Group #26134871 — subgroups

Subgroups — #26134871

Order 12032.

334 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15689525	C₂	1	1	yes	maximal, characteristic
2	#20245144	C₂	1	1	yes	maximal, characteristic
3	#20241498	C₂	1	4	yes	maximal
4	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	—	2	1	no	—
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689525	—	2	2	no	—
8	#15689525	C₄	1	4	yes	—
9	C₂×#13093958	C₂²	1	2	yes	—
10	C₂×#13093956	—	2	1	no	—
11	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
12	C₂×#108×C₄₇	C₂²	1	2	yes	—
13	C₄×M₁₆×C₄₇	—	2	1	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#341	—	2	1	no	—
16	#13094016	—	2	4	no	—
17	#13094016	—	4	1	no	—
18	#13094016	—	4	1	no	—
19	C₂×C₄⋊Dic₄₇	—	2	2	no	—
20	C₂×Dic₄₇⋊C₄	—	4	2	no	—
21	#13093964	—	2	4	no	—
22	C₂×C₄₇⋊C₁₆	—	2	2	no	—
23	#13093958	—	2	2	no	—
24	#13093958	C₂×C₄	1	4	yes	—
25	#13093956	—	2	4	no	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	#108×C₄₇	—	2	2	no	—
31	#108×C₄₇	Q₈	1	4	yes	—
32	#97×C₄₇	—	4	1	no	—
33	C₄×C₈×C₄₇	—	4	1	no	—
34	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₂²×C₄×C₄₇	SD₁₆	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	4	2	no	—
39	M₁₆×C₄₇	—	2	4	no	—
40	M₁₆×C₄₇	—	4	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
47	C₂×C₈×C₄₇	#41	1	2	yes	—
48	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₄²×C₄₇	D₈	1	1	yes	characteristic
50	C₄²×C₄₇	D₈	1	1	yes	characteristic
51	C₄⋊Dic₄₇	—	2	4	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	Dic₄₇⋊C₄	—	4	4	no	—
54	Dic₄₇⋊C₄	—	4	4	no	—
55	C₄₇⋊C₁₆	—	4	2	no	—
56	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	#102	1	2	yes	—
59	C₂×C₄×C₄₇	#102	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#137	1	2	yes	—
64	C₂×Dic₄₇	—	4	4	no	—
65	C₂×Dic₄₇	—	8	4	no	—
66	C₈×C₄₇	—	2	2	no	—
67	C₈×C₄₇	—	4	4	no	—
68	C₈×C₄₇	C₄×Q₈	1	2	yes	—
69	C₈×C₄₇	#124	1	1	yes	characteristic
70	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
71	#13280	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
74	C₂²×C₄₇	#475	1	2	yes	—
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	#391	1	1	yes	characteristic
78	C₄×C₄₇	#419	1	1	yes	characteristic
79	C₄×C₄₇	#474	1	2	yes	—
80	Dic₄₇	—	8	4	no	—
81	C₂×#499	—	47	1	no	—
82	#2150	—	47	1	no	—
83	C₄×#108	D₄₇	1	1	yes	characteristic
84	#1387	—	47	4	no	—
85	C₂×C₄₇	—	2	2	no	—
86	C₂×C₄₇	#1988	1	1	yes	characteristic
87	C₂×C₄₇	#2148	1	2	yes	—
88	C₂×#125	—	94	1	no	—
89	C₂×M₃₂	—	47	2	no	—
90	#499	—	47	4	no	—
91	#499	—	94	2	no	—
92	C₂×#108	C₂×D₄₇	1	2	yes	—
93	C₂×#107	—	47	2	no	—
94	C₂×#106	—	94	1	no	—
95	C₄×M₁₆	—	2	1	no	—
96	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
97	C₄⋊C₁₆	—	94	2	no	—
98	#341	—	2	1	no	—
99	C₄₇	#13280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
103	#125	—	94	4	no	—
104	#125	—	188	1	no	—
105	#125	—	188	1	no	—
106	C₂×C₄⋊C₄	—	94	2	no	—
107	C₂×C₄⋊C₄	—	188	2	no	—
108	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
109	M₃₂	—	94	4	no	—
110	C₂×C₁₆	—	94	2	no	—
111	#108	—	2	2	no	—
112	#108	Dic₉₄	1	4	yes	—
113	#107	—	47	4	no	—
114	#107	—	94	2	no	—
115	#106	—	94	4	no	—
116	#97	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	C₄×D₄₇	1	2	yes	—
119	C₄×C₈	D₁₈₈	1	2	yes	—
120	C₂²×C₄	—	188	2	no	—
121	C₂²×C₄	#10625336	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
123	M₁₆	—	2	4	no	—
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
131	C₂×C₈	#10625350	1	2	yes	—
132	C₄⋊C₄	—	94	4	no	—
133	C₄⋊C₄	—	188	4	no	—
134	C₄⋊C₄	—	188	4	no	—
135	C₄⋊C₄	—	188	2	no	—
136	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
137	C₄²	#10625334	1	1	yes	characteristic
138	C₄²	#10625334	1	1	yes	characteristic
139	C₁₆	—	188	2	no	—
140	C₂³	C₂×#10625336	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	376	4	no	—
143	C₂×C₄	—	188	4	no	—
144	C₂×C₄	#13093944	1	2	yes	—
145	C₂×C₄	#13093944	1	2	yes	—
146	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
147	C₂×C₄	#13094003	1	2	yes	—
148	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
149	C₂×C₄	#13094054	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	4	4	no	—
152	C₈	C₄×Dic₉₄	1	2	yes	—
153	C₈	#13094012	1	1	yes	characteristic
154	C₈	#13094024	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#13093944	1	1	yes	characteristic
157	C₂²	#15689408	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	376	4	no	—
161	C₄	#15689029	1	1	yes	characteristic
162	C₄	#15689131	1	1	yes	characteristic
163	C₄	#15689403	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#20244340	1	1	yes	characteristic
166	C₂	#20245135	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689525

