Group #25013013 — subgroups

Subgroups — #25013013

Order 11008.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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×#14582719	C₂	1	4	yes	maximal
2	#19136351	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	C₄₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063265	C₂²	1	2	yes	—
6	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₄₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
9	C₄²×Dic₄₃	C₄	1	1	yes	characteristic
10	#14582719	—	2	8	no	—
11	C₄⋊(C₄₃⋊C₁₆)	—	2	2	no	—
12	C₂×M₃₂×C₄₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₄₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₄₃	—	2	2	no	—
17	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
18	#13063265	—	2	4	no	—
19	C₂×C₄₃⋊C₁₆	—	2	2	no	—
20	C₄×C₄₃⋊C₈	C₈	1	2	yes	—
21	C₄×C₄₃⋊C₈	C₈	1	2	yes	—
22	C₈×Dic₄₃	—	2	4	no	—
23	C₈×Dic₄₃	D₄	1	4	yes	—
24	C₈×Dic₄₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₄₃	—	2	4	no	—
29	C₂×C₁₆×C₄₃	—	2	2	no	—
30	C₄×C₈×C₄₃	D₄	1	2	yes	—
31	C₄×C₈×C₄₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
33	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₄₃	—	2	2	no	—
35	C₂×C₈×C₄₃	—	2	2	no	—
36	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
41	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
44	C₄×Dic₄₃	—	2	8	no	—
45	C₄×Dic₄₃	—	2	2	no	—
46	C₄×Dic₄₃	C₄⋊C₄	1	4	yes	—
47	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₄₃⋊C₈	C₂×C₈	1	4	yes	—
49	C₂×C₄₃⋊C₈	M₁₆	1	4	yes	—
50	C₁₆×C₄₃	—	4	2	no	—
51	C₄₃⋊C₁₆	—	4	2	no	—
52	C₂³×C₄₃	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₄₃	—	2	4	no	—
54	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₄₃	—	2	8	no	—
61	C₂×Dic₄₃	—	2	4	no	—
62	C₈×C₄₃	—	2	2	no	—
63	C₈×C₄₃	#127	1	1	yes	characteristic
64	C₈×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₄₃	C₄⋊Q₈	1	2	yes	—
66	C₄₃⋊C₈	C₄⋊C₈	1	8	yes	—
67	#12900	—	43	1	no	maximal, Sylow
68	C₂²×C₄₃	—	2	2	no	—
69	C₂²×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₄₃	#423	1	2	yes	—
71	C₄×C₄₃	—	2	2	no	—
72	C₄×C₄₃	#364	1	4	yes	complemented
73	C₄×C₄₃	#384	1	1	yes	characteristic
74	C₄×C₄₃	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₄₃	#423	1	1	yes	characteristic
76	Dic₄₃	—	2	8	no	—
77	C₂×#364	—	43	4	no	—
78	#2150	—	43	1	no	—
79	#2150	D₄₃	1	1	yes	characteristic
80	C₄²×C₈	—	43	1	no	—
81	C₂×C₄₃	—	2	2	no	—
82	C₂×C₄₃	#1842	1	1	yes	characteristic
83	C₂×C₄₃	C₂×#364	1	2	yes	—
84	C₂×M₃₂	—	43	2	no	—
85	C₂×M₃₂	C₂×D₄₃	1	2	yes	—
86	C₂×C₄×C₈	—	43	2	no	—
87	C₂×C₄×C₈	—	43	1	no	—
88	C₂×C₄×C₈	—	43	2	no	—
89	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
90	C₄³	—	43	1	no	—
91	#364	—	86	8	no	—
92	C₄⋊C₁₆	—	2	2	no	—
93	C₄⋊C₁₆	—	86	2	no	—
94	C₄₃	#12900	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	43	2	no	—
96	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
98	C₂×C₄²	—	43	2	no	—
99	C₂×C₄²	—	86	2	no	—
100	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	86	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	86	2	no	—
105	C₄×C₈	—	43	4	no	—
106	C₄×C₈	—	43	4	no	—
107	C₄×C₈	—	43	2	no	—
108	C₄×C₈	—	43	2	no	—
109	C₄×C₈	—	86	4	no	—
110	C₄×C₈	Dic₈₆	1	2	yes	—
111	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
112	C₂²×C₄	—	86	2	no	—
113	C₂²×C₄	#10621872	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	43	4	no	—
117	C₂×C₈	—	43	4	no	—
118	C₂×C₈	—	43	4	no	—
119	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
120	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
121	C₂×C₈	D₄×D₄₃	1	2	yes	—
122	C₂×C₈	Q₈×D₄₃	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
124	C₄²	—	43	4	no	—
125	C₄²	—	86	8	no	—
126	C₄²	—	86	2	no	—
127	C₄²	C₈×D₄₃	1	1	yes	characteristic
128	C₄²	C₈×D₄₃	1	1	yes	characteristic
129	C₄²	Dic₄₃⋊C₄	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	172	2	no	—
132	C₂³	C₂×#10621872	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	86	8	no	—
135	C₂×C₄	—	86	4	no	—
136	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
137	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
139	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	43	8	no	—
144	C₈	#13063321	1	1	yes	characteristic
145	C₈	#13063323	1	1	yes	characteristic
146	C₈	#13063324	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×Dic₄₃⋊C₈	1	1	yes	characteristic
149	C₂²	#14582921	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	86	8	no	—
152	C₄	#14582719	1	4	yes	complemented
153	C₄	#14582768	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₄₃	1	2	yes	—
155	C₄	#14582922	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#19134746	1	1	yes	characteristic
158	C₂	C₂×#14582719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582719

