Subgroups — #36442110

Order 22784.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018386	C₂	1	1	yes	maximal, characteristic
2	#26018326	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1372	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667918	C₄	1	1	yes	characteristic
5	C₂×#15667918	C₄	1	1	yes	characteristic
6	#19151351	—	2	1	no	—
7	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
8	#335×C₈₉	—	2	1	no	—
9	C₄⋊#10623494	—	4	1	no	—
10	#15667918	—	2	1	no	—
11	#15667918	C₈	1	2	yes	complemented
12	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	—	4	1	no	—
15	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
16	C₂×#10623494	—	8	1	no	—
17	C₂×C₈×C₈₉	—	8	1	no	—
18	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
20	C₄²×C₈₉	D₈	1	1	yes	characteristic
21	C₁₆×C₈₉	—	2	1	no	—
22	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
25	C₂×C₄×C₈₉	SD₃₂	1	1	yes	characteristic
26	C₂×C₄×C₈₉	SD₃₂	1	1	yes	characteristic
27	C₈×C₈₉	—	2	1	no	—
28	C₈×C₈₉	—	16	1	no	—
29	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₈₉	#108	1	1	yes	characteristic
31	#10623494	—	16	1	no	—
32	C₂²×C₈₉	#367	1	1	yes	characteristic
33	C₄×C₈₉	—	2	1	no	—
34	C₄×C₈₉	—	2	1	no	—
35	C₄×C₈₉	#335	1	1	yes	characteristic
36	C₄×C₈₉	#368	1	1	yes	characteristic
37	#7463	—	89	1	no	maximal, Sylow
38	C₂×C₈₉	#1372	1	1	yes	characteristic
39	C₂×C₈₉	#1426	1	1	yes	characteristic
40	C₂×C₈₉	#1426	1	1	yes	characteristic
41	#1397	—	89	1	no	—
42	#1372	—	89	1	no	—
43	#1372	D₈₉	1	1	yes	characteristic
44	C₈₉	#7463	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃₂	—	89	1	no	—
46	C₂×C₃₂	—	89	1	no	—
47	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
48	#335	—	2	1	no	—
49	#335	—	178	1	no	—
50	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
51	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
52	C₄⋊C₈	—	4	1	no	—
53	C₄⋊C₈	—	356	1	no	—
54	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
55	C₃₂	—	89	2	no	—
56	C₃₂	—	178	1	no	—
57	C₂×C₈	—	8	1	no	—
58	C₂×C₈	—	712	1	no	—
59	C₂×C₈	#13083553	1	1	yes	characteristic
60	C₂×C₈	#13083564	1	1	yes	characteristic
61	C₄²	#13083559	1	1	yes	characteristic
62	C₁₆	—	2	1	no	—
63	C₁₆	C₈×D₈₉	1	1	yes	characteristic
64	C₁₆	#13083557	1	1	yes	characteristic
65	C₂×C₄	#15667945	1	1	yes	characteristic
66	C₂×C₄	#15667955	1	1	yes	characteristic
67	C₂×C₄	#15667956	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	16	1	no	—
70	C₈	—	1424	1	no	—
71	C₈	#15667941	1	1	yes	characteristic
72	C₈	#15667947	1	1	yes	characteristic
73	C₂²	#19151425	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#19151350	1	1	yes	characteristic
77	C₄	#19151427	1	1	yes	characteristic
78	C₂	#26018325	1	1	yes	characteristic
79	C₂	#26018457	1	1	yes	characteristic
80	C₂	#26018459	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151425
commutator	C₁₆×C₈₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×#1372	C₂
socle, prime radical	C₂²×C₈₉	#367
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	#1372	D₈₉
hypocenter	C₈₉	#7463
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

Complements

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

N	H ≅ G/N	type
11. #15667918	C₈	semidirect
44. C₈₉	#7463	semidirect

Maximal subgroups

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

subgroup	index
#26018386	2
#26018326	2
C₈₉×#1372	2
#7463	89