Group #26110065 — subgroups

Subgroups — #26110065

Order 12032.

438 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 203 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	#20243548	C₂	1	1	yes	maximal, characteristic
2	#20243547	C₂	1	2	yes	maximal
3	#20243515	C₂	1	2	yes	maximal
4	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093958	—	2	2	no	—
7	C₂×#13093957	—	2	4	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688993	C₂²	1	2	yes	—
10	#15688885	—	2	4	no	—
11	#15688844	C₄	1	4	yes	complemented
12	C₂×#106×C₄₇	—	2	2	no	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#384	C₂²	1	1	yes	characteristic
15	C₄₇×#384	C₂²	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
17	C₂×C₄⋊Dic₄₇	—	2	2	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	#13093958	—	4	4	no	—
21	#13093957	—	4	8	no	—
22	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
23	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
24	C₂²×C₈×C₄₇	D₄	1	2	yes	—
25	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
28	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	#106×C₄₇	—	4	4	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	Q₈	1	2	yes	—
33	C₂²×C₄×C₄₇	—	4	1	no	—
34	C₂²×C₄×C₄₇	—	4	1	no	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
37	C₂²×Dic₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₄₇	D₈	1	4	yes	—
40	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
41	C₄⋊C₄×C₄₇	—	4	2	no	—
42	C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₄⋊C₄×C₄₇	—	4	4	no	—
45	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
48	C₄⋊Dic₄₇	—	4	4	no	—
49	C₄⋊Dic₄₇	—	4	4	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	2	4	no	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
54	C₂³×C₄₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	8	1	no	—
56	C₂×C₄×C₄₇	—	8	1	no	—
57	C₂×C₄×C₄₇	—	8	2	no	—
58	C₂×C₄×C₄₇	—	8	2	no	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
65	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
66	C₂×C₄×C₄₇	#137	1	4	yes	—
67	C₂×Dic₄₇	—	8	2	no	—
68	C₂×Dic₄₇	—	8	2	no	—
69	C₂×Dic₄₇	—	8	2	no	—
70	C₈×C₄₇	#107	1	8	yes	complemented
71	C₄₇⋊C₈	—	2	8	no	—
72	#8820	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	#384	1	1	yes	characteristic
74	C₂²×C₄₇	#493	1	1	yes	characteristic
75	C₂²×C₄₇	#494	1	1	yes	characteristic
76	C₂²×C₄₇	#496	1	2	yes	—
77	C₂²×C₄₇	#497	1	2	yes	—
78	C₄×C₄₇	—	16	1	no	—
79	C₄×C₄₇	—	16	1	no	—
80	C₄×C₄₇	C₂×#107	1	2	yes	—
81	C₄×C₄₇	#428	1	4	yes	—
82	C₄×C₄₇	#500	1	2	yes	—
83	C₄×C₄₇	#501	1	4	yes	—
84	Dic₄₇	—	16	2	no	—
85	#1845	—	47	1	no	—
86	#1845	—	47	2	no	—
87	#1844	D₄₇	1	1	yes	characteristic
88	#1834	—	47	2	no	—
89	C₂×#322	—	47	1	no	—
90	C₂×C₄₇	#1721	1	1	yes	characteristic
91	C₂×C₄₇	#1722	1	1	yes	characteristic
92	C₂×C₄₇	#1723	1	2	yes	—
93	C₂×C₄₇	#1845	1	1	yes	characteristic
94	C₂×C₄₇	#1847	1	2	yes	—
95	C₂×#107	—	94	4	no	—
96	C₂×#107	—	94	2	no	—
97	C₂×#106	—	2	2	no	—
98	C₂×C₄×C₈	—	47	2	no	—
99	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
100	#384	—	47	2	no	—
101	#384	C₂×D₄₇	1	1	yes	characteristic
102	#384	C₂×D₄₇	1	1	yes	characteristic
103	#340	—	94	4	no	—
104	#322	—	47	4	no	—
105	C₄₇	#8820	1	1	yes	characteristic, complemented, Sylow
106	C₂²×C₈	—	47	4	no	—
107	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
108	C₂×C₄⋊C₄	—	2	2	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄⋊C₄	—	2	1	no	—
111	C₂×C₄⋊C₄	—	2	1	no	—
112	C₂×C₄⋊C₄	—	94	2	no	—
113	C₂×C₄⋊C₄	—	94	2	no	—
114	C₂×C₄⋊C₄	—	94	2	no	—
115	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	#107	—	188	8	no	—
117	#107	—	188	4	no	—
118	#106	—	4	4	no	—
119	C₄×C₈	—	47	4	no	—
120	C₄×C₈	—	47	4	no	—
121	C₄×C₈	Dic₉₄	1	2	yes	—
122	C₄×C₈	C₄×D₄₇	1	2	yes	—
123	C₂²×C₄	—	4	1	no	—
124	C₂²×C₄	—	4	1	no	—
125	C₂²×C₄	—	188	2	no	—
126	C₂²×C₄	D₄×D₄₇	1	2	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
128	C₂×C₈	—	47	8	no	—
129	C₂×C₈	—	94	4	no	—
130	C₂×C₈	—	94	4	no	—
131	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
132	C₂×C₈	#10625334	1	4	yes	—
133	C₂×C₈	#10625337	1	4	yes	—
134	C₄⋊C₄	—	4	2	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄⋊C₄	—	4	4	no	—
137	C₄⋊C₄	—	4	4	no	—
138	C₄⋊C₄	—	188	4	no	—
139	C₄⋊C₄	—	188	4	no	—
140	C₄⋊C₄	—	188	4	no	—
141	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
142	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
143	C₄²	Q₈×D₄₇	1	2	yes	—
144	C₂³	#13094020	1	1	yes	characteristic
145	C₂×C₄	—	8	1	no	—
146	C₂×C₄	—	8	1	no	—
147	C₂×C₄	—	8	2	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	376	2	no	—
150	C₂×C₄	—	376	2	no	—
151	C₂×C₄	—	376	2	no	—
152	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
153	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#13094022	1	1	yes	characteristic
155	C₂×C₄	#13094023	1	2	yes	—
156	C₂×C₄	C₂×#10625334	1	2	yes	—
157	C₂×C₄	C₂×#10625337	1	2	yes	—
158	C₂×C₄	#13094051	1	4	yes	—
159	C₂×C₄	#13094053	1	4	yes	—
160	C₈	—	94	8	no	—
161	C₈	#13093957	1	8	yes	complemented
162	C₂²	#15688992	1	1	yes	characteristic
163	C₂²	#15689499	1	1	yes	characteristic
164	C₂²	#15689502	1	1	yes	characteristic
165	C₂²	#15689509	1	2	yes	—
166	C₂²	#15689514	1	2	yes	—
167	C₄	—	16	1	no	—
168	C₄	—	16	1	no	—
169	C₄	—	752	2	no	—
170	C₄	C₂×#13093957	1	2	yes	—
171	C₄	#15689162	1	4	yes	—
172	C₄	#15689526	1	2	yes	—
173	C₄	#15689530	1	4	yes	—
174	C₂	#20243052	1	1	yes	characteristic
175	C₂	#20243057	1	1	yes	characteristic
176	C₂	#20243062	1	2	yes	—
177	C₂	#20243547	1	1	yes	characteristic
178	C₂	#20243553	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243548

