Subgroups — #36441043

Order 22784.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083549	C₂	1	1	yes	maximal, characteristic
2	C₈×#13083549	C₂	1	1	yes	maximal, characteristic
3	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₄×#13083549	C₄	1	2	yes	—
5	C₄×#13083549	C₄	1	2	yes	—
6	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
7	C₈²×C₈₉	C₂²	1	1	yes	characteristic
8	C₂×#13083549	C₈	1	4	yes	—
9	C₂×#13083549	C₈	1	4	yes	—
10	C₂×C₁₆×C₈₉	C₈	1	4	yes	—
11	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
14	#13083549	—	2	4	no	—
15	#13083549	C₁₆	1	8	yes	complemented
16	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
17	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
18	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
19	C₄²×C₈₉	C₄²	1	1	yes	characteristic
20	C₁₆×C₈₉	—	2	4	no	—
21	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
22	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
23	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
24	C₈×C₈₉	C₂×C₁₆	1	4	yes	—
25	C₈×C₈₉	M₃₂	1	4	yes	—
26	C₈×C₈₉	M₃₂	1	4	yes	—
27	C₂²×C₈₉	C₈²	1	1	yes	characteristic
28	C₄×C₈₉	C₄×C₁₆	1	2	yes	—
29	C₄×C₈₉	#346	1	2	yes	—
30	C₄×C₈₉	#346	1	2	yes	—
31	#7052	—	89	1	no	maximal, Sylow
32	C₂×C₈₉	C₈×C₁₆	1	1	yes	characteristic
33	C₂×C₈₉	#1311	1	1	yes	characteristic
34	C₂×C₈₉	#1311	1	1	yes	characteristic
35	C₈×C₁₆	—	89	1	no	—
36	C₈×C₁₆	—	89	1	no	—
37	C₈×C₁₆	D₈₉	1	1	yes	characteristic
38	C₈₉	#7052	1	1	yes	characteristic, complemented, Sylow
39	C₄×C₁₆	—	89	2	no	—
40	C₄×C₁₆	—	89	2	no	—
41	C₄×C₁₆	Dic₈₉	1	2	yes	—
42	C₈²	C₂×D₈₉	1	1	yes	characteristic
43	C₂×C₁₆	—	89	4	no	—
44	C₂×C₁₆	—	89	4	no	—
45	C₂×C₁₆	#10623494	1	4	yes	—
46	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
47	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
48	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
49	C₂×C₈	C₈×D₈₉	1	2	yes	—
50	C₂×C₈	C₈×D₈₉	1	2	yes	—
51	C₂×C₈	C₂×#10623494	1	2	yes	—
52	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
53	C₁₆	—	2	4	no	—
54	C₁₆	—	89	8	no	—
55	C₁₆	—	178	4	no	—
56	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
57	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
58	C₂×C₄	C₄×#10623494	1	1	yes	characteristic
59	C₈	C₁₆×D₈₉	1	4	yes	—
60	C₈	#15667951	1	4	yes	—
61	C₈	#15667952	1	4	yes	—
62	C₂²	C₈×#10623494	1	1	yes	characteristic
63	C₄	C₁₆×Dic₈₉	1	2	yes	—
64	C₄	#19151373	1	2	yes	—
65	C₄	#19151374	1	2	yes	—
66	C₂	C₁₆×#10623494	1	1	yes	characteristic
67	C₂	#26018168	1	1	yes	characteristic
68	C₂	#26018169	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈²	C₂×D₈₉
commutator	C₂×C₈₉	C₈×C₁₆
Fitting	C₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₈²
cosocle	C₈²×C₈₉	C₂²
hypercenter	C₈×C₁₆	D₈₉
hypocenter	C₈₉	#7052
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
15. #13083549	C₁₆	semidirect
38. C₈₉	#7052	semidirect

Maximal subgroups

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

subgroup	index
C₈×#13083549	2
C₈×C₁₆×C₈₉	2
#7052	89