Group #25010890 — subgroups

Subgroups — #25010890

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	#19136734	C₂	1	2	yes	maximal
2	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
3	#19132665	C₂	1	1	yes	maximal, characteristic
4	#19132650	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	#14582726	—	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	1	no	—
26	C₂×Q₁₆×C₄₃	—	4	1	no	—
27	C₄⋊Q₈×C₄₃	—	2	2	no	—
28	#107×C₄₃	—	2	2	no	—
29	#107×C₄₃	—	2	2	no	—
30	#107×C₄₃	—	4	2	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	2	no	—
37	Q₁₆×C₄₃	—	8	2	no	—
38	C₂×C₈×C₄₃	—	2	1	no	—
39	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
41	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
43	C₄⋊C₄×C₄₃	—	4	2	no	—
44	C₄⋊C₄×C₄₃	—	4	2	no	—
45	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊Dic₄₃	—	2	1	no	—
47	Dic₄₃⋊C₄	—	4	1	no	—
48	C₄×Dic₄₃	—	4	1	no	—
49	C₂×C₄₃⋊C₈	—	4	1	no	—
50	Dic₁₇₂	—	2	1	no	—
51	Dic₁₇₂	Q₁₆	1	2	yes	complemented
52	C₄₃⋊C₁₆	—	4	2	no	—
53	Q₈×C₄₃	—	8	2	no	—
54	Q₈×C₄₃	—	8	2	no	—
55	C₂×C₄×C₄₃	—	8	2	no	—
56	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂×Dic₄₃	—	4	1	no	—
60	C₂×Dic₄₃	—	4	1	no	—
61	Dic₈₆	—	2	2	no	—
62	Dic₈₆	—	4	1	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	—	2	1	no	—
65	C₈×C₄₃	—	2	1	no	—
66	C₈×C₄₃	SD₃₂	1	2	yes	—
67	C₈×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₈×C₄₃	#135	1	1	yes	characteristic
69	C₄₃⋊C₈	—	8	1	no	—
70	#12446	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	#459	1	1	yes	characteristic
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	16	2	no	—
74	C₄×C₄₃	#467	1	1	yes	characteristic
75	C₄×C₄₃	#473	1	1	yes	characteristic
76	C₄×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
77	C₄×C₄₃	#509	1	1	yes	characteristic
78	Dic₄₃	—	4	2	no	—
79	Dic₄₃	—	8	1	no	—
80	#2225	—	43	2	no	—
81	#1713	D₄₃	1	1	yes	characteristic
82	C₈×Q₁₆	—	43	1	no	—
83	#1370	—	43	1	no	—
84	#1364	—	43	2	no	—
85	C₂×C₄₃	#1669	1	1	yes	characteristic
86	C₂×C₄₃	#2207	1	1	yes	characteristic
87	C₂×C₄₃	#2214	1	1	yes	characteristic
88	#500	—	2	1	no	—
89	#500	C₂×D₄₃	1	2	yes	—
90	#494	—	2	1	no	—
91	C₈×Q₈	—	86	1	no	—
92	C₄×Q₁₆	—	43	1	no	—
93	#367	—	86	2	no	—
94	C₄⋊C₁₆	—	43	2	no	—
95	#358	—	86	2	no	—
96	#335	—	43	1	no	—
97	Q₈⋊C₈	—	86	1	no	—
98	C₈²	C₂×D₄₃	1	1	yes	characteristic
99	C₄₃	#12446	1	1	yes	characteristic, complemented, Sylow
100	C₂×Q₁₆	—	4	1	no	—
101	C₂×Q₁₆	—	4	1	no	—
102	C₂×Q₁₆	—	43	1	no	—
103	C₄⋊Q₈	—	2	2	no	—
104	C₄×Q₈	—	86	1	no	—
105	C₂×C₁₆	—	86	2	no	—
106	#107	—	2	2	no	—
107	#107	—	2	2	no	—
108	#107	—	4	2	no	—
109	#107	—	43	1	no	—
110	C₄⋊C₈	—	86	1	no	—
111	C₄⋊C₈	—	172	1	no	—
112	#103	—	86	1	no	—
113	C₄×C₈	—	172	1	no	—
114	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
115	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
116	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
117	C₂×Q₈	—	4	2	no	—
118	C₂×Q₈	—	86	1	no	—
119	Q₁₆	—	8	2	no	—
120	Q₁₆	—	8	2	no	—
121	Q₁₆	—	43	2	no	—
122	Q₁₆	—	86	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	172	1	no	—
125	C₂×C₈	#10621882	1	1	yes	characteristic
126	C₂×C₈	#10621882	1	1	yes	characteristic
127	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
128	C₂×C₈	#10621900	1	1	yes	characteristic
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	86	1	no	—
132	C₄⋊C₄	—	172	1	no	—
133	C₄²	—	172	1	no	—
134	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
135	C₁₆	—	172	2	no	—
136	Q₈	—	8	2	no	—
137	Q₈	—	8	2	no	—
138	Q₈	—	86	2	no	—
139	Q₈	—	172	1	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	172	1	no	—
142	C₂×C₄	—	172	1	no	—
143	C₂×C₄	#13063296	1	1	yes	characteristic
144	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
145	C₂×C₄	#13063351	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	344	1	no	—
150	C₈	#13063269	1	2	yes	—
151	C₈	Q₁₆×D₄₃	1	1	yes	characteristic
152	C₈	#13063348	1	1	yes	characteristic
153	C₂²	#14583091	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	16	2	no	—
156	C₄	—	344	1	no	—
157	C₄	—	172	2	no	—
158	C₄	#14583137	1	1	yes	characteristic
159	C₄	#14583171	1	1	yes	characteristic
160	C₄	C₂×#13063269	1	1	yes	characteristic
161	C₄	#14583335	1	1	yes	characteristic
162	C₂	#19133957	1	1	yes	characteristic
163	C₂	#19136632	1	1	yes	characteristic
164	C₂	#19136673	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136734

