Group #26103090 — subgroups

Subgroups — #26103090

Order 12032.

236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20241466	C₂	1	2	yes	maximal
2	C₄₇×#1984	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	—	2	2	no	—
4	C₂×C₄×Q₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₈×C₄₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#336	—	2	2	no	—
8	C₂×C₄₇⋊C₁₆	—	2	4	no	—
9	C₂×C₄₇⋊C₁₆	—	4	2	no	—
10	C₂×C₄₇⋊C₁₆	—	4	2	no	—
11	C₂²×Q₈×C₄₇	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₄₇	—	2	1	no	—
13	C₂²×C₈×C₄₇	D₄	1	2	yes	—
14	C₄×Q₈×C₄₇	—	4	1	no	—
15	C₄×Q₈×C₄₇	—	4	1	no	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	—	2	1	no	—
19	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	—	4	1	no	—
21	C₄×C₈×C₄₇	—	4	1	no	—
22	C₂×Q₈×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	4	1	no	—
24	C₂×Q₈×C₄₇	—	4	1	no	—
25	C₂×Q₈×C₄₇	C₁₆	1	2	yes	complemented
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	M₁₆	1	1	yes	characteristic
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	4	1	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
37	C₄⋊C₄×C₄₇	—	4	2	no	—
38	C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₄²×C₄₇	—	2	1	no	—
40	C₄²×C₄₇	—	2	1	no	—
41	C₄²×C₄₇	—	4	1	no	—
42	C₄²×C₄₇	—	4	1	no	—
43	C₄₇⋊C₁₆	—	4	4	no	—
44	C₂³×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
45	Q₈×C₄₇	—	4	2	no	—
46	Q₈×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	2	1	no	—
49	C₂×C₄×C₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	4	1	no	—
53	C₂×C₄×C₄₇	—	4	1	no	—
54	C₂×C₄×C₄₇	—	4	1	no	—
55	C₂×C₄×C₄₇	—	4	2	no	—
56	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#103	1	2	yes	—
59	C₂×C₄×C₄₇	#104	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
61	C₂×C₄×C₄₇	M₃₂	1	1	yes	characteristic
62	C₈×C₄₇	—	2	2	no	—
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	—	8	1	no	—
65	#7128	—	47	1	no	maximal, Sylow
66	C₂²×C₄₇	#323	1	1	yes	characteristic
67	C₂²×C₄₇	Q₈⋊C₈	1	2	yes	—
68	C₂²×C₄₇	C₂²⋊C₁₆	1	1	yes	characteristic
69	C₂²×C₄₇	#349	1	1	yes	characteristic
70	C₂²×C₄₇	#350	1	2	yes	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	4	2	no	—
74	C₄×C₄₇	—	4	1	no	—
75	C₄×C₄₇	—	8	1	no	—
76	C₄×C₄₇	#328	1	2	yes	—
77	C₄×C₄₇	#329	1	1	yes	characteristic
78	C₄×C₄₇	#333	1	1	yes	characteristic
79	#1984	D₄₇	1	1	yes	characteristic
80	#1374	—	47	2	no	—
81	C₂×C₄₇	#1272	1	1	yes	characteristic
82	C₂×C₄₇	#1314	1	1	yes	characteristic
83	C₂×C₄₇	#1315	1	1	yes	characteristic
84	C₂×C₄₇	Q₈⋊C₁₆	1	2	yes	—
85	C₂×C₄₇	#1338	1	2	yes	—
86	C₂×C₄×Q₈	Dic₄₇	1	1	yes	characteristic
87	C₂²×C₁₆	—	94	2	no	—
88	C₂×C₄⋊C₈	Dic₄₇	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
90	#336	—	2	2	no	—
91	C₄₇	#7128	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	C₄₇⋊C₈	1	1	yes	characteristic
93	C₂²×C₈	—	2	1	no	—
94	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
95	C₄×Q₈	—	4	1	no	—
96	C₄×Q₈	—	4	1	no	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₄⋊C₄	C₄₇⋊C₈	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
101	C₂×C₁₆	—	94	4	no	—
102	C₂×C₁₆	—	188	2	no	—
103	C₂×C₁₆	—	188	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄×C₈	—	4	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	C₄₇⋊C₁₆	1	2	yes	complemented
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
113	C₂²×C₄	#10625329	1	1	yes	characteristic
114	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	#10625336	1	2	yes	—
120	C₂×C₈	#10625337	1	2	yes	—
121	C₄⋊C₄	—	4	1	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₁₆	—	188	4	no	—
128	C₂³	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
129	Q₈	—	4	2	no	—
130	Q₈	—	4	2	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	#13093939	1	1	yes	characteristic
141	C₂×C₄	#13093941	1	1	yes	characteristic
142	C₂×C₄	#13093948	1	2	yes	—
143	C₂×C₄	#13093951	1	2	yes	—
144	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
145	C₂×C₄	#13093964	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	2	no	—
148	C₈	—	8	1	no	—
149	C₂²	#15688847	1	1	yes	characteristic
150	C₂²	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
151	C₂²	C₂²⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
152	C₂²	#15688905	1	1	yes	characteristic
153	C₂²	#15688907	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	4	1	no	—
158	C₄	—	8	1	no	—
159	C₄	#15688860	1	2	yes	—
160	C₄	#15688863	1	1	yes	characteristic
161	C₄	#15688872	1	1	yes	characteristic
162	C₂	#20241214	1	1	yes	characteristic
163	C₂	#20241312	1	1	yes	characteristic
164	C₂	#20241314	1	1	yes	characteristic
165	C₂	Q₈⋊(C₄₇⋊C₁₆)	1	2	yes	—
166	C₂	#20241370	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20241466

