Group #26130466 — subgroups

Subgroups — #26130466

Order 12032.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20245439	C₂	1	2	yes	maximal
2	#20243006	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
4	#20241438	C₂	1	2	yes	maximal
5	C₄₇×#1370	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	2	no	—
7	#15689528	—	2	1	no	—
8	#15689528	—	2	1	no	—
9	#15689503	C₂²	1	2	yes	—
10	C₈×Dic₉₄	—	2	1	no	—
11	C₄×Dic₁₈₈	C₂²	1	1	yes	characteristic
12	#15688928	—	2	2	no	—
13	#15688876	C₂²	1	1	yes	characteristic
14	#15688855	—	2	1	no	—
15	C₄₇×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	2	no	—
19	C₂×Dic₁₈₈	D₄	1	1	yes	characteristic
20	#13094024	—	2	2	no	—
21	C₄×Dic₉₄	—	2	1	no	—
22	Dic₃₇₆	—	4	4	no	—
23	#13093958	—	4	2	no	—
24	#13093958	—	4	1	no	—
25	#13093958	—	4	1	no	—
26	#13093958	D₄	1	1	yes	characteristic
27	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
28	Dic₄₇⋊C₈	—	4	1	no	—
29	#13093949	—	2	1	no	—
30	C₈×Dic₄₇	—	4	1	no	—
31	C₂×C₁₆×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
34	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×Dic₉₄	—	2	1	no	—
36	C₂×Dic₉₄	—	4	2	no	—
37	C₂×C₈×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₄₇	—	2	1	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	C₄⋊Dic₄₇	—	4	2	no	—
46	Dic₄₇⋊C₄	—	4	1	no	—
47	C₄×Dic₄₇	—	4	1	no	—
48	C₂×C₄₇⋊C₈	—	4	1	no	—
49	Dic₁₈₈	—	2	1	no	—
50	Dic₁₈₈	—	4	2	no	—
51	Dic₁₈₈	—	4	2	no	—
52	Dic₁₈₈	—	8	2	no	—
53	Dic₁₈₈	D₈	1	2	yes	—
54	C₁₆×C₄₇	—	4	2	no	—
55	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₂×Dic₄₇	—	4	1	no	—
59	C₂×Dic₄₇	—	4	1	no	—
60	C₂×Dic₄₇	—	8	2	no	—
61	Dic₉₄	—	2	2	no	—
62	Dic₉₄	—	4	1	no	—
63	Dic₉₄	—	8	2	no	—
64	Dic₉₄	—	8	2	no	—
65	C₈×C₄₇	—	2	2	no	—
66	C₈×C₄₇	—	2	1	no	—
67	C₈×C₄₇	—	2	1	no	—
68	C₈×C₄₇	Q₃₂	1	2	yes	complemented
69	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
70	C₈×C₄₇	#135	1	1	yes	characteristic
71	C₄₇⋊C₈	—	8	1	no	—
72	#12427	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	#459	1	1	yes	characteristic
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	#466	1	1	yes	characteristic
76	C₄×C₄₇	#472	1	1	yes	characteristic
77	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
78	C₄×C₄₇	#510	1	1	yes	characteristic
79	Dic₄₇	—	4	2	no	—
80	Dic₄₇	—	8	1	no	—
81	Dic₄₇	—	16	2	no	—
82	#2209	—	47	2	no	—
83	#1713	—	47	1	no	—
84	C₈×Q₁₆	—	47	1	no	—
85	#1370	D₄₇	1	1	yes	characteristic
86	#1362	—	47	2	no	—
87	C₂×C₄₇	#1667	1	1	yes	characteristic
88	C₂×C₄₇	#2209	1	1	yes	characteristic
89	C₂×C₄₇	#2214	1	1	yes	characteristic
90	C₂×Q₃₂	—	94	2	no	—
91	#500	—	94	1	no	—
92	#500	—	94	1	no	—
93	#494	—	47	2	no	—
94	C₈×Q₈	—	94	1	no	—
95	C₄×Q₁₆	—	47	1	no	—
96	C₄⋊C₁₆	C₂×D₄₇	1	2	yes	—
97	#358	—	94	2	no	—
98	#335	—	47	1	no	—
99	Q₈⋊C₈	—	94	1	no	—
100	C₈²	C₂×D₄₇	1	1	yes	characteristic
101	C₄₇	#12427	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	47	1	no	—
103	C₂×Q₁₆	—	94	2	no	—
104	C₂×Q₁₆	—	94	2	no	—
105	C₄⋊Q₈	—	94	2	no	—
106	C₄×Q₈	—	94	1	no	—
107	Q₃₂	—	188	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#107	—	47	1	no	—
110	#107	—	188	2	no	—
111	#107	—	188	1	no	—
112	#107	—	188	1	no	—
113	C₄⋊C₈	—	94	1	no	—
114	C₄⋊C₈	—	188	1	no	—
115	#103	—	94	1	no	—
116	C₄×C₈	—	188	1	no	—
117	C₄×C₈	D₁₈₈	1	1	yes	characteristic
118	C₄×C₈	D₁₈₈	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
120	C₂×Q₈	—	94	1	no	—
121	C₂×Q₈	—	188	2	no	—
122	Q₁₆	—	47	2	no	—
123	Q₁₆	—	94	1	no	—
124	Q₁₆	—	376	2	no	—
125	Q₁₆	—	188	2	no	—
126	Q₁₆	—	188	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	188	1	no	—
129	C₂×C₈	D₃₇₆	1	1	yes	characteristic
130	C₂×C₈	D₃₇₆	1	1	yes	characteristic
131	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
132	C₂×C₈	#10625354	1	1	yes	characteristic
133	C₄⋊C₄	—	94	1	no	—
134	C₄⋊C₄	—	188	2	no	—
135	C₄⋊C₄	—	188	2	no	—
136	C₄⋊C₄	—	188	1	no	—
137	C₄²	—	188	1	no	—
138	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	Q₈	—	94	2	no	—
141	Q₈	—	376	2	no	—
142	Q₈	—	376	2	no	—
143	Q₈	—	188	1	no	—
144	C₂×C₄	—	376	2	no	—
145	C₂×C₄	—	188	1	no	—
146	C₂×C₄	—	188	1	no	—
147	C₂×C₄	#13093999	1	1	yes	characteristic
148	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
149	C₂×C₄	#13094052	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	376	1	no	—
154	C₈	Dic₃₇₆	1	2	yes	complemented
155	C₈	D₈×D₄₇	1	1	yes	characteristic
156	C₈	#13094047	1	1	yes	characteristic
157	C₂²	#15689319	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	376	1	no	—
160	C₄	—	188	2	no	—
161	C₄	—	752	2	no	—
162	C₄	#15689360	1	1	yes	characteristic
163	C₄	#15689394	1	1	yes	characteristic
164	C₄	C₂×Dic₃₇₆	1	1	yes	characteristic
165	C₄	#15689566	1	1	yes	characteristic
166	C₂	#20242744	1	1	yes	characteristic
167	C₂	#20245439	1	1	yes	characteristic
168	C₂	#20245470	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245439

