Group #26132694 — subgroups

Subgroups — #26132694

Order 12032.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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×#15688943	C₂	1	4	yes	maximal
2	#20245144	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₄₇	C₄	1	2	yes	—
8	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
9	C₄²×Dic₄₇	C₄	1	1	yes	characteristic
10	#15688943	—	2	8	no	—
11	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₄₇	—	2	2	no	—
17	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
18	#13093964	—	2	4	no	—
19	C₂×C₄₇⋊C₁₆	—	2	2	no	—
20	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
21	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
22	C₈×Dic₄₇	—	2	4	no	—
23	C₈×Dic₄₇	D₄	1	4	yes	—
24	C₈×Dic₄₇	Q₈	1	4	yes	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₄₇	—	2	4	no	—
29	C₂×C₁₆×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	D₄	1	2	yes	—
31	C₄×C₈×C₄₇	Q₈	1	2	yes	—
32	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
41	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
44	C₄×Dic₄₇	—	2	8	no	—
45	C₄×Dic₄₇	—	2	2	no	—
46	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
49	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
50	C₁₆×C₄₇	—	4	2	no	—
51	C₄₇⋊C₁₆	—	4	2	no	—
52	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₄₇	—	2	8	no	—
61	C₂×Dic₄₇	—	2	4	no	—
62	C₈×C₄₇	—	2	2	no	—
63	C₈×C₄₇	#127	1	1	yes	characteristic
64	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
66	C₄₇⋊C₈	C₄⋊C₈	1	8	yes	—
67	#12900	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₄₇	#423	1	2	yes	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	#364	1	4	yes	complemented
73	C₄×C₄₇	#384	1	1	yes	characteristic
74	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₄₇	#423	1	1	yes	characteristic
76	Dic₄₇	—	2	8	no	—
77	C₂×#364	—	47	4	no	—
78	#2150	—	47	1	no	—
79	#2150	D₄₇	1	1	yes	characteristic
80	C₄²×C₈	—	47	1	no	—
81	C₂×C₄₇	—	2	2	no	—
82	C₂×C₄₇	#1842	1	1	yes	characteristic
83	C₂×C₄₇	C₂×#364	1	2	yes	—
84	C₂×M₃₂	—	47	2	no	—
85	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
86	C₂×C₄×C₈	—	47	2	no	—
87	C₂×C₄×C₈	—	47	1	no	—
88	C₂×C₄×C₈	—	47	2	no	—
89	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
90	C₄³	—	47	1	no	—
91	#364	—	94	8	no	—
92	C₄⋊C₁₆	—	2	2	no	—
93	C₄⋊C₁₆	—	94	2	no	—
94	C₄₇	#12900	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	47	2	no	—
96	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	C₂×C₄²	—	47	2	no	—
99	C₂×C₄²	—	94	2	no	—
100	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	94	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	94	2	no	—
105	C₄×C₈	—	47	4	no	—
106	C₄×C₈	—	47	4	no	—
107	C₄×C₈	—	47	2	no	—
108	C₄×C₈	—	47	2	no	—
109	C₄×C₈	—	94	4	no	—
110	C₄×C₈	Dic₉₄	1	2	yes	—
111	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
112	C₂²×C₄	—	94	2	no	—
113	C₂²×C₄	#10625324	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	47	4	no	—
117	C₂×C₈	—	47	4	no	—
118	C₂×C₈	—	47	4	no	—
119	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
120	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
121	C₂×C₈	D₄×D₄₇	1	2	yes	—
122	C₂×C₈	Q₈×D₄₇	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄²	—	47	4	no	—
125	C₄²	—	94	8	no	—
126	C₄²	—	94	2	no	—
127	C₄²	C₈×D₄₇	1	1	yes	characteristic
128	C₄²	C₈×D₄₇	1	1	yes	characteristic
129	C₄²	Dic₄₇⋊C₄	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	188	2	no	—
132	C₂³	C₂×#10625324	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	94	8	no	—
135	C₂×C₄	—	94	4	no	—
136	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
137	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
139	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	47	8	no	—
144	C₈	#13094020	1	1	yes	characteristic
145	C₈	#13094022	1	1	yes	characteristic
146	C₈	#13094023	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
149	C₂²	#15689145	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	94	8	no	—
152	C₄	#15688943	1	4	yes	complemented
153	C₄	#15688992	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
155	C₄	#15689146	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20243539	1	1	yes	characteristic
158	C₂	C₂×#15688943	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688943

