Group #35324058 — subgroups

Subgroups — #35324058

Order 21248.

324 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 105 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×#19128103	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127524	C₂	1	2	yes	maximal
3	#24908211	C₂	1	1	yes	maximal, characteristic
4	#24905569	C₂	1	2	yes	maximal
5	C₈₃×#1376	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581784	—	2	1	no	—
7	#19128103	—	2	4	no	—
8	C₂×#14581728	C₂²	1	1	yes	characteristic
9	#19127739	C₂²	1	1	yes	characteristic
10	C₂×#14581726	—	2	1	no	—
11	C₂×#14581724	C₂²	1	2	yes	—
12	#19127727	—	2	1	no	—
13	#19127524	—	2	4	no	—
14	#19127461	—	2	1	no	—
15	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×#13050574	—	2	1	no	—
18	#14581784	—	2	4	no	—
19	#14581784	—	4	2	no	—
20	#14581751	—	2	1	no	—
21	C₂×C₄⋊Dic₈₃	—	2	2	no	—
22	C₂×Dic₈₃⋊C₄	—	4	2	no	—
23	#14581728	—	2	2	no	—
24	#14581726	—	2	4	no	—
25	#14581726	Q₈	1	2	yes	—
26	#14581724	—	2	2	no	—
27	#14581724	—	2	1	no	—
28	#14581724	D₄	1	4	yes	—
29	Dic₈₃⋊C₈	—	4	2	no	—
30	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
33	M₃₂×C₈₃	—	2	2	no	—
34	M₃₂×C₈₃	Q₈	1	4	yes	—
35	C₂×C₁₆×C₈₃	—	2	2	no	—
36	C₄×C₈×C₈₃	—	2	2	no	—
37	C₂²×Dic₈₃	—	4	2	no	—
38	#13050574	—	4	1	no	—
39	#13050574	—	4	1	no	—
40	C₂×C₈₃⋊C₈	—	4	1	no	—
41	C₄⋊Dic₈₃	—	2	4	no	—
42	C₄⋊Dic₈₃	—	2	2	no	—
43	C₄⋊Dic₈₃	—	4	2	no	—
44	Dic₈₃⋊C₄	—	4	8	no	—
45	C₂²⋊Dic₈₃	—	4	1	no	—
46	C₄×Dic₈₃	—	4	1	no	—
47	C₂²×C₄×C₈₃	—	2	1	no	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	—	2	1	no	—
50	C₂×C₈×C₈₃	—	2	1	no	—
51	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
52	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
53	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
56	C₄²×C₈₃	—	2	1	no	—
57	C₄²×C₈₃	SD₁₆	1	2	yes	—
58	C₁₆×C₈₃	—	2	4	no	—
59	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	4	2	no	—
62	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
68	C₂×Dic₈₃	—	4	4	no	—
69	C₂×Dic₈₃	—	4	2	no	—
70	C₂×Dic₈₃	—	8	4	no	—
71	C₈×C₈₃	—	2	2	no	—
72	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
73	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
74	C₈₃⋊C₈	—	8	1	no	—
75	C₂²×C₈₃	—	2	2	no	—
76	C₂²×C₈₃	#450	1	1	yes	characteristic
77	C₂²×C₈₃	#476	1	1	yes	characteristic
78	C₂²×C₈₃	#501	1	1	yes	characteristic
79	C₄×C₈₃	—	4	2	no	—
80	C₄×C₈₃	#395	1	1	yes	characteristic
81	C₄×C₈₃	#448	1	1	yes	characteristic
82	C₄×C₈₃	#475	1	1	yes	characteristic
83	C₄×C₈₃	#498	1	1	yes	characteristic
84	Dic₈₃	—	8	4	no	—
85	Dic₈₃	—	8	2	no	—
86	#13458	—	83	1	no	maximal, Sylow
87	C₂×C₈₃	—	2	1	no	—
88	C₂×C₈₃	—	2	1	no	—
89	C₂×C₈₃	#2058	1	1	yes	characteristic
90	C₂×C₈₃	#2231	1	2	yes	—
