Group #25010856 — subgroups

Subgroups — #25010856

Order 11008.

216 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₁₆×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	#19133962	C₂	1	2	yes	maximal
3	#19132667	C₂	1	1	yes	maximal, characteristic
4	#19132644	C₂	1	2	yes	maximal
5	C₄₃×#2247	C₂	1	1	yes	maximal, characteristic
6	#14583181	—	2	2	no	—
7	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
8	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₄₃⋊C₁₆	—	2	1	no	—
10	C₁₆×Dic₄₃	—	2	1	no	—
11	#14582652	C₂²	1	2	yes	—
12	#14582645	—	2	2	no	—
13	C₂×Q₃₂×C₄₃	—	2	1	no	—
14	C₂×Q₃₂×C₄₃	—	2	1	no	—
15	C₄₃×#494	C₂²	1	2	yes	—
16	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₈₆	D₄	1	1	yes	characteristic
18	C₂×C₄₃⋊C₁₆	—	2	1	no	—
19	#13063256	—	4	2	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₄₃⋊C₈	—	2	1	no	—
23	#13063248	—	4	2	no	—
24	C₈×Dic₄₃	—	2	1	no	—
25	C₂×Q₁₆×C₄₃	—	2	2	no	—
26	C₂×Q₁₆×C₄₃	—	2	2	no	—
27	C₄⋊Q₈×C₄₃	—	2	2	no	—
28	Q₃₂×C₄₃	—	4	2	no	—
29	Q₃₂×C₄₃	—	4	2	no	—
30	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₄₃	—	4	2	no	—
34	C₂×Dic₈₆	D₈	1	1	yes	characteristic
35	Q₁₆×C₄₃	—	4	2	no	—
36	Q₁₆×C₄₃	—	4	2	no	—
37	Q₁₆×C₄₃	—	8	2	no	—
38	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
40	C₄⋊C₄×C₄₃	—	4	2	no	—
41	C₄⋊C₄×C₄₃	—	4	2	no	—
42	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
44	Dic₄₃⋊C₄	—	2	1	no	—
45	C₄×Dic₄₃	—	2	1	no	—
46	C₂×C₄₃⋊C₈	—	2	1	no	—
47	C₂×C₄₃⋊C₈	—	4	2	no	—
48	C₁₆×C₄₃	—	2	1	no	—
49	C₁₆×C₄₃	SD₁₆	1	2	yes	—
50	C₄₃⋊C₁₆	—	4	1	no	—
51	Q₈×C₄₃	—	8	2	no	—
52	Q₈×C₄₃	—	8	2	no	—
53	C₂×C₄×C₄₃	—	8	2	no	—
54	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
57	C₂×Dic₄₃	—	2	1	no	—
58	C₂×Dic₄₃	—	2	1	no	—
59	Dic₈₆	—	2	1	no	—
60	Dic₈₆	Q₃₂	1	2	yes	complemented
61	C₈×C₄₃	—	2	1	no	—
62	C₈×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄₃⋊C₈	—	4	1	no	—
65	C₄₃⋊C₈	—	8	2	no	—
66	#12440	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	#466	1	1	yes	characteristic
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	—	16	2	no	—
70	C₄×C₄₃	#460	1	1	yes	characteristic
71	C₄×C₄₃	#472	1	1	yes	characteristic
72	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
73	C₄×C₄₃	#508	1	1	yes	characteristic
74	Dic₄₃	—	2	2	no	—
75	Dic₄₃	—	4	1	no	—
76	#2247	D₄₃	1	1	yes	characteristic
77	Q₈×C₁₆	—	43	1	no	—
78	#1670	—	43	2	no	—
79	#1371	—	43	1	no	—
80	#1362	—	43	2	no	—
81	C₂×C₄₃	#1668	1	1	yes	characteristic
82	C₂×C₄₃	#2216	1	1	yes	characteristic
83	C₂×C₄₃	#2222	1	1	yes	characteristic
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	—	2	1	no	—
86	#494	C₂×D₄₃	1	2	yes	—
87	#475	—	86	2	no	—
88	C₈×Q₈	—	43	1	no	—
89	C₄⋊C₁₆	—	43	1	no	—
90	C₄⋊C₁₆	—	86	1	no	—
91	C₄×C₁₆	—	86	1	no	—
92	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
93	#335	—	43	2	no	—
94	#332	—	86	2	no	—
95	C₄₃	#12440	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	—	2	2	no	—
97	C₂×Q₁₆	—	2	2	no	—
98	C₄⋊Q₈	—	2	2	no	—
99	C₄×Q₈	—	43	1	no	—
100	Q₃₂	—	4	2	no	—
101	Q₃₂	—	4	2	no	—
102	C₂×C₁₆	—	86	1	no	—
103	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
104	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
105	#106	—	172	2	no	—
106	C₄⋊C₈	—	43	1	no	—
107	C₄⋊C₈	—	86	1	no	—
108	C₄⋊C₈	—	86	2	no	—
109	#103	—	172	2	no	—
110	C₄×C₈	—	86	1	no	—
111	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	43	1	no	—
114	Q₁₆	—	4	2	no	—
115	Q₁₆	—	4	2	no	—
116	Q₁₆	—	8	2	no	—
117	C₂×C₈	—	86	1	no	—
118	C₂×C₈	—	172	2	no	—
119	C₂×C₈	#10621900	1	1	yes	characteristic
120	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	43	1	no	—
124	C₄⋊C₄	—	86	1	no	—
125	C₄²	—	86	1	no	—
126	C₄²	D₄×D₄₃	1	1	yes	characteristic
127	C₁₆	—	2	1	no	—
128	C₁₆	—	172	1	no	—
129	C₁₆	#10621883	1	2	yes	—
130	Q₈	—	8	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	43	2	no	—
133	Q₈	—	86	1	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	86	1	no	—
136	C₂×C₄	—	86	1	no	—
137	C₂×C₄	#13063296	1	1	yes	characteristic
138	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
139	C₂×C₄	#13063344	1	1	yes	characteristic
140	C₈	—	2	1	no	—
141	C₈	—	344	2	no	—
142	C₈	—	172	1	no	—
143	C₈	C₂×#10621883	1	1	yes	characteristic
144	C₈	#13063351	1	1	yes	characteristic
145	C₂²	#14583132	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	16	2	no	—
148	C₄	—	86	2	no	—
149	C₄	—	172	1	no	—
150	C₄	#14583096	1	1	yes	characteristic
151	C₄	#14583166	1	1	yes	characteristic
152	C₄	Q₃₂×D₄₃	1	1	yes	characteristic
153	C₄	#14583332	1	1	yes	characteristic
154	C₂	#19133952	1	1	yes	characteristic
155	C₂	#19136683	1	1	yes	characteristic
156	C₂	#19136717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₈₆

