Group #26020800 — subgroups

Subgroups — #26020800

Order 11392.

340 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19152365	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151582	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#15667929	C₂	1	2	yes	maximal, complemented
4	#19151368	C₂	1	2	yes	maximal, complemented
5	#535×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668019	—	2	4	no	—
7	C₂×#13083558	—	2	2	no	—
8	C₂×#13083558	C₂²	1	2	yes	—
9	C₂×#13083558	C₂²	1	2	yes	—
10	#15667931	—	2	1	no	—
11	#15667929	—	2	4	no	—
12	#15667929	—	2	1	no	—
13	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	#127×C₈₉	—	2	1	no	—
16	#124×C₈₉	—	2	1	no	—
17	#121×C₈₉	—	2	2	no	—
18	#120×C₈₉	C₄	1	4	yes	—
19	#117×C₈₉	C₂²	1	1	yes	characteristic
20	#13083558	—	2	4	no	—
21	#13083558	—	2	2	no	—
22	#13083558	—	2	2	no	—
23	#13083558	—	2	2	no	—
24	#13083558	D₄	1	4	yes	complemented
25	C₂×#10623494	—	2	4	no	—
26	C₂×#10623494	—	2	2	no	—
27	C₂×#10623494	—	2	2	no	—
28	C₂⁴×C₈₉	C₂×C₄	1	2	yes	—
29	#41×C₈₉	—	2	2	no	—
30	#41×C₈₉	—	2	2	no	—
31	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
32	C₂×D₄×C₈₉	—	2	4	no	—
33	C₂×D₄×C₈₉	—	2	1	no	—
34	C₂×D₄×C₈₉	—	2	1	no	—
35	C₂×D₄×C₈₉	—	2	2	no	—
36	C₂×D₄×C₈₉	—	2	2	no	—
37	C₂×D₄×C₈₉	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
39	C₂²×C₄×C₈₉	D₄	1	2	yes	—
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₄×C₈₉	—	2	1	no	—
42	C₂²⋊C₄×C₈₉	—	2	4	no	—
43	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
44	C₄²×C₈₉	—	2	1	no	—
45	C₂³×C₈₉	—	2	2	no	—
46	C₂³×C₈₉	—	2	4	no	—
47	C₂³×C₈₉	—	8	2	no	—
48	C₂³×C₈₉	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	Q₈×C₈₉	—	2	2	no	—
51	D₄×C₈₉	—	2	4	no	—
52	D₄×C₈₉	—	2	2	no	—
53	D₄×C₈₉	—	4	1	no	—
54	D₄×C₈₉	—	4	2	no	—
55	D₄×C₈₉	—	4	1	no	—
56	D₄×C₈₉	—	8	4	no	—
57	C₂×C₄×C₈₉	—	2	2	no	—
58	C₂×C₄×C₈₉	—	2	2	no	—
59	C₂×C₄×C₈₉	—	2	2	no	—
60	C₂×C₄×C₈₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₈₉	#41	1	2	yes	—
65	#10623494	—	2	4	no	—
66	#10623494	—	4	2	no	—
67	C₂²×C₈₉	—	2	2	no	—
68	C₂²×C₈₉	—	4	1	no	—
69	C₂²×C₈₉	—	4	2	no	—
70	C₂²×C₈₉	—	4	4	no	—
71	C₂²×C₈₉	—	4	2	no	—
72	C₂²×C₈₉	—	8	4	no	—
73	C₂²×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₈₉	C₄×D₄	1	2	yes	—
75	C₄×C₈₉	—	2	2	no	—
76	C₄×C₈₉	—	4	2	no	—
77	C₄×C₈₉	#120	1	1	yes	characteristic
78	C₄×C₈₉	#121	1	2	yes	—
79	C₄×C₈₉	#123	1	1	yes	characteristic
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	—	2	2	no	—
