Subgroups — #66995

Order 352.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#4339	C₂	1	1	yes	maximal, characteristic
3	#4326	C₂	1	2	yes	maximal, complemented
4	#4325	C₂	1	2	yes	maximal, complemented
5	#4318	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₁₁	—	2	1	no	—
7	Q₈×C₁₁	C₂²	1	2	yes	—
8	C₂×C₄×C₁₁	—	2	1	no	—
9	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂²⋊D₁₁	—	2	1	no	—
11	D₄₄	C₂²	1	1	yes	characteristic
12	C₄×D₁₁	—	2	1	no	—
13	Dic₂₂	C₂²	1	1	yes	characteristic
14	C₁₁⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₁₁	D₄	1	1	yes	characteristic
16	C₂×D₁₁	—	2	1	no	—
17	C₄×C₁₁	—	2	2	no	—
18	C₄×C₁₁	D₄	1	1	yes	characteristic
19	C₄×C₁₁	C₂³	1	1	yes	characteristic
20	Dic₁₁	—	2	1	no	—
21	#137	—	11	1	no	maximal, Sylow
22	C₂×C₁₁	—	2	1	no	—
23	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
24	D₁₁	—	4	1	no	—
25	#41	—	11	1	no	—
26	C₂×Q₈	D₁₁	1	1	yes	characteristic, complemented
27	Q₁₆	—	11	2	no	—
28	SD₁₆	—	11	2	no	—
29	M₁₆	—	11	1	no	—
30	C₁₁	#137	1	1	yes	characteristic, complemented, Sylow
31	Q₈	—	2	1	no	—
32	Q₈	—	11	1	no	—
33	Q₈	C₂×D₁₁	1	2	yes	—
34	D₄	—	11	1	no	—
35	D₄	—	22	1	no	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	—	22	1	no	—
38	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
39	C₈	—	11	2	no	—
40	C₂²	—	22	1	no	—
41	C₂²	C₂²⋊D₁₁	1	1	yes	characteristic
42	C₄	—	2	2	no	—
43	C₄	—	22	1	no	—
44	C₄	C₂²⋊D₁₁	1	1	yes	characteristic
45	C₄	C₂²×D₁₁	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	44	1	no	—
48	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×Q₈×C₁₁	C₂
socle	C₂×C₁₁	C₂×D₄
prime radical	#4339	C₂
hypercenter	C₂×Q₈	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₁₁	C₂	semidirect
3. #4326	C₂	semidirect
4. #4325	C₂	semidirect
5. #4318	C₂	semidirect
26. C₂×Q₈	D₁₁	semidirect
30. C₁₁	#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₁	2
#4339	2
#4326	2
#4325	2
#4318	2
#137	11