Subgroups — #36441765

Order 22784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018388	C₂	1	1	yes	maximal, characteristic
2	#26018172	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1398	C₂	1	1	yes	maximal, characteristic
4	#19151432	C₄	1	2	yes	—
5	#19151374	C₄	1	1	yes	characteristic
6	#19151319	C₄	1	1	yes	characteristic
7	M₆₄×C₈₉	C₄	1	2	yes	—
8	#346×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083549	—	2	1	no	—
10	C₄×#10623494	—	2	1	no	—
11	#15667918	C₈	1	4	yes	complemented
12	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₈₉	—	2	2	no	—
16	C₂×#10623494	—	4	1	no	—
17	#13083549	—	4	1	no	—
18	C₂×C₈×C₈₉	C₄²	1	1	yes	characteristic
19	C₂×C₈×C₈₉	M₁₆	1	1	yes	characteristic
20	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
21	C₁₆×C₈₉	C₂×C₈	1	2	yes	—
22	C₁₆×C₈₉	M₁₆	1	2	yes	—
23	C₂×C₄×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
25	C₈×C₈₉	—	2	1	no	—
26	C₈×C₈₉	C₄×C₈	1	1	yes	characteristic
27	C₈×C₈₉	#97	1	1	yes	characteristic
28	#10623494	—	8	1	no	—
29	C₂²×C₈₉	#347	1	1	yes	characteristic
30	C₄×C₈₉	—	4	1	no	—
31	C₄×C₈₉	#322	1	1	yes	characteristic
32	C₄×C₈₉	#347	1	1	yes	characteristic
33	#7333	—	89	1	no	maximal, Sylow
34	C₂×C₈₉	—	2	1	no	—
35	C₂×C₈₉	#1313	1	1	yes	characteristic
36	#1398	—	89	1	no	—
37	#1398	D₈₉	1	1	yes	characteristic
38	#1313	—	89	1	no	—
39	C₈₉	#7333	1	1	yes	characteristic, complemented, Sylow
40	M₆₄	—	89	2	no	—
41	M₆₄	Dic₈₉	1	2	yes	—
42	#346	—	89	1	no	—
43	#346	C₂×D₈₉	1	1	yes	characteristic
44	#322	—	89	1	no	—
45	C₂×C₁₆	—	178	1	no	—
46	C₂×C₁₆	C₄×D₈₉	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₈₉	1	1	yes	characteristic
48	C₄×C₈	—	178	1	no	—
49	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
50	C₃₂	—	2	2	no	—
51	C₃₂	—	89	4	no	—
52	C₂×C₈	—	356	1	no	—
53	C₂×C₈	C₄×Dic₈₉	1	1	yes	characteristic
54	C₂×C₈	#13083557	1	1	yes	characteristic
55	C₄²	#13083557	1	1	yes	characteristic
56	C₁₆	—	356	1	no	—
57	C₁₆	C₈×D₈₉	1	2	yes	—
58	C₁₆	#13083558	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	#15667922	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	712	1	no	—
63	C₈	C₈×Dic₈₉	1	1	yes	characteristic
64	C₈	#15667923	1	1	yes	characteristic
65	C₂²	#19151375	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#19151320	1	1	yes	characteristic
68	C₄	#19151375	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#26018174	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19151320
commutator	C₈×C₈₉	C₄×C₈
Frattini	#346	C₂×D₈₉
Fitting	C₈₉×#1398	C₂
socle	C₂×C₈₉	#1313
prime radical	C₂²×C₈₉	#347
cosocle	#346×C₈₉	C₂²
hypercenter	#1398	D₈₉
hypocenter	C₈₉	#7333
LCS	G ▷ C₈×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
11. #15667918	C₈	semidirect
39. C₈₉	#7333	semidirect

Maximal subgroups

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

subgroup	index
#26018388	2
#26018172	2
C₈₉×#1398	2
#7333	89