Subgroups — #66977

Order 352.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₁₁	C₂	1	2	yes	maximal
4	C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₁⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₁	—	2	2	no	—
8	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₁	C₂²	1	2	yes	—
11	C₂×Dic₁₁	C₂²	1	2	yes	—
12	Dic₂₂	—	2	2	no	—
13	C₂²×C₁₁	—	2	1	no	—
14	C₂²×C₁₁	—	2	1	no	—
15	C₂²×C₁₁	Q₈	1	2	yes	complemented
16	C₂²×C₁₁	C₂³	1	1	yes	characteristic
17	C₄×C₁₁	—	2	1	no	—
18	C₄×C₁₁	D₄	1	2	yes	—
19	Dic₁₁	—	2	2	no	—
20	Dic₁₁	—	2	2	no	—
21	C₂²⋊Q₈	—	11	1	no	maximal, Sylow
22	C₂×C₁₁	—	2	2	no	—
23	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
24	C₂×C₁₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₁₁	#41	1	1	yes	characteristic
26	C₂×Q₈	—	11	1	no	—
27	C₂²×C₄	D₁₁	1	1	yes	characteristic
28	C₄⋊C₄	—	11	2	no	—
29	C₄⋊C₄	—	11	1	no	—
30	C₂²⋊C₄	—	11	2	no	—
31	C₁₁	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
32	C₂³	C₂×D₁₁	1	1	yes	characteristic
33	Q₈	—	22	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	—	11	2	no	—
36	C₂×C₄	—	11	2	no	—
37	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
38	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
39	C₂²	—	2	1	no	—
40	C₂²	—	2	1	no	—
41	C₂²	Dic₂₂	1	2	yes	complemented
42	C₂²	C₂²×D₁₁	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	22	2	no	—
45	C₄	—	22	2	no	—
46	C₄	C₂²⋊D₁₁	1	2	yes	—
47	C₂	—	2	2	no	—
48	C₂	C₂×Dic₂₂	1	1	yes	characteristic
49	C₂	#4339	1	1	yes	characteristic
50	C₂	C₂×C₂²⋊D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator, socle, cosocle	C₂²×C₁₁	C₂³
Fitting	C₂²×C₄×C₁₁	C₂
prime radical	C₂³×C₁₁	C₂²
hypercenter	C₂²×C₄	D₁₁
hypocenter	C₁₁	C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₂₂	C₂	semidirect
4. C₄⋊Dic₁₁	C₂	semidirect
5. Dic₁₁⋊C₄	C₂	semidirect
15. C₂²×C₁₁	Q₈	semidirect
31. C₁₁	C₂²⋊Q₈	semidirect
41. C₂²	Dic₂₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₁₁	2
C₂×Dic₂₂	2
C₂²⋊Dic₁₁	2
C₄⋊Dic₁₁	2
Dic₁₁⋊C₄	2
C₂²⋊Q₈	11