C₂

yes

maximal

C₈×Dic₁₇₂

C₂

yes

maximal, characteristic

#19132665

C₂

yes

maximal, characteristic

#19132650

C₂

yes

maximal

C₄₃×#1713

C₂

yes

maximal, characteristic

C₈×Dic₈₆

—

C₄×Dic₁₇₂

C₂²

yes

characteristic

#14582726

—

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

—

C₂×Q₁₆×C₄₃

—

C₄⋊Q₈×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

—

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

—

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

SD₃₂

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×SD₃₂

yes

characteristic

C₄×C₄₃

#509

yes

characteristic

Dic₄₃

—

Dic₄₃

—

#2225

—

#1713

D₄₃

yes

characteristic

C₈×Q₁₆

—

#1370

—

#1364

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×D₄₃

yes

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

—

#358

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#12446

yes

characteristic, complemented, Sylow

100

C₂×Q₁₆

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₄⋊Q₈

—

104

C₄×Q₈

—

105

C₂×C₁₆

—

106

#107

—

107

#107

—

108

#107

—

109

#107

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

172

—

112

#103

—

113

C₄×C₈

—

172

—

114

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

115

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

116

C₄×C₈

C₂²×D₄₃

yes

characteristic

117

C₂×Q₈

—

118

C₂×Q₈

—

119

Q₁₆

—

120

Q₁₆

—

121

Q₁₆

—

122

Q₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

172

—

125

C₂×C₈

#10621882

yes

characteristic

126

C₂×C₈

#10621882

yes

characteristic

127

C₂×C₈

D₄×D₄₃

yes

characteristic

128

C₂×C₈

#10621900

yes

characteristic

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

172

—

133

C₄²

—

172

—

134

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

135

C₁₆

—

172

—

136

Q₈

—

137

Q₈

—

138

Q₈

—

139

Q₈

—

172

—

140

C₂×C₄

—

141

C₂×C₄

—

172

—

142

C₂×C₄

—

172

—

143

C₂×C₄

#13063296

yes

characteristic

144

C₂×C₄

C₂×#10621882

yes

characteristic

145

C₂×C₄

#13063351

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

344

—

150

C₈

#13063269

yes

—

151

C₈

Q₁₆×D₄₃

yes

characteristic

152

C₈

#13063348

yes

characteristic

153

C₂²

#14583091

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

344

—

157

C₄

—

172

—

158

C₄

#14583137

yes

characteristic

159

C₄

#14583171

yes

characteristic

160

C₄

C₂×#13063269

yes

characteristic

161

C₄

#14583335

yes

characteristic

162

C₂

#19133957

yes

characteristic

163

C₂

#19136632

yes

characteristic

164

C₂

#19136673

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	#12446
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
51. Dic₁₇₂	Q₁₆	semidirect
99. C₄₃	#12446	semidirect

H ≅ G/N

type

51. Dic₁₇₂

Q₁₆

semidirect

99. C₄₃

#12446

semidirect

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

subgroup	index
#19136734	2
C₈×Dic₁₇₂	2
#19132665	2
#19132650	2
C₄₃×#1713	2
#12446	43

subgroup

index

C₈×Dic₁₇₂

C₄₃×#1713

Subgroups — #25010890

Conjugacy classes of subgroups

Complements

Maximal subgroups