Group #26137294 — subgroups

Subgroups — #26137294

Order 12032.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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×Dic₇₅₂	C₂	1	2	yes	maximal, complemented
2	C₂×#15688962	C₂	1	2	yes	maximal
3	#20245649	C₂	1	1	yes	maximal, characteristic
4	#20245641	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₇₆	—	2	1	no	—
7	C₂×Dic₃₇₆	C₂²	1	2	yes	—
8	C₂×Dic₃₇₆	C₂²	1	1	yes	characteristic
9	C₂×#13093969	—	2	1	no	—
10	C₂×D₇₅₂	C₂²	1	1	yes	characteristic
11	#15689507	—	2	1	no	—
12	#15689503	—	2	1	no	—
13	Dic₇₅₂	—	2	4	no	—
14	#15688962	—	2	4	no	—
15	C₂×C₃₂×C₄₇	C₂²	1	2	yes	—
16	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	1	no	—
19	C₂×Dic₁₈₈	—	4	1	no	—
20	C₂×#10625325	—	4	1	no	—
21	C₂×D₃₇₆	—	2	1	no	—
22	#13094024	—	4	1	no	—
23	#13094012	—	4	1	no	—
24	Dic₃₇₆	—	2	2	no	—
25	Dic₃₇₆	—	2	2	no	—
26	Dic₃₇₆	—	2	2	no	—
27	Dic₃₇₆	—	4	2	no	—
28	#13093969	—	4	2	no	—
29	D₇₅₂	—	2	2	no	—
30	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
33	C₃₂×C₄₇	D₄	1	4	yes	—
34	C₂×D₁₈₈	—	4	1	no	—
35	C₂×Dic₉₄	—	4	1	no	—
36	C₂×Dic₉₄	—	4	2	no	—
37	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₄₇	D₈	1	1	yes	characteristic
40	(C₂×D₄₇)⋊C₄	—	8	2	no	—
41	C₄⋊Dic₄₇	—	8	2	no	—
42	Dic₁₈₈	—	4	2	no	—
43	Dic₁₈₈	—	4	1	no	—
44	Dic₁₈₈	—	4	1	no	—
45	Dic₁₈₈	—	4	2	no	—
46	Dic₁₈₈	—	8	2	no	—
47	D₃₇₆	—	4	1	no	—
48	D₃₇₆	—	4	1	no	—
49	#10625325	—	8	2	no	—
50	C₁₆×C₄₇	D₈	1	2	yes	—
51	C₁₆×C₄₇	C₂×D₄	1	2	yes	—
52	C₂²×D₄₇	—	8	1	no	—
53	C₂×C₄×C₄₇	D₁₆	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₄₇	—	8	2	no	—
56	C₂×Dic₄₇	—	8	1	no	—
57	D₁₈₈	—	8	1	no	—
58	D₁₈₈	—	8	1	no	—
59	Dic₉₄	—	8	1	no	—
60	Dic₉₄	—	8	2	no	—
61	Dic₉₄	—	8	1	no	—
62	Dic₉₄	—	8	2	no	—
63	C₈×C₄₇	D₁₆	1	2	yes	—
64	C₈×C₄₇	#127	1	1	yes	characteristic
65	C₈×C₄₇	C₂×D₈	1	1	yes	characteristic
66	#13730	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
68	C₂×D₄₇	—	16	1	no	—
69	C₂×D₄₇	—	16	2	no	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	#493	1	1	yes	characteristic
72	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
73	Dic₄₇	—	16	1	no	—
74	Dic₄₇	—	16	2	no	—
75	C₂×Q₆₄	—	47	2	no	—
76	C₂×SD₆₄	—	47	2	no	—
