Subgroups — #66946

Order 352.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₁₁	C₂	1	2	yes	maximal, complemented
3	C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	(C₂×D₁₁)⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₁	C₂²	1	2	yes	—
7	C₂×C₄×C₁₁	C₂²	1	2	yes	—
8	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
11	D₄₄	C₄	1	4	yes	complemented
12	C₄×D₁₁	—	2	4	no	—
13	C₂²×C₁₁	C₂³	1	1	yes	characteristic
14	C₂×D₁₁	—	2	2	no	—
15	C₂×D₁₁	—	2	2	no	—
16	C₂×D₁₁	C₂×C₄	1	4	yes	complemented
17	C₄×C₁₁	—	2	2	no	—
18	C₄×C₁₁	C₂×C₄	1	2	yes	complemented
19	Dic₁₁	—	2	1	no	—
20	Dic₁₁	D₄	1	2	yes	complemented
21	C₄×D₄	—	11	1	no	maximal, Sylow
22	C₂×C₁₁	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
24	C₂×C₁₁	#41	1	1	yes	characteristic
25	D₁₁	—	2	4	no	—
26	C₂×D₄	—	11	1	no	—
27	C₂²×C₄	—	11	2	no	—
28	C₄⋊C₄	D₁₁	1	1	yes	characteristic, complemented
29	C₂²⋊C₄	—	11	2	no	—
30	C₄²	—	11	1	no	—
31	C₁₁	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
32	C₂³	—	11	2	no	—
33	D₄	—	11	4	no	—
34	C₂×C₄	—	11	1	no	—
35	C₂×C₄	—	11	1	no	—
36	C₂×C₄	—	22	4	no	—
37	C₂×C₄	C₂×D₁₁	1	2	yes	—
38	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
39	C₂²	—	11	4	no	—
40	C₂²	—	22	2	no	—
41	C₂²	—	22	2	no	—
42	C₂²	C₂²×D₁₁	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	11	2	no	—
45	C₄	—	22	1	no	—
46	C₄	C₄×D₁₁	1	2	yes	complemented
47	C₂	—	22	4	no	—
48	C₂	C₂×C₄×D₁₁	1	1	yes	characteristic
49	C₂	D₄×D₁₁	1	1	yes	characteristic
50	C₂	#4343	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator	C₂×C₁₁	C₂²×C₄
Fitting	C₁₁×C₄⋊C₄	C₂
socle, cosocle	C₂²×C₁₁	C₂³
prime radical	C₂×D₄₄	C₂
hypercenter	C₄⋊C₄	D₁₁
hypocenter	C₁₁	C₄×D₄
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₁₁	C₂	semidirect
3. C₁₁×C₄⋊C₄	C₂	semidirect
4. (C₂×D₁₁)⋊C₄	C₂	semidirect
5. C₄×Dic₁₁	C₂	semidirect
11. D₄₄	C₄	semidirect
16. C₂×D₁₁	C₂×C₄	semidirect
18. C₄×C₁₁	C₂×C₄	semidirect
20. Dic₁₁	D₄	semidirect
28. C₄⋊C₄	D₁₁	semidirect
31. C₁₁	C₄×D₄	semidirect
46. C₄	C₄×D₁₁	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
C₄×Dic₁₁	2
C₄×D₄	11