Group #26132697 — subgroups

Subgroups — #26132697

Order 12032.

342 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20243074	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093964	C₂²	1	2	yes	—
6	C₂×#13093935	C₄	1	1	yes	characteristic
7	C₂×#13093934	C₄	1	2	yes	—
8	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
9	#15688969	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	C₂×C₄	1	2	yes	—
17	#13093964	—	2	4	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	#13093935	C₈	1	4	yes	—
20	#13093934	—	2	4	no	—
21	C₈×Dic₄₇	D₄	1	4	yes	—
22	C₈×Dic₄₇	Q₈	1	4	yes	—
23	C₄₇⋊#95	—	2	2	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	2	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
40	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₄²×C₄₇	C₂×C₈	1	2	yes	—
42	C₄×Dic₄₇	—	2	2	no	—
43	C₄×Dic₄₇	C₄⋊C₄	1	4	yes	—
44	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
45	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
46	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
47	C₁₆×C₄₇	—	4	2	no	—
48	C₄₇⋊C₁₆	—	4	2	no	—
49	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
52	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
56	C₂×Dic₄₇	—	2	4	no	—
57	C₂×Dic₄₇	—	4	4	no	—
58	C₈×C₄₇	—	2	2	no	—
59	C₈×C₄₇	#127	1	1	yes	characteristic
60	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
61	C₈×C₄₇	C₄⋊Q₈	1	2	yes	—
62	C₄₇⋊C₈	C₄⋊C₈	1	8	yes	—
63	#12901	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	2	2	no	—
65	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
66	C₂²×C₄₇	#423	1	2	yes	—
67	C₄×C₄₇	—	2	4	no	—
68	C₄×C₄₇	#384	1	1	yes	characteristic
69	C₄×C₄₇	C₂×C₄⋊C₈	1	2	yes	—
70	C₄×C₄₇	#423	1	1	yes	characteristic
71	Dic₄₇	—	4	4	no	—
72	C₂×#364	—	47	4	no	—
73	#2150	—	47	1	no	—
74	#2150	D₄₇	1	1	yes	characteristic
75	#1726	—	47	1	no	—
76	C₂×C₄₇	—	2	2	no	—
77	C₂×C₄₇	#1842	1	1	yes	characteristic
78	C₂×C₄₇	#2153	1	2	yes	—
79	C₂×M₃₂	—	47	2	no	—
80	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
81	C₂×#97	—	47	1	no	—
82	C₂×#97	—	47	2	no	—
83	C₂×C₄×C₈	—	47	2	no	—
84	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
85	#376	—	47	1	no	—
86	#364	—	94	8	no	—
87	C₄⋊C₁₆	—	2	2	no	—
88	C₄⋊C₁₆	—	94	2	no	—
89	C₄₇	#12901	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	47	2	no	—
91	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
92	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
93	C₂×C₄²	—	47	2	no	—
94	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	M₃₂	—	94	4	no	—
97	C₂×C₁₆	—	2	2	no	—
98	C₂×C₁₆	—	94	2	no	—
99	#97	—	47	4	no	—
100	#97	—	94	4	no	—
101	C₄×C₈	—	47	4	no	—
102	C₄×C₈	—	47	4	no	—
103	C₄×C₈	Dic₉₄	1	2	yes	—
104	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
105	#95	—	94	2	no	—
106	C₂²×C₄	—	94	2	no	—
107	C₂²×C₄	#10625324	1	2	yes	—
108	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	47	4	no	—
111	C₂×C₈	—	47	4	no	—
112	C₂×C₈	—	47	4	no	—
113	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
114	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
115	C₂×C₈	D₄×D₄₇	1	2	yes	—
116	C₂×C₈	Q₈×D₄₇	1	2	yes	—
117	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
118	C₄²	—	47	4	no	—
119	C₄²	—	94	2	no	—
120	C₄²	C₈×D₄₇	1	2	yes	—
121	C₄²	Dic₄₇⋊C₄	1	2	yes	—
122	C₁₆	—	4	2	no	—
123	C₁₆	—	188	2	no	—
124	C₂³	C₂×#10625324	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	94	4	no	—
127	C₂×C₄	—	188	4	no	—
128	C₂×C₄	Dic₄₇⋊C₈	1	4	yes	—
129	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
130	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
132	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
133	C₈	—	2	2	no	—
134	C₈	—	47	8	no	—
135	C₈	#13094020	1	1	yes	characteristic
136	C₈	#13094022	1	1	yes	characteristic
137	C₈	#13094023	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
140	C₂²	#15689145	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	188	4	no	—
143	C₄	#15688992	1	1	yes	characteristic
144	C₄	C₄⋊C₈×D₄₇	1	2	yes	—
145	C₄	#15689146	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#20243539	1	1	yes	characteristic
148	C₂	#20245153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688943

