Group #25017613 — subgroups

Subgroups — #25017613

Order 11008.

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×Dic₆₈₈	C₂	1	2	yes	maximal, complemented
2	C₂×#14582738	C₂	1	2	yes	maximal
3	#19136856	C₂	1	1	yes	maximal, characteristic
4	#19136848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₄₄	—	2	1	no	—
7	C₂×Dic₃₄₄	C₂²	1	2	yes	—
8	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
9	C₂×#13063270	—	2	1	no	—
10	C₂×D₆₈₈	C₂²	1	1	yes	characteristic
11	#14583283	—	2	1	no	—
12	#14583279	—	2	1	no	—
13	Dic₆₈₈	—	2	4	no	—
14	#14582738	—	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×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	—	4	1	no	—
20	C₂×#10621873	—	4	1	no	—
21	C₂×D₃₄₄	—	2	1	no	—
22	#13063325	—	4	1	no	—
23	#13063313	—	4	1	no	—
24	Dic₃₄₄	—	2	2	no	—
25	Dic₃₄₄	—	2	2	no	—
26	Dic₃₄₄	—	2	2	no	—
27	Dic₃₄₄	—	4	2	no	—
28	#13063270	—	4	2	no	—
29	D₆₈₈	—	2	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	—
34	C₂×D₁₇₂	—	4	1	no	—
35	C₂×Dic₈₆	—	4	1	no	—
36	C₂×Dic₈₆	—	4	2	no	—
37	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₄₃	D₈	1	1	yes	characteristic
40	(C₂×D₄₃)⋊C₄	—	8	2	no	—
41	C₄⋊Dic₄₃	—	8	2	no	—
42	Dic₁₇₂	—	4	2	no	—
43	Dic₁₇₂	—	4	1	no	—
44	Dic₁₇₂	—	4	1	no	—
45	Dic₁₇₂	—	4	2	no	—
46	Dic₁₇₂	—	8	2	no	—
47	D₃₄₄	—	4	1	no	—
48	D₃₄₄	—	4	1	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	C₂×C₄×C₄₃	D₁₆	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₄₃	—	8	2	no	—
56	C₂×Dic₄₃	—	8	1	no	—
57	D₁₇₂	—	8	1	no	—
58	D₁₇₂	—	8	1	no	—
59	Dic₈₆	—	8	1	no	—
60	Dic₈₆	—	8	2	no	—
61	Dic₈₆	—	8	1	no	—
62	Dic₈₆	—	8	2	no	—
63	C₈×C₄₃	D₁₆	1	2	yes	—
64	C₈×C₄₃	#127	1	1	yes	characteristic
65	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
66	#13730	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
68	C₂×D₄₃	—	16	1	no	—
69	C₂×D₄₃	—	16	2	no	—
70	C₄×C₄₃	—	2	2	no	—
71	C₄×C₄₃	#493	1	1	yes	characteristic
72	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
73	Dic₄₃	—	16	1	no	—
74	Dic₄₃	—	16	2	no	—
75	C₂×Q₆₄	—	43	2	no	—
76	C₂×SD₆₄	—	43	2	no	—
77	#2249	—	43	1	no	—
78	#2247	—	43	1	no	—
