Subgroups — #37589925

Order 24608.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27134381	C₂	1	1	yes	maximal, characteristic, complemented
2	#27134369	C₂	1	2	yes	maximal, complemented
3	#27134366	C₂	1	2	yes	maximal, complemented
4	#27134365	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×C₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₅₃₈	C₂²	1	1	yes	characteristic, complemented
7	D₃₀₇₆	C₂²	1	1	yes	characteristic, complemented
8	C₂²⋊D₇₆₉	—	2	1	no	—
9	C₄×D₇₆₉	—	2	1	no	—
10	#20259670	C₂²	1	2	yes	complemented
11	C₂³×C₇₆₉	—	2	1	no	—
12	D₄×C₇₆₉	—	2	1	no	—
13	D₄×C₇₆₉	C₂²	1	2	yes	—
14	C₂×C₄×C₇₆₉	C₂²	1	1	yes	characteristic
15	C₂²×C₇₆₉	—	2	2	no	—
16	C₂²×C₇₆₉	—	4	2	no	—
17	C₂²×C₇₆₉	D₄	1	1	yes	characteristic
18	C₂×D₇₆₉	—	2	1	no	—
19	C₄×C₇₆₉	D₄	1	1	yes	characteristic
20	C₄×C₇₆₉	C₂³	1	1	yes	characteristic
21	Dic₇₆₉	—	2	1	no	—
22	C₂×C₇₆₉	—	2	1	no	—
23	C₂×C₇₆₉	—	4	2	no	—
24	C₂×C₇₆₉	C₂×D₄	1	1	yes	characteristic
25	D₇₆₉	—	4	1	no	—
26	C₇₆₉	AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
27	AGL(1,ℤ/8)	—	769	1	no	maximal, Sylow
28	#41	—	769	1	no	—
29	C₂×D₄	D₇₆₉	1	1	yes	characteristic, complemented
30	SD₁₆	—	769	2	no	—
31	D₈	—	769	2	no	—
32	M₁₆	—	769	1	no	—
33	C₂³	—	2	1	no	—
34	Q₈	—	769	1	no	—
35	D₄	—	769	1	no	—
36	D₄	—	2	1	no	—
37	D₄	—	1538	1	no	—
38	D₄	C₂×D₇₆₉	1	2	yes	—
39	C₂×C₄	—	1538	1	no	—
40	C₂×C₄	C₂×D₇₆₉	1	1	yes	characteristic
41	C₈	—	769	2	no	—
42	C₂²	—	2	2	no	—
43	C₂²	—	1538	1	no	—
44	C₂²	—	4	2	no	—
45	C₂²	C₂²⋊D₇₆₉	1	1	yes	characteristic
46	C₄	—	1538	1	no	—
47	C₄	C₂²⋊D₇₆₉	1	1	yes	characteristic
48	C₄	C₂²×D₇₆₉	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	2	no	—
51	C₂	—	3076	1	no	—
52	C₂	C₂×C₂²⋊D₇₆₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #27134381	C₂	semidirect
2. #27134369	C₂	semidirect
3. #27134366	C₂	semidirect
4. #27134365	C₂	semidirect
5. C₂×D₄×C₇₆₉	C₂	semidirect
6. Dic₁₅₃₈	C₂²	semidirect
7. D₃₀₇₆	C₂²	semidirect
10. #20259670	C₂²	semidirect
26. C₇₆₉	AGL(1,ℤ/8)	semidirect
29. C₂×D₄	D₇₆₉	semidirect

Maximal subgroups

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

subgroup	index
#27134381	2
#27134369	2
#27134366	2
#27134365	2
C₂×D₄×C₇₆₉	2
AGL(1,ℤ/8)	769