77	#2249	—	47	1	no	—
78	#2247	—	47	1	no	—
79	#1397	D₄₇	1	1	yes	characteristic, complemented
80	C₂×C₄₇	#2246	1	1	yes	characteristic
81	C₂×C₄₇	#2264	1	2	yes	—
82	D₄₇	—	32	1	no	—
83	C₂×Q₃₂	—	47	2	no	—
84	C₂×Q₃₂	—	47	1	no	—
85	C₂×Q₃₂	—	94	1	no	—
86	C₂×SD₃₂	—	94	1	no	—
87	C₂×D₁₆	—	47	1	no	—
88	#495	—	94	1	no	—
89	#494	—	94	1	no	—
90	Q₆₄	—	94	4	no	—
91	SD₆₄	—	94	4	no	—
92	C₂×C₃₂	C₂×D₄₇	1	2	yes	—
93	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
94	C₄₇	#13730	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₁₆	—	94	2	no	—
96	C₂×Q₁₆	—	94	1	no	—
97	C₂×Q₁₆	—	188	1	no	—
98	C₂×SD₁₆	—	188	1	no	—
99	C₂×D₈	—	94	1	no	—
100	C₄⋊Q₈	—	188	1	no	—
101	#124	—	188	1	no	—
102	Q₃₂	—	94	2	no	—
103	Q₃₂	—	94	2	no	—
104	Q₃₂	—	94	2	no	—
105	Q₃₂	—	188	2	no	—
106	SD₃₂	—	188	2	no	—
107	D₁₆	—	94	2	no	—
108	C₂×C₁₆	D₁₈₈	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
110	C₄×C₈	D₁₈₈	1	1	yes	characteristic
111	C₃₂	D₁₈₈	1	4	yes	—
112	C₂×Q₈	—	188	1	no	—
113	C₂×Q₈	—	188	2	no	—
114	C₂×D₄	—	188	1	no	—
115	Q₁₆	—	376	2	no	—
116	Q₁₆	—	188	2	no	—
117	Q₁₆	—	188	1	no	—
118	Q₁₆	—	188	1	no	—
119	Q₁₆	—	188	2	no	—
120	SD₁₆	—	376	2	no	—
121	D₈	—	188	1	no	—
122	D₈	—	188	1	no	—
123	C₂×C₈	D₃₇₆	1	1	yes	characteristic
124	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
125	C₄⋊C₄	—	376	2	no	—
126	C₂²⋊C₄	—	376	2	no	—
127	C₄²	D₃₇₆	1	1	yes	characteristic
128	C₁₆	D₃₇₆	1	2	yes	—
129	C₁₆	C₂×D₁₈₈	1	2	yes	—
130	C₂³	—	376	1	no	—
131	Q₈	—	376	1	no	—
132	Q₈	—	376	2	no	—
133	Q₈	—	376	1	no	—
134	Q₈	—	376	2	no	—
135	D₄	—	376	1	no	—
136	D₄	—	376	1	no	—
137	C₂×C₄	—	376	1	no	—
138	C₂×C₄	—	376	2	no	—
139	C₂×C₄	D₇₅₂	1	2	yes	—
140	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
141	C₈	D₇₅₂	1	2	yes	—
142	C₈	#13094021	1	1	yes	characteristic
143	C₈	C₂×D₃₇₆	1	1	yes	characteristic
144	C₂²	—	752	1	no	—
145	C₂²	—	752	2	no	—
146	C₂²	C₂×D₇₅₂	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	752	1	no	—
149	C₄	—	752	2	no	—
150	C₄	#15689500	1	1	yes	characteristic
151	C₄	C₂×D₇₅₂	1	1	yes	characteristic
152	C₂	—	1504	1	no	—
153	C₂	#20245638	1	1	yes	characteristic
154	C₂	#20245708	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₇₅₂

