Group #25104999 — subgroups

Subgroups — #25104999

Order 11008.

1154 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 271 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×#14582976	C₂	1	6	yes	maximal, complemented
2	#19137876	C₂	1	1	yes	maximal, characteristic
3	#19135392	C₂	1	6	yes	maximal, complemented
4	#19134282	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#10621872	C₂²	1	4	yes	—
7	C₂×C₄×D₁₇₂	C₂²	1	6	yes	—
8	#14582976	—	2	24	no	—
9	C₂×(C₂×D₄₃)⋊C₈	C₂²	1	12	yes	—
10	C₂×#13063236	C₂²	1	3	yes	—
11	C₂×C₄×C₄₃⋊C₈	C₂²	1	3	yes	—
12	#14582836	C₄	1	4	yes	—
13	#14582809	C₄	1	4	yes	—
14	C₂×C₄⋊C₈×C₄₃	C₂²	1	6	yes	—
15	C₄₃×#376	C₂²	1	1	yes	characteristic
16	C₂²×D₁₇₂	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₄₃	C₂³	1	4	yes	—
18	C₂×#10621872	—	2	24	no	—
19	C₂²×C₄₃⋊C₈	C₂³	1	3	yes	—
20	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
21	C₄×D₁₇₂	—	2	12	no	—
22	C₄×D₁₇₂	—	2	12	no	—
23	C₂×C₄⋊Dic₄₃	C₂×C₄	1	6	yes	—
24	C₂×(C₂×D₄₃)⋊C₄	C₂×C₄	1	12	yes	—
25	(C₂×D₄₃)⋊C₈	—	2	24	no	—
26	#13063236	—	2	6	no	—
27	C₄×C₄₃⋊C₈	D₄	1	12	yes	complemented
28	C₂²×C₈×C₄₃	C₂³	1	4	yes	—
29	C₂×C₄²×C₄₃	C₂³	1	3	yes	—
30	C₄⋊C₈×C₄₃	—	2	12	no	—
31	#95×C₄₃	C₂×C₄	1	4	yes	—
32	C₂³×D₄₃	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
35	C₂²×Dic₄₃	C₂²×C₄	1	4	yes	—
36	C₂×D₁₇₂	—	2	12	no	—
37	C₂×D₁₇₂	—	4	12	no	—
38	C₂×C₄×D₄₃	—	2	12	no	—
39	C₂×C₄×D₄₃	—	2	12	no	—
40	C₂×C₈×C₄₃	—	2	12	no	—
41	C₄²×C₄₃	C₂×D₄	1	6	yes	—
42	C₄²×C₄₃	#41	1	6	yes	—
43	(C₂×D₄₃)⋊C₄	—	2	24	no	—
44	C₄⋊Dic₄₃	—	2	12	no	—
45	C₂×C₄₃⋊C₈	—	2	6	no	—
46	C₂×C₄₃⋊C₈	C₂×D₄	1	6	yes	—
47	C₂×C₄₃⋊C₈	C₂×D₄	1	6	yes	—
48	#10621872	—	4	16	no	—
49	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²×D₄₃	—	2	12	no	—
51	C₂²×D₄₃	—	2	4	no	—
52	C₂²×D₄₃	—	2	12	no	—
53	C₂×C₄×C₄₃	—	2	12	no	—
54	C₂×C₄×C₄₃	C₄×D₄	1	12	yes	—
55	C₂×C₄×C₄₃	C₂²×D₄	1	3	yes	—
56	C₂×C₄×C₄₃	C₂×#41	1	3	yes	—
57	C₂×Dic₄₃	—	2	12	no	—
58	D₁₇₂	—	4	24	no	—
59	C₄×D₄₃	—	4	4	no	—
60	C₄×D₄₃	—	4	12	no	—
61	C₈×C₄₃	—	4	4	no	—
62	C₄₃⋊C₈	—	2	6	no	—
63	C₄₃⋊C₈	#127	1	4	yes	—
64	#21684	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	C₂×C₄×D₄	1	3	yes	—
66	C₂²×C₄₃	#568	1	4	yes	—
67	C₂×D₄₃	—	4	4	no	—
68	C₂×D₄₃	—	4	12	no	—
69	C₂×D₄₃	—	4	12	no	—
70	C₄×C₄₃	—	2	12	no	—
