Group #26102884 — subgroups

Subgroups — #26102884

Order 12032.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20243150	C₂	1	2	yes	maximal
2	C₄₇×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
4	C₂×#13093935	—	2	2	no	—
5	#15688878	C₄	1	4	yes	—
6	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
8	C₄₇×#336	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	2	4	no	—
10	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
11	#13093935	—	4	4	no	—
12	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	—	2	2	no	—
16	C₂×C₄²×C₄₇	—	4	2	no	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₇	D₄	1	2	yes	—
19	C₂²⋊C₈×C₄₇	—	4	2	no	—
20	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	—	2	2	no	—
22	C₂²×C₄×C₄₇	—	2	2	no	—
23	C₂²×C₄×C₄₇	—	2	4	no	—
24	C₂²×C₄×C₄₇	—	4	1	no	—
25	C₂²×C₄×C₄₇	—	4	1	no	—
26	C₂²×C₄×C₄₇	—	4	2	no	—
27	C₂²×C₄×C₄₇	—	4	1	no	—
28	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₄₇	—	4	4	no	—
33	C₂×C₈×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	—	4	2	no	—
36	C₂×C₄₇⋊C₈	—	2	8	no	—
37	C₂×C₄₇⋊C₈	—	4	4	no	—
38	C₂×C₄₇⋊C₈	—	4	4	no	—
39	C₂³×C₄₇	—	2	1	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	—	2	1	no	—
42	C₂³×C₄₇	—	2	2	no	—
43	C₂³×C₄₇	#95	1	1	yes	characteristic
44	C₂³×C₄₇	#108	1	2	yes	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	4	4	no	—
54	C₂×C₄×C₄₇	—	4	4	no	—
55	C₂×C₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	#99	1	2	yes	—
57	C₂×C₄×C₄₇	#104	1	4	yes	—
58	C₈×C₄₇	—	8	2	no	—
59	C₄₇⋊C₈	—	4	8	no	—
60	#7046	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	2	4	no	—
63	C₂²×C₄₇	—	4	2	no	—
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	—	4	1	no	—
66	C₂²×C₄₇	#338	1	1	yes	characteristic
67	C₂²×C₄₇	#339	1	2	yes	—
68	C₂²×C₄₇	#341	1	1	yes	characteristic
69	C₂²×C₄₇	#342	1	1	yes	characteristic
70	C₂²×C₄₇	#344	1	2	yes	—
71	C₄×C₄₇	—	2	4	no	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	4	2	no	—
74	C₄×C₄₇	—	4	1	no	—
75	#1817	D₄₇	1	1	yes	characteristic
76	#1749	—	47	2	no	—
77	C₂×C₄₇	—	4	2	no	—
78	C₂×C₄₇	#1291	1	2	yes	—
79	C₂×C₄₇	#1293	1	1	yes	characteristic
80	C₂×C₄₇	#1297	1	1	yes	characteristic
81	C₂×C₄₇	#1302	1	2	yes	—
82	C₂×C₄₇	#1306	1	1	yes	characteristic
83	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
84	C₂×C₂²⋊C₈	—	94	2	no	—
85	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	—
86	C₂×#97	—	94	2	no	—
87	#336	—	2	2	no	—
88	#336	—	47	4	no	—
89	C₄₇	#7046	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
91	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
92	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	94	4	no	—
95	C₂×C₄²	—	4	2	no	—
96	C₂×C₄²	C₄×D₄₇	1	2	yes	—
97	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
98	C₂²⋊C₈	—	4	2	no	—
99	C₂²⋊C₈	—	94	8	no	—
100	#97	—	188	4	no	—
101	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	4	1	no	—
106	C₂²×C₄	—	4	1	no	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
110	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
111	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
112	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	94	8	no	—
116	C₂×C₈	—	188	4	no	—
117	C₂×C₈	—	188	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	—	4	2	no	—
120	C₂³	—	2	1	no	—
121	C₂³	—	2	2	no	—
122	C₂³	—	2	1	no	—
123	C₂³	—	2	2	no	—
124	C₂³	C₄₇⋊#95	1	1	yes	characteristic
125	C₂³	#13093960	1	2	yes	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	#13093939	1	2	yes	—
138	C₂×C₄	#13093950	1	4	yes	—
139	C₈	—	8	2	no	—
140	C₈	—	188	8	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	C₄₇⋊#338	1	1	yes	characteristic
147	C₂²	#15688884	1	2	yes	—
148	C₂²	#15688888	1	1	yes	characteristic
149	C₂²	#15688890	1	1	yes	characteristic
150	C₂²	#15688893	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	1	no	—
155	C₂	—	4	2	no	—
156	C₂	#20241261	1	2	yes	—
157	C₂	#20241265	1	1	yes	characteristic
158	C₂	#20241273	1	1	yes	characteristic
159	C₂	#20241288	1	2	yes	—
160	C₂	#20241296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243150

C₂

yes

maximal

C₄₇×#1817

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

—

C₂×#13093935

—

#15688878

C₄

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₄₇

C₄

yes

—

C₄₇×#336

—

C₂²×C₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

characteristic

C₂³×C₄×C₄₇

Q₈

yes

characteristic

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#95

yes

characteristic

C₂³×C₄₇

#108

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#99

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7046

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#338

yes

characteristic

C₂²×C₄₇

#339

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1817

D₄₇

yes

characteristic

#1749

—

C₂×C₄₇

—

C₂×C₄₇

#1291

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1306

yes

characteristic

C₂²×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

Dic₄₇

yes

—

C₂×#97

—

#336

—

#336

—

C₄₇

#7046

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₉₄

yes

characteristic

C₂³×C₄

D₁₈₈

yes

characteristic

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂²⋊D₄₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#97

—

188

—

101

C₂⁴

C₄⋊Dic₄₇

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

110

C₂²×C₄

Dic₄₇⋊C₄

yes

—

111

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

112

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

188

—

117

C₂×C₈

—

188

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

C₄₇⋊#95

yes

characteristic

125

C₂³

#13093960

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13093939

yes

—

138

C₂×C₄

#13093950

yes

—

139

C₈

—

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₄₇⋊#338

yes

characteristic

147

C₂²

#15688884

yes

—

148

C₂²

#15688888

yes

characteristic

149

C₂²

#15688890

yes

characteristic

150

C₂²

#15688893

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₂

—

156

C₂

#20241261

yes

—

157

C₂

#20241265

yes

characteristic

158

C₂

#20241273

yes

characteristic

159

C₂

#20241288

yes

—

160

C₂

#20241296

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₇⋊#95
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₂²×C₄²	C₂×D₄₇
Fitting	C₄₇×#1817	C₂
socle	C₂³×C₄₇	#95
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₇	C₂²
hypercenter	#1817	D₄₇
hypocenter	C₄₇	#7046
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

subgroup

G/H

center

C₂³

C₄₇⋊#95

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₂²×C₄²

C₂×D₄₇

Fitting

C₄₇×#1817

C₂

socle

C₂³×C₄₇

#95

prime radical

C₂⁴×C₄₇

C₄⋊C₄

cosocle

C₂²×C₄²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

N	H ≅ G/N	type
89. C₄₇	#7046	semidirect

H ≅ G/N

type

89. C₄₇

#7046

semidirect

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

subgroup	index
#20243150	2
C₄₇×#1817	2
#7046	47

subgroup

index

#20243150

C₄₇×#1817

#7046

Subgroups — #26102884

Conjugacy classes of subgroups

Complements

Maximal subgroups