Group #25010900 — subgroups

Subgroups — #25010900

Order 11008.

218 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 71 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	#19136744	C₂	1	2	yes	maximal
2	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19132665	C₂	1	1	yes	maximal, characteristic
4	#19132653	C₂	1	2	yes	maximal
5	C₄₃×#1713	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₈₆	—	2	1	no	—
7	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
8	#14582724	—	2	2	no	—
9	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
10	#14582702	—	2	2	no	—
11	#14582652	C₂²	1	1	yes	characteristic
12	#14582631	—	2	1	no	—
13	C₄₃×#500	—	2	1	no	—
14	C₄₃×#500	C₂²	1	2	yes	—
15	C₄₃×#494	—	2	1	no	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	D₄	1	1	yes	characteristic
18	C₄×Dic₈₆	—	2	1	no	—
19	C₂×C₄₃⋊C₁₆	—	2	2	no	—
20	#13063259	D₄	1	1	yes	characteristic
21	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
22	Dic₄₃⋊C₈	—	4	1	no	—
23	#13063250	—	2	1	no	—
24	C₈×Dic₄₃	—	4	1	no	—
25	C₂×Q₁₆×C₄₃	—	4	2	no	—
26	C₄⋊Q₈×C₄₃	—	2	2	no	—
27	#107×C₄₃	—	2	2	no	—
28	#107×C₄₃	—	2	2	no	—
29	#107×C₄₃	—	4	1	no	—
30	#107×C₄₃	—	4	1	no	—
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×Q₈×C₄₃	—	4	2	no	—
35	C₂×Dic₈₆	—	2	1	no	—
36	Q₁₆×C₄₃	—	8	4	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
42	C₄⋊C₄×C₄₃	—	4	2	no	—
43	C₄⋊C₄×C₄₃	—	4	2	no	—
44	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₄₃	—	2	1	no	—
46	Dic₄₃⋊C₄	—	4	1	no	—
47	C₄×Dic₄₃	—	4	1	no	—
48	C₂×C₄₃⋊C₈	—	4	1	no	—
49	Dic₁₇₂	—	2	1	no	—
50	Dic₁₇₂	Q₁₆	1	2	yes	complemented
51	C₄₃⋊C₁₆	—	4	2	no	—
52	Q₈×C₄₃	—	8	4	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	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₂×Dic₄₃	—	4	1	no	—
58	C₂×Dic₄₃	—	4	1	no	—
59	Dic₈₆	—	2	2	no	—
60	Dic₈₆	—	4	1	no	—
61	C₈×C₄₃	—	2	2	no	—
62	C₈×C₄₃	—	2	1	no	—
63	C₈×C₄₃	—	2	1	no	—
64	C₈×C₄₃	Q₃₂	1	2	yes	—
65	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₈×C₄₃	#135	1	1	yes	characteristic
67	C₄₃⋊C₈	—	8	1	no	—
68	#12448	—	43	1	no	maximal, Sylow
69	C₂²×C₄₃	#459	1	1	yes	characteristic
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	16	2	no	—
72	C₄×C₄₃	#467	1	1	yes	characteristic
73	C₄×C₄₃	#473	1	1	yes	characteristic
74	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
75	C₄×C₄₃	#510	1	1	yes	characteristic
76	Dic₄₃	—	4	2	no	—
77	Dic₄₃	—	8	1	no	—
78	#2227	—	43	2	no	—
79	#1713	D₄₃	1	1	yes	characteristic
80	C₈×Q₁₆	—	43	1	no	—
81	#1370	—	43	1	no	—
82	#1365	—	43	2	no	—
83	C₂×C₄₃	#1669	1	1	yes	characteristic
84	C₂×C₄₃	#2209	1	1	yes	characteristic
85	C₂×C₄₃	#2213	1	1	yes	characteristic
86	#500	—	2	1	no	—
87	#500	C₂×D₄₃	1	2	yes	—
88	#494	—	2	1	no	—
89	C₈×Q₈	—	86	1	no	—
90	C₄×Q₁₆	—	43	1	no	—
91	#366	—	86	2	no	—
92	C₄⋊C₁₆	—	43	2	no	—
93	#358	—	86	2	no	—
94	#335	—	43	1	no	—
95	Q₈⋊C₈	—	86	1	no	—
96	C₈²	C₂×D₄₃	1	1	yes	characteristic
97	C₄₃	#12448	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₁₆	—	4	2	no	—
99	C₂×Q₁₆	—	43	1	no	—
100	C₄⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	86	1	no	—
102	C₂×C₁₆	—	86	2	no	—
103	#107	—	2	2	no	—
104	#107	—	2	2	no	—
105	#107	—	4	1	no	—
106	#107	—	4	1	no	—
107	#107	—	43	1	no	—
108	C₄⋊C₈	—	86	1	no	—
109	C₄⋊C₈	—	172	1	no	—
110	#103	—	86	1	no	—
111	C₄×C₈	—	172	1	no	—
112	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
114	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
115	C₂×Q₈	—	4	2	no	—
116	C₂×Q₈	—	86	1	no	—
117	Q₁₆	—	8	4	no	—
118	Q₁₆	—	43	2	no	—
119	Q₁₆	—	86	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	172	1	no	—
122	C₂×C₈	#10621882	1	1	yes	characteristic
123	C₂×C₈	#10621882	1	1	yes	characteristic
124	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
125	C₂×C₈	#10621900	1	1	yes	characteristic
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	86	1	no	—
129	C₄⋊C₄	—	172	1	no	—
130	C₄²	—	172	1	no	—
131	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
132	C₁₆	—	172	2	no	—
133	Q₈	—	8	4	no	—
134	Q₈	—	86	2	no	—
135	Q₈	—	172	1	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	172	1	no	—
138	C₂×C₄	—	172	1	no	—
139	C₂×C₄	#13063296	1	1	yes	characteristic
140	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
141	C₂×C₄	#13063351	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	344	1	no	—
146	C₈	#13063271	1	2	yes	—
147	C₈	Q₁₆×D₄₃	1	1	yes	characteristic
148	C₈	#13063348	1	1	yes	characteristic
149	C₂²	#14583091	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	16	2	no	—
152	C₄	—	344	1	no	—
153	C₄	—	172	2	no	—
154	C₄	#14583137	1	1	yes	characteristic
155	C₄	#14583171	1	1	yes	characteristic
156	C₄	C₂×#13063271	1	1	yes	characteristic
157	C₄	#14583340	1	1	yes	characteristic
158	C₂	#19133957	1	1	yes	characteristic
159	C₂	#19136642	1	1	yes	characteristic
160	C₂	#19136668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136744

