Subgroups — #36442310

Order 22784.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄₂₄	C₂	1	1	yes	maximal, characteristic
2	#26018452	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₇₁₂	—	2	1	no	—
5	Dic₁₄₂₄	—	2	1	no	—
6	Dic₁₄₂₄	C₄	1	2	yes	complemented
7	#19151424	—	2	1	no	—
8	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
9	C₆₄×C₈₉	—	2	1	no	—
10	C₂×Dic₃₅₆	—	4	1	no	—
11	Dic₇₁₂	—	2	2	no	—
12	Dic₇₁₂	—	4	1	no	—
13	#15667946	—	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	C₂×Dic₁₇₈	—	8	1	no	—
18	Dic₃₅₆	—	4	2	no	—
19	Dic₃₅₆	—	8	1	no	—
20	C₄⋊Dic₈₉	—	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	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
25	C₂×Dic₈₉	—	16	1	no	—
26	C₂×Dic₈₉	—	16	1	no	—
27	Dic₁₇₈	—	8	2	no	—
28	Dic₁₇₈	—	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	#357	1	1	yes	characteristic
33	C₄×C₈₉	SD₆₄	1	1	yes	characteristic
34	Dic₈₉	—	16	2	no	—
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	—	89	1	no	—
41	C₂×C₆₄	D₈₉	1	1	yes	characteristic
42	#1423	—	89	1	no	—
43	C₈₉	#7538	1	1	yes	characteristic, complemented, Sylow
44	C₂×Q₃₂	—	178	1	no	—
45	Q₆₄	—	89	2	no	—
46	Q₆₄	—	178	1	no	—
47	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
48	#366	—	178	1	no	—
49	C₆₄	—	2	1	no	—
50	C₂×Q₁₆	—	356	1	no	—
51	Q₃₂	—	356	1	no	—
52	Q₃₂	—	178	2	no	—
53	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
54	#107	—	356	1	no	—
55	C₃₂	C₄×D₈₉	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
57	C₂×Q₈	—	712	1	no	—
58	Q₁₆	—	356	2	no	—
59	Q₁₆	—	712	1	no	—
60	C₂×C₈	D₇₁₂	1	1	yes	characteristic
61	C₄⋊C₄	—	712	1	no	—
62	C₁₆	#13083551	1	1	yes	characteristic
63	C₁₆	#13083563	1	1	yes	characteristic
64	Q₈	—	1424	1	no	—
65	Q₈	—	712	2	no	—
66	C₂×C₄	—	1424	1	no	—
67	C₂×C₄	—	1424	1	no	—
68	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
69	C₈	#15667934	1	1	yes	characteristic
70	C₈	#15667957	1	1	yes	characteristic
71	C₂²	D₂₈₄₈	1	1	yes	characteristic
72	C₄	—	2848	1	no	—
73	C₄	—	1424	2	no	—
74	C₄	#19151400	1	1	yes	characteristic
75	C₄	#19151437	1	1	yes	characteristic
76	C₂	#26018431	1	1	yes	characteristic
77	C₂	#26018468	1	1	yes	characteristic
78	C₂	Dic₂₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₈₄₈
commutator	C₃₂×C₈₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₂×C₆₄×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	D₃₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₂×C₆₄	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×C₆₄

Complements

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

N	H ≅ G/N	type
6. Dic₁₄₂₄	C₄	semidirect
43. C₈₉	#7538	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₄₂₄	2
#26018452	2
C₂×C₆₄×C₈₉	2
#7538	89