Group #25027740 — subgroups

Subgroups — #25027740

Order 11008.

1182 conjugacy classes of subgroups; only the 371 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 371 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×#14582769	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582767	C₂	1	1	yes	maximal, characteristic
3	#19132847	C₂	1	12	yes	maximal
4	C₂×C₄₃×#383	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₄₃	C₂²	1	6	yes	—
6	C₂²×Dic₄₃⋊C₄	C₂²	1	4	yes	—
7	#14582769	C₄	1	8	yes	complemented
8	#14582767	—	2	4	no	—
9	C₂×C₄₃⋊#95	C₂²	1	6	yes	—
10	C₂×C₄₃⋊#95	C₂²	1	12	yes	—
11	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#383	—	2	4	no	—
13	C₂×#95×C₄₃	C₂²	1	6	yes	—
14	C₂³×Dic₄₃	C₂³	1	4	yes	—
15	C₂³×Dic₄₃	C₂³	1	4	yes	—
16	C₂×C₄⋊Dic₄₃	—	4	6	no	—
17	C₂×C₄⋊Dic₄₃	—	4	6	no	—
18	C₂×C₄⋊Dic₄₃	C₂×C₄	1	24	yes	—
19	C₂×Dic₄₃⋊C₄	—	4	12	no	—
20	C₄₃⋊#95	—	2	12	no	—
21	C₄₃⋊#95	—	4	12	no	—
22	C₂³×C₄×C₄₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₄₃	C₂³	1	3	yes	—
24	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
25	C₂×C₄²×C₄₃	D₄	1	4	yes	—
26	C₂×C₄²×C₄₃	Q₈	1	4	yes	—
27	#95×C₄₃	—	2	12	no	—
28	C₂⁴×C₄₃	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	—	2	4	no	—
30	C₂²×C₄×C₄₃	—	2	6	no	—
31	C₂²×C₄×C₄₃	—	2	12	no	—
32	C₂²×C₄×C₄₃	—	2	6	no	—
33	C₂²×C₄×C₄₃	—	2	12	no	—
34	C₂²×C₄×C₄₃	C₂²×C₄	1	6	yes	—
35	C₂²×C₄×C₄₃	C₂×D₄	1	6	yes	—
36	C₂²×C₄×C₄₃	C₂×Q₈	1	6	yes	—
37	C₂²×Dic₄₃	—	2	4	no	—
38	C₂²×Dic₄₃	—	2	12	no	—
39	C₂²×Dic₄₃	—	2	12	no	—
40	C₂²×Dic₄₃	—	2	12	no	—
41	C₂²×Dic₄₃	—	2	12	no	—
42	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
43	C₄²×C₄₃	C₄⋊C₄	1	16	yes	complemented
44	C₄⋊Dic₄₃	—	4	24	no	—
45	Dic₄₃⋊C₄	—	4	16	no	—
46	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
47	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
48	C₂³×C₄₃	C₂²×Q₈	1	1	yes	characteristic
49	C₂³×C₄₃	H(5,𝔽₂)	1	6	yes	—
50	C₂³×C₄₃	2₋¹⁺⁴	1	6	yes	—
51	C₂×C₄×C₄₃	—	2	6	no	—
52	C₂×C₄×C₄₃	—	2	6	no	—
53	C₂×C₄×C₄₃	—	2	12	no	—
54	C₂×C₄×C₄₃	—	2	12	no	—
55	C₂×C₄×C₄₃	—	4	12	no	—
56	C₂×C₄×C₄₃	—	4	4	no	—
57	C₂×C₄×C₄₃	—	4	12	no	—
58	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	12	yes	—
59	C₂×Dic₄₃	—	2	24	no	—
60	C₂×Dic₄₃	—	4	4	no	—
61	C₂×Dic₄₃	—	4	12	no	—
62	C₂×Dic₄₃	—	4	12	no	—
63	C₂×Dic₄₃	—	4	4	no	—
64	C₂×Dic₄₃	—	4	12	no	—
65	#14676	—	43	1	no	maximal, Sylow
66	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₃	#518	1	3	yes	—
68	C₂²×C₄₃	#519	1	3	yes	—
69	C₂²×C₄₃	#537	1	6	yes	—
70	C₂²×C₄₃	#544	1	6	yes	—
71	C₂²×C₄₃	#560	1	4	yes	—
72	C₂²×C₄₃	#563	1	12	yes	—
73	C₄×C₄₃	—	2	12	no	—
