Group #26103197 — subgroups

Subgroups — #26103197

Order 12032.

246 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 127 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×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#20242136	C₂	1	1	yes	maximal, characteristic
3	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₁₆	—	2	1	no	—
5	#15689176	—	2	1	no	—
6	C₄×C₄₇⋊C₁₆	—	2	1	no	—
7	C₄×C₄₇⋊C₁₆	C₄	1	2	yes	—
8	#15688876	C₄	1	2	yes	—
9	#15688874	—	2	1	no	—
10	C₂×#108×C₄₇	C₄	1	2	yes	—
11	C₄×M₁₆×C₄₇	—	2	1	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#341	—	2	1	no	—
14	C₂×C₄₇⋊C₁₆	—	2	4	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₄₇⋊C₈)	—	4	1	no	—
18	C₄×C₄₇⋊C₈	—	4	1	no	—
19	C₂×M₁₆×C₄₇	—	2	2	no	—
20	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
23	#108×C₄₇	—	2	2	no	—
24	#108×C₄₇	C₈	1	4	yes	complemented
25	#97×C₄₇	—	4	1	no	—
26	C₄×C₈×C₄₇	—	4	1	no	—
27	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
34	M₁₆×C₄₇	—	2	4	no	—
35	M₁₆×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
44	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
46	C₄²×C₄₇	D₈	1	1	yes	characteristic
47	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
48	C₂×C₄₇⋊C₈	—	4	2	no	—
49	C₄₇⋊C₁₆	—	2	4	no	—
50	C₂³×C₄₇	#103	1	1	yes	characteristic
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#104	1	2	yes	—
59	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	—	2	1	no	—
63	C₈×C₄₇	—	4	4	no	—
64	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
65	C₈×C₄₇	#97	1	1	yes	characteristic
66	C₈×C₄₇	C₄⋊C₈	1	2	yes	—
67	C₄₇⋊C₈	—	8	2	no	—
68	#7172	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	—	2	1	no	—
70	C₂²×C₄₇	—	2	1	no	—
71	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
72	C₂²×C₄₇	#340	1	1	yes	characteristic
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	D₄⋊C₈	1	2	yes	—
76	C₄×C₄₇	#336	1	1	yes	characteristic
77	C₄×C₄₇	#339	1	1	yes	characteristic
78	C₄×C₄₇	#359	1	2	yes	—
79	C₄×C₄₇	#368	1	2	yes	—
80	C₂×C₄×C₁₆	—	47	1	no	—
81	C₄×#108	D₄₇	1	1	yes	characteristic
82	#1564	—	47	1	no	—
83	C₂×C₄₇	—	2	2	no	—
84	C₂×C₄₇	#1279	1	1	yes	characteristic
85	C₂×C₄₇	#1381	1	1	yes	characteristic
86	C₂×C₄₇	#1382	1	1	yes	characteristic
87	C₂²×C₁₆	—	94	1	no	—
88	#431	—	94	1	no	—
89	C₂×#108	Dic₄₇	1	2	yes	—
90	C₄×M₁₆	—	2	1	no	—
91	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
92	C₄×C₁₆	—	47	2	no	—
93	C₄×C₁₆	—	94	1	no	—
94	#341	—	2	1	no	—
95	#335	—	47	2	no	—
96	#334	—	94	1	no	—
97	C₄₇	#7172	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	2	no	—
99	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
100	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
101	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
102	C₂×C₁₆	—	94	4	no	—
103	C₂×C₁₆	—	94	2	no	—
104	#108	—	2	2	no	—
105	#108	C₄₇⋊C₈	1	4	yes	complemented
106	C₄⋊C₈	—	94	2	no	—
107	C₂²⋊C₈	—	188	1	no	—
108	#97	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₄×C₈	—	188	1	no	—
111	C₄×C₈	Dic₉₄	1	1	yes	characteristic
112	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
113	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
114	C₄×C₈	D₁₈₈	1	1	yes	characteristic
115	C₂²×C₄	#10625336	1	1	yes	characteristic
116	C₂²×C₄	#10625337	1	1	yes	characteristic
117	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
118	M₁₆	—	2	4	no	—
119	M₁₆	—	4	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	—	4	2	no	—
122	C₂×C₈	—	188	2	no	—
123	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
124	C₂×C₈	#10625329	1	2	yes	—
125	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
126	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
127	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
128	C₂×C₈	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₄²	Dic₄₇⋊C₄	1	1	yes	characteristic
130	C₄²	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
131	C₄²	#10625334	1	1	yes	characteristic
132	C₄²	#10625335	1	1	yes	characteristic
133	C₁₆	—	94	4	no	—
134	C₂³	#13093948	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
139	C₂×C₄	#13093944	1	1	yes	characteristic
140	C₂×C₄	#13093945	1	1	yes	characteristic
141	C₂×C₄	#13093947	1	1	yes	characteristic
142	C₂×C₄	#13093951	1	2	yes	—
143	C₂×C₄	#13093955	1	1	yes	characteristic
144	C₂×C₄	#13093957	1	1	yes	characteristic
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	4	4	no	—
148	C₈	—	376	2	no	—
149	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
150	C₈	#13093935	1	1	yes	characteristic
151	C₈	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
152	C₂²	—	2	1	no	—
153	C₂²	—	2	1	no	—
154	C₂²	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
155	C₂²	#15688885	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	D₄⋊(C₄₇⋊C₈)	1	2	yes	—
159	C₄	#15688878	1	1	yes	characteristic
160	C₄	#15688882	1	1	yes	characteristic
161	C₄	#15688929	1	2	yes	—
162	C₄	#15688952	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	#20241229	1	1	yes	characteristic
165	C₂	#20241481	1	1	yes	characteristic
166	C₂	#20241485	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

