Group #35307514 — subgroups

Subgroups — #35307514

Order 21248.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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	#24908440	C₂	1	1	yes	maximal, characteristic
2	#24906026	C₂	1	2	yes	maximal
3	#24905321	C₂	1	2	yes	maximal
4	#24905296	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#2864	C₂	1	1	yes	maximal, characteristic
6	#19128258	C₂²	1	1	yes	characteristic
7	#19127754	C₂²	1	2	yes	—
8	#19127725	C₂²	1	1	yes	characteristic
9	#19127461	—	2	2	no	—
10	#19127437	—	2	4	no	—
11	#389×C₈₃	—	2	2	no	—
12	#387×C₈₃	—	2	4	no	—
13	#382×C₈₃	C₂²	1	2	yes	—
14	C₄³×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#13050574	—	2	1	no	—
16	#14581792	Q₈	1	2	yes	—
17	#14581784	—	2	1	no	—
18	#14581771	—	2	2	no	—
19	C₄×Dic₁₆₆	—	2	2	no	—
20	#14581751	—	2	1	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄×C₈₃⋊C₈	—	2	2	no	—
25	C₂×C₄⋊C₄×C₈₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
27	C₂×C₄²×C₈₃	—	2	2	no	—
28	C₂×C₄²×C₈₃	D₄	1	2	yes	—
29	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
30	#95×C₈₃	—	2	4	no	—
31	#95×C₈₃	—	2	2	no	—
32	#95×C₈₃	D₄	1	4	yes	—
33	C₂×Dic₁₆₆	—	2	2	no	—
34	#13050574	—	4	2	no	—
35	C₂×C₈₃⋊C₈	—	4	1	no	—
36	C₂×C₈₃⋊C₈	—	4	2	no	—
37	C₄⋊Dic₈₃	—	2	2	no	—
38	C₄⋊Dic₈₃	—	2	2	no	—
39	Dic₈₃⋊C₄	—	4	2	no	—
40	C₂²⋊Dic₈₃	—	4	1	no	—
41	C₄×Dic₈₃	—	4	1	no	—
42	C₂²×C₄×C₈₃	—	2	2	no	—
43	C₂²×C₄×C₈₃	—	2	4	no	—
44	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
46	C₄⋊C₄×C₈₃	—	4	8	no	—
47	C₄⋊C₄×C₈₃	—	4	2	no	—
48	C₄⋊C₄×C₈₃	—	4	2	no	—
49	C₄²×C₈₃	—	2	2	no	—
50	C₄²×C₈₃	—	2	2	no	—
51	C₄²×C₈₃	—	4	4	no	—
52	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₈₃	#41	1	1	yes	characteristic
54	C₄²×C₈₃	#41	1	2	yes	—
55	C₂³×C₈₃	#120	1	1	yes	characteristic
56	C₂×C₄×C₈₃	—	2	4	no	—
57	C₂×C₄×C₈₃	—	4	2	no	—
58	C₂×C₄×C₈₃	—	4	2	no	—
59	C₂×C₄×C₈₃	—	4	2	no	—
60	C₂×C₄×C₈₃	—	4	8	no	—
61	C₂×C₄×C₈₃	—	4	4	no	—
62	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#124	1	2	yes	—
65	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#135	1	2	yes	—
67	C₂×C₄×C₈₃	#137	1	2	yes	—
68	C₂×Dic₈₃	—	4	2	no	—
69	Dic₁₆₆	—	8	2	no	—
70	C₈₃⋊C₈	—	8	2	no	—
71	C₈₃⋊C₈	—	8	1	no	—
72	C₂²×C₈₃	—	2	2	no	—
73	C₂²×C₈₃	#393	1	1	yes	characteristic
74	C₂²×C₈₃	#452	1	2	yes	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	—	2	2	no	—
77	C₄×C₈₃	—	4	4	no	—
78	C₄×C₈₃	—	8	4	no	—
79	C₄×C₈₃	#479	1	2	yes	—
80	C₄×C₈₃	#488	1	1	yes	characteristic
81	C₄×C₈₃	#491	1	1	yes	characteristic
82	Dic₈₃	—	8	2	no	—
83	#10367	—	83	1	no	maximal, Sylow
84	C₂×C₈₃	—	2	2	no	—
85	C₂×C₈₃	#2026	1	1	yes	characteristic
86	C₂×C₈₃	#2031	1	2	yes	—
87	#2864	D₈₃	1	1	yes	characteristic
88	#1994	—	83	1	no	—
89	#1518	—	83	2	no	—
90	#1286	—	83	2	no	—
91	#1276	—	83	1	no	—
92	C₈₃	#10367	1	1	yes	characteristic, complemented, Sylow
93	#533	—	83	1	no	—
94	#431	—	83	2	no	—
95	#423	—	83	1	no	—
96	#389	—	2	2	no	—
97	#387	—	2	4	no	—
98	#382	C₂×D₈₃	1	2	yes	—
99	C₄³	C₂×D₈₃	1	1	yes	characteristic
100	#339	—	166	2	no	—
101	#328	—	166	4	no	—
102	C₂×M₁₆	—	166	1	no	—
103	C₄⋊Q₈	—	83	2	no	—
104	#125	—	166	1	no	—
105	C₂²⋊Q₈	—	166	2	no	—
106	C₄×Q₈	—	166	2	no	—
107	#117	—	166	1	no	—
108	C₂×C₄⋊C₄	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
112	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
113	C₄⋊C₈	—	166	2	no	—
114	C₄⋊C₈	—	166	2	no	—
115	C₂²⋊C₈	—	166	2	no	—
116	C₄×C₈	—	166	2	no	—
117	#95	—	2	4	no	—
118	#95	—	2	2	no	—
119	#95	C₂²⋊D₈₃	1	4	yes	—
120	C₂×Q₈	—	166	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	C₂×C₂²⋊D₈₃	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
125	M₁₆	—	332	2	no	—
126	C₂×C₈	—	332	2	no	—
127	C₂×C₈	—	332	1	no	—
128	C₄⋊C₄	—	4	8	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	332	2	no	—
132	C₄⋊C₄	—	166	2	no	—
133	C₄⋊C₄	—	166	2	no	—
134	C₂²⋊C₄	—	332	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	332	1	no	—
139	C₄²	Q₈×D₈₃	1	1	yes	characteristic
140	C₄²	#13050591	1	1	yes	characteristic
141	C₄²	#13050591	1	2	yes	—
142	C₂³	#14581761	1	1	yes	characteristic
143	Q₈	—	664	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	4	8	no	—
149	C₂×C₄	—	4	4	no	—
150	C₂×C₄	—	332	2	no	—
151	C₂×C₄	#14581768	1	2	yes	—
152	C₂×C₄	#14581768	1	1	yes	characteristic
153	C₂×C₄	#14581781	1	2	yes	—
154	C₂×C₄	#14581781	1	1	yes	characteristic
155	C₂×C₄	#14581812	1	2	yes	—
156	C₂×C₄	#14581822	1	2	yes	—
157	C₈	—	664	2	no	—
158	C₈	—	664	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#19127612	1	1	yes	characteristic
161	C₂²	#19127863	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	4	no	—
166	C₄	—	664	2	no	—
167	C₄	#19128005	1	2	yes	—
168	C₄	#19128058	1	1	yes	characteristic
169	C₄	#19128071	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#24908611	1	1	yes	characteristic
172	C₂	#24908632	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908440

