Group #35318592 — subgroups

Subgroups — #35318592

Order 21248.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#24909514	C₂	1	2	yes	maximal
2	#24907081	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic
4	#24905513	C₂	1	2	yes	maximal
5	C₈₃×#1370	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₄	—	2	2	no	—
7	#19128106	—	2	1	no	—
8	#19128106	—	2	1	no	—
9	#19128081	C₂²	1	2	yes	—
10	C₈×Dic₁₆₆	—	2	1	no	—
11	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
12	#19127506	—	2	2	no	—
13	#19127454	C₂²	1	1	yes	characteristic
14	#19127433	—	2	1	no	—
15	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	C₂×Dic₃₃₂	D₄	1	1	yes	characteristic
20	#14581792	—	2	2	no	—
21	C₄×Dic₁₆₆	—	2	1	no	—
22	Dic₆₆₄	—	4	4	no	—
23	#14581726	—	4	2	no	—
24	#14581726	—	4	1	no	—
25	#14581726	—	4	1	no	—
26	#14581726	D₄	1	1	yes	characteristic
27	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
28	Dic₈₃⋊C₈	—	4	1	no	—
29	#14581717	—	2	1	no	—
30	C₈×Dic₈₃	—	4	1	no	—
31	C₂×C₁₆×C₈₃	—	2	2	no	—
32	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₁₆₆	—	2	1	no	—
36	C₂×Dic₁₆₆	—	4	2	no	—
37	Dic₃₃₂	—	2	1	no	—
38	Dic₃₃₂	—	4	2	no	—
39	Dic₃₃₂	—	4	2	no	—
40	Dic₃₃₂	—	8	2	no	—
41	Dic₃₃₂	D₈	1	2	yes	—
42	C₂×C₈₃⋊C₈	—	4	1	no	—
43	C₄⋊Dic₈₃	—	2	1	no	—
44	C₄⋊Dic₈₃	—	4	2	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	Dic₈₃⋊C₄	—	4	1	no	—
47	C₄×Dic₈₃	—	4	1	no	—
48	C₂×C₈×C₈₃	—	2	1	no	—
49	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
50	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
53	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
54	C₁₆×C₈₃	—	4	2	no	—
55	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₂×Dic₈₃	—	4	1	no	—
59	C₂×Dic₈₃	—	4	1	no	—
60	C₂×Dic₈₃	—	8	2	no	—
61	Dic₁₆₆	—	2	2	no	—
62	Dic₁₆₆	—	4	1	no	—
63	Dic₁₆₆	—	8	2	no	—
64	Dic₁₆₆	—	8	2	no	—
65	C₈×C₈₃	—	2	2	no	—
66	C₈×C₈₃	—	2	1	no	—
67	C₈×C₈₃	—	2	1	no	—
68	C₈×C₈₃	Q₃₂	1	2	yes	complemented
69	C₈×C₈₃	C₂×D₈	1	1	yes	characteristic
70	C₈×C₈₃	#135	1	1	yes	characteristic
71	C₈₃⋊C₈	—	8	1	no	—
72	C₂²×C₈₃	#459	1	1	yes	characteristic
73	C₄×C₈₃	—	2	1	no	—
74	C₄×C₈₃	#466	1	1	yes	characteristic
75	C₄×C₈₃	#472	1	1	yes	characteristic
76	C₄×C₈₃	C₂×Q₃₂	1	1	yes	characteristic
77	C₄×C₈₃	#510	1	1	yes	characteristic
78	Dic₈₃	—	4	2	no	—
79	Dic₈₃	—	8	1	no	—
80	Dic₈₃	—	16	2	no	—
81	#12427	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	#1667	1	1	yes	characteristic
83	C₂×C₈₃	#2209	1	1	yes	characteristic
