Group #26021476 — subgroups

Subgroups — #26021476

Order 11392.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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	#19152179	C₂	1	1	yes	maximal, characteristic
2	C₂×#15667940	C₂	1	2	yes	maximal, complemented
3	#19151597	C₂	1	2	yes	maximal, complemented
4	#19151367	C₂	1	1	yes	maximal, characteristic, complemented
5	#384×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13083574	C₂²	1	2	yes	—
7	C₂×#13083558	C₂²	1	2	yes	—
8	#15667992	—	2	2	no	—
9	#15667987	—	2	2	no	—
10	#15667986	—	2	2	no	—
11	#15667940	—	2	4	no	—
12	#15667935	—	2	2	no	—
13	#15667932	—	2	2	no	—
14	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
15	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
18	#13083574	—	2	2	no	—
19	#13083574	—	2	2	no	—
20	#13083574	—	2	2	no	—
21	#13083574	—	2	2	no	—
22	C₂×Dic₁₇₈	D₄	1	2	yes	—
23	C₂×D₃₅₆	—	2	2	no	—
24	C₂×D₃₅₆	D₄	1	2	yes	—
25	C₂×C₂²⋊D₈₉	—	2	2	no	—
26	C₂×C₄×D₈₉	—	2	2	no	—
27	#13083558	—	2	2	no	—
28	#13083558	—	2	2	no	—
29	C₂×#10623494	—	2	2	no	—
30	C₄⋊Dic₈₉	—	2	2	no	—
31	#13083551	—	2	4	no	—
32	C₂²×C₄×C₈₉	—	2	1	no	—
33	C₂²×C₄×C₈₉	—	2	1	no	—
34	C₂²×C₄×C₈₉	D₄	1	2	yes	—
35	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₈₉	—	2	2	no	—
37	C₄⋊C₄×C₈₉	—	2	2	no	—
38	C₄⋊C₄×C₈₉	—	4	4	no	—
39	C₄²×C₈₉	—	2	1	no	—
40	C₄²×C₈₉	D₄	1	2	yes	complemented
41	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₈₉	—	2	2	no	—
43	C₂×C₄×C₈₉	—	2	2	no	—
44	C₂×C₄×C₈₉	—	2	2	no	—
45	C₂×C₄×C₈₉	—	2	2	no	—
46	C₂×C₄×C₈₉	—	2	2	no	—
47	C₂×C₄×C₈₉	—	4	2	no	—
48	C₂×C₄×C₈₉	—	4	2	no	—
49	C₂×C₄×C₈₉	—	4	1	no	—
50	C₂×C₄×C₈₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
53	C₂×C₄×C₈₉	#41	1	2	yes	—
54	C₂²⋊D₈₉	—	4	2	no	—
55	C₂²⋊D₈₉	—	4	2	no	—
56	C₂×Dic₈₉	—	2	2	no	—
57	D₃₅₆	—	2	2	no	—
58	D₃₅₆	—	2	2	no	—
59	D₃₅₆	—	8	2	no	—
60	C₄×D₈₉	—	4	2	no	—
61	C₄×D₈₉	—	4	2	no	—
62	Dic₁₇₈	—	2	2	no	—
63	Dic₁₇₈	—	2	2	no	—
64	#10623494	—	4	2	no	—
65	C₂²×C₈₉	—	2	1	no	—
66	C₂²×C₈₉	—	2	1	no	—
67	C₂²×C₈₉	#120	1	1	yes	characteristic
68	C₂²×C₈₉	#121	1	1	yes	characteristic
69	C₂²×C₈₉	#124	1	1	yes	characteristic
70	C₂×D₈₉	—	4	2	no	—
71	C₂×D₈₉	—	4	2	no	—
72	C₂×D₈₉	—	4	2	no	—
73	C₄×C₈₉	—	2	4	no	—
74	C₄×C₈₉	—	4	2	no	—
75	C₄×C₈₉	—	8	1	no	—
76	C₄×C₈₉	#121	1	2	yes	—
77	C₄×C₈₉	#123	1	2	yes	—
78	Dic₈₉	—	4	2	no	—
79	C₂×C₈₉	—	2	1	no	—
80	C₂×C₈₉	—	2	1	no	—
