Subgroups — #36442107

Order 22784.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018324	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151351	—	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	C₄⋊#10623494	—	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	complemented
13	C₂×#10623494	—	8	2	no	—
14	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₈₉	Q₁₆	1	1	yes	characteristic
16	C₄²×C₈₉	D₈	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	#107	1	1	yes	characteristic
21	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
22	C₂×C₄×C₈₉	Q₃₂	1	1	yes	characteristic
23	C₈×C₈₉	—	2	1	no	—
24	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₈₉	#108	1	1	yes	characteristic
26	#10623494	—	16	2	no	—
27	C₂²×C₈₉	#366	1	1	yes	characteristic
28	C₄×C₈₉	—	2	1	no	—
29	C₄×C₈₉	#335	1	1	yes	characteristic
30	C₄×C₈₉	#368	1	1	yes	characteristic
31	C₄×C₈₉	SD₆₄	1	2	yes	—
32	#7461	—	89	1	no	maximal, Sylow
33	C₂×C₈₉	#1371	1	1	yes	characteristic
34	C₂×C₈₉	#1424	1	1	yes	characteristic
35	C₂×C₈₉	#1425	1	1	yes	characteristic
36	C₄×C₃₂	D₈₉	1	1	yes	characteristic
37	#1371	—	89	2	no	—
38	C₈₉	#7461	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
40	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
41	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
42	#335	—	178	2	no	—
43	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
44	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
45	C₄⋊C₈	—	356	2	no	—
46	C₄×C₈	D₃₅₆	1	1	yes	characteristic
47	C₃₂	—	2	1	no	—
48	C₃₂	#10623494	1	2	yes	complemented
49	C₂×C₈	—	712	2	no	—
50	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
51	C₂×C₈	Dic₃₅₆	1	1	yes	characteristic
52	C₄²	D₇₁₂	1	1	yes	characteristic
53	C₁₆	—	2	1	no	—
54	C₁₆	C₂×#10623494	1	1	yes	characteristic
55	C₁₆	#13083558	1	1	yes	characteristic
56	C₂×C₄	#15667946	1	1	yes	characteristic
57	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
58	C₂×C₄	Dic₇₁₂	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	1424	2	no	—
61	C₈	C₄⋊#10623494	1	1	yes	characteristic
62	C₈	#15667948	1	1	yes	characteristic
63	C₂²	#19151424	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	#19151351	1	1	yes	characteristic
66	C₄	#19151428	1	1	yes	characteristic
67	C₄	#19151437	1	2	yes	—
68	C₂	#26018324	1	1	yes	characteristic
69	C₂	#26018454	1	1	yes	characteristic
70	C₂	#26018456	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₄₂₄
commutator	C₁₆×C₈₉	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₄×C₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	#366
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	C₄×C₃₂	D₈₉
hypocenter	C₈₉	#7461
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₈₉	C₈	semidirect
38. C₈₉	#7461	semidirect
48. C₃₂	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26018324	2
C₄×C₃₂×C₈₉	2
#7461	89