Subgroups — #37589928

Order 24608.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27134380	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃₀₇₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#27134371	C₂	1	1	yes	maximal, characteristic, complemented
5	#27134369	C₂	1	1	yes	maximal, characteristic, complemented
6	#27134364	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₅₃₈	—	2	1	no	—
9	Dic₁₅₃₈	C₂²	1	1	yes	characteristic
10	Dic₁₅₃₈	C₂²	1	1	yes	characteristic
11	C₂²⋊D₇₆₉	—	2	1	no	—
12	C₂×Dic₇₆₉	—	2	1	no	—
13	C₄×D₇₆₉	—	2	1	no	—
14	C₄×D₇₆₉	C₂²	1	1	yes	characteristic
15	#20259670	C₂²	1	1	yes	characteristic
16	Q₈×C₇₆₉	C₂²	1	1	yes	characteristic
17	D₄×C₇₆₉	C₂²	1	1	yes	characteristic
18	C₈×C₇₆₉	C₂²	1	1	yes	characteristic
19	C₂²×C₇₆₉	—	2	1	no	—
20	C₂×D₇₆₉	D₄	1	1	yes	characteristic
21	C₄×C₇₆₉	—	2	1	no	—
22	C₄×C₇₆₉	C₂³	1	1	yes	characteristic
23	Dic₇₆₉	—	2	1	no	—
24	Dic₇₆₉	—	2	1	no	—
25	Dic₇₆₉	D₄	1	1	yes	characteristic
26	C₂×C₇₆₉	—	4	1	no	—
27	C₂×C₇₆₉	C₂×D₄	1	1	yes	characteristic
28	D₇₆₉	—	2	1	no	—
29	C₇₆₉	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	769	1	no	maximal, Sylow
31	#41	—	769	1	no	—
32	C₂×Q₈	—	769	1	no	—
33	Q₁₆	—	769	1	no	—
34	Q₁₆	—	769	1	no	—
35	SD₁₆	—	769	1	no	—
36	SD₁₆	D₇₆₉	1	1	yes	characteristic, complemented
37	M₁₆	—	769	1	no	—
38	Q₈	—	769	1	no	—
39	Q₈	—	769	1	no	—
40	Q₈	—	1538	1	no	—
41	Q₈	C₂×D₇₆₉	1	1	yes	characteristic
42	D₄	—	1538	1	no	—
43	D₄	C₂×D₇₆₉	1	1	yes	characteristic
44	C₂×C₄	—	769	1	no	—
45	C₂×C₄	—	1538	1	no	—
46	C₂×C₄	—	1538	1	no	—
47	C₈	—	769	1	no	—
48	C₈	C₂×D₇₆₉	1	1	yes	characteristic
49	C₂²	—	769	1	no	—
50	C₂²	—	2	1	no	—
51	C₄	—	769	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	1538	1	no	—
54	C₄	—	1538	1	no	—
55	C₄	C₂²×D₇₆₉	1	1	yes	characteristic
56	C₂	—	1538	1	no	—
57	C₂	—	4	1	no	—
58	C₂	D₄×D₇₆₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₇₆₉
commutator, cosocle	C₄×C₇₆₉	C₂³
Frattini	C₄	C₂²×D₇₆₉
Fitting	SD₁₆×C₇₆₉	C₂
socle	C₂×C₇₆₉	C₂×D₄
prime radical	#27134380	C₂
hypercenter	SD₁₆	D₇₆₉
hypocenter	C₇₆₉	#137
LCS	G ▷ C₄×C₇₆₉ ▷ C₂×C₇₆₉ ▷ C₇₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₇₆₉	C₂	semidirect
3. Dic₃₀₇₆	C₂	semidirect
4. #27134371	C₂	semidirect
5. #27134369	C₂	semidirect
6. #27134364	C₂	semidirect
7. SD₁₆×C₇₆₉	C₂	semidirect
29. C₇₆₉	#137	semidirect
36. SD₁₆	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
Q₈×D₇₆₉	2
Dic₃₀₇₆	2
#27134371	2
#27134369	2
#27134364	2
SD₁₆×C₇₆₉	2
#137	769