Subgroups — #36442323

Order 22784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151317	C₂	1	1	yes	maximal, characteristic
2	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₄×C₈₉	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	C₃₂	1	1	yes	characteristic
23	C₂×Dic₈₉	C₃₂	1	1	yes	characteristic
24	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₈₉	M₃₂	1	1	yes	characteristic
26	#10623494	—	2	1	no	—
27	C₂²×C₈₉	C₂×C₃₂	1	1	yes	characteristic
28	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
29	C₄×C₈₉	M₆₄	1	1	yes	characteristic
30	Dic₈₉	—	2	1	no	—
31	Dic₈₉	C₆₄	1	2	yes	complemented
32	C₄⋊C₆₄	—	89	1	no	maximal, Sylow
33	C₂×C₈₉	C₄⋊C₃₂	1	1	yes	characteristic
34	C₂×C₈₉	C₂×C₆₄	1	1	yes	characteristic
35	C₂×C₈₉	M₁₂₈	1	1	yes	characteristic
36	C₂×C₆₄	—	89	1	no	—
37	C₂×C₆₄	D₈₉	1	1	yes	characteristic
38	C₄×C₃₂	—	89	1	no	—
39	C₈₉	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₃₂	—	89	1	no	—
41	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
42	C₄×C₁₆	—	89	1	no	—
43	C₆₄	—	2	1	no	—
44	C₆₄	—	178	1	no	—
45	C₂×C₁₆	—	89	1	no	—
46	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
47	C₄×C₈	—	89	1	no	—
48	C₃₂	—	178	1	no	—
49	C₃₂	Dic₁₇₈	1	1	yes	characteristic
50	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
51	C₂×C₈	—	89	1	no	—
52	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
53	C₄²	—	89	1	no	—
54	C₁₆	—	178	1	no	—
55	C₁₆	#13083553	1	1	yes	characteristic
56	C₁₆	#13083557	1	1	yes	characteristic
57	C₂×C₄	—	89	1	no	—
58	C₂×C₄	—	89	1	no	—
59	C₂×C₄	C₁₆×D₈₉	1	1	yes	characteristic
60	C₈	—	178	1	no	—
61	C₈	#15667941	1	1	yes	characteristic
62	C₈	#15667951	1	1	yes	characteristic
63	C₂²	C₃₂×D₈₉	1	1	yes	characteristic
64	C₄	—	89	2	no	—
65	C₄	—	178	1	no	—
66	C₄	#19151416	1	1	yes	characteristic
67	C₄	#19151431	1	1	yes	characteristic
68	C₂	#26018447	1	1	yes	characteristic
69	C₂	C₆₄×D₈₉	1	1	yes	characteristic
70	C₂	#26018462	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
31. Dic₈₉	C₆₄	semidirect
39. C₈₉	C₄⋊C₆₄	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
C₃₂×Dic₈₉	2
C₂×C₆₄×C₈₉	2
C₄⋊C₆₄	89