Group #26131696 — subgroups

Subgroups — #26131696

Order 12032.

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

normal subgroups only

This group has 306 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	C₂×#15688953	C₂	1	1	yes	maximal, characteristic
2	C₂×#15688898	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₄₇	C₂	1	2	yes	maximal
4	#20241481	C₂	1	2	yes	maximal
5	C₂×C₄₇×#368	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	M₁₆×Dic₄₇	—	2	2	no	—
9	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688953	C₄	1	4	yes	complemented
11	#15688898	C₄	1	4	yes	—
12	#15688887	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	2	yes	—
15	C₄₇×#368	C₄	1	4	yes	complemented
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
19	C₂×C₄₇⋊C₁₆	—	2	1	no	—
20	C₂×C₄₇⋊C₁₆	—	2	1	no	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
22	#13093960	—	2	2	no	—
23	#13093960	C₂×C₄	1	4	yes	—
24	#13093934	—	4	2	no	—
25	C₈×Dic₄₇	—	4	2	no	—
26	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
27	C₈×Dic₄₇	Q₈	1	2	yes	—
28	C₂×M₁₆×C₄₇	—	2	2	no	—
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
31	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
33	#108×C₄₇	—	2	2	no	—
34	#108×C₄₇	C₂×C₄	1	4	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₇	Q₁₆	1	2	yes	—
37	M₁₆×C₄₇	—	2	4	no	—
38	M₁₆×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
46	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
47	C₄×Dic₄₇	D₈	1	2	yes	—
48	#10625329	—	2	4	no	—
49	#10625329	—	4	2	no	—
50	C₂×C₄₇⋊C₈	—	2	1	no	—
51	C₂×C₄₇⋊C₈	—	2	1	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₁₆×C₄₇	C₄²	1	4	yes	—
55	C₄₇⋊C₁₆	—	2	2	no	—
56	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#135	1	1	yes	characteristic
63	C₂×Dic₄₇	—	4	2	no	—
64	C₂×Dic₄₇	#107	1	4	yes	—
65	C₈×C₄₇	—	4	4	no	—
66	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₄₇	#117	1	1	yes	characteristic
68	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
69	C₈×C₄₇	C₄×Q₈	1	1	yes	characteristic
70	C₄₇⋊C₈	—	2	2	no	—
71	C₄₇⋊C₈	—	4	4	no	—
72	#12676	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
75	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
76	C₂²×C₄₇	C₄×Q₁₆	1	1	yes	characteristic
77	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
78	C₄×C₄₇	#427	1	1	yes	characteristic
79	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
80	C₄×C₄₇	C₄×D₈	1	1	yes	characteristic
81	Dic₄₇	—	2	4	no	—
82	C₂×#368	—	47	1	no	—
83	C₂×#368	D₄₇	1	1	yes	characteristic
84	C₂×#346	—	47	1	no	—
85	C₄×#108	—	47	2	no	—
86	#1381	—	47	2	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
90	C₂×C₄₇	#2161	1	2	yes	—
91	C₂²×C₁₆	—	47	1	no	—
92	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
93	C₂×#108	—	47	2	no	—
94	C₂×#108	C₂×D₄₇	1	2	yes	—
95	C₄×M₁₆	—	94	2	no	—
96	C₂×C₄×C₈	—	47	1	no	—
97	#368	—	47	4	no	—
98	#368	Dic₄₇	1	4	yes	complemented
99	#346	—	47	4	no	—
100	#341	—	94	2	no	—
101	C₄₇	#12676	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	2	no	—
103	C₂×M₁₆	—	94	2	no	—
104	C₂²×C₈	—	47	1	no	—
105	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
106	C₂×C₄²	—	47	1	no	—
107	C₂×C₁₆	—	47	2	no	—
108	C₂×C₁₆	—	94	1	no	—
109	C₂×C₁₆	—	94	1	no	—
110	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
111	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
113	#108	—	2	2	no	—
114	#108	—	47	4	no	—
115	#108	—	94	2	no	—
116	#108	C₂×Dic₄₇	1	4	yes	—
117	#97	—	188	2	no	—
118	C₄×C₈	—	47	2	no	—
119	C₄×C₈	—	47	2	no	—
120	C₄×C₈	—	188	2	no	—
121	C₂²×C₄	—	47	2	no	—
122	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	M₁₆	—	94	4	no	—
126	M₁₆	—	188	2	no	—
127	C₂×C₈	—	4	2	no	—
128	C₂×C₈	—	47	2	no	—
129	C₂×C₈	—	94	1	no	—
130	C₂×C₈	—	94	1	no	—
131	C₂×C₈	—	188	2	no	—
132	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
134	C₂×C₈	#10625352	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
136	C₂×C₈	#10625354	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
138	C₄²	—	47	2	no	—
139	C₄²	—	47	2	no	—
140	C₁₆	—	94	2	no	—
141	C₁₆	C₄×Dic₄₇	1	4	yes	—
142	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
143	C₂×C₄	—	47	4	no	—
144	C₂×C₄	—	188	2	no	—
145	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
146	C₂×C₄	#13093986	1	1	yes	characteristic
147	C₂×C₄	D₄×Dic₄₇	1	1	yes	characteristic
148	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
149	C₂×C₄	#13094043	1	1	yes	characteristic
150	C₂×C₄	#13094047	1	1	yes	characteristic
151	C₈	—	4	4	no	—
152	C₈	—	94	2	no	—
153	C₈	—	188	4	no	—
154	C₈	C₂×C₄×Dic₄₇	1	1	yes	characteristic
155	C₈	#13093982	1	1	yes	characteristic
156	C₈	#13093989	1	1	yes	characteristic
157	C₈	Q₈×Dic₄₇	1	1	yes	characteristic
158	C₂²	—	2	2	no	—
159	C₂²	#107×D₄₇	1	1	yes	characteristic
160	C₂²	#15689217	1	1	yes	characteristic
161	C₂²	Q₁₆×Dic₄₇	1	1	yes	characteristic
162	C₄	—	94	4	no	—
163	C₄	C₄⋊C₄×Dic₄₇	1	1	yes	characteristic
164	C₄	#15689160	1	1	yes	characteristic
165	C₄	#15689205	1	1	yes	characteristic
166	C₄	D₈×Dic₄₇	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#107×Dic₄₇	1	1	yes	characteristic
170	C₂	#20245181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688953

