Group #25001618 — subgroups

Subgroups — #25001618

Order 11008.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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	#19135576	C₂	1	1	yes	maximal, characteristic
2	#19134749	C₂	1	1	yes	maximal, characteristic
3	#19133370	C₂	1	2	yes	maximal
4	#19132999	C₂	1	2	yes	maximal
5	C₄₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	1	yes	characteristic
7	#14582923	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
10	#14582659	—	2	2	no	—
11	#14582652	—	2	4	no	—
12	#14582631	—	2	4	no	—
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₄₃×#376	C₂²	1	1	yes	characteristic
15	C₂×#10621877	—	2	1	no	—
16	C₂²×C₄₃⋊C₈	—	2	2	no	—
17	#13063325	Q₈	1	2	yes	—
18	#13063317	—	2	1	no	—
19	#13063304	—	2	2	no	—
20	C₄×Dic₈₆	—	2	2	no	—
21	#13063284	—	2	1	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₄₃⋊C₈)	—	4	4	no	—
26	C₂×C₄²×C₄₃	D₄	1	2	yes	—
27	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₃	—	2	2	no	—
29	C₂²⋊C₈×C₄₃	—	2	2	no	—
30	C₄×C₈×C₄₃	D₄	1	4	yes	—
31	#95×C₄₃	—	2	2	no	—
32	C₂²×C₄×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₃	#41	1	2	yes	—
35	C₂×Dic₈₆	—	2	2	no	—
36	C₂×C₈×C₄₃	—	2	4	no	—
37	C₄²×C₄₃	—	2	2	no	—
38	C₄²×C₄₃	Q₁₆	1	4	yes	—
39	C₄²×C₄₃	C₂×D₄	1	2	yes	—
40	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₄₃	#41	1	1	yes	characteristic
42	C₂²⋊Dic₄₃	—	4	1	no	—
43	C₄⋊Dic₄₃	—	2	2	no	—
44	C₄⋊Dic₄₃	—	2	2	no	—
45	Dic₄₃⋊C₄	—	4	2	no	—
46	C₄×Dic₄₃	—	4	1	no	—
47	#10621877	—	4	2	no	—
48	C₂×C₄₃⋊C₈	—	4	2	no	—
49	C₂×C₄₃⋊C₈	—	4	1	no	—
50	C₂×C₄₃⋊C₈	—	4	4	no	—
51	C₂³×C₄₃	#123	1	1	yes	characteristic
52	C₂×C₄×C₄₃	—	2	4	no	—
53	C₂×C₄×C₄₃	—	4	2	no	—
54	C₂×C₄×C₄₃	—	4	4	no	—
55	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
56	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#123	1	2	yes	—
58	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
60	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×Dic₄₃	—	4	2	no	—
62	Dic₈₆	—	4	4	no	—
63	C₈×C₄₃	—	4	4	no	—
64	C₄₃⋊C₈	—	8	2	no	—
65	C₄₃⋊C₈	—	8	1	no	—
66	#10668	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	—	2	2	no	—
68	C₂²×C₄₃	#399	1	1	yes	characteristic
69	C₂²×C₄₃	#484	1	2	yes	—
70	C₄×C₄₃	—	2	2	no	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	—	4	4	no	—
73	C₄×C₄₃	#477	1	2	yes	—
74	C₄×C₄₃	#494	1	1	yes	characteristic
75	C₄×C₄₃	#496	1	1	yes	characteristic
76	Dic₄₃	—	8	2	no	—
77	#1995	—	43	1	no	—
78	#1843	—	43	1	no	—
79	#1571	—	43	2	no	—
80	#1481	—	43	2	no	—
81	#1274	D₄₃	1	1	yes	characteristic
82	C₂×C₄₃	—	2	2	no	—
83	C₂×C₄₃	#2095	1	1	yes	characteristic
84	C₂×C₄₃	#2098	1	2	yes	—
85	#533	—	43	1	no	—
86	#431	C₂×D₄₃	1	2	yes	—
87	#423	—	43	1	no	—
88	C₂×C₄⋊C₈	—	43	2	no	—
89	C₂×C₄⋊C₈	—	86	2	no	—
90	#376	C₂×D₄₃	1	1	yes	characteristic
91	#339	—	86	2	no	—
92	#335	—	86	4	no	—
93	Q₈⋊C₈	—	86	4	no	—
94	C₄₃	#10668	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	86	1	no	—
96	C₂²×C₈	—	86	2	no	—
97	C₄⋊Q₈	—	43	2	no	—
98	#125	—	86	1	no	—
99	C₂²⋊Q₈	—	86	2	no	—
100	C₄×Q₈	—	86	2	no	—
101	#117	—	86	1	no	—
102	C₂×C₄²	D₁₇₂	1	2	yes	—
103	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	86	2	no	—
106	C₄⋊C₈	—	86	2	no	—
107	C₄⋊C₈	—	86	2	no	—
108	C₄⋊C₈	—	172	4	no	—
109	C₂²⋊C₈	—	2	2	no	—
110	C₄×C₈	D₁₇₂	1	4	yes	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	86	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
115	C₂²×C₄	#10621900	1	2	yes	—
116	M₁₆	—	172	2	no	—
117	C₂×C₈	—	2	4	no	—
118	C₂×C₈	—	172	2	no	—
119	C₂×C₈	—	172	1	no	—
120	C₂×C₈	—	172	4	no	—
121	C₄⋊C₄	—	86	2	no	—
122	C₄⋊C₄	—	86	2	no	—
123	C₄⋊C₄	—	172	2	no	—
124	C₂²⋊C₄	—	172	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	172	1	no	—
127	C₄²	Dic₁₇₂	1	4	yes	—
128	C₄²	C₂×D₁₇₂	1	2	yes	—
129	C₄²	#10621900	1	1	yes	characteristic
130	C₄²	Q₈×D₄₃	1	1	yes	characteristic
131	C₂³	#13063309	1	1	yes	characteristic
132	Q₈	—	172	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	172	2	no	—
137	C₂×C₄	#13063305	1	2	yes	—
138	C₂×C₄	#13063305	1	1	yes	characteristic
139	C₂×C₄	#13063309	1	2	yes	—
140	C₂×C₄	#13063322	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₁₇₂	1	2	yes	—
142	C₂×C₄	#13063353	1	2	yes	—
143	C₈	—	4	4	no	—
144	C₈	—	344	2	no	—
145	C₈	—	344	1	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#14582836	1	1	yes	characteristic
148	C₂²	#14583235	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	344	2	no	—
153	C₄	#14583195	1	2	yes	—
154	C₄	#14583279	1	1	yes	characteristic
155	C₄	#14583288	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#19136132	1	1	yes	characteristic
158	C₂	#19136149	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135576