C₂

yes

maximal, characteristic

#24906026

C₂

yes

maximal

#24905321

C₂

yes

maximal

#24905296

C₂

yes

maximal, characteristic

C₈₃×#2864

C₂

yes

maximal, characteristic

#19128258

C₂²

yes

characteristic

#19127754

C₂²

yes

—

#19127725

C₂²

yes

characteristic

#19127461

—

#19127437

—

#389×C₈₃

—

#387×C₈₃

—

#382×C₈₃

C₂²

yes

—

C₄³×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

#14581792

Q₈

yes

—

#14581784

—

#14581771

—

C₄×Dic₁₆₆

—

#14581751

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

—

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#95×C₈₃

—

#95×C₈₃

—

#95×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

C₂×Q₈

yes

C₄²×C₈₃

yes

C₄²×C₈₃

yes

—

C₂³×C₈₃

#120

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

#124

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

#135

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#393

yes

characteristic

C₂²×C₈₃

#452

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#479

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#10367

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#2026

yes

characteristic

C₂×C₈₃

#2031

yes

—

#2864

D₈₃

yes

—

—

—

—

C₈₃

#10367

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×D₈₃

yes

—

C₄³

C₂×D₈₃

yes

characteristic

100

#339

—

166

—

101

#328

—

166

—

102

C₂×M₁₆

—

166

—

103

C₄⋊Q₈

—

104

#125

—

166

—

105

C₂²⋊Q₈

—

166

—

106

C₄×Q₈

—

166

—

107

#117

—

166

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂²⋊D₈₃

yes

—

112

C₂×C₄²

C₂²×D₈₃

yes

characteristic

113

C₄⋊C₈

—

166

—

114

C₄⋊C₈

—

166

—

115

C₂²⋊C₈

—

166

—

116

C₄×C₈

—

166

—

117

#95

—

118

#95

—

119

#95

C₂²⋊D₈₃

yes

—

120

C₂×Q₈

—

166

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

125

M₁₆

—

332

—

126

C₂×C₈

—

332

—

127

C₂×C₈

—

332

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

332

—

132

C₄⋊C₄

—

166

—

133

C₄⋊C₄

—

166

—

134

C₂²⋊C₄

—

332

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

332

—

139

C₄²

Q₈×D₈₃

yes

characteristic

140

C₄²

#13050591

yes

characteristic

141

C₄²

#13050591

yes

—

142

C₂³

#14581761

yes

characteristic

143

Q₈

—

664

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

332

—

151

C₂×C₄

#14581768

yes

—

152

C₂×C₄

#14581768

yes

characteristic

153

C₂×C₄

#14581781

yes

—

154

C₂×C₄

#14581781

yes

characteristic

155

C₂×C₄

#14581812

yes

—

156

C₂×C₄

#14581822

yes

—

157

C₈

—

664

—

158

C₈

—

664

—

159

C₂²

—

160

C₂²

#19127612

yes

characteristic

161

C₂²

#19127863

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

664

—

167

C₄

#19128005

yes

—

168

C₄

#19128058

yes

characteristic

169

C₄

#19128071

yes

characteristic

170

C₂

—

171

C₂

#24908611

yes

characteristic

172

C₂

#24908632

yes

—

	subgroup	G/H
center	C₂²	#19127612
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#2864	C₂
socle	C₂²×C₈₃	#393
prime radical	C₂³×C₈₃	#120
hypercenter	#2864	D₈₃
hypocenter	C₈₃	#10367
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

subgroup

G/H

center

C₂²

#19127612

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#2864

C₂

C₂²×C₈₃

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2864

N	H ≅ G/N	type
92. C₈₃	#10367	semidirect

H ≅ G/N

type

92. C₈₃

#10367

semidirect

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

subgroup	index
#24908440	2
#24906026	2
#24905321	2
#24905296	2
C₈₃×#2864	2
#10367	83

subgroup

index

C₈₃×#2864

Subgroups — #35307514

Conjugacy classes of subgroups

Complements

Maximal subgroups