Group #26132638 — subgroups

Subgroups — #26132638

Order 12032.

342 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 143 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	#20245144	C₂	1	1	yes	maximal, characteristic
2	C₄×C₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241469	C₂	1	4	yes	maximal
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	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
11	C₂×M₃₂×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₄₇	—	2	2	no	—
16	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
17	#13093964	—	2	4	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
20	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
21	C₈×Dic₄₇	—	2	2	no	—
22	C₈×Dic₄₇	—	2	2	no	—
23	C₈×Dic₄₇	—	2	4	no	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	M₃₂×C₄₇	—	2	4	no	—
28	C₂×C₁₆×C₄₇	—	2	2	no	—
29	C₄×C₈×C₄₇	D₄	1	2	yes	—
30	C₄×C₈×C₄₇	Q₈	1	2	yes	—
31	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	#41	1	2	yes	—
39	C₂×C₈×C₄₇	#41	1	2	yes	—
40	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
43	C₄×Dic₄₇	—	2	8	no	—
44	C₄×Dic₄₇	—	2	2	no	—
45	C₄×Dic₄₇	M₁₆	1	4	yes	—
46	C₂×C₄₇⋊C₈	—	2	2	no	—
47	C₂×C₄₇⋊C₈	—	2	2	no	—
48	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
49	C₁₆×C₄₇	—	4	2	no	—
50	C₄₇⋊C₁₆	—	4	2	no	—
51	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₄×C₄₇	—	2	4	no	—
53	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#117	1	2	yes	—
57	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
59	C₂×Dic₄₇	—	2	8	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₈×C₄₇	—	2	2	no	—
62	C₈×C₄₇	#124	1	1	yes	characteristic
63	C₈×C₄₇	#124	1	1	yes	characteristic
64	C₈×C₄₇	#125	1	2	yes	—
65	C₄₇⋊C₈	—	2	4	no	—
66	#12883	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₄₇	#432	1	2	yes	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	#364	1	4	yes	—
72	C₄×C₄₇	#382	1	1	yes	characteristic
73	C₄×C₄₇	#423	1	1	yes	characteristic
74	C₄×C₄₇	#431	1	2	yes	—
75	Dic₄₇	—	2	8	no	—
76	#2150	—	47	1	no	—
77	#2150	D₄₇	1	1	yes	characteristic
78	C₄²×C₈	—	47	1	no	—
79	#1376	—	47	4	no	—
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	#1831	1	1	yes	characteristic
82	C₂×C₄₇	C₂×#364	1	2	yes	—
83	C₂×M₃₂	—	47	2	no	—
84	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
85	C₂×C₄×C₈	—	47	1	no	—
86	C₂×C₄×C₈	—	47	2	no	—
87	C₂×C₄×C₈	—	47	2	no	—
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	C₄³	—	47	1	no	—
90	C₄⋊C₁₆	—	2	2	no	—
91	C₄⋊C₁₆	—	94	2	no	—
92	C₄₇	#12883	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	47	2	no	—
94	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
96	C₂×C₄²	—	47	2	no	—
97	C₂×C₄²	—	94	2	no	—
98	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
99	M₃₂	—	2	4	no	—
100	M₃₂	—	94	4	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	—	94	2	no	—
103	C₄×C₈	—	47	2	no	—
104	C₄×C₈	—	47	2	no	—
105	C₄×C₈	—	94	4	no	—
106	C₄×C₈	—	94	2	no	—
107	C₄×C₈	—	94	2	no	—
108	C₄×C₈	Dic₉₄	1	2	yes	—
109	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
110	C₂²×C₄	—	94	2	no	—
111	C₂²×C₄	#10625324	1	2	yes	—
112	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	47	4	no	—
115	C₂×C₈	—	94	2	no	—
116	C₂×C₈	—	94	2	no	—
117	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
118	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
119	C₂×C₈	#10625352	1	2	yes	—
120	C₂×C₈	#10625354	1	2	yes	—
121	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
122	C₄²	—	47	4	no	—
123	C₄²	—	94	8	no	—
124	C₄²	—	94	2	no	—
125	C₄²	C₈×D₄₇	1	1	yes	characteristic
126	C₄²	C₈×D₄₇	1	1	yes	characteristic
127	C₄²	Dic₄₇⋊C₄	1	2	yes	—
128	C₁₆	—	4	2	no	—
129	C₁₆	—	188	2	no	—
130	C₂³	C₂×#10625324	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	94	8	no	—
133	C₂×C₄	—	94	4	no	—
134	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
135	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
136	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
137	C₂×C₄	#13093982	1	2	yes	—
138	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
139	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
140	C₈	—	2	2	no	—
141	C₈	—	94	4	no	—
142	C₈	#13094011	1	1	yes	characteristic
143	C₈	#13094013	1	1	yes	characteristic
144	C₈	#13094015	1	2	yes	—
145	C₂²	—	2	2	no	—
146	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
147	C₂²	#15689179	1	2	yes	—
148	C₄	—	2	2	no	—
149	C₄	—	94	8	no	—
150	C₄	#15688943	1	4	yes	—
151	C₄	#15688988	1	1	yes	characteristic
152	C₄	#15689146	1	1	yes	characteristic
153	C₄	#15689175	1	2	yes	—
154	C₂	—	2	2	no	—
155	C₂	#20243504	1	1	yes	characteristic
156	C₂	C₂×#15688943	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245144