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

C₈×Dic₁₈₈

C₂

yes

maximal, characteristic

#20241438

C₂

yes

maximal

C₄₇×#1370

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

#15689528

—

#15689528

—

#15689503

C₂²

yes

—

C₈×Dic₉₄

—

C₄×Dic₁₈₈

C₂²

yes

characteristic

#15688928

—

#15688876

C₂²

yes

characteristic

#15688855

—

C₄₇×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

D₄

yes

characteristic

#13094024

—

C₄×Dic₉₄

—

Dic₃₇₆

—

—

—

—

D₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093949

—

C₈×Dic₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

D₈

yes

—

C₁₆×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

Q₃₂

yes

complemented

C₈×C₄₇

C₂×D₈

yes

characteristic

C₈×C₄₇

#135

yes

characteristic

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂×Q₃₂

yes

characteristic

C₄×C₄₇

#510

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2209

—

#1713

—

C₈×Q₁₆

—

#1370

D₄₇

yes

characteristic

#1362

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

#500

—

#500

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

C₂×D₄₇

yes

—

#358

—

#335

—

Q₈⋊C₈

—

100

C₈²

C₂×D₄₇

yes

characteristic

101

C₄₇

#12427

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₄⋊Q₈

—

106

C₄×Q₈

—

107

Q₃₂

—

188

—

108

C₂×C₁₆

—

109

#107

—

110

#107

—

188

—

111

#107

—

188

—

112

#107

—

188

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

188

—

115

#103

—

116

C₄×C₈

—

188

—

117

C₄×C₈

D₁₈₈

yes

characteristic

118

C₄×C₈

D₁₈₈

yes

characteristic

119

C₄×C₈

C₂²×D₄₇

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

188

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

376

—

125

Q₁₆

—

188

—

126

Q₁₆

—

188

—

127

C₂×C₈

—

128

C₂×C₈

—

188

—

129

C₂×C₈

D₃₇₆

yes

characteristic

130

C₂×C₈

D₃₇₆

yes

characteristic

131

C₂×C₈

D₄×D₄₇

yes

characteristic

132

C₂×C₈

#10625354

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

188

—

135

C₄⋊C₄

—

188

—

136

C₄⋊C₄

—

188

—

137

C₄²

—

188

—

138

C₄²

C₂×D₁₈₈

yes

characteristic

139

C₁₆

—

140

Q₈

—

141

Q₈

—

376

—

142

Q₈

—

376

—

143

Q₈

—

188

—

144

C₂×C₄

—

376

—

145

C₂×C₄

—

188

—

146

C₂×C₄

—

188

—

147

C₂×C₄

#13093999

yes

characteristic

148

C₂×C₄

C₂×D₃₇₆

yes

characteristic

149

C₂×C₄

#13094052

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

376

—

154

C₈

Dic₃₇₆

yes

complemented

155

C₈

D₈×D₄₇

yes

characteristic

156

C₈

#13094047

yes

characteristic

157

C₂²

#15689319

yes

characteristic

158

C₄

—

159

C₄

—

376

—

160

C₄

—

188

—

161

C₄

—

752

—

162

C₄

#15689360

yes

characteristic

163

C₄

#15689394

yes

characteristic

164

C₄

C₂×Dic₃₇₆

yes

characteristic

165

C₄

#15689566

yes

characteristic

166

C₂

#20242744

yes

characteristic

167

C₂

#20245439

yes

characteristic

168

C₂

#20245470

yes

characteristic

	subgroup	G/H
center	C₂²	#15689319
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1370	C₂
socle, prime radical	C₂²×C₄₇	#459
hypercenter	#1370	D₄₇
hypocenter	C₄₇	#12427
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#15689319

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1370

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
68. C₈×C₄₇	Q₃₂	semidirect
101. C₄₇	#12427	semidirect
154. C₈	Dic₃₇₆	semidirect

H ≅ G/N

type

68. C₈×C₄₇

Q₃₂

semidirect

101. C₄₇

#12427

semidirect

154. C₈

Dic₃₇₆

semidirect

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

subgroup	index
#20245439	2
#20243006	2
C₈×Dic₁₈₈	2
#20241438	2
C₄₇×#1370	2
#12427	47

subgroup

index

C₈×Dic₁₈₈

C₄₇×#1370

Subgroups — #26130466

Conjugacy classes of subgroups

Complements

Maximal subgroups