Group #26021757 — subgroups

Subgroups — #26021757

Order 11392.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151641	C₂	1	2	yes	maximal
2	C₂×#15667946	C₂	1	1	yes	maximal, characteristic
3	#19151366	C₂	1	2	yes	maximal
4	#19151363	C₂	1	1	yes	maximal, characteristic
5	#424×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
8	#15667946	—	2	2	no	—
9	#15667945	—	2	2	no	—
10	#15667943	—	2	2	no	—
11	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	#117×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	—	2	2	no	—
15	C₂²×Dic₈₉	—	2	2	no	—
16	#13083558	—	2	2	no	—
17	#13083558	Q₈	1	4	yes	—
18	C₂×#10623494	—	2	2	no	—
19	C₄⋊Dic₈₉	—	2	2	no	—
20	C₄⋊Dic₈₉	—	2	2	no	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	—	2	2	no	—
24	C₂×C₈×C₈₉	—	2	1	no	—
25	C₂×C₈×C₈₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₈₉	—	2	1	no	—
29	C₄²×C₈₉	—	2	1	no	—
30	C₂³×C₈₉	#41	1	1	yes	characteristic
31	C₂×C₄×C₈₉	—	2	1	no	—
32	C₂×C₄×C₈₉	—	2	1	no	—
33	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₈₉	#41	1	2	yes	—
37	C₂×Dic₈₉	—	4	2	no	—
38	C₂×Dic₈₉	—	8	2	no	—
39	C₈×C₈₉	—	2	2	no	—
40	C₈×C₈₉	—	4	1	no	—
41	#10623494	—	2	4	no	—
42	C₂²×C₈₉	—	2	1	no	—
43	C₂²×C₈₉	—	2	1	no	—
44	C₂²×C₈₉	#121	1	1	yes	characteristic
45	C₂²×C₈₉	#125	1	2	yes	—
46	C₄×C₈₉	—	4	1	no	—
47	C₄×C₈₉	—	4	1	no	—
48	C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₈₉	#123	1	1	yes	characteristic
50	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₈₉	—	8	2	no	—
52	C₂×C₈₉	—	2	2	no	—
53	C₂×C₈₉	#398	1	1	yes	characteristic
54	C₂×C₈₉	#472	1	1	yes	characteristic
55	C₂×C₈₉	#473	1	1	yes	characteristic
56	#2089	—	89	1	no	maximal, Sylow
57	C₈₉	#2089	1	1	yes	characteristic, complemented, Sylow
58	#428	—	89	2	no	—
59	C₂×#107	—	89	1	no	—
60	#424	D₈₉	1	1	yes	characteristic
61	#343	—	89	2	no	—
62	#341	—	89	1	no	—
63	C₂×M₁₆	—	89	2	no	—
64	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
65	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
66	#117	C₂×D₈₉	1	1	yes	characteristic
67	C₂×C₄⋊C₄	—	89	2	no	—
68	#107	—	178	2	no	—
69	#107	—	178	2	no	—
70	#106	—	178	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₂²×C₄	—	178	2	no	—
73	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
74	M₁₆	—	2	2	no	—
75	M₁₆	—	89	4	no	—
76	M₁₆	—	178	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	178	2	no	—
80	C₂×C₈	C₂²⋊D₈₉	1	2	yes	—
81	C₄⋊C₄	—	178	2	no	—
82	C₄⋊C₄	—	178	2	no	—
83	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
84	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
85	C₂²⋊C₄	—	2	1	no	—
86	C₄²	—	2	1	no	—
87	C₂³	#13083574	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	356	2	no	—
91	C₂×C₄	—	712	2	no	—
92	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
93	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
94	C₂×C₄	Q₈×D₈₉	1	2	yes	—
95	C₂×C₄	#13083573	1	2	yes	—
96	C₈	—	2	2	no	—
97	C₈	—	4	1	no	—
98	C₈	—	178	4	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	#15667986	1	1	yes	characteristic
102	C₂²	#15668002	1	2	yes	—
103	C₄	—	4	1	no	—
104	C₄	—	4	1	no	—
105	C₄	—	712	2	no	—
106	C₄	#15667992	1	2	yes	—
107	C₄	#15667993	1	1	yes	characteristic
108	C₄	#15668010	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#19151528	1	1	yes	characteristic
111	C₂	#19151868	1	1	yes	characteristic
112	C₂	#19151873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151641

C₂

yes

maximal

C₂×#15667946

C₂

yes

maximal, characteristic

#19151366

C₂

yes

maximal

#19151363

C₂

yes

maximal, characteristic

#424×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

#15667946

—

#15667945

—

#15667943

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

—

C₂²×Dic₈₉

—

#13083558

—

#13083558

Q₈

yes

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#121

yes

characteristic

C₂²×C₈₉

#125

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#123

yes

characteristic

C₄×C₈₉

C₄⋊Q₈

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#428

—

C₂×#107

—

#424

D₈₉

yes

characteristic

#343

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

178

—

#107

—

178

—

#106

—

178

—

C₄⋊C₈

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₂×C₈

C₂²⋊D₈₉

yes

—

C₄⋊C₄

—

178

—

C₄⋊C₄

—

178

—

C₄⋊C₄

D₃₅₆

yes

characteristic

C₄⋊C₄

D₃₅₆

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₂³

#13083574

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

—

712

—

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

C₂×C₄

C₂×D₃₅₆

yes

characteristic

C₂×C₄

Q₈×D₈₉

yes

—

C₂×C₄

#13083573

yes

—

C₈

—

C₈

—

C₈

—

178

—

C₂²

—

100

C₂²

—

101

C₂²

#15667986

yes

characteristic

102

C₂²

#15668002

yes

—

103

C₄

—

104

C₄

—

105

C₄

—

712

—

106

C₄

#15667992

yes

—

107

C₄

#15667993

yes

characteristic

108

C₄

#15668010

yes

characteristic

109

C₂

—

110

C₂

#19151528

yes

characteristic

111

C₂

#19151868

yes

characteristic

112

C₂

#19151873

yes

characteristic

	subgroup	G/H
center	C₂²	#15667986
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#424×C₈₉	C₂
socle	C₂²×C₈₉	#121
prime radical	C₂³×C₈₉	#41
hypercenter	#424	D₈₉
hypocenter	C₈₉	#2089
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#15667986

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#424×C₈₉

C₂

C₂²×C₈₉

C₂³×C₈₉

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
57. C₈₉	#2089	semidirect

H ≅ G/N

type

57. C₈₉

#2089

semidirect

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

subgroup	index
#19151641	2
C₂×#15667946	2
#19151366	2
#19151363	2
#424×C₈₉	2
#2089	89

subgroup

index

#424×C₈₉

Subgroups — #26021757

Conjugacy classes of subgroups

Complements

Maximal subgroups