Subgroups — #67042

Order 352.

166 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	#4343	C₂	1	2	yes	maximal, complemented
3	D₄×D₁₁	C₂	1	6	yes	maximal, complemented
4	#4339	C₂	1	3	yes	maximal, complemented
5	C₂×D₄₄	C₂	1	3	yes	maximal, complemented
6	C₂²×D₁₁	C₂²	1	6	yes	complemented
7	Q₈×C₁₁	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₁₁	C₂²	1	3	yes	complemented
9	C₂×C₄×C₁₁	C₂²	1	3	yes	complemented
10	C₂²⋊D₁₁	C₂²	1	6	yes	complemented
11	D₄₄	C₂²	1	3	yes	complemented
12	D₄₄	C₂²	1	6	yes	complemented
13	C₄×D₁₁	C₂²	1	6	yes	complemented
14	Dic₂₂	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₁₁	C₂³	1	3	yes	—
16	C₂×D₁₁	—	4	6	no	—
17	C₂×D₁₁	C₂³	1	6	yes	—
18	C₄×C₁₁	C₂³	1	1	yes	characteristic
19	C₄×C₁₁	C₂³	1	3	yes	—
20	Dic₁₁	C₂³	1	2	yes	—
21	H(5,𝔽₂)	—	11	1	no	maximal, Sylow
22	C₂×C₁₁	—	2	3	no	—
23	C₂×C₁₁	C₂⁴	1	1	yes	characteristic
24	D₁₁	—	2	6	no	—
25	#41	—	11	2	no	—
26	#41	—	11	3	no	—
27	#41	D₁₁	1	1	yes	characteristic, complemented
28	C₂×D₄	—	11	6	no	—
29	C₂×D₄	—	11	3	no	—
30	C₁₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
31	C₂³	—	11	6	no	—
32	Q₈	—	11	1	no	—
33	Q₈	C₂×D₁₁	1	1	yes	characteristic, complemented
34	D₄	—	11	6	no	—
35	D₄	—	11	6	no	—
36	D₄	—	11	3	no	—
37	D₄	C₂×D₁₁	1	3	yes	complemented
38	C₂×C₄	—	11	6	no	—
39	C₂×C₄	C₂×D₁₁	1	3	yes	complemented
40	C₂²	—	11	6	no	—
41	C₂²	—	44	6	no	—
42	C₂²	C₂²×D₁₁	1	3	yes	—
43	C₄	—	11	2	no	—
44	C₄	C₂²×D₁₁	1	3	yes	—
45	C₄	C₂²×D₁₁	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	22	6	no	—
48	C₂	C₂³×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₁
commutator, socle, cosocle	C₂×C₁₁	C₂⁴
Fitting	C₁₁×#41	C₂
hypercenter	#41	D₁₁
hypocenter	C₁₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₁₁×#41	C₂	semidirect
2. #4343	C₂	semidirect
3. D₄×D₁₁	C₂	semidirect
4. #4339	C₂	semidirect
5. C₂×D₄₄	C₂	semidirect
6. C₂²×D₁₁	C₂²	semidirect
7. Q₈×C₁₁	C₂²	semidirect
8. D₄×C₁₁	C₂²	semidirect
9. C₂×C₄×C₁₁	C₂²	semidirect
10. C₂²⋊D₁₁	C₂²	semidirect
11. D₄₄	C₂²	semidirect
12. D₄₄	C₂²	semidirect
13. C₄×D₁₁	C₂²	semidirect
14. Dic₂₂	C₂²	semidirect
27. #41	D₁₁	semidirect
30. C₁₁	H(5,𝔽₂)	semidirect
33. Q₈	C₂×D₁₁	semidirect
37. D₄	C₂×D₁₁	semidirect
39. C₂×C₄	C₂×D₁₁	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
#4343	2
D₄×D₁₁	2
#4339	2
C₂×D₄₄	2
H(5,𝔽₂)	11