Subgroups — #36442109

Order 22784.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018328	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151419	—	2	2	no	—
4	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
7	#15667952	—	4	2	no	—
8	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
11	C₃₂×C₈₉	—	2	1	no	—
12	C₃₂×C₈₉	C₈	1	2	yes	—
13	#13083549	—	8	2	no	—
14	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
16	C₄²×C₈₉	Q₁₆	1	1	yes	characteristic
17	C₁₆×C₈₉	—	2	1	no	—
18	C₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₈₉	—	2	1	no	—
21	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
22	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₈₉	#108	1	1	yes	characteristic
24	C₈×C₈₉	D₁₆	1	1	yes	characteristic
25	C₈×C₈₉	Q₃₂	1	1	yes	characteristic
26	C₂²×C₈₉	#368	1	1	yes	characteristic
27	C₄×C₈₉	—	2	2	no	—
28	C₄×C₈₉	#335	1	1	yes	characteristic
29	C₄×C₈₉	#366	1	1	yes	characteristic
30	#7462	—	89	1	no	maximal, Sylow
31	C₂×C₈₉	—	2	1	no	—
32	C₂×C₈₉	#1371	1	1	yes	characteristic
33	C₄×C₃₂	D₈₉	1	1	yes	characteristic
34	#1373	—	89	2	no	—
35	C₈₉	#7462	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
37	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
38	#364	—	178	2	no	—
39	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
40	M₃₂	—	356	2	no	—
41	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
42	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
43	C₄×C₈	D₃₅₆	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	#10623494	1	2	yes	—
46	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
47	C₂×C₈	D₇₁₂	1	1	yes	characteristic
48	C₄²	Dic₃₅₆	1	1	yes	characteristic
49	C₁₆	—	2	1	no	—
50	C₁₆	—	712	2	no	—
51	C₁₆	C₂×#10623494	1	1	yes	characteristic
52	C₁₆	#13083558	1	1	yes	characteristic
53	C₂×C₄	—	2	1	no	—
54	C₂×C₄	#15667946	1	1	yes	characteristic
55	C₈	C₄⋊#10623494	1	1	yes	characteristic
56	C₈	#15667948	1	1	yes	characteristic
57	C₈	D₁₄₂₄	1	1	yes	characteristic
58	C₈	Dic₇₁₂	1	1	yes	characteristic
59	C₂²	#19151428	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#19151351	1	1	yes	characteristic
62	C₄	#19151424	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#26018324	1	1	yes	characteristic

	subgroup	G/H
center	C₈	D₁₄₂₄
commutator	C₁₆×C₈₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₄×C₃₂×C₈₉	C₂
socle	C₂×C₈₉	#1371
prime radical	C₂²×C₈₉	#368
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	C₄×C₃₂	D₈₉
hypocenter	C₈₉	#7462
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
35. C₈₉	#7462	semidirect

Maximal subgroups

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

subgroup	index
#26018328	2
C₄×C₃₂×C₈₉	2
#7462	89