C₂

yes

maximal, characteristic

#19134749

C₂

yes

maximal, characteristic

#19133370

C₂

yes

maximal

#19132999

C₂

yes

maximal

C₄₃×#1274

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

characteristic

#14582923

C₂²

yes

characteristic

C₂×C₄⋊(C₄₃⋊C₈)

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

#14582659

—

#14582652

—

#14582631

—

C₄₃×#431

C₂²

yes

—

C₄₃×#376

C₂²

yes

characteristic

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

#13063325

Q₈

yes

—

#13063317

—

#13063304

—

C₄×Dic₈₆

—

#13063284

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#123

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#10668

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#399

yes

characteristic

C₂²×C₄₃

#484

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#477

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

#2095

yes

characteristic

C₂×C₄₃

#2098

yes

—

#533

—

#431

C₂×D₄₃

yes

—

#423

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

C₂×D₄₃

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₄₃

#10668

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

D₁₇₂

yes

—

103

C₂×C₄²

C₂²×D₄₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

172

—

109

C₂²⋊C₈

—

110

C₄×C₈

D₁₇₂

yes

—

111

#95

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

115

C₂²×C₄

#10621900

yes

—

116

M₁₆

—

172

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₂×C₈

—

172

—

120

C₂×C₈

—

172

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

172

—

124

C₂²⋊C₄

—

172

—

125

C₄²

—

126

C₄²

—

172

—

127

C₄²

Dic₁₇₂

yes

—

128

C₄²

C₂×D₁₇₂

yes

—

129

C₄²

#10621900

yes

characteristic

130

C₄²

Q₈×D₄₃

yes

characteristic

131

C₂³

#13063309

yes

characteristic

132

Q₈

—

172

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

172

—

137

C₂×C₄

#13063305

yes

—

138

C₂×C₄

#13063305

yes

characteristic

139

C₂×C₄

#13063309

yes

—

140

C₂×C₄

#13063322

yes

characteristic

141

C₂×C₄

C₂×Dic₁₇₂

yes

—

142

C₂×C₄

#13063353

yes

—

143

C₈

—

144

C₈

—

344

—

145

C₈

—

344

—

146

C₂²

—

147

C₂²

#14582836

yes

characteristic

148

C₂²

#14583235

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

344

—

153

C₄

#14583195

yes

—

154

C₄

#14583279

yes

characteristic

155

C₄

#14583288

yes

characteristic

156

C₂

—

157

C₂

#19136132

yes

characteristic

158

C₂

#19136149

yes

—

	subgroup	G/H
center	C₂×C₄	#13063322
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1274	C₂
socle	C₂²×C₄₃	#399
prime radical	C₂³×C₄₃	#123
hypercenter	#1274	D₄₃
hypocenter	C₄₃	#10668
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

subgroup

G/H

center

C₂×C₄

#13063322

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1274

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1274

N	H ≅ G/N	type
94. C₄₃	#10668	semidirect

H ≅ G/N

type

94. C₄₃

#10668

semidirect

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

subgroup	index
#19135576	2
#19134749	2
#19133370	2
#19132999	2
C₄₃×#1274	2
#10668	43

subgroup

index

C₄₃×#1274

Subgroups — #25001618

Conjugacy classes of subgroups

Complements

Maximal subgroups