Subgroups — #36442338

Order 22784.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₄×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26018071	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₈₉	C₄	1	1	yes	characteristic
5	#19151317	C₄	1	1	yes	characteristic
6	C₆₄×C₈₉	C₂²	1	1	yes	characteristic
7	C₁₆×D₈₉	C₈	1	1	yes	characteristic
8	#15667918	C₈	1	1	yes	characteristic
9	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
10	C₈×D₈₉	C₁₆	1	1	yes	characteristic
11	#13083549	C₁₆	1	1	yes	characteristic
12	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
13	C₄×D₈₉	C₃₂	1	1	yes	characteristic
14	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
15	#10623494	C₃₂	1	1	yes	characteristic
16	C₂×D₈₉	C₆₄	1	1	yes	characteristic
17	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
18	Dic₈₉	C₆₄	1	1	yes	characteristic
19	M₂₅₆	—	89	1	no	maximal, Sylow
20	C₂×C₈₉	C₂×C₆₄	1	1	yes	characteristic
21	D₈₉	—	2	1	no	—
22	C₂×C₆₄	—	89	1	no	—
23	C₁₂₈	—	89	1	no	—
24	C₁₂₈	D₈₉	1	1	yes	characteristic, complemented
25	C₈₉	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃₂	—	89	1	no	—
27	C₆₄	—	89	1	no	—
28	C₆₄	C₂×D₈₉	1	1	yes	characteristic
29	C₂×C₁₆	—	89	1	no	—
30	C₃₂	—	89	1	no	—
31	C₃₂	C₄×D₈₉	1	1	yes	characteristic
32	C₂×C₈	—	89	1	no	—
33	C₁₆	—	89	1	no	—
34	C₁₆	C₈×D₈₉	1	1	yes	characteristic
35	C₂×C₄	—	89	1	no	—
36	C₈	—	89	1	no	—
37	C₈	C₁₆×D₈₉	1	1	yes	characteristic
38	C₂²	—	89	1	no	—
39	C₄	—	89	1	no	—
40	C₄	C₃₂×D₈₉	1	1	yes	characteristic
41	C₂	—	178	1	no	—
42	C₂	C₆₄×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₈₉
commutator, socle	C₂×C₈₉	C₂×C₆₄
Fitting	C₈₉×C₁₂₈	C₂
prime radical	C₂×D₈₉	C₆₄
cosocle	C₆₄×C₈₉	C₂²
hypercenter	C₁₂₈	D₈₉
hypocenter	C₈₉	M₂₅₆
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

Complements

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

N	H ≅ G/N	type
2. #26018071	C₂	semidirect
3. C₈₉×C₁₂₈	C₂	semidirect
24. C₁₂₈	D₈₉	semidirect
25. C₈₉	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₆₄×D₈₉	2
#26018071	2
C₈₉×C₁₂₈	2
M₂₅₆	89