71	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
72	C₄×C₄₃	#532	1	3	yes	—
73	Dic₄₃	—	4	4	no	—
74	C₂×#436	—	43	6	no	—
75	#2397	—	43	1	no	—
76	#1955	—	43	6	no	—
77	#1843	D₄₃	1	1	yes	characteristic, complemented
78	#1726	—	43	1	no	—
79	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
80	C₂×C₄₃	#2930	1	6	yes	—
81	D₄₃	—	8	4	no	—
82	C₂²×M₁₆	—	43	4	no	—
83	C₂×C₄×D₄	—	43	6	no	—
84	#436	—	86	24	no	—
85	C₂×C₄⋊C₈	C₂×D₄₃	1	6	yes	—
86	C₂×C₂²⋊C₈	—	43	12	no	—
87	C₂×#97	—	43	3	no	—
88	C₂×C₄×C₈	—	43	3	no	—
89	#399	—	43	4	no	—
90	#392	—	43	4	no	—
91	#376	C₂×D₄₃	1	1	yes	characteristic
92	C₄₃	#21684	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	43	6	no	—
94	C₂³×C₄	—	43	4	no	—
95	C₂×M₁₆	—	86	24	no	—
96	C₂²×C₈	—	43	1	no	—
97	C₂²×C₈	—	43	3	no	—
98	C₂²×C₈	C₂²×D₄₃	1	4	yes	—
99	C₄×D₄	—	86	12	no	—
100	C₄×D₄	—	86	12	no	—
101	C₂×C₄⋊C₄	—	43	6	no	—
102	C₂×C₂²⋊C₄	—	43	12	no	—
103	C₂×C₄²	C₂²×D₄₃	1	3	yes	—
104	C₄⋊C₈	—	2	12	no	—
105	C₂²⋊C₈	—	86	24	no	—
106	#97	—	86	6	no	—
107	C₄×C₈	—	43	12	no	—
108	#95	C₄×D₄₃	1	4	yes	—
109	C₂⁴	—	43	4	no	—
110	C₂×D₄	—	86	12	no	—
111	C₂×D₄	—	172	12	no	—
112	C₂²×C₄	—	43	4	no	—
113	C₂²×C₄	—	86	12	no	—
114	C₂²×C₄	—	86	12	no	—
115	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
116	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
117	M₁₆	—	172	16	no	—
118	C₂×C₈	—	2	12	no	—
119	C₂×C₈	—	43	6	no	—
120	C₂×C₈	—	43	6	no	—
121	C₂×C₈	—	86	6	no	—
122	C₄⋊C₄	—	86	12	no	—
123	C₂²⋊C₄	—	86	24	no	—
124	C₄²	D₄×D₄₃	1	6	yes	—
125	C₄²	#10621902	1	6	yes	—
126	C₂³	—	86	12	no	—
127	C₂³	—	86	4	no	—
128	C₂³	—	86	12	no	—
129	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
130	D₄	—	172	24	no	—
131	C₂×C₄	—	2	12	no	—
132	C₂×C₄	—	86	12	no	—
133	C₂×C₄	—	172	4	no	—
134	C₂×C₄	—	172	12	no	—
135	C₂×C₄	#13063287	1	12	yes	—
136	C₂×C₄	C₂×D₄×D₄₃	1	3	yes	—
137	C₂×C₄	C₂×#10621902	1	3	yes	—
138	C₈	—	4	4	no	—
139	C₈	—	43	4	no	—
140	C₈	—	86	6	no	—
141	C₂²	—	172	12	no	—
142	C₂²	—	172	12	no	—
143	C₂²	—	172	4	no	—
144	C₂²	C₂×#13063287	1	3	yes	—
145	C₂²	#14583664	1	4	yes	—
146	C₄	—	2	12	no	—
147	C₄	—	172	4	no	—
148	C₄	#127×D₄₃	1	1	yes	characteristic
149	C₄	#14583450	1	3	yes	—
150	C₂	—	344	4	no	—
151	C₂	#19137157	1	1	yes	characteristic
152	C₂	#19143297	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582976

