Group #26110091 — subgroups

Subgroups — #26110091

Order 12032.

338 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20243559	C₂	1	2	yes	maximal
2	#20242147	C₂	1	1	yes	maximal, characteristic
3	#20242146	C₂	1	2	yes	maximal
4	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1567	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093959	—	2	4	no	—
7	#15689147	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	#15688876	—	2	2	no	—
11	#15688875	—	2	4	no	—
12	#15688845	C₄	1	4	yes	—
13	C₄₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#335	—	2	2	no	—
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
19	#13093959	—	4	8	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
25	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	D₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	M₁₆×C₄₇	—	4	4	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
40	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	#10625329	—	4	4	no	—
45	C₂×C₄₇⋊C₈	—	4	2	no	—
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂×C₄₇⋊C₈	D₈	1	4	yes	—
50	C₂×C₄₇⋊C₈	Q₁₆	1	4	yes	—
51	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
62	C₈×C₄₇	—	2	4	no	—
63	C₈×C₄₇	—	8	2	no	—
64	C₄₇⋊C₈	—	2	4	no	—
65	C₄₇⋊C₈	—	8	2	no	—
66	C₄₇⋊C₈	#107	1	8	yes	—
67	#8826	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	—	2	1	no	—
70	C₂²×C₄₇	#384	1	1	yes	characteristic
71	C₂²×C₄₇	#500	1	2	yes	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	C₂×#107	1	2	yes	—
75	C₄×C₄₇	#430	1	2	yes	—
76	C₄×C₄₇	#430	1	2	yes	—
77	C₄×C₄₇	#493	1	1	yes	characteristic
78	C₄×C₄₇	#494	1	1	yes	characteristic
79	#1848	—	47	2	no	—
80	#1567	—	47	2	no	—
81	#1567	—	47	1	no	—
82	#1567	D₄₇	1	1	yes	characteristic
83	C₂×#322	—	47	1	no	—
84	C₂×C₄₇	—	2	2	no	—
85	C₂×C₄₇	#1845	1	1	yes	characteristic
86	C₂×C₄₇	#1849	1	1	yes	characteristic
87	C₂×C₄₇	#1849	1	1	yes	characteristic
88	C₂×#108	—	94	4	no	—
89	#423	—	47	2	no	—
90	#423	C₂×D₄₇	1	2	yes	—
91	C₂×C₄×C₈	—	47	1	no	—
92	C₂×C₄×C₈	—	47	1	no	—
93	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
94	#335	—	2	2	no	—
95	#335	—	94	4	no	—
96	#335	—	94	2	no	—
97	#322	—	47	4	no	—
98	C₄₇	#8826	1	1	yes	characteristic, complemented, Sylow
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	94	2	no	—
101	C₂²×C₈	—	47	2	no	—
102	C₂²×C₈	—	47	2	no	—
103	C₂²×C₈	Dic₉₄	1	2	yes	—
104	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
105	#108	—	188	8	no	—
106	C₄⋊C₈	—	2	2	no	—
107	C₄⋊C₈	—	2	2	no	—
108	C₄⋊C₈	—	94	2	no	—
109	C₄⋊C₈	—	94	2	no	—
110	C₄×C₈	—	47	2	no	—
111	C₄×C₈	—	47	2	no	—
112	C₄×C₈	—	47	2	no	—
113	C₄×C₈	—	47	2	no	—
114	C₄×C₈	C₄×D₄₇	1	2	yes	—
115	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
116	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
117	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
119	M₁₆	—	4	4	no	—
120	M₁₆	—	188	4	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	4	2	no	—
123	C₂×C₈	—	47	4	no	—
124	C₂×C₈	—	47	4	no	—
125	C₂×C₈	—	47	4	no	—
126	C₂×C₈	—	47	4	no	—
127	C₂×C₈	—	188	2	no	—
128	C₂×C₈	—	188	2	no	—
129	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
130	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
131	C₄²	D₄×D₄₇	1	1	yes	characteristic
132	C₄²	Q₈×D₄₇	1	1	yes	characteristic
133	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂³	#13094023	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
139	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
140	C₂×C₄	#13094020	1	1	yes	characteristic
141	C₂×C₄	#13094022	1	1	yes	characteristic
142	C₂×C₄	#13094023	1	1	yes	characteristic
143	C₂×C₄	D₈×D₄₇	1	2	yes	—
144	C₂×C₄	Q₁₆×D₄₇	1	2	yes	—
145	C₈	—	2	4	no	—
146	C₈	—	8	2	no	—
147	C₈	—	47	8	no	—
148	C₈	—	94	4	no	—
149	C₈	—	376	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#15688992	1	1	yes	characteristic
153	C₂²	#15689527	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	#107×D₄₇	1	2	yes	—
157	C₄	#15689170	1	2	yes	—
158	C₄	#15689171	1	2	yes	—
159	C₄	#15689498	1	1	yes	characteristic
160	C₄	#15689501	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	#20243546	1	1	yes	characteristic
163	C₂	#20243561	1	1	yes	characteristic
164	C₂	#20243562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243559

