Subgroups — #36442309

Order 22784.

90 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	#26018452	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19151424	—	2	1	no	—
5	#19151317	—	2	1	no	—
6	C₂×Q₃₂×C₈₉	—	2	1	no	—
7	Q₆₄×C₈₉	—	2	1	no	—
8	Q₆₄×C₈₉	C₄	1	2	yes	complemented
9	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	#15667946	—	4	1	no	—
11	C₂×Q₁₆×C₈₉	—	4	1	no	—
12	Q₃₂×C₈₉	—	2	2	no	—
13	Q₃₂×C₈₉	—	4	1	no	—
14	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
15	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₈₉	D₄	1	1	yes	characteristic
17	C₄⋊Dic₈₉	—	8	1	no	—
18	C₂×Q₈×C₈₉	—	8	1	no	—
19	Q₁₆×C₈₉	—	4	2	no	—
20	Q₁₆×C₈₉	—	8	1	no	—
21	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
22	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₈₉	SD₁₆	1	1	yes	characteristic
24	Q₈×C₈₉	—	8	2	no	—
25	Q₈×C₈₉	—	16	1	no	—
26	C₂×C₄×C₈₉	—	16	1	no	—
27	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
28	C₂×Dic₈₉	—	16	1	no	—
29	C₈×C₈₉	#102	1	1	yes	characteristic
30	C₈×C₈₉	SD₃₂	1	1	yes	characteristic
31	C₂²×C₈₉	D₃₂	1	1	yes	characteristic
32	C₄×C₈₉	—	16	2	no	—
33	C₄×C₈₉	#357	1	1	yes	characteristic
34	C₄×C₈₉	SD₆₄	1	1	yes	characteristic
35	Dic₈₉	—	32	1	no	—
36	#7538	—	89	1	no	maximal, Sylow
37	C₂×C₈₉	#1415	1	1	yes	characteristic
38	C₂×C₈₉	SD₁₂₈	1	1	yes	characteristic
39	C₂×C₈₉	Q₁₂₈	1	1	yes	characteristic
40	C₂×Q₆₄	D₈₉	1	1	yes	characteristic
41	C₂×C₆₄	—	89	1	no	—
42	#1423	—	89	1	no	—
43	C₈₉	#7538	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	2	1	no	—
45	Q₆₄	—	2	1	no	—
46	Q₆₄	Dic₈₉	1	2	yes	complemented
47	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
48	#366	—	178	1	no	—
49	C₆₄	—	178	1	no	—
50	C₂×Q₁₆	—	4	1	no	—
51	Q₃₂	—	2	2	no	—
52	Q₃₂	—	4	1	no	—
53	C₂×C₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
54	#107	—	356	1	no	—
55	C₃₂	C₂×Dic₈₉	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
57	C₂×Q₈	—	8	1	no	—
58	Q₁₆	—	4	2	no	—
59	Q₁₆	—	8	1	no	—
60	C₂×C₈	#13083559	1	1	yes	characteristic
61	C₄⋊C₄	—	712	1	no	—
62	C₁₆	C₂²⋊Dic₈₉	1	1	yes	characteristic
63	C₁₆	#13083562	1	1	yes	characteristic
64	Q₈	—	8	2	no	—
65	Q₈	—	16	1	no	—
66	C₂×C₄	—	16	1	no	—
67	C₂×C₄	—	1424	1	no	—
68	C₂×C₄	#15667953	1	1	yes	characteristic
69	C₈	#15667933	1	1	yes	characteristic
70	C₈	#15667956	1	1	yes	characteristic
71	C₂²	#19151433	1	1	yes	characteristic
72	C₄	—	16	2	no	—
73	C₄	—	2848	1	no	—
74	C₄	#19151399	1	1	yes	characteristic
75	C₄	#19151436	1	1	yes	characteristic
76	C₂	#26018430	1	1	yes	characteristic
77	C₂	#26018466	1	1	yes	characteristic
78	C₂	#26018469	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151433
commutator	C₃₂×C₈₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₂×Q₆₄×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	D₃₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₂×Q₆₄	D₈₉
hypocenter	C₈₉	#7538
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×Q₆₄

Complements

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

N	H ≅ G/N	type
8. Q₆₄×C₈₉	C₄	semidirect
43. C₈₉	#7538	semidirect
46. Q₆₄	Dic₈₉	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
#26018452	2
C₂×Q₆₄×C₈₉	2
#7538	89