Group #26025635 — subgroups

Subgroups — #26025635

Order 11392.

468 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 185 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	C₂×#15668004	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#19151530	C₂	1	2	yes	maximal
4	#19151519	C₂	1	2	yes	maximal
5	#19151497	C₂	1	2	yes	maximal
6	#19151497	C₂	1	2	yes	maximal
7	#19151496	C₂	1	4	yes	maximal
8	C₄×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₁₇₈	C₂²	1	1	yes	characteristic
10	#15668004	—	2	4	no	—
11	C₄×Dic₁₇₈	—	2	4	no	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₈₉	C₂²	1	4	yes	—
17	C₂×#13083553	C₂²	1	2	yes	—
18	C₂×#13083553	C₂²	1	2	yes	—
19	C₂×#13083553	C₂²	1	2	yes	—
20	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
21	#15667919	C₂²	1	4	yes	—
22	#15667919	C₂²	1	4	yes	—
23	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
26	#95×C₈₉	C₂²	1	2	yes	—
27	C₂×Dic₁₇₈	—	2	2	no	—
28	C₂×Dic₁₇₈	—	2	4	no	—
29	C₂²×Dic₈₉	C₂³	1	2	yes	—
30	C₂²×Dic₈₉	C₂³	1	2	yes	—
31	C₂²×Dic₈₉	C₂³	1	2	yes	—
32	C₂²×Dic₈₉	C₂³	1	2	yes	—
33	C₄⋊Dic₈₉	—	2	2	no	—
34	C₄⋊Dic₈₉	—	2	2	no	—
35	C₄⋊Dic₈₉	—	2	2	no	—
36	C₄⋊Dic₈₉	—	2	8	no	—
37	C₄⋊Dic₈₉	—	4	4	no	—
38	#13083553	—	2	4	no	—
39	#13083553	—	2	4	no	—
40	#13083553	—	2	4	no	—
41	C₄×Dic₈₉	—	2	4	no	—
42	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
45	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
46	C₄⋊C₄×C₈₉	Q₈	1	8	yes	complemented
47	C₄²×C₈₉	—	4	2	no	—
48	C₄²×C₈₉	D₄	1	4	yes	—
49	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
50	C₂×C₄×C₈₉	—	2	4	no	—
51	C₂×C₄×C₈₉	—	2	2	no	—
52	C₂×C₄×C₈₉	—	2	1	no	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	—	2	1	no	—
55	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
58	C₂×C₄×C₈₉	C₂×Q₈	1	8	yes	—
59	C₂×C₄×C₈₉	C₂×Q₈	1	4	yes	—
60	C₂×Dic₈₉	—	2	2	no	—
61	C₂×Dic₈₉	—	2	2	no	—
62	C₂×Dic₈₉	—	2	2	no	—
63	C₂×Dic₈₉	—	2	2	no	—
64	C₂×Dic₈₉	—	2	2	no	—
65	C₂×Dic₈₉	—	2	4	no	—
66	C₂×Dic₈₉	—	2	2	no	—
67	C₂×Dic₈₉	—	2	2	no	—
68	C₂×Dic₈₉	—	2	2	no	—
69	C₂×Dic₈₉	—	2	2	no	—
70	C₂×Dic₈₉	#41	1	4	yes	—
71	Dic₁₇₈	—	2	8	no	—
72	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
73	C₂²×C₈₉	C₂²×Q₈	1	2	yes	—
74	C₂²×C₈₉	C₂×#41	1	2	yes	—
75	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
76	C₂²×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
77	C₄×C₈₉	—	2	2	no	—
78	C₄×C₈₉	—	2	2	no	—
79	C₄×C₈₉	—	2	8	no	—
80	C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
81	Dic₈₉	—	2	4	no	—
82	Dic₈₉	—	4	2	no	—
83	Dic₈₉	—	4	2	no	—
84	Dic₈₉	—	4	2	no	—
85	C₂×C₈₉	#525	1	1	yes	characteristic
86	C₂×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₈₉	#537	1	1	yes	characteristic