91	C₂×#499	—	83	1	no	—
92	C₂×#365	—	83	2	no	—
93	#1945	—	83	1	no	—
94	#1386	—	83	2	no	—
95	#1376	D₈₃	1	1	yes	characteristic
96	C₈₃	#13458	1	1	yes	characteristic, complemented, Sylow
97	C₂×#125	—	166	1	no	—
98	C₂×M₃₂	C₂×D₈₃	1	2	yes	—
99	#499	—	166	4	no	—
100	C₂×#108	—	83	1	no	—
101	#427	—	83	1	no	—
102	C₂×#107	—	166	1	no	—
103	C₂×#106	—	83	2	no	—
104	#424	—	166	1	no	—
105	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
106	#365	—	166	4	no	—
107	#339	—	166	1	no	—
108	C₂×M₁₆	—	166	1	no	—
109	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
111	#125	—	332	2	no	—
112	#125	—	166	4	no	—
113	#117	—	166	1	no	—
114	C₂×C₄⋊C₄	—	332	2	no	—
115	C₂×C₄⋊C₄	—	166	2	no	—
116	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
117	M₃₂	—	2	2	no	—
118	M₃₂	Dic₁₆₆	1	4	yes	—
119	C₂×C₁₆	—	2	2	no	—
120	#108	—	166	2	no	—
121	#107	—	83	2	no	—
122	#107	—	166	4	no	—
123	#106	—	83	4	no	—
124	#106	—	166	2	no	—
125	#106	—	166	1	no	—
126	C₄⋊C₈	—	332	2	no	—
127	C₄×C₈	—	2	2	no	—
128	C₂²×C₄	—	2	1	no	—
129	C₂²×C₄	—	332	2	no	—
130	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
131	M₁₆	—	332	1	no	—
132	M₁₆	—	332	1	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	332	1	no	—
136	C₂×C₈	#13050579	1	2	yes	—
137	C₂×C₈	D₄×D₈₃	1	2	yes	—
138	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
139	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
140	C₂×C₈	#13050589	1	1	yes	characteristic
141	C₄⋊C₄	—	332	8	no	—
142	C₄⋊C₄	—	332	2	no	—
143	C₄⋊C₄	—	166	4	no	—
144	C₄⋊C₄	—	166	2	no	—
145	C₂²⋊C₄	—	332	1	no	—
146	C₄²	—	2	1	no	—
147	C₄²	—	332	1	no	—
148	C₄²	#13050579	1	2	yes	—
149	C₁₆	—	2	4	no	—
150	C₂³	#14581820	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	332	4	no	—
154	C₂×C₄	—	332	2	no	—
155	C₂×C₄	—	664	4	no	—
156	C₂×C₄	#14581762	1	1	yes	characteristic
157	C₂×C₄	#14581772	1	1	yes	characteristic
158	C₂×C₄	#14581792	1	1	yes	characteristic
159	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
160	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
161	C₂×C₄	#14581825	1	1	yes	characteristic
162	C₈	—	2	2	no	—
163	C₈	—	664	1	no	—
164	C₈	#14581769	1	2	yes	—
165	C₈	#14581791	1	2	yes	—
166	C₂²	—	2	2	no	—
167	C₂²	#19127854	1	1	yes	characteristic
168	C₂²	#19127992	1	1	yes	characteristic
169	C₂²	#19128110	1	1	yes	characteristic
170	C₄	—	4	2	no	—
171	C₄	—	664	4	no	—
172	C₄	—	664	2	no	—
173	C₄	#19127622	1	1	yes	characteristic
174	C₄	#19127842	1	1	yes	characteristic
175	C₄	#19127987	1	1	yes	characteristic
176	C₄	#19128099	1	1	yes	characteristic
177	C₂	—	2	1	no	—
178	C₂	—	2	1	no	—
179	C₂	#24908797	1	1	yes	characteristic
180	C₂	#24909640	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128103

