Group #26029193 — subgroups

Subgroups — #26029193

Order 11392.

348 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 221 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	#19151729	C₂	1	8	yes	maximal, complemented
2	#19151625	C₂	1	2	yes	maximal
3	#19151622	C₂	1	1	yes	maximal, characteristic
4	#19151554	C₂	1	2	yes	maximal
5	C₂×#15667923	C₂	1	1	yes	maximal, characteristic
6	#533×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13083558	C₂²	1	2	yes	—
8	C₂²×#10623494	C₂²	1	1	yes	characteristic
9	C₂²×#10623494	C₂²	1	1	yes	characteristic
10	C₄⋊#10623494	C₂²	1	4	yes	—
11	C₄⋊#10623494	C₂²	1	8	yes	—
12	#15667923	C₂²	1	4	yes	—
13	#15667923	C₂²	1	4	yes	—
14	C₄×#10623494	C₂²	1	4	yes	—
15	C₄⋊Q₈×C₈₉	C₄	1	2	yes	—
16	#125×C₈₉	C₄	1	2	yes	—
17	C₂²⋊Q₈×C₈₉	C₄	1	4	yes	—
18	C₄×Q₈×C₈₉	C₂²	1	4	yes	—
19	#117×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	#13083558	—	2	4	no	—
22	C₂×#10623494	—	2	2	no	—
23	C₂×#10623494	Q₈	1	4	yes	complemented
24	C₂×#10623494	C₂³	1	2	yes	—
25	C₂×#10623494	C₂³	1	4	yes	—
26	C₂×#10623494	C₂³	1	2	yes	—
27	C₂×Q₈×C₈₉	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
31	C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
32	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
34	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₈₉	C₂³	1	2	yes	—
37	C₄²×C₈₉	C₂³	1	4	yes	—
38	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
39	Q₈×C₈₉	—	4	4	no	—
40	C₂×C₄×C₈₉	—	4	2	no	—
41	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	8	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₈₉	#41	1	2	yes	—
47	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
48	#10623494	—	2	2	no	—
49	#10623494	—	2	4	no	—
50	#10623494	C₂×Q₈	1	4	yes	—
51	C₂²×C₈₉	—	2	2	no	—
52	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
53	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₈₉	—	2	2	no	—
55	C₄×C₈₉	—	2	2	no	—
56	C₄×C₈₉	—	2	8	no	—
57	C₄×C₈₉	C₄×Q₈	1	2	yes	—
58	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
60	C₂×C₈₉	—	2	2	no	—
61	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₈₉	#568	1	2	yes	—
63	#2961	—	89	1	no	maximal, Sylow
64	C₈₉	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	D₈₉	1	1	yes	characteristic
66	#446	—	89	8	no	—
67	#424	—	89	2	no	—
68	#423	—	89	1	no	—
69	#405	—	89	2	no	—
70	C₂×#97	—	89	1	no	—
71	C₂×M₁₆	—	89	2	no	—
72	C₂²×C₈	—	89	1	no	—
73	C₂²×C₈	—	89	1	no	—
74	C₄⋊Q₈	Dic₈₉	1	2	yes	—
75	#125	Dic₈₉	1	2	yes	—
76	C₂²⋊Q₈	Dic₈₉	1	4	yes	—
77	C₄×Q₈	C₂×D₈₉	1	4	yes	—
78	#117	C₂×D₈₉	1	2	yes	—
79	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
80	C₄⋊C₈	—	89	4	no	—
81	C₄⋊C₈	—	89	8	no	—
82	#97	—	89	4	no	—
83	#97	—	89	4	no	—
84	C₄×C₈	—	89	4	no	—
85	C₂×Q₈	C₂×Dic₈₉	1	4	yes	—
86	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
87	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
88	M₁₆	—	178	4	no	—
89	C₂×C₈	—	89	4	no	—
90	C₂×C₈	—	89	2	no	—
91	C₂×C₈	—	89	4	no	—
92	C₂×C₈	—	89	2	no	—
93	C₂×C₈	—	178	2	no	—
94	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
95	C₄⋊C₄	C₂×Dic₈₉	1	8	yes	—
96	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
98	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
99	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
100	C₄²	C₂²×D₈₉	1	4	yes	—
101	C₄²	C₂²×D₈₉	1	2	yes	—
102	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
103	Q₈	—	4	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
106	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
107	C₂×C₄	C₂²×Dic₈₉	1	8	yes	—
108	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
109	C₂×C₄	Q₈×D₈₉	1	2	yes	—
110	C₂×C₄	#13083575	1	2	yes	—
111	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
112	C₈	—	89	4	no	—
113	C₈	—	178	2	no	—
114	C₈	—	178	4	no	—
115	C₂²	—	2	2	no	—
116	C₂²	Q₈×Dic₈₉	1	2	yes	—
117	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	8	no	—
121	C₄	Q₈×Dic₈₉	1	2	yes	—
122	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
123	C₄	C₂×#13083575	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×Q₈×Dic₈₉	1	1	yes	characteristic
126	C₂	#19152365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151729

