Subgroups — #66968

Order 352.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#4341	C₂	1	1	yes	maximal, characteristic
3	C₁₁×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#4326	C₂	1	1	yes	maximal, characteristic, complemented
5	#4324	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₄₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#4313	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	D₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂²⋊D₁₁	—	2	1	no	—
11	C₂×Dic₁₁	—	2	1	no	—
12	C₄×D₁₁	—	2	1	no	—
13	C₄×D₁₁	C₂²	1	1	yes	characteristic
14	Dic₂₂	—	2	1	no	—
15	Dic₂₂	C₂²	1	1	yes	characteristic
16	Dic₂₂	C₂²	1	1	yes	characteristic
17	C₈×C₁₁	C₂²	1	1	yes	characteristic
18	C₁₁⋊C₈	C₂²	1	1	yes	characteristic
19	C₂²×C₁₁	—	2	1	no	—
20	C₂×D₁₁	D₄	1	1	yes	characteristic
21	C₄×C₁₁	—	2	1	no	—
22	C₄×C₁₁	C₂³	1	1	yes	characteristic
23	Dic₁₁	—	2	1	no	—
24	Dic₁₁	—	2	1	no	—
25	Dic₁₁	D₄	1	1	yes	characteristic
26	#137	—	11	1	no	maximal, Sylow
27	C₂×C₁₁	—	4	1	no	—
28	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
29	D₁₁	—	2	1	no	—
30	#41	—	11	1	no	—
31	C₂×Q₈	—	11	1	no	—
32	Q₁₆	—	11	1	no	—
33	Q₁₆	—	11	1	no	—
34	SD₁₆	—	11	1	no	—
35	SD₁₆	D₁₁	1	1	yes	characteristic, complemented
36	M₁₆	—	11	1	no	—
37	C₁₁	#137	1	1	yes	characteristic, complemented, Sylow
38	Q₈	—	11	1	no	—
39	Q₈	—	11	1	no	—
40	Q₈	—	22	1	no	—
41	Q₈	C₂×D₁₁	1	1	yes	characteristic
42	D₄	—	22	1	no	—
43	D₄	C₂×D₁₁	1	1	yes	characteristic
44	C₂×C₄	—	11	1	no	—
45	C₂×C₄	—	22	1	no	—
46	C₂×C₄	—	22	1	no	—
47	C₈	—	11	1	no	—
48	C₈	C₂×D₁₁	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	11	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	11	1	no	—
53	C₄	—	22	1	no	—
54	C₄	—	22	1	no	—
55	C₄	C₂²×D₁₁	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	22	1	no	—
58	C₂	D₄×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₁
commutator, cosocle	C₄×C₁₁	C₂³
Frattini	C₄	C₂²×D₁₁
Fitting	C₁₁×SD₁₆	C₂
socle	C₂×C₁₁	C₂×D₄
prime radical	#4341	C₂
hypercenter	SD₁₆	D₁₁
hypocenter	C₁₁	#137
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
1. Q₈×D₁₁	C₂	semidirect
3. C₁₁×SD₁₆	C₂	semidirect
4. #4326	C₂	semidirect
5. #4324	C₂	semidirect
6. Dic₄₄	C₂	semidirect
7. #4313	C₂	semidirect
35. SD₁₆	D₁₁	semidirect
37. C₁₁	#137	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₁₁	2
#4341	2
C₁₁×SD₁₆	2
#4326	2
#4324	2
Dic₄₄	2
#4313	2
#137	11