Subgroups — #36475483

Order 22784.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022562	C₂	1	2	yes	maximal, complemented
2	#26018466	C₂	1	2	yes	maximal, complemented
3	D₅₆₉₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₄×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×Q₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152031	—	2	2	no	—
7	#19151435	—	2	2	no	—
8	D₂₈₄₈	C₂²	1	2	yes	—
9	C₃₂×D₈₉	C₂²	1	1	yes	characteristic
10	#19151317	C₂²	1	1	yes	characteristic
11	Q₆₄×C₈₉	C₂²	1	2	yes	—
12	C₆₄×C₈₉	C₂²	1	1	yes	characteristic
13	#15668035	—	4	2	no	—
14	#15667955	—	4	2	no	—
15	D₁₄₂₄	—	2	2	no	—
16	C₁₆×D₈₉	D₄	1	1	yes	characteristic
17	#15667918	D₄	1	1	yes	characteristic
18	Q₃₂×C₈₉	—	2	2	no	—
19	C₃₂×C₈₉	C₂³	1	1	yes	characteristic
20	#13083575	—	8	2	no	—
21	#13083561	—	8	2	no	—
22	D₇₁₂	—	4	2	no	—
23	C₈×D₈₉	D₈	1	1	yes	characteristic
24	#13083549	D₈	1	1	yes	characteristic
25	Q₁₆×C₈₉	—	4	2	no	—
26	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
27	Q₈×C₈₉	—	8	2	no	—
28	D₃₅₆	—	8	2	no	—
29	D₃₅₆	—	16	2	no	—
30	C₄×D₈₉	—	16	2	no	—
31	C₄×D₈₉	D₁₆	1	1	yes	characteristic
32	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
33	#10623494	D₁₆	1	1	yes	characteristic
34	C₂×D₈₉	—	16	2	no	—
35	C₂×D₈₉	D₃₂	1	1	yes	characteristic
36	C₄×C₈₉	—	16	2	no	—
37	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
38	Dic₈₉	D₃₂	1	1	yes	characteristic
39	#13741	—	89	1	no	maximal, Sylow
40	C₂×C₈₉	C₂×D₃₂	1	1	yes	characteristic
41	D₈₉	—	2	1	no	—
42	D₈₉	—	32	2	no	—
43	#2262	—	89	2	no	—
44	Q₁₂₈	D₈₉	1	1	yes	characteristic, complemented
45	SD₁₂₈	—	89	2	no	—
46	D₆₄	—	89	1	no	—
47	C₂×C₆₄	—	89	1	no	—
48	C₈₉	#13741	1	1	yes	characteristic, complemented, Sylow
49	#508	—	178	2	no	—
50	Q₆₄	C₂×D₈₉	1	2	yes	—
51	SD₆₄	—	178	2	no	—
52	D₃₂	—	89	2	no	—
53	C₂×C₃₂	—	89	1	no	—
54	C₆₄	—	89	1	no	—
55	C₆₄	C₂×D₈₉	1	1	yes	characteristic
56	#135	—	356	2	no	—
57	Q₃₂	—	2	2	no	—
58	SD₃₂	—	356	2	no	—
59	D₁₆	—	178	2	no	—
60	C₂×C₁₆	—	89	1	no	—
61	C₃₂	—	89	1	no	—
62	C₃₂	C₂²×D₈₉	1	1	yes	characteristic
63	#41	—	712	2	no	—
64	Q₁₆	—	4	2	no	—
65	SD₁₆	—	712	2	no	—
66	D₈	—	356	2	no	—
67	C₂×C₈	—	89	1	no	—
68	C₁₆	—	89	1	no	—
69	C₁₆	D₄×D₈₉	1	1	yes	characteristic
70	Q₈	—	8	2	no	—
71	D₄	—	1424	2	no	—
72	D₄	—	712	2	no	—
73	C₂×C₄	—	89	1	no	—
74	C₂×C₄	—	1424	2	no	—
75	C₈	—	89	1	no	—
76	C₈	D₈×D₈₉	1	1	yes	characteristic
77	C₂²	—	89	1	no	—
78	C₂²	—	1424	2	no	—
79	C₄	—	16	2	no	—
80	C₄	—	89	1	no	—
81	C₄	D₁₆×D₈₉	1	1	yes	characteristic
82	C₂	—	2848	2	no	—
83	C₂	—	178	1	no	—
84	C₂	D₃₂×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃₂×D₈₉
commutator, cosocle	C₃₂×C₈₉	C₂³
Frattini	C₃₂	C₂²×D₈₉
Fitting	C₈₉×Q₁₂₈	C₂
socle	C₂×C₈₉	C₂×D₃₂
hypercenter	Q₁₂₈	D₈₉
hypocenter	C₈₉	#13741
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂ ◁ Q₁₂₈

Complements

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

N	H ≅ G/N	type
1. #26022562	C₂	semidirect
2. #26018466	C₂	semidirect
3. D₅₆₉₆	C₂	semidirect
4. C₆₄×D₈₉	C₂	semidirect
5. C₈₉×Q₁₂₈	C₂	semidirect
44. Q₁₂₈	D₈₉	semidirect
48. C₈₉	#13741	semidirect

Maximal subgroups

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

subgroup	index
#26022562	2
#26018466	2
D₅₆₉₆	2
C₆₄×D₈₉	2
C₈₉×Q₁₂₈	2
#13741	89