Group #35303889 — subgroups

Subgroups — #35303889

Order 21248.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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×#19128099	C₂	1	1	yes	maximal, characteristic
2	#24907601	C₂	1	1	yes	maximal, characteristic
3	#24906147	C₂	1	2	yes	maximal
4	#24906143	C₂	1	2	yes	maximal
5	C₈₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581792	C₂²	1	2	yes	—
7	#19128099	—	2	2	no	—
8	#19128099	C₄	1	4	yes	—
9	C₂×#14581724	—	2	2	no	—
10	#19127725	C₂²	1	1	yes	characteristic
11	#19127725	C₂²	1	1	yes	characteristic
12	#19127466	—	2	2	no	—
13	#19127449	—	2	4	no	—
14	#19127448	—	2	4	no	—
15	#431×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#322×C₈₃	—	2	2	no	—
18	C₂²×Dic₁₆₆	—	2	2	no	—
19	C₂×#13050574	—	2	1	no	—
20	C₂×#13050574	—	2	1	no	—
21	#14581792	—	2	2	no	—
22	#14581792	—	2	2	no	—
23	#14581792	—	2	2	no	—
24	#14581792	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₈₃	—	2	2	no	—
26	C₂×C₄⋊Dic₈₃	—	2	2	no	—
27	#14581724	—	2	8	no	—
28	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
29	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
30	C₂²×C₈×C₈₃	D₄	1	2	yes	—
31	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₈₃	D₄	1	4	yes	—
33	C₂²⋊C₈×C₈₃	—	2	2	no	—
34	C₄×C₈×C₈₃	—	2	2	no	—
35	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₃	D₄	1	2	yes	—
37	C₂×Dic₁₆₆	—	2	4	no	—
38	C₂×Dic₁₆₆	—	4	2	no	—
39	C₂×Dic₁₆₆	—	8	4	no	—
40	C₂²×Dic₈₃	—	4	2	no	—
41	#13050574	—	4	2	no	—
42	#13050574	—	4	2	no	—
43	C₂×C₈₃⋊C₈	—	4	1	no	—
44	C₂×C₈₃⋊C₈	—	4	1	no	—
45	C₄⋊Dic₈₃	—	2	4	no	—
46	C₄⋊Dic₈₃	—	2	4	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	C₄⋊Dic₈₃	—	4	2	no	—
49	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₃	#41	1	2	yes	—
51	C₂×C₈×C₈₃	—	2	2	no	—
52	C₂×C₈×C₈₃	—	2	2	no	—
53	C₂×C₈×C₈₃	—	2	4	no	—
54	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
55	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₈₃	C₂×D₄	1	2	yes	—
58	C₂³×C₈₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₃	—	2	4	no	—
60	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₈₃	#121	1	2	yes	—
63	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#137	1	2	yes	—
65	C₂×C₄×C₈₃	#137	1	2	yes	—
66	C₂×Dic₈₃	—	4	4	no	—
67	C₂×Dic₈₃	—	8	4	no	—
68	Dic₁₆₆	—	8	4	no	—
69	Dic₁₆₆	—	8	4	no	—
70	C₈×C₈₃	—	2	4	no	—
71	C₈×C₈₃	—	4	4	no	—
72	C₈₃⋊C₈	—	8	1	no	—
73	C₈₃⋊C₈	—	8	1	no	—
74	C₂²×C₈₃	—	2	2	no	—
75	C₂²×C₈₃	#390	1	1	yes	characteristic
76	C₂²×C₈₃	#491	1	2	yes	—
77	C₄×C₈₃	—	2	2	no	—
78	C₄×C₈₃	#419	1	2	yes	—
79	C₄×C₈₃	#471	1	4	yes	—
80	C₄×C₈₃	#497	1	1	yes	characteristic
81	C₄×C₈₃	#497	1	1	yes	characteristic
82	Dic₈₃	—	8	4	no	—
