Group #26025426 — subgroups

Subgroups — #26025426

Order 11392.

564 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₄×D₃₅₆	C₂	1	1	yes	maximal, characteristic
2	#19151509	C₂	1	4	yes	maximal, complemented
3	#19151499	C₂	1	2	yes	maximal
4	#19151463	C₂	1	2	yes	maximal, complemented
5	#19151455	C₂	1	4	yes	maximal, complemented
6	C₄×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
10	C₄×D₃₅₆	—	2	4	no	—
11	C₂×C₄⋊Dic₈₉	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
13	C₂×#13083551	C₂²	1	2	yes	—
14	C₂×#13083551	C₂²	1	8	yes	—
15	C₂×C₄×Dic₈₉	C₂²	1	2	yes	—
16	#15667919	C₂²	1	2	yes	—
17	#15667919	C₂²	1	8	yes	—
18	C₂²×Q₈×C₈₉	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₈₉	—	2	4	no	—
20	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂³×D₈₉	C₂³	1	2	yes	—
24	C₂×D₃₅₆	—	2	2	no	—
25	C₂×D₃₅₆	—	2	2	no	—
26	C₂×D₃₅₆	Q₈	1	4	yes	complemented
27	C₂²×Dic₈₉	C₂³	1	2	yes	—
28	C₂²×Dic₈₉	C₂³	1	4	yes	—
29	C₂×C₄×D₈₉	—	2	4	no	—
30	C₂×C₄×D₈₉	—	2	8	no	—
31	C₄⋊Dic₈₉	—	2	2	no	—
32	C₄⋊Dic₈₉	—	2	4	no	—
33	#13083551	—	2	4	no	—
34	#13083551	—	4	8	no	—
35	C₄×Dic₈₉	—	2	4	no	—
36	C₂×Q₈×C₈₉	—	2	4	no	—
37	C₂×Q₈×C₈₉	—	4	4	no	—
38	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
39	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₄⋊C₄×C₈₉	—	2	4	no	—
43	C₄⋊C₄×C₈₉	—	2	4	no	—
44	C₄²×C₈₉	D₄	1	4	yes	—
45	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
46	C₂²×D₈₉	—	2	2	no	—
47	C₂²×D₈₉	—	2	4	no	—
48	C₂²×D₈₉	—	2	2	no	—
49	C₂²×D₈₉	—	2	4	no	—
50	C₂²×D₈₉	C₂×Q₈	1	4	yes	—
51	Q₈×C₈₉	—	4	8	no	—
52	C₂×C₄×C₈₉	—	2	2	no	—
53	C₂×C₄×C₈₉	—	2	4	no	—
54	C₂×C₄×C₈₉	—	2	8	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	2	yes	—
59	C₂×C₄×C₈₉	#41	1	2	yes	—
60	C₂×C₄×C₈₉	#41	1	4	yes	—
61	C₂×Dic₈₉	—	2	2	no	—
62	C₂×Dic₈₉	—	2	4	no	—
63	C₂×Dic₈₉	—	2	4	no	—
64	C₂×Dic₈₉	—	2	4	no	—
65	C₂×Dic₈₉	—	2	4	no	—
66	D₃₅₆	—	2	8	no	—
67	C₄×D₈₉	—	4	8	no	—
68	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
69	C₂²×C₈₉	C₂²×Q₈	1	1	yes	characteristic
70	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
71	C₂²×C₈₉	C₂×#41	1	2	yes	—
72	C₂²×C₈₉	H(5,𝔽₂)	1	2	yes	—
73	C₂×D₈₉	—	2	4	no	—
74	C₂×D₈₉	—	2	8	no	—
75	C₂×D₈₉	—	4	2	no	—
76	C₂×D₈₉	—	4	2	no	—
77	C₂×D₈₉	—	4	4	no	—
78	C₄×C₈₉	—	2	2	no	—
79	C₄×C₈₉	—	4	4	no	—
80	C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
81	C₄×C₈₉	#124	1	4	yes	—
82	Dic₈₉	—	4	2	no	—
83	Dic₈₉	—	4	4	no	—
84	C₂×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
85	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
86	C₂×C₈₉	#535	1	1	yes	characteristic
87	C₂×C₈₉	#554	1	2	yes	—
88	C₂×C₈₉	#555	1	2	yes	—
89	D₈₉	—	4	4	no	—
90	#2563	—	89	1	no	maximal, Sylow
91	C₈₉	#2563	1	1	yes	characteristic, complemented, Sylow
92	C₂×C₄⋊Q₈	D₈₉	1	1	yes	characteristic, complemented
93	C₂×C₄×D₄	—	89	1	no	—
94	#393	—	89	4	no	—
95	#390	—	89	2	no	—
96	#382	—	89	2	no	—
97	#380	—	89	4	no	—
98	C₄×C₄⋊C₄	—	89	1	no	—
99	C₂²×Q₈	C₂×D₈₉	1	2	yes	—
100	C₂²×D₄	—	89	1	no	—
101	C₂³×C₄	—	89	2	no	—
102	C₄⋊Q₈	—	2	4	no	—
103	C₄×D₄	—	178	4	no	—
104	C₂×C₄⋊C₄	—	89	1	no	—
105	C₂×C₄⋊C₄	—	89	2	no	—
106	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
107	C₂×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
108	C₂×C₂²⋊C₄	—	89	8	no	—
109	C₂×C₂²⋊C₄	—	89	2	no	—
110	C₂×C₄²	—	89	2	no	—
111	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
112	#95	—	89	2	no	—
113	#95	—	89	8	no	—
114	C₂⁴	—	89	2	no	—
115	C₂×Q₈	—	2	4	no	—
116	C₂×Q₈	—	4	4	no	—
117	C₂×D₄	—	89	4	no	—
118	C₂×D₄	—	178	2	no	—
119	C₂×D₄	—	178	2	no	—
120	C₂²×C₄	—	89	2	no	—
121	C₂²×C₄	—	89	4	no	—
122	C₂²×C₄	—	178	4	no	—
123	C₂²×C₄	—	178	8	no	—
124	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
125	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
127	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
128	C₄⋊C₄	—	2	4	no	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	178	2	no	—
131	C₄⋊C₄	—	178	4	no	—
132	C₂²⋊C₄	—	356	8	no	—
133	C₂²⋊C₄	—	178	4	no	—
134	C₄²	—	178	4	no	—
135	C₄²	C₂²⋊D₈₉	1	4	yes	—
136	C₂³	—	89	4	no	—
137	C₂³	—	178	2	no	—
138	C₂³	—	178	4	no	—
139	C₂³	—	178	2	no	—
140	C₂³	—	178	4	no	—
141	C₂³	C₂³×D₈₉	1	1	yes	characteristic
142	Q₈	—	4	8	no	—
143	D₄	—	178	8	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	—	2	8	no	—
147	C₂×C₄	—	356	8	no	—
148	C₂×C₄	—	178	2	no	—
149	C₂×C₄	—	178	4	no	—
150	C₂×C₄	—	178	4	no	—
151	C₂×C₄	—	178	4	no	—
152	C₂×C₄	—	178	4	no	—
153	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
154	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
155	C₂×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
156	C₂×C₄	Q₈×D₈₉	1	2	yes	—
157	C₂×C₄	#13083575	1	2	yes	—
158	C₂×C₄	#13083575	1	4	yes	—
159	C₂²	—	356	2	no	—
160	C₂²	—	356	4	no	—
161	C₂²	—	356	2	no	—
162	C₂²	—	178	8	no	—
163	C₂²	—	178	4	no	—
164	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
165	C₂²	C₂×Q₈×D₈₉	1	1	yes	characteristic
166	C₂²	C₂×#13083575	1	2	yes	—
167	C₂²	C₂×#13083575	1	1	yes	characteristic
168	C₂²	#15668057	1	2	yes	—
169	C₄	—	2	2	no	—
170	C₄	—	4	4	no	—
171	C₄	—	356	4	no	—
172	C₄	—	356	2	no	—
173	C₄	#15667988	1	4	yes	—
174	C₄	#15668001	1	4	yes	—
175	C₂	—	356	4	no	—
176	C₂	C₂×#15667988	1	1	yes	characteristic
177	C₂	C₂×#15668001	1	1	yes	characteristic
178	C₂	#19152165	1	1	yes	characteristic
179	C₂	#19152309	1	2	yes	—
180	C₂	#19152314	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₃₅₆