79	#1397	D₄₃	1	1	yes	characteristic, complemented
80	C₂×C₄₃	#2246	1	1	yes	characteristic
81	C₂×C₄₃	#2264	1	2	yes	—
82	D₄₃	—	32	1	no	—
83	C₂×Q₃₂	—	43	2	no	—
84	C₂×Q₃₂	—	43	1	no	—
85	C₂×Q₃₂	—	86	1	no	—
86	C₂×SD₃₂	—	86	1	no	—
87	C₂×D₁₆	—	43	1	no	—
88	#495	—	86	1	no	—
89	#494	—	86	1	no	—
90	Q₆₄	—	86	4	no	—
91	SD₆₄	—	86	4	no	—
92	C₂×C₃₂	C₂×D₄₃	1	2	yes	—
93	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
94	C₄₃	#13730	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₁₆	—	86	2	no	—
96	C₂×Q₁₆	—	86	1	no	—
97	C₂×Q₁₆	—	172	1	no	—
98	C₂×SD₁₆	—	172	1	no	—
99	C₂×D₈	—	86	1	no	—
100	C₄⋊Q₈	—	172	1	no	—
101	#124	—	172	1	no	—
102	Q₃₂	—	86	2	no	—
103	Q₃₂	—	86	2	no	—
104	Q₃₂	—	86	2	no	—
105	Q₃₂	—	172	2	no	—
106	SD₃₂	—	172	2	no	—
107	D₁₆	—	86	2	no	—
108	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
110	C₄×C₈	D₁₇₂	1	1	yes	characteristic
111	C₃₂	D₁₇₂	1	4	yes	—
112	C₂×Q₈	—	172	1	no	—
113	C₂×Q₈	—	172	2	no	—
114	C₂×D₄	—	172	1	no	—
115	Q₁₆	—	344	2	no	—
116	Q₁₆	—	172	2	no	—
117	Q₁₆	—	172	1	no	—
118	Q₁₆	—	172	1	no	—
119	Q₁₆	—	172	2	no	—
120	SD₁₆	—	344	2	no	—
121	D₈	—	172	1	no	—
122	D₈	—	172	1	no	—
123	C₂×C₈	D₃₄₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
125	C₄⋊C₄	—	344	2	no	—
126	C₂²⋊C₄	—	344	2	no	—
127	C₄²	D₃₄₄	1	1	yes	characteristic
128	C₁₆	D₃₄₄	1	2	yes	—
129	C₁₆	C₂×D₁₇₂	1	2	yes	—
130	C₂³	—	344	1	no	—
131	Q₈	—	344	1	no	—
132	Q₈	—	344	2	no	—
133	Q₈	—	344	1	no	—
134	Q₈	—	344	2	no	—
135	D₄	—	344	1	no	—
136	D₄	—	344	1	no	—
137	C₂×C₄	—	344	1	no	—
138	C₂×C₄	—	344	2	no	—
139	C₂×C₄	D₆₈₈	1	2	yes	—
140	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
141	C₈	D₆₈₈	1	2	yes	—
142	C₈	#13063322	1	1	yes	characteristic
143	C₈	C₂×D₃₄₄	1	1	yes	characteristic
144	C₂²	—	688	1	no	—
145	C₂²	—	688	2	no	—
146	C₂²	C₂×D₆₈₈	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	688	1	no	—
149	C₄	—	688	2	no	—
150	C₄	#14583276	1	1	yes	characteristic
151	C₄	C₂×D₆₈₈	1	1	yes	characteristic
152	C₂	—	1376	1	no	—
153	C₂	#19136845	1	1	yes	characteristic
154	C₂	#19136915	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₆₈₈

