Subgroups — #37589932

Order 24608.

60 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	C₂×Dic₁₅₃₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₃₀₇₆	C₂	1	2	yes	maximal, complemented
4	#27134370	C₂	1	2	yes	maximal, complemented
5	M₁₆×C₇₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₅₃₈	—	2	1	no	—
7	Dic₁₅₃₈	C₂²	1	2	yes	—
8	Dic₁₅₃₈	C₂²	1	1	yes	characteristic
9	D₃₀₇₆	C₂²	1	1	yes	characteristic
10	C₂²⋊D₇₆₉	—	2	1	no	—
11	C₂×Dic₇₆₉	—	2	1	no	—
12	C₄×D₇₆₉	—	2	1	no	—
13	C₂×C₄×C₇₆₉	C₂²	1	1	yes	characteristic
14	C₈×C₇₆₉	C₂²	1	2	yes	—
15	C₂²×C₇₆₉	D₄	1	1	yes	characteristic
16	C₂×D₇₆₉	—	2	1	no	—
17	C₄×C₇₆₉	D₄	1	1	yes	characteristic
18	C₄×C₇₆₉	C₂³	1	1	yes	characteristic
19	Dic₇₆₉	—	2	1	no	—
20	Dic₇₆₉	—	2	2	no	—
21	C₂×C₇₆₉	—	2	1	no	—
22	C₂×C₇₆₉	C₂×D₄	1	1	yes	characteristic
23	D₇₆₉	—	4	1	no	—
24	C₇₆₉	#137	1	1	yes	characteristic, complemented, Sylow
25	#137	—	769	1	no	maximal, Sylow
26	#41	—	769	1	no	—
27	C₂×Q₈	—	769	1	no	—
28	Q₁₆	—	769	2	no	—
29	SD₁₆	—	769	2	no	—
30	M₁₆	D₇₆₉	1	1	yes	characteristic, complemented
31	Q₈	—	769	2	no	—
32	Q₈	—	769	1	no	—
33	Q₈	—	1538	1	no	—
34	D₄	—	769	1	no	—
35	D₄	—	1538	1	no	—
36	C₂×C₄	—	1538	1	no	—
37	C₂×C₄	—	1538	1	no	—
38	C₂×C₄	C₂×D₇₆₉	1	1	yes	characteristic
39	C₈	C₂×D₇₆₉	1	2	yes	—
40	C₂²	—	1538	1	no	—
41	C₂²	D₃₀₇₆	1	1	yes	characteristic
42	C₄	—	1538	1	no	—
43	C₄	—	1538	2	no	—
44	C₄	D₃₀₇₆	1	1	yes	characteristic
45	C₄	C₂²×D₇₆₉	1	1	yes	characteristic
46	C₂	—	2	1	no	—
47	C₂	—	3076	1	no	—
48	C₂	C₂×D₃₀₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₃₀₇₆
commutator, cosocle	C₄×C₇₆₉	C₂³
Frattini	C₄	C₂²×D₇₆₉
Fitting	M₁₆×C₇₆₉	C₂
socle	C₂×C₇₆₉	C₂×D₄
prime radical	#27134381	C₂
hypercenter	M₁₆	D₇₆₉
hypocenter	C₇₆₉	#137
LCS	G ▷ C₄×C₇₆₉ ▷ C₂×C₇₆₉ ▷ C₇₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₅₃₈	C₂	semidirect
3. Dic₃₀₇₆	C₂	semidirect
4. #27134370	C₂	semidirect
5. M₁₆×C₇₆₉	C₂	semidirect
24. C₇₆₉	#137	semidirect
30. M₁₆	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
C₂×Dic₁₅₃₈	2
Dic₃₀₇₆	2
#27134370	2
M₁₆×C₇₆₉	2
#137	769