Subgroups — #36441045

Order 22784.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018170	C₂	1	1	yes	maximal, characteristic
2	#26018169	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1312	C₂	1	1	yes	maximal, characteristic
4	C₄×#13083549	C₄	1	2	yes	—
5	C₄×#13083549	C₄	1	1	yes	characteristic
6	C₄×#13083549	C₄	1	1	yes	characteristic
7	C₄×C₁₆×C₈₉	C₄	1	2	yes	—
8	C₈²×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#13083549	—	2	2	no	—
10	C₂×#13083549	C₈	1	2	yes	—
11	C₂×#13083549	C₈	1	2	yes	—
12	C₂×C₁₆×C₈₉	—	2	2	no	—
13	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	#13083549	—	2	2	no	—
17	#13083549	—	4	2	no	—
18	#13083549	C₁₆	1	4	yes	complemented
19	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
20	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
21	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
22	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
23	C₄²×C₈₉	C₄²	1	1	yes	characteristic
24	C₁₆×C₈₉	—	4	2	no	—
25	C₂×C₄×C₈₉	C₄×C₈	1	1	yes	characteristic
26	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
27	C₂×C₄×C₈₉	#97	1	1	yes	characteristic
28	C₈×C₈₉	—	2	2	no	—
29	C₈×C₈₉	—	2	2	no	—
30	C₈×C₈₉	C₂×C₁₆	1	2	yes	—
31	C₈×C₈₉	M₃₂	1	2	yes	—
32	C₂²×C₈₉	#322	1	1	yes	characteristic
33	C₄×C₈₉	C₄×C₁₆	1	1	yes	characteristic
34	C₄×C₈₉	#346	1	1	yes	characteristic
35	C₄×C₈₉	#347	1	2	yes	—
36	C₄×C₈₉	#347	1	2	yes	—
37	#7053	—	89	1	no	maximal, Sylow
38	C₂×C₈₉	#1312	1	1	yes	characteristic
39	C₂×C₈₉	#1313	1	1	yes	characteristic
40	C₂×C₈₉	#1313	1	1	yes	characteristic
41	#1312	—	89	1	no	—
42	#1312	D₈₉	1	1	yes	characteristic
43	#1311	—	89	1	no	—
44	C₈₉	#7053	1	1	yes	characteristic, complemented, Sylow
45	C₄×C₁₆	—	89	1	no	—
46	C₄×C₁₆	—	89	1	no	—
47	C₄×C₁₆	—	89	2	no	—
48	C₄×C₁₆	Dic₈₉	1	2	yes	—
49	C₈²	C₂×D₈₉	1	1	yes	characteristic
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	—	89	2	no	—
52	C₂×C₁₆	—	89	2	no	—
53	C₂×C₁₆	—	178	2	no	—
54	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
55	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
56	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
57	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
58	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
59	C₂×C₈	#13083557	1	2	yes	—
60	C₂×C₈	#13083558	1	2	yes	—
61	C₄²	C₄×Dic₈₉	1	1	yes	characteristic
62	C₁₆	—	4	2	no	—
63	C₁₆	—	89	4	no	—
64	C₁₆	—	356	2	no	—
65	C₁₆	—	178	2	no	—
66	C₂×C₄	C₈×Dic₈₉	1	1	yes	characteristic
67	C₂×C₄	#15667922	1	1	yes	characteristic
68	C₂×C₄	#15667923	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	2	2	no	—
71	C₈	C₁₆×D₈₉	1	2	yes	—
72	C₈	#15667951	1	2	yes	—
73	C₂²	#19151320	1	1	yes	characteristic
74	C₄	C₁₆×Dic₈₉	1	1	yes	characteristic
75	C₄	#19151373	1	1	yes	characteristic
76	C₄	#19151375	1	2	yes	—
77	C₄	#19151376	1	2	yes	—
78	C₂	#26018171	1	1	yes	characteristic
79	C₂	#26018173	1	1	yes	characteristic
80	C₂	#26018174	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×Dic₈₉
commutator	C₄×C₈₉	C₄×C₁₆
Frattini	C₈²	C₂×D₈₉
Fitting	C₈₉×#1312	C₂
socle, prime radical	C₂²×C₈₉	#322
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1312	D₈₉
hypocenter	C₈₉	#7053
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ #1312

Complements

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

N	H ≅ G/N	type
18. #13083549	C₁₆	semidirect
44. C₈₉	#7053	semidirect

Maximal subgroups

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

subgroup	index
#26018170	2
#26018169	2
C₈₉×#1312	2
#7053	89