Group #25005280 — subgroups

Subgroups — #25005280

Order 11008.

686 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 235 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	#19143185	C₂	1	3	yes	maximal
2	#19134293	C₂	1	3	yes	maximal
3	C₂×C₄₃×#338	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
5	C₂×#13063284	C₂²	1	3	yes	—
6	#14582941	—	2	12	no	—
7	#14582937	—	2	6	no	—
8	#14582653	—	2	6	no	—
9	C₂²×M₁₆×C₄₃	C₂²	1	3	yes	—
10	C₄₃×#338	C₄	1	4	yes	complemented
11	C₂²×C₄₃⋊C₈	—	2	3	no	—
12	C₂²×C₄₃⋊C₈	—	2	3	no	—
13	C₂²×C₄₃⋊C₈	Q₈	1	2	yes	—
14	#13063284	—	2	6	no	—
15	#13063284	—	2	6	no	—
16	C₂×C₄⋊Dic₄₃	D₄	1	6	yes	—
17	C₂×C₂²⋊Dic₄₃	—	2	3	no	—
18	C₂×C₄×Dic₄₃	—	2	3	no	—
19	#13063259	—	2	12	no	—
20	#13063257	—	2	12	no	—
21	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₄₃	—	2	6	no	—
23	C₂×M₁₆×C₄₃	—	2	6	no	—
24	C₂×M₁₆×C₄₃	C₂×C₄	1	6	yes	—
25	C₂×M₁₆×C₄₃	Q₈	1	6	yes	—
26	C₂²×C₈×C₄₃	D₄	1	6	yes	—
27	C₂⁴×C₄₃	#41	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
30	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
31	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	C₂×Q₈	1	3	yes	—
33	C₂²×C₄×C₄₃	#41	1	3	yes	—
34	C₂²×Dic₄₃	—	2	6	no	—
35	M₁₆×C₄₃	—	2	12	no	—
36	M₁₆×C₄₃	—	2	12	no	—
37	C₂×C₈×C₄₃	—	2	6	no	—
38	C₂×C₈×C₄₃	—	2	6	no	—
39	C₂×C₈×C₄₃	C₄⋊C₄	1	12	yes	—
40	C₂²⋊Dic₄₃	—	4	6	no	—
41	C₄⋊Dic₄₃	—	2	12	no	—
42	C₄×Dic₄₃	—	4	3	no	—
43	C₄×Dic₄₃	—	4	3	no	—
44	C₂×C₄₃⋊C₈	—	2	6	no	—
45	C₂×C₄₃⋊C₈	—	4	3	no	—
46	C₂×C₄₃⋊C₈	—	4	1	no	—
47	C₂³×C₄₃	—	4	2	no	—
48	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₄₃	#121	1	3	yes	—
50	C₂³×C₄₃	#125	1	3	yes	—
51	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	3	yes	—
52	C₂×C₄×C₄₃	C₄×D₄	1	3	yes	—
53	C₂×C₄×C₄₃	#120	1	3	yes	—
54	C₂×C₄×C₄₃	C₂²⋊Q₈	1	6	yes	—
55	C₂×C₄×C₄₃	C₂²⋊Q₈	1	3	yes	—
56	C₂×C₄×C₄₃	#123	1	3	yes	—
57	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₄⋊Q₈	1	3	yes	—
59	C₂×C₄×C₄₃	C₄⋊Q₈	1	3	yes	—
60	C₂×Dic₄₃	—	4	12	no	—
61	C₂×Dic₄₃	—	4	6	no	—
62	C₈×C₄₃	—	2	12	no	—
63	C₄₃⋊C₈	—	4	2	no	—
64	#11337	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	—	2	6	no	—
66	C₂²×C₄₃	—	4	1	no	—
67	C₂²×C₄₃	—	4	3	no	—
68	C₂²×C₄₃	#389	1	3	yes	—
69	C₂²×C₄₃	#392	1	1	yes	characteristic
70	C₂²×C₄₃	#398	1	3	yes	—
71	C₄×C₄₃	#384	1	1	yes	characteristic
72	C₄×C₄₃	#385	1	3	yes	—
73	C₄×C₄₃	#395	1	3	yes	—
74	C₄×C₄₃	#399	1	1	yes	characteristic
