Group #35304170 — subgroups

Subgroups — #35304170

Order 21248.

558 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 199 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×#19128081	C₂	1	1	yes	maximal, characteristic
2	#24908423	C₂	1	4	yes	maximal
3	C₂×C₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₃₂	—	2	2	no	—
6	C₂×#14581792	C₂²	1	2	yes	—
7	#19128081	—	2	2	no	—
8	#19128081	C₄	1	4	yes	complemented
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
10	#19127463	—	2	8	no	—
11	C₈×C₈₃⋊C₈	—	2	2	no	—
12	C₂×#107×C₈₃	—	2	2	no	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
14	#384×C₈₃	C₂²	1	2	yes	—
15	C₂²×Dic₁₆₆	—	2	2	no	—
16	C₂×Dic₃₃₂	—	2	8	no	—
17	C₂×Dic₃₃₂	—	4	4	no	—
18	C₂×Dic₃₃₂	—	4	4	no	—
19	C₂²×C₈₃⋊C₈	D₄	1	4	yes	—
20	#14581792	—	2	2	no	—
21	#14581792	—	2	4	no	—
22	#14581792	C₂×C₄	1	4	yes	—
23	C₂×C₄⋊Dic₈₃	—	2	4	no	—
24	C₄×C₈₃⋊C₈	—	2	2	no	—
25	C₄×C₈₃⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₈₃	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	#107×C₈₃	—	4	4	no	—
31	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₈₃	D₄	1	2	yes	—
33	C₂×Dic₁₆₆	—	2	4	no	—
34	C₂×Dic₁₆₆	—	4	4	no	—
35	C₂×Dic₁₆₆	—	4	4	no	—
36	C₂×Dic₁₆₆	—	4	2	no	—
37	C₂²×Dic₈₃	—	4	2	no	—
38	Dic₃₃₂	—	4	16	no	—
39	C₂×C₈₃⋊C₈	—	2	4	no	—
40	C₂×C₈₃⋊C₈	—	2	4	no	—
41	C₂×C₈₃⋊C₈	Q₁₆	1	8	yes	complemented
42	C₄⋊Dic₈₃	—	2	8	no	—
43	C₄⋊Dic₈₃	—	4	4	no	—
44	C₂²×C₄×C₈₃	—	4	2	no	—
45	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
48	C₂×C₈×C₈₃	D₈	1	4	yes	—
49	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
50	C₄⋊C₄×C₈₃	—	4	4	no	—
51	C₄⋊C₄×C₈₃	—	4	8	no	—
52	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₈₃	#41	1	2	yes	—
55	C₂³×C₈₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₈₃	—	8	2	no	—
57	C₂×C₄×C₈₃	—	8	4	no	—
58	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₈₃	#121	1	2	yes	—
61	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
63	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
64	C₂×C₄×C₈₃	C₂×Q₁₆	1	4	yes	—
65	C₂×C₄×C₈₃	#135	1	4	yes	—
66	C₂×Dic₈₃	—	4	4	no	—
67	C₂×Dic₈₃	—	8	2	no	—
68	C₂×Dic₈₃	—	8	2	no	—
69	Dic₁₆₆	—	4	8	no	—
70	Dic₁₆₆	—	8	4	no	—
71	C₈×C₈₃	#102	1	8	yes	—
72	C₈₃⋊C₈	—	2	8	no	—
73	C₂²×C₈₃	#390	1	1	yes	characteristic
74	C₂²×C₈₃	#492	1	1	yes	characteristic
75	C₂²×C₈₃	#493	1	1	yes	characteristic
76	C₂²×C₈₃	#494	1	2	yes	—
77	C₂²×C₈₃	#495	1	2	yes	—
78	C₄×C₈₃	—	16	2	no	—
79	C₄×C₈₃	C₂×#102	1	2	yes	—
80	C₄×C₈₃	C₄×Q₁₆	1	4	yes	—
81	C₄×C₈₃	#467	1	4	yes	—
82	C₄×C₈₃	#486	1	2	yes	—
83	Dic₈₃	—	8	4	no	—
84	#9825	—	83	1	no	maximal, Sylow
85	C₂×C₈₃	#1707	1	2	yes	—
86	C₂×C₈₃	#1709	1	1	yes	characteristic
87	C₂×C₈₃	#1713	1	1	yes	characteristic
88	C₂×C₈₃	#1957	1	1	yes	characteristic