C₂

yes

maximal, complemented

C₂×#15688962

C₂

yes

maximal

#20245649

C₂

yes

maximal, characteristic

#20245641

C₂

yes

maximal, characteristic, complemented

C₄₇×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₇₆

—

C₂×Dic₃₇₆

C₂²

yes

—

C₂×Dic₃₇₆

C₂²

yes

characteristic

C₂×#13093969

—

C₂×D₇₅₂

C₂²

yes

characteristic

#15689507

—

#15689503

—

Dic₇₅₂

—

#15688962

—

C₂×C₃₂×C₄₇

C₂²

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×#10625325

—

C₂×D₃₇₆

—

#13094024

—

#13094012

—

Dic₃₇₆

—

Dic₃₇₆

—

Dic₃₇₆

—

Dic₃₇₆

—

#13093969

—

D₇₅₂

—

C₂×C₁₆×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₃₂×C₄₇

D₄

yes

—

C₂×D₁₈₈

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

D₃₇₆

—

D₃₇₆

—

#10625325

—

C₁₆×C₄₇

D₈

yes

—

C₁₆×C₄₇

C₂×D₄

yes

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

D₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

D₁₆

yes

—

C₈×C₄₇

#127

yes

characteristic

C₈×C₄₇

C₂×D₈

yes

characteristic

#13730

—

maximal, Sylow

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#493

yes

characteristic

C₄×C₄₇

C₂×D₁₆

yes

characteristic

Dic₄₇

—

Dic₄₇

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

—

#2247

—

#1397

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#2246

yes

characteristic

C₂×C₄₇

#2264

yes

—

D₄₇

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

#494

—

Q₆₄

—

SD₆₄

—

C₂×C₃₂

C₂×D₄₇

yes

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13730

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

188

—

C₂×SD₁₆

—

188

—

C₂×D₈

—

100

C₄⋊Q₈

—

188

—

101

#124

—

188

—

102

Q₃₂

—

103

Q₃₂

—

104

Q₃₂

—

105

Q₃₂

—

188

—

106

SD₃₂

—

188

—

107

D₁₆

—

108

C₂×C₁₆

D₁₈₈

yes

characteristic

109

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

110

C₄×C₈

D₁₈₈

yes

characteristic

111

C₃₂

D₁₈₈

yes

—

112

C₂×Q₈

—

188

—

113

C₂×Q₈

—

188

—

114

C₂×D₄

—

188

—

115

Q₁₆

—

376

—

116

Q₁₆

—

188

—

117

Q₁₆

—

188

—

118

Q₁₆

—

188

—

119

Q₁₆

—

188

—

120

SD₁₆

—

376

—

121

D₈

—

188

—

122

D₈

—

188

—

123

C₂×C₈

D₃₇₆

yes

characteristic

124

C₂×C₈

C₂×D₁₈₈

yes

characteristic

125

C₄⋊C₄

—

376

—

126

C₂²⋊C₄

—

376

—

127

C₄²

D₃₇₆

yes

characteristic

128

C₁₆

D₃₇₆

yes

—

129

C₁₆

C₂×D₁₈₈

yes

—

130

C₂³

—

376

—

131

Q₈

—

376

—

132

Q₈

—

376

—

133

Q₈

—

376

—

134

Q₈

—

376

—

135

D₄

—

376

—

136

D₄

—

376

—

137

C₂×C₄

—

376

—

138

C₂×C₄

—

376

—

139

C₂×C₄

D₇₅₂

yes

—

140

C₂×C₄

C₂×D₃₇₆

yes

characteristic

141

C₈

D₇₅₂

yes

—

142

C₈

#13094021

yes

characteristic

143

C₈

C₂×D₃₇₆

yes

characteristic

144

C₂²

—

752

—

145

C₂²

—

752

—

146

C₂²

C₂×D₇₅₂

yes

characteristic

147

C₄

—

148

C₄

—

752

—

149

C₄

—

752

—

150

C₄

#15689500

yes

characteristic

151

C₄

C₂×D₇₅₂

yes

characteristic

152

C₂

—

1504

—

153

C₂

#20245638

yes

characteristic

154

C₂

#20245708

yes

—

	subgroup	G/H
center	C₂²	C₂×D₇₅₂
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1397	C₂
socle	C₂²×C₄₇	C₂×D₁₆
prime radical	C₂×D₇₅₂	C₂²
hypercenter	#1397	D₄₇
hypocenter	C₄₇	#13730
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₇₅₂

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄₇×#1397

C₂

socle

C₂²×C₄₇

C₂×D₁₆

prime radical

C₂×D₇₅₂

C₂²

D₄₇

C₄₇

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. C₂×Dic₇₅₂	C₂	semidirect
4. #20245641	C₂	semidirect
5. C₄₇×#1397	C₂	semidirect
79. #1397	D₄₇	semidirect
94. C₄₇	#13730	semidirect

H ≅ G/N

type

1. C₂×Dic₇₅₂

C₂

semidirect

4. #20245641

C₂

semidirect

5. C₄₇×#1397

C₂

semidirect

79. #1397

D₄₇

semidirect

94. C₄₇

#13730

semidirect

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

subgroup	index
C₂×Dic₇₅₂	2
C₂×#15688962	2
#20245649	2
#20245641	2
C₄₇×#1397	2
#13730	47

subgroup

index

C₂×Dic₇₅₂

C₄₇×#1397

Subgroups — #26137294

Conjugacy classes of subgroups

Complements

Maximal subgroups