Subgroups — #36442237

Order 22784.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151317	C₂	1	2	yes	maximal, complemented
2	C₂²×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151317	—	2	2	no	—
4	C₂²×C₁₆×C₈₉	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₈₉	—	2	2	no	—
6	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₈₉	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₈₉	—	2	1	no	—
10	C₂×C₁₆×C₈₉	—	2	1	no	—
11	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₃₂×C₈₉	—	2	1	no	—
14	C₃₂×C₈₉	D₄	1	2	yes	—
15	C₂²×C₄×C₈₉	C₁₆	1	1	yes	characteristic
16	C₂×C₈×C₈₉	—	2	1	no	—
17	C₂×C₈×C₈₉	—	2	1	no	—
18	C₂×C₈×C₈₉	C₁₆	1	1	yes	characteristic
19	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
20	C₁₆×C₈₉	—	2	1	no	—
21	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
23	C₂³×C₈₉	C₃₂	1	1	yes	characteristic
24	C₂×C₄×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈₉	—	2	1	no	—
26	C₂×C₄×C₈₉	C₃₂	1	1	yes	characteristic
27	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₈₉	—	2	1	no	—
29	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₈×C₈₉	M₃₂	1	1	yes	characteristic
31	C₂²×C₈₉	—	2	1	no	—
32	C₂²×C₈₉	—	2	1	no	—
33	C₂²×C₈₉	C₆₄	1	2	yes	complemented
34	C₂²×C₈₉	C₂×C₃₂	1	1	yes	characteristic
35	C₄×C₈₉	—	2	1	no	—
36	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
37	C₄×C₈₉	M₆₄	1	1	yes	characteristic
38	C₂²⋊C₆₄	—	89	1	no	maximal, Sylow
39	C₂×C₈₉	—	2	2	no	—
40	C₂×C₈₉	C₂²⋊C₃₂	1	1	yes	characteristic
41	C₂×C₈₉	C₂×C₆₄	1	1	yes	characteristic
42	C₂×C₈₉	M₁₂₈	1	1	yes	characteristic
43	C₂²×C₃₂	D₈₉	1	1	yes	characteristic
44	C₂×C₆₄	—	89	2	no	—
45	C₈₉	C₂²⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₁₆	Dic₈₉	1	1	yes	characteristic
47	C₂×C₃₂	—	2	2	no	—
48	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
49	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
50	C₆₄	—	178	2	no	—
51	C₂²×C₈	#10623494	1	1	yes	characteristic
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	#10623494	1	1	yes	characteristic
55	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
56	C₃₂	—	2	1	no	—
57	C₃₂	C₂²⋊D₈₉	1	2	yes	—
58	C₂²×C₄	#13083549	1	1	yes	characteristic
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	#13083549	1	1	yes	characteristic
62	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
63	C₁₆	—	2	1	no	—
64	C₁₆	C₂²⋊Dic₈₉	1	1	yes	characteristic
65	C₁₆	#13083558	1	1	yes	characteristic
66	C₂³	#15667918	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	#15667918	1	1	yes	characteristic
70	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	C₂²⋊#10623494	1	1	yes	characteristic
73	C₈	#15667952	1	1	yes	characteristic
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	#19151317	1	2	yes	complemented
77	C₂²	C₂×#15667918	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	C₂²⋊#13083549	1	1	yes	characteristic
80	C₄	#19151432	1	1	yes	characteristic
81	C₂	—	2	2	no	—
82	C₂	C₂²⋊#15667918	1	1	yes	characteristic
83	C₂	C₂×#19151317	1	1	yes	characteristic
84	C₂	#26018463	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₈₉
commutator	C₂×C₈₉	C₂×C₆₄
Fitting	C₂²×C₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₃₂
prime radical	C₂³×C₈₉	C₃₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₂²×C₃₂	D₈₉
hypocenter	C₈₉	C₂²⋊C₆₄
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#19151317	C₂	semidirect
33. C₂²×C₈₉	C₆₄	semidirect
45. C₈₉	C₂²⋊C₆₄	semidirect
76. C₂²	#19151317	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151317	2
C₂²×C₃₂×C₈₉	2
C₂²⋊C₆₄	89