Subgroups — #36442292

Order 22784.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018421	C₂	1	2	yes	maximal
2	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic
3	#19151397	—	2	2	no	—
4	#500×C₈₉	—	2	2	no	—
5	#494×C₈₉	C₄	1	1	yes	characteristic
6	#494×C₈₉	C₂²	1	1	yes	characteristic
7	C₈²×C₈₉	C₄	1	1	yes	characteristic
8	#15667931	—	4	2	no	—
9	C₂×Q₁₆×C₈₉	—	4	1	no	—
10	C₂×Q₁₆×C₈₉	—	4	1	no	—
11	C₄⋊Q₈×C₈₉	—	2	1	no	—
12	C₄⋊Q₈×C₈₉	D₄	1	2	yes	—
13	#107×C₈₉	—	4	4	no	—
14	C₄×C₈×C₈₉	—	2	1	no	—
15	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	#13083558	—	8	2	no	—
17	C₂×Q₈×C₈₉	—	2	2	no	—
18	C₂×Q₈×C₈₉	—	4	1	no	—
19	Q₁₆×C₈₉	—	8	2	no	—
20	Q₁₆×C₈₉	—	8	2	no	—
21	C₂×C₈×C₈₉	—	2	2	no	—
22	C₂×C₈×C₈₉	—	2	1	no	—
23	C₄⋊C₄×C₈₉	—	4	2	no	—
24	C₄⋊C₄×C₈₉	—	4	2	no	—
25	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
26	Q₈×C₈₉	—	8	2	no	—
27	Q₈×C₈₉	—	8	2	no	—
28	C₂×C₄×C₈₉	—	2	1	no	—
29	C₂×C₄×C₈₉	—	4	2	no	—
30	C₂×C₄×C₈₉	—	8	1	no	—
31	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
32	C₈×C₈₉	—	2	4	no	—
33	C₈×C₈₉	—	2	2	no	—
34	#10623494	—	16	2	no	—
35	C₂²×C₈₉	#353	1	1	yes	characteristic
36	C₄×C₈₉	—	2	2	no	—
37	C₄×C₈₉	—	2	1	no	—
38	C₄×C₈₉	—	8	2	no	—
39	C₄×C₈₉	—	16	1	no	—
40	#7532	—	89	1	no	maximal, Sylow
41	C₂×C₈₉	—	2	1	no	—
42	C₂×C₈₉	#1408	1	1	yes	characteristic
43	#1713	D₈₉	1	1	yes	characteristic
44	#1411	—	89	2	no	—
45	C₈₉	#7532	1	1	yes	characteristic, complemented, Sylow
46	#500	—	2	2	no	—
47	#494	Dic₈₉	1	1	yes	characteristic
48	#494	C₂×D₈₉	1	1	yes	characteristic
49	#356	—	178	2	no	—
50	C₈²	Dic₈₉	1	1	yes	characteristic
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	4	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	C₂²⋊D₈₉	1	2	yes	—
55	#107	—	4	4	no	—
56	#101	—	356	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	4	1	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	8	2	no	—
63	M₁₆	—	712	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	8	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	8	1	no	—
74	C₂×C₄	#15667927	1	1	yes	characteristic
75	C₈	—	2	4	no	—
76	C₈	—	2	2	no	—
77	C₈	—	1424	2	no	—
78	C₂²	#19151391	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	8	2	no	—
82	C₄	—	16	1	no	—
83	C₂	—	2	1	no	—
84	C₂	#26018415	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26018415
commutator	C₄×C₈×C₈₉	C₂×C₄
Frattini	#494	C₂×D₈₉
Fitting	C₈₉×#1713	C₂
socle	C₂×C₈₉	#1408
prime radical	C₂²×C₈₉	#353
cosocle	#494×C₈₉	C₂²
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#7532
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1713

Complements

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

N	H ≅ G/N	type
45. C₈₉	#7532	semidirect

Maximal subgroups

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

subgroup	index
#26018421	2
C₈₉×#1713	2
#7532	89