C₂

yes

maximal, complemented

#19151625

C₂

yes

maximal

#19151622

C₂

yes

maximal, characteristic

#19151554

C₂

yes

maximal

C₂×#15667923

C₂

yes

maximal, characteristic

#533×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₄⋊#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667923

C₂²

yes

—

#15667923

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₄

yes

—

#125×C₈₉

C₄

yes

—

C₂²⋊Q₈×C₈₉

C₄

yes

—

C₄×Q₈×C₈₉

C₂²

yes

—

#117×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

—

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

—

—

C₂×Q₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×Q₈

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄×Q₈

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₈₉

#568

yes

—

#2961

—

maximal, Sylow

C₈₉

#2961

yes

characteristic, complemented, Sylow

#533

D₈₉

yes

—

—

—

—

C₂×#97

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₈₉

yes

—

#125

Dic₈₉

yes

—

C₂²⋊Q₈

Dic₈₉

yes

—

C₄×Q₈

C₂×D₈₉

yes

—

#117

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₂²⋊C₄

C₂×Dic₈₉

yes

—

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

100

C₄²

C₂²×D₈₉

yes

—

101

C₄²

C₂²×D₈₉

yes

—

102

C₂³

C₂²×Dic₈₉

yes

characteristic

103

Q₈

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₈₉

yes

—

106

C₂×C₄

C₂²×Dic₈₉

yes

—

107

C₂×C₄

C₂²×Dic₈₉

yes

—

108

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

109

C₂×C₄

Q₈×D₈₉

yes

—

110

C₂×C₄

#13083575

yes

—

111

C₂×C₄

C₂³×D₈₉

yes

characteristic

112

C₈

—

113

C₈

—

178

—

114

C₈

—

178

—

115

C₂²

—

116

C₂²

Q₈×Dic₈₉

yes

—

117

C₂²

C₂³×Dic₈₉

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

Q₈×Dic₈₉

yes

—

122

C₄

C₂×Q₈×D₈₉

yes

characteristic

123

C₄

C₂×#13083575

yes

characteristic

124

C₂

—

125

C₂

C₂×Q₈×Dic₈₉

yes

characteristic

126

C₂

#19152365

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#533×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#533	D₈₉
hypocenter	C₈₉	#2961
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#533×C₈₉

C₂

prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #19151729	C₂	semidirect
23. C₂×#10623494	Q₈	semidirect
64. C₈₉	#2961	semidirect

H ≅ G/N

type

1. #19151729

C₂

semidirect

23. C₂×#10623494

Q₈

semidirect

64. C₈₉

#2961

semidirect

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

subgroup	index
#19151729	2
#19151625	2
#19151622	2
#19151554	2
C₂×#15667923	2
#533×C₈₉	2
#2961	89

subgroup

index

#533×C₈₉

Subgroups — #26029193

Conjugacy classes of subgroups

Complements

Maximal subgroups