C₂

yes

maximal, complemented

C₂×#14582738

C₂

yes

maximal

#19136856

C₂

yes

maximal, characteristic

#19136848

C₂

yes

maximal, characteristic, complemented

C₄₃×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₄₄

—

C₂×Dic₃₄₄

C₂²

yes

—

C₂×Dic₃₄₄

C₂²

yes

characteristic

C₂×#13063270

—

C₂×D₆₈₈

C₂²

yes

characteristic

#14583283

—

#14583279

—

Dic₆₈₈

—

#14582738

—

C₂×C₃₂×C₄₃

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×#10621873

—

C₂×D₃₄₄

—

#13063325

—

#13063313

—

Dic₃₄₄

—

Dic₃₄₄

—

Dic₃₄₄

—

Dic₃₄₄

—

#13063270

—

D₆₈₈

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₃₂×C₄₃

D₄

yes

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

D₈

yes

characteristic

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

#10621873

—

C₁₆×C₄₃

D₈

yes

—

C₁₆×C₄₃

C₂×D₄

yes

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

D₁₆

yes

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

D₁₆

yes

—

C₈×C₄₃

#127

yes

characteristic

C₈×C₄₃

C₂×D₈

yes

characteristic

#13730

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#493

yes

characteristic

C₄×C₄₃

C₂×D₁₆

yes

characteristic

Dic₄₃

—

Dic₄₃

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

—

#2247

—

#1397

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#2246

yes

characteristic

C₂×C₄₃

#2264

yes

—

D₄₃

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#495

—

#494

—

Q₆₄

—

SD₆₄

—

C₂×C₃₂

C₂×D₄₃

yes

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13730

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

172

—

C₂×SD₁₆

—

172

—

C₂×D₈

—

100

C₄⋊Q₈

—

172

—

101

#124

—

172

—

102

Q₃₂

—

103

Q₃₂

—

104

Q₃₂

—

105

Q₃₂

—

172

—

106

SD₃₂

—

172

—

107

D₁₆

—

108

C₂×C₁₆

D₁₇₂

yes

characteristic

109

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

110

C₄×C₈

D₁₇₂

yes

characteristic

111

C₃₂

D₁₇₂

yes

—

112

C₂×Q₈

—

172

—

113

C₂×Q₈

—

172

—

114

C₂×D₄

—

172

—

115

Q₁₆

—

344

—

116

Q₁₆

—

172

—

117

Q₁₆

—

172

—

118

Q₁₆

—

172

—

119

Q₁₆

—

172

—

120

SD₁₆

—

344

—

121

D₈

—

172

—

122

D₈

—

172

—

123

C₂×C₈

D₃₄₄

yes

characteristic

124

C₂×C₈

C₂×D₁₇₂

yes

characteristic

125

C₄⋊C₄

—

344

—

126

C₂²⋊C₄

—

344

—

127

C₄²

D₃₄₄

yes

characteristic

128

C₁₆

D₃₄₄

yes

—

129

C₁₆

C₂×D₁₇₂

yes

—

130

C₂³

—

344

—

131

Q₈

—

344

—

132

Q₈

—

344

—

133

Q₈

—

344

—

134

Q₈

—

344

—

135

D₄

—

344

—

136

D₄

—

344

—

137

C₂×C₄

—

344

—

138

C₂×C₄

—

344

—

139

C₂×C₄

D₆₈₈

yes

—

140

C₂×C₄

C₂×D₃₄₄

yes

characteristic

141

C₈

D₆₈₈

yes

—

142

C₈

#13063322

yes

characteristic

143

C₈

C₂×D₃₄₄

yes

characteristic

144

C₂²

—

688

—

145

C₂²

—

688

—

146

C₂²

C₂×D₆₈₈

yes

characteristic

147

C₄

—

148

C₄

—

688

—

149

C₄

—

688

—

150

C₄

#14583276

yes

characteristic

151

C₄

C₂×D₆₈₈

yes

characteristic

152

C₂

—

1376

—

153

C₂

#19136845

yes

characteristic

154

C₂

#19136915

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
prime radical	C₂×D₆₈₈	C₂²
hypercenter	#1397	D₄₃
hypocenter	C₄₃	#13730
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

prime radical

C₂×D₆₈₈

C₂²

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×Dic₆₈₈	C₂	semidirect
4. #19136848	C₂	semidirect
5. C₄₃×#1397	C₂	semidirect
79. #1397	D₄₃	semidirect
94. C₄₃	#13730	semidirect

H ≅ G/N

type

1. C₂×Dic₆₈₈

C₂

semidirect

4. #19136848

C₂

semidirect

5. C₄₃×#1397

C₂

semidirect

79. #1397

D₄₃

semidirect

94. C₄₃

#13730

semidirect

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

subgroup	index
C₂×Dic₆₈₈	2
C₂×#14582738	2
#19136856	2
#19136848	2
C₄₃×#1397	2
#13730	43

subgroup

index

C₂×Dic₆₈₈

C₄₃×#1397

Subgroups — #25017613

Conjugacy classes of subgroups

Complements

Maximal subgroups