²⋊D₁₁	—	8	2	no	—
23	D₄₄	—	4	2	no	—
24	C₄×D₁₁	—	8	2	no	—
25	Dic₂₂	—	4	2	no	—
26	C₈×C₁₁	D₈	1	1	yes	characteristic
27	C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
28	#508	—	11	2	no	—
29	Q₆₄	—	11	1	no	—
30	SD₆₄	—	11	2	no	—
31	D₃₂	—	11	1	no	—
32	C₂×C₃₂	D₁₁	1	1	yes	characteristic, complemented
33	C₂²×C₁₁	D₁₆	1	1	yes	characteristic
34	C₂×D₁₁	—	8	2	no	—
35	C₄×C₁₁	D₁₆	1	1	yes	characteristic
36	C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
37	Dic₁₁	—	8	2	no	—
38	#135	—	22	2	no	—
39	Q₃₂	—	11	2	no	—
40	SD₃₂	—	22	2	no	—
41	D₁₆	—	11	2	no	—
42	C₂×C₁₆	C₂×D₁₁	1	1	yes	characteristic
43	C₃₂	C₂×D₁₁	1	1	yes	characteristic
44	C₃₂	C₂×D₁₁	1	1	yes	characteristic
45	C₂×C₁₁	—	2	1	no	—
46	C₂×C₁₁	C₂×D₁₆	1	1	yes	characteristic
47	D₁₁	—	16	2	no	—
48	#41	—	44	2	no	—
49	Q₁₆	—	22	2	no	—
50	SD₁₆	—	44	2	no	—
51	D₈	—	22	2	no	—
52	C₂×C₈	D₄₄	1	1	yes	characteristic
53	C₁₆	D₄₄	1	1	yes	characteristic
54	C₁₆	C₂²×D₁₁	1	1	yes	characteristic
55	C₁₁	#2262	1	1	yes	characteristic, complemented, Sylow
56	Q₈	—	44	2	no	—
57	D₄	—	44	2	no	—
58	D₄	—	88	2	no	—
59	C₂×C₄	—	88	2	no	—
60	C₂×C₄	D₈₈	1	1	yes	characteristic
61	C₈	D₈₈	1	1	yes	characteristic
62	C₈	C₂×D₄₄	1	1	yes	characteristic
63	C₂²	—	88	2	no	—
64	C₂²	D₁₇₆	1	1	yes	characteristic
65	C₄	—	88	2	no	—
66	C₄	D₁₇₆	1	1	yes	characteristic
67	C₄	C₂×D₈₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	176	2	no	—
70	C₂	C₂×D₁₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₇₆
commutator, cosocle	C₁₁×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₁₁
Fitting	C₂×C₁₁×C₃₂	C₂
socle	C₂×C₁₁	C₂×D₁₆
hypercenter	C₂×C₃₂	D₁₁
hypocenter	C₁₁	#2262
LCS	G ▷ C₁₁×C₁₆ ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #10622517	C₂	semidirect
2. C₂×C₁₁×C₃₂	C₂	semidirect
3. Dic₁₇₆	C₂	semidirect
4. #10622087	C₂	semidirect
5. D₃₅₂	C₂	semidirect
32. C₂×C₃₂	D₁₁	semidirect
55. C₁₁	#2262	semidirect

Maximal subgroups

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

subgroup	index
#10622517	2
C₂×C₁₁×C₃₂	2
Dic₁₇₆	2
#10622087	2
D₃₅₂	2
#2262	11