Group #35298189 — subgroups

Subgroups — #35298189

Order 21248.

438 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 227 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	#24907623	C₂	1	1	yes	maximal, characteristic
2	#24907620	C₂	1	1	yes	maximal, characteristic
3	#24907620	C₂	1	2	yes	maximal
4	#24907591	C₂	1	2	yes	maximal
5	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	—	2	2	no	—
7	C₂×#14581724	—	2	4	no	—
8	C₂×#14581724	—	2	2	no	—
9	#19127571	C₂²	1	1	yes	characteristic
10	#19127571	C₂²	1	2	yes	—
11	#19127571	C₂²	1	1	yes	characteristic
12	#19127464	—	2	4	no	—
13	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	#322×C₈₃	C₄	1	4	yes	complemented
16	C₂×C₄⋊Dic₈₃	—	2	2	no	—
17	C₂×C₄⋊Dic₈₃	—	2	2	no	—
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	C₂×C₄⋊Dic₈₃	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	—	2	1	no	—
22	C₂×C₄⋊Dic₈₃	—	2	1	no	—
23	#14581726	—	4	4	no	—
24	#14581724	—	4	8	no	—
25	#14581724	—	4	4	no	—
26	C₂²×C₈×C₈₃	D₄	1	4	yes	—
27	C₂²×C₈×C₈₃	D₄	1	2	yes	—
28	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₃	Q₈	1	4	yes	—
32	C₄×C₈×C₈₃	Q₈	1	2	yes	—
33	C₂²×Dic₈₃	—	4	1	no	—
34	C₂²×Dic₈₃	—	4	1	no	—
35	C₂²×Dic₈₃	—	4	2	no	—
36	C₄⋊Dic₈₃	—	4	4	no	—
37	C₄⋊Dic₈₃	—	4	2	no	—
38	C₄⋊Dic₈₃	—	4	2	no	—
39	C₄⋊Dic₈₃	—	4	4	no	—
40	C₄⋊Dic₈₃	—	4	4	no	—
41	C₄⋊Dic₈₃	—	4	4	no	—
42	C₄⋊Dic₈₃	—	4	4	no	—
43	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₈₃	—	2	4	no	—
46	C₂×C₈×C₈₃	—	2	4	no	—
47	C₂×C₈×C₈₃	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
49	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
50	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
51	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
53	C₄²×C₈₃	C₂×Q₈	1	2	yes	—
54	C₂³×C₈₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
62	C₂×C₄×C₈₃	#137	1	4	yes	—
63	C₂×Dic₈₃	—	8	2	no	—
64	C₂×Dic₈₃	—	8	1	no	—
65	C₂×Dic₈₃	—	8	1	no	—
66	C₂×Dic₈₃	—	8	2	no	—
67	C₂×Dic₈₃	—	8	2	no	—
68	C₂×Dic₈₃	—	8	2	no	—
69	C₂×Dic₈₃	—	8	2	no	—
70	C₈×C₈₃	—	2	8	no	—
71	C₈×C₈₃	#106	1	8	yes	complemented
72	C₂²×C₈₃	#384	1	1	yes	characteristic
73	C₂²×C₈₃	#492	1	1	yes	characteristic
74	C₂²×C₈₃	#492	1	1	yes	characteristic
75	C₂²×C₈₃	#496	1	2	yes	—
76	C₂²×C₈₃	#497	1	2	yes	—
77	C₄×C₈₃	C₂×#106	1	2	yes	—
78	C₄×C₈₃	#428	1	4	yes	—
79	C₄×C₈₃	#498	1	2	yes	—
80	C₄×C₈₃	#501	1	4	yes	—
81	Dic₈₃	—	16	2	no	—
82	Dic₈₃	—	16	1	no	—
83	Dic₈₃	—	16	1	no	—
84	#8819	—	83	1	no	maximal, Sylow
85	C₂×C₈₃	#1714	1	2	yes	—
86	C₂×C₈₃	#1715	1	1	yes	characteristic
87	C₂×C₈₃	#1716	1	1	yes	characteristic
88	C₂×C₈₃	#1844	1	1	yes	characteristic
89	C₂×C₈₃	#1847	1	2	yes	—
90	#1845	—	83	1	no	—
91	#1844	—	83	1	no	—
92	#1844	—	83	2	no	—
93	#1834	—	83	2	no	—
