Subgroups — #36442313

Order 22784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022561	C₂	1	1	yes	maximal, characteristic
2	#26018456	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152030	—	2	1	no	—
5	Dic₁₄₂₄	C₄	1	1	yes	characteristic
6	#19151437	—	2	1	no	—
7	D₂₈₄₈	C₄	1	1	yes	characteristic
8	#19151428	—	2	1	no	—
9	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₆₄×C₈₉	—	2	1	no	—
11	#15668034	—	4	1	no	—
12	Dic₇₁₂	—	2	1	no	—
13	#15667957	—	4	1	no	—
14	D₁₄₂₄	—	2	1	no	—
15	#15667948	—	4	1	no	—
16	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
17	C₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₈₉	D₄	1	1	yes	characteristic
19	#13083574	—	8	1	no	—
20	Dic₃₅₆	—	4	1	no	—
21	#13083563	—	8	1	no	—
22	D₇₁₂	—	4	1	no	—
23	#13083558	—	8	1	no	—
24	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
25	C₁₆×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₈₉	SD₁₆	1	1	yes	characteristic
27	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₈₉	—	16	1	no	—
29	D₃₅₆	—	8	1	no	—
30	C₄×D₈₉	—	16	1	no	—
31	Dic₁₇₈	—	8	1	no	—
32	C₈×C₈₉	#102	1	1	yes	characteristic
33	C₈×C₈₉	SD₃₂	1	1	yes	characteristic
34	#10623494	—	16	1	no	—
35	C₂²×C₈₉	SD₆₄	1	1	yes	characteristic
36	C₂×D₈₉	—	16	1	no	—
37	C₄×C₈₉	#357	1	1	yes	characteristic
38	C₄×C₈₉	D₃₂	1	1	yes	characteristic
39	Dic₈₉	—	16	1	no	—
40	#7539	—	89	1	no	maximal, Sylow
41	C₂×C₈₉	—	2	1	no	—
42	C₂×C₈₉	#1415	1	1	yes	characteristic
43	D₈₉	—	32	1	no	—
44	#2262	—	89	1	no	—
45	C₂×C₆₄	D₈₉	1	1	yes	characteristic, complemented
46	#1425	—	89	1	no	—
47	C₈₉	#7539	1	1	yes	characteristic, complemented, Sylow
48	#508	—	178	1	no	—
49	Q₆₄	—	89	1	no	—
50	SD₆₄	—	178	1	no	—
51	D₃₂	—	89	1	no	—
52	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
53	#368	—	178	1	no	—
54	C₆₄	—	2	1	no	—
55	#135	—	356	1	no	—
56	Q₃₂	—	178	1	no	—
57	SD₃₂	—	356	1	no	—
58	D₁₆	—	178	1	no	—
59	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
60	#108	—	356	1	no	—
61	C₃₂	C₄×D₈₉	1	1	yes	characteristic
62	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
63	#41	—	712	1	no	—
64	Q₁₆	—	356	1	no	—
65	SD₁₆	—	712	1	no	—
66	D₈	—	356	1	no	—
67	M₁₆	—	712	1	no	—
68	C₂×C₈	D₇₁₂	1	1	yes	characteristic
69	C₁₆	#13083551	1	1	yes	characteristic
70	C₁₆	#13083563	1	1	yes	characteristic
71	Q₈	—	712	1	no	—
72	D₄	—	1424	1	no	—
73	D₄	—	712	1	no	—
74	C₂×C₄	—	1424	1	no	—
75	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
76	C₈	—	1424	1	no	—
77	C₈	#15667934	1	1	yes	characteristic
78	C₈	#15667957	1	1	yes	characteristic
79	C₂²	—	1424	1	no	—
80	C₂²	#19151437	1	1	yes	characteristic
81	C₄	—	1424	1	no	—
82	C₄	#19151400	1	1	yes	characteristic
83	C₄	D₂₈₄₈	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2848	1	no	—
86	C₂	#26018431	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₈₄₈
commutator	C₃₂×C₈₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₂×C₆₄×C₈₉	C₂
socle	C₂×C₈₉	#1415
prime radical	#26022561	C₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₂×C₆₄	D₈₉
hypocenter	C₈₉	#7539
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ 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
2. #26018456	C₂	semidirect
3. C₂×C₆₄×C₈₉	C₂	semidirect
45. C₂×C₆₄	D₈₉	semidirect
47. C₈₉	#7539	semidirect

Maximal subgroups

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

subgroup	index
#26022561	2
#26018456	2
C₂×C₆₄×C₈₉	2
#7539	89