Subgroups — #37589905

Order 24608.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27134381	C₂	1	1	yes	maximal, characteristic
2	#27134365	C₂	1	1	yes	maximal, characteristic, complemented
3	#27134364	C₂	1	2	yes	maximal, complemented
4	C₈×D₇₆₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×C₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₅₃₈	C₄	1	1	yes	characteristic
7	D₃₀₇₆	C₄	1	1	yes	characteristic
8	C₂²⋊D₇₆₉	C₄	1	2	yes	—
9	C₄×D₇₆₉	C₂²	1	2	yes	—
10	#20259670	C₂²	1	2	yes	—
11	C₂×C₄×C₇₆₉	C₂²	1	1	yes	characteristic
12	C₈×C₇₆₉	C₂²	1	1	yes	characteristic
13	C₈×C₇₆₉	C₂²	1	1	yes	characteristic
14	C₂²×C₇₆₉	C₂×C₄	1	1	yes	characteristic
15	C₂×D₇₆₉	C₂×C₄	1	2	yes	—
16	C₄×C₇₆₉	C₂×C₄	1	1	yes	characteristic
17	C₄×C₇₆₉	C₂³	1	1	yes	characteristic
18	Dic₇₆₉	C₂×C₄	1	2	yes	—
19	C₂×C₇₆₉	—	2	1	no	—
20	C₂×C₇₆₉	C₂²×C₄	1	1	yes	characteristic
21	D₇₆₉	—	2	2	no	—
22	C₇₆₉	#131	1	1	yes	characteristic, complemented, Sylow
23	#131	—	769	1	no	maximal, Sylow
24	#41	—	769	1	no	—
25	M₁₆	—	769	1	no	—
26	M₁₆	—	769	2	no	—
27	C₂×C₈	—	769	2	no	—
28	C₂×C₈	D₇₆₉	1	1	yes	characteristic, complemented
29	Q₈	—	769	1	no	—
30	D₄	—	769	2	no	—
31	D₄	—	769	1	no	—
32	C₂×C₄	—	769	2	no	—
33	C₂×C₄	C₂×D₇₆₉	1	1	yes	characteristic
34	C₈	—	769	2	no	—
35	C₈	C₂×D₇₆₉	1	1	yes	characteristic
36	C₈	C₂×D₇₆₉	1	1	yes	characteristic
37	C₂²	—	769	2	no	—
38	C₂²	C₄×D₇₆₉	1	1	yes	characteristic
39	C₄	—	769	2	no	—
40	C₄	C₄×D₇₆₉	1	1	yes	characteristic
41	C₄	C₂²×D₇₆₉	1	1	yes	characteristic
42	C₂	—	2	1	no	—
43	C₂	—	1538	2	no	—
44	C₂	C₂×C₄×D₇₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₇₆₉
commutator, socle	C₂×C₇₆₉	C₂²×C₄
Frattini	C₄	C₂²×D₇₆₉
Fitting	C₂×C₈×C₇₆₉	C₂
prime radical	#27134381	C₂
cosocle	C₄×C₇₆₉	C₂³
hypercenter	C₂×C₈	D₇₆₉
hypocenter	C₇₆₉	#131
LCS	G ▷ C₂×C₇₆₉ ▷ C₇₆₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #27134365	C₂	semidirect
3. #27134364	C₂	semidirect
4. C₈×D₇₆₉	C₂	semidirect
5. C₂×C₈×C₇₆₉	C₂	semidirect
22. C₇₆₉	#131	semidirect
28. C₂×C₈	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
#27134365	2
#27134364	2
C₈×D₇₆₉	2
C₂×C₈×C₇₆₉	2
#131	769