94	C₂×#322	D₈₃	1	1	yes	characteristic
95	C₈₃	#8819	1	1	yes	characteristic, complemented, Sylow
96	C₂×#107	—	166	2	no	—
97	C₂×#106	—	166	4	no	—
98	C₂×#106	—	166	2	no	—
99	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
101	#384	—	83	1	no	—
102	#384	—	83	2	no	—
103	#384	—	83	1	no	—
104	#340	—	166	4	no	—
105	#322	Dic₈₃	1	4	yes	complemented
106	C₂²×C₈	D₃₃₂	1	4	yes	—
107	C₂²×C₈	D₃₃₂	1	2	yes	—
108	C₂×C₄⋊C₄	—	166	2	no	—
109	C₂×C₄⋊C₄	—	166	2	no	—
110	C₂×C₄⋊C₄	—	166	2	no	—
111	C₂×C₄⋊C₄	—	166	2	no	—
112	C₂×C₄⋊C₄	—	166	2	no	—
113	C₂×C₄⋊C₄	—	166	1	no	—
114	C₂×C₄⋊C₄	—	166	1	no	—
115	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
116	#107	—	332	4	no	—
117	#106	—	332	8	no	—
118	#106	—	332	4	no	—
119	C₄×C₈	Dic₁₆₆	1	4	yes	—
120	C₄×C₈	Dic₁₆₆	1	2	yes	—
121	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
122	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
123	C₂²×C₄	—	332	1	no	—
124	C₂²×C₄	—	332	1	no	—
125	C₂²×C₄	—	332	2	no	—
126	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
127	C₂²×C₄	C₂×D₃₃₂	1	2	yes	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	C₄⋊Dic₈₃	1	8	yes	—
131	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
132	C₂×C₈	#13050579	1	4	yes	—
133	C₂×C₈	#13050579	1	4	yes	—
134	C₄⋊C₄	—	332	4	no	—
135	C₄⋊C₄	—	332	2	no	—
136	C₄⋊C₄	—	332	2	no	—
137	C₄⋊C₄	—	332	4	no	—
138	C₄⋊C₄	—	332	4	no	—
139	C₄⋊C₄	—	332	4	no	—
140	C₄⋊C₄	—	332	4	no	—
141	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
142	C₄²	C₂×Dic₁₆₆	1	2	yes	—
143	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
144	C₂³	#14581789	1	1	yes	characteristic
145	C₂×C₄	—	664	2	no	—
146	C₂×C₄	—	664	1	no	—
147	C₂×C₄	—	664	1	no	—
148	C₂×C₄	—	664	2	no	—
149	C₂×C₄	—	664	2	no	—
150	C₂×C₄	—	664	2	no	—
151	C₂×C₄	—	664	2	no	—
152	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
154	C₂×C₄	#14581792	1	1	yes	characteristic
155	C₂×C₄	#14581792	1	2	yes	—
156	C₂×C₄	C₂×#13050579	1	2	yes	—
157	C₂×C₄	C₂×#13050579	1	2	yes	—
158	C₂×C₄	#14581820	1	4	yes	—
159	C₂×C₄	#14581825	1	4	yes	—
160	C₈	—	2	8	no	—
161	C₈	#14581724	1	8	yes	complemented
162	C₂²	#19127571	1	1	yes	characteristic
163	C₂²	#19128074	1	1	yes	characteristic
164	C₂²	#19128074	1	1	yes	characteristic
165	C₂²	#19128090	1	2	yes	—
166	C₂²	#19128095	1	2	yes	—
167	C₄	—	1328	2	no	—
168	C₄	—	1328	1	no	—
169	C₄	—	1328	1	no	—
170	C₄	C₂×#14581724	1	2	yes	—
171	C₄	#19127743	1	4	yes	—
172	C₄	#19128099	1	2	yes	—
173	C₄	#19128110	1	4	yes	—
174	C₂	#24907085	1	2	yes	—
175	C₂	#24907091	1	1	yes	characteristic
176	C₂	#24907096	1	1	yes	characteristic
177	C₂	#24907620	1	1	yes	characteristic
178	C₂	#24907631	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907623