C₂

yes

maximal

#20242147

C₂

yes

maximal, characteristic

#20242146

C₂

yes

maximal

C₂×#15688845

C₂

yes

maximal, characteristic

C₄₇×#1567

C₂

yes

maximal, characteristic

C₂×#13093959

—

#15689147

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

—

C₄

yes

—

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂²×C₄₇⋊C₈

Q₈

yes

—

#13093959

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

D₈

yes

—

C₂×C₄₇⋊C₈

Q₁₆

yes

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#107

yes

—

#8826

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#500

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#107

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

C₂×#322

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

—

C₄₇

#8826

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₉₄

yes

—

104

C₂×C₄²

C₂²×D₄₇

yes

characteristic

105

#108

—

188

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₄×C₈

C₄×D₄₇

yes

—

115

C₄×C₈

C₂²⋊D₄₇

yes

—

116

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

117

C₂²×C₄

D₄×D₄₇

yes

characteristic

118

C₂²×C₄

Q₈×D₄₇

yes

characteristic

119

M₁₆

—

120

M₁₆

—

188

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

188

—

128

C₂×C₈

—

188

—

129

C₂×C₈

Dic₄₇⋊C₄

yes

—

130

C₄²

C₂×C₄×D₄₇

yes

characteristic

131

C₄²

D₄×D₄₇

yes

characteristic

132

C₄²

Q₈×D₄₇

yes

characteristic

133

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

134

C₂³

#13094023

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

138

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

139

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

140

C₂×C₄

#13094020

yes

characteristic

141

C₂×C₄

#13094022

yes

characteristic

142

C₂×C₄

#13094023

yes

characteristic

143

C₂×C₄

D₈×D₄₇

yes

—

144

C₂×C₄

Q₁₆×D₄₇

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

376

—

150

C₂²

—

151

C₂²

—

152

C₂²

#15688992

yes

characteristic

153

C₂²

#15689527

yes

—

154

C₄

—

155

C₄

—

156

C₄

#107×D₄₇

yes

—

157

C₄

#15689170

yes

—

158

C₄

#15689171

yes

—

159

C₄

#15689498

yes

characteristic

160

C₄

#15689501

yes

characteristic

161

C₂

—

162

C₂

#20243546

yes

characteristic

163

C₂

#20243561

yes

characteristic

164

C₂

#20243562

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1567	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1567	D₄₇
hypocenter	C₄₇	#8826
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1567

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
98. C₄₇	#8826	semidirect

H ≅ G/N

type

98. C₄₇

#8826

semidirect

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

subgroup	index
#20243559	2
#20242147	2
#20242146	2
C₂×#15688845	2
C₄₇×#1567	2
#8826	47

subgroup

index

C₄₇×#1567

Subgroups — #26110091

Conjugacy classes of subgroups

Complements

Maximal subgroups