Group #25011371 — subgroups

Subgroups — #25011371

Order 11008.

318 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19136851	C₂	1	2	yes	maximal, complemented
2	#19136819	C₂	1	2	yes	maximal, complemented
3	#19134210	C₂	1	1	yes	maximal, characteristic
4	#19134192	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063270	—	2	4	no	—
7	#14583304	C₂²	1	1	yes	characteristic
8	#14583297	—	2	1	no	—
9	#14583279	C₂²	1	1	yes	characteristic
10	#14583276	—	2	1	no	—
11	#14583276	—	2	1	no	—
12	#14583276	C₂²	1	2	yes	—
13	#14583250	—	2	1	no	—
14	#14582701	—	2	4	no	—
15	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×D₃₄₄	—	2	2	no	—
19	C₂×D₃₄₄	—	2	2	no	—
20	C₂×D₃₄₄	—	4	2	no	—
21	C₂×D₃₄₄	—	4	2	no	—
22	#13063325	—	2	1	no	—
23	#13063325	—	2	1	no	—
24	#13063322	—	2	2	no	—
25	#13063270	—	4	8	no	—
26	#13063259	—	2	2	no	—
27	#13063257	—	4	2	no	—
28	#13063250	—	4	2	no	—
29	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
30	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×D₁₇₂	—	4	2	no	—
34	C₂×D₁₇₂	—	4	2	no	—
35	C₂×D₁₇₂	—	4	2	no	—
36	C₂×Dic₈₆	—	4	1	no	—
37	C₂×Dic₈₆	—	4	1	no	—
38	C₂×C₈×C₄₃	D₈	1	2	yes	—
39	C₂×C₈×C₄₃	D₈	1	2	yes	—
40	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
41	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	4	1	no	—
43	C₄⋊Dic₄₃	—	4	1	no	—
44	C₄⋊Dic₄₃	—	4	1	no	—
45	C₄⋊Dic₄₃	—	4	1	no	—
46	Dic₁₇₂	—	4	4	no	—
47	D₃₄₄	—	4	4	no	—
48	D₃₄₄	—	4	4	no	—
49	D₃₄₄	—	8	4	no	—
50	D₃₄₄	—	8	4	no	—
51	C₁₆×C₄₃	—	2	4	no	—
52	C₂²×D₄₃	—	8	2	no	—
53	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₄₃	—	8	1	no	—
57	C₂×Dic₄₃	—	8	1	no	—
58	D₁₇₂	—	8	2	no	—
59	D₁₇₂	—	8	2	no	—
60	D₁₇₂	—	8	4	no	—
61	D₁₇₂	—	8	4	no	—
62	Dic₈₆	—	8	2	no	—
63	Dic₈₆	—	8	2	no	—
64	C₈×C₄₃	—	2	2	no	—
65	C₈×C₄₃	SD₃₂	1	4	yes	complemented
66	C₈×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
67	#12529	—	43	1	no	maximal, Sylow
68	C₂²×C₄₃	#493	1	1	yes	characteristic
69	C₂×D₄₃	—	16	2	no	—
70	C₂×D₄₃	—	16	2	no	—
71	C₂×D₄₃	—	16	2	no	—
72	C₄×C₄₃	#496	1	2	yes	—
73	C₄×C₄₃	C₂×SD₃₂	1	2	yes	—
74	C₄×C₄₃	#509	1	2	yes	—
75	Dic₄₃	—	16	1	no	—
76	Dic₄₃	—	16	1	no	—
77	#2248	—	43	2	no	—
78	#2240	—	43	2	no	—
79	#1712	—	43	1	no	—
80	#1707	—	43	1	no	—
81	#1312	D₄₃	1	1	yes	characteristic, complemented
