Group #35319154 — subgroups

Subgroups — #35319154

Order 21248.

340 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 87 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	#24909712	C₂	1	2	yes	maximal, complemented
2	#24909688	C₂	1	2	yes	maximal, complemented
3	#24907078	C₂	1	1	yes	maximal, characteristic, complemented
4	#24905381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581733	—	2	4	no	—
7	#19128078	—	2	1	no	—
8	#19128078	—	2	1	no	—
9	#19128078	C₂²	1	2	yes	—
10	#19127501	—	2	4	no	—
11	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
12	#500×C₈₃	C₂²	1	1	yes	characteristic
13	#493×C₈₃	C₂²	1	1	yes	characteristic
14	#492×C₈₃	—	2	1	no	—
15	#486×C₈₃	—	2	1	no	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×D₆₆₄	—	2	2	no	—
18	C₂×D₆₆₄	—	2	2	no	—
19	C₂×D₆₆₄	—	4	2	no	—
20	C₂×D₆₆₄	—	4	2	no	—
21	#14581789	—	2	2	no	—
22	#14581733	—	4	8	no	—
23	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
24	C₂×SD₁₆×C₈₃	—	4	2	no	—
25	C₂×D₈×C₈₃	—	2	2	no	—
26	C₄⋊Q₈×C₈₃	—	2	1	no	—
27	#127×C₈₃	—	2	1	no	—
28	#107×C₈₃	—	2	2	no	—
29	#102×C₈₃	—	4	2	no	—
30	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×D₃₃₂	—	4	2	no	—
34	C₂×D₃₃₂	—	4	2	no	—
35	C₂×D₃₃₂	—	4	2	no	—
36	D₆₆₄	—	4	4	no	—
37	D₆₆₄	—	4	4	no	—
38	D₆₆₄	—	8	4	no	—
39	D₆₆₄	—	8	4	no	—
40	C₈₃⋊C₁₆	—	2	4	no	—
41	C₂×Q₈×C₈₃	—	4	1	no	—
42	C₂×D₄×C₈₃	—	4	1	no	—
43	C₂×D₄×C₈₃	—	4	1	no	—
44	C₂×D₄×C₈₃	—	4	1	no	—
45	SD₁₆×C₈₃	—	8	4	no	—
46	D₈×C₈₃	—	4	4	no	—
47	C₂×C₈×C₈₃	D₈	1	2	yes	—
48	C₂×C₈×C₈₃	D₈	1	2	yes	—
49	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
50	C₄⋊C₄×C₈₃	—	4	1	no	—
51	C₄⋊C₄×C₈₃	—	4	1	no	—
52	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂³×C₈₃	—	8	1	no	—
54	C₂²×D₈₃	—	8	2	no	—
55	Q₈×C₈₃	—	8	2	no	—
56	D₄×C₈₃	—	8	2	no	—
57	D₄×C₈₃	—	8	2	no	—
58	D₄×C₈₃	—	8	2	no	—
59	C₂×C₄×C₈₃	—	8	1	no	—
60	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
63	D₃₃₂	—	8	2	no	—
64	D₃₃₂	—	8	2	no	—
65	D₃₃₂	—	8	4	no	—
66	D₃₃₂	—	8	4	no	—
67	C₈×C₈₃	—	2	2	no	—
68	C₈×C₈₃	D₁₆	1	4	yes	complemented
69	C₈×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
70	C₂²×C₈₃	—	16	1	no	—
71	C₂²×C₈₃	—	16	1	no	—
72	C₂²×C₈₃	—	16	1	no	—
73	C₂²×C₈₃	#493	1	1	yes	characteristic
74	C₂×D₈₃	—	16	2	no	—
75	C₂×D₈₃	—	16	2	no	—
76	C₂×D₈₃	—	16	2	no	—
77	C₄×C₈₃	—	16	1	no	—
78	C₄×C₈₃	#496	1	2	yes	—
79	C₄×C₈₃	C₂×D₁₆	1	2	yes	—
80	C₄×C₈₃	#509	1	2	yes	—
81	#12525	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	32	1	no	—