74	C₄×C₄₃	—	8	4	no	—
75	Dic₄₃	—	4	8	no	—
76	Dic₄₃	—	8	4	no	—
77	C₂×#384	—	43	1	no	—
78	C₂×#383	—	43	1	no	—
79	C₂×#383	D₄₃	1	1	yes	characteristic
80	#1441	—	43	12	no	—
81	C₂×C₄₃	#2316	1	3	yes	—
82	C₂×C₄₃	#2380	1	1	yes	characteristic
83	C₂×C₄₃	#2381	1	3	yes	—
84	C₂×C₄₃	#2823	1	4	yes	—
85	C₂×C₄₃	#2843	1	4	yes	—
86	C₂²×C₄⋊C₄	—	43	6	no	—
87	C₂²×C₄⋊C₄	—	43	4	no	—
88	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
89	#384	—	43	8	no	—
90	#383	—	2	4	no	—
91	#383	—	86	4	no	—
92	C₂×#95	—	43	6	no	—
93	C₂×#95	—	43	12	no	—
94	C₂×#95	C₂×D₄₃	1	6	yes	—
95	C₄₃	#14676	1	1	yes	characteristic, complemented, Sylow
96	C₂³×C₄	—	43	4	no	—
97	C₂³×C₄	—	43	4	no	—
98	C₂³×C₄	C₂²×D₄₃	1	4	yes	—
99	C₂³×C₄	C₂²×D₄₃	1	3	yes	—
100	C₂×C₄⋊C₄	—	43	24	no	—
101	C₂×C₄⋊C₄	—	172	12	no	—
102	C₂×C₄⋊C₄	—	172	6	no	—
103	C₂×C₄⋊C₄	—	172	6	no	—
104	C₂×C₄²	Dic₈₆	1	4	yes	—
105	C₂×C₄²	C₄×D₄₃	1	4	yes	—
106	C₂×C₄²	C₂²⋊D₄₃	1	4	yes	—
107	#95	—	2	12	no	—
108	#95	—	86	12	no	—
109	#95	—	172	12	no	—
110	C₂⁴	C₂³×D₄₃	1	1	yes	characteristic
111	C₂²×C₄	—	2	6	no	—
112	C₂²×C₄	—	2	6	no	—
113	C₂²×C₄	—	2	4	no	—
114	C₂²×C₄	—	2	12	no	—
115	C₂²×C₄	—	2	12	no	—
116	C₂²×C₄	—	43	8	no	—
117	C₂²×C₄	—	86	12	no	—
118	C₂²×C₄	—	86	12	no	—
119	C₂²×C₄	—	86	4	no	—
120	C₂²×C₄	—	86	12	no	—
121	C₂²×C₄	—	86	12	no	—
122	C₂²×C₄	C₂×Dic₈₆	1	6	yes	—
123	C₂²×C₄	C₂×C₄×D₄₃	1	6	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
125	C₄⋊C₄	—	172	24	no	—
126	C₄⋊C₄	—	172	16	no	—
127	C₄²	Dic₄₃⋊C₄	1	16	yes	complemented
128	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
129	C₂³	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
130	C₂³	C₂²×Dic₈₆	1	1	yes	characteristic
131	C₂³	#13063371	1	6	yes	—
132	C₂³	#13063372	1	6	yes	—
133	C₂×C₄	—	2	6	no	—
134	C₂×C₄	—	2	6	no	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	—	2	12	no	—
137	C₂×C₄	—	4	4	no	—
138	C₂×C₄	—	4	12	no	—
139	C₂×C₄	—	4	12	no	—
140	C₂×C₄	—	86	24	no	—
141	C₂×C₄	—	172	4	no	—
142	C₂×C₄	—	172	12	no	—
143	C₂×C₄	—	172	12	no	—
144	C₂×C₄	—	172	4	no	—
145	C₂×C₄	—	172	12	no	—
146	C₂×C₄	C₂×Dic₄₃⋊C₄	1	12	yes	—
147	C₂²	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
148	C₂²	#14583373	1	3	yes	—
149	C₂²	#14583376	1	3	yes	—
150	C₂²	#14583477	1	6	yes	—
151	C₂²	#14583527	1	6	yes	—
152	C₂²	#14583640	1	4	yes	—
153	C₂²	#14583651	1	12	yes	—
154	C₄	—	2	12	no	—
155	C₄	—	8	4	no	—
156	C₄	—	344	4	no	—
157	C₄	—	172	8	no	—
158	C₂	#19137211	1	3	yes	—
159	C₂	#19137771	1	1	yes	characteristic
160	C₂	#19137775	1	3	yes	—
161	C₂	#19142667	1	4	yes	—
162	C₂	#19142862	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582769