82	C₂×C₈₉	—	4	2	no	—
83	C₂×C₈₉	—	8	2	no	—
84	C₂×C₈₉	#386	1	1	yes	characteristic
85	#1914	—	89	1	no	maximal, Sylow
86	C₈₉	#1914	1	1	yes	characteristic, complemented, Sylow
87	#568	—	89	1	no	—
88	#535	D₈₉	1	1	yes	characteristic
89	#413	—	89	1	no	—
90	C₂×#100	—	89	2	no	—
91	#344	—	89	2	no	—
92	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
93	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
94	#131	—	178	4	no	—
95	C₂×M₁₆	—	89	2	no	—
96	C₂×M₁₆	—	89	2	no	—
97	C₂×M₁₆	—	178	2	no	—
98	#127	—	2	1	no	—
99	#124	—	2	1	no	—
100	#121	—	2	2	no	—
101	#120	Dic₈₉	1	4	yes	—
102	#117	C₂×D₈₉	1	1	yes	characteristic
103	#101	—	178	1	no	—
104	#100	—	178	4	no	—
105	#100	—	178	1	no	—
106	C₂⁴	C₂×Dic₈₉	1	2	yes	—
107	#41	—	2	2	no	—
108	#41	—	2	2	no	—
109	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
110	C₂×D₄	—	2	4	no	—
111	C₂×D₄	—	2	2	no	—
112	C₂×D₄	—	2	1	no	—
113	C₂×D₄	—	2	1	no	—
114	C₂×D₄	—	2	2	no	—
115	C₂×D₄	C₂×Dic₈₉	1	2	yes	—
116	C₂×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
118	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
119	M₁₆	—	89	4	no	—
120	M₁₆	—	178	2	no	—
121	M₁₆	—	178	2	no	—
122	M₁₆	—	178	4	no	—
123	M₁₆	—	178	2	no	—
124	C₂×C₈	—	178	2	no	—
125	C₂×C₈	—	178	4	no	—
126	C₂×C₈	—	178	2	no	—
127	C₄⋊C₄	—	2	1	no	—
128	C₂²⋊C₄	—	2	4	no	—
129	C₂²⋊C₄	C₂×Dic₈₉	1	2	yes	—
130	C₄²	—	2	1	no	—
131	C₂³	—	2	2	no	—
132	C₂³	—	2	4	no	—
133	C₂³	—	8	2	no	—
134	C₂³	C₂²⋊Dic₈₉	1	2	yes	—
135	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
136	Q₈	—	2	2	no	—
137	D₄	—	2	4	no	—
138	D₄	—	2	2	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	2	no	—
141	D₄	—	4	1	no	—
142	D₄	—	8	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	C₂²⋊Dic₈₉	1	2	yes	—
147	C₂×C₄	D₄×D₈₉	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
149	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
150	C₂×C₄	#13083573	1	2	yes	—
151	C₈	—	356	2	no	—
152	C₈	—	178	4	no	—
153	C₂²	—	2	2	no	—
154	C₂²	—	4	1	no	—
155	C₂²	—	4	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	8	4	no	—
159	C₂²	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
160	C₂²	D₄×Dic₈₉	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	#15667980	1	1	yes	characteristic
164	C₄	#15667984	1	2	yes	—
165	C₄	#15667993	1	1	yes	characteristic
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	—	4	2	no	—
169	C₂	—	8	2	no	—
170	C₂	#19151478	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152365