83	C₂×C₈₃	#1721	1	1	yes	characteristic
84	C₂×C₈₃	#2246	1	1	yes	characteristic
85	C₂×C₈₃	#2250	1	1	yes	characteristic
86	D₈₃	—	32	2	no	—
87	#2246	—	83	2	no	—
88	#2240	—	83	2	no	—
89	#1712	—	83	1	no	—
90	#1706	D₈₃	1	1	yes	characteristic, complemented
91	#1312	—	83	1	no	—
92	C₈₃	#12525	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₁₆	—	166	4	no	—
94	#500	C₂×D₈₃	1	1	yes	characteristic
95	#493	—	83	2	no	—
96	#493	—	166	1	no	—
97	#493	—	166	1	no	—
98	#493	C₂×D₈₃	1	1	yes	characteristic
99	#492	—	2	1	no	—
100	#486	—	2	1	no	—
101	#357	—	166	4	no	—
102	C₄×C₁₆	—	83	2	no	—
103	C₈²	C₂×D₈₃	1	1	yes	characteristic
104	C₂×SD₁₆	—	4	2	no	—
105	C₂×D₈	—	2	2	no	—
106	C₂×D₈	—	332	2	no	—
107	C₂×D₈	—	332	2	no	—
108	C₂×D₈	—	166	2	no	—
109	C₂×D₈	—	166	2	no	—
110	C₄⋊Q₈	—	2	1	no	—
111	#127	—	2	1	no	—
112	#127	—	166	2	no	—
113	D₁₆	—	332	8	no	—
114	C₂×C₁₆	—	83	4	no	—
115	#107	—	2	2	no	—
116	#102	—	4	2	no	—
117	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
118	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
120	C₂×Q₈	—	4	1	no	—
121	C₂×D₄	—	4	1	no	—
122	C₂×D₄	—	4	1	no	—
123	C₂×D₄	—	4	1	no	—
124	C₂×D₄	—	332	2	no	—
125	C₂×D₄	—	332	2	no	—
126	C₂×D₄	—	332	2	no	—
127	SD₁₆	—	8	4	no	—
128	D₈	—	4	4	no	—
129	D₈	—	332	4	no	—
130	D₈	—	332	4	no	—
131	D₈	—	664	4	no	—
132	D₈	—	664	4	no	—
133	C₂×C₈	#13050575	1	2	yes	—
134	C₂×C₈	#13050575	1	2	yes	—
135	C₂×C₈	D₄×D₈₃	1	2	yes	—
136	C₄⋊C₄	—	4	1	no	—
137	C₄⋊C₄	—	4	1	no	—
138	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
139	C₁₆	—	166	4	no	—
140	C₂³	—	8	1	no	—
141	C₂³	—	664	2	no	—
142	Q₈	—	8	2	no	—
143	D₄	—	8	2	no	—
144	D₄	—	8	2	no	—
145	D₄	—	8	2	no	—
146	D₄	—	664	2	no	—
147	D₄	—	664	2	no	—
148	D₄	—	664	4	no	—
149	D₄	—	664	4	no	—
150	C₂×C₄	—	8	1	no	—
151	C₂×C₄	#14581788	1	1	yes	characteristic
152	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
153	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
154	C₈	—	2	2	no	—
155	C₈	#14581733	1	4	yes	complemented
156	C₈	#14581819	1	4	yes	—
157	C₂²	—	16	1	no	—
158	C₂²	—	16	1	no	—
159	C₂²	—	16	1	no	—
160	C₂²	—	1328	2	no	—
161	C₂²	—	1328	2	no	—
162	C₂²	—	1328	2	no	—
163	C₂²	#19128077	1	1	yes	characteristic
164	C₄	—	16	1	no	—
165	C₄	#19128087	1	2	yes	—
166	C₄	C₂×#14581733	1	2	yes	—
167	C₄	#19128136	1	2	yes	—
168	C₂	—	32	1	no	—
169	C₂	—	2656	2	no	—
170	C₂	#24907127	1	1	yes	characteristic
171	C₂	#24909712	1	1	yes	characteristic
172	C₂	#24909730	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909712

