Group #35324300 — subgroups

Subgroups — #35324300

Order 21248.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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	#24909078	C₂	1	1	yes	maximal, characteristic
2	#24907518	C₂	1	2	yes	maximal
3	#24905565	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905560	C₂	1	2	yes	maximal, complemented
5	C₈₃×#2161	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581798	C₂²	1	1	yes	characteristic
7	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
8	#19127739	C₂²	1	2	yes	—
9	#19127717	—	2	2	no	—
10	(C₂×D₈₃)⋊C₁₆	—	2	2	no	—
11	#19127472	—	2	2	no	—
12	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×#108×C₈₃	C₂²	1	2	yes	—
14	#368×C₈₃	—	2	2	no	—
15	C₂×#13050590	D₄	1	1	yes	characteristic
16	#14581798	—	2	2	no	—
17	#14581798	—	2	2	no	—
18	C₂×#13050574	D₄	1	1	yes	characteristic
19	C₂×#13050573	—	2	1	no	—
20	C₂×C₈×D₈₃	—	2	1	no	—
21	#14581751	—	2	2	no	—
22	C₂×C₈₃⋊C₁₆	—	2	1	no	—
23	C₂×C₈₃⋊C₁₆	—	2	2	no	—
24	#14581726	—	2	2	no	—
25	#14581724	—	2	2	no	—
26	#14581718	—	4	4	no	—
27	C₂×M₁₆×C₈₃	—	2	2	no	—
28	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	M₃₂×C₈₃	—	2	2	no	—
30	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
31	#108×C₈₃	—	2	2	no	—
32	#108×C₈₃	—	2	2	no	—
33	#13050590	—	2	2	no	—
34	#13050590	—	2	2	no	—
35	C₂×Dic₁₆₆	Q₁₆	1	1	yes	characteristic
36	C₂×D₃₃₂	Q₁₆	1	1	yes	characteristic
37	C₂×C₂²⋊D₈₃	—	2	1	no	—
38	C₂×C₄×D₈₃	—	2	1	no	—
39	#13050574	—	2	2	no	—
40	#13050573	—	4	2	no	—
41	C₂×C₈₃⋊C₈	—	2	1	no	—
42	C₈×D₈₃	—	4	1	no	—
43	C₈×D₈₃	—	4	1	no	—
44	C₄⋊Dic₈₃	—	4	2	no	—
45	C₂²⋊Dic₈₃	—	4	2	no	—
46	C₄×Dic₈₃	—	4	2	no	—
47	C₈₃⋊C₁₆	—	4	1	no	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₈₃	—	4	2	no	—
50	M₁₆×C₈₃	—	4	2	no	—
51	C₂×C₈×C₈₃	—	4	2	no	—
52	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₃	#41	1	2	yes	—
55	C₂×C₈×C₈₃	#41	1	2	yes	—
56	C₁₆×C₈₃	—	2	2	no	—
57	C₂³×C₈₃	#135	1	1	yes	characteristic
58	C₂²×D₈₃	—	2	1	no	—
59	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
60	C₂×C₄×C₈₃	#123	1	2	yes	—
61	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₂²⋊D₈₃	—	4	2	no	—
64	C₂×Dic₈₃	—	2	1	no	—
65	C₂×Dic₈₃	—	8	2	no	—
66	D₃₃₂	—	2	2	no	—
67	C₄×D₈₃	—	4	1	no	—
68	C₄×D₈₃	—	4	1	no	—
69	Dic₁₆₆	—	2	2	no	—
70	C₈×C₈₃	—	8	2	no	—
71	C₈×C₈₃	#123	1	1	yes	characteristic
72	C₈×C₈₃	#124	1	1	yes	characteristic
73	C₈×C₈₃	#126	1	2	yes	—
74	C₈₃⋊C₈	—	4	1	no	—
75	C₂²×C₈₃	—	4	1	no	—
76	C₂²×C₈₃	#467	1	1	yes	characteristic
77	C₂²×C₈₃	#482	1	2	yes	—
78	C₂×D₈₃	—	4	2	no	—
79	C₂×D₈₃	—	4	1	no	—
80	C₄×C₈₃	#397	1	1	yes	characteristic
81	C₄×C₈₃	#469	1	1	yes	characteristic
82	C₄×C₈₃	#484	1	2	yes	—
83	Dic₈₃	—	4	1	no	—
84	Dic₈₃	—	16	2	no	—
85	#13499	—	83	1	no	maximal, Sylow
86	C₂×C₈₃	—	2	2	no	—
87	C₂×C₈₃	—	2	1	no	—
88	C₂×C₈₃	#2076	1	1	yes	characteristic
89	D₈₃	—	8	1	no	—
90	#2161	D₈₃	1	1	yes	characteristic, complemented
91	#2113	—	83	1	no	—
92	#1814	—	83	2	no	—
93	#1384	—	83	1	no	—
94	#1382	—	83	2	no	—
95	C₈₃	#13499	1	1	yes	characteristic, complemented, Sylow
96	C₂×#131	—	83	1	no	—
97	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
98	C₂²×C₁₆	—	83	1	no	—
99	C₂×#108	C₂×D₈₃	1	2	yes	—
100	#427	—	83	2	no	—
101	#421	—	166	2	no	—
102	#368	—	2	2	no	—
103	C₂²⋊C₁₆	—	166	2	no	—
104	#344	—	166	2	no	—
105	C₂×#41	—	83	1	no	—
106	#131	—	166	2	no	—
107	#131	—	166	2	no	—
108	C₂×M₁₆	—	2	2	no	—
109	C₂×M₁₆	—	83	1	no	—
110	C₂×M₁₆	—	166	1	no	—
111	C₂²×C₈	—	166	1	no	—
112	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
113	#117	—	166	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	166	1	no	—
116	C₂×C₁₆	—	166	2	no	—
117	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
118	#108	—	2	2	no	—
119	#108	—	2	2	no	—
120	#107	—	166	2	no	—
121	#106	—	166	2	no	—
122	#104	—	332	4	no	—
123	#41	—	166	2	no	—
124	#41	—	166	2	no	—
125	C₂×Q₈	—	83	1	no	—
126	C₂×D₄	—	83	1	no	—
127	C₂×D₄	—	166	1	no	—
128	C₂²×C₄	—	166	1	no	—
129	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
130	M₁₆	—	4	2	no	—
131	M₁₆	—	4	2	no	—
132	M₁₆	—	332	2	no	—
133	M₁₆	—	166	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	332	1	no	—
136	C₂×C₈	—	332	1	no	—
137	C₂×C₈	—	166	1	no	—
138	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
139	C₂×C₈	#13050589	1	1	yes	characteristic
140	C₂×C₈	#13050590	1	2	yes	—
141	C₂×C₈	#13050591	1	2	yes	—
142	C₄⋊C₄	—	332	2	no	—
143	C₂²⋊C₄	—	332	2	no	—
144	C₄²	—	332	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	332	1	no	—
147	C₂³	—	166	1	no	—
148	C₂³	#14581815	1	1	yes	characteristic
149	Q₈	—	166	2	no	—
150	D₄	—	332	2	no	—
151	D₄	—	166	2	no	—
152	C₂×C₄	—	332	1	no	—
153	C₂×C₄	—	332	1	no	—
154	C₂×C₄	—	664	2	no	—
155	C₂×C₄	—	166	1	no	—
156	C₂×C₄	#14581763	1	1	yes	characteristic
157	C₂×C₄	#14581777	1	2	yes	—
158	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
159	C₂×C₄	#14581816	1	2	yes	—
160	C₈	—	8	2	no	—
161	C₈	—	332	1	no	—
162	C₈	#14581775	1	1	yes	characteristic
163	C₈	#14581781	1	1	yes	characteristic
164	C₈	#14581787	1	2	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	332	2	no	—
167	C₂²	—	332	1	no	—
168	C₂²	#19127939	1	1	yes	characteristic
169	C₂²	#19128028	1	2	yes	—
170	C₄	—	1328	2	no	—
171	C₄	—	332	1	no	—
172	C₄	#19127627	1	1	yes	characteristic
173	C₄	#19127951	1	1	yes	characteristic
174	C₄	#19128038	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	664	1	no	—
178	C₂	#24908892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909078

