Subgroups — #37566924

Order 24352.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131648	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₇₆₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27131638	C₂	1	1	yes	maximal, characteristic, complemented
4	#27131637	C₂	1	1	yes	maximal, characteristic, complemented
5	#27131634	C₂	1	1	yes	maximal, characteristic, complemented
6	#27131632	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₇₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₆₁	—	2	1	no	—
9	Dic₁₅₂₂	C₂²	1	1	yes	characteristic, complemented
10	D₃₀₄₄	C₂²	1	1	yes	characteristic
11	C₂²⋊D₇₆₁	—	2	1	no	—
12	C₂²⋊D₇₆₁	—	2	1	no	—
13	C₂×Dic₇₆₁	—	2	1	no	—
14	C₄×D₇₆₁	C₂²	1	1	yes	characteristic
15	#20259004	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₇₆₁	C₂²	1	1	yes	characteristic
17	D₄×C₇₆₁	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₇₆₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₇₆₁	—	2	1	no	—
20	C₂²×C₇₆₁	—	2	1	no	—
21	C₂×D₇₆₁	—	2	1	no	—
22	C₂×D₇₆₁	—	4	2	no	—
23	C₂×D₇₆₁	D₄	1	1	yes	characteristic
24	C₄×C₇₆₁	C₂³	1	1	yes	characteristic
25	Dic₇₆₁	—	2	1	no	—
26	Dic₇₆₁	D₄	1	1	yes	characteristic
27	C₂×C₇₆₁	—	4	1	no	—
28	C₂×C₇₆₁	—	4	1	no	—
29	C₂×C₇₆₁	C₂×D₄	1	1	yes	characteristic
30	D₇₆₁	—	2	1	no	—
31	D₇₆₁	—	4	1	no	—
32	C₇₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	761	1	no	maximal, Sylow
34	#41	—	761	1	no	—
35	C₂×D₄	—	761	1	no	—
36	SD₁₆	—	761	1	no	—
37	SD₁₆	—	761	1	no	—
38	D₈	—	761	1	no	—
39	D₈	D₇₆₁	1	1	yes	characteristic, complemented
40	M₁₆	—	761	1	no	—
41	C₂³	—	1522	1	no	—
42	Q₈	—	761	1	no	—
43	D₄	—	1522	1	no	—
44	D₄	—	1522	1	no	—
45	D₄	—	761	1	no	—
46	D₄	C₂×D₇₆₁	1	1	yes	characteristic
47	D₄	C₂×D₇₆₁	1	1	yes	characteristic, complemented
48	C₂×C₄	—	1522	1	no	—
49	C₂×C₄	—	761	1	no	—
50	C₈	—	761	1	no	—
51	C₈	C₂×D₇₆₁	1	1	yes	characteristic, complemented
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	3044	2	no	—
55	C₂²	—	1522	1	no	—
56	C₂²	—	761	1	no	—
57	C₄	—	1522	1	no	—
58	C₄	—	761	1	no	—
59	C₄	C₂²×D₇₆₁	1	1	yes	characteristic
60	C₂	—	4	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	3044	1	no	—
63	C₂	—	1522	1	no	—
64	C₂	D₄×D₇₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₇₆₁
commutator, cosocle	C₄×C₇₆₁	C₂³
Frattini	C₄	C₂²×D₇₆₁
Fitting	D₈×C₇₆₁	C₂
socle	C₂×C₇₆₁	C₂×D₄
hypercenter	D₈	D₇₆₁
hypocenter	C₇₆₁	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₆₁ ▷ C₂×C₇₆₁ ▷ C₇₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #27131648	C₂	semidirect
2. D₄×D₇₆₁	C₂	semidirect
3. #27131638	C₂	semidirect
4. #27131637	C₂	semidirect
5. #27131634	C₂	semidirect
6. #27131632	C₂	semidirect
7. D₈×C₇₆₁	C₂	semidirect
9. Dic₁₅₂₂	C₂²	semidirect
15. #20259004	C₂²	semidirect
17. D₄×C₇₆₁	C₂²	semidirect
18. C₈×C₇₆₁	C₂²	semidirect
32. C₇₆₁	AGL(1,ℤ/8)	semidirect
39. D₈	D₇₆₁	semidirect
47. D₄	C₂×D₇₆₁	semidirect
51. C₈	C₂×D₇₆₁	semidirect

Maximal subgroups

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

subgroup	index
#27131648	2
D₄×D₇₆₁	2
#27131638	2
#27131637	2
#27131634	2
#27131632	2
D₈×C₇₆₁	2
AGL(1,ℤ/8)	761