C₂

yes

maximal, complemented

#24909688

C₂

yes

maximal, complemented

#24907078

C₂

yes

maximal, characteristic, complemented

#24905381

C₂

yes

maximal, characteristic, complemented

C₈₃×#1706

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

#19127501

—

C₄×C₈₃⋊C₁₆

C₂²

yes

—

#500×C₈₃

C₂²

yes

characteristic

#493×C₈₃

C₂²

yes

characteristic

#492×C₈₃

—

#486×C₈₃

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

C₂×D₆₆₄

—

#14581789

—

#14581733

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₂×SD₁₆×C₈₃

—

C₂×D₈×C₈₃

—

C₄⋊Q₈×C₈₃

—

#127×C₈₃

—

#107×C₈₃

—

#102×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

D₆₆₄

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

C₂×D₄×C₈₃

—

SD₁₆×C₈₃

—

D₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₂³×C₈₃

—

C₂²×D₈₃

—

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

D₃₃₂

—

C₈×C₈₃

—

C₈×C₈₃

D₁₆

yes

complemented

C₈×C₈₃

AGL(1,ℤ/8)

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#493

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#496

yes

—

C₄×C₈₃

C₂×D₁₆

yes

—

C₄×C₈₃

#509

yes

—

#12525

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

—

—

—

—

D₈₃

yes

characteristic, complemented

#1312

—

C₈₃

#12525

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

166

—

#500

C₂×D₈₃

yes

characteristic

#493

—

#493

—

166

—

#493

—

166

—

#493

C₂×D₈₃

yes

characteristic

#492

—

100

#486

—

101

#357

—

166

—

102

C₄×C₁₆

—

103

C₈²

C₂×D₈₃

yes

characteristic

104

C₂×SD₁₆

—

105

C₂×D₈

—

106

C₂×D₈

—

332

—

107

C₂×D₈

—

332

—

108

C₂×D₈

—

166

—

109

C₂×D₈

—

166

—

110

C₄⋊Q₈

—

111

#127

—

112

#127

—

166

—

113

D₁₆

—

332

—

114

C₂×C₁₆

—

115

#107

—

116

#102

—

117

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

118

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

119

C₄×C₈

C₂²×D₈₃

yes

characteristic

120

C₂×Q₈

—

121

C₂×D₄

—

122

C₂×D₄

—

123

C₂×D₄

—

124

C₂×D₄

—

332

—

125

C₂×D₄

—

332

—

126

C₂×D₄

—

332

—

127

SD₁₆

—

128

D₈

—

129

D₈

—

332

—

130

D₈

—

332

—

131

D₈

—

664

—

132

D₈

—

664

—

133

C₂×C₈

#13050575

yes

—

134

C₂×C₈

#13050575

yes

—

135

C₂×C₈

D₄×D₈₃

yes

—

136

C₄⋊C₄

—

137

C₄⋊C₄

—

138

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

139

C₁₆

—

166

—

140

C₂³

—

141

C₂³

—

664

—

142

Q₈

—

143

D₄

—

144

D₄

—

145

D₄

—

146

D₄

—

664

—

147

D₄

—

664

—

148

D₄

—

664

—

149

D₄

—

664

—

150

C₂×C₄

—

151

C₂×C₄

#14581788

yes

characteristic

152

C₂×C₄

C₂×#13050575

yes

characteristic

153

C₂×C₄

C₂×#13050575

yes

characteristic

154

C₈

—

155

C₈

#14581733

yes

complemented

156

C₈

#14581819

yes

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

1328

—

161

C₂²

—

1328

—

162

C₂²

—

1328

—

163

C₂²

#19128077

yes

characteristic

164

C₄

—

165

C₄

#19128087

yes

—

166

C₄

C₂×#14581733

yes

—

167

C₄

#19128136

yes

—

168

C₂

—

169

C₂

—

2656

—

170

C₂

#24907127

yes

characteristic

171

C₂

#24909712

yes

characteristic

172

C₂

#24909730

yes

characteristic

	subgroup	G/H
center	C₂²	#19128077
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1706	C₂
socle	C₂²×C₈₃	#493
hypercenter	#1706	D₈₃
hypocenter	C₈₃	#12525
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

subgroup

G/H

center

C₂²

#19128077

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1706

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1706

N	H ≅ G/N	type
1. #24909712	C₂	semidirect
2. #24909688	C₂	semidirect
3. #24907078	C₂	semidirect
4. #24905381	C₂	semidirect
5. C₈₃×#1706	C₂	semidirect
68. C₈×C₈₃	D₁₆	semidirect
90. #1706	D₈₃	semidirect
92. C₈₃	#12525	semidirect
155. C₈	#14581733	semidirect

H ≅ G/N

type

1. #24909712

C₂

semidirect

2. #24909688

C₂

semidirect

3. #24907078

C₂

semidirect

4. #24905381

C₂

semidirect

5. C₈₃×#1706

C₂

semidirect

68. C₈×C₈₃

D₁₆

semidirect

90. #1706

D₈₃

semidirect

92. C₈₃

#12525

semidirect

155. C₈

#14581733

semidirect

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

subgroup	index
#24909712	2
#24909688	2
#24907078	2
#24905381	2
C₈₃×#1706	2
#12525	83

subgroup

index

C₈₃×#1706

Subgroups — #35319154

Conjugacy classes of subgroups

Complements

Maximal subgroups