C₂

yes

maximal, characteristic

#19151509

C₂

yes

maximal, complemented

#19151499

C₂

yes

maximal

#19151463

C₂

yes

maximal, complemented

#19151455

C₂

yes

maximal, complemented

C₄×C₄⋊Dic₈₉

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊Q₈×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₃₅₆

C₂²

yes

characteristic

C₂²×C₄×D₈₉

C₂²

yes

—

C₄×D₃₅₆

—

C₂×C₄⋊Dic₈₉

C₂²

yes

—

C₂×C₄⋊Dic₈₉

C₂²

yes

characteristic

C₂×#13083551

C₂²

yes

—

C₂×#13083551

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

—

#15667919

C₂²

yes

—

#15667919

C₂²

yes

—

C₂²×Q₈×C₈₉

C₂²

yes

—

C₄⋊Q₈×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂³×D₈₉

C₂³

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

C₂×D₃₅₆

Q₈

yes

complemented

C₂²×Dic₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₂³

yes

—

C₂×C₄×D₈₉

—

C₂×C₄×D₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083551

—

#13083551

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

—

C₂³×C₈₉

C₂⁴

yes

characteristic

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

—

C₂²×D₈₉

C₂×Q₈

yes

—

Q₈×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

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

C₄×D₈₉

—

C₂²×C₈₉

C₂²×D₄

yes

characteristic

C₂²×C₈₉

C₂²×Q₈

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

characteristic

C₂²×C₈₉

C₂×#41

yes

—

C₂²×C₈₉

H(5,𝔽₂)

