Subgroups — #37589923

Order 24608.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27134380	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×D₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#27134370	C₂	1	1	yes	maximal, characteristic, complemented
4	#27134369	C₂	1	1	yes	maximal, characteristic, complemented
5	#27134366	C₂	1	1	yes	maximal, characteristic, complemented
6	#27134364	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×C₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₆₉	—	2	1	no	—
9	Dic₁₅₃₈	C₂²	1	1	yes	characteristic, complemented
10	D₃₀₇₆	C₂²	1	1	yes	characteristic
11	C₂²⋊D₇₆₉	—	2	1	no	—
12	C₂²⋊D₇₆₉	—	2	1	no	—
13	C₂×Dic₇₆₉	—	2	1	no	—
14	C₄×D₇₆₉	C₂²	1	1	yes	characteristic
15	#20259670	C₂²	1	1	yes	characteristic, complemented
16	D₄×C₇₆₉	C₂²	1	1	yes	characteristic
17	D₄×C₇₆₉	C₂²	1	1	yes	characteristic, complemented
18	C₈×C₇₆₉	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₇₆₉	—	2	1	no	—
20	C₂²×C₇₆₉	—	2	1	no	—
21	C₂×D₇₆₉	—	2	1	no	—
22	C₂×D₇₆₉	—	4	2	no	—
23	C₂×D₇₆₉	D₄	1	1	yes	characteristic
24	C₄×C₇₆₉	C₂³	1	1	yes	characteristic
25	Dic₇₆₉	—	2	1	no	—
26	Dic₇₆₉	D₄	1	1	yes	characteristic
27	C₂×C₇₆₉	—	4	1	no	—
28	C₂×C₇₆₉	—	4	1	no	—
29	C₂×C₇₆₉	C₂×D₄	1	1	yes	characteristic
30	D₇₆₉	—	2	1	no	—
31	D₇₆₉	—	4	1	no	—
32	C₇₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
33	AGL(1,ℤ/8)	—	769	1	no	maximal, Sylow
34	#41	—	769	1	no	—
35	C₂×D₄	—	769	1	no	—
36	SD₁₆	—	769	1	no	—
37	SD₁₆	—	769	1	no	—
38	D₈	—	769	1	no	—
39	D₈	D₇₆₉	1	1	yes	characteristic, complemented
40	M₁₆	—	769	1	no	—
41	C₂³	—	1538	1	no	—
42	Q₈	—	769	1	no	—
43	D₄	—	769	1	no	—
44	D₄	—	1538	1	no	—
45	D₄	—	1538	1	no	—
46	D₄	C₂×D₇₆₉	1	1	yes	characteristic
47	D₄	C₂×D₇₆₉	1	1	yes	characteristic, complemented
48	C₂×C₄	—	769	1	no	—
49	C₂×C₄	—	1538	1	no	—
50	C₈	—	769	1	no	—
51	C₈	C₂×D₇₆₉	1	1	yes	characteristic, complemented
52	C₂²	—	769	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	1538	1	no	—
56	C₂²	—	3076	2	no	—
57	C₄	—	769	1	no	—
58	C₄	—	1538	1	no	—
59	C₄	C₂²×D₇₆₉	1	1	yes	characteristic
60	C₂	—	1538	1	no	—
61	C₂	—	4	1	no	—
62	C₂	—	4	1	no	—
63	C₂	—	3076	1	no	—
64	C₂	D₄×D₇₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₇₆₉
commutator, cosocle	C₄×C₇₆₉	C₂³
Frattini	C₄	C₂²×D₇₆₉
Fitting	D₈×C₇₆₉	C₂
socle	C₂×C₇₆₉	C₂×D₄
hypercenter	D₈	D₇₆₉
hypocenter	C₇₆₉	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇₆₉ ▷ C₂×C₇₆₉ ▷ C₇₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #27134380	C₂	semidirect
2. D₄×D₇₆₉	C₂	semidirect
3. #27134370	C₂	semidirect
4. #27134369	C₂	semidirect
5. #27134366	C₂	semidirect
6. #27134364	C₂	semidirect
7. D₈×C₇₆₉	C₂	semidirect
9. Dic₁₅₃₈	C₂²	semidirect
15. #20259670	C₂²	semidirect
17. D₄×C₇₆₉	C₂²	semidirect
18. C₈×C₇₆₉	C₂²	semidirect
32. C₇₆₉	AGL(1,ℤ/8)	semidirect
39. D₈	D₇₆₉	semidirect
47. D₄	C₂×D₇₆₉	semidirect
51. C₈	C₂×D₇₆₉	semidirect

Maximal subgroups

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

subgroup	index
#27134380	2
D₄×D₇₆₉	2
#27134370	2
#27134369	2
#27134366	2
#27134364	2
D₈×C₇₆₉	2
AGL(1,ℤ/8)	769