88	C₂×C₈₉	#554	1	2	yes	—
89	C₂×C₈₉	#557	1	2	yes	—
90	#2584	—	89	1	no	maximal, Sylow
91	C₈₉	#2584	1	1	yes	characteristic, complemented, Sylow
92	C₂×#125	—	89	1	no	—
93	C₂×C₄×Q₈	—	89	1	no	—
94	#398	—	89	2	no	—
95	#395	—	89	2	no	—
96	#389	—	89	2	no	—
97	#389	—	89	2	no	—
98	#389	—	89	4	no	—
99	C₄×C₄⋊C₄	D₈₉	1	1	yes	characteristic
100	C₂²×Q₈	—	89	1	no	—
101	#125	—	178	4	no	—
102	C₄×Q₈	—	178	4	no	—
103	C₂×C₄⋊C₄	—	89	4	no	—
104	C₂×C₄⋊C₄	—	89	2	no	—
105	C₂×C₄⋊C₄	—	89	1	no	—
106	C₂×C₄⋊C₄	—	89	2	no	—
107	C₂×C₄⋊C₄	—	89	1	no	—
108	C₂×C₄⋊C₄	—	89	1	no	—
109	C₂×C₄⋊C₄	—	89	2	no	—
110	C₂×C₄⋊C₄	—	89	4	no	—
111	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
112	C₂×C₄²	—	89	2	no	—
113	C₂×C₄²	C₂×D₈₉	1	2	yes	—
114	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
115	#95	—	89	4	no	—
116	#95	—	89	4	no	—
117	#95	C₂×D₈₉	1	2	yes	—
118	C₂×Q₈	—	178	4	no	—
119	C₂×Q₈	—	178	2	no	—
120	C₂²×C₄	—	89	2	no	—
121	C₂²×C₄	—	89	2	no	—
122	C₂²×C₄	—	89	2	no	—
123	C₂²×C₄	—	89	2	no	—
124	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
125	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
127	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₄	—	356	4	no	—
129	C₄⋊C₄	—	178	8	no	—
130	C₄⋊C₄	—	178	4	no	—
131	C₄⋊C₄	—	178	2	no	—
132	C₄⋊C₄	—	178	4	no	—
133	C₄⋊C₄	—	178	4	no	—
134	C₄⋊C₄	—	178	2	no	—
135	C₄⋊C₄	—	178	2	no	—
136	C₄⋊C₄	Dic₁₇₈	1	8	yes	complemented
137	C₄²	—	4	2	no	—
138	C₄²	—	178	4	no	—
139	C₄²	D₃₅₆	1	4	yes	—
140	C₂³	C₂³×D₈₉	1	1	yes	characteristic
141	Q₈	—	178	8	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	89	4	no	—
148	C₂×C₄	—	178	2	no	—
149	C₂×C₄	—	178	2	no	—
150	C₂×C₄	—	178	2	no	—
151	C₂×C₄	—	178	4	no	—
152	C₂×C₄	—	178	2	no	—
153	C₂×C₄	—	178	2	no	—
154	C₂×C₄	—	178	2	no	—
155	C₂×C₄	—	178	2	no	—
156	C₂×C₄	—	178	2	no	—
157	C₂×C₄	—	178	2	no	—
158	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
159	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
160	C₂×C₄	C₂×D₃₅₆	1	2	yes	—
161	C₂×C₄	C₂×Dic₁₇₈	1	8	yes	—
162	C₂×C₄	C₂×Dic₁₇₈	1	4	yes	—
163	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
164	C₂²	C₂²×Dic₁₇₈	1	2	yes	—
165	C₂²	#41×D₈₉	1	2	yes	—
166	C₂²	#15668057	1	1	yes	characteristic
167	C₂²	#15668060	1	1	yes	characteristic
168	C₄	—	2	2	no	—
169	C₄	—	2	8	no	—
170	C₄	—	2	2	no	—
171	C₄	—	356	2	no	—
172	C₄	—	356	2	no	—
173	C₄	—	356	2	no	—
174	C₄	—	178	4	no	—
175	C₄	#15668012	1	4	yes	—
176	C₂	#19152110	1	1	yes	characteristic
177	C₂	C₂×#15668012	1	1	yes	characteristic
178	C₂	#19152179	1	1	yes	characteristic
179	C₂	#19152307	1	2	yes	—
180	C₂	#19152324	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15668004