yes

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂²⋊Q₈

yes

—

C₄×C₈₉

#124

yes

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

C₂×C₂²⋊Q₈

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

C₂×C₈₉

#555

yes

—

D₈₉

—

#2563

—

maximal, Sylow

C₈₉

#2563

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₈₉

yes

characteristic, complemented

C₂×C₄×D₄

—

—

—

—

—

C₄×C₄⋊C₄

—

C₂²×Q₈

C₂×D₈₉

yes

—

100

C₂²×D₄

—

101

C₂³×C₄

—

102

C₄⋊Q₈

—

103

C₄×D₄

—

178

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

107

C₂×C₄⋊C₄

C₂×D₈₉

yes

—

108

C₂×C₂²⋊C₄

—

109

C₂×C₂²⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂×D₈₉

yes

characteristic

112

#95

—

113

#95

—

114

C₂⁴

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×D₄

—

118

C₂×D₄

—

178

—

119

C₂×D₄

—

178

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

178

—

123

C₂²×C₄

—

178

—

124

C₂²×C₄

C₂²×D₈₉

yes

—

125

C₂²×C₄

C₂²×D₈₉

yes

characteristic

126

C₂²×C₄

C₂²×D₈₉

yes

characteristic

127

C₂²×C₄

C₂²×D₈₉

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

178

—

131

C₄⋊C₄

—

178

—

132

C₂²⋊C₄

—

356

—

133

C₂²⋊C₄

—

178

—

134

C₄²

—

178

—

135

C₄²

C₂²⋊D₈₉

yes

—

136

C₂³

—

137

C₂³

—

178

—

138

C₂³

—

178

—

139

C₂³

—

178

—

140

C₂³

—

178

—

141

C₂³

C₂³×D₈₉

yes

characteristic

142

Q₈

—

143

D₄

—

178

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

356

—

148

C₂×C₄

—

178

—

149

C₂×C₄

—

178

—

150

C₂×C₄

—

178

—

151

C₂×C₄

—

178

—

152

C₂×C₄

—

178

—

153

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

154

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

155

C₂×C₄

C₂×C₂²⋊D₈₉

yes

—

156

C₂×C₄

Q₈×D₈₉

yes

—

157

C₂×C₄

#13083575

yes

—

158

C₂×C₄

#13083575

yes

—

159

C₂²

—

356

—

160

C₂²

—

356

—

161

C₂²

—

356

—

162

C₂²

—

178

—

163

C₂²

—

178

—

164

C₂²

C₂²×C₂²⋊D₈₉

yes

characteristic

165

C₂²

C₂×Q₈×D₈₉

yes

characteristic

166

C₂²

C₂×#13083575

yes

—

167

C₂²

C₂×#13083575

yes

characteristic

168

C₂²

#15668057

yes

—

169

C₄

—

170

C₄

—

171

C₄

—

356

—

172

C₄

—

356

—

173

C₄

#15667988

yes

—

174

C₄

#15668001

yes

—

175

C₂

—

356

—

176

C₂

C₂×#15667988

yes

characteristic

177

C₂

C₂×#15668001

yes

characteristic

178

C₂

#19152165

yes

characteristic

179

C₂

#19152309

yes

—

180

C₂

#19152314

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₈₉	C₂
prime radical	C₂²×D₃₅₆	C₂²
hypercenter	C₂×C₄⋊Q₈	D₈₉
hypocenter	C₈₉	#2563
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₈₉

commutator, socle, cosocle

C₂³×C₈₉

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₈₉

C₂

prime radical

C₂²×D₃₅₆

C₂²

hypercenter

C₂×C₄⋊Q₈

D₈₉

C₈₉

G ▷ C₂³×C₈₉ ▷ C₈₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #19151509	C₂	semidirect
4. #19151463	C₂	semidirect
5. #19151455	C₂	semidirect
6. C₄×C₄⋊Dic₈₉	C₂	semidirect
7. C₂×C₄⋊Q₈×C₈₉	C₂	semidirect
26. C₂×D₃₅₆	Q₈	semidirect
91. C₈₉	#2563	semidirect
92. C₂×C₄⋊Q₈	D₈₉	semidirect

H ≅ G/N

type

2. #19151509

C₂

semidirect

4. #19151463

C₂

semidirect

5. #19151455

C₂

semidirect

6. C₄×C₄⋊Dic₈₉

C₂

semidirect

7. C₂×C₄⋊Q₈×C₈₉

C₂

semidirect

26. C₂×D₃₅₆

Q₈

semidirect

91. C₈₉

#2563

semidirect

92. C₂×C₄⋊Q₈

D₈₉

semidirect

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

subgroup	index
C₂×C₄×D₃₅₆	2
#19151509	2
#19151499	2
#19151463	2
#19151455	2
C₄×C₄⋊Dic₈₉	2
C₂×C₄⋊Q₈×C₈₉	2
#2563	89

subgroup

index

C₂×C₄×D₃₅₆

C₄×C₄⋊Dic₈₉

C₂×C₄⋊Q₈×C₈₉

#2563

Subgroups — #26025426

Conjugacy classes of subgroups

Complements

Maximal subgroups