Group #25017608 — subgroups

Subgroups — #25017608

Order 11008.

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

normal subgroups only

This group has 252 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×#14582738	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₃₇₆	C₂	1	2	yes	maximal, complemented
3	#19136856	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
7	C₂×#13063270	—	2	1	no	—
8	C₂×D₆₈₈	—	2	1	no	—
9	C₂×D₆₈₈	C₂²	1	2	yes	—
10	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
11	#14583283	—	2	1	no	—
12	#14583276	—	2	1	no	—
13	#14582738	—	2	4	no	—
14	D₁₃₇₆	—	2	4	no	—
15	C₂×C₃₂×C₄₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	1	no	—
18	C₂×#10621873	—	4	1	no	—
19	C₂×D₃₄₄	—	2	2	no	—
20	C₂×D₃₄₄	—	2	1	no	—
21	C₂×D₃₄₄	—	4	1	no	—
22	#13063322	—	4	1	no	—
23	#13063313	—	4	1	no	—
24	Dic₃₄₄	—	2	2	no	—
25	#13063270	—	4	2	no	—
26	D₆₈₈	—	2	2	no	—
27	D₆₈₈	—	2	2	no	—
28	D₆₈₈	—	2	2	no	—
29	D₆₈₈	—	4	2	no	—
30	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₃₂×C₄₃	D₄	1	4	yes	complemented
34	C₂×D₁₇₂	—	4	1	no	—
35	C₂×D₁₇₂	—	4	2	no	—
36	C₂×D₁₇₂	—	8	2	no	—
37	C₂×Dic₈₆	—	4	1	no	—
38	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₄₃	D₈	1	1	yes	characteristic
41	(C₂×D₄₃)⋊C₄	—	8	2	no	—
42	Dic₁₇₂	—	4	1	no	—
43	Dic₁₇₂	—	4	1	no	—
44	D₃₄₄	—	4	1	no	—
45	D₃₄₄	—	4	2	no	—
46	D₃₄₄	—	4	2	no	—
47	D₃₄₄	—	4	1	no	—
48	D₃₄₄	—	8	2	no	—
49	#10621873	—	8	2	no	—
50	C₁₆×C₄₃	D₈	1	2	yes	—
51	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
52	C₂²×D₄₃	—	8	1	no	—
53	C₂²×D₄₃	—	8	2	no	—
54	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
55	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₄₃	—	8	1	no	—
57	D₁₇₂	—	8	2	no	—
58	D₁₇₂	—	8	1	no	—
59	D₁₇₂	—	8	1	no	—
60	D₁₇₂	—	8	2	no	—
61	D₁₇₂	—	16	4	no	—
62	Dic₈₆	—	8	1	no	—
63	Dic₈₆	—	8	1	no	—
64	C₈×C₄₃	D₁₆	1	2	yes	—
65	C₈×C₄₃	#127	1	1	yes	characteristic
66	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
67	#13729	—	43	1	no	maximal, Sylow
68	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
69	C₂×D₄₃	—	16	1	no	—
70	C₂×D₄₃	—	16	2	no	—
71	C₂×D₄₃	—	16	2	no	—
72	C₂×D₄₃	—	16	4	no	—
73	C₄×C₄₃	—	2	2	no	—
74	C₄×C₄₃	#493	1	1	yes	characteristic
75	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
76	Dic₄₃	—	16	1	no	—
77	C₂×SD₆₄	—	43	2	no	—
