Group #24996564 — subgroups

Subgroups — #24996564

Order 11008.

406 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 151 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	#19145109	C₂	1	1	yes	maximal, characteristic
2	#19133049	C₂	1	4	yes	maximal
3	C₂×#14582620	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#1848	C₂	1	1	yes	maximal, characteristic
5	#14583456	C₂²	1	2	yes	—
6	#14583304	C₄	1	2	yes	—
7	#14583301	—	2	2	no	—
8	#14583297	C₄	1	2	yes	—
9	#14582937	—	2	2	no	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
11	#14582629	—	2	8	no	—
12	#14582620	—	2	2	no	—
13	C₂×#108×C₄₃	—	2	2	no	—
14	C₄₃×#423	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₈	—	2	2	no	—
17	#13063325	C₂×C₄	1	4	yes	—
18	#13063317	—	2	2	no	—
19	#13063304	—	2	4	no	—
20	C₄×Dic₈₆	—	2	4	no	—
21	#13063284	—	2	2	no	—
22	#13063259	—	2	4	no	—
23	#13063257	—	2	4	no	—
24	C₄×C₄₃⋊C₈	—	2	2	no	—
25	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
26	C₂×M₁₆×C₄₃	—	2	2	no	—
27	C₂²×C₈×C₄₃	D₄	1	2	yes	—
28	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	#108×C₄₃	—	4	4	no	—
30	C₄⋊C₈×C₄₃	—	2	4	no	—
31	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₃	D₄	1	2	yes	—
34	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	#41	1	2	yes	—
36	C₂×Dic₈₆	—	2	4	no	—
37	M₁₆×C₄₃	—	4	4	no	—
38	C₂×C₈×C₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
42	C₂×C₈×C₄₃	Q₁₆	1	4	yes	—
43	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₄₃	C₂×D₄	1	2	yes	—
46	C₂²⋊Dic₄₃	—	4	2	no	—
47	C₄⋊Dic₄₃	—	2	4	no	—
48	C₄⋊Dic₄₃	—	2	4	no	—
49	Dic₄₃⋊C₄	—	4	4	no	—
50	C₄×Dic₄₃	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	2	4	no	—
52	C₂×C₄₃⋊C₈	—	4	4	no	—
53	C₂³×C₄₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₄₃	—	2	4	no	—
55	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₄₃	#121	1	2	yes	—
58	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
61	C₂×Dic₄₃	—	4	4	no	—
62	Dic₈₆	—	4	8	no	—
63	C₈×C₄₃	—	8	2	no	—
64	C₈×C₄₃	#103	1	4	yes	—
65	C₈×C₄₃	#103	1	4	yes	—
66	C₄₃⋊C₈	—	4	4	no	—
67	#9861	—	43	1	no	maximal, Sylow
68	C₂²×C₄₃	—	2	2	no	—
69	C₂²×C₄₃	#390	1	1	yes	characteristic
70	C₂²×C₄₃	#487	1	2	yes	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	C₂×#103	1	2	yes	—
73	C₄×C₄₃	#444	1	4	yes	—
74	C₄×C₄₃	#492	1	1	yes	characteristic
75	C₄×C₄₃	#494	1	1	yes	characteristic
76	Dic₄₃	—	8	4	no	—
77	#3160	—	43	1	no	—
78	#1848	D₄₃	1	1	yes	characteristic
79	#1493	—	43	4	no	—
80	C₂×#322	—	43	1	no	—
81	C₂×C₄₃	—	2	2	no	—
82	C₂×C₄₃	#1956	1	1	yes	characteristic
83	C₂×C₄₃	#1975	1	2	yes	—
84	#533	—	43	2	no	—
85	#500	—	43	2	no	—
86	#499	—	86	2	no	—
87	#498	—	43	2	no	—
88	C₂×#108	—	2	2	no	—
89	#427	—	86	2	no	—
90	#423	C₂×D₄₃	1	2	yes	—
91	C₂×C₄×C₈	—	43	2	no	—
92	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
93	Q₈⋊C₈	—	86	8	no	—
94	#322	—	86	2	no	—
95	C₄₃	#9861	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	2	2	no	—
97	C₂²×C₈	—	86	2	no	—
98	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
99	C₄⋊Q₈	—	43	4	no	—
100	#125	—	86	2	no	—
101	C₂²⋊Q₈	—	86	4	no	—
102	C₄×Q₈	—	86	4	no	—
103	#117	—	86	2	no	—
104	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
105	#108	—	4	4	no	—
106	#107	—	86	4	no	—
107	#106	—	86	4	no	—
108	C₄⋊C₈	—	2	4	no	—
109	C₄×C₈	—	43	4	no	—
110	C₄×C₈	—	86	2	no	—
111	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
112	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
113	C₄×C₈	D₁₇₂	1	2	yes	—
114	C₂×Q₈	—	86	4	no	—
115	C₂²×C₄	#10621902	1	2	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	86	4	no	—
120	C₂×C₈	—	172	4	no	—
121	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
122	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
123	C₂×C₈	#10621883	1	4	yes	—
124	C₂×C₈	#10621885	1	4	yes	—
125	C₄⋊C₄	—	86	4	no	—
126	C₄⋊C₄	—	86	4	no	—
127	C₄⋊C₄	—	172	4	no	—
128	C₂²⋊C₄	—	172	2	no	—
129	C₄²	—	172	2	no	—
130	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
131	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
132	C₄²	D₄×D₄₃	1	2	yes	—
133	C₂³	#13063314	1	1	yes	characteristic
134	Q₈	—	172	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	172	4	no	—
137	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	#13063287	1	2	yes	—
139	C₂×C₄	#13063300	1	2	yes	—
140	C₂×C₄	#13063321	1	1	yes	characteristic
141	C₂×C₄	C₂×#10621883	1	2	yes	—
142	C₂×C₄	C₂×#10621885	1	2	yes	—
143	C₈	—	8	2	no	—
144	C₈	—	172	4	no	—
145	C₈	#13063248	1	4	yes	—
146	C₈	#13063248	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#14582800	1	1	yes	characteristic
149	C₂²	#14583251	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	344	4	no	—
152	C₄	C₂×#13063248	1	2	yes	—
153	C₄	#14583020	1	4	yes	—
154	C₄	#14583271	1	1	yes	characteristic
155	C₄	#14583278	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#19135396	1	1	yes	characteristic
158	C₂	#19135486	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145109

