Subgroups — #36442324

Order 22784.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151317	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151317	—	2	2	no	—
4	C₂×C₃₂×C₈₉	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
6	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₈₉	C₈	1	1	yes	characteristic
10	C₃₂×C₈₉	—	2	1	no	—
11	C₃₂×C₈₉	D₄	1	1	yes	characteristic
12	C₃₂×C₈₉	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₈₉	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₈₉	C₁₆	1	1	yes	characteristic
16	C₁₆×C₈₉	—	2	1	no	—
17	C₁₆×C₈₉	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₈₉	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₈₉	C₃₂	1	1	yes	characteristic
20	C₂×C₄×C₈₉	C₃₂	1	1	yes	characteristic
21	C₂×C₄×C₈₉	C₂×C₁₆	1	1	yes	characteristic
22	C₈×C₈₉	—	2	1	no	—
23	C₈×C₈₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₈₉	M₃₂	1	1	yes	characteristic
25	C₂²×C₈₉	C₂×C₃₂	1	1	yes	characteristic
26	C₄×C₈₉	—	2	1	no	—
27	C₄×C₈₉	C₆₄	1	2	yes	complemented
28	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
29	C₄×C₈₉	M₆₄	1	1	yes	characteristic
30	C₄⋊C₆₄	—	89	1	no	maximal, Sylow
31	C₂×C₈₉	C₄⋊C₃₂	1	1	yes	characteristic
32	C₂×C₈₉	C₂×C₆₄	1	1	yes	characteristic
33	C₂×C₈₉	M₁₂₈	1	1	yes	characteristic
34	C₂×C₆₄	—	89	2	no	—
35	C₄×C₃₂	D₈₉	1	1	yes	characteristic
36	C₈₉	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃₂	Dic₈₉	1	1	yes	characteristic
38	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
39	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
40	C₆₄	—	178	2	no	—
41	C₂×C₁₆	#10623494	1	1	yes	characteristic
42	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
43	C₄×C₈	#10623494	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	Dic₁₇₈	1	1	yes	characteristic
46	C₃₂	D₃₅₆	1	1	yes	characteristic
47	C₂×C₈	#13083549	1	1	yes	characteristic
48	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
49	C₄²	#13083549	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₄⋊Dic₈₉	1	1	yes	characteristic
52	C₁₆	#13083558	1	1	yes	characteristic
53	C₂×C₄	#15667918	1	1	yes	characteristic
54	C₂×C₄	#15667918	1	1	yes	characteristic
55	C₂×C₄	C₂×#13083549	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄⋊#10623494	1	1	yes	characteristic
58	C₈	#15667952	1	1	yes	characteristic
59	C₂²	C₂×#15667918	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	#19151317	1	2	yes	complemented
62	C₄	C₄⋊#13083549	1	1	yes	characteristic
63	C₄	#19151432	1	1	yes	characteristic
64	C₂	C₄⋊#15667918	1	1	yes	characteristic
65	C₂	C₂×#19151317	1	1	yes	characteristic
66	C₂	#26018463	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₈₉
commutator	C₂×C₈₉	C₂×C₆₄
Fitting	C₄×C₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×C₃₂
cosocle	C₂×C₃₂×C₈₉	C₂²
hypercenter	C₄×C₃₂	D₈₉
hypocenter	C₈₉	C₄⋊C₆₄
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₃₂ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
27. C₄×C₈₉	C₆₄	semidirect
36. C₈₉	C₄⋊C₆₄	semidirect
61. C₄	#19151317	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151317	2
C₄×C₃₂×C₈₉	2
C₄⋊C₆₄	89