C₂

yes

maximal, characteristic

#20243547

C₂

yes

maximal

#20243515

C₂

yes

maximal

C₂×#15688844

C₂

yes

maximal, characteristic

C₄₇×#1844

C₂

yes

maximal, characteristic

C₂×#13093958

—

C₂×#13093957

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

—

#15688885

—

#15688844

C₄

yes

complemented

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

#13093957

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#107

yes

complemented

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#497

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#107

yes

—

C₄×C₄₇

#428

yes

—

C₄×C₄₇

#500

yes

—

C₄×C₄₇

#501

yes

—

Dic₄₇

—

#1845

—

#1845

—

#1844

D₄₇

yes

characteristic

#1834

—

C₂×#322

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#1847

yes

—

C₂×#107

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

100

#384

—

101

#384

C₂×D₄₇

yes

characteristic

102

#384

C₂×D₄₇

yes

characteristic

103

#340

—

104

#322

—

105

C₄₇

#8820

yes

characteristic, complemented, Sylow

106

C₂²×C₈

—

107

C₂²×C₈

C₂²⋊D₄₇

yes

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²×D₄₇

yes

characteristic

116

#107

—

188

—

117

#107

—

188

—

118

#106

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

Dic₉₄

yes

—

122

C₄×C₈

C₄×D₄₇

yes

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

—

188

—

126

C₂²×C₄

D₄×D₄₇

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

Dic₄₇⋊C₄

yes

—

132

C₂×C₈

#10625334

yes

—

133

C₂×C₈

#10625337

yes

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄⋊C₄

—

188

—

139

C₄⋊C₄

—

188

—

140

C₄⋊C₄

—

188

—

141

C₄²

C₂×Dic₉₄

yes

characteristic

142

C₄²

C₂×C₄×D₄₇

yes

characteristic

143

C₄²

Q₈×D₄₇

yes

—

144

C₂³

#13094020

yes

characteristic

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

376

—

150

C₂×C₄

—

376

—

151

C₂×C₄

—

376

—

152

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

153

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

154

C₂×C₄

#13094022

yes

characteristic

155

C₂×C₄

#13094023

yes

—

156

C₂×C₄

C₂×#10625334

yes

—

157

C₂×C₄

C₂×#10625337

yes

—

158

C₂×C₄

#13094051

yes

—

159

C₂×C₄

#13094053

yes

—

160

C₈

—

161

C₈

#13093957

yes

complemented

162

C₂²

#15688992

yes

characteristic

163

C₂²

#15689499

yes

characteristic

164

C₂²

#15689502

yes

characteristic

165

C₂²

#15689509

yes

—

166

C₂²

#15689514

yes

—

167

C₄

—

168

C₄

—

169

C₄

—

752

—

170

C₄

C₂×#13093957

yes

—

171

C₄

#15689162

yes

—

172

C₄

#15689526

yes

—

173

C₄

#15689530

yes

—

174

C₂

#20243052

yes

characteristic

175

C₂

#20243057

yes

characteristic

176

C₂

#20243062

yes

—

177

C₂

#20243547

yes

characteristic

178

C₂

#20243553

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1844	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1844	D₄₇
hypocenter	C₄₇	#8820
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1844

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
11. #15688844	C₄	semidirect
70. C₈×C₄₇	#107	semidirect
105. C₄₇	#8820	semidirect
161. C₈	#13093957	semidirect

H ≅ G/N

type

11. #15688844

C₄

semidirect

70. C₈×C₄₇

#107

semidirect

105. C₄₇

#8820

semidirect

161. C₈

#13093957

semidirect

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

subgroup	index
#20243548	2
#20243547	2
#20243515	2
C₂×#15688844	2
C₄₇×#1844	2
#8820	47

subgroup

index

C₄₇×#1844

Subgroups — #26110065

Conjugacy classes of subgroups

Complements

Maximal subgroups