78	C₂×D₃₂	—	43	2	no	—
79	#2249	—	43	1	no	—
80	#2246	—	43	1	no	—
81	#1397	D₄₃	1	1	yes	characteristic, complemented
82	C₂×C₄₃	#2246	1	1	yes	characteristic
83	C₂×C₄₃	#2263	1	2	yes	—
84	D₄₃	—	32	1	no	—
85	D₄₃	—	32	2	no	—
86	C₂×Q₃₂	—	43	1	no	—
87	C₂×SD₃₂	—	86	1	no	—
88	C₂×D₁₆	—	43	2	no	—
89	C₂×D₁₆	—	43	1	no	—
90	C₂×D₁₆	—	86	1	no	—
91	#495	—	86	1	no	—
92	#493	—	86	1	no	—
93	SD₆₄	—	86	4	no	—
94	D₃₂	—	86	4	no	—
95	C₂×C₃₂	C₂×D₄₃	1	2	yes	—
96	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
97	C₄₃	#13729	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	86	1	no	—
99	C₂×SD₁₆	—	172	1	no	—
100	C₂×D₈	—	86	2	no	—
101	C₂×D₈	—	86	1	no	—
102	C₂×D₈	—	172	1	no	—
103	#127	—	172	1	no	—
104	#124	—	172	1	no	—
105	Q₃₂	—	86	2	no	—
106	SD₃₂	—	172	2	no	—
107	D₁₆	—	86	2	no	—
108	D₁₆	—	86	2	no	—
109	D₁₆	—	86	2	no	—
110	D₁₆	—	172	2	no	—
111	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
112	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
113	C₄×C₈	D₁₇₂	1	1	yes	characteristic
114	C₃₂	D₁₇₂	1	4	yes	complemented
115	C₂×Q₈	—	172	1	no	—
116	C₂×D₄	—	344	2	no	—
117	C₂×D₄	—	172	1	no	—
118	C₂×D₄	—	172	2	no	—
119	Q₁₆	—	172	1	no	—
120	Q₁₆	—	172	1	no	—
121	SD₁₆	—	344	2	no	—
122	D₈	—	344	2	no	—
123	D₈	—	172	1	no	—
124	D₈	—	172	2	no	—
125	D₈	—	172	2	no	—
126	D₈	—	172	1	no	—
127	C₂×C₈	D₃₄₄	1	1	yes	characteristic
128	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
129	C₂²⋊C₄	—	344	2	no	—
130	C₄²	D₃₄₄	1	1	yes	characteristic
131	C₁₆	D₃₄₄	1	2	yes	—
132	C₁₆	C₂×D₁₇₂	1	2	yes	—
133	C₂³	—	344	1	no	—
134	C₂³	—	344	2	no	—
135	Q₈	—	344	1	no	—
136	Q₈	—	344	1	no	—
137	D₄	—	344	2	no	—
138	D₄	—	344	1	no	—
139	D₄	—	344	2	no	—
140	D₄	—	344	1	no	—
141	D₄	—	688	4	no	—
142	C₂×C₄	—	344	1	no	—
143	C₂×C₄	D₆₈₈	1	2	yes	—
144	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
145	C₈	D₆₈₈	1	2	yes	—
146	C₈	#13063322	1	1	yes	characteristic
147	C₈	C₂×D₃₄₄	1	1	yes	characteristic
148	C₂²	—	688	2	no	—
149	C₂²	—	688	1	no	—
150	C₂²	—	688	4	no	—
151	C₂²	—	688	2	no	—
152	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	688	1	no	—
155	C₄	#14583276	1	1	yes	characteristic
156	C₄	C₂×D₆₈₈	1	1	yes	characteristic
157	C₂	—	1376	2	no	—
158	C₂	—	1376	1	no	—
159	C₂	#19136845	1	1	yes	characteristic
160	C₂	#19136910	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582738