C₂

yes

maximal, characteristic

C₂×#14582767

C₂

yes

maximal, characteristic

#19132847

C₂

yes

maximal

C₂×C₄₃×#383

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₄₃

C₂²

yes

—

C₂²×Dic₄₃⋊C₄

C₂²

yes

—

#14582769

C₄

yes

complemented

#14582767

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂×C₄₃⋊#95

C₂²

yes

—

C₂²×C₄²×C₄₃

C₂²

yes

characteristic

C₄₃×#383

—

C₂×#95×C₄₃

C₂²

yes

—

C₂³×Dic₄₃

C₂³

yes

—

C₂³×Dic₄₃

C₂³

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

C₂×C₄

yes

—

C₂×Dic₄₃⋊C₄

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₂³×C₄×C₄₃

C₂³

yes

—

C₂³×C₄×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

Q₈

yes

—

#95×C₄₃

—

C₂⁴×C₄₃

C₂⁴

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×Q₈

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₄⋊C₄

yes

complemented

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

C₂³×C₄₃

C₂²×D₄

yes

characteristic

C₂³×C₄₃

C₂²×Q₈

yes

characteristic

C₂³×C₄₃

H(5,𝔽₂)

yes

—

C₂³×C₄₃

2₋¹⁺⁴

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

#14676

—

maximal, Sylow

C₂²×C₄₃

C₂²×C₄⋊C₄

yes

characteristic

C₂²×C₄₃

#518

yes

—

C₂²×C₄₃

#519

yes

—

C₂²×C₄₃

#537

yes

—

C₂²×C₄₃

#544

yes

—

C₂²×C₄₃

#560

yes

—

C₂²×C₄₃

#563

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

Dic₄₃

—

C₂×#384

—

C₂×#383

—

C₂×#383

D₄₃

yes

characteristic

#1441

—

C₂×C₄₃

#2316

yes

—

C₂×C₄₃

#2380

yes

characteristic

C₂×C₄₃

#2381

yes

—

C₂×C₄₃

#2823

yes

—

C₂×C₄₃

#2843

yes

—

C₂²×C₄⋊C₄

—

C₂²×C₄⋊C₄

—

C₂²×C₄²

C₂×D₄₃

yes

—

—

—

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₄₃

yes

—

C₄₃

#14676

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₄₃

yes

—

C₂³×C₄

C₂²×D₄₃

yes

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

172

—

102

C₂×C₄⋊C₄

—

172

—

103

C₂×C₄⋊C₄

—

172

—

104

C₂×C₄²

Dic₈₆

yes

—

105

C₂×C₄²

C₄×D₄₃

yes

—

106

C₂×C₄²

C₂²⋊D₄₃

yes

—

107

#95

—

108

#95

—

109

#95

—

172

—

110

C₂⁴

C₂³×D₄₃

yes

characteristic

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂×Dic₈₆

yes

—

123

C₂²×C₄

C₂×C₄×D₄₃

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

125

C₄⋊C₄

—

172

—

126

C₄⋊C₄

—

172

—

127

C₄²

Dic₄₃⋊C₄

yes

complemented

128

C₂³

C₂²×C₄×D₄₃

yes

characteristic

129

C₂³

C₂²×C₂²⋊D₄₃

yes

characteristic

130

C₂³

C₂²×Dic₈₆

yes

characteristic

131

C₂³

#13063371

yes

—

132

C₂³

#13063372

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

172

—

142

C₂×C₄

—

172

—

143

C₂×C₄

—

172

—

144

C₂×C₄

—

172

—

145

C₂×C₄

—

172

—

146

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

—

147

C₂²

C₂²×Dic₄₃⋊C₄

yes

characteristic

148

C₂²

#14583373

yes

—

149

C₂²

#14583376

yes

—

150

C₂²

#14583477

yes

—

151

C₂²

#14583527

yes

—

152

C₂²

#14583640

yes

—

153

C₂²

#14583651

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

344

—

157

C₄

—

172

—

158

C₂

#19137211

yes

—

159

C₂

#19137771

yes

characteristic

160

C₂

#19137775

yes

—

161

C₂

#19142667

yes

—

162

C₂

#19142862

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₄₃
commutator	C₂³×C₄₃	C₂³×C₄
Fitting	C₂×C₄₃×#383	C₂
socle, prime radical, cosocle	C₂⁴×C₄₃	C₂⁴
hypercenter	C₂×#383	D₄₃
hypocenter	C₄₃	#14676
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#383

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₄₃

commutator

C₂³×C₄₃

C₂³×C₄

Fitting

C₂×C₄₃×#383

C₂

socle, prime radical, cosocle

C₂⁴×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂⁴ ◁ C₂×#383

N	H ≅ G/N	type
7. #14582769	C₄	semidirect
43. C₄²×C₄₃	C₄⋊C₄	semidirect
95. C₄₃	#14676	semidirect
127. C₄²	Dic₄₃⋊C₄	semidirect

H ≅ G/N

type

7. #14582769

C₄

semidirect

43. C₄²×C₄₃

C₄⋊C₄

semidirect

95. C₄₃

#14676

semidirect

127. C₄²

Dic₄₃⋊C₄

semidirect

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

subgroup	index
C₂×#14582769	2
C₂×#14582767	2
#19132847	2
C₂×C₄₃×#383	2
#14676	43

subgroup

index

C₂×C₄₃×#383

Subgroups — #25027740

Conjugacy classes of subgroups

Complements

Maximal subgroups