89	C₂×C₈₃	#1969	1	2	yes	—
90	C₂×#494	—	83	1	no	—
91	#1990	—	83	4	no	—
92	#1845	D₈₃	1	1	yes	characteristic
93	C₂×C₈²	—	83	1	no	—
94	C₈₃	#9825	1	1	yes	characteristic, complemented, Sylow
95	C₂²×Q₁₆	—	166	2	no	—
96	C₂×C₄⋊Q₈	—	83	2	no	—
97	#494	—	83	4	no	—
98	#494	—	166	2	no	—
99	C₂×#107	—	2	2	no	—
100	C₂×C₄×C₈	—	83	2	no	—
101	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
102	#384	C₂×D₈₃	1	2	yes	—
103	#340	—	166	8	no	—
104	C₈²	—	166	2	no	—
105	C₂²×Q₈	—	166	2	no	—
106	C₂×Q₁₆	—	332	4	no	—
107	C₂×Q₁₆	—	332	4	no	—
108	C₂×Q₁₆	—	166	8	no	—
109	C₂²×C₈	—	83	4	no	—
110	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
111	C₄⋊Q₈	—	83	4	no	—
112	C₄⋊Q₈	—	166	2	no	—
113	C₄⋊Q₈	—	166	4	no	—
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	166	4	no	—
117	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
118	#107	—	4	4	no	—
119	C₄×C₈	—	166	2	no	—
120	C₄×C₈	—	166	2	no	—
121	C₄×C₈	C₄×D₈₃	1	2	yes	—
122	C₄×C₈	D₃₃₂	1	2	yes	—
123	C₂×Q₈	—	332	2	no	—
124	C₂×Q₈	—	332	4	no	—
125	C₂×Q₈	—	332	4	no	—
126	C₂×Q₈	—	166	4	no	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	332	2	no	—
129	C₂²×C₄	D₄×D₈₃	1	2	yes	—
130	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
131	Q₁₆	—	332	16	no	—
132	C₂×C₈	—	83	8	no	—
133	C₂×C₈	—	166	4	no	—
134	C₂×C₈	—	166	4	no	—
135	C₂×C₈	(C₂×D₈₃)⋊C₄	1	4	yes	—
136	C₂×C₈	#13050575	1	4	yes	—
137	C₂×C₈	#13050577	1	4	yes	—
138	C₄⋊C₄	—	4	4	no	—
139	C₄⋊C₄	—	4	8	no	—
140	C₄⋊C₄	—	332	4	no	—
141	C₄⋊C₄	—	166	8	no	—
142	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
143	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
144	C₄²	#13050591	1	2	yes	—
145	C₂³	#14581788	1	1	yes	characteristic
146	Q₈	—	332	8	no	—
147	Q₈	—	664	4	no	—
148	C₂×C₄	—	8	2	no	—
149	C₂×C₄	—	8	4	no	—
150	C₂×C₄	—	332	4	no	—
151	C₂×C₄	—	664	2	no	—
152	C₂×C₄	—	664	2	no	—
153	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#14581754	1	2	yes	—
155	C₂×C₄	#14581767	1	2	yes	—
156	C₂×C₄	#14581781	1	1	yes	characteristic
157	C₂×C₄	C₂×#13050575	1	2	yes	—
158	C₂×C₄	C₂×#13050577	1	2	yes	—
159	C₂×C₄	Q₁₆×D₈₃	1	4	yes	—
160	C₂×C₄	#14581811	1	4	yes	—
161	C₈	—	166	8	no	—
162	C₈	#14581712	1	8	yes	—
163	C₂²	#19127602	1	1	yes	characteristic
164	C₂²	#19128075	1	1	yes	characteristic
165	C₂²	#19128077	1	1	yes	characteristic
166	C₂²	#19128079	1	2	yes	—
167	C₂²	#19128084	1	2	yes	—
168	C₄	—	16	2	no	—
169	C₄	—	664	4	no	—
170	C₄	C₂×#14581712	1	2	yes	—
171	C₄	#19127795	1	4	yes	—
172	C₄	#19127943	1	4	yes	—
173	C₄	#19128049	1	2	yes	—
174	C₂	#24907061	1	2	yes	—
175	C₂	#24907068	1	1	yes	characteristic
176	C₂	#24907079	1	1	yes	characteristic
177	C₂	#24908268	1	1	yes	characteristic
178	C₂	#24908322	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128081

