Group #26107447 — subgroups

Subgroups — #26107447

Order 12032.

366 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄×#13093935	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal
3	#20241681	C₂	1	2	yes	maximal
4	#20241221	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₄	1	2	yes	—
8	C₂×#13093935	C₄	1	2	yes	—
9	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
11	#15688875	C₄	1	4	yes	complemented
12	#15688873	C₄	1	4	yes	complemented
13	#15688845	—	2	2	no	—
14	#15688844	—	2	2	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₄³×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
21	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
22	#13093935	—	2	4	no	—
23	#13093935	—	2	4	no	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
27	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
28	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
29	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄²×C₄₇	—	2	2	no	—
31	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	—	2	2	no	—
36	C₂²×C₄×C₄₇	—	2	2	no	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	#41	1	2	yes	—
39	C₂×C₈×C₄₇	—	2	4	no	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	—	4	4	no	—
42	C₄²×C₄₇	M₁₆	1	4	yes	—
43	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
44	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	—	2	2	no	—
47	C₂×C₄₇⋊C₈	—	2	2	no	—
48	C₂×C₄₇⋊C₈	—	2	2	no	—
49	C₂×C₄₇⋊C₈	—	2	4	no	—
50	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
51	C₂×C₄₇⋊C₈	C₄²	1	1	yes	characteristic
52	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
53	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
54	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
55	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
56	C₂³×C₄₇	#117	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	4	4	no	—
60	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
63	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
66	C₈×C₄₇	—	4	4	no	—
67	C₄₇⋊C₈	—	2	1	no	—
68	C₄₇⋊C₈	—	2	1	no	—
69	C₄₇⋊C₈	—	2	2	no	—
70	C₄₇⋊C₈	—	2	2	no	—
71	C₄₇⋊C₈	—	4	4	no	—
72	C₄₇⋊C₈	#97	1	4	yes	—
73	#8269	—	47	1	no	maximal, Sylow
74	C₂²×C₄₇	—	2	2	no	—
75	C₂²×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₂²×C₄₇	#432	1	2	yes	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	—	4	4	no	—
79	C₄×C₄₇	C₂×#97	1	1	yes	characteristic
80	C₄×C₄₇	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₄₇	#423	1	2	yes	—
82	C₄×C₄₇	#444	1	4	yes	—
83	C₄×#97	—	47	1	no	—
84	#1564	—	47	2	no	—
85	#1452	—	47	2	no	—
86	#1276	—	47	1	no	—
87	#1276	D₄₇	1	1	yes	characteristic
88	C₂×C₄₇	—	2	2	no	—
89	C₂×C₄₇	#1767	1	1	yes	characteristic
90	C₂×C₄₇	#1773	1	2	yes	—
91	#431	—	47	2	no	—
92	#431	C₂×D₄₇	1	2	yes	—
93	C₂×#97	—	47	2	no	—
94	C₂×#97	—	47	2	no	—
95	C₂×C₄×C₈	—	47	1	no	—
96	C₂×C₄×C₈	—	47	1	no	—
97	C₄³	C₂×D₄₇	1	1	yes	characteristic
98	#335	—	47	4	no	—
99	#334	—	47	4	no	—
100	#322	—	94	2	no	—
101	#322	—	94	2	no	—
102	C₄₇	#8269	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	—	47	1	no	—
104	C₂²×C₈	—	47	1	no	—
105	C₂²×C₈	—	47	2	no	—
106	C₂×C₄²	—	2	2	no	—
107	C₂×C₄²	C₄×D₄₇	1	2	yes	—
108	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
109	C₄⋊C₈	—	47	4	no	—
110	C₄⋊C₈	C₄×D₄₇	1	4	yes	—
111	C₂²⋊C₈	—	2	2	no	—
112	C₂²⋊C₈	—	94	2	no	—
113	#97	—	94	4	no	—
114	#97	—	94	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	47	2	no	—
117	C₄×C₈	—	47	2	no	—
118	C₄×C₈	—	47	1	no	—
119	C₄×C₈	—	47	1	no	—
120	C₄×C₈	—	47	2	no	—
121	C₄×C₈	—	94	2	no	—
122	C₂²×C₄	—	2	2	no	—
123	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
124	C₂²×C₄	#10625352	1	2	yes	—
125	C₂×C₈	—	2	4	no	—
126	C₂×C₈	—	47	2	no	—
127	C₂×C₈	—	47	4	no	—
128	C₂×C₈	—	47	1	no	—
129	C₂×C₈	—	47	1	no	—
130	C₂×C₈	—	47	2	no	—
131	C₂×C₈	—	47	2	no	—
132	C₂×C₈	—	94	2	no	—
133	C₂×C₈	—	94	2	no	—
134	C₂×C₈	—	94	2	no	—
135	C₂×C₈	—	94	4	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	#10625324	1	4	yes	—
139	C₄²	C₂×C₄×D₄₇	1	2	yes	—
140	C₄²	D₄×D₄₇	1	1	yes	characteristic
141	C₄²	Q₈×D₄₇	1	1	yes	characteristic
142	C₂³	#13093980	1	1	yes	characteristic
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
147	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
148	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
149	C₂×C₄	#13093989	1	2	yes	—
150	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
151	C₂×C₄	C₂×#10625324	1	2	yes	—
152	C₈	—	4	4	no	—
153	C₈	—	47	4	no	—
154	C₈	—	94	1	no	—
155	C₈	—	94	1	no	—
156	C₈	—	94	2	no	—
157	C₈	—	94	2	no	—
158	C₈	—	188	4	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#15688974	1	1	yes	characteristic
161	C₂²	#15689177	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	4	4	no	—
164	C₄	#97×D₄₇	1	1	yes	characteristic
165	C₄	C₄×#10625324	1	1	yes	characteristic
166	C₄	#15689145	1	2	yes	—
167	C₄	#15689242	1	4	yes	—
168	C₂	—	2	2	no	—
169	C₂	#20243240	1	1	yes	characteristic
170	C₂	#20243271	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093935