C₂

yes

maximal, characteristic

#24907518

C₂

yes

maximal

#24905565

C₂

yes

maximal, characteristic, complemented

#24905560

C₂

yes

maximal, complemented

C₈₃×#2161

C₂

yes

maximal, characteristic, complemented

C₂×#14581798

C₂²

yes

characteristic

C₂²×C₈₃⋊C₁₆

C₂²

yes

characteristic

#19127739

C₂²

yes

—

#19127717

—

(C₂×D₈₃)⋊C₁₆

—

#19127472

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

C₂×#108×C₈₃

C₂²

yes

—

#368×C₈₃

—

C₂×#13050590

D₄

yes

—

—

D₄

yes

characteristic

C₂×#13050573

—

C₂×C₈×D₈₃

—

#14581751

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

—

#14581726

—

#14581724

—

#14581718

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#108×C₈₃

—

#108×C₈₃

—

#13050590

—

#13050590

—

C₂×Dic₁₆₆

Q₁₆

yes

characteristic

C₂×D₃₃₂

Q₁₆

yes

characteristic

C₂×C₂²⋊D₈₃

—

C₂×C₄×D₈₃

—

#13050574

—

#13050573

—

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₈×D₈₃

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

C₂×C₈×C₈₃

yes

C₂×C₈×C₈₃

yes

—

C₂×C₈×C₈₃

#41

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

#135

yes

characteristic

C₂²×D₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂²⋊D₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