C₂

yes

maximal, complemented

C₂×D₁₃₇₆

C₂

yes

maximal, complemented

#19136856

C₂

yes

maximal, characteristic, complemented

#19136845

C₂

yes

maximal, characteristic, complemented

C₄₃×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₄₄

C₂²

yes

characteristic

C₂×#13063270

—

C₂×D₆₈₈

—

C₂×D₆₈₈

C₂²

yes

—

C₂×D₆₈₈

C₂²

yes

—

—

—

D₁₃₇₆

—

C₂×C₃₂×C₄₃

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×#10621873

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

#13063322

—

#13063313

—

Dic₃₄₄

—

#13063270

—

D₆₈₈

—

D₆₈₈

—

D₆₈₈

—

D₆₈₈

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₃₂×C₄₃

D₄

yes

complemented

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

(C₂×D₄₃)⋊C₄

—

Dic₁₇₂

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

#10621873

—

C₁₆×C₄₃

D₈

yes

—

C₁₆×C₄₃

C₂×D₄

yes

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

D₁₆

yes

—

C₈×C₄₃

#127

yes

characteristic

C₈×C₄₃

C₂×D₈

yes

characteristic

#13729

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#493

yes

characteristic

C₄×C₄₃

C₂×D₁₆

yes

characteristic

Dic₄₃

—

C₂×SD₆₄

—

C₂×D₃₂

—

#2249

—

#2246

—

#1397

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#2246

yes

characteristic

C₂×C₄₃

#2263

yes

—

D₄₃

—

D₄₃

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

#495

—

#493

—

SD₆₄

—

D₃₂

—

C₂×C₃₂

C₂×D₄₃

yes

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13729

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×SD₁₆

—

172

—

100

C₂×D₈

—

101

C₂×D₈

—

102

C₂×D₈

—

172

—

103

#127

—

172

—

104

#124

—

172

—

105

Q₃₂

—

106

SD₃₂

—

172

—

107

D₁₆

—

108

D₁₆

—

109

D₁₆

—

110

D₁₆

—

172

—

111

C₂×C₁₆

D₁₇₂

yes

characteristic

112

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

113

C₄×C₈

D₁₇₂

yes

characteristic

114

C₃₂

D₁₇₂

yes

complemented

115

C₂×Q₈

—

172

—

116

C₂×D₄

—

344

—

117

C₂×D₄

—

172

—

118

C₂×D₄

—

172

—

119

Q₁₆

—

172

—

120

Q₁₆

—

172

—

121

SD₁₆

—

344

—

122

D₈

—

344

—

123

D₈

—

172

—

124

D₈

—

172

—

125

D₈

—

172

—

126

D₈

—

172

—

127

C₂×C₈

D₃₄₄

yes

characteristic

128

C₂×C₈

C₂×D₁₇₂

yes

characteristic

129

C₂²⋊C₄

—

344

—

130

C₄²

D₃₄₄

yes

characteristic

131

C₁₆

D₃₄₄

yes

—

132

C₁₆

C₂×D₁₇₂

yes

—

133

C₂³

—

344

—

134

C₂³

—

344

—

135

Q₈

—

344

—

136

Q₈

—

344

—

137

D₄

—

344

—

138

D₄

—

344

—

139

D₄

—

344

—

140

D₄

—

344

—

141

D₄

—

688

—

142

C₂×C₄

—

344

—

143

C₂×C₄

D₆₈₈

yes

—

144

C₂×C₄

C₂×D₃₄₄

yes

characteristic

145

C₈

D₆₈₈

yes

—

146

C₈

#13063322

yes

characteristic

147

C₈

C₂×D₃₄₄

yes

characteristic

148

C₂²

—

688

—

149

C₂²

—

688

—

150

C₂²

—

688

—

151

C₂²

—

688

—

152

C₂²

C₂×D₆₈₈

yes

characteristic

153

C₄

—

154

C₄

—

688

—

155

C₄

#14583276

yes

characteristic

156

C₄

C₂×D₆₈₈

yes

characteristic

157

C₂

—

1376

—

158

C₂

—

1376

—

159

C₂

#19136845

yes

characteristic

160

C₂

#19136910

yes

—

	subgroup	G/H
center	C₂²	C₂×D₆₈₈
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1397	C₂
socle	C₂²×C₄₃	C₂×D₁₆
hypercenter	#1397	D₄₃
hypocenter	C₄₃	#13729
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₆₈₈

commutator, cosocle

C₂×C₁₆×C₄₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄₃×#1397

C₂

socle

C₂²×C₄₃

C₂×D₁₆

D₄₃

C₄₃

G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

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

N	H ≅ G/N	type
1. C₂×#14582738	C₂	semidirect
2. C₂×D₁₃₇₆	C₂	semidirect
3. #19136856	C₂	semidirect
4. #19136845	C₂	semidirect
5. C₄₃×#1397	C₂	semidirect
33. C₃₂×C₄₃	D₄	semidirect
81. #1397	D₄₃	semidirect
97. C₄₃	#13729	semidirect
114. C₃₂	D₁₇₂	semidirect

H ≅ G/N

type

1. C₂×#14582738

C₂

semidirect

2. C₂×D₁₃₇₆

C₂

semidirect

3. #19136856

C₂

semidirect

4. #19136845

C₂

semidirect

5. C₄₃×#1397

C₂

semidirect

33. C₃₂×C₄₃

D₄

semidirect

81. #1397

D₄₃

semidirect

97. C₄₃

#13729

semidirect

114. C₃₂

D₁₇₂

semidirect

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

subgroup	index
C₂×#14582738	2
C₂×D₁₃₇₆	2
#19136856	2
#19136845	2
C₄₃×#1397	2
#13729	43

subgroup

index

C₂×D₁₃₇₆

C₄₃×#1397

Subgroups — #25017608

Conjugacy classes of subgroups

Complements

Maximal subgroups