Subgroups — #36442241

Order 22784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018463	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151317	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2258	C₂	1	1	yes	maximal, characteristic
4	#19151317	—	2	1	no	—
5	#19151317	—	2	1	no	—
6	#504×C₈₉	C₄	1	1	yes	characteristic
7	M₆₄×C₈₉	—	2	1	no	—
8	M₆₄×C₈₉	C₄	1	1	yes	characteristic
9	C₂×C₃₂×C₈₉	—	2	1	no	—
10	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	#131×C₈₉	C₈	1	1	yes	characteristic
12	M₃₂×C₈₉	—	2	1	no	—
13	M₃₂×C₈₉	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	—	2	1	no	—
15	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₈₉	—	2	1	no	—
17	C₃₂×C₈₉	D₄	1	1	yes	characteristic
18	C₃₂×C₈₉	D₄	1	1	yes	characteristic
19	#41×C₈₉	C₁₆	1	1	yes	characteristic
20	M₁₆×C₈₉	—	2	1	no	—
21	M₁₆×C₈₉	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₈₉	—	2	1	no	—
23	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₈₉	—	2	1	no	—
25	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
27	Q₈×C₈₉	C₃₂	1	1	yes	characteristic
28	D₄×C₈₉	—	2	1	no	—
29	D₄×C₈₉	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₈₉	—	2	1	no	—
33	C₈×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₈₉	M₃₂	1	1	yes	characteristic
35	C₂²×C₈₉	—	2	1	no	—
36	C₂²×C₈₉	M₆₄	1	1	yes	characteristic
37	C₄×C₈₉	—	2	1	no	—
38	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₈₉	C₂×C₃₂	1	1	yes	characteristic
40	#7514	—	89	1	no	maximal, Sylow
41	C₂×C₈₉	—	2	1	no	—
42	C₂×C₈₉	—	4	1	no	—
43	C₂×C₈₉	C₂²⋊C₃₂	1	1	yes	characteristic
44	#2258	D₈₉	1	1	yes	characteristic
45	M₁₂₈	—	89	1	no	—
46	C₂×C₆₄	—	89	1	no	—
47	C₈₉	#7514	1	1	yes	characteristic, complemented, Sylow
48	#504	Dic₈₉	1	1	yes	characteristic
49	M₆₄	—	2	1	no	—
50	M₆₄	Dic₈₉	1	1	yes	characteristic
51	C₂×C₃₂	—	2	1	no	—
52	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
53	C₆₄	—	178	1	no	—
54	C₆₄	—	178	1	no	—
55	#131	#10623494	1	1	yes	characteristic
56	M₃₂	—	2	1	no	—
57	M₃₂	#10623494	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
60	C₃₂	—	2	1	no	—
61	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
63	#41	#13083549	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	#13083549	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	C₂²⋊Dic₈₉	1	1	yes	characteristic
70	C₁₆	#13083558	1	1	yes	characteristic
71	Q₈	#15667918	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	#15667918	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊#10623494	1	1	yes	characteristic
78	C₈	#15667952	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	#19151432	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊#13083549	1	1	yes	characteristic
83	C₄	C₂×#15667918	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₂²⋊#15667918	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊D₈₉
commutator	C₄×C₈₉	C₂×C₃₂
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₈₉×#2258	C₂
socle	C₂×C₈₉	C₂²⋊C₃₂
prime radical	#41×C₈₉	C₁₆
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	#2258	D₈₉
hypocenter	C₈₉	#7514
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₃₂ ◁ C₂×C₃₂ ◁ #2258

Complements

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

N	H ≅ G/N	type
1. #26018463	C₂	semidirect
2. C₂×#19151317	C₂	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
#26018463	2
C₂×#19151317	2
C₈₉×#2258	2
#7514	89