Subgroups — #67037

Order 352.

166 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	C₂×C₂²⋊D₁₁	C₂	1	4	yes	maximal, complemented
3	#4341	C₂	1	4	yes	maximal, complemented
4	D₄×D₁₁	C₂	1	4	yes	maximal, complemented
5	#4339	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁	C₂²	1	2	yes	complemented
7	C₂²×D₁₁	C₂²	1	4	yes	complemented
8	D₄×C₁₁	C₂²	1	4	yes	complemented
9	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic, complemented
10	C₂²⋊D₁₁	C₂²	1	4	yes	complemented
11	C₂²⋊D₁₁	C₂²	1	8	yes	complemented
12	C₂×Dic₁₁	C₂²	1	4	yes	complemented
13	D₄₄	C₂²	1	2	yes	complemented
14	C₄×D₁₁	C₂²	1	4	yes	complemented
15	Dic₂₂	C₂²	1	2	yes	complemented
16	C₂²×C₁₁	—	4	2	no	—
17	C₂²×C₁₁	C₂³	1	1	yes	characteristic
18	C₂²×C₁₁	C₂³	1	4	yes	—
19	C₂×D₁₁	—	4	4	no	—
20	C₂×D₁₁	C₂³	1	4	yes	—
21	C₄×C₁₁	C₂³	1	2	yes	—
22	Dic₁₁	C₂³	1	4	yes	—
23	H(5,𝔽₂)	—	11	1	no	maximal, Sylow
24	C₂×C₁₁	—	2	4	no	—
25	C₂×C₁₁	—	2	1	no	—
26	C₂×C₁₁	C₂⁴	1	1	yes	characteristic
27	D₁₁	—	2	4	no	—
28	#41	—	11	4	no	—
29	#41	—	11	2	no	—
30	C₂×D₄	—	11	4	no	—
31	C₂×D₄	—	11	4	no	—
32	C₂×D₄	D₁₁	1	1	yes	characteristic, complemented
33	C₁₁	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
34	C₂³	—	11	4	no	—
35	C₂³	C₂×D₁₁	1	2	yes	complemented
36	Q₈	—	11	2	no	—
37	D₄	—	11	8	no	—
38	D₄	—	11	4	no	—
39	D₄	—	11	2	no	—
40	D₄	C₂×D₁₁	1	4	yes	complemented
41	C₂×C₄	—	11	4	no	—
42	C₂×C₄	—	11	4	no	—
43	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic, complemented
44	C₂²	—	4	2	no	—
45	C₂²	—	11	4	no	—
46	C₂²	—	44	4	no	—
47	C₂²	C₂²×D₁₁	1	4	yes	—
48	C₂²	C₂²×D₁₁	1	1	yes	characteristic
49	C₄	—	11	4	no	—
50	C₄	C₂²×D₁₁	1	2	yes	—
51	C₂	—	2	1	no	—
52	C₂	—	2	4	no	—
53	C₂	—	22	4	no	—
54	C₂	C₂³×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₁
commutator, socle, cosocle	C₂×C₁₁	C₂⁴
Fitting	C₂×D₄×C₁₁	C₂
hypercenter	C₂×D₄	D₁₁
hypocenter	C₁₁	H(5,𝔽₂)
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ 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. C₂×C₂²⋊D₁₁	C₂	semidirect
3. #4341	C₂	semidirect
4. D₄×D₁₁	C₂	semidirect
5. #4339	C₂	semidirect
6. C₂³×C₁₁	C₂²	semidirect
7. C₂²×D₁₁	C₂²	semidirect
8. D₄×C₁₁	C₂²	semidirect
9. C₂×C₄×C₁₁	C₂²	semidirect
10. C₂²⋊D₁₁	C₂²	semidirect
11. C₂²⋊D₁₁	C₂²	semidirect
12. C₂×Dic₁₁	C₂²	semidirect
13. D₄₄	C₂²	semidirect
14. C₄×D₁₁	C₂²	semidirect
15. Dic₂₂	C₂²	semidirect
32. C₂×D₄	D₁₁	semidirect
33. C₁₁	H(5,𝔽₂)	semidirect
35. C₂³	C₂×D₁₁	semidirect
40. D₄	C₂×D₁₁	semidirect
43. 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
C₂×D₄×C₁₁	2
C₂×C₂²⋊D₁₁	2
#4341	2
D₄×D₁₁	2
#4339	2
H(5,𝔽₂)	11