Group #26103157 — subgroups

Subgroups — #26103157

Order 12032.

222 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#13093964	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	C₄	1	2	yes	—
5	#15689176	—	2	1	no	—
6	#15688899	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	—	2	1	no	—
8	#15688876	C₄	1	2	yes	—
9	#15688874	—	2	1	no	—
10	C₂×C₄⋊C₈×C₄₇	—	2	1	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#334	—	2	1	no	—
13	C₄₇×#334	C₄	1	2	yes	—
14	#13093964	—	2	2	no	—
15	#13093964	C₈	1	4	yes	complemented
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
19	C₄×C₄₇⋊C₈	—	4	1	no	—
20	C₂²×C₈×C₄₇	—	4	1	no	—
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₄₇	—	2	2	no	—
25	C₄⋊C₈×C₄₇	—	4	1	no	—
26	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
33	C₂×C₈×C₄₇	—	2	2	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	—	8	2	no	—
36	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
42	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₄₇⋊C₁₆	—	2	4	no	—
48	C₂³×C₄₇	#107	1	1	yes	characteristic
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#108	1	2	yes	—
59	C₈×C₄₇	—	2	1	no	—
60	C₈×C₄₇	—	2	1	no	—
61	C₈×C₄₇	—	8	2	no	—
62	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
63	C₈×C₄₇	#97	1	1	yes	characteristic
64	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
65	C₄₇⋊C₈	—	8	2	no	—
66	#7158	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	#335	1	2	yes	—
70	C₂²×C₄₇	#340	1	1	yes	characteristic
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	#334	1	2	yes	—
75	C₄×C₄₇	#336	1	1	yes	characteristic
76	C₄×C₄₇	#341	1	1	yes	characteristic
77	C₄×C₄₇	#360	1	2	yes	—
78	C₄×M₃₂	—	47	1	no	—
79	#1564	—	47	1	no	—
80	C₂×#334	D₄₇	1	1	yes	characteristic
81	C₂×C₄₇	—	2	2	no	—
82	C₂×C₄₇	#1277	1	1	yes	characteristic
83	C₂×C₄₇	#1386	1	1	yes	characteristic
84	C₂×C₄₇	#1388	1	1	yes	characteristic
85	C₂×M₃₂	—	47	2	no	—
86	#431	—	94	1	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	#346	—	94	1	no	—
90	C₄×C₁₆	—	94	1	no	—
91	#335	—	47	2	no	—
92	#334	—	2	1	no	—
93	#334	—	94	1	no	—
94	#334	Dic₄₇	1	2	yes	—
95	C₄₇	#7158	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	4	1	no	—
97	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
98	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
100	M₃₂	—	47	4	no	—
101	M₃₂	—	94	2	no	—
102	C₂×C₁₆	—	94	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	94	2	no	—
106	C₂²⋊C₈	—	188	1	no	—
107	C₄×C₈	—	188	1	no	—
108	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
109	C₄×C₈	D₁₈₈	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
112	C₂²×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	#10625334	1	1	yes	characteristic
114	C₂²×C₄	#10625337	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	—	188	2	no	—
119	C₂×C₈	C₈×D₄₇	1	2	yes	—
120	C₂×C₈	#10625324	1	2	yes	—
121	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
122	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
124	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
125	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
126	C₄²	#10625335	1	1	yes	characteristic
127	C₄²	#10625336	1	1	yes	characteristic
128	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
129	C₁₆	—	94	4	no	—
130	C₂³	#13093957	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
134	C₂×C₄	#13093944	1	1	yes	characteristic
135	C₂×C₄	#13093945	1	1	yes	characteristic
136	C₂×C₄	#13093947	1	1	yes	characteristic
137	C₂×C₄	#13093948	1	1	yes	characteristic
138	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
139	C₂×C₄	#13093955	1	1	yes	characteristic
140	C₂×C₄	#13093959	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	2	no	—
144	C₈	—	376	2	no	—
145	C₈	C₈×Dic₄₇	1	1	yes	characteristic
146	C₈	#13093934	1	1	yes	characteristic
147	C₈	(C₂×D₄₇)⋊C₈	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#15688875	1	2	yes	—
151	C₂²	#15688885	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#15688873	1	2	yes	—
156	C₄	#15688877	1	1	yes	characteristic
157	C₄	#15688887	1	1	yes	characteristic
158	C₄	#15688932	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#20241222	1	1	yes	characteristic
161	C₂	#20241495	1	1	yes	characteristic
162	C₂	#20241501	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093964

