Group #26029349 — subgroups

Subgroups — #26029349

Order 11392.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#19151726	C₂	1	4	yes	maximal, complemented
3	#19151725	C₂	1	2	yes	maximal, complemented
4	#19151722	C₂	1	2	yes	maximal, complemented
5	#19151656	C₂	1	4	yes	maximal
6	#19151651	C₂	1	1	yes	maximal, characteristic
7	#432×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#15668012	C₄	1	1	yes	characteristic
9	#15668012	C₄	1	1	yes	characteristic
10	#15668004	C₄	1	2	yes	—
11	#15667991	C₄	1	2	yes	—
12	#15667991	C₄	1	2	yes	—
13	C₄×Dic₁₇₈	C₂²	1	2	yes	—
14	C₄×Dic₁₇₈	C₂²	1	2	yes	—
15	#15667971	C₂²	1	2	yes	—
16	C₄⋊#10623494	C₂²	1	2	yes	—
17	#15667941	C₂²	1	4	yes	—
18	#15667941	C₂²	1	4	yes	—
19	C₂²⋊#10623494	C₂²	1	2	yes	—
20	#15667922	C₂²	1	4	yes	—
21	C₄×#10623494	C₂²	1	1	yes	characteristic
22	C₄×#10623494	C₂²	1	1	yes	characteristic
23	C₈×Dic₈₉	C₂²	1	4	yes	—
24	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
26	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
27	#97×C₈₉	C₂²	1	2	yes	—
28	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₇₈	C₂×C₄	1	2	yes	—
30	C₂×#10623494	C₂³	1	2	yes	—
31	C₂×#10623494	C₂³	1	2	yes	—
32	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
36	#13083553	C₂×C₄	1	4	yes	—
37	#13083553	C₂×C₄	1	4	yes	—
38	C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
39	C₄×Dic₈₉	C₂³	1	4	yes	—
40	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₈₉	C₂³	1	4	yes	—
44	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄²×C₈₉	C₂³	1	1	yes	characteristic
48	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₈₉	—	2	2	no	—
50	C₂×C₄×C₈₉	—	2	2	no	—
51	C₂×C₄×C₈₉	—	4	1	no	—
52	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
59	C₂×Dic₈₉	C₂²×C₄	1	4	yes	—
60	Dic₁₇₈	—	2	4	no	—
61	Dic₁₇₈	—	4	2	no	—
62	C₈×C₈₉	—	2	4	no	—
63	#10623494	—	2	2	no	—
64	#10623494	—	2	2	no	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	—	2	1	no	—
68	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₈₉	—	2	1	no	—
70	C₄×C₈₉	—	2	1	no	—
71	C₄×C₈₉	—	2	1	no	—
72	C₄×C₈₉	—	2	1	no	—
73	C₄×C₈₉	#131	1	2	yes	—
74	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
75	Dic₈₉	—	2	4	no	—
76	Dic₈₉	—	2	4	no	—
77	C₂×C₈₉	—	4	1	no	—
78	C₂×C₈₉	#519	1	1	yes	characteristic
79	C₂×C₈₉	C₂×#131	1	1	yes	characteristic
80	C₂×C₈₉	#568	1	1	yes	characteristic
81	#2985	—	89	1	no	maximal, Sylow
82	C₈₉	#2985	1	1	yes	characteristic, complemented, Sylow
83	#533	—	89	1	no	—
84	#446	—	89	2	no	—
85	#446	—	89	4	no	—
86	C₈×Q₈	—	89	2	no	—
87	#433	—	89	4	no	—
88	#432	D₈₉	1	1	yes	characteristic
89	#431	—	89	1	no	—
90	C₄⋊Q₈	—	89	1	no	—
91	C₄⋊Q₈	—	89	1	no	—
92	#125	—	89	2	no	—
93	C₂²⋊Q₈	—	89	2	no	—
94	C₂²⋊Q₈	—	89	2	no	—
95	C₄×Q₈	—	89	2	no	—
96	C₄×Q₈	—	89	2	no	—
97	#117	—	89	2	no	—
98	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
99	C₄⋊C₈	—	89	4	no	—
100	C₄⋊C₈	—	89	4	no	—
101	C₄⋊C₈	—	89	2	no	—
102	C₄⋊C₈	C₂×D₈₉	1	2	yes	—
103	C₂²⋊C₈	—	89	2	no	—
104	C₂²⋊C₈	C₂×D₈₉	1	2	yes	—
105	#97	—	89	4	no	—
106	#97	C₂×D₈₉	1	2	yes	—
107	C₄×C₈	—	89	4	no	—
108	C₄×C₈	—	89	1	no	—
109	C₄×C₈	—	89	1	no	—
110	C₂×Q₈	—	89	2	no	—
111	C₂×Q₈	—	89	2	no	—
112	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₈₉	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
115	C₂×C₈	—	89	2	no	—
116	C₂×C₈	—	89	2	no	—
117	C₂×C₈	C₂²×D₈₉	1	4	yes	—
118	C₄⋊C₄	—	89	2	no	—
119	C₄⋊C₄	—	89	2	no	—
120	C₄⋊C₄	—	89	4	no	—
121	C₄⋊C₄	—	89	4	no	—
122	C₄⋊C₄	—	89	2	no	—
123	C₄⋊C₄	—	89	2	no	—
124	C₂²⋊C₄	—	89	4	no	—
125	C₄²	—	89	4	no	—
126	C₄²	C₄×D₈₉	1	1	yes	characteristic
127	C₄²	C₄×D₈₉	1	1	yes	characteristic
128	C₄²	C₂²×D₈₉	1	1	yes	characteristic
129	C₄²	C₂²×D₈₉	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₈₉	1	1	yes	characteristic
131	Q₈	—	356	2	no	—
132	Q₈	—	178	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	89	4	no	—
137	C₂×C₄	—	89	4	no	—
138	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	178	2	no	—
146	C₈	—	178	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	178	4	no	—
156	C₄	—	178	4	no	—
157	C₄	#15668020	1	2	yes	—
158	C₄	#15668059	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#19152075	1	1	yes	characteristic
161	C₂	C₂×#15668020	1	1	yes	characteristic
162	C₂	#19152364	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152155

