Subgroups — #66932

Order 352.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×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	—	2	1	no	—
16	C₂²×C₁₁	—	2	1	no	—
17	C₂²×C₁₁	—	2	1	no	—
18	C₂²×C₁₁	C₂³	1	1	yes	characteristic
19	C₄×C₁₁	—	2	1	no	—
20	C₄×C₁₁	—	2	1	no	—
21	Dic₁₁	—	2	1	no	—
22	Dic₁₁	—	2	1	no	—
23	Dic₁₁	—	2	1	no	—
24	Dic₁₁	—	2	1	no	—
25	#126	—	11	1	no	maximal, Sylow
26	C₂×C₁₁	—	4	1	no	—
27	C₂×C₁₁	#41	1	1	yes	characteristic
28	C₂×C₁₁	#41	1	1	yes	characteristic
29	C₂×C₁₁	#41	1	1	yes	characteristic
30	C₄⋊C₄	—	11	1	no	—
31	C₄⋊C₄	—	11	1	no	—
32	C₄⋊C₄	—	11	1	no	—
33	C₂²⋊C₄	—	11	1	no	—
34	C₂²⋊C₄	—	11	1	no	—
35	C₂²⋊C₄	D₁₁	1	1	yes	characteristic
36	C₄²	—	11	1	no	—
37	C₁₁	#126	1	1	yes	characteristic, complemented, Sylow
38	C₂³	C₂×D₁₁	1	1	yes	characteristic
39	C₂×C₄	—	11	1	no	—
40	C₂×C₄	—	11	1	no	—
41	C₂×C₄	—	11	1	no	—
42	C₂×C₄	—	11	1	no	—
43	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
44	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
45	C₂²	—	2	1	no	—
46	C₂²	—	2	1	no	—
47	C₂²	—	2	1	no	—
48	C₂²	C₂²×D₁₁	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	22	1	no	—
52	C₄	—	22	1	no	—
53	C₄	—	22	1	no	—
54	C₄	—	22	1	no	—
55	C₂	—	4	1	no	—
56	C₂	#4339	1	1	yes	characteristic
57	C₂	#4341	1	1	yes	characteristic
58	C₂	#4341	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₁₁
commutator, socle, cosocle	C₂²×C₁₁	C₂³
Fitting	C₁₁×C₂²⋊C₄	C₂
prime radical	C₂³×C₁₁	C₂²
hypercenter	C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#126
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₄⋊Dic₁₁	C₂	semidirect
5. Dic₁₁⋊C₄	C₂	semidirect
6. Dic₁₁⋊C₄	C₂	semidirect
7. C₄×Dic₁₁	C₂	semidirect
37. C₁₁	#126	semidirect

Maximal subgroups

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

subgroup	index
C₁₁×C₂²⋊C₄	2
C₂²⋊Dic₁₁	2
C₄⋊Dic₁₁	2
Dic₁₁⋊C₄	2
C₄×Dic₁₁	2
#126	11