C₂

yes

maximal, characteristic

C₂×#15688898

C₂

yes

maximal, characteristic

#108×Dic₄₇

C₂

yes

maximal

#20241481

C₂

yes

maximal

C₂×C₄₇×#368

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

M₁₆×Dic₄₇

—

C₂×C₈×Dic₄₇

C₂²

yes

characteristic

#15688953

C₄

yes

complemented

#15688898

C₄

yes

—

#15688887

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₄₇×#368

C₄

yes

complemented

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093960

—

#13093960

C₂×C₄

yes

—

#13093934

—

C₈×Dic₄₇

—

C₈×Dic₄₇

C₂×C₄

yes

—

C₈×Dic₄₇

Q₈

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#108×C₄₇

—

#108×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

Q₁₆

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

C₂×C₈×C₄₇

yes

C₂×C₈×C₄₇

yes

C₄×Dic₄₇

C₄⋊C₄

yes

—

C₄×Dic₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₄₇

C₄²

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄²

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

C₄×Q₁₆

yes

characteristic

C₂²×C₄₇

C₄×Q₁₆

yes

characteristic

C₄×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₇

#427

yes

characteristic

C₄×C₄₇

C₄×D₈

yes

characteristic

C₄×C₄₇

C₄×D₈

yes

characteristic

Dic₄₇

—

C₂×#368

—

C₂×#368

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₄×#107

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₄₇

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#368

—

#368

Dic₄₇

yes

complemented

#346

—

100

#341

—

101

C₄₇

#12676

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

C₂²×C₈

C₂²×D₄₇

yes

characteristic

106

C₂×C₄²

—

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

—

110

C₂×C₁₆

C₄×D₄₇

yes

—

111

C₂×C₁₆

C₄×D₄₇

yes

—

112

C₂×C₁₆

C₂×Dic₄₇

yes

—

113

#108

—

114

#108

—

115

#108

—

116

#108

C₂×Dic₄₇

yes

—

117

#97

—

188

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

188

—

121

C₂²×C₄

—

122

C₂²×C₄

D₄×D₄₇

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

M₁₆

—

126

M₁₆

—

188

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

188

—

132

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

133

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

134

C₂×C₈

#10625352

yes

characteristic

135

C₂×C₈

Q₈×D₄₇

yes

characteristic

136

C₂×C₈

#10625354

yes

characteristic

137

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

138

C₄²

—

139

C₄²

—

140

C₁₆

—

141

C₁₆

C₄×Dic₄₇

yes

—

142

C₂³

Q₁₆×D₄₇

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

188

—

145

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

146

C₂×C₄

#13093986

yes

characteristic

147

C₂×C₄

D₄×Dic₄₇

yes

characteristic

148

C₂×C₄

D₈×D₄₇

yes

characteristic

149

C₂×C₄

#13094043

yes

characteristic

150

C₂×C₄

#13094047

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

188

—

154

C₈

C₂×C₄×Dic₄₇

yes

characteristic

155

C₈

#13093982

yes

characteristic

156

C₈

#13093989

yes

characteristic

157

C₈

Q₈×Dic₄₇

yes

characteristic

158

C₂²

—

159

C₂²

#107×D₄₇

yes

characteristic

160

C₂²

#15689217

yes

characteristic

161

C₂²

Q₁₆×Dic₄₇

yes

characteristic

162

C₄

—

163

C₄

C₄⋊C₄×Dic₄₇

yes

characteristic

164

C₄

#15689160

yes

characteristic

165

C₄

#15689205

yes

characteristic

166

C₄

D₈×Dic₄₇

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#107×Dic₄₇

yes

characteristic

170

C₂

#20245181

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#368	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#368	D₄₇
hypocenter	C₄₇	#12676
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₈×C₄₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#368

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×C₄₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

N	H ≅ G/N	type
10. #15688953	C₄	semidirect
15. C₄₇×#368	C₄	semidirect
98. #368	Dic₄₇	semidirect
101. C₄₇	#12676	semidirect

H ≅ G/N

type

10. #15688953

C₄

semidirect

15. C₄₇×#368

C₄

semidirect

98. #368

Dic₄₇

semidirect

101. C₄₇

#12676

semidirect

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

subgroup	index
C₂×#15688953	2
C₂×#15688898	2
#108×Dic₄₇	2
#20241481	2
C₂×C₄₇×#368	2
#12676	47

subgroup

index

#108×Dic₄₇

C₂×C₄₇×#368

Subgroups — #26131696

Conjugacy classes of subgroups

Complements

Maximal subgroups