84	C₂×C₈₃	#2214	1	1	yes	characteristic
85	#2209	—	83	2	no	—
86	#1713	—	83	1	no	—
87	C₈×Q₁₆	—	83	1	no	—
88	#1370	D₈₃	1	1	yes	characteristic
89	#1362	—	83	2	no	—
90	C₈₃	#12427	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₃₂	—	166	2	no	—
92	#500	—	166	1	no	—
93	#500	—	166	1	no	—
94	#494	—	83	2	no	—
95	C₈×Q₈	—	166	1	no	—
96	C₄×Q₁₆	—	83	1	no	—
97	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
98	#358	—	166	2	no	—
99	#335	—	83	1	no	—
100	Q₈⋊C₈	—	166	1	no	—
101	C₈²	C₂×D₈₃	1	1	yes	characteristic
102	C₂×Q₁₆	—	83	1	no	—
103	C₂×Q₁₆	—	166	2	no	—
104	C₂×Q₁₆	—	166	2	no	—
105	C₄⋊Q₈	—	166	2	no	—
106	C₄×Q₈	—	166	1	no	—
107	Q₃₂	—	332	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#107	—	332	2	no	—
110	#107	—	332	1	no	—
111	#107	—	332	1	no	—
112	#107	—	83	1	no	—
113	C₄⋊C₈	—	332	1	no	—
114	C₄⋊C₈	—	166	1	no	—
115	#103	—	166	1	no	—
116	C₄×C₈	—	332	1	no	—
117	C₄×C₈	D₃₃₂	1	1	yes	characteristic
118	C₄×C₈	D₃₃₂	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
120	C₂×Q₈	—	332	2	no	—
121	C₂×Q₈	—	166	1	no	—
122	Q₁₆	—	332	2	no	—
123	Q₁₆	—	332	2	no	—
124	Q₁₆	—	83	2	no	—
125	Q₁₆	—	664	2	no	—
126	Q₁₆	—	166	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	332	1	no	—
129	C₂×C₈	D₆₆₄	1	1	yes	characteristic
130	C₂×C₈	D₆₆₄	1	1	yes	characteristic
131	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
132	C₂×C₈	#13050591	1	1	yes	characteristic
133	C₄⋊C₄	—	332	1	no	—
134	C₄⋊C₄	—	332	2	no	—
135	C₄⋊C₄	—	332	2	no	—
136	C₄⋊C₄	—	166	1	no	—
137	C₄²	—	332	1	no	—
138	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	Q₈	—	332	1	no	—
141	Q₈	—	664	2	no	—
142	Q₈	—	664	2	no	—
143	Q₈	—	166	2	no	—
144	C₂×C₄	—	332	1	no	—
145	C₂×C₄	—	332	1	no	—
146	C₂×C₄	—	664	2	no	—
147	C₂×C₄	#14581767	1	1	yes	characteristic
148	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
149	C₂×C₄	#14581820	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	664	1	no	—
154	C₈	Dic₆₆₄	1	2	yes	complemented
155	C₈	D₈×D₈₃	1	1	yes	characteristic
156	C₈	#14581815	1	1	yes	characteristic
157	C₂²	#19127897	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	1328	2	no	—
160	C₄	—	332	2	no	—
161	C₄	—	664	1	no	—
162	C₄	#19127938	1	1	yes	characteristic
163	C₄	#19127972	1	1	yes	characteristic
164	C₄	C₂×Dic₆₆₄	1	1	yes	characteristic
165	C₄	#19128144	1	1	yes	characteristic
166	C₂	#24906819	1	1	yes	characteristic
167	C₂	#24909514	1	1	yes	characteristic
168	C₂	#24909545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909514

