Group #26029182 — subgroups

Subgroups — #26029182

Order 11392.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#19152155	C₂	1	1	yes	maximal, characteristic
2	#19151725	C₂	1	4	yes	maximal, complemented
3	#19151722	C₂	1	4	yes	maximal, complemented
4	#19151626	C₂	1	2	yes	maximal
5	#19151556	C₂	1	2	yes	maximal
6	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic
7	#423×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂²×#10623494	C₂²	1	1	yes	characteristic
9	C₂²×#10623494	C₂²	1	1	yes	characteristic
10	#15668012	C₄	1	2	yes	—
11	#15668004	C₄	1	2	yes	—
12	#15667991	C₄	1	4	yes	—
13	C₄×Dic₁₇₈	C₂²	1	4	yes	—
14	#15667971	C₂²	1	2	yes	—
15	#15667941	C₂²	1	8	yes	—
16	#15667922	C₂²	1	4	yes	—
17	C₄×#10623494	C₂²	1	2	yes	—
18	C₄×#10623494	C₂²	1	2	yes	—
19	C₈×Dic₈₉	C₂²	1	4	yes	—
20	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₈₉	C₂²	1	4	yes	—
23	C₂×Dic₁₇₈	C₂×C₄	1	4	yes	—
24	C₂×#10623494	—	2	2	no	—
25	C₂×#10623494	Q₈	1	4	yes	complemented
26	C₂×#10623494	C₂³	1	2	yes	—
27	C₂×#10623494	C₂³	1	1	yes	characteristic
28	C₂×#10623494	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
31	#13083553	C₂×C₄	1	8	yes	—
32	C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
33	C₄×Dic₈₉	C₂³	1	4	yes	—
34	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	M₁₆×C₈₉	—	2	4	no	—
37	C₂×C₈×C₈₉	C₂³	1	4	yes	—
38	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₈₉	C₂³	1	2	yes	—
41	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₈₉	—	2	4	no	—
43	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₈₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₈₉	#41	1	2	yes	—
48	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₈₉	C₂²×C₄	1	8	yes	—
50	Dic₁₇₈	—	2	8	no	—
51	C₈×C₈₉	—	2	4	no	—
52	#10623494	—	2	2	no	—
53	#10623494	C₂×Q₈	1	2	yes	—
54	#10623494	C₂×Q₈	1	2	yes	—
55	C₂²×C₈₉	—	2	2	no	—
56	C₂²×C₈₉	C₄×Q₈	1	2	yes	—
57	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₈₉	—	2	2	no	—
59	C₄×C₈₉	C₄×Q₈	1	2	yes	—
60	C₄×C₈₉	#131	1	4	yes	—
61	C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
63	Dic₈₉	—	2	8	no	—
64	C₂×C₈₉	—	2	2	no	—
65	C₂×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₈₉	C₂×#131	1	2	yes	—
67	#2960	—	89	1	no	maximal, Sylow
68	C₈₉	#2960	1	1	yes	characteristic, complemented, Sylow
69	#533	—	89	1	no	—
70	#446	—	89	4	no	—
71	C₈×Q₈	—	89	4	no	—
72	#424	—	89	2	no	—
73	#423	D₈₉	1	1	yes	characteristic
74	#405	—	89	2	no	—
75	C₂×C₄×C₈	—	89	1	no	—
76	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
77	C₂²×C₈	—	89	1	no	—
78	C₂²×C₈	—	89	1	no	—
79	C₄⋊Q₈	—	89	2	no	—
80	#125	—	89	2	no	—
81	C₂²⋊Q₈	—	89	4	no	—
82	C₄×Q₈	—	89	4	no	—
83	#117	—	89	2	no	—
84	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
85	C₄⋊C₈	—	89	8	no	—
86	C₄⋊C₈	C₂×D₈₉	1	4	yes	—
87	#97	—	89	4	no	—
88	C₄×C₈	—	89	4	no	—
89	C₄×C₈	—	89	2	no	—
90	C₄×C₈	—	89	2	no	—
91	C₂×Q₈	—	89	4	no	—
92	C₂²×C₄	C₄×D₈₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	C₂×C₈	—	89	4	no	—
96	C₂×C₈	—	89	1	no	—
97	C₂×C₈	—	89	1	no	—
98	C₂×C₈	—	89	2	no	—
99	C₂×C₈	—	178	2	no	—
100	C₂×C₈	C₂²×D₈₉	1	4	yes	—
101	C₄⋊C₄	—	89	4	no	—
102	C₄⋊C₄	—	89	8	no	—
103	C₄⋊C₄	—	89	4	no	—
104	C₂²⋊C₄	—	89	4	no	—
105	C₄²	—	89	4	no	—
106	C₄²	C₄×D₈₉	1	1	yes	characteristic
107	C₄²	C₄×D₈₉	1	1	yes	characteristic
108	C₄²	C₂²×D₈₉	1	2	yes	—
109	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
110	Q₈	—	178	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	89	8	no	—
113	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
116	C₂×C₄	Q₈×D₈₉	1	2	yes	—
117	C₂×C₄	#13083573	1	2	yes	—
118	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	89	2	no	—
121	C₈	—	89	2	no	—
122	C₈	—	178	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#15667977	1	2	yes	—
125	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	178	8	no	—
128	C₄	#15667977	1	2	yes	—
129	C₄	#15668020	1	4	yes	—
130	C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
131	C₄	C₂×#13083573	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#15667977	1	1	yes	characteristic
134	C₂	C₂×#15668020	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19152155