C₂

yes

maximal

C₄₇×#1984

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

—

C₂×C₄×Q₈×C₄₇

C₄

yes

characteristic

C₂×C₄⋊C₈×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#336

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂²×Q₈×C₄₇

C₈

yes

characteristic

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₈

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₁₆

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²⋊C₈

yes

characteristic

Q₈×C₄₇

—

Q₈×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

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₂×C₄×C₄₇

C₂×C₁₆

yes

characteristic

C₂×C₄×C₄₇

M₃₂

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

Q₈⋊C₈

yes

—

C₂²×C₄₇

C₂²⋊C₁₆

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#328

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1374

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

Q₈⋊C₁₆

yes

—

C₂×C₄₇

#1338

yes

—

C₂×C₄×Q₈

Dic₄₇

yes

characteristic

C₂²×C₁₆

—

C₂×C₄⋊C₈

Dic₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#336

—

C₄₇

#7128

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₄₇⋊C₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄₇⋊C₈

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

188

—

103

C₂×C₁₆

—

188

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

C₄₇⋊C₁₆

yes

complemented

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂×C₄₇⋊C₈

yes

characteristic

113

C₂²×C₄

#10625329

yes

characteristic

114

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#10625336

yes

—

120

C₂×C₈

#10625337

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₁₆

—

188

—

128

C₂³

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

129

Q₈

—

130

Q₈

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#13093939

yes

characteristic

141

C₂×C₄

#13093941

yes

characteristic

142

C₂×C₄

#13093948

yes

—

143

C₂×C₄

#13093951

yes

—

144

C₂×C₄

C₂×C₄₇⋊C₁₆

yes

characteristic

145

C₂×C₄

#13093964

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₂²

#15688847

yes

characteristic

150

C₂²

Q₈⋊(C₄₇⋊C₈)

yes

—

151

C₂²

C₂²⋊(C₄₇⋊C₁₆)

yes

characteristic

152

C₂²

#15688905

yes

characteristic

153

C₂²

#15688907

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#15688860

yes

—

160

C₄

#15688863

yes

characteristic

161

C₄

#15688872

yes

characteristic

162

C₂

#20241214

yes

characteristic

163

C₂

#20241312

yes

characteristic

164

C₂

#20241314

yes

characteristic

165

C₂

Q₈⋊(C₄₇⋊C₁₆)

yes

—

166

C₂

#20241370

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₁₆
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1984	C₂
socle, prime radical	C₂³×C₄₇	C₂²⋊C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1984	D₄₇
hypocenter	C₄₇	#7128
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1984

subgroup

G/H

center

C₂²×C₄

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₁₆

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#1984

C₂

socle, prime radical

C₂³×C₄₇

C₂²⋊C₈

cosocle

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1984

N	H ≅ G/N	type
25. C₂×Q₈×C₄₇	C₁₆	semidirect
91. C₄₇	#7128	semidirect
109. C₂×Q₈	C₄₇⋊C₁₆	semidirect

H ≅ G/N

type

25. C₂×Q₈×C₄₇

C₁₆

semidirect

91. C₄₇

#7128

semidirect

109. C₂×Q₈

C₄₇⋊C₁₆

semidirect

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

subgroup	index
#20241466	2
C₄₇×#1984	2
#7128	47

subgroup

index

#20241466

C₄₇×#1984

#7128

Subgroups — #26103090

Conjugacy classes of subgroups

Complements

Maximal subgroups