C₂

yes

maximal

#24907081

C₂

yes

maximal, characteristic

C₈×Dic₃₃₂

C₂

yes

maximal, characteristic

#24905513

C₂

yes

maximal

C₈₃×#1370

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

—

#19128106

—

#19128106

—

#19128081

C₂²

yes

—

C₈×Dic₁₆₆

—

C₄×Dic₃₃₂

C₂²

yes

characteristic

#19127506

—

#19127454

C₂²

yes

characteristic

#19127433

—

C₄⋊C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

D₄

yes

characteristic

#14581792

—

C₄×Dic₁₆₆

—

Dic₆₆₄

—

—

—

—

D₄

yes

characteristic

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

#14581717

—

C₈×Dic₈₃

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

D₈

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

Q₃₂

yes

complemented

C₈×C₈₃

C₂×D₈

yes

characteristic

C₈×C₈₃

#135

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

#459

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×Q₃₂

yes

characteristic

C₄×C₈₃

#510

yes

characteristic

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

C₈×Q₁₆

—

#1370

D₈₃

yes

characteristic

#1362

—

C₈₃

#12427

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

166

—

#500

—

166

—

#500

—

166

—

#494

—

C₈×Q₈

—

166

—

C₄×Q₁₆

—

C₄⋊C₁₆

C₂×D₈₃

yes

—

#358

—

166

—

#335

—

100

Q₈⋊C₈

—

166

—

101

C₈²

C₂×D₈₃

yes

characteristic

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

166

—

104

C₂×Q₁₆

—

166

—

105

C₄⋊Q₈

—

166

—

106

C₄×Q₈

—

166

—

107

Q₃₂

—

332

—

108

C₂×C₁₆

—

109

#107

—

332

—

110

#107

—

332

—

111

#107

—

332

—

112

#107

—

113

C₄⋊C₈

—

332

—

114

C₄⋊C₈

—

166

—

115

#103

—

166

—

116

C₄×C₈

—

332

—

117

C₄×C₈

D₃₃₂

yes

characteristic

118

C₄×C₈

D₃₃₂

yes

characteristic

119

C₄×C₈

C₂²×D₈₃

yes

characteristic

120

C₂×Q₈

—

332

—

121

C₂×Q₈

—

166

—

122

Q₁₆

—

332

—

123

Q₁₆

—

332

—

124

Q₁₆

—

125

Q₁₆

—

664

—

126

Q₁₆

—

166

—

127

C₂×C₈

—

128

C₂×C₈

—

332

—

129

C₂×C₈

D₆₆₄

yes

characteristic

130

C₂×C₈

D₆₆₄

yes

characteristic

131

C₂×C₈

D₄×D₈₃

yes

characteristic

132

C₂×C₈

#13050591

yes

characteristic

133

C₄⋊C₄

—

332

—

134

C₄⋊C₄

—

332

—

135

C₄⋊C₄

—

332

—

136

C₄⋊C₄

—

166

—

137

C₄²

—

332

—

138

C₄²

C₂×D₃₃₂

yes

characteristic

139

C₁₆

—

140

Q₈

—

332

—

141

Q₈

—

664

—

142

Q₈

—

664

—

143

Q₈

—

166

—

144

C₂×C₄

—

332

—

145

C₂×C₄

—

332

—

146

C₂×C₄

—

664

—

147

C₂×C₄

#14581767

yes

characteristic

148

C₂×C₄

C₂×D₆₆₄

yes

characteristic

149

C₂×C₄

#14581820

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

664

—

154

C₈

Dic₆₆₄

yes

complemented

155

C₈

D₈×D₈₃

yes

characteristic

156

C₈

#14581815

yes

characteristic

157

C₂²

#19127897

yes

characteristic

158

C₄

—

159

C₄

—

1328

—

160

C₄

—

332

—

161

C₄

—

664

—

162

C₄

#19127938

yes

characteristic

163

C₄

#19127972

yes

characteristic

164

C₄

C₂×Dic₆₆₄

yes

characteristic

165

C₄

#19128144

yes

characteristic

166

C₂

#24906819

yes

characteristic

167

C₂

#24909514

yes

characteristic

168

C₂

#24909545

yes

characteristic

	subgroup	G/H
center	C₂²	#19127897
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1370	C₂
socle, prime radical	C₂²×C₈₃	#459
hypercenter	#1370	D₈₃
hypocenter	C₈₃	#12427
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#19127897

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1370

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
68. C₈×C₈₃	Q₃₂	semidirect
90. C₈₃	#12427	semidirect
154. C₈	Dic₆₆₄	semidirect

H ≅ G/N

type

68. C₈×C₈₃

Q₃₂

semidirect

90. C₈₃

#12427

semidirect

154. C₈

Dic₆₆₄

semidirect

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

subgroup	index
#24909514	2
#24907081	2
C₈×Dic₃₃₂	2
#24905513	2
C₈₃×#1370	2
#12427	83

subgroup

index

C₈×Dic₃₃₂

C₈₃×#1370

Subgroups — #35318592

Conjugacy classes of subgroups

Complements

Maximal subgroups