Subgroups — #29567923

Order 15616.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₄×D₆₁	C₂	1	1	yes	maximal, characteristic
2	#21443638	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₁×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₆₁	C₄	1	1	yes	characteristic
5	#16839481	C₄	1	1	yes	characteristic
6	C₆₁×C₆₄	C₂²	1	1	yes	characteristic
7	C₁₆×D₆₁	C₈	1	1	yes	characteristic
8	#14519683	C₈	1	1	yes	characteristic
9	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
10	C₈×D₆₁	C₁₆	1	1	yes	characteristic
11	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
12	#11758185	C₁₆	1	1	yes	characteristic
13	C₄×D₆₁	C₃₂	1	1	yes	characteristic
14	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
15	#92418	C₃₂	1	1	yes	characteristic
16	M₂₅₆	—	61	1	no	maximal, Sylow
17	C₂×D₆₁	C₆₄	1	1	yes	characteristic
18	C₄×C₆₁	C₂×C₃₂	1	1	yes	characteristic
19	Dic₆₁	C₆₄	1	1	yes	characteristic
20	C₂×C₆₄	—	61	1	no	—
21	C₁₂₈	—	61	1	no	—
22	C₁₂₈	D₆₁	1	1	yes	characteristic, complemented
23	C₂×C₆₁	C₂×C₆₄	1	1	yes	characteristic
24	D₆₁	—	2	1	no	—
25	C₂×C₃₂	—	61	1	no	—
26	C₆₄	—	61	1	no	—
27	C₆₄	C₂×D₆₁	1	1	yes	characteristic
28	C₆₁	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₁₆	—	61	1	no	—
30	C₃₂	—	61	1	no	—
31	C₃₂	C₄×D₆₁	1	1	yes	characteristic
32	C₂×C₈	—	61	1	no	—
33	C₁₆	—	61	1	no	—
34	C₁₆	C₈×D₆₁	1	1	yes	characteristic
35	C₂×C₄	—	61	1	no	—
36	C₈	—	61	1	no	—
37	C₈	C₁₆×D₆₁	1	1	yes	characteristic
38	C₂²	—	61	1	no	—
39	C₄	—	61	1	no	—
40	C₄	C₃₂×D₆₁	1	1	yes	characteristic
41	C₂	—	122	1	no	—
42	C₂	C₆₄×D₆₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₆₁
commutator, socle	C₂×C₆₁	C₂×C₆₄
Fitting	C₆₁×C₁₂₈	C₂
prime radical	C₂×D₆₁	C₆₄
cosocle	C₆₁×C₆₄	C₂²
hypercenter	C₁₂₈	D₆₁
hypocenter	C₆₁	M₂₅₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

Complements

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

N	H ≅ G/N	type
2. #21443638	C₂	semidirect
3. C₆₁×C₁₂₈	C₂	semidirect
22. C₁₂₈	D₆₁	semidirect
28. C₆₁	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₆₄×D₆₁	2
#21443638	2
C₆₁×C₁₂₈	2
M₂₅₆	61