81	C₂×C₈₉	#394	1	1	yes	characteristic
82	C₂×C₈₉	#453	1	1	yes	characteristic
83	C₂×C₈₉	#456	1	1	yes	characteristic
84	D₈₉	—	8	2	no	—
85	#2039	—	89	1	no	maximal, Sylow
86	C₈₉	#2039	1	1	yes	characteristic, complemented, Sylow
87	#537	—	89	1	no	—
88	C₂×#104	—	89	2	no	—
89	#417	—	89	2	no	—
90	#384	D₈₉	1	1	yes	characteristic, complemented
91	#343	—	89	1	no	—
92	C₂×#41	—	89	2	no	—
93	C₂×M₁₆	—	89	2	no	—
94	C₂²⋊Q₈	—	178	2	no	—
95	#121	—	178	2	no	—
96	#121	—	178	2	no	—
97	C₂×C₄⋊C₄	—	2	2	no	—
98	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
100	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
101	#104	—	178	4	no	—
102	#103	—	178	2	no	—
103	#102	—	178	2	no	—
104	#41	—	178	2	no	—
105	#41	—	178	2	no	—
106	#41	—	178	2	no	—
107	#41	—	178	2	no	—
108	C₂×Q₈	—	89	2	no	—
109	C₂×D₄	—	89	2	no	—
110	C₂×D₄	—	178	2	no	—
111	C₂×D₄	—	178	2	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	2	1	no	—
114	C₂²×C₄	—	178	2	no	—
115	C₂²×C₄	C₂²⋊D₈₉	1	2	yes	—
116	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
117	M₁₆	—	178	2	no	—
118	M₁₆	—	178	2	no	—
119	C₂×C₈	—	178	2	no	—
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	178	2	no	—
124	C₂²⋊C₄	—	178	4	no	—
125	C₄²	—	2	1	no	—
126	C₄²	D₃₅₆	1	2	yes	complemented
127	C₂³	—	178	2	no	—
128	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
129	Q₈	—	178	2	no	—
130	Q₈	—	178	2	no	—
131	D₄	—	356	2	no	—
132	D₄	—	356	2	no	—
133	D₄	—	178	2	no	—
134	D₄	—	178	2	no	—
135	D₄	—	712	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	356	2	no	—
144	C₂×C₄	—	356	2	no	—
145	C₂×C₄	—	178	2	no	—
146	C₂×C₄	D₄×D₈₉	1	2	yes	—
147	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
148	C₂×C₄	#13083574	1	1	yes	characteristic
149	C₂×C₄	#13083575	1	2	yes	—
150	C₈	—	356	2	no	—
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	—	356	2	no	—
154	C₂²	—	356	2	no	—
155	C₂²	—	356	2	no	—
156	C₂²	#15667980	1	1	yes	characteristic
157	C₂²	#15667986	1	1	yes	characteristic
158	C₂²	#15668001	1	1	yes	characteristic
159	C₄	—	2	4	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	356	2	no	—
163	C₄	#15667987	1	2	yes	—
164	C₄	#15667997	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	—	712	2	no	—
168	C₂	#19151515	1	1	yes	characteristic
169	C₂	#19151765	1	1	yes	characteristic
170	C₂	#19151780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19152179