yes

C₈×C₈₃

yes

C₈×C₈₃

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#467

yes

characteristic

C₂²×C₈₃

#482

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

Dic₈₃

—

#13499

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#2076

yes

characteristic

D₈₃

—

#2161

D₈₃

yes

characteristic, complemented

—

—

—

—

C₈₃

#13499

yes

characteristic, complemented, Sylow

C₂×#131

—

C₂×M₃₂

C₂×D₈₃

yes

characteristic

C₂²×C₁₆

—

C₂×#108

C₂×D₈₃

yes

—

100

#427

—

101

#421

—

166

—

102

#368

—

103

C₂²⋊C₁₆

—

166

—

104

#344

—

166

—

105

C₂×#41

—

106

#131

—

166

—

107

#131

—

166

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

166

—

111

C₂²×C₈

—

166

—

112

C₂²×C₈

C₂²×D₈₃

yes

characteristic

113

#117

—

166

—

114

M₃₂

—

115

C₂×C₁₆

—

166

—

116

C₂×C₁₆

—

166

—

117

C₂×C₁₆

C₂²⋊D₈₃

yes

—

118

#108

—

119

#108

—

120

#107

—

166

—

121

#106

—

166

—

122

#104

—

332

—

123

#41

—

166

—

124

#41

—

166

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

166

—

128

C₂²×C₄

—

166

—

129

C₂²×C₄

D₄×D₈₃

yes

characteristic

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

332

—

133

M₁₆

—

166

—

134

C₂×C₈

—

135

C₂×C₈

—

332

—

136

C₂×C₈

—

332

—

137

C₂×C₈

—

166

—

138

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

139

C₂×C₈

#13050589

yes

characteristic

140

C₂×C₈

#13050590

yes

—

141

C₂×C₈

#13050591

yes

—

142

C₄⋊C₄

—

332

—

143

C₂²⋊C₄

—

332

—

144

C₄²

—

332

—

145

C₁₆

—

146

C₁₆

—

332

—

147

C₂³

—

166

—

148

C₂³

#14581815

yes

characteristic

149

Q₈

—

166

—

150

D₄

—

332

—

151

D₄

—

166

—

152

C₂×C₄

—

332

—

153

C₂×C₄

—

332

—

154

C₂×C₄

—

664

—

155

C₂×C₄

—

166

—

156

C₂×C₄

#14581763

yes

characteristic

157

C₂×C₄

#14581777

yes

—

158

C₂×C₄

Q₁₆×D₈₃

yes

characteristic

159

C₂×C₄

#14581816

yes

—

160

C₈

—

161

C₈

—

332

—

162

C₈

#14581775

yes

characteristic

163

C₈

#14581781

yes

characteristic

164

C₈

#14581787

yes

—

165

C₂²

—

166

C₂²

—

332

—

167

C₂²

—

332

—

168

C₂²

#19127939

yes

characteristic

169

C₂²

#19128028

yes

—

170

C₄

—

1328

—

171

C₄

—

332

—

172

C₄

#19127627

yes

characteristic

173

C₄

#19127951

yes

characteristic

174

C₄

#19128038

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

664

—

178

C₂

#24908892

yes

characteristic

	subgroup	G/H
center	C₄	#19127951
commutator, cosocle	C₂²×C₈×C₈₃	C₂³
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#2161	C₂
socle	C₂×C₈₃	#2076
prime radical	C₂×#13050590	D₄
hypercenter	#2161	D₈₃
hypocenter	C₈₃	#13499
LCS	G ▷ C₂²×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

subgroup

G/H

center

C₄

#19127951

commutator, cosocle

C₂²×C₈×C₈₃

C₂³

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#2161

C₂

C₂×C₈₃

D₄

D₈₃

C₈₃

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2161

N	H ≅ G/N	type
3. #24905565	C₂	semidirect
4. #24905560	C₂	semidirect
5. C₈₃×#2161	C₂	semidirect
90. #2161	D₈₃	semidirect
95. C₈₃	#13499	semidirect

H ≅ G/N

type

3. #24905565

C₂

semidirect

4. #24905560

C₂

semidirect

5. C₈₃×#2161

C₂

semidirect

90. #2161

D₈₃

semidirect

95. C₈₃

#13499

semidirect

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

subgroup	index
#24909078	2
#24907518	2
#24905565	2
#24905560	2
C₈₃×#2161	2
#13499	83

subgroup

index

C₈₃×#2161

Subgroups — #35324300

Conjugacy classes of subgroups

Complements

Maximal subgroups