75	Dic₄₃	—	8	6	no	—
76	#2911	—	43	3	no	—
77	C₂×#338	D₄₃	1	1	yes	characteristic
78	#1729	—	43	3	no	—
79	C₂×C₄₃	—	2	3	no	—
80	C₂×C₄₃	—	2	3	no	—
81	C₂×C₄₃	#1444	1	1	yes	characteristic
82	C₂×C₄₃	#2017	1	2	yes	—
83	C₂²×M₁₆	C₂×D₄₃	1	3	yes	—
84	C₂³×C₈	—	43	1	no	—
85	C₂×#117	—	43	3	no	—
86	#428	—	86	12	no	—
87	#427	—	86	6	no	—
88	#338	Dic₄₃	1	4	yes	complemented
89	#336	—	86	6	no	—
90	C₄₃	#11337	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
92	C₂×M₁₆	—	2	6	no	—
93	C₂×M₁₆	—	2	6	no	—
94	C₂×M₁₆	Dic₈₆	1	6	yes	—
95	C₂×M₁₆	C₂×Dic₄₃	1	6	yes	—
96	C₂²×C₈	—	43	2	no	—
97	C₂²×C₈	—	86	3	no	—
98	C₂²×C₈	—	86	3	no	—
99	C₂²×C₈	D₁₇₂	1	6	yes	—
100	#117	—	86	6	no	—
101	#117	—	86	6	no	—
102	C₂×C₄⋊C₄	—	43	6	no	—
103	C₂×C₂²⋊C₄	—	86	3	no	—
104	C₂×C₄²	—	86	3	no	—
105	#107	—	86	12	no	—
106	#106	—	86	12	no	—
107	C₂⁴	#10621902	1	1	yes	characteristic
108	C₂²×C₄	—	86	6	no	—
109	C₂²×C₄	C₂×Dic₈₆	1	3	yes	—
110	C₂²×C₄	C₂×D₁₇₂	1	3	yes	—
111	C₂²×C₄	D₄×D₄₃	1	3	yes	—
112	C₂²×C₄	#10621900	1	3	yes	—
113	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
114	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
115	M₁₆	—	2	12	no	—
116	M₁₆	—	2	12	no	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	86	6	no	—
120	C₂×C₈	—	172	3	no	—
121	C₂×C₈	—	172	1	no	—
122	C₂×C₈	C₄⋊Dic₄₃	1	12	yes	—
123	C₄⋊C₄	—	86	12	no	—
124	C₂²⋊C₄	—	172	6	no	—
125	C₄²	—	172	3	no	—
126	C₄²	—	172	3	no	—
127	C₂³	—	4	2	no	—
128	C₂³	Q₈×Dic₄₃	1	1	yes	characteristic
129	C₂³	#13063300	1	3	yes	—
130	C₂³	#13063316	1	3	yes	—
131	C₂×C₄	—	172	12	no	—
132	C₂×C₄	—	172	6	no	—
133	C₂×C₄	C₂×C₄⋊Dic₄₃	1	3	yes	—
134	C₂×C₄	D₄×Dic₄₃	1	3	yes	—
135	C₂×C₄	#13063295	1	3	yes	—
136	C₂×C₄	#13063302	1	6	yes	—
137	C₂×C₄	#13063305	1	3	yes	—
138	C₂×C₄	#13063309	1	3	yes	—
139	C₂×C₄	#13063322	1	1	yes	characteristic
140	C₂×C₄	#13063324	1	3	yes	—
141	C₂×C₄	#13063325	1	3	yes	—
142	C₈	—	2	12	no	—
143	C₈	—	172	2	no	—
144	C₂²	—	2	6	no	—
145	C₂²	—	4	1	no	—
146	C₂²	—	4	3	no	—
147	C₂²	#14582798	1	3	yes	—
148	C₂²	#14582809	1	1	yes	characteristic
149	C₂²	#14582831	1	3	yes	—
150	C₄	—	344	6	no	—
151	C₄	#14582769	1	1	yes	characteristic
152	C₄	#14582774	1	3	yes	—
153	C₄	#14582820	1	3	yes	—
154	C₄	#14582836	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	3	no	—
157	C₂	#19132860	1	1	yes	characteristic
158	C₂	#19135700	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19143185