82	C₂×C₄₃	#1721	1	1	yes	characteristic
83	C₂×C₄₃	#2248	1	1	yes	characteristic
84	C₂×C₄₃	#2250	1	1	yes	characteristic
85	D₄₃	—	32	2	no	—
86	C₂×SD₃₂	—	86	4	no	—
87	#500	—	43	1	no	—
88	#498	—	86	1	no	—
89	#494	—	43	1	no	—
90	#493	—	43	2	no	—
91	#493	—	86	1	no	—
92	#493	—	86	1	no	—
93	#487	—	86	1	no	—
94	#357	—	86	4	no	—
95	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
96	C₈²	C₂×D₄₃	1	1	yes	characteristic
97	C₄₃	#12529	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	86	2	no	—
99	C₂×D₈	—	86	2	no	—
100	C₂×D₈	—	86	2	no	—
101	C₂×D₈	—	172	2	no	—
102	C₂×D₈	—	172	2	no	—
103	C₄⋊Q₈	—	86	1	no	—
104	C₄⋊Q₈	—	86	1	no	—
105	#127	—	86	2	no	—
106	SD₃₂	—	172	8	no	—
107	C₂×C₁₆	D₁₇₂	1	4	yes	—
108	#107	—	86	2	no	—
109	#106	—	172	2	no	—
110	#103	—	172	2	no	—
111	C₄×C₈	D₁₇₂	1	1	yes	characteristic
112	C₄×C₈	D₁₇₂	1	1	yes	characteristic
113	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
114	C₂×Q₈	—	172	1	no	—
115	C₂×Q₈	—	172	1	no	—
116	C₂×D₄	—	172	2	no	—
117	C₂×D₄	—	172	2	no	—
118	C₂×D₄	—	172	2	no	—
119	Q₁₆	—	172	4	no	—
120	D₈	—	344	4	no	—
121	D₈	—	344	4	no	—
122	D₈	—	172	4	no	—
123	D₈	—	172	4	no	—
124	C₂×C₈	D₃₄₄	1	2	yes	—
125	C₂×C₈	D₃₄₄	1	2	yes	—
126	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
127	C₄⋊C₄	—	172	1	no	—
128	C₄⋊C₄	—	172	1	no	—
129	C₄⋊C₄	—	172	1	no	—
130	C₄⋊C₄	—	172	1	no	—
131	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
132	C₁₆	—	2	4	no	—
133	C₂³	—	344	2	no	—
134	Q₈	—	344	2	no	—
135	Q₈	—	344	2	no	—
136	D₄	—	344	4	no	—
137	D₄	—	344	4	no	—
138	D₄	—	344	2	no	—
139	D₄	—	344	2	no	—
140	C₂×C₄	—	344	1	no	—
141	C₂×C₄	—	344	1	no	—
142	C₂×C₄	#13063322	1	1	yes	characteristic
143	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
144	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	#13063270	1	4	yes	complemented
147	C₈	#13063353	1	4	yes	—
148	C₂²	—	688	2	no	—
149	C₂²	—	688	2	no	—
150	C₂²	—	688	2	no	—
151	C₂²	#14583276	1	1	yes	characteristic
152	C₄	—	688	1	no	—
153	C₄	—	688	1	no	—
154	C₄	#14583288	1	2	yes	—
155	C₄	C₂×#13063270	1	2	yes	—
156	C₄	#14583337	1	2	yes	—
157	C₂	—	1376	2	no	—
158	C₂	#19134261	1	1	yes	characteristic
159	C₂	#19136851	1	1	yes	characteristic
160	C₂	#19136861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136851