C₂

yes

maximal

#20245144

C₂

yes

maximal, characteristic

#20243074

C₂

yes

maximal, characteristic

C₄₇×#2150

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂×#13093935

C₄

yes

characteristic

C₂×#13093934

C₄

yes

—

C₂×C₈×Dic₄₇

C₂²

yes

—

#15688969

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

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

#13093935

C₈

yes

—

#13093934

—

C₈×Dic₄₇

D₄

yes

—

C₈×Dic₄₇

Q₈

yes

—

C₄₇⋊#95

—

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

—

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

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

—

#12901

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#423

yes

—

C₄×C₄₇

—

C₄×C₄₇

#384

yes

characteristic

C₄×C₄₇

C₂×C₄⋊C₈

yes

—

C₄×C₄₇

#423

yes

characteristic

Dic₄₇

—

C₂×#364

—

#2150

—

#2150

D₄₇

yes

characteristic

#1726

—

C₂×C₄₇

—

C₂×C₄₇

#1842

yes

characteristic

C₂×C₄₇

#2153

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₄₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#376

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄₇

#12901

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₄×D₄₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

Dic₉₄

yes

—

104

C₄×C₈

C₂²⋊D₄₇

yes

—

105

#95

—

106

C₂²×C₄

—

107

C₂²×C₄

#10625324

yes

—

108

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

Dic₄₇⋊C₄

yes

—

114

C₂×C₈

C₂×Dic₉₄

yes

characteristic

115

C₂×C₈

D₄×D₄₇

yes

—

116

C₂×C₈

Q₈×D₄₇

yes

—

117

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

118

C₄²

—

119

C₄²

—

120

C₄²

C₈×D₄₇

yes

—

121

C₄²

Dic₄₇⋊C₄

yes

—

122

C₁₆

—

123

C₁₆

—

188

—

124

C₂³

C₂×#10625324

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

Dic₄₇⋊C₈

yes

—

129

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

130

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

131

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

132

C₂×C₄

M₁₆×D₄₇

yes

—

133

C₈

—

134

C₈

—

135

C₈

#13094020

yes

characteristic

136

C₈

#13094022

yes

characteristic

137

C₈

#13094023

yes

—

138

C₂²

—

139

C₂²

C₂×Dic₄₇⋊C₈

yes

characteristic

140

C₂²

#15689145

yes

—

141

C₄

—

142

C₄

—

188

—

143

C₄

#15688992

yes

characteristic

144

C₄

C₄⋊C₈×D₄₇

yes

—

145

C₄

#15689146

yes

characteristic

146

C₂

—

147

C₂

#20243539

yes

characteristic

148

C₂

#20245153

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	#12901
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
89. C₄₇	#12901	semidirect

H ≅ G/N

type

89. C₄₇

#12901

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
#20243074	2
C₄₇×#2150	2
#12901	47

subgroup

index

C₄₇×#2150

Subgroups — #26132697

Conjugacy classes of subgroups

Complements

Maximal subgroups