Subgroups — #67003

Order 352.

62 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, complemented
2	#4339	C₂	1	1	yes	maximal, characteristic, complemented
3	#4326	C₂	1	1	yes	maximal, characteristic, complemented
4	#4325	C₂	1	1	yes	maximal, characteristic, complemented
5	#4324	C₂	1	1	yes	maximal, characteristic, complemented
6	#4323	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	D₄×C₁₁	—	2	1	no	—
10	D₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₁	—	2	1	no	—
12	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²⋊D₁₁	—	2	1	no	—
14	D₄₄	C₂²	1	1	yes	characteristic
15	C₄×D₁₁	—	2	1	no	—
16	Dic₂₂	C₂²	1	1	yes	characteristic
17	C₁₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₁₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₁	—	2	1	no	—
20	C₂²×C₁₁	D₄	1	1	yes	characteristic
21	C₂×D₁₁	—	2	1	no	—
22	C₄×C₁₁	—	2	1	no	—
23	C₄×C₁₁	D₄	1	1	yes	characteristic
24	C₄×C₁₁	C₂³	1	1	yes	characteristic
25	Dic₁₁	—	2	1	no	—
26	#135	—	11	1	no	maximal, Sylow
27	C₂×C₁₁	—	2	1	no	—
28	C₂×C₁₁	—	4	1	no	—
29	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
30	D₁₁	—	4	1	no	—
31	#41	—	11	1	no	—
32	#41	D₁₁	1	1	yes	characteristic, complemented
33	Q₁₆	—	11	1	no	—
34	SD₁₆	—	11	1	no	—
35	SD₁₆	—	11	1	no	—
36	D₈	—	11	1	no	—
37	C₂×C₈	—	11	1	no	—
38	C₁₁	#135	1	1	yes	characteristic, complemented, Sylow
39	Q₈	—	11	1	no	—
40	Q₈	C₂×D₁₁	1	1	yes	characteristic
41	D₄	—	2	1	no	—
42	D₄	—	11	1	no	—
43	D₄	—	22	1	no	—
44	D₄	C₂×D₁₁	1	1	yes	characteristic
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	22	1	no	—
47	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
48	C₈	—	11	1	no	—
49	C₈	—	11	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	—	22	1	no	—
52	C₂²	C₂²⋊D₁₁	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	22	1	no	—
55	C₄	C₂²⋊D₁₁	1	1	yes	characteristic
56	C₄	C₂²×D₁₁	1	1	yes	characteristic
57	C₂	—	2	1	no	—
58	C₂	—	4	1	no	—
59	C₂	—	44	1	no	—
60	C₂	C₂×C₂²⋊D₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₁₁×#41	C₂
socle	C₂×C₁₁	C₂×D₄
hypercenter	#41	D₁₁
hypocenter	C₁₁	#135
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₁₁×#41	C₂	semidirect
2. #4339	C₂	semidirect
3. #4326	C₂	semidirect
4. #4325	C₂	semidirect
5. #4324	C₂	semidirect
6. #4323	C₂	semidirect
7. C₂×C₁₁⋊C₈	C₂	semidirect
32. #41	D₁₁	semidirect
38. C₁₁	#135	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
#4339	2
#4326	2
#4325	2
#4324	2
#4323	2
C₂×C₁₁⋊C₈	2
#135	11