C₂

yes

maximal, complemented

#19137876

C₂

yes

maximal, characteristic

#19135392

C₂

yes

maximal, complemented

#19134282

C₂

yes

maximal, characteristic, complemented

C₄₃×#1843

C₂

yes

maximal, characteristic, complemented

C₂²×#10621872

C₂²

yes

—

C₂×C₄×D₁₇₂

C₂²

yes

—

#14582976

—

C₂×(C₂×D₄₃)⋊C₈

C₂²

yes

—

C₂×#13063236

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582836

C₄

yes

—

#14582809

C₄

yes

—

C₂×C₄⋊C₈×C₄₃

C₂²

yes

—

C₄₃×#376

C₂²

yes

characteristic

C₂²×D₁₇₂

C₂×C₄

yes

—

C₂²×C₄×D₄₃

C₂³

yes

—

C₂×#10621872

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

characteristic

C₄×D₁₇₂

—

C₄×D₁₇₂

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×(C₂×D₄₃)⋊C₄

C₂×C₄

yes

—

(C₂×D₄₃)⋊C₈

—

#13063236

—

C₄×C₄₃⋊C₈

D₄

yes

complemented

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

—

C₄⋊C₈×C₄₃

—

#95×C₄₃

C₂×C₄

yes

—

C₂³×D₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

C₂²×C₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×C₄×D₄₃

—

C₂×C₄×D₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

(C₂×D₄₃)⋊C₄

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

#10621872

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

—

C₂×C₄×C₄₃

C₂×#41

yes

—

C₂×Dic₄₃

—

D₁₇₂

—

C₄×D₄₃

—

C₄×D₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

#127

yes

—

#21684

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×D₄

yes

—

C₂²×C₄₃

#568

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×#127

yes

characteristic

C₄×C₄₃

#532

yes

—

Dic₄₃

—

—

—

—

D₄₃

yes

characteristic, complemented

#1726

—

C₂×C₄₃

C₄×#127

yes

characteristic

C₂×C₄₃

#2930

yes

—

D₄₃

—

C₂²×M₁₆

—

C₂×C₄×D₄

—

#436

—

C₂×C₄⋊C₈

C₂×D₄₃

yes

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#399

—

#392

—

#376

C₂×D₄₃

yes

characteristic

C₄₃

#21684

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₃

yes

—

C₄×D₄

—

100

C₄×D₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂²×D₄₃

yes

—

104

C₄⋊C₈

—

105

C₂²⋊C₈

—

106

#97

—

107

C₄×C₈

—

108

#95

C₄×D₄₃

yes

—

109

C₂⁴

—

110

C₂×D₄

—

111

C₂×D₄

—

172

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

C₂×C₄×D₄₃

yes

—

116

C₂²×C₄

C₂³×D₄₃

yes

characteristic

117

M₁₆

—

172

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

D₄×D₄₃

yes

—

125

C₄²

#10621902

yes

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂²×C₄×D₄₃

yes

characteristic

130

D₄

—

172

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

172

—

134

C₂×C₄

—

172

—

135

C₂×C₄

#13063287

yes

—

136

C₂×C₄

C₂×D₄×D₄₃

yes

—

137

C₂×C₄

C₂×#10621902

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

172

—

142

C₂²

—

172

—

143

C₂²

—

172

—

144

C₂²

C₂×#13063287

yes

—

145

C₂²

#14583664

yes

—

146

C₄

—

147

C₄

—

172

—

148

C₄

#127×D₄₃

yes

characteristic

149

C₄

#14583450

yes

—

150

C₂

—

344

—

151

C₂

#19137157

yes

characteristic

152

C₂

#19143297

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#1843	C₂
prime radical	#19137876	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1843	D₄₃
hypocenter	C₄₃	#21684
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1843

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₃

commutator, socle

C₂³×C₄₃

C₂³×C₄

Fitting

C₄₃×#1843

C₂

prime radical

#19137876

C₂

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ #1843

N	H ≅ G/N	type
1. C₂×#14582976	C₂	semidirect
3. #19135392	C₂	semidirect
4. #19134282	C₂	semidirect
5. C₄₃×#1843	C₂	semidirect
27. C₄×C₄₃⋊C₈	D₄	semidirect
77. #1843	D₄₃	semidirect
92. C₄₃	#21684	semidirect

H ≅ G/N

type

1. C₂×#14582976

C₂

semidirect

3. #19135392

C₂

semidirect

4. #19134282

C₂

semidirect

5. C₄₃×#1843

C₂

semidirect

27. C₄×C₄₃⋊C₈

D₄

semidirect

77. #1843

D₄₃

semidirect

92. C₄₃

#21684

semidirect

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

subgroup	index
C₂×#14582976	2
#19137876	2
#19135392	2
#19134282	2
C₄₃×#1843	2
#21684	43

subgroup

index

C₄₃×#1843

Subgroups — #25104999

Conjugacy classes of subgroups

Complements

Maximal subgroups