Subgroups — #36442238

Order 22784.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151431	C₂	1	1	yes	maximal, characteristic
2	#26018463	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×D₈₉	C₄	1	1	yes	characteristic
5	#19151431	—	2	1	no	—
6	#19151431	—	2	1	no	—
7	C₂×#15667918	C₄	1	1	yes	characteristic
8	#19151317	—	2	1	no	—
9	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₆₄×C₈₉	—	2	1	no	—
11	C₂×C₈×D₈₉	C₈	1	1	yes	characteristic
12	C₂×#13083549	C₈	1	1	yes	characteristic
13	C₁₆×D₈₉	—	2	1	no	—
14	C₁₆×D₈₉	—	2	1	no	—
15	#15667918	—	2	1	no	—
16	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
17	C₃₂×C₈₉	D₄	1	1	yes	characteristic
18	C₃₂×C₈₉	D₄	1	1	yes	characteristic
19	C₂×C₄×D₈₉	C₁₆	1	1	yes	characteristic
20	C₂×#10623494	C₁₆	1	1	yes	characteristic
21	C₈×D₈₉	—	2	1	no	—
22	C₈×D₈₉	—	2	1	no	—
23	#13083549	—	2	1	no	—
24	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
25	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
27	C₂²×D₈₉	C₃₂	1	1	yes	characteristic
28	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
29	C₂×Dic₈₉	C₃₂	1	1	yes	characteristic
30	C₄×D₈₉	—	2	1	no	—
31	C₄×D₈₉	—	2	1	no	—
32	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
33	C₈×C₈₉	M₃₂	1	1	yes	characteristic
34	#10623494	—	2	1	no	—
35	C₂²×C₈₉	C₂×C₃₂	1	1	yes	characteristic
36	C₂×D₈₉	—	2	1	no	—
37	C₂×D₈₉	—	4	1	no	—
38	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₈₉	M₆₄	1	1	yes	characteristic
40	Dic₈₉	—	2	1	no	—
41	#7513	—	89	1	no	maximal, Sylow
42	C₂×C₈₉	—	2	1	no	—
43	C₂×C₈₉	C₂²⋊C₃₂	1	1	yes	characteristic
44	D₈₉	—	4	1	no	—
45	C₂×M₆₄	—	89	1	no	—
46	M₁₂₈	—	89	1	no	—
47	M₁₂₈	D₈₉	1	1	yes	characteristic, complemented
48	C₈₉	#7513	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₁₆	—	89	1	no	—
50	M₆₄	—	178	1	no	—
51	M₆₄	—	178	1	no	—
52	C₂×C₃₂	—	89	1	no	—
53	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
54	C₆₄	—	2	1	no	—
55	C₆₄	—	178	1	no	—
56	C₂²×C₈	—	89	1	no	—
57	C₂×C₁₆	—	89	1	no	—
58	C₂×C₁₆	—	178	1	no	—
59	C₂×C₁₆	—	178	1	no	—
60	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
61	C₃₂	—	178	1	no	—
62	C₃₂	D₃₅₆	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
64	C₂²×C₄	—	89	1	no	—
65	C₂×C₈	—	89	1	no	—
66	C₂×C₈	—	178	1	no	—
67	C₂×C₈	—	178	1	no	—
68	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
69	C₁₆	—	178	1	no	—
70	C₁₆	#13083551	1	1	yes	characteristic
71	C₁₆	#13083557	1	1	yes	characteristic
72	C₂³	—	89	1	no	—
73	C₂×C₄	—	89	1	no	—
74	C₂×C₄	—	178	1	no	—
75	C₂×C₄	—	178	1	no	—
76	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
77	C₈	—	178	1	no	—
78	C₈	#15667924	1	1	yes	characteristic
79	C₈	#15667951	1	1	yes	characteristic
80	C₂²	—	356	1	no	—
81	C₂²	—	178	1	no	—
82	C₂²	C₃₂×D₈₉	1	1	yes	characteristic
83	C₄	—	178	1	no	—
84	C₄	#19151377	1	1	yes	characteristic
85	C₄	#19151431	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	356	1	no	—
88	C₂	#26018389	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#13083551
commutator	C₂²×C₈₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₈₉×M₁₂₈	C₂
socle	C₂×C₈₉	C₂²⋊C₃₂
prime radical	C₂²×D₈₉	C₃₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	M₁₂₈	D₈₉
hypocenter	C₈₉	#7513
LCS	G ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #26018463	C₂	semidirect
3. C₈₉×M₁₂₈	C₂	semidirect
47. M₁₂₈	D₈₉	semidirect
48. C₈₉	#7513	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151431	2
#26018463	2
C₈₉×M₁₂₈	2
#7513	89