C₂

yes

maximal, characteristic

C₂×#15667940

C₂

yes

maximal, complemented

#19151597

C₂

yes

maximal, complemented

#19151367

C₂

yes

maximal, characteristic, complemented

#384×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×#13083574

C₂²

yes

—

C₂×#13083558

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

—

—

—

—

C₂×Dic₁₇₈

D₄

yes

—

C₂×D₃₅₆

—

C₂×D₃₅₆

D₄

yes

—

C₂×C₂²⋊D₈₉

—

C₂×C₄×D₈₉

—

#13083558

—

#13083558

—

C₂×#10623494

—

C₄⋊Dic₈₉

—

#13083551

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

D₄

yes

complemented

C₂³×C₈₉

C₂×D₄

yes

characteristic

C₂²×D₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

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

yes

C₂×C₄×C₈₉

yes

—

C₂²⋊D₈₉

—

C₂²⋊D₈₉

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

D₃₅₆

—

C₄×D₈₉

—

C₄×D₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂²×C₈₉

yes

C₂×D₈₉

—

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#121

yes

—

C₄×C₈₉

#123

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

D₈₉

—

#2039

—

maximal, Sylow

C₈₉

#2039

yes

characteristic, complemented, Sylow

—

—

—

D₈₉

yes

characteristic, complemented

#343

—

C₂×#41

—

C₂×M₁₆

—

C₂²⋊Q₈

—

178

—

#121

—

178

—

#121

—

178

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

C₂×C₄⋊C₄

C₂×D₈₉

yes

characteristic

100

C₂×C₄²

C₂×D₈₉

yes

characteristic

101

#104

—

178

—

102

#103

—

178

—

103

#102

—

178

—

104

#41

—

178

—

105

#41

—

178

—

106

#41

—

178

—

107

#41

—

178

—

108

C₂×Q₈

—

109

C₂×D₄

—

110

C₂×D₄

—

178

—

111

C₂×D₄

—

178

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

178

—

115

C₂²×C₄

C₂²⋊D₈₉

yes

—

116

C₂²×C₄

C₂²×D₈₉

yes

characteristic

117

M₁₆

—

178

—

118

M₁₆

—

178

—

119

C₂×C₈

—

178

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

178

—

124

C₂²⋊C₄

—

178

—

125

C₄²

—

126

C₄²

D₃₅₆

yes

complemented

127

C₂³

—

178

—

128

C₂³

C₂×C₂²⋊D₈₉

yes

characteristic

129

Q₈

—

178

—

130

Q₈

—

178

—

131

D₄

—

356

—

132

D₄

—

356

—

133

D₄

—

178

—

134

D₄

—

178

—

135

D₄

—

712

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

356

—

144

C₂×C₄

—

356

—

145

C₂×C₄

—

178

—

146

C₂×C₄

D₄×D₈₉

yes

—

147

C₂×C₄

C₂×D₃₅₆

yes

characteristic

148

C₂×C₄

#13083574

yes

characteristic

149

C₂×C₄

#13083575

yes

—

150

C₈

—

356

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

356

—

154

C₂²

—

356

—

155

C₂²

—

356

—

156

C₂²

#15667980

yes

characteristic

157

C₂²

#15667986

yes

characteristic

158

C₂²

#15668001

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

356

—

163

C₄

#15667987

yes

—

164

C₄

#15667997

yes

—

165

C₂

—

166

C₂

—

167

C₂

—

712

—

168

C₂

#19151515

yes

characteristic

169

C₂

#19151765

yes

characteristic

170

C₂

#19151780

yes

characteristic

	subgroup	G/H
center	C₂²	#15667980
commutator, cosocle	C₂²×C₄×C₈₉	C₂³
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#384×C₈₉	C₂
socle	C₂²×C₈₉	#120
prime radical	#19152179	C₂
hypercenter	#384	D₈₉
hypocenter	C₈₉	#2039
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #384

subgroup

G/H

center

C₂²

#15667980

commutator, cosocle

C₂²×C₄×C₈₉

C₂³

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#384×C₈₉

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #384

N	H ≅ G/N	type
2. C₂×#15667940	C₂	semidirect
3. #19151597	C₂	semidirect
4. #19151367	C₂	semidirect
5. #384×C₈₉	C₂	semidirect
40. C₄²×C₈₉	D₄	semidirect
86. C₈₉	#2039	semidirect
90. #384	D₈₉	semidirect
126. C₄²	D₃₅₆	semidirect

H ≅ G/N

type

2. C₂×#15667940

C₂

semidirect

3. #19151597

C₂

semidirect

4. #19151367

C₂

semidirect

5. #384×C₈₉

C₂

semidirect

40. C₄²×C₈₉

D₄

semidirect

86. C₈₉

#2039

semidirect

90. #384

D₈₉

semidirect

126. C₄²

D₃₅₆

semidirect

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

subgroup	index
#19152179	2
C₂×#15667940	2
#19151597	2
#19151367	2
#384×C₈₉	2
#2039	89

subgroup

index

#384×C₈₉

Subgroups — #26021476

Conjugacy classes of subgroups

Complements

Maximal subgroups