C₂

yes

maximal, characteristic, complemented

#19151582

C₂

yes

maximal, characteristic, complemented

C₂×#15667929

C₂

yes

maximal, complemented

#19151368

C₂

yes

maximal, complemented

#535×C₈₉

C₂

yes

maximal, characteristic

#15668019

—

C₂×#13083558

—

C₂×#13083558

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

#15667931

—

#15667929

—

#15667929

—

C₂×#41×C₈₉

C₂²

yes

characteristic

C₂²×D₄×C₈₉

C₂²

yes

characteristic

#127×C₈₉

—

#124×C₈₉

—

#121×C₈₉

—

#120×C₈₉

C₄

yes

—

#117×C₈₉

C₂²

yes

—

—

—

—

D₄

yes

—

—

—

C₂⁴×C₈₉

C₂×C₄

yes

—

#41×C₈₉

—

#41×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

C₂×C₄

yes

—

C₂×D₄×C₈₉

D₄

yes

characteristic

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

—

C₂³×C₈₉

C₂²⋊C₄

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

#10623494

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₉

C₄×D₄

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#120

yes

characteristic

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#123

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#568

—

#535

D₈₉

yes

—

—

—

C₂×#41

C₂×D₈₉

yes

characteristic

C₂²×D₄

C₂×D₈₉

yes

characteristic

#131

—

178

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

178

—

#127

—

#124

—

100

#121

—

101

#120

Dic₈₉

yes

—

102

#117

C₂×D₈₉

yes

characteristic

103

#101

—

178

—

104

#100

—

178

—

105

#100

—

178

—

106

C₂⁴

C₂×Dic₈₉

yes

—

107

#41

—

108

#41

—

109

C₂×Q₈

C₂²⋊D₈₉

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

C₂×Dic₈₉

yes

—

116

C₂×D₄

C₂²⋊D₈₉

yes

characteristic

117

C₂²×C₄

C₂²⋊D₈₉

yes

—

118

C₂²×C₄

C₂²×D₈₉

yes

characteristic

119

M₁₆

—

120

M₁₆

—

178

—

121

M₁₆

—

178

—

122

M₁₆

—

178

—

123

M₁₆

—

178

—

124

C₂×C₈

—

178

—

125

C₂×C₈

—

178

—

126

C₂×C₈

—

178

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

C₂×Dic₈₉

yes

—

130

C₄²

—

131

C₂³

—

132

C₂³

—

133

C₂³

—

134

C₂³

C₂²⋊Dic₈₉

yes

—

135

C₂³

C₂²×Dic₈₉

yes

characteristic

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊Dic₈₉

yes

—

147

C₂×C₄

D₄×D₈₉

yes

—

148

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

149

C₂×C₄

C₂×C₂²⋊D₈₉

yes

characteristic

150

C₂×C₄

#13083573

yes

—

151

C₈

—

356

—

152

C₈

—

178

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

C₂×C₂²⋊Dic₈₉

yes

characteristic

160

C₂²

D₄×Dic₈₉

yes

—

161

C₄

—

162

C₄

—

163

C₄

#15667980

yes

characteristic

164

C₄

#15667984

yes

—

165

C₄

#15667993

yes

characteristic

166

C₂

—

167

C₂

—

168

C₂

—

169

C₂

—

170

C₂

#19151478

yes

characteristic

	subgroup	G/H
center	C₂	#19151478
commutator	C₂³×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting, prime radical	#535×C₈₉	C₂
socle	C₂×C₈₉	#386
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#535	D₈₉
hypocenter	C₈₉	#1914
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

subgroup

G/H

center

C₂

#19151478

commutator

C₂³×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting, prime radical

#535×C₈₉

C₂

socle

C₂×C₈₉

#386

cosocle

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #535

N	H ≅ G/N	type
1. #19152365	C₂	semidirect
2. #19151582	C₂	semidirect
3. C₂×#15667929	C₂	semidirect
4. #19151368	C₂	semidirect
24. #13083558	D₄	semidirect
86. C₈₉	#1914	semidirect

H ≅ G/N

type

1. #19152365

C₂

semidirect

2. #19151582

C₂

semidirect

3. C₂×#15667929

C₂

semidirect

4. #19151368

C₂

semidirect

24. #13083558

D₄

semidirect

86. C₈₉

#1914

semidirect

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

subgroup	index
#19152365	2
#19151582	2
C₂×#15667929	2
#19151368	2
#535×C₈₉	2
#1914	89

subgroup

index

#535×C₈₉

Subgroups — #26020800

Conjugacy classes of subgroups

Complements

Maximal subgroups