C₂

yes

maximal, characteristic

#19151726

C₂

yes

maximal, complemented

#19151725

C₂

yes

maximal, complemented

#19151722

C₂

yes

maximal, complemented

#19151656

C₂

yes

maximal

#19151651

C₂

yes

maximal, characteristic

#432×C₈₉

C₂

yes

maximal, characteristic

#15668012

C₄

yes

characteristic

#15668012

C₄

yes

characteristic

#15668004

C₄

yes

—

#15667991

C₄

yes

—

#15667991

C₄

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

C₄×Dic₁₇₈

C₂²

yes

—

#15667971

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

#15667941

C₂²

yes

—

#15667941

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

#15667922

C₂²

yes

—

C₄×#10623494

C₂²

yes

characteristic

C₄×#10623494

C₂²

yes

characteristic

C₈×Dic₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

C₂²⋊C₈×C₈₉

C₂²

yes

—

#97×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×Dic₁₇₈

C₂×C₄

yes

—

C₂×#10623494

C₂³

yes

—

C₂×#10623494

C₂³

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

#13083553

C₂×C₄

yes

—

C₂²⋊Dic₈₉

C₂×C₄

yes

—

C₄×Dic₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₄²×C₈₉

C₂³

yes

characteristic

C₂³×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

C₂×Dic₈₉

C₂²×C₄

yes

—

C₂×Dic₈₉

C₂²×C₄

yes

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#131

yes

—

C₄×C₈₉

2₋¹⁺⁴

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#533

—

#446

—

#446

—

C₈×Q₈

—

#433

—

#432

D₈₉

yes

characteristic

#431

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₈₉

yes

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

C₂×D₈₉

yes

—

105

#97

—

106

#97

C₂×D₈₉

yes

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₈₉

yes

characteristic

113

C₂²×C₄

C₄×D₈₉

yes

characteristic

114

C₂²×C₄

C₂²×D₈₉

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₈₉

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₈₉

yes

characteristic

127

C₄²

C₄×D₈₉

yes

characteristic

128

C₄²

C₂²×D₈₉

yes

characteristic

129

C₄²

C₂²×D₈₉

yes

characteristic

130

C₂³

C₂×C₄×D₈₉

yes

characteristic

131

Q₈

—

356

—

132

Q₈

—

178

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

139

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

140

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

141

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

142

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

143

C₂×C₄

C₂³×D₈₉

yes

characteristic

144

C₈

—

145

C₈

—

178

—

146

C₈

—

178

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₈₉

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

178

—

156

C₄

—

178

—

157

C₄

#15668020

yes

—

158

C₄

#15668059

yes

—

159

C₂

—

160

C₂

#19152075

yes

characteristic

161

C₂

C₂×#15668020

yes

characteristic

162

C₂

#19152364

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#432×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#432	D₈₉
hypocenter	C₈₉	#2985
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#432×C₈₉

C₂

prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #19151726	C₂	semidirect
3. #19151725	C₂	semidirect
4. #19151722	C₂	semidirect
82. C₈₉	#2985	semidirect

H ≅ G/N

type

2. #19151726

C₂

semidirect

3. #19151725

C₂

semidirect

4. #19151722

C₂

semidirect

82. C₈₉

#2985

semidirect

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

subgroup	index
#19152155	2
#19151726	2
#19151725	2
#19151722	2
#19151656	2
#19151651	2
#432×C₈₉	2
#2985	89

subgroup

index

#432×C₈₉

Subgroups — #26029349

Conjugacy classes of subgroups

Complements

Maximal subgroups