Subgroups — #66936

Order 352.

78 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	C₂×C₄×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₁₁)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₁₁⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₁₁	C₂²	1	1	yes	characteristic
9	C₂²×D₁₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₁	C₂²	1	1	yes	characteristic
12	C₂²⋊D₁₁	—	2	2	no	—
13	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₁	C₂²	1	1	yes	characteristic
16	C₄×D₁₁	—	2	2	no	—
17	C₂²×C₁₁	—	2	1	no	—
18	C₂²×C₁₁	—	2	1	no	—
19	C₂²×C₁₁	—	2	1	no	—
20	C₂²×C₁₁	C₂³	1	1	yes	characteristic
21	C₂×D₁₁	—	2	1	no	—
22	C₂×D₁₁	—	2	1	no	—
23	C₂×D₁₁	D₄	1	2	yes	—
24	C₄×C₁₁	—	2	1	no	—
25	C₄×C₁₁	—	2	1	no	—
26	Dic₁₁	—	2	1	no	—
27	Dic₁₁	—	2	1	no	—
28	Dic₁₁	—	2	1	no	—
29	#123	—	11	1	no	maximal, Sylow
30	C₂×C₁₁	—	4	1	no	—
31	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₁₁	#41	1	1	yes	characteristic
33	C₂×C₁₁	#41	1	1	yes	characteristic
34	D₁₁	—	2	2	no	—
35	C₂×D₄	—	11	1	no	—
36	C₂²×C₄	—	11	1	no	—
37	C₄⋊C₄	—	11	1	no	—
38	C₄⋊C₄	—	11	1	no	—
39	C₂²⋊C₄	—	11	1	no	—
40	C₂²⋊C₄	—	11	1	no	—
41	C₂²⋊C₄	D₁₁	1	1	yes	characteristic, complemented
42	C₁₁	#123	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	11	1	no	—
44	C₂³	C₂×D₁₁	1	1	yes	characteristic
45	D₄	—	22	2	no	—
46	C₂×C₄	—	11	1	no	—
47	C₂×C₄	—	11	1	no	—
48	C₂×C₄	—	11	1	no	—
49	C₂×C₄	—	22	2	no	—
50	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
51	C₂×C₄	C₂×D₁₁	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	11	2	no	—
56	C₂²	—	22	1	no	—
57	C₂²	—	22	1	no	—
58	C₂²	C₂²×D₁₁	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	22	1	no	—
62	C₄	—	22	1	no	—
63	C₄	—	22	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	22	2	no	—
66	C₂	#4339	1	1	yes	characteristic
67	C₂	D₄×D₁₁	1	1	yes	characteristic
68	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	C₂×C₂²⋊D₁₁	C₂
hypercenter	C₂²⋊C₄	D₁₁
hypocenter	C₁₁	#123
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
2. C₂×C₄×D₁₁	C₂	semidirect
3. C₁₁×C₂²⋊C₄	C₂	semidirect
4. C₂²⋊Dic₁₁	C₂	semidirect
5. (C₂×D₁₁)⋊C₄	C₂	semidirect
6. C₄⋊Dic₁₁	C₂	semidirect
7. Dic₁₁⋊C₄	C₂	semidirect
41. C₂²⋊C₄	D₁₁	semidirect
42. C₁₁	#123	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
C₂×C₄×D₁₁	2
C₁₁×C₂²⋊C₄	2
C₂²⋊Dic₁₁	2
(C₂×D₁₁)⋊C₄	2
C₄⋊Dic₁₁	2
Dic₁₁⋊C₄	2
#123	11