Subgroups — #36442316

Order 22784.

90 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	#26018454	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#19152030	—	2	1	no	—
5	Dic₁₄₂₄	C₄	1	1	yes	characteristic, complemented
6	#19151437	—	2	1	no	—
7	D₂₈₄₈	C₄	1	1	yes	characteristic, complemented
8	#19151424	—	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	#15667946	—	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	C₄⋊Dic₈₉	—	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	C₂×Dic₈₉	—	16	1	no	—
30	D₃₅₆	—	8	1	no	—
31	C₄×D₈₉	—	16	1	no	—
32	Dic₁₇₈	—	8	1	no	—
33	C₈×C₈₉	#102	1	1	yes	characteristic
34	C₈×C₈₉	SD₃₂	1	1	yes	characteristic
35	C₂²×C₈₉	D₃₂	1	1	yes	characteristic
36	C₂×D₈₉	—	16	1	no	—
37	C₄×C₈₉	#357	1	1	yes	characteristic
38	C₄×C₈₉	SD₆₄	1	1	yes	characteristic
39	Dic₈₉	—	16	1	no	—
40	Dic₈₉	—	32	1	no	—
41	#7540	—	89	1	no	maximal, Sylow
42	C₂×C₈₉	—	2	1	no	—
43	C₂×C₈₉	#1415	1	1	yes	characteristic
44	D₈₉	—	32	1	no	—
45	#2262	—	89	1	no	—
46	M₁₂₈	D₈₉	1	1	yes	characteristic, complemented
47	#1424	—	89	1	no	—
48	C₈₉	#7540	1	1	yes	characteristic, complemented, Sylow
49	#508	—	178	1	no	—
50	Q₆₄	—	89	1	no	—
51	SD₆₄	—	178	1	no	—
52	D₃₂	—	89	1	no	—
53	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
54	#366	—	178	1	no	—
55	C₆₄	—	2	1	no	—
56	#135	—	356	1	no	—
57	Q₃₂	—	178	1	no	—
58	SD₃₂	—	356	1	no	—
59	D₁₆	—	178	1	no	—
60	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
61	#107	—	356	1	no	—
62	C₃₂	C₄×D₈₉	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₈₉	1	1	yes	characteristic
64	#41	—	712	1	no	—
65	Q₁₆	—	356	1	no	—
66	SD₁₆	—	712	1	no	—
67	D₈	—	356	1	no	—
68	C₂×C₈	D₇₁₂	1	1	yes	characteristic
69	C₄⋊C₄	—	712	1	no	—
70	C₁₆	#13083551	1	1	yes	characteristic
71	C₁₆	#13083563	1	1	yes	characteristic
72	Q₈	—	712	1	no	—
73	D₄	—	1424	1	no	—
74	D₄	—	712	1	no	—
75	C₂×C₄	—	1424	1	no	—
76	C₂×C₄	—	1424	1	no	—
77	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
78	C₈	#15667934	1	1	yes	characteristic
79	C₈	#15667957	1	1	yes	characteristic
80	C₂²	—	1424	1	no	—
81	C₂²	D₂₈₄₈	1	1	yes	characteristic
82	C₄	—	2848	1	no	—
83	C₄	—	1424	1	no	—
84	C₄	#19151400	1	1	yes	characteristic
85	C₄	#19151437	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2848	1	no	—
88	C₂	#26018431	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26018431
commutator	C₃₂×C₈₉	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₈₉
Fitting	C₈₉×M₁₂₈	C₂
socle	C₂×C₈₉	#1415
prime radical	#26022561	C₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	M₁₂₈	D₈₉
hypocenter	C₈₉	#7540
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #26018454	C₂	semidirect
3. C₈₉×M₁₂₈	C₂	semidirect
5. Dic₁₄₂₄	C₄	semidirect
7. D₂₈₄₈	C₄	semidirect
46. M₁₂₈	D₈₉	semidirect
48. C₈₉	#7540	semidirect

Maximal subgroups

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

subgroup	index
#26022561	2
#26018454	2
C₈₉×M₁₂₈	2
#7540	89