Subgroups — #36442096

Order 22784.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018450	C₂	1	2	yes	maximal
2	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic
3	#19151432	—	2	2	no	—
4	C₄×C₁₆×C₈₉	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
6	C₈²×C₈₉	C₄	1	1	yes	characteristic
7	#15667918	—	4	2	no	—
8	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₈₉	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₈₉	Q₈	1	1	yes	characteristic
12	C₄×C₈×C₈₉	—	2	1	no	—
13	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₈₉	—	2	2	no	—
15	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
18	C₁₆×C₈₉	—	2	1	no	—
19	C₁₆×C₈₉	—	2	1	no	—
20	C₁₆×C₈₉	C₁₆	1	2	yes	—
21	C₁₆×C₈₉	SD₁₆	1	2	yes	—
22	C₂×C₄×C₈₉	—	2	1	no	—
23	C₂×C₄×C₈₉	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₈₉	—	4	2	no	—
25	C₈×C₈₉	#106	1	1	yes	characteristic
26	C₈×C₈₉	#108	1	1	yes	characteristic
27	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₈₉	M₃₂	1	1	yes	characteristic
29	C₂²×C₈₉	#364	1	1	yes	characteristic
30	C₄×C₈₉	—	2	2	no	—
31	C₄×C₈₉	#334	1	1	yes	characteristic
32	C₄×C₈₉	C₄⋊C₁₆	1	1	yes	characteristic
33	#7456	—	89	1	no	maximal, Sylow
34	C₂×C₈₉	—	2	1	no	—
35	C₂×C₈₉	#1367	1	1	yes	characteristic
36	#1422	—	89	2	no	—
37	#1311	D₈₉	1	1	yes	characteristic
38	C₈₉	#7456	1	1	yes	characteristic, complemented, Sylow
39	M₆₄	—	178	2	no	—
40	C₄×C₁₆	Dic₈₉	1	1	yes	characteristic
41	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
42	C₈²	Dic₈₉	1	1	yes	characteristic
43	C₂×C₁₆	#10623494	1	1	yes	characteristic
44	C₂×C₁₆	#10623494	1	1	yes	characteristic
45	C₂×C₁₆	Dic₁₇₈	1	1	yes	characteristic
46	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
49	C₃₂	—	356	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄⋊Dic₈₉	1	1	yes	characteristic
52	C₂×C₈	C₂×#10623494	1	1	yes	characteristic
53	C₄²	#13083558	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	#13083549	1	2	yes	—
57	C₁₆	#13083563	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊#10623494	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#15667944	1	1	yes	characteristic
62	C₈	#15667948	1	1	yes	characteristic
63	C₈	C₂×#13083549	1	1	yes	characteristic
64	C₈	#15667952	1	1	yes	characteristic
65	C₂²	#19151419	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#19151349	1	1	yes	characteristic
68	C₄	C₄⋊#13083549	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#26018316	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#15667944
commutator	C₈×C₈₉	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₈₉
Fitting	C₈₉×#1311	C₂
socle	C₂×C₈₉	#1367
prime radical	C₂²×C₈₉	#364
cosocle	C₄×C₁₆×C₈₉	C₂²
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#7456
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1311

Complements

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

N	H ≅ G/N	type
38. C₈₉	#7456	semidirect

Maximal subgroups

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

subgroup	index
#26018450	2
C₈₉×#1311	2
#7456	89