C₂

yes

maximal, characteristic

#24907620

C₂

yes

maximal, characteristic

#24907620

C₂

yes

maximal

#24907591

C₂

yes

maximal

C₂×#322×C₈₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#19127571

C₂²

yes

—

#19127571

C₂²

yes

characteristic

#19127464

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₄

yes

complemented

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581726

—

#14581724

—

#14581724

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

—

C₂³×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

#497

yes

—

C₄×C₈₃

C₂×#106

yes

—

C₄×C₈₃

#428

yes

—

C₄×C₈₃

#498

yes

—

C₄×C₈₃

#501

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

#8819

—

maximal, Sylow

C₂×C₈₃

#1714

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

—

—

—

D₈₃

yes

characteristic

C₈₃

#8819

yes

characteristic, complemented, Sylow

C₂×#107

—

166

—

C₂×#106

—

166

—

C₂×#106

—

166

—

C₂×C₄×C₈

C₂×D₈₃

yes

—

100

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

101

#384

—

102

#384

—

103

#384

—

104

#340

—

166

—

105

#322

Dic₈₃

yes

complemented

106

C₂²×C₈

D₃₃₂

yes

—

107

C₂²×C₈

D₃₃₂

yes

—

108

C₂×C₄⋊C₄

—

166

—

109

C₂×C₄⋊C₄

—

166

—

110

C₂×C₄⋊C₄

—

166

—

111

C₂×C₄⋊C₄

—

166

—

112

C₂×C₄⋊C₄

—

166

—

113

C₂×C₄⋊C₄

—

166

—

114

C₂×C₄⋊C₄

—

166

—

115

C₂×C₄²

C₂²×D₈₃

yes

characteristic

116

#107

—

332

—

117

#106

—

332

—

118

#106

—

332

—

119

C₄×C₈

Dic₁₆₆

yes

—

120

C₄×C₈

Dic₁₆₆

yes

—

121

C₄×C₈

C₂×Dic₈₃

yes

—

122

C₄×C₈

C₂×Dic₈₃

yes

—

123

C₂²×C₄

—

332

—

124

C₂²×C₄

—

332

—

125

C₂²×C₄

—

332

—

126

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

127

C₂²×C₄

C₂×D₃₃₂

yes

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₄⋊Dic₈₃

yes

—

131

C₂×C₈

C₄⋊Dic₈₃

yes

—

132

C₂×C₈

#13050579

yes

—

133

C₂×C₈

#13050579

yes

—

134

C₄⋊C₄

—

332

—

135

C₄⋊C₄

—

332

—

136

C₄⋊C₄

—

332

—

137

C₄⋊C₄

—

332

—

138

C₄⋊C₄

—

332

—

139

C₄⋊C₄

—

332

—

140

C₄⋊C₄

—

332

—

141

C₄²

C₂²×Dic₈₃

yes

characteristic

142

C₄²

C₂×Dic₁₆₆

yes

—

143

C₄²

C₂×Dic₁₆₆

yes

characteristic

144

C₂³

#14581789

yes

characteristic

145

C₂×C₄

—

664

—

146

C₂×C₄

—

664

—

147

C₂×C₄

—

664

—

148

C₂×C₄

—

664

—

149

C₂×C₄

—

664

—

150

C₂×C₄

—

664

—

151

C₂×C₄

—

664

—

152

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

153

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

154

C₂×C₄

#14581792

yes

characteristic

155

C₂×C₄

#14581792

yes

—

156

C₂×C₄

C₂×#13050579

yes

—

157

C₂×C₄

C₂×#13050579

yes

—

158

C₂×C₄

#14581820

yes

—

159

C₂×C₄

#14581825

yes

—

160

C₈

—

161

C₈

#14581724

yes

complemented

162

C₂²

#19127571

yes

characteristic

163

C₂²

#19128074

yes

characteristic

164

C₂²

#19128074

yes

characteristic

165

C₂²

#19128090

yes

—

166

C₂²

#19128095

yes

—

167

C₄

—

1328

—

168

C₄

—

1328

—

169

C₄

—

1328

—

170

C₄

C₂×#14581724

yes

—

171

C₄

#19127743

yes

—

172

C₄

#19128099

yes

—

173

C₄

#19128110

yes

—

174

C₂

#24907085

yes

—

175

C₂

#24907091

yes

characteristic

176

C₂

#24907096

yes

characteristic

177

C₂

#24907620

yes

characteristic

178

C₂

#24907631

yes

—

	subgroup	G/H
center	C₂³	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#322×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#127
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#322	D₈₃
hypocenter	C₈₃	#8819
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#14581789

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#322×C₈₃

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 ◁ C₂×#322

N	H ≅ G/N	type
15. #322×C₈₃	C₄	semidirect
71. C₈×C₈₃	#106	semidirect
95. C₈₃	#8819	semidirect
105. #322	Dic₈₃	semidirect
161. C₈	#14581724	semidirect

H ≅ G/N

type

15. #322×C₈₃

C₄

semidirect

71. C₈×C₈₃

#106

semidirect

95. C₈₃

#8819

semidirect

105. #322

Dic₈₃

semidirect

161. C₈

#14581724

semidirect

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

subgroup	index
#24907623	2
#24907620	2
#24907591	2
C₂×#322×C₈₃	2
#8819	83

subgroup

index

C₂×#322×C₈₃

Subgroups — #35298189

Conjugacy classes of subgroups

Complements

Maximal subgroups