Subgroups — #36442325

Order 22784.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018463	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667918	C₄	1	1	yes	characteristic
5	C₁₆×Dic₈₉	C₄	1	1	yes	characteristic
6	#19151317	—	2	1	no	—
7	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
8	C₆₄×C₈₉	—	2	1	no	—
9	C₂×#13083549	C₈	1	1	yes	characteristic
10	C₈×Dic₈₉	C₈	1	1	yes	characteristic
11	#15667918	—	2	1	no	—
12	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₈₉	D₄	1	1	yes	characteristic
14	C₃₂×C₈₉	Q₈	1	1	yes	characteristic
15	C₂×#10623494	C₁₆	1	1	yes	characteristic
16	C₄×Dic₈₉	C₁₆	1	1	yes	characteristic
17	#13083549	—	2	1	no	—
18	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
20	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
21	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
22	C₂×Dic₈₉	—	2	1	no	—
23	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₈₉	M₃₂	1	1	yes	characteristic
25	#10623494	C₃₂	1	2	yes	—
26	C₂²×C₈₉	M₆₄	1	1	yes	characteristic
27	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
28	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
29	Dic₈₉	—	2	2	no	—
30	#7544	—	89	1	no	maximal, Sylow
31	C₂×C₈₉	—	2	1	no	—
32	C₂×C₈₉	C₄⋊C₃₂	1	1	yes	characteristic
33	M₁₂₈	—	89	1	no	—
34	M₁₂₈	D₈₉	1	1	yes	characteristic
35	C₄×C₃₂	—	89	1	no	—
36	C₈₉	#7544	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃₂	—	89	1	no	—
38	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
39	C₄×C₁₆	—	89	1	no	—
40	C₆₄	—	2	1	no	—
41	C₆₄	—	178	1	no	—
42	C₂×C₁₆	—	89	1	no	—
43	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
44	C₄×C₈	—	89	1	no	—
45	C₃₂	—	178	1	no	—
46	C₃₂	Dic₁₇₈	1	1	yes	characteristic
47	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
48	C₂×C₈	—	89	1	no	—
49	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
50	C₄²	—	89	1	no	—
51	C₁₆	—	178	1	no	—
52	C₁₆	#13083553	1	1	yes	characteristic
53	C₁₆	#13083557	1	1	yes	characteristic
54	C₂×C₄	—	178	1	no	—
55	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
56	C₈	—	89	2	no	—
57	C₈	#15667941	1	1	yes	characteristic
58	C₈	#15667951	1	1	yes	characteristic
59	C₂²	#19151431	1	1	yes	characteristic
60	C₄	—	178	2	no	—
61	C₄	#19151416	1	1	yes	characteristic
62	C₄	C₃₂×D₈₉	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#26018447	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₈₉×M₁₂₈	C₂
socle	C₂×C₈₉	C₄⋊C₃₂
prime radical	C₂²×C₈₉	M₆₄
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	M₁₂₈	D₈₉
hypocenter	C₈₉	#7544
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
36. C₈₉	#7544	semidirect

Maximal subgroups

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

subgroup	index
#26018463	2
C₃₂×Dic₈₉	2
C₈₉×M₁₂₈	2
#7544	89