Subgroups — #36475454

Order 22784.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022526	C₂	1	2	yes	maximal
2	#26018454	C₂	1	1	yes	maximal, characteristic
3	#26018453	C₂	1	2	yes	maximal
4	C₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1424	C₂	1	1	yes	maximal, characteristic
6	#19152002	—	2	2	no	—
7	C₂×#15667918	C₂²	1	1	yes	characteristic
8	#19151424	C₂²	1	2	yes	—
9	#19151423	—	2	2	no	—
10	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
12	#366×C₈₉	C₂²	1	2	yes	—
13	#15668011	—	4	2	no	—
14	C₂×#13083549	D₄	1	1	yes	characteristic
15	#15667946	—	2	2	no	—
16	#15667945	—	4	2	no	—
17	C₈×Dic₈₉	D₄	1	1	yes	characteristic
18	#15667918	Q₈	1	2	yes	complemented
19	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
20	#107×C₈₉	—	2	2	no	—
21	C₃₂×C₈₉	Q₈	1	2	yes	complemented
22	C₂×Dic₁₇₈	—	8	2	no	—
23	C₂×#10623494	D₈	1	1	yes	characteristic
24	C₄⋊Dic₈₉	—	4	2	no	—
25	#13083553	—	8	2	no	—
26	C₄×Dic₈₉	D₈	1	1	yes	characteristic
27	#13083549	Q₁₆	1	2	yes	—
28	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₈₉	—	4	2	no	—
30	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
31	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	8	2	no	—
33	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₈₉	—	8	2	no	—
35	C₂×Dic₈₉	D₁₆	1	2	yes	—
36	Dic₁₇₈	—	16	4	no	—
37	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
38	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
39	#10623494	Q₃₂	1	2	yes	—
40	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
41	C₄×C₈₉	—	16	2	no	—
42	C₄×C₈₉	#500	1	1	yes	characteristic
43	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
44	Dic₈₉	—	16	2	no	—
45	Dic₈₉	SD₆₄	1	4	yes	—
46	#13733	—	89	1	no	maximal, Sylow
47	C₂×C₈₉	#2252	1	1	yes	characteristic
48	C₂×C₈₉	C₂×SD₆₄	1	2	yes	—
49	#2252	—	89	2	no	—
50	#1424	—	89	1	no	—
51	#1424	—	89	2	no	—
52	#1424	D₈₉	1	1	yes	characteristic
53	C₄×C₃₂	—	89	1	no	—
54	C₈₉	#13733	1	1	yes	characteristic, complemented, Sylow
55	#500	—	178	2	no	—
56	C₂×C₃₂	—	89	1	no	—
57	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
58	#366	—	89	2	no	—
59	#366	—	178	2	no	—
60	#366	C₂×D₈₉	1	2	yes	—
61	C₄×C₁₆	—	89	1	no	—
62	C₄⋊Q₈	—	356	2	no	—
63	C₂×C₁₆	—	89	1	no	—
64	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
65	#107	—	2	2	no	—
66	#107	—	356	2	no	—
67	#107	—	178	2	no	—
68	C₄×C₈	—	89	1	no	—
69	C₃₂	—	89	2	no	—
70	C₃₂	Dic₁₇₈	1	2	yes	complemented
71	C₂×Q₈	—	712	2	no	—
72	C₂×C₈	—	89	1	no	—
73	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	356	2	no	—
76	C₄⋊C₄	—	712	2	no	—
77	C₄²	—	89	1	no	—
78	C₁₆	—	89	2	no	—
79	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
80	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
81	Q₈	—	1424	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	89	2	no	—
84	C₂×C₄	—	712	2	no	—
85	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
86	C₈	—	89	2	no	—
87	C₈	#15668011	1	1	yes	characteristic
88	C₈	Q₁₆×D₈₉	1	1	yes	characteristic
89	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
90	C₄	—	16	2	no	—
91	C₄	—	89	4	no	—
92	C₄	—	1424	2	no	—
93	C₄	#19152002	1	1	yes	characteristic
94	C₄	Q₃₂×D₈₉	1	1	yes	characteristic
95	C₂	#26022526	1	1	yes	characteristic
96	C₂	SD₆₄×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₁₆×D₈₉
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₈₉×#1424	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	#1424	D₈₉
hypocenter	C₈₉	#13733
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
18. #15667918	Q₈	semidirect
21. C₃₂×C₈₉	Q₈	semidirect
54. C₈₉	#13733	semidirect
70. C₃₂	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26022526	2
#26018454	2
#26018453	2
C₃₂×Dic₈₉	2
C₈₉×#1424	2
#13733	89