83	#9776	—	83	1	no	maximal, Sylow
84	C₂×C₈₃	—	2	2	no	—
85	C₂×C₈₃	#1960	1	1	yes	characteristic
86	C₂×C₈₃	#1977	1	2	yes	—
87	C₂×#498	—	83	1	no	—
88	#1837	—	83	1	no	—
89	#1549	—	83	2	no	—
90	#1548	—	83	2	no	—
91	#1452	D₈₃	1	1	yes	characteristic
92	C₈₃	#9776	1	1	yes	characteristic, complemented, Sylow
93	C₂×C₄⋊Q₈	—	83	2	no	—
94	#498	—	83	4	no	—
95	#498	—	166	2	no	—
96	#431	C₂×D₈₃	1	2	yes	—
97	C₂×#106	—	166	2	no	—
98	#423	—	83	1	no	—
99	#423	—	83	1	no	—
100	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
101	#341	—	166	2	no	—
102	#333	—	166	4	no	—
103	#332	—	166	4	no	—
104	#322	—	2	2	no	—
105	C₂²×Q₈	—	166	2	no	—
106	C₂×M₁₆	—	166	1	no	—
107	C₂×M₁₆	—	166	1	no	—
108	C₂²×C₈	D₃₃₂	1	2	yes	—
109	C₄⋊Q₈	—	83	4	no	—
110	C₄⋊Q₈	—	166	2	no	—
111	C₄⋊Q₈	—	166	2	no	—
112	C₄⋊Q₈	—	166	2	no	—
113	C₂×C₄⋊C₄	—	166	2	no	—
114	C₂×C₄⋊C₄	—	166	2	no	—
115	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
116	#106	—	166	8	no	—
117	C₄⋊C₈	—	166	2	no	—
118	C₄⋊C₈	—	166	2	no	—
119	C₄⋊C₈	D₃₃₂	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₈₃	1	2	yes	—
123	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
124	C₂×Q₈	—	332	2	no	—
125	C₂×Q₈	—	664	4	no	—
126	C₂×Q₈	—	166	4	no	—
127	C₂²×C₄	—	332	2	no	—
128	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
129	C₂²×C₄	#13050590	1	2	yes	—
130	M₁₆	—	332	2	no	—
131	M₁₆	—	332	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	332	1	no	—
136	C₂×C₈	—	332	1	no	—
137	C₂×C₈	(C₂×D₈₃)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	332	2	no	—
139	C₄⋊C₄	—	332	2	no	—
140	C₄⋊C₄	—	166	4	no	—
141	C₄⋊C₄	—	166	4	no	—
142	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
143	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
144	C₄²	C₂×D₃₃₂	1	2	yes	—
145	C₂³	#14581780	1	1	yes	characteristic
146	Q₈	—	664	4	no	—
147	Q₈	—	664	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	332	4	no	—
150	C₂×C₄	—	664	4	no	—
151	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×D₃₃₂	1	2	yes	—
153	C₂×C₄	#14581766	1	2	yes	—
154	C₂×C₄	#14581789	1	1	yes	characteristic
155	C₂×C₄	#14581825	1	2	yes	—
156	C₂×C₄	#14581825	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	664	1	no	—
160	C₈	—	664	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#19127603	1	1	yes	characteristic
163	C₂²	#19128070	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	664	4	no	—
166	C₄	#19127711	1	2	yes	—
167	C₄	#19127964	1	4	yes	—
168	C₄	#19128095	1	1	yes	characteristic
169	C₄	#19128095	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#24908283	1	1	yes	characteristic
172	C₂	#24908366	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128099