C₂

yes

maximal, characteristic

#19133049

C₂

yes

maximal

C₂×#14582620

C₂

yes

maximal, characteristic

C₄₃×#1848

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583304

C₄

yes

—

#14583301

—

#14583297

C₄

yes

—

#14582937

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582629

—

#14582620

—

C₂×#108×C₄₃

—

C₄₃×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

—

#13063325

C₂×C₄

yes

—

#13063317

—

#13063304

—

C₄×Dic₈₆

—

#13063284

—

#13063259

—

#13063257

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#108×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#121

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

—

C₂×C₄×C₄₃

C₂×Q₁₆

yes

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

#103

yes

—

C₈×C₄₃

#103

yes

—

C₄₃⋊C₈

—

#9861

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#390

yes

characteristic

C₂²×C₄₃

#487

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#103

yes

—

C₄×C₄₃

#444

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

#3160

—

#1848

D₄₃

yes

characteristic

#1493

—

C₂×#322

—

C₂×C₄₃

—

C₂×C₄₃

#1956

yes

characteristic

C₂×C₄₃

#1975

yes

—

—

—

—

—

—

—

C₂×D₄₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

Q₈⋊C₈

—

#322

—

C₄₃

#9861

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₃

yes

—

C₄⋊Q₈

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₄₃

yes

characteristic

105

#108

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₄×D₄₃

yes

characteristic

112

C₄×C₈

C₄×D₄₃

yes

characteristic

113

C₄×C₈

D₁₇₂

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

#10621902

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

172

—

121

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

122

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

123

C₂×C₈

#10621883

yes

—

124

C₂×C₈

#10621885

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

172

—

128

C₂²⋊C₄

—

172

—

129

C₄²

—

172

—

130

C₄²

C₂×C₄×D₄₃

yes

characteristic

131

C₄²

C₂×D₁₇₂

yes

characteristic

132

C₄²

D₄×D₄₃

yes

—

133

C₂³

#13063314

yes

characteristic

134

Q₈

—

172

—

135

C₂×C₄

—

136

C₂×C₄

—

172

—

137

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

138

C₂×C₄

#13063287

yes

—

139

C₂×C₄

#13063300

yes

—

140

C₂×C₄

#13063321

yes

characteristic

141

C₂×C₄

C₂×#10621883

yes

—

142

C₂×C₄

C₂×#10621885

yes

—

143

C₈

—

144

C₈

—

172

—

145

C₈

#13063248

yes

—

146

C₈

#13063248

yes

—

147

C₂²

—

148

C₂²

#14582800

yes

characteristic

149

C₂²

#14583251

yes

—

150

C₄

—

151

C₄

—

344

—

152

C₄

C₂×#13063248

yes

—

153

C₄

#14583020

yes

—

154

C₄

#14583271

yes

characteristic

155

C₄

#14583278

yes

characteristic

156

C₂

—

157

C₂

#19135396

yes

characteristic

158

C₂

#19135486

yes

—

	subgroup	G/H
center	C₂×C₄	#13063321
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1848	C₂
socle	C₂²×C₄₃	#390
prime radical	C₂³×C₄₃	#124
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1848	D₄₃
hypocenter	C₄₃	#9861
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#13063321

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1848

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
95. C₄₃	#9861	semidirect

H ≅ G/N

type

95. C₄₃

#9861

semidirect

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

subgroup	index
#19145109	2
#19133049	2
C₂×#14582620	2
C₄₃×#1848	2
#9861	43

subgroup

index

C₄₃×#1848

Subgroups — #24996564

Conjugacy classes of subgroups

Complements

Maximal subgroups