Subgroups — #36441042

Order 22784.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#13083549	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₄×#13083549	C₄	1	4	yes	—
4	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
5	C₈²×C₈₉	C₂²	1	1	yes	characteristic
6	C₂×#13083549	C₈	1	8	yes	—
7	C₂×C₁₆×C₈₉	C₈	1	4	yes	—
8	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
9	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
10	#13083549	—	2	8	no	—
11	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
12	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
13	C₄²×C₈₉	C₄²	1	1	yes	characteristic
14	C₁₆×C₈₉	C₁₆	1	8	yes	complemented
15	C₂×C₄×C₈₉	C₄×C₈	1	2	yes	—
16	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
17	C₈×C₈₉	C₂×C₁₆	1	4	yes	—
18	C₈×C₈₉	M₃₂	1	8	yes	—
19	C₂²×C₈₉	C₈²	1	1	yes	characteristic
20	C₄×C₈₉	C₄×C₁₆	1	2	yes	—
21	C₄×C₈₉	#346	1	4	yes	—
22	#7052	—	89	1	no	maximal, Sylow
23	C₂×C₈₉	C₈×C₁₆	1	1	yes	characteristic
24	C₂×C₈₉	#1311	1	2	yes	—
25	C₈×C₁₆	—	89	2	no	—
26	C₈×C₁₆	D₈₉	1	1	yes	characteristic
27	C₈₉	#7052	1	1	yes	characteristic, complemented, Sylow
28	C₄×C₁₆	—	89	4	no	—
29	C₄×C₁₆	Dic₈₉	1	2	yes	—
30	C₈²	C₂×D₈₉	1	1	yes	characteristic
31	C₂×C₁₆	—	89	8	no	—
32	C₂×C₁₆	#10623494	1	4	yes	—
33	C₄×C₈	C₄×D₈₉	1	2	yes	—
34	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
35	C₂×C₈	C₈×D₈₉	1	4	yes	—
36	C₂×C₈	C₂×#10623494	1	2	yes	—
37	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
38	C₁₆	—	178	8	no	—
39	C₁₆	#13083549	1	8	yes	complemented
40	C₂×C₄	C₈×Dic₈₉	1	2	yes	—
41	C₂×C₄	C₄×#10623494	1	1	yes	characteristic
42	C₈	C₂×#13083549	1	4	yes	—
43	C₈	#15667951	1	8	yes	—
44	C₂²	C₈×#10623494	1	1	yes	characteristic
45	C₄	C₄×#13083549	1	2	yes	—
46	C₄	#19151373	1	4	yes	—
47	C₂	C₈×#13083549	1	1	yes	characteristic
48	C₂	#26018168	1	2	yes	—

	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
14. C₁₆×C₈₉	C₁₆	semidirect
27. C₈₉	#7052	semidirect
39. C₁₆	#13083549	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