Group #35298107 — subgroups

Subgroups — #35298107

Order 21248.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#24907627	C₂	1	1	yes	maximal, characteristic
2	#24907601	C₂	1	1	yes	maximal, characteristic
3	#24905322	C₂	1	2	yes	maximal
4	#24905322	C₂	1	2	yes	maximal
5	C₈₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	—	2	1	no	—
7	C₂×#14581724	—	2	1	no	—
8	#19127725	C₂²	1	1	yes	characteristic
9	#19127725	C₂²	1	1	yes	characteristic
10	#19127567	C₂²	1	2	yes	—
11	#19127466	—	2	1	no	—
12	#19127466	—	2	1	no	—
13	#431×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×#13050574	—	2	1	no	—
17	C₂×#13050574	—	2	1	no	—
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	#14581726	—	4	2	no	—
20	#14581724	—	4	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
23	C₈₃⋊#95	—	2	4	no	—
24	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₈₃	—	2	2	no	—
28	C₂²⋊C₈×C₈₃	—	2	2	no	—
29	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₈₃	Q₈	1	2	yes	—
32	C₂²×Dic₈₃	—	4	2	no	—
33	#13050574	—	4	1	no	—
34	#13050574	—	4	1	no	—
35	#13050574	—	4	1	no	—
36	#13050574	—	4	1	no	—
37	C₂×C₈₃⋊C₈	—	4	1	no	—
38	C₂×C₈₃⋊C₈	—	4	1	no	—
39	C₄⋊Dic₈₃	—	4	2	no	—
40	C₄⋊Dic₈₃	—	4	2	no	—
41	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₈₃	#41	1	2	yes	—
43	C₂×C₈×C₈₃	—	2	1	no	—
44	C₂×C₈×C₈₃	—	2	1	no	—
45	C₂×C₈×C₈₃	—	2	1	no	—
46	C₂×C₈×C₈₃	—	2	1	no	—
47	C₂×C₈×C₈₃	—	2	4	no	—
48	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
50	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₄²×C₈₃	#41	1	2	yes	—
53	C₂³×C₈₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	4	2	no	—
55	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#117	1	2	yes	—
57	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#125	1	2	yes	—
59	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
63	C₂×Dic₈₃	—	8	4	no	—
64	C₂×Dic₈₃	—	8	2	no	—
65	C₈×C₈₃	—	2	2	no	—
66	C₈×C₈₃	—	2	2	no	—
67	C₈×C₈₃	—	4	4	no	—
68	C₈₃⋊C₈	—	8	1	no	—
69	C₈₃⋊C₈	—	8	1	no	—
70	C₂²×C₈₃	—	2	2	no	—
71	C₂²×C₈₃	#382	1	1	yes	characteristic
72	C₂²×C₈₃	#490	1	1	yes	characteristic
73	C₂²×C₈₃	#491	1	1	yes	characteristic
74	C₄×C₈₃	—	2	2	no	—
75	C₄×C₈₃	—	2	2	no	—
76	C₄×C₈₃	#428	1	1	yes	characteristic
77	C₄×C₈₃	#428	1	1	yes	characteristic
78	C₄×C₈₃	#489	1	1	yes	characteristic
79	C₄×C₈₃	#489	1	1	yes	characteristic
80	Dic₈₃	—	16	2	no	—
81	#8795	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	2	1	no	—
83	C₂×C₈₃	—	2	1	no	—
84	C₂×C₈₃	#1833	1	2	yes	—
85	C₂×C₈₃	#1836	1	1	yes	characteristic
86	#1846	—	83	1	no	—
87	#1837	—	83	1	no	—
88	#1452	D₈₃	1	1	yes	characteristic
89	#1286	—	83	2	no	—
90	#1286	—	83	2	no	—
91	C₈₃	#8795	1	1	yes	characteristic, complemented, Sylow
92	#431	C₂×D₈₃	1	2	yes	—
93	C₂×#107	—	166	1	no	—
94	C₂×#106	—	166	1	no	—
95	#423	—	83	1	no	—
96	#423	—	83	1	no	—
97	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
98	#382	—	83	2	no	—
99	#341	—	166	1	no	—
100	#341	—	166	1	no	—
101	#322	Dic₈₃	1	4	yes	complemented
102	C₂×M₁₆	—	166	1	no	—
103	C₂×M₁₆	—	166	1	no	—
104	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
105	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	166	2	no	—
107	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
108	#107	—	332	2	no	—
109	#106	—	332	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	166	2	no	—
112	C₄⋊C₈	—	166	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₁₆₆	1	2	yes	—
115	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
116	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
117	#95	—	166	4	no	—
118	C₂²×C₄	—	332	2	no	—
119	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
120	C₂²×C₄	#13050590	1	2	yes	—
121	M₁₆	—	332	1	no	—
122	M₁₆	—	332	1	no	—
123	M₁₆	—	332	1	no	—
124	M₁₆	—	332	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	332	1	no	—
131	C₂×C₈	—	332	1	no	—
132	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
134	C₄⋊C₄	—	332	2	no	—
135	C₄⋊C₄	—	332	2	no	—
136	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
137	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
138	C₄²	#13050590	1	2	yes	—
139	C₂³	#14581780	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	664	4	no	—
142	C₂×C₄	—	664	2	no	—
143	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
144	C₂×C₄	#14581751	1	2	yes	—
145	C₂×C₄	#14581780	1	1	yes	characteristic
146	C₂×C₄	#14581784	1	2	yes	—
147	C₂×C₄	#14581820	1	1	yes	characteristic
148	C₂×C₄	#14581820	1	1	yes	characteristic
149	C₂×C₄	#14581825	1	1	yes	characteristic
150	C₂×C₄	#14581825	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	4	no	—
154	C₈	—	664	1	no	—
155	C₈	—	664	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#19127567	1	1	yes	characteristic
158	C₂²	#19128066	1	1	yes	characteristic
159	C₂²	#19128070	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	1328	2	no	—
163	C₄	#19127743	1	1	yes	characteristic
164	C₄	#19127743	1	1	yes	characteristic
165	C₄	#19128061	1	1	yes	characteristic
166	C₄	#19128061	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#24907587	1	2	yes	—
170	C₂	#24907597	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907627