C₂

yes

maximal, complemented

#19136819

C₂

yes

maximal, complemented

#19134210

C₂

yes

maximal, characteristic

#19134192

C₂

yes

maximal, characteristic, complemented

C₄₃×#1312

C₂

yes

maximal, characteristic, complemented

C₂×#13063270

—

#14583304

C₂²

yes

characteristic

#14583297

—

#14583279

C₂²

yes

—

—

C₂²

yes

—

#14583250

—

#14582701

—

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

C₂×D₃₄₄

—

—

—

—

—

—

—

—

C₂×C₁₆×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

D₈

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

D₃₄₄

—

C₁₆×C₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

SD₃₂

yes

complemented

C₈×C₄₃

AGL(1,ℤ/8)

yes

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

#496

yes

—

C₄×C₄₃

C₂×SD₃₂

yes

—

C₄×C₄₃

#509

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

C₂×SD₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12529

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

100

C₂×D₈

—

101

C₂×D₈

—

172

—

102

C₂×D₈

—

172

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

#127

—

106

SD₃₂

—

172

—

107

C₂×C₁₆

D₁₇₂

yes

—

108

#107

—

109

#106

—

172

—

110

#103

—

172

—

111

C₄×C₈

D₁₇₂

yes

characteristic

112

C₄×C₈

D₁₇₂

yes

characteristic

113

C₄×C₈

C₂²×D₄₃

yes

characteristic

114

C₂×Q₈

—

172

—

115

C₂×Q₈

—

172

—

116

C₂×D₄

—

172

—

117

C₂×D₄

—

172

—

118

C₂×D₄

—

172

—

119

Q₁₆

—

172

—

120

D₈

—

344

—

121

D₈

—

344

—

122

D₈

—

172

—

123

D₈

—

172

—

124

C₂×C₈

D₃₄₄

yes

—

125

C₂×C₈

D₃₄₄

yes

—

126

C₂×C₈

C₂×D₁₇₂

yes

—

127

C₄⋊C₄

—

172

—

128

C₄⋊C₄

—

172

—

129

C₄⋊C₄

—

172

—

130

C₄⋊C₄

—

172

—

131

C₄²

C₂×D₁₇₂

yes

characteristic

132

C₁₆

—

133

C₂³

—

344

—

134

Q₈

—

344

—

135

Q₈

—

344

—

136

D₄

—

344

—

137

D₄

—

344

—

138

D₄

—

344

—

139

D₄

—

344

—

140

C₂×C₄

—

344

—

141

C₂×C₄

—

344

—

142

C₂×C₄

#13063322

yes

characteristic

143

C₂×C₄

C₂×D₃₄₄

yes

characteristic

144

C₂×C₄

C₂×D₃₄₄

yes

characteristic

145

C₈

—

146

C₈

#13063270

yes

complemented

147

C₈

#13063353

yes

—

148

C₂²

—

688

—

149

C₂²

—

688

—

150

C₂²

—

688

—

151

C₂²

#14583276

yes

characteristic

152

C₄

—

688

—

153

C₄

—

688

—

154

C₄

#14583288

yes

—

155

C₄

C₂×#13063270

yes

—

156

C₄

#14583337

yes

—

157

C₂

—

1376

—

158

C₂

#19134261

yes

characteristic

159

C₂

#19136851

yes

characteristic

160

C₂

#19136861

yes

characteristic

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1312	C₂
socle	C₂²×C₄₃	#493
prime radical	#19134210	C₂
hypercenter	#1312	D₄₃
hypocenter	C₄₃	#12529
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#14583276

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1312

C₂

C₂²×C₄₃

C₂

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
1. #19136851	C₂	semidirect
2. #19136819	C₂	semidirect
4. #19134192	C₂	semidirect
5. C₄₃×#1312	C₂	semidirect
65. C₈×C₄₃	SD₃₂	semidirect
81. #1312	D₄₃	semidirect
97. C₄₃	#12529	semidirect
146. C₈	#13063270	semidirect

H ≅ G/N

type

1. #19136851

C₂

semidirect

2. #19136819

C₂

semidirect

4. #19134192

C₂

semidirect

5. C₄₃×#1312

C₂

semidirect

65. C₈×C₄₃

SD₃₂

semidirect

81. #1312

D₄₃

semidirect

97. C₄₃

#12529

semidirect

146. C₈

#13063270

semidirect

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

subgroup	index
#19136851	2
#19136819	2
#19134210	2
#19134192	2
C₄₃×#1312	2
#12529	43

subgroup

index

C₄₃×#1312

Subgroups — #25011371

Conjugacy classes of subgroups

Complements

Maximal subgroups