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₇₈

C₂

yes

maximal, characteristic

#19151530

C₂

yes

maximal

#19151519

C₂

yes

maximal

#19151497

C₂

yes

maximal

#19151497

C₂

yes

maximal

#19151496

C₂

yes

maximal

C₄×C₄⋊C₄×C₈₉

C₂

yes

maximal, characteristic

C₂²×Dic₁₇₈

C₂²

yes

characteristic

#15668004

—

C₄×Dic₁₇₈

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×#13083553

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#95×C₈₉

C₂²

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

#13083553

—

#13083553

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

Q₈

yes

complemented

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×C₄×C₈₉

C₂×Q₈

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

#41

yes

—

Dic₁₇₈

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

—

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

characteristic

C₂²×C₈₉

2₋¹⁺⁴

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₄⋊Q₈

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

#525

yes

characteristic

C₂×C₈₉

C₂×C₄⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#557

yes

—

#2584

—

maximal, Sylow

C₈₉

#2584

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×Q₈

—

—

—

—

—

—

C₄×C₄⋊C₄

D₈₉

yes

characteristic

100

C₂²×Q₈

—

101

#125

—

178

—

102

C₄×Q₈

—

178

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

112

C₂×C₄²

—

113

C₂×C₄²

C₂×D₈₉

yes

—

114

C₂×C₄²

C₂×D₈₉

yes

characteristic

115

#95

—

116

#95

—

117

#95

C₂×D₈₉

yes

—

118

C₂×Q₈

—

178

—

119

C₂×Q₈

—

178

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂²×D₈₉

yes

characteristic

125

C₂²×C₄

C₂²×D₈₉

yes

characteristic

126

C₂²×C₄

C₂²×D₈₉

yes

—

127

C₂²×C₄

C₂²×D₈₉

yes

characteristic

128

C₄⋊C₄

—

356

—

129

C₄⋊C₄

—

178

—

130

C₄⋊C₄

—

178

—

131

C₄⋊C₄

—

178

—

132

C₄⋊C₄

—

178

—

133

C₄⋊C₄

—

178

—

134

C₄⋊C₄

—

178

—

135

C₄⋊C₄

—

178

—

136

C₄⋊C₄

Dic₁₇₈

yes

complemented

137

C₄²

—

138

C₄²

—

178

—

139

C₄²

D₃₅₆

yes

—

140

C₂³

C₂³×D₈₉

yes

characteristic

141

Q₈

—

178

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

178

—

149

C₂×C₄

—

178

—

150

C₂×C₄

—

178

—

151

C₂×C₄

—

178

—

152

C₂×C₄

—

178

—

153

C₂×C₄

—

178

—

154

C₂×C₄

—

178

—

155

C₂×C₄

—

178

—

156

C₂×C₄

—

178

—

157

C₂×C₄

—

178

—

158

C₂×C₄

C₂×D₃₅₆

yes

—

159

C₂×C₄

C₂×D₃₅₆

yes

—

160

C₂×C₄

C₂×D₃₅₆

yes

—

161

C₂×C₄

C₂×Dic₁₇₈

yes

—

162

C₂×C₄

C₂×Dic₁₇₈

yes

—

163

C₂²

C₂²×D₃₅₆

yes

characteristic

164

C₂²

C₂²×Dic₁₇₈

yes

—

165

C₂²

#41×D₈₉

yes

—

166

C₂²

#15668057

yes

characteristic

167

C₂²

#15668060

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

356

—

172

C₄

—

356

—

173

C₄

—

356

—

174

C₄

—

178

—

175

C₄

#15668012

yes

—

176

C₂

#19152110

yes

characteristic

177

C₂

C₂×#15668012

yes

characteristic

178

C₂

#19152179

yes

characteristic

179

C₂

#19152307

yes

—

180

C₂

#19152324

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, prime radical, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₄×C₄⋊C₄×C₈₉	C₂
hypercenter	C₄×C₄⋊C₄	D₈₉
hypocenter	C₈₉	#2584
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, prime radical, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₄×C₄⋊C₄×C₈₉

C₂

hypercenter

C₄×C₄⋊C₄

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
46. C₄⋊C₄×C₈₉	Q₈	semidirect
91. C₈₉	#2584	semidirect
136. C₄⋊C₄	Dic₁₇₈	semidirect

H ≅ G/N

type

46. C₄⋊C₄×C₈₉

Q₈

semidirect

91. C₈₉

#2584

semidirect

136. C₄⋊C₄

Dic₁₇₈

semidirect

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

subgroup	index
C₂×#15668004	2
C₂×C₄×Dic₁₇₈	2
#19151530	2
#19151519	2
#19151497	2
#19151496	2
C₄×C₄⋊C₄×C₈₉	2
#2584	89

subgroup

index

C₂×C₄×Dic₁₇₈

C₄×C₄⋊C₄×C₈₉

Subgroups — #26025635

Conjugacy classes of subgroups

Complements

Maximal subgroups