Subgroups — #66931

Order 352.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₁	—	2	2	no	—
12	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
16	Dic₂₂	—	2	2	no	—
17	C₂²×C₁₁	—	2	1	no	—
18	C₂²×C₁₁	—	2	1	no	—
19	C₂²×C₁₁	Q₈	1	2	yes	complemented
20	C₂²×C₁₁	C₂³	1	1	yes	characteristic
21	C₄×C₁₁	—	2	1	no	—
22	C₄×C₁₁	—	2	1	no	—
23	Dic₁₁	—	2	1	no	—
24	Dic₁₁	—	2	1	no	—
25	Dic₁₁	—	2	1	no	—
26	Dic₁₁	D₄	1	2	yes	—
27	C₂²⋊Q₈	—	11	1	no	maximal, Sylow
28	C₂×C₁₁	—	2	2	no	—
29	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₁₁	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₁₁	#41	1	1	yes	characteristic
32	C₂×Q₈	—	11	1	no	—
33	C₂²×C₄	—	11	1	no	—
34	C₄⋊C₄	—	11	1	no	—
35	C₄⋊C₄	—	11	1	no	—
36	C₄⋊C₄	—	11	1	no	—
37	C₂²⋊C₄	—	11	1	no	—
38	C₂²⋊C₄	D₁₁	1	1	yes	characteristic
39	C₁₁	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂³	C₂×D₁₁	1	1	yes	characteristic
41	Q₈	—	22	2	no	—
42	C₂×C₄	—	11	1	no	—
43	C₂×C₄	—	11	1	no	—
44	C₂×C₄	—	11	1	no	—
45	C₂×C₄	—	11	1	no	—
46	C₂×C₄	—	22	2	no	—
47	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
48	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	2	1	no	—
51	C₂²	Dic₂₂	1	2	yes	complemented
52	C₂²	C₂²×D₁₁	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	2	1	no	—
55	C₄	—	11	2	no	—
56	C₄	—	22	1	no	—
57	C₄	—	22	1	no	—
58	C₄	—	22	1	no	—
59	C₂	—	2	2	no	—
60	C₂	C₂×Dic₂₂	1	1	yes	characteristic
61	C₂	D₄×D₁₁	1	1	yes	characteristic
62	C₂	#4341	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator, socle, cosocle	C₂²×C₁₁	C₂³
Fitting	C₁₁×C₂²⋊C₄	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
5. C₄⋊Dic₁₁	C₂	semidirect
6. Dic₁₁⋊C₄	C₂	semidirect
7. Dic₁₁⋊C₄	C₂	semidirect
19. C₂²×C₁₁	Q₈	semidirect
39. C₁₁	C₂²⋊Q₈	semidirect
51. 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×Dic₁₁	2
C₂×Dic₂₂	2
C₁₁×C₂²⋊C₄	2
C₂²⋊Dic₁₁	2
C₄⋊Dic₁₁	2
Dic₁₁⋊C₄	2
C₂²⋊Q₈	11