C₂

yes

maximal, characteristic

#19151725

C₂

yes

maximal, complemented

#19151722

C₂

yes

maximal, complemented

#19151626

C₂

yes

maximal

#19151556

C₂

yes

maximal

C₂×C₄×#10623494

C₂

yes

maximal, characteristic

#423×C₈₉

C₂

yes

maximal, characteristic

C₂²×#10623494

C₂²

yes

characteristic

C₂²×#10623494

C₂²

yes

characteristic

#15668012

C₄

yes

—

#15668004

C₄

yes

—

#15667991

C₄

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667971

C₂²

yes

—

#15667941

C₂²

yes

—

#15667922

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₈×Dic₈₉

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×#10623494

—

C₂×#10623494

Q₈

yes

complemented

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

characteristic

C₂×#10623494

C₂³

yes

characteristic

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

C₂²⋊Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

C₂²×C₄

yes

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

C₂×Q₈

yes

—

#10623494

C₂×Q₈

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

C₄×Q₈

yes

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₄×Q₈

yes

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

C₂²×Q₈

yes

characteristic

C₄×C₈₉

C₂×#41

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₈₉

C₂×#131

yes

—

#2960

—

maximal, Sylow

C₈₉

#2960

yes

characteristic, complemented, Sylow

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

D₈₉

yes

characteristic

#405

—

C₂×C₄×C₈

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₈₉

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

178

—

100

C₂×C₈

C₂²×D₈₉

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₈₉

yes

characteristic

107

C₄²

C₄×D₈₉

yes

characteristic

108

C₄²

C₂²×D₈₉

yes

—

109

C₂³

C₂×C₄×D₈₉

yes

characteristic

110

Q₈

—

178

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₈₉

yes

—

114

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

115

C₂×C₄

C₂×C₄×D₈₉

yes

—

116

C₂×C₄

Q₈×D₈₉

yes

—

117

C₂×C₄

#13083573

yes

—

118

C₂×C₄

C₂³×D₈₉

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

178

—

123

C₂²

—

124

C₂²

#15667977

yes

—

125

C₂²

C₂²×C₄×D₈₉

yes

characteristic

126

C₄

—

127

C₄

—

178

—

128

C₄

#15667977

yes

—

129

C₄

#15668020

yes

—

130

C₄

C₂×Q₈×D₈₉

yes

characteristic

131

C₄

C₂×#13083573

yes

characteristic

132

C₂

—

133

C₂

C₂×#15667977

yes

characteristic

134

C₂

C₂×#15668020

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#423×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#423	D₈₉
hypocenter	C₈₉	#2960
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#423×C₈₉

C₂

prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #19151725	C₂	semidirect
3. #19151722	C₂	semidirect
25. C₂×#10623494	Q₈	semidirect
68. C₈₉	#2960	semidirect

H ≅ G/N

type

2. #19151725

C₂

semidirect

3. #19151722

C₂

semidirect

25. C₂×#10623494

Q₈

semidirect

68. C₈₉

#2960

semidirect

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

subgroup	index
#19152155	2
#19151725	2
#19151722	2
#19151626	2
#19151556	2
C₂×C₄×#10623494	2
#423×C₈₉	2
#2960	89

subgroup

index

#423×C₈₉

Subgroups — #26029182

Conjugacy classes of subgroups

Complements

Maximal subgroups