C₂

yes

maximal

#20245144

C₂

yes

maximal, characteristic

C₄×C₈×Dic₄₇

C₂

yes

maximal, characteristic

C₄₇×#2150

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₄₇

C₄

yes

—

C₂×C₈×Dic₄₇

C₂²

yes

—

C₄²×Dic₄₇

C₄

yes

characteristic

#15688943

—

C₄⋊(C₄₇⋊C₁₆)

—

C₂×M₃₂×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₄×C₄₇⋊C₈

C₈

yes

—

C₄×C₄₇⋊C₈

C₈

yes

—

C₈×Dic₄₇

—

C₈×Dic₄₇

D₄

yes

—

C₈×Dic₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#127

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

—

C₄₇⋊C₈

C₄⋊C₈

yes

—

#12900

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#423

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×C₄⋊C₈

yes

—

C₄×C₄₇

#423

yes

characteristic

Dic₄₇

—

C₂×#364

—

#2150

—

#2150

D₄₇

yes

characteristic

C₄²×C₈

—

C₂×C₄₇

—

C₂×C₄₇

#1842

yes

characteristic

C₂×C₄₇

C₂×#364

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄³

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄₇

#12900

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₄×D₄₇

yes

characteristic

101

M₃₂

—

102

M₃₂

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

Dic₉₄

yes

—

111

C₄×C₈

C₂²⋊D₄₇

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

#10625324

yes

—

114

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

Dic₄₇⋊C₄

yes

—

120

C₂×C₈

C₂×Dic₉₄

yes

characteristic

121

C₂×C₈

D₄×D₄₇

yes

—

122

C₂×C₈

Q₈×D₄₇

yes

—

123

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

C₈×D₄₇

yes

characteristic

128

C₄²

C₈×D₄₇

yes

characteristic

129

C₄²

Dic₄₇⋊C₄

yes

—

130

C₁₆

—

131

C₁₆

—

188

—

132

C₂³

C₂×#10625324

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

Dic₄₇⋊C₈

yes

—

137

C₂×C₄

Dic₄₇⋊C₈

yes

—

138

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

139

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

140

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

141

C₂×C₄

M₁₆×D₄₇

yes

—

142

C₈

—

143

C₈

—

144

C₈

#13094020

yes

characteristic

145

C₈

#13094022

yes

characteristic

146

C₈

#13094023

yes

—

147

C₂²

—

148

C₂²

C₂×Dic₄₇⋊C₈

yes

characteristic

149

C₂²

#15689145

yes

—

150

C₄

—

151

C₄

—

152

C₄

#15688943

yes

complemented

153

C₄

#15688992

yes

characteristic

154

C₄

C₄⋊C₈×D₄₇

yes

—

155

C₄

#15689146

yes

characteristic

156

C₂

—

157

C₂

#20243539

yes

characteristic

158

C₂

C₂×#15688943

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#2150	C₂
socle	C₂²×C₄₇	C₂×C₄⋊C₈
prime radical	C₂³×C₄₇	C₂×M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#2150	D₄₇
hypocenter	C₄₇	#12900
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#2150

C₂

socle

C₂²×C₄₇

C₂×C₄⋊C₈

prime radical

C₂³×C₄₇

C₂×M₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
72. C₄×C₄₇	#364	semidirect
94. C₄₇	#12900	semidirect
152. C₄	#15688943	semidirect

H ≅ G/N

type

72. C₄×C₄₇

#364

semidirect

94. C₄₇

#12900

semidirect

152. C₄

#15688943

semidirect

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

subgroup	index
C₂×#15688943	2
#20245144	2
C₄×C₈×Dic₄₇	2
C₄₇×#2150	2
#12900	47

subgroup

index

C₂×#15688943

#20245144

C₄×C₈×Dic₄₇

C₄₇×#2150

#12900

Subgroups — #26132694

Conjugacy classes of subgroups

Complements

Maximal subgroups