Subgroups — #66967

Order 352.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4343	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₁×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#4325	C₂	1	1	yes	maximal, characteristic, complemented
5	#4323	C₂	1	1	yes	maximal, characteristic, complemented
6	D₈₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#4313	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₁₁	—	2	1	no	—
9	Q₈×C₁₁	C₂²	1	1	yes	characteristic, complemented
10	D₄×C₁₁	C₂²	1	1	yes	characteristic
11	C₂²⋊D₁₁	—	2	1	no	—
12	D₄₄	—	2	1	no	—
13	D₄₄	C₂²	1	1	yes	characteristic
14	D₄₄	C₂²	1	1	yes	characteristic, complemented
15	C₄×D₁₁	—	2	1	no	—
16	C₄×D₁₁	C₂²	1	1	yes	characteristic
17	C₈×C₁₁	C₂²	1	1	yes	characteristic, complemented
18	C₁₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₁₁	—	2	1	no	—
20	C₂×D₁₁	—	2	1	no	—
21	C₂×D₁₁	—	2	1	no	—
22	C₂×D₁₁	—	4	2	no	—
23	C₂×D₁₁	D₄	1	1	yes	characteristic
24	C₄×C₁₁	—	2	1	no	—
25	C₄×C₁₁	C₂³	1	1	yes	characteristic
26	Dic₁₁	D₄	1	1	yes	characteristic
27	AGL(1,ℤ/8)	—	11	1	no	maximal, Sylow
28	C₂×C₁₁	—	4	1	no	—
29	C₂×C₁₁	C₂×D₄	1	1	yes	characteristic
30	D₁₁	—	2	1	no	—
31	D₁₁	—	4	1	no	—
32	D₁₁	—	4	1	no	—
33	#41	—	11	1	no	—
34	C₂×D₄	—	11	1	no	—
35	SD₁₆	—	11	1	no	—
36	SD₁₆	D₁₁	1	1	yes	characteristic, complemented
37	D₈	—	11	1	no	—
38	D₈	—	11	1	no	—
39	M₁₆	—	11	1	no	—
40	C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
41	C₂³	—	22	1	no	—
42	Q₈	C₂×D₁₁	1	1	yes	characteristic, complemented
43	D₄	—	11	1	no	—
44	D₄	—	11	1	no	—
45	D₄	—	22	1	no	—
46	D₄	—	22	1	no	—
47	D₄	C₂×D₁₁	1	1	yes	characteristic
48	C₂×C₄	—	11	1	no	—
49	C₂×C₄	—	22	1	no	—
50	C₈	—	11	1	no	—
51	C₈	C₂×D₁₁	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	11	1	no	—
54	C₂²	—	22	1	no	—
55	C₂²	—	22	1	no	—
56	C₂²	—	44	2	no	—
57	C₄	—	2	1	no	—
58	C₄	—	11	1	no	—
59	C₄	C₂²×D₁₁	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	22	1	no	—
62	C₂	—	44	1	no	—
63	C₂	—	44	1	no	—
64	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
hypercenter	SD₁₆	D₁₁
hypocenter	C₁₁	AGL(1,ℤ/8)
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. #4343	C₂	semidirect
2. D₄×D₁₁	C₂	semidirect
3. C₁₁×SD₁₆	C₂	semidirect
4. #4325	C₂	semidirect
5. #4323	C₂	semidirect
6. D₈₈	C₂	semidirect
7. #4313	C₂	semidirect
9. Q₈×C₁₁	C₂²	semidirect
14. D₄₄	C₂²	semidirect
17. C₈×C₁₁	C₂²	semidirect
18. C₁₁⋊C₈	C₂²	semidirect
36. SD₁₆	D₁₁	semidirect
40. C₁₁	AGL(1,ℤ/8)	semidirect
42. Q₈	C₂×D₁₁	semidirect
51. C₈	C₂×D₁₁	semidirect

Maximal subgroups

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

subgroup	index
#4343	2
D₄×D₁₁	2
C₁₁×SD₁₆	2
#4325	2
#4323	2
D₈₈	2
#4313	2
AGL(1,ℤ/8)	11