#20242136

C₂

yes

maximal, characteristic

C₄×#108×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

—

#15689176

—

C₄×C₄₇⋊C₁₆

—

C₄×C₄₇⋊C₁₆

C₄

yes

—

#15688876

C₄

yes

—

#15688874

—

C₂×#108×C₄₇

C₄

yes

—

C₄×M₁₆×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

#108×C₄₇

—

#108×C₄₇

C₈

yes

complemented

#97×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

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

M₁₆×C₄₇

—

M₁₆×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

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#103

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₄×C₈

yes

characteristic

C₈×C₄₇

#97

yes

characteristic

C₈×C₄₇

C₄⋊C₈

yes

—

C₄₇⋊C₈

—

#7172

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

Q₈⋊C₈

yes

—

C₂²×C₄₇

#340

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

D₄⋊C₈

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#368

yes

—

C₂×C₄×C₁₆

—

C₄×#108

D₄₇

yes

characteristic

#1564

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₁₆

—

#431

—

C₂×#108

Dic₄₇

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

#341

—

#335

—

#334

—

C₄₇

#7172

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

100

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

101

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

#108

—

105

#108

C₄₇⋊C₈

yes

complemented

106

C₄⋊C₈

—

107

C₂²⋊C₈

—

188

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

188

—

111

C₄×C₈

Dic₉₄

yes

characteristic

112

C₄×C₈

C₄×D₄₇

yes

characteristic

113

C₄×C₈

C₄×D₄₇

yes

characteristic

114

C₄×C₈

D₁₈₈

yes

characteristic

115

C₂²×C₄

#10625336

yes

characteristic

116

C₂²×C₄

#10625337

yes

characteristic

117

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

118

M₁₆

—

119

M₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

188

—

123

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

124

C₂×C₈

#10625329

yes

—

125

C₂×C₈

C₄×Dic₄₇

yes

characteristic

126

C₂×C₈

Dic₄₇⋊C₄

yes

characteristic

127

C₂×C₈

C₄⋊Dic₄₇

yes

characteristic

128

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

characteristic

129

C₄²

Dic₄₇⋊C₄

yes

characteristic

130

C₄²

(C₂×D₄₇)⋊C₄

yes

characteristic

131

C₄²

#10625334

yes

characteristic

132

C₄²

#10625335

yes

characteristic

133

C₁₆

—

134

C₂³

#13093948

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₄₇⋊#95

yes

characteristic

138

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

139

C₂×C₄

#13093944

yes

characteristic

140

C₂×C₄

#13093945

yes

characteristic

141

C₂×C₄

#13093947

yes

characteristic

142

C₂×C₄

#13093951

yes

—

143

C₂×C₄

#13093955

yes

characteristic

144

C₂×C₄

#13093957

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

376

—

149

C₈

C₄×C₄₇⋊C₈

yes

characteristic

150

C₈

#13093935

yes

characteristic

151

C₈

C₄⋊(C₄₇⋊C₈)

yes

—

152

C₂²

—

153

C₂²

—

154

C₂²

Q₈⋊(C₄₇⋊C₈)

yes

—

155

C₂²

#15688885

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

D₄⋊(C₄₇⋊C₈)

yes

—

159

C₄

#15688878

yes

characteristic

160

C₄

#15688882

yes

characteristic

161

C₄

#15688929

yes

—

162

C₄

#15688952

yes

—

163

C₂

—

164

C₂

#20241229

yes

characteristic

165

C₂

#20241481

yes

characteristic

166

C₂

#20241485

yes

characteristic

	subgroup	G/H
center	C₄²	#10625334
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	#340
prime radical	C₂³×C₄₇	#103
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	#7172
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₄²

#10625334

commutator

C₈×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄×#108×C₄₇

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
24. #108×C₄₇	C₈	semidirect
97. C₄₇	#7172	semidirect
105. #108	C₄₇⋊C₈	semidirect

H ≅ G/N

type

24. #108×C₄₇

C₈

semidirect

97. C₄₇

#7172

semidirect

105. #108

C₄₇⋊C₈

semidirect

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
#20242136	2
C₄×#108×C₄₇	2
#7172	47

subgroup

index

C₂×C₄×C₄₇⋊C₁₆

#20242136

C₄×#108×C₄₇

#7172

Subgroups — #26103197

Conjugacy classes of subgroups

Complements

Maximal subgroups