Subgroups — #66945

Order 352.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic
2	C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₁₁)⋊C₄	C₂	1	2	yes	maximal
4	C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×Dic₁₁	C₂	1	2	yes	maximal, complemented
6	C₂²×D₁₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₁	C₂²	1	2	yes	—
8	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×Dic₁₁	C₂²	1	2	yes	—
10	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
11	C₄×D₁₁	C₄	1	4	yes	complemented
12	C₂²×C₁₁	C₂³	1	1	yes	characteristic
13	C₂×D₁₁	—	2	1	no	—
14	C₂×D₁₁	—	2	1	no	—
15	C₂×D₁₁	C₂×C₄	1	2	yes	—
16	C₄×C₁₁	—	2	2	no	—
17	C₄×C₁₁	C₂×C₄	1	2	yes	—
18	Dic₁₁	—	2	2	no	—
19	Dic₁₁	C₂×C₄	1	2	yes	—
20	#117	—	11	1	no	maximal, Sylow
21	C₂×C₁₁	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₁₁	#41	1	1	yes	characteristic
23	C₂×C₁₁	#41	1	1	yes	characteristic
24	D₁₁	—	2	2	no	—
25	C₂²×C₄	—	11	1	no	—
26	C₄⋊C₄	—	11	1	no	—
27	C₄⋊C₄	D₁₁	1	1	yes	characteristic, complemented
28	C₂²⋊C₄	—	11	2	no	—
29	C₄²	—	11	2	no	—
30	C₁₁	#117	1	1	yes	characteristic, complemented, Sylow
31	C₂³	—	11	1	no	—
32	C₂×C₄	—	11	1	no	—
33	C₂×C₄	—	11	4	no	—
34	C₂×C₄	—	11	2	no	—
35	C₂×C₄	C₂×D₁₁	1	2	yes	—
36	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
37	C₂²	—	11	2	no	—
38	C₂²	—	22	1	no	—
39	C₂²	—	22	1	no	—
40	C₂²	C₂²×D₁₁	1	1	yes	characteristic
41	C₄	—	2	2	no	—
42	C₄	—	11	2	no	—
43	C₄	—	22	2	no	—
44	C₄	C₄×D₁₁	1	2	yes	—
45	C₂	—	22	2	no	—
46	C₂	C₂×C₄×D₁₁	1	1	yes	characteristic
47	C₂	#4341	1	1	yes	characteristic
48	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	C₂²×D₁₁	C₂²
hypercenter	C₄⋊C₄	D₁₁
hypocenter	C₁₁	#117
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. C₁₁×C₄⋊C₄	C₂	semidirect
4. C₄⋊Dic₁₁	C₂	semidirect
5. C₄×Dic₁₁	C₂	semidirect
11. C₄×D₁₁	C₄	semidirect
27. C₄⋊C₄	D₁₁	semidirect
30. C₁₁	#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₁	2
C₁₁×C₄⋊C₄	2
(C₂×D₁₁)⋊C₄	2
C₄⋊Dic₁₁	2
C₄×Dic₁₁	2
#117	11