C₂

yes

maximal, characteristic

C₂×#19127524

C₂

yes

maximal

#24908211

C₂

yes

maximal, characteristic

#24905569

C₂

yes

maximal

C₈₃×#1376

C₂

yes

maximal, characteristic

C₂×#14581784

—

#19128103

—

C₂×#14581728

C₂²

yes

characteristic

#19127739

C₂²

yes

characteristic

C₂×#14581726

—

C₂×#14581724

C₂²

yes

—

#19127727

—

#19127524

—

#19127461

—

C₂×M₃₂×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

—

—

—

C₂×C₄⋊Dic₈₃

—

C₂×Dic₈₃⋊C₄

—

#14581728

—

#14581726

—

#14581726

Q₈

yes

—

#14581724

—

#14581724

—

#14581724

D₄

yes

—

Dic₈₃⋊C₈

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

C₂³

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

characteristic

M₃₂×C₈₃

—

M₃₂×C₈₃

Q₈

yes

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

—

C₂²×Dic₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

—

C₄²×C₈₃

SD₁₆

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#120

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₈₃

#137

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂²⋊Q₈

yes

—

C₈×C₈₃

C₄⋊Q₈

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

#13458

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#2058

yes

characteristic

C₂×C₈₃

#2231

yes

—

—

—

—

—

D₈₃

yes

characteristic

C₈₃

#13458

yes

characteristic, complemented, Sylow

C₂×#125

—

166

—

C₂×M₃₂

C₂×D₈₃

yes

—

#499

—

166

—

100

C₂×#108

—

101

#427

—

102

C₂×#107

—

166

—

103

C₂×#106

—

104

#424

—

166

—

105

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

106

#365

—

166

—

107

#339

—

166

—

108

C₂×M₁₆

—

166

—

109

C₂²×C₈

D₃₃₂

yes

characteristic

110

C₂²×C₈

C₂²×D₈₃

yes

characteristic

111

#125

—

332

—

112

#125

—

166

—

113

#117

—

166

—

114

C₂×C₄⋊C₄

—

332

—

115

C₂×C₄⋊C₄

—

166

—

116

C₂×C₄²

D₃₃₂

yes

characteristic

117

M₃₂

—

118

M₃₂

Dic₁₆₆

yes

—

119

C₂×C₁₆

—

120

#108

—

166

—

121

#107

—

122

#107

—

166

—

123

#106

—

124

#106

—

166

—

125

#106

—

166

—

126

C₄⋊C₈

—

332

—

127

C₄×C₈

—

128

C₂²×C₄

—

129

C₂²×C₄

—

332

—

130

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

131

M₁₆

—

332

—

132

M₁₆

—

332

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

332

—

136

C₂×C₈

#13050579

yes

—

137

C₂×C₈

D₄×D₈₃

yes

—

138

C₂×C₈

Q₈×D₈₃

yes

characteristic

139

C₂×C₈

C₂×Dic₁₆₆

yes

—

140

C₂×C₈

#13050589

yes

characteristic

141

C₄⋊C₄

—

332

—

142

C₄⋊C₄

—

332

—

143

C₄⋊C₄

—

166

—

144

C₄⋊C₄

—

166

—

145

C₂²⋊C₄

—

332

—

146

C₄²

—

147

C₄²

—

332

—

148

C₄²

#13050579

yes

—

149

C₁₆

—

150

C₂³

#14581820

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

332

—

154

C₂×C₄

—

332

—

155

C₂×C₄

—

664

—

156

C₂×C₄

#14581762

yes

characteristic

157

C₂×C₄

#14581772

yes

characteristic

158

C₂×C₄

#14581792

yes

characteristic

159

C₂×C₄

C₂×#13050579

yes

characteristic

160

C₂×C₄

C₂×#13050579

yes

characteristic

161

C₂×C₄

#14581825

yes

characteristic

162

C₈

—

163

C₈

—

664

—

164

C₈

#14581769

yes

—

165

C₈

#14581791

yes

—

166

C₂²

—

167

C₂²

#19127854

yes

characteristic

168

C₂²

#19127992

yes

characteristic

169

C₂²

#19128110

yes

characteristic

170

C₄

—

171

C₄

—

664

—

172

C₄

—

664

—

173

C₄

#19127622

yes

characteristic

174

C₄

#19127842

yes

characteristic

175

C₄

#19127987

yes

characteristic

176

C₄

#19128099

yes

characteristic

177

C₂

—

178

C₂

—

179

C₂

#24908797

yes

characteristic

180

C₂

#24909640

yes

—

	subgroup	G/H
center	C₂²	#19127992
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1376	C₂
socle	C₂²×C₈₃	#476
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
hypercenter	#1376	D₈₃
hypocenter	C₈₃	#13458
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

subgroup

G/H

center

C₂²

#19127992

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1376

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1376

N	H ≅ G/N	type
96. C₈₃	#13458	semidirect

H ≅ G/N

type

96. C₈₃

#13458

semidirect

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

subgroup	index
C₂×#19128103	2
C₂×#19127524	2
#24908211	2
#24905569	2
C₈₃×#1376	2
#13458	83

subgroup

index

C₈₃×#1376

Subgroups — #35324058

Conjugacy classes of subgroups

Complements

Maximal subgroups