C₂

yes

maximal, characteristic

#19133962

C₂

yes

maximal

#19132667

C₂

yes

maximal, characteristic

#19132644

C₂

yes

maximal

C₄₃×#2247

C₂

yes

maximal, characteristic

#14583181

—

C₈×Dic₈₆

C₂²

yes

characteristic

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

—

C₁₆×Dic₄₃

—

#14582652

C₂²

yes

—

#14582645

—

C₂×Q₃₂×C₄₃

—

C₂×Q₃₂×C₄₃

—

C₄₃×#494

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₄×Dic₈₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

—

#13063256

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

D₄

yes

characteristic

Dic₄₃⋊C₈

—

#13063248

—

C₈×Dic₄₃

—

C₂×Q₁₆×C₄₃

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

—

Q₃₂×C₄₃

—

Q₃₂×C₄₃

—

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

D₈

yes

characteristic

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

D₈

yes

characteristic

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₁₆×C₄₃

SD₁₆

yes

—

C₄₃⋊C₁₆

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

#135

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

Q₃₂

yes

complemented

C₈×C₄₃

—

C₈×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#508

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2247

D₄₃

yes

characteristic

Q₈×C₁₆

—

#1670

—

#1371

—

#1362

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₄₃

yes

—

#475

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

#335

—

#332

—

C₄₃

#12440

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

100

Q₃₂

—

101

Q₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

104

C₂×C₁₆

C₂²⋊D₄₃

yes

characteristic

105

#106

—

172

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

#103

—

172

—

110

C₄×C₈

—

111

C₄×C₈

C₂²×D₄₃

yes

characteristic

112

C₂×Q₈

—

113

C₂×Q₈

—

114

Q₁₆

—

115

Q₁₆

—

116

Q₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₂×C₈

#10621900

yes

characteristic

120

C₂×C₈

C₂×C₂²⋊D₄₃

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₄²

D₄×D₄₃

yes

characteristic

127

C₁₆

—

128

C₁₆

—

172

—

129

C₁₆

#10621883

yes

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13063296

yes

characteristic

138

C₂×C₄

D₈×D₄₃

yes

characteristic

139

C₂×C₄

#13063344

yes

characteristic

140

C₈

—

141

C₈

—

344

—

142

C₈

—

172

—

143

C₈

C₂×#10621883

yes

characteristic

144

C₈

#13063351

yes

characteristic

145

C₂²

#14583132

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

172

—

150

C₄

#14583096

yes

characteristic

151

C₄

#14583166

yes

characteristic

152

C₄

Q₃₂×D₄₃

yes

characteristic

153

C₄

#14583332

yes

characteristic

154

C₂

#19133952

yes

characteristic

155

C₂

#19136683

yes

characteristic

156

C₂

#19136717

yes

characteristic

	subgroup	G/H
center	C₂²	#14583132
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#2247	C₂
socle, prime radical	C₂²×C₄₃	#466
hypercenter	#2247	D₄₃
hypocenter	C₄₃	#12440
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#14583132

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#2247

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
60. Dic₈₆	Q₃₂	semidirect
95. C₄₃	#12440	semidirect

H ≅ G/N

type

60. Dic₈₆

Q₃₂

semidirect

95. C₄₃

#12440

semidirect

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

subgroup	index
C₁₆×Dic₈₆	2
#19133962	2
#19132667	2
#19132644	2
C₄₃×#2247	2
#12440	43

subgroup

index

C₁₆×Dic₈₆

C₄₃×#2247

Subgroups — #25010856

Conjugacy classes of subgroups

Complements

Maximal subgroups