C₂

yes

maximal

#19136351

C₂

yes

maximal, characteristic

C₄×C₈×Dic₄₃

C₂

yes

maximal, characteristic

C₄₃×#2150

C₂

yes

maximal, characteristic

C₂×#13063265

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₄

yes

characteristic

C₂×C₈×Dic₄₃

C₄

yes

—

C₂×C₈×Dic₄₃

C₂²

yes

—

C₄²×Dic₄₃

C₄

yes

characteristic

#14582719

—

C₄⋊(C₄₃⋊C₁₆)

—

C₂×M₃₂×C₄₃

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₂×C₄

yes

—

#13063265

—

C₂×C₄₃⋊C₁₆

—

C₄×C₄₃⋊C₈

C₈

yes

—

C₄×C₄₃⋊C₈

C₈

yes

—

C₈×Dic₄₃

—

C₈×Dic₄₃

D₄

yes

—

C₈×Dic₄₃

Q₈

yes

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

M₃₂×C₄₃

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

M₁₆

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₄²×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

C₂×C₈

yes

characteristic

C₄²×C₄₃

C₂×C₈

yes

characteristic

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₄⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₃⋊C₈

C₂×C₈

yes

—

C₂×C₄₃⋊C₈

M₁₆

yes

—

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#127

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

characteristic

C₈×C₄₃

C₄⋊Q₈

yes

—

C₄₃⋊C₈

C₄⋊C₈

yes

—

#12900

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₃

#423

yes

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×C₄⋊C₈

yes

—

C₄×C₄₃

#423

yes

characteristic

Dic₄₃

—

C₂×#364

—

#2150

—

#2150

D₄₃

yes

characteristic

C₄²×C₈

—

C₂×C₄₃

—

C₂×C₄₃

#1842

yes

characteristic

C₂×C₄₃

C₂×#364

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₄₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄³

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄₃

#12900

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₃

yes

characteristic

C₂²×C₈

C₂²×D₄₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₄×D₄₃

yes

characteristic

101

M₃₂

—

102

M₃₂

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

Dic₈₆

yes

—

111

C₄×C₈

C₂²⋊D₄₃

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

#10621872

yes

—

114

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

Dic₄₃⋊C₄

yes

—

120

C₂×C₈

C₂×Dic₈₆

yes

characteristic

121

C₂×C₈

D₄×D₄₃

yes

—

122

C₂×C₈

Q₈×D₄₃

yes

—

123

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

C₈×D₄₃

yes

characteristic

128

C₄²

C₈×D₄₃

yes

characteristic

129

C₄²

Dic₄₃⋊C₄

yes

—

130

C₁₆

—

131

C₁₆

—

172

—

132

C₂³

C₂×#10621872

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

Dic₄₃⋊C₈

yes

—

137

C₂×C₄

Dic₄₃⋊C₈

yes

—

138

C₂×C₄

C₄⋊C₄×D₄₃

yes

—

139

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

140

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

141

C₂×C₄

M₁₆×D₄₃

yes

—

142

C₈

—

143

C₈

—

144

C₈

#13063321

yes

characteristic

145

C₈

#13063323

yes

characteristic

146

C₈

#13063324

yes

—

147

C₂²

—

148

C₂²

C₂×Dic₄₃⋊C₈

yes

characteristic

149

C₂²

#14582921

yes

—

150

C₄

—

151

C₄

—

152

C₄

#14582719

yes

complemented

153

C₄

#14582768

yes

characteristic

154

C₄

C₄⋊C₈×D₄₃

yes

—

155

C₄

#14582922

yes

characteristic

156

C₂

—

157

C₂

#19134746

yes

characteristic

158

C₂

C₂×#14582719

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2150	C₂
socle	C₂²×C₄₃	C₂×C₄⋊C₈
prime radical	C₂³×C₄₃	C₂×M₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2150	D₄₃
hypocenter	C₄₃	#12900
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₄₃

commutator

C₂×C₄×C₄₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#2150

C₂

socle

C₂²×C₄₃

C₂×C₄⋊C₈

prime radical

C₂³×C₄₃

C₂×M₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
72. C₄×C₄₃	#364	semidirect
94. C₄₃	#12900	semidirect
152. C₄	#14582719	semidirect

H ≅ G/N

type

72. C₄×C₄₃

#364

semidirect

94. C₄₃

#12900

semidirect

152. C₄

#14582719

semidirect

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

subgroup	index
C₂×#14582719	2
#19136351	2
C₄×C₈×Dic₄₃	2
C₄₃×#2150	2
#12900	43

subgroup

index

C₂×#14582719

#19136351

C₄×C₈×Dic₄₃

C₄₃×#2150

#12900

Subgroups — #25013013

Conjugacy classes of subgroups

Complements

Maximal subgroups