C₂

yes

maximal, characteristic

C₄×C₈×Dic₄₇

C₂

yes

maximal, characteristic

#20241469

C₂

yes

maximal

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

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

—

C₈×Dic₄₇

—

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

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

M₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₈

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

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

#12883

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#432

yes

—

C₄×C₄₇

—

C₄×C₄₇

#364

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2150

—

#2150

D₄₇

yes

characteristic

C₄²×C₈

—

#1376

—

C₂×C₄₇

—

C₂×C₄₇

#1831

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

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄₇

#12883

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄×D₄₇

yes

characteristic

M₃₂

—

100

M₃₂

—

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

Dic₉₄

yes

—

109

C₄×C₈

C₂²⋊D₄₇

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

#10625324

yes

—

112

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

Dic₄₇⋊C₄

yes

—

118

C₂×C₈

C₂×Dic₉₄

yes

characteristic

119

C₂×C₈

#10625352

yes

—

120

C₂×C₈

#10625354

yes

—

121

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₈×D₄₇

yes

characteristic

126

C₄²

C₈×D₄₇

yes

characteristic

127

C₄²

Dic₄₇⋊C₄

yes

—

128

C₁₆

—

129

C₁₆

—

188

—

130

C₂³

C₂×#10625324

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

Dic₄₇⋊C₈

yes

—

135

C₂×C₄

Dic₄₇⋊C₈

yes

—

136

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

137

C₂×C₄

#13093982

yes

—

138

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

139

C₂×C₄

M₁₆×D₄₇

yes

—

140

C₈

—

141

C₈

—

142

C₈

#13094011

yes

characteristic

143

C₈

#13094013

yes

characteristic

144

C₈

#13094015

yes

—

145

C₂²

—

146

C₂²

C₂×Dic₄₇⋊C₈

yes

characteristic

147

C₂²

#15689179

yes

—

148

C₄

—

149

C₄

—

150

C₄

#15688943

yes

—

151

C₄

#15688988

yes

characteristic

152

C₄

#15689146

yes

characteristic

153

C₄

#15689175

yes

—

154

C₂

—

155

C₂

#20243504

yes

characteristic

156

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	#12883
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
92. C₄₇	#12883	semidirect

H ≅ G/N

type

92. C₄₇

#12883

semidirect

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

subgroup	index
#20245144	2
C₄×C₈×Dic₄₇	2
#20241469	2
C₄₇×#2150	2
#12883	47

subgroup

index

#20245144

C₄×C₈×Dic₄₇

#20241469

C₄₇×#2150

#12883

Subgroups — #26132638

Conjugacy classes of subgroups

Complements

Maximal subgroups