Subgroups — #66985

Order 352.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#4339	C₂	1	1	yes	maximal, characteristic, complemented
3	#4324	C₂	1	2	yes	maximal, complemented
4	#4323	C₂	1	2	yes	maximal, complemented
5	#4318	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₁₁	—	2	1	no	—
7	D₄×C₁₁	—	2	1	no	—
8	D₄×C₁₁	C₂²	1	2	yes	—
9	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂²⋊D₁₁	—	2	1	no	—
11	D₄₄	C₂²	1	1	yes	characteristic, complemented
12	C₄×D₁₁	—	2	1	no	—
13	Dic₂₂	C₂²	1	1	yes	characteristic, complemented
14	C₁₁⋊C₈	C₂²	1	2	yes	complemented
15	C₂²×C₁₁	—	2	2	no	—
16	C₂²×C₁₁	—	4	2	no	—
17	C₂²×C₁₁	D₄	1	1	yes	characteristic
18	C₂×D₁₁	—	2	1	no	—
19	C₄×C₁₁	D₄	1	1	yes	characteristic
20	C₄×C₁₁	C₂³	1	1	yes	characteristic
21	Dic₁₁	—	2	1	no	—
22	AGL(1,ℤ/8)	—	11	1	no	maximal, Sylow
23	C₂×C₁₁	—	2	1	no	—
24	C₂×C₁₁	—	4	2	no	—
25	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
26	D₁₁	—	4	1	no	—
27	#41	—	11	1	no	—
28	C₂×D₄	D₁₁	1	1	yes	characteristic, complemented
29	SD₁₆	—	11	2	no	—
30	D₈	—	11	2	no	—
31	M₁₆	—	11	1	no	—
32	C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	C₂³	—	2	1	no	—
34	Q₈	—	11	1	no	—
35	D₄	—	2	1	no	—
36	D₄	—	11	1	no	—
37	D₄	—	22	1	no	—
38	D₄	C₂×D₁₁	1	2	yes	—
39	C₂×C₄	—	22	1	no	—
40	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
41	C₈	—	11	2	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	4	2	no	—
44	C₂²	—	22	1	no	—
45	C₂²	C₂²⋊D₁₁	1	1	yes	characteristic
46	C₄	—	22	1	no	—
47	C₄	C₂²⋊D₁₁	1	1	yes	characteristic
48	C₄	C₂²×D₁₁	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	2	no	—
51	C₂	—	44	1	no	—
52	C₂	C₂×C₂²⋊D₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₂×D₄×C₁₁	C₂
socle	C₂×C₁₁	C₂×D₄
hypercenter	C₂×D₄	D₁₁
hypocenter	C₁₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×C₁₁	C₂	semidirect
2. #4339	C₂	semidirect
3. #4324	C₂	semidirect
4. #4323	C₂	semidirect
5. #4318	C₂	semidirect
11. D₄₄	C₂²	semidirect
13. Dic₂₂	C₂²	semidirect
14. C₁₁⋊C₈	C₂²	semidirect
28. C₂×D₄	D₁₁	semidirect
32. C₁₁	AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×C₁₁	2
#4339	2
#4324	2
#4323	2
#4318	2
AGL(1,ℤ/8)	11