Subgroups — #66942

Order 352.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic
4	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic
5	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic
6	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic
7	C₄×Dic₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×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	C₂²	1	1	yes	characteristic
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	C₂²×C₁₁	C₂³	1	1	yes	characteristic
16	C₄×C₁₁	—	2	1	no	—
17	C₄×C₁₁	—	2	1	no	—
18	C₄×C₁₁	—	2	1	no	—
19	Dic₁₁	—	2	1	no	—
20	Dic₁₁	—	2	1	no	—
21	Dic₁₁	—	2	1	no	—
22	Dic₁₁	Q₈	1	2	yes	—
23	#125	—	11	1	no	maximal, Sylow
24	C₂×C₁₁	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₁₁	#41	1	1	yes	characteristic
26	C₂×C₁₁	#41	1	1	yes	characteristic
27	C₄⋊C₄	—	11	1	no	—
28	C₄⋊C₄	—	11	1	no	—
29	C₄⋊C₄	—	11	1	no	—
30	C₄⋊C₄	—	11	1	no	—
31	C₄⋊C₄	—	11	1	no	—
32	C₄⋊C₄	D₁₁	1	1	yes	characteristic
33	C₄²	—	11	1	no	—
34	C₁₁	#125	1	1	yes	characteristic, complemented, Sylow
35	C₂×C₄	—	11	1	no	—
36	C₂×C₄	—	11	1	no	—
37	C₂×C₄	—	11	1	no	—
38	C₂×C₄	—	11	1	no	—
39	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
40	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
41	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
42	C₂²	C₂²×D₁₁	1	1	yes	characteristic
43	C₄	—	2	1	no	—
44	C₄	—	2	1	no	—
45	C₄	—	2	1	no	—
46	C₄	—	11	2	no	—
47	C₄	—	22	1	no	—
48	C₄	—	22	1	no	—
49	C₄	—	22	1	no	—
50	C₂	#4339	1	1	yes	characteristic
51	C₂	#4341	1	1	yes	characteristic
52	C₂	Q₈×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator, socle, prime radical, cosocle	C₂²×C₁₁	C₂³
Fitting	C₁₁×C₄⋊C₄	C₂
hypercenter	C₄⋊C₄	D₁₁
hypocenter	C₁₁	#125
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
34. C₁₁	#125	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×C₄⋊C₄	2
C₄⋊Dic₁₁	2
Dic₁₁⋊C₄	2
C₄×Dic₁₁	2
#125	11