C₂

yes

maximal, characteristic

#24908423

C₂

yes

maximal

C₂×C₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic

C₈₃×#1845

C₂

yes

maximal, characteristic

C₂²×Dic₃₃₂

—

C₂×#14581792

C₂²

yes

—

#19128081

—

#19128081

C₄

yes

complemented

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127463

—

C₈×C₈₃⋊C₈

—

C₂×#107×C₈₃

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#384×C₈₃

C₂²

yes

—

C₂²×Dic₁₆₆

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂²×C₈₃⋊C₈

D₄

yes

—

#14581792

—

#14581792

—

#14581792

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₃

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#107×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

Q₁₆

yes

complemented

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#121

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

#102

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#495

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#102

yes

—

C₄×C₈₃

C₄×Q₁₆

yes

—

C₄×C₈₃

#467

yes

—

C₄×C₈₃

#486

yes

—

Dic₈₃

—

#9825

—

maximal, Sylow

C₂×C₈₃

#1707

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×#494

—

#1990

—

#1845

D₈₃

yes

characteristic

C₂×C₈²

—

C₈₃

#9825

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

166

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

166

—

C₂×#107

—

100

C₂×C₄×C₈

—

101

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

102

#384

C₂×D₈₃

yes

—

103

#340

—

166

—

104

C₈²

—

166

—

105

C₂²×Q₈

—

166

—

106

C₂×Q₁₆

—

332

—

107

C₂×Q₁₆

—

332

—

108

C₂×Q₁₆

—

166

—

109

C₂²×C₈

—

110

C₂²×C₈

C₂²⋊D₈₃

yes

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

166

—

113

C₄⋊Q₈

—

166

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄⋊C₄

—

116

C₂×C₄⋊C₄

—

166

—

117

C₂×C₄²

C₂²×D₈₃

yes

characteristic

118

#107

—

119

C₄×C₈

—

166

—

120

C₄×C₈

—

166

—

121

C₄×C₈

C₄×D₈₃

yes

—

122

C₄×C₈

D₃₃₂

yes

—

123

C₂×Q₈

—

332

—

124

C₂×Q₈

—

332

—

125

C₂×Q₈

—

332

—

126

C₂×Q₈

—

166

—

127

C₂²×C₄

—

128

C₂²×C₄

—

332

—

129

C₂²×C₄

D₄×D₈₃

yes

—

130

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

131

Q₁₆

—

332

—

132

C₂×C₈

—

133

C₂×C₈

—

166

—

134

C₂×C₈

—

166

—

135

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

136

C₂×C₈

#13050575

yes

—

137

C₂×C₈

#13050577

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

332

—

141

C₄⋊C₄

—

166

—

142

C₄²

C₂×C₄×D₈₃

yes

characteristic

143

C₄²

C₂×D₃₃₂

yes

characteristic

144

C₄²

#13050591

yes

—

145

C₂³

#14581788

yes

characteristic

146

Q₈

—

332

—

147

Q₈

—

664

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

332

—

151

C₂×C₄

—

664

—

152

C₂×C₄

—

664

—

153

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

154

C₂×C₄

#14581754

yes

—

155

C₂×C₄

#14581767

yes

—

156

C₂×C₄

#14581781

yes

characteristic

157

C₂×C₄

C₂×#13050575

yes

—

158

C₂×C₄

C₂×#13050577

yes

—

159

C₂×C₄

Q₁₆×D₈₃

yes

—

160

C₂×C₄

#14581811

yes

—

161

C₈

—

166

—

162

C₈

#14581712

yes

—

163

C₂²

#19127602

yes

characteristic

164

C₂²

#19128075

yes

characteristic

165

C₂²

#19128077

yes

characteristic

166

C₂²

#19128079

yes

—

167

C₂²

#19128084

yes

—

168

C₄

—

169

C₄

—

664

—

170

C₄

C₂×#14581712

yes

—

171

C₄

#19127795

yes

—

172

C₄

#19127943

yes

—

173

C₄

#19128049

yes

—

174

C₂

#24907061

yes

—

175

C₂

#24907068

yes

characteristic

176

C₂

#24907079

yes

characteristic

177

C₂

#24908268

yes

characteristic

178

C₂

#24908322

yes

—

	subgroup	G/H
center	C₂³	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1845	C₂
socle, prime radical	C₂³×C₈₃	#127
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1845	D₈₃
hypocenter	C₈₃	#9825
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#14581788

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1845

C₂

socle, prime radical

C₂³×C₈₃

#127

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
8. #19128081	C₄	semidirect
41. C₂×C₈₃⋊C₈	Q₁₆	semidirect
94. C₈₃	#9825	semidirect

H ≅ G/N

type

8. #19128081

C₄

semidirect

41. C₂×C₈₃⋊C₈

Q₁₆

semidirect

94. C₈₃

#9825

semidirect

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

subgroup	index
C₂×#19128081	2
#24908423	2
C₂×C₈×C₈₃⋊C₈	2
C₈₃×#1845	2
#9825	83

subgroup

index

C₂×#19128081

#24908423

C₂×C₈×C₈₃⋊C₈

C₈₃×#1845

#9825

Subgroups — #35304170

Conjugacy classes of subgroups

Complements

Maximal subgroups