Subgroups — #66948

Order 352.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₄	C₂	1	2	yes	maximal, complemented
2	C₂×D₄₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₁)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₁	C₂²	1	2	yes	—
7	C₂²×D₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₁	C₂²	1	2	yes	—
9	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
11	D₄₄	—	2	4	no	—
12	D₄₄	—	2	2	no	—
13	C₄×D₁₁	—	2	2	no	—
14	C₂²×C₁₁	C₂³	1	1	yes	characteristic
15	C₂×D₁₁	—	2	1	no	—
16	C₂×D₁₁	—	2	1	no	—
17	C₂×D₁₁	—	2	2	no	—
18	C₂×D₁₁	—	2	2	no	—
19	C₂×D₁₁	—	2	2	no	—
20	C₂×D₁₁	D₄	1	2	yes	complemented
21	C₄×C₁₁	—	2	2	no	—
22	C₄×C₁₁	D₄	1	2	yes	complemented
23	Dic₁₁	—	2	1	no	—
24	#121	—	11	1	no	maximal, Sylow
25	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
26	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₁₁	#41	1	1	yes	characteristic
28	D₁₁	—	2	2	no	—
29	D₁₁	—	4	2	no	—
30	C₂×D₄	—	11	2	no	—
31	C₂×D₄	—	11	1	no	—
32	C₂²×C₄	—	11	1	no	—
33	C₄⋊C₄	D₁₁	1	1	yes	characteristic, complemented
34	C₂²⋊C₄	—	11	2	no	—
35	C₁₁	#121	1	1	yes	characteristic, complemented, Sylow
36	C₂³	—	11	1	no	—
37	C₂³	—	11	2	no	—
38	D₄	—	22	4	no	—
39	D₄	—	22	2	no	—
40	C₂×C₄	—	11	1	no	—
41	C₂×C₄	—	22	2	no	—
42	C₂×C₄	C₂×D₁₁	1	2	yes	—
43	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
44	C₂²	—	11	2	no	—
45	C₂²	—	22	1	no	—
46	C₂²	—	22	1	no	—
47	C₂²	—	22	2	no	—
48	C₂²	—	22	2	no	—
49	C₂²	—	22	2	no	—
50	C₂²	C₂²×D₁₁	1	1	yes	characteristic
51	C₄	—	2	2	no	—
52	C₄	—	22	1	no	—
53	C₄	D₄₄	1	2	yes	complemented
54	C₂	—	22	2	no	—
55	C₂	—	44	2	no	—
56	C₂	C₂×D₄₄	1	1	yes	characteristic
57	C₂	D₄×D₁₁	1	1	yes	characteristic
58	C₂	#4343	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator, socle, cosocle	C₂²×C₁₁	C₂³
Fitting	C₁₁×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₁₁
hypocenter	C₁₁	#121
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₄₄	C₂	semidirect
2. C₂×D₄₄	C₂	semidirect
3. C₂×C₄×D₁₁	C₂	semidirect
4. C₁₁×C₄⋊C₄	C₂	semidirect
5. (C₂×D₁₁)⋊C₄	C₂	semidirect
20. C₂×D₁₁	D₄	semidirect
22. C₄×C₁₁	D₄	semidirect
33. C₄⋊C₄	D₁₁	semidirect
35. C₁₁	#121	semidirect
53. C₄	D₄₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₄	2
C₂×C₄×D₁₁	2
C₁₁×C₄⋊C₄	2
(C₂×D₁₁)⋊C₄	2
#121	11