Group #26132641 — subgroups

Subgroups — #26132641

Order 12032.

302 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20243074	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×#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	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	C₂×C₄	1	2	yes	—
16	#13093964	—	2	4	no	—
17	C₂×C₄₇⋊C₁₆	—	2	2	no	—
18	#13093935	C₈	1	4	yes	—
19	#13093934	—	2	4	no	—
20	C₈×Dic₄₇	—	2	2	no	—
21	C₈×Dic₄₇	—	2	2	no	—
22	C₄₇⋊#95	—	2	2	no	—
23	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
26	M₃₂×C₄₇	—	2	4	no	—
27	C₂×C₁₆×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	D₄	1	2	yes	—
29	C₄×C₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×Dic₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	#41	1	2	yes	—
38	C₂×C₈×C₄₇	#41	1	2	yes	—
39	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
40	C₄²×C₄₇	C₂×C₈	1	2	yes	—
41	C₄×Dic₄₇	—	2	2	no	—
42	C₄×Dic₄₇	M₁₆	1	4	yes	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
46	C₁₆×C₄₇	—	4	2	no	—
47	C₄₇⋊C₁₆	—	4	2	no	—
48	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#117	1	2	yes	—
53	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	4	4	no	—
57	C₈×C₄₇	—	2	2	no	—
58	C₈×C₄₇	#124	1	1	yes	characteristic
59	C₈×C₄₇	#124	1	1	yes	characteristic
60	C₈×C₄₇	#125	1	2	yes	—
61	C₄₇⋊C₈	—	2	4	no	—
62	#12884	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂²×C₄₇	#432	1	2	yes	—
66	C₄×C₄₇	—	2	4	no	—
67	C₄×C₄₇	#382	1	1	yes	characteristic
68	C₄×C₄₇	#423	1	1	yes	characteristic
69	C₄×C₄₇	#431	1	2	yes	—
70	Dic₄₇	—	4	4	no	—
71	#2150	—	47	1	no	—
72	#2150	D₄₇	1	1	yes	characteristic
73	#1726	—	47	1	no	—
74	#1376	—	47	4	no	—
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	#1831	1	1	yes	characteristic
77	C₂×C₄₇	#2153	1	2	yes	—
78	C₂×M₃₂	—	47	2	no	—
79	C₂×M₃₂	C₂×D₄₇	1	2	yes	—
80	C₂×#97	—	47	1	no	—
81	C₂×#97	—	47	2	no	—
82	C₂×C₄×C₈	—	47	2	no	—
83	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
84	#376	—	47	1	no	—
85	C₄⋊C₁₆	—	2	2	no	—
86	C₄⋊C₁₆	—	94	2	no	—
87	C₄₇	#12884	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	47	2	no	—
89	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
90	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×C₄²	—	47	2	no	—
92	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
93	M₃₂	—	2	4	no	—
94	M₃₂	—	94	4	no	—
95	C₂×C₁₆	—	2	2	no	—
96	C₂×C₁₆	—	94	2	no	—
97	#97	—	47	4	no	—
98	#97	—	94	4	no	—
99	C₄×C₈	—	94	2	no	—
100	C₄×C₈	—	94	2	no	—
101	C₄×C₈	Dic₉₄	1	2	yes	—
102	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
103	#95	—	94	2	no	—
104	C₂²×C₄	—	94	2	no	—
105	C₂²×C₄	#10625324	1	2	yes	—
106	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	47	4	no	—
109	C₂×C₈	—	94	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
112	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
113	C₂×C₈	#10625352	1	2	yes	—
114	C₂×C₈	#10625354	1	2	yes	—
115	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
116	C₄²	—	47	4	no	—
117	C₄²	—	94	2	no	—
118	C₄²	C₈×D₄₇	1	2	yes	—
119	C₄²	Dic₄₇⋊C₄	1	2	yes	—
120	C₁₆	—	4	2	no	—
121	C₁₆	—	188	2	no	—
122	C₂³	C₂×#10625324	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	94	4	no	—
125	C₂×C₄	—	188	4	no	—
126	C₂×C₄	Dic₄₇⋊C₈	1	4	yes	—
127	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#13093982	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
130	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	94	4	no	—
133	C₈	#13094011	1	1	yes	characteristic
134	C₈	#13094013	1	1	yes	characteristic
135	C₈	#13094015	1	2	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
138	C₂²	#15689179	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	188	4	no	—
141	C₄	#15688988	1	1	yes	characteristic
142	C₄	#15689146	1	1	yes	characteristic
143	C₄	#15689175	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#20243504	1	1	yes	characteristic
146	C₂	#20245153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245144

C₂

yes

maximal, characteristic

#20243074

C₂

yes

maximal, characteristic

#20241469

C₂

yes

maximal

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

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

—

C₈×Dic₄₇

—

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

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

—

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

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

—

#12884

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#432

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

#2150

—

#2150

D₄₇

yes

characteristic

#1726

—

#1376

—

C₂×C₄₇

—

C₂×C₄₇

#1831

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

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄₇

#12884

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

—

#97

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

Dic₉₄

yes

—

102

C₄×C₈

C₂²⋊D₄₇

yes

—

103

#95

—

104

C₂²×C₄

—

105

C₂²×C₄

#10625324

yes

—

106

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

Dic₄₇⋊C₄

yes

—

112

C₂×C₈

C₂×Dic₉₄

yes

characteristic

113

C₂×C₈

#10625352

yes

—

114

C₂×C₈

#10625354

yes

—

115

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

116

C₄²

—

117

C₄²

—

118

C₄²

C₈×D₄₇

yes

—

119

C₄²

Dic₄₇⋊C₄

yes

—

120

C₁₆

—

121

C₁₆

—

188

—

122

C₂³

C₂×#10625324

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

188

—

126

C₂×C₄

Dic₄₇⋊C₈

yes

—

127

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

128

C₂×C₄

#13093982

yes

—

129

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

130

C₂×C₄

M₁₆×D₄₇

yes

—

131

C₈

—

132

C₈

—

133

C₈

#13094011

yes

characteristic

134

C₈

#13094013

yes

characteristic

135

C₈

#13094015

yes

—

136

C₂²

—

137

C₂²

C₂×Dic₄₇⋊C₈

yes

characteristic

138

C₂²

#15689179

yes

—

139

C₄

—

140

C₄

—

188

—

141

C₄

#15688988

yes

characteristic

142

C₄

#15689146

yes

characteristic

143

C₄

#15689175

yes

—

144

C₂

—

145

C₂

#20243504

yes

characteristic

146

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	#12884
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
87. C₄₇	#12884	semidirect

H ≅ G/N

type

87. C₄₇

#12884

semidirect

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

subgroup	index
#20245144	2
#20243074	2
#20241469	2
C₄₇×#2150	2
#12884	47

subgroup

index

C₄₇×#2150

Subgroups — #26132641

Conjugacy classes of subgroups

Complements

Maximal subgroups