C₂

yes

maximal

#19134293

C₂

yes

maximal

C₂×C₄₃×#338

C₂

yes

maximal, characteristic

C₂³×C₄₃⋊C₈

C₂²

yes

characteristic

C₂×#13063284

C₂²

yes

—

#14582941

—

#14582937

—

#14582653

—

C₂²×M₁₆×C₄₃

C₂²

yes

—

C₄₃×#338

C₄

yes

complemented

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

Q₈

yes

—

#13063284

—

#13063284

—

C₂×C₄⋊Dic₄₃

D₄

yes

—

C₂×C₂²⋊Dic₄₃

—

C₂×C₄×Dic₄₃

—

#13063259

—

#13063257

—

C₂³×C₄×C₄₃

C₂³

yes

characteristic

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

—

C₂×M₁₆×C₄₃

C₂×C₄

yes

—

C₂×M₁₆×C₄₃

Q₈

yes

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂⁴×C₄₃

#41

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

—

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

C₄×Q₈

yes

characteristic

C₂³×C₄₃

#121

yes

—

C₂³×C₄₃

#125

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

#120

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#123

yes

—

C₂×C₄×C₄₃

#127

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×C₄×C₄₃

C₄⋊Q₈

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#11337

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#389

yes

—

C₂²×C₄₃

#392

yes

characteristic

C₂²×C₄₃

#398

yes

—

C₄×C₄₃

#384

yes

characteristic

C₄×C₄₃

#385

yes

—

C₄×C₄₃

#395

yes

—

C₄×C₄₃

#399

yes

characteristic

Dic₄₃

—

#2911

—

C₂×#338

D₄₃

yes

characteristic

#1729

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1444

yes

characteristic

C₂×C₄₃

#2017

yes

—

C₂²×M₁₆

C₂×D₄₃

yes

—

C₂³×C₈

—

—

—

—

Dic₄₃

yes

complemented

#336

—

C₄₃

#11337

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂²×D₄₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₈₆

yes

—

C₂×M₁₆

C₂×Dic₄₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₁₇₂

yes

—

100

#117

—

101

#117

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

—

105

#107

—

106

#106

—

107

C₂⁴

#10621902

yes

characteristic

108

C₂²×C₄

—

109

C₂²×C₄

C₂×Dic₈₆

yes

—

110

C₂²×C₄

C₂×D₁₇₂

yes

—

111

C₂²×C₄

D₄×D₄₃

yes

—

112

C₂²×C₄

#10621900

yes

—

113

C₂²×C₄

Q₈×D₄₃

yes

characteristic

114

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

172

—

121

C₂×C₈

—

172

—

122

C₂×C₈

C₄⋊Dic₄₃

yes

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

172

—

125

C₄²

—

172

—

126

C₄²

—

172

—

127

C₂³

—

128

C₂³

Q₈×Dic₄₃

yes

characteristic

129

C₂³

#13063300

yes

—

130

C₂³

#13063316

yes

—

131

C₂×C₄

—

172

—

132

C₂×C₄

—

172

—

133

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

—

134

C₂×C₄

D₄×Dic₄₃

yes

—

135

C₂×C₄

#13063295

yes

—

136

C₂×C₄

#13063302

yes

—

137

C₂×C₄

#13063305

yes

—

138

C₂×C₄

#13063309

yes

—

139

C₂×C₄

#13063322

yes

characteristic

140

C₂×C₄

#13063324

yes

—

141

C₂×C₄

#13063325

yes

—

142

C₈

—

143

C₈

—

172

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#14582798

yes

—

148

C₂²

#14582809

yes

characteristic

149

C₂²

#14582831

yes

—

150

C₄

—

344

—

151

C₄

#14582769

yes

characteristic

152

C₄

#14582774

yes

—

153

C₄

#14582820

yes

—

154

C₄

#14582836

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#19132860

yes

characteristic

158

C₂

#19135700

yes

—

	subgroup	G/H
center	C₂×C₄	#13063322
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#338	C₂
socle	C₂²×C₄₃	#392
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#338	D₄₃
hypocenter	C₄₃	#11337
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#13063322

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₃

Fitting

C₂×C₄₃×#338

C₂

C₂²×C₄₃

C₂⁴×C₄₃

C₂³×C₄×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. C₄₃×#338	C₄	semidirect
88. #338	Dic₄₃	semidirect
90. C₄₃	#11337	semidirect

H ≅ G/N

type

10. C₄₃×#338

C₄

semidirect

88. #338

Dic₄₃

semidirect

90. C₄₃

#11337

semidirect

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

subgroup	index
#19143185	2
#19134293	2
C₂×C₄₃×#338	2
#11337	43

subgroup

index

#19143185

#19134293

C₂×C₄₃×#338

#11337

Subgroups — #25005280

Conjugacy classes of subgroups

Complements

Maximal subgroups