C₂

yes

maximal, characteristic

#20242136

C₂

yes

maximal, characteristic

C₂×C₄₇×#334

C₂

yes

maximal, characteristic

C₂×#13093964

C₄

yes

—

#15689176

—

#15688899

—

C₄×C₄₇⋊C₁₆

—

#15688876

C₄

yes

—

#15688874

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#334

—

C₄₇×#334

C₄

yes

—

#13093964

—

#13093964

C₈

yes

complemented

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#107

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

#108

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₄×C₈

yes

characteristic

C₈×C₄₇

#97

yes

characteristic

C₈×C₄₇

C₂²⋊C₈

yes

—

C₄₇⋊C₈

—

#7158

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#335

yes

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#334

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×M₃₂

—

#1564

—

C₂×#334

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×M₃₂

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#346

—

C₄×C₁₆

—

—

—

—

Dic₄₇

yes

—

C₄₇

#7158

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

188

—

107

C₄×C₈

—

188

—

108

C₄×C₈

C₄×D₄₇

yes

characteristic

109

C₄×C₈

D₁₈₈

yes

characteristic

110

C₄×C₈

C₂×Dic₄₇

yes

characteristic

111

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

112

C₂²×C₄

Dic₄₇⋊C₄

yes

characteristic

113

C₂²×C₄

#10625334

yes

characteristic

114

C₂²×C₄

#10625337

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

188

—

119

C₂×C₈

C₈×D₄₇

yes

—

120

C₂×C₈

#10625324

yes

—

121

C₂×C₈

C₄×Dic₄₇

yes

characteristic

122

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

123

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

124

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

characteristic

125

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

126

C₄²

#10625335

yes

characteristic

127

C₄²

#10625336

yes

characteristic

128

C₄²

C₂²⋊Dic₄₇

yes

characteristic

129

C₁₆

—

130

C₂³

#13093957

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄₇⋊#95

yes

characteristic

134

C₂×C₄

#13093944

yes

characteristic

135

C₂×C₄

#13093945

yes

characteristic

136

C₂×C₄

#13093947

yes

characteristic

137

C₂×C₄

#13093948

yes

characteristic

138

C₂×C₄

Dic₄₇⋊C₈

yes

—

139

C₂×C₄

#13093955

yes

characteristic

140

C₂×C₄

#13093959

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

376

—

145

C₈

C₈×Dic₄₇

yes

characteristic

146

C₈

#13093934

yes

characteristic

147

C₈

(C₂×D₄₇)⋊C₈

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

#15688875

yes

—

151

C₂²

#15688885

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#15688873

yes

—

156

C₄

#15688877

yes

characteristic

157

C₄

#15688887

yes

characteristic

158

C₄

#15688932

yes

—

159

C₂

—

160

C₂

#20241222

yes

characteristic

161

C₂

#20241495

yes

characteristic

162

C₂

#20241501

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093944
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#107
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#7158
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#13093944

commutator

C₈×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄₇×#334

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

N	H ≅ G/N	type
15. #13093964	C₈	semidirect
95. C₄₇	#7158	semidirect

H ≅ G/N

type

15. #13093964

C₈

semidirect

95. C₄₇

#7158

semidirect

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

subgroup	index
C₄×#13093964	2
#20242136	2
C₂×C₄₇×#334	2
#7158	47

subgroup

index

C₄×#13093964

#20242136

C₂×C₄₇×#334

#7158

Subgroups — #26103157

Conjugacy classes of subgroups

Complements

Maximal subgroups