C₂

yes

maximal, characteristic

#20242134

C₂

yes

maximal

#20241681

C₂

yes

maximal

#20241221

C₂

yes

maximal, characteristic

C₄₇×#1276

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×#13093935

C₄

yes

—

C₂×#13093935

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688875

C₄

yes

complemented

#15688873

C₄

yes

complemented

#15688845

—

#15688844

—

C₄₇×#431

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂×C₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

C₂×C₄

yes

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄²

yes

characteristic

C₂×C₄₇⋊C₈

C₄²

yes

characteristic

C₂×C₄₇⋊C₈

C₄²

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

#97

yes

—

#8269

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#432

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#97

yes

characteristic

C₄×C₄₇

C₄×M₁₆

yes

characteristic

C₄×C₄₇

#423

yes

—

C₄×C₄₇

#444

yes

—

C₄×#97

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1767

yes

characteristic

C₂×C₄₇

#1773

yes

—

#431

—

#431

C₂×D₄₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

C₂×D₄₇

yes

characteristic

#335

—

#334

—

100

#322

—

101

#322

—

102

C₄₇

#8269

yes

characteristic, complemented, Sylow

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂×C₄²

—

107

C₂×C₄²

C₄×D₄₇

yes

—

108

C₂×C₄²

C₂²×D₄₇

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

C₄×D₄₇

yes

—

111

C₂²⋊C₈

—

112

C₂²⋊C₈

—

113

#97

—

114

#97

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₄×C₈

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

124

C₂²×C₄

#10625352

yes

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

136

C₄²

—

137

C₄²

—

138

C₄²

#10625324

yes

—

139

C₄²

C₂×C₄×D₄₇

yes

—

140

C₄²

D₄×D₄₇

yes

characteristic

141

C₄²

Q₈×D₄₇

yes

characteristic

142

C₂³

#13093980

yes

characteristic

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₄²×D₄₇

yes

characteristic

147

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

148

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

149

C₂×C₄

#13093989

yes

—

150

C₂×C₄

M₁₆×D₄₇

yes

—

151

C₂×C₄

C₂×#10625324

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

188

—

159

C₂²

—

160

C₂²

#15688974

yes

characteristic

161

C₂²

#15689177

yes

—

162

C₄

—

163

C₄

—

164

C₄

#97×D₄₇

yes

characteristic

165

C₄

C₄×#10625324

yes

characteristic

166

C₄

#15689145

yes

—

167

C₄

#15689242

yes

—

168

C₂

—

169

C₂

#20243240

yes

characteristic

170

C₂

#20243271

yes

—

	subgroup	G/H
center	C₄²	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1276	C₂
socle	C₂²×C₄₇	C₄×C₄⋊C₄
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1276	D₄₇
hypocenter	C₄₇	#8269
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₄²

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1276

C₂

socle

C₂²×C₄₇

C₄×C₄⋊C₄

prime radical

C₂³×C₄₇

#117

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
11. #15688875	C₄	semidirect
12. #15688873	C₄	semidirect
102. C₄₇	#8269	semidirect

H ≅ G/N

type

11. #15688875

C₄

semidirect

12. #15688873

C₄

semidirect

102. C₄₇

#8269

semidirect

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

subgroup	index
C₄×#13093935	2
#20242134	2
#20241681	2
#20241221	2
C₄₇×#1276	2
#8269	47

subgroup

index

C₄₇×#1276

Subgroups — #26107447

Conjugacy classes of subgroups

Complements

Maximal subgroups