C₂

yes

maximal, characteristic

#24907601

C₂

yes

maximal, characteristic

#24906147

C₂

yes

maximal

#24906143

C₂

yes

maximal

C₈₃×#1452

C₂

yes

maximal, characteristic

C₂×#14581792

C₂²

yes

—

#19128099

—

#19128099

C₄

yes

—

C₂×#14581724

—

#19127725

C₂²

yes

characteristic

#19127725

C₂²

yes

—

—

—

#431×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

—

C₂²×Dic₁₆₆

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581724

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

D₄

yes

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²⋊C₄

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₂³×C₈₃

#124

yes

characteristic

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

#127

yes

characteristic

C₂×C₄×C₈₃

#137

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#390

yes

characteristic

C₂²×C₈₃

#491

yes

—

C₄×C₈₃

—

C₄×C₈₃

#419

yes

—

C₄×C₈₃

#471

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#9776

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1960

yes

characteristic

C₂×C₈₃

#1977

yes

—

—

—

—

—

D₈₃

yes

characteristic

C₈₃

#9776

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#498

—

#498

—

166

—

#431

C₂×D₈₃

yes

—

C₂×#106

—

166

—

#423

—

#423

—

100

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

101

#341

—

166

—

102

#333

—

166

—

103

#332

—

166

—

104

#322

—

105

C₂²×Q₈

—

166

—

106

C₂×M₁₆

—

166

—

107

C₂×M₁₆

—

166

—

108

C₂²×C₈

D₃₃₂

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

166

—

111

C₄⋊Q₈

—

166

—

112

C₄⋊Q₈

—

166

—

113

C₂×C₄⋊C₄

—

166

—

114

C₂×C₄⋊C₄

—

166

—

115

C₂×C₄²

C₂²×D₈₃

yes

characteristic

116

#106

—

166

—

117

C₄⋊C₈

—

166

—

118

C₄⋊C₈

—

166

—

119

C₄⋊C₈

D₃₃₂

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₄×D₈₃

yes

—

123

C₄×C₈

C₂²⋊D₈₃

yes

—

124

C₂×Q₈

—

332

—

125

C₂×Q₈

—

664

—

126

C₂×Q₈

—

166

—

127

C₂²×C₄

—

332

—

128

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

129

C₂²×C₄

#13050590

yes

—

130

M₁₆

—

332

—

131

M₁₆

—

332

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

332

—

136

C₂×C₈

—

332

—

137

C₂×C₈

(C₂×D₈₃)⋊C₄

yes

—

138

C₄⋊C₄

—

332

—

139

C₄⋊C₄

—

332

—

140

C₄⋊C₄

—

166

—

141

C₄⋊C₄

—

166

—

142

C₄²

C₂×C₄×D₈₃

yes

characteristic

143

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

144

C₄²

C₂×D₃₃₂

yes

—

145

C₂³

#14581780

yes

characteristic

146

Q₈

—

664

—

147

Q₈

—

664

—

148

C₂×C₄

—

149

C₂×C₄

—

332

—

150

C₂×C₄

—

664

—

151

C₂×C₄

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

152

C₂×C₄

C₄×D₃₃₂

yes

—

153

C₂×C₄

#14581766

yes

—

154

C₂×C₄

#14581789

yes

characteristic

155

C₂×C₄

#14581825

yes

—

156

C₂×C₄

#14581825

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

664

—

160

C₈

—

664

—

161

C₂²

—

162

C₂²

#19127603

yes

characteristic

163

C₂²

#19128070

yes

—

164

C₄

—

165

C₄

—

664

—

166

C₄

#19127711

yes

—

167

C₄

#19127964

yes

—

168

C₄

#19128095

yes

characteristic

169

C₄

#19128095

yes

characteristic

170

C₂

—

171

C₂

#24908283

yes

characteristic

172

C₂

#24908366

yes

—

	subgroup	G/H
center	C₂²	#19127603
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1452	C₂
socle	C₂²×C₈₃	#390
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1452	D₈₃
hypocenter	C₈₃	#9776
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#19127603

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1452

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
92. C₈₃	#9776	semidirect

H ≅ G/N

type

92. C₈₃

#9776

semidirect

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

subgroup	index
C₂×#19128099	2
#24907601	2
#24906147	2
#24906143	2
C₈₃×#1452	2
#9776	83

subgroup

index

C₈₃×#1452

Subgroups — #35303889

Conjugacy classes of subgroups

Complements

Maximal subgroups