Subgroups — #36442240

Order 22784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022545	C₂	1	1	yes	maximal, characteristic
2	#26018463	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152014	C₄	1	1	yes	characteristic
5	#19151432	C₄	1	1	yes	characteristic
6	#19151431	—	2	1	no	—
7	C₃₂×D₈₉	—	2	1	no	—
8	#19151317	—	2	1	no	—
9	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₆₄×C₈₉	—	2	1	no	—
11	#15668018	C₈	1	1	yes	characteristic
12	#15667952	C₈	1	1	yes	characteristic
13	#15667951	—	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	#13083574	C₁₆	1	1	yes	characteristic
20	#13083558	C₁₆	1	1	yes	characteristic
21	#13083557	—	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	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
28	C₂²⋊D₈₉	—	2	1	no	—
29	D₃₅₆	C₃₂	1	1	yes	characteristic
30	C₄×D₈₉	—	2	1	no	—
31	Dic₁₇₈	C₃₂	1	1	yes	characteristic
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	M₆₄	1	1	yes	characteristic
36	C₂×D₈₉	—	2	1	no	—
37	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
39	Dic₈₉	—	2	1	no	—
40	#7514	—	89	1	no	maximal, Sylow
41	C₂×C₈₉	—	2	1	no	—
42	C₂×C₈₉	C₂²⋊C₃₂	1	1	yes	characteristic
43	D₈₉	—	4	1	no	—
44	#2258	—	89	1	no	—
45	M₁₂₈	—	89	1	no	—
46	C₂×C₆₄	D₈₉	1	1	yes	characteristic, complemented
47	C₈₉	#7514	1	1	yes	characteristic, complemented, Sylow
48	#504	—	89	1	no	—
49	M₆₄	—	89	1	no	—
50	M₆₄	—	178	1	no	—
51	C₂×C₃₂	—	178	1	no	—
52	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
53	C₆₄	—	2	1	no	—
54	C₆₄	—	178	1	no	—
55	#131	—	89	1	no	—
56	M₃₂	—	89	1	no	—
57	M₃₂	—	178	1	no	—
58	C₂×C₁₆	—	178	1	no	—
59	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
60	C₃₂	—	178	1	no	—
61	C₃₂	D₃₅₆	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
63	#41	—	89	1	no	—
64	M₁₆	—	89	1	no	—
65	M₁₆	—	178	1	no	—
66	C₂×C₈	—	178	1	no	—
67	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
68	C₁₆	—	178	1	no	—
69	C₁₆	#13083551	1	1	yes	characteristic
70	C₁₆	#13083557	1	1	yes	characteristic
71	Q₈	—	89	1	no	—
72	D₄	—	89	1	no	—
73	D₄	—	178	1	no	—
74	C₂×C₄	—	178	1	no	—
75	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
76	C₈	—	178	1	no	—
77	C₈	#15667924	1	1	yes	characteristic
78	C₈	#15667951	1	1	yes	characteristic
79	C₂²	—	178	1	no	—
80	C₂²	#19151431	1	1	yes	characteristic
81	C₄	—	178	1	no	—
82	C₄	#19151377	1	1	yes	characteristic
83	C₄	C₃₂×D₈₉	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	356	1	no	—
86	C₂	#26018389	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	D₃₅₆
commutator	C₄×C₈₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₂×C₆₄×C₈₉	C₂
socle	C₂×C₈₉	C₂²⋊C₃₂
prime radical	#13083574	C₁₆
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₂×C₆₄	D₈₉
hypocenter	C₈₉	#7514
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #26018463	C₂	semidirect
3. C₂×C₆₄×C₈₉	C₂	semidirect
46. C₂×C₆₄	D₈₉	semidirect
47. C₈₉	#7514	semidirect

Maximal subgroups

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

subgroup	index
#26022545	2
#26018463	2
C₂×C₆₄×C₈₉	2
#7514	89