Subgroups — #36441159

Order 22784.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26018322	C₂	1	2	yes	maximal, complemented
2	C₈₉×#1712	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13083549	—	2	2	no	—
4	#493×C₈₉	—	2	2	no	—
5	#493×C₈₉	C₄	1	2	yes	—
6	C₈²×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×#13083549	—	4	2	no	—
8	C₂×D₈×C₈₉	—	4	2	no	—
9	C₂×D₈×C₈₉	—	4	2	no	—
10	C₂×D₈×C₈₉	—	4	2	no	—
11	#127×C₈₉	—	2	2	no	—
12	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₈₉	D₄	1	2	yes	—
14	#13083549	—	8	2	no	—
15	C₂×D₄×C₈₉	—	4	2	no	—
16	C₂×D₄×C₈₉	—	4	4	no	—
17	D₈×C₈₉	—	8	4	no	—
18	D₈×C₈₉	—	8	4	no	—
19	D₈×C₈₉	—	8	2	no	—
20	D₈×C₈₉	—	8	2	no	—
21	C₂×C₈×C₈₉	—	2	1	no	—
22	C₂×C₈×C₈₉	D₈	1	2	yes	—
23	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
24	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₂³×C₈₉	—	8	2	no	—
26	D₄×C₈₉	—	8	4	no	—
27	D₄×C₈₉	—	8	4	no	—
28	D₄×C₈₉	—	8	4	no	—
29	C₂×C₄×C₈₉	#100	1	1	yes	characteristic
30	C₂×C₄×C₈₉	#102	1	2	yes	—
31	C₈×C₈₉	—	2	2	no	—
32	C₈×C₈₉	—	2	2	no	—
33	C₈×C₈₉	D₁₆	1	2	yes	—
34	C₈×C₈₉	SD₃₂	1	2	yes	—
35	C₂²×C₈₉	—	16	2	no	—
36	C₂²×C₈₉	—	16	4	no	—
37	C₂²×C₈₉	#331	1	1	yes	characteristic
38	C₄×C₈₉	—	2	1	no	—
39	C₄×C₈₉	#357	1	2	yes	—
40	C₄×C₈₉	#361	1	2	yes	—
41	#7096	—	89	1	no	maximal, Sylow
42	C₂×C₈₉	—	32	2	no	—
43	C₂×C₈₉	#1357	1	1	yes	characteristic
44	C₂×C₈₉	#1361	1	2	yes	—
45	#1712	D₈₉	1	1	yes	characteristic
46	#1370	—	89	2	no	—
47	C₈₉	#7096	1	1	yes	characteristic, complemented, Sylow
48	#493	—	2	2	no	—
49	#493	Dic₈₉	1	2	yes	—
50	C₄⋊C₁₆	—	178	2	no	—
51	C₈²	C₂×D₈₉	1	1	yes	characteristic
52	C₂×D₈	—	4	2	no	—
53	C₂×D₈	—	4	2	no	—
54	C₂×D₈	—	4	2	no	—
55	#127	—	2	2	no	—
56	C₂×C₁₆	—	356	2	no	—
57	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
58	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
59	C₂×D₄	—	4	2	no	—
60	C₂×D₄	—	4	4	no	—
61	D₈	—	8	4	no	—
62	D₈	—	8	4	no	—
63	D₈	—	8	2	no	—
64	D₈	—	8	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	#13083559	1	2	yes	—
67	C₂×C₈	#13083562	1	2	yes	—
68	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
69	C₁₆	—	712	2	no	—
70	C₂³	—	8	2	no	—
71	D₄	—	8	4	no	—
72	D₄	—	8	4	no	—
73	D₄	—	8	4	no	—
74	C₂×C₄	#15667929	1	1	yes	characteristic
75	C₂×C₄	#15667933	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	#15667953	1	2	yes	—
79	C₈	#15667956	1	2	yes	—
80	C₂²	—	16	2	no	—
81	C₂²	—	16	4	no	—
82	C₂²	#19151342	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#19151399	1	2	yes	—
85	C₄	#19151411	1	2	yes	—
86	C₂	—	32	2	no	—
87	C₂	#26018289	1	1	yes	characteristic
88	C₂	#26018298	1	2	yes	—

	subgroup	G/H
center	C₂²	#19151342
commutator	C₄×C₈×C₈₉	C₂×C₄
Frattini	C₈²	C₂×D₈₉
Fitting, prime radical	C₈₉×#1712	C₂
socle	C₂²×C₈₉	#331
cosocle	C₈²×C₈₉	C₂²
hypercenter	#1712	D₈₉
hypocenter	C₈₉	#7096
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

Complements

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

N	H ≅ G/N	type
1. #26018322	C₂	semidirect
47. C₈₉	#7096	semidirect

Maximal subgroups

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

subgroup	index
#26018322	2
C₈₉×#1712	2
#7096	89