Subgroups — #66870

Order 352.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₁₁	C₂²	1	1	yes	characteristic
5	C₂²×D₁₁	C₄	1	1	yes	characteristic, complemented
6	C₂×C₄×C₁₁	—	2	1	no	—
7	C₂²⋊D₁₁	—	2	1	no	—
8	C₂²⋊D₁₁	—	2	1	no	—
9	C₂×Dic₁₁	—	2	1	no	—
10	C₂×Dic₁₁	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₁₁	—	4	1	no	—
12	C₂²×C₁₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₁₁	D₄	1	1	yes	characteristic
14	C₂²×C₁₁	D₄	1	1	yes	characteristic
15	C₂×D₁₁	—	2	1	no	—
16	C₂×D₁₁	—	4	1	no	—
17	C₄×C₁₁	—	4	1	no	—
18	Dic₁₁	—	2	1	no	—
19	Dic₁₁	—	4	1	no	—
20	#99	—	11	1	no	maximal, Sylow
21	C₂×C₁₁	—	2	1	no	—
22	C₂×C₁₁	—	2	1	no	—
23	C₂×C₁₁	—	2	1	no	—
24	C₂×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
25	D₁₁	—	4	1	no	—
26	C₂×D₄	—	11	1	no	—
27	C₂²⋊C₄	—	11	1	no	—
28	C₂²⋊C₄	D₁₁	1	1	yes	characteristic, complemented
29	C₁₁	#99	1	1	yes	characteristic, complemented, Sylow
30	C₂³	—	11	1	no	—
31	C₂³	C₂×D₁₁	1	1	yes	characteristic
32	D₄	—	22	1	no	—
33	D₄	—	22	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	11	1	no	—
36	C₂×C₄	—	22	1	no	—
37	C₂²	—	4	1	no	—
38	C₂²	—	22	1	no	—
39	C₂²	—	44	1	no	—
40	C₂²	C₄×D₁₁	1	1	yes	characteristic
41	C₂²	D₄₄	1	1	yes	characteristic
42	C₂²	C₂²⋊D₁₁	1	1	yes	characteristic
43	C₄	—	4	1	no	—
44	C₄	—	22	1	no	—
45	C₄	—	44	1	no	—
46	C₂	—	2	1	no	—
47	C₂	—	2	1	no	—
48	C₂	—	2	1	no	—
49	C₂	—	44	1	no	—
50	C₂	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×D₁₁)⋊C₄
commutator	C₂²×C₁₁	C₂×C₄
Frattini	C₂³	C₂×D₁₁
Fitting	C₁₁×C₂²⋊C₄	C₂
socle	C₂×C₁₁	C₂²⋊C₄
prime radical	C₂×C₂²⋊D₁₁	C₂
cosocle	C₂³×C₁₁	C₂²
hypercenter	C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#99
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₁₁×C₂²⋊C₄	C₂	semidirect
3. C₂²⋊Dic₁₁	C₂	semidirect
5. C₂²×D₁₁	C₄	semidirect
10. C₂×Dic₁₁	C₄	semidirect
28. C₂²⋊C₄	D₁₁	semidirect
29. C₁₁	#99	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₁₁	2
C₁₁×C₂²⋊C₄	2
C₂²⋊Dic₁₁	2
#99	11