Subgroups — #67004

Order 352.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#4326	C₂	1	2	yes	maximal, complemented
4	#4324	C₂	1	2	yes	maximal, complemented
5	#4318	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×C₁₁	C₂²	1	1	yes	characteristic
7	D₄×C₁₁	—	2	1	no	—
8	D₄×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₁	—	2	1	no	—
10	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₁	—	2	1	no	—
12	Dic₂₂	—	2	1	no	—
13	Dic₂₂	C₂²	1	2	yes	—
14	C₁₁⋊C₈	C₂²	1	2	yes	—
15	C₂²×C₁₁	—	2	1	no	—
16	C₂²×C₁₁	D₄	1	1	yes	characteristic
17	C₄×C₁₁	—	2	1	no	—
18	C₄×C₁₁	D₄	1	1	yes	characteristic
19	C₄×C₁₁	C₂³	1	1	yes	characteristic
20	Dic₁₁	—	2	2	no	—
21	#137	—	11	1	no	maximal, Sylow
22	C₂×C₁₁	—	2	1	no	—
23	C₂×C₁₁	—	4	1	no	—
24	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
25	#41	D₁₁	1	1	yes	characteristic
26	C₂×Q₈	—	11	1	no	—
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	—	11	2	no	—
32	Q₈	—	22	1	no	—
33	Q₈	C₂×D₁₁	1	1	yes	characteristic
34	D₄	—	2	1	no	—
35	D₄	C₂×D₁₁	1	1	yes	characteristic
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	—	2	1	no	—
41	C₂²	C₂²⋊D₁₁	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	—	22	2	no	—
44	C₄	C₂²⋊D₁₁	1	1	yes	characteristic
45	C₄	C₂²×D₁₁	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	4	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, prime radical	C₁₁×#41	C₂
socle	C₂×C₁₁	C₂×D₄
hypercenter	#41	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₂	C₂	semidirect
3. #4326	C₂	semidirect
4. #4324	C₂	semidirect
5. #4318	C₂	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
C₁₁×#41	2
C₂×Dic₂₂	2
#4326	2
#4324	2
#4318	2
#137	11