C₂

yes

maximal

C₈×Dic₁₇₂

C₂

yes

maximal, characteristic

#19132665

C₂

yes

maximal, characteristic

#19132653

C₂

yes

maximal

C₄₃×#1713

C₂

yes

maximal, characteristic

C₈×Dic₈₆

—

C₄×Dic₁₇₂

C₂²

yes

characteristic

#14582724

—

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

—

#14582702

—

#14582652

C₂²

yes

characteristic

#14582631

—

C₄₃×#500

—

C₄₃×#500

C₂²

yes

—

C₄₃×#494

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

D₄

yes

characteristic

C₄×Dic₈₆

—

C₂×C₄₃⋊C₁₆

—

#13063259

D₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063250

—

C₈×Dic₄₃

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

#107×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

D₈

yes

characteristic

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

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

Q₁₆

yes

complemented

C₄₃⋊C₁₆

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

Q₃₂

yes

—

C₈×C₄₃

C₂×Q₁₆

yes

characteristic

C₈×C₄₃

#135

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#510

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2227

—

#1713

D₄₃

yes

characteristic

C₈×Q₁₆

—

#1370

—

#1365

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×D₄₃

yes

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

#366

—

C₄⋊C₁₆

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12448

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

100

C₄⋊Q₈

—

101

C₄×Q₈

—

102

C₂×C₁₆

—

103

#107

—

104

#107

—

105

#107

—

106

#107

—

107

#107

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

172

—

110

#103

—

111

C₄×C₈

—

172

—

112

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

113

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

114

C₄×C₈

C₂²×D₄₃

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

—

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

—

121

C₂×C₈

—

172

—

122

C₂×C₈

#10621882

yes

characteristic

123

C₂×C₈

#10621882

yes

characteristic

124

C₂×C₈

D₄×D₄₃

yes

characteristic

125

C₂×C₈

#10621900

yes

characteristic

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

172

—

130

C₄²

—

172

—

131

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

132

C₁₆

—

172

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

172

—

136

C₂×C₄

—

137

C₂×C₄

—

172

—

138

C₂×C₄

—

172

—

139

C₂×C₄

#13063296

yes

characteristic

140

C₂×C₄

C₂×#10621882

yes

characteristic

141

C₂×C₄

#13063351

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

344

—

146

C₈

#13063271

yes

—

147

C₈

Q₁₆×D₄₃

yes

characteristic

148

C₈

#13063348

yes

characteristic

149

C₂²

#14583091

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

344

—

153

C₄

—

172

—

154

C₄

#14583137

yes

characteristic

155

C₄

#14583171

yes

characteristic

156

C₄

C₂×#13063271

yes

characteristic

157

C₄

#14583340

yes

characteristic

158

C₂

#19133957

yes

characteristic

159

C₂

#19136642

yes

characteristic

160

C₂

#19136668

yes

characteristic

	subgroup	G/H
center	C₂²	#14583091
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1713	C₂
socle, prime radical	C₂²×C₄₃	#459
hypercenter	#1713	D₄₃
hypocenter	C₄₃	#12448
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#14583091

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1713

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
50. Dic₁₇₂	Q₁₆	semidirect
97. C₄₃	#12448	semidirect

H ≅ G/N

type

50. Dic₁₇₂

Q₁₆

semidirect

97. C₄₃

#12448

semidirect

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

subgroup	index
#19136744	2
C₈×Dic₁₇₂	2
#19132665	2
#19132653	2
C₄₃×#1713	2
#12448	43

subgroup

index

C₈×Dic₁₇₂

C₄₃×#1713

Subgroups — #25010900

Conjugacy classes of subgroups

Complements

Maximal subgroups