C₂

yes

maximal, characteristic

#20245144

C₂

yes

maximal, characteristic

#20241498

C₂

yes

maximal

C₄×#108×C₄₇

C₂

yes

maximal, characteristic

C₂×#13094016

—

C₂×#13093964

C₂²

yes

—

#15689525

—

#15689525

C₄

yes

—

C₂×#13093958

C₂²

yes

—

C₂×#13093956

—

C₄⋊(C₄₇⋊C₁₆)

—

C₂×#108×C₄₇

C₂²

yes

—

C₄×M₁₆×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

#13094016

—

#13094016

—

#13094016

—

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093958

—

#13093958

C₂×C₄

yes

—

#13093956

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

#108×C₄₇

—

#108×C₄₇

Q₈

yes

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₄×Q₈

yes

—

C₈×C₄₇

#124

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

#13280

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#102

yes

characteristic

C₂²×C₄₇

#475

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×#499

—

#2150

—

C₄×#108

D₄₇

yes

characteristic

#1387

—

C₂×C₄₇

—

C₂×C₄₇

#1988

yes

characteristic

C₂×C₄₇

#2148

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

—

C₂×#108

C₂×D₄₇

yes

—

C₂×#107

—

C₂×#106

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

#341

—

C₄₇

#13280

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

102

C₂²×C₈

C₂²×D₄₇

yes

characteristic

103

#125

—

104

#125

—

188

—

105

#125

—

188

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

188

—

108

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

109

M₃₂

—

110

C₂×C₁₆

—

111

#108

—

112

#108

Dic₉₄

yes

—

113

#107

—

114

#107

—

115

#106

—

116

#97

—

117

C₄×C₈

—

118

C₄×C₈

C₄×D₄₇

yes

—

119

C₄×C₈

D₁₈₈

yes

—

120

C₂²×C₄

—

188

—

121

C₂²×C₄

#10625336

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

123

M₁₆

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

128

C₂×C₈

C₂×Dic₉₄

yes

—

129

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

130

C₂×C₈

C₂×D₁₈₈

yes

characteristic

131

C₂×C₈

#10625350

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

188

—

134

C₄⋊C₄

—

188

—

135

C₄⋊C₄

—

188

—

136

C₄²

(C₂×D₄₇)⋊C₄

yes

—

137

C₄²

#10625334

yes

characteristic

138

C₄²

#10625334

yes

characteristic

139

C₁₆

—

188

—

140

C₂³

C₂×#10625336

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

376

—

143

C₂×C₄

—

188

—

144

C₂×C₄

#13093944

yes

—

145

C₂×C₄

#13093944

yes

—

146

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

147

C₂×C₄

#13094003

yes

—

148

C₂×C₄

C₂×#10625334

yes

characteristic

149

C₂×C₄

#13094054

yes

—

150

C₈

—

151

C₈

—

152

C₈

C₄×Dic₉₄

yes

—

153

C₈

#13094012

yes

characteristic

154

C₈

#13094024

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#13093944

yes

characteristic

157

C₂²

#15689408

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

376

—

161

C₄

#15689029

yes

characteristic

162

C₄

#15689131

yes

characteristic

163

C₄

#15689403

yes

—

164

C₂

—

165

C₂

#20244340

yes

characteristic

166

C₂

#20245135

yes

—

	subgroup	G/H
center	C₂²	C₂×#13093944
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#102
prime radical	C₂³×C₄₇	C₂×SD₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#13280
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₂²

C₂×#13093944

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄×#108×C₄₇

C₂

socle

C₂²×C₄₇

C₂×#102

prime radical

C₂³×C₄₇

C₂×SD₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
99. C₄₇	#13280	semidirect

H ≅ G/N

type

99. C₄₇

#13280

semidirect

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

subgroup	index
C₂×#15689525	2
#20245144	2
#20241498	2
C₄×#108×C₄₇	2
#13280	47

subgroup

index

C₄×#108×C₄₇

Subgroups — #26134871

Conjugacy classes of subgroups

Complements

Maximal subgroups