C₂

yes

maximal, characteristic

#24907601

C₂

yes

maximal, characteristic

#24905322

C₂

yes

maximal

#24905322

C₂

yes

maximal

C₈₃×#1452

C₂

yes

maximal, characteristic

C₂×#14581726

—

C₂×#14581724

—

#19127725

C₂²

yes

characteristic

#19127725

C₂²

yes

characteristic

#19127567

C₂²

yes

—

#19127466

—

#19127466

—

#431×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₄

yes

complemented

C₂×#13050574

—

C₂×#13050574

—

C₂×C₄⋊Dic₈₃

—

#14581726

—

#14581724

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂²×C₈×C₈₃

D₄

yes

characteristic

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

C₂²×Dic₈₃

—

—

—

—

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

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

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

#125

yes

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

#8795

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1833

yes

—

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

#1286

—

#1286

—

C₈₃

#8795

yes

characteristic, complemented, Sylow

#431

C₂×D₈₃

yes

—

C₂×#107

—

166

—

C₂×#106

—

166

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#382

—

#341

—

166

—

100

#341

—

166

—

101

#322

Dic₈₃

yes

complemented

102

C₂×M₁₆

—

166

—

103

C₂×M₁₆

—

166

—

104

C₂²×C₈

D₃₃₂

yes

characteristic

105

C₂²×C₈

D₃₃₂

yes

characteristic

106

C₂×C₄⋊C₄

—

166

—

107

C₂×C₄²

C₂²×D₈₃

yes

characteristic

108

#107

—

332

—

109

#106

—

332

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

166

—

112

C₄⋊C₈

—

166

—

113

C₂²⋊C₈

—

114

C₄×C₈

Dic₁₆₆

yes

—

115

C₄×C₈

C₂×Dic₈₃

yes

—

116

C₄×C₈

C₂×Dic₈₃

yes

—

117

#95

—

166

—

118

C₂²×C₄

—

332

—

119

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

120

C₂²×C₄

#13050590

yes

—

121

M₁₆

—

332

—

122

M₁₆

—

332

—

123

M₁₆

—

332

—

124

M₁₆

—

332

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

332

—

131

C₂×C₈

—

332

—

132

C₂×C₈

C₄⋊Dic₈₃

yes

—

133

C₂×C₈

C₄⋊Dic₈₃

yes

—

134

C₄⋊C₄

—

332

—

135

C₄⋊C₄

—

332

—

136

C₄²

C₂²×Dic₈₃

yes

characteristic

137

C₄²

C₂×Dic₁₆₆

yes

characteristic

138

C₄²

#13050590

yes

—

139

C₂³

#14581780

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

664

—

142

C₂×C₄

—

664

—

143

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

144

C₂×C₄

#14581751

yes

—

145

C₂×C₄

#14581780

yes

characteristic

146

C₂×C₄

#14581784

yes

—

147

C₂×C₄

#14581820

yes

characteristic

148

C₂×C₄

#14581820

yes

characteristic

149

C₂×C₄

#14581825

yes

characteristic

150

C₂×C₄

#14581825

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

664

—

155

C₈

—

664

—

156

C₂²

—

157

C₂²

#19127567

yes

characteristic

158

C₂²

#19128066

yes

characteristic

159

C₂²

#19128070

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

1328

—

163

C₄

#19127743

yes

characteristic

164

C₄

#19127743

yes

characteristic

165

C₄

#19128061

yes

characteristic

166

C₄

#19128061

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#24907587

yes

—

170

C₂

#24907597

yes

characteristic

	subgroup	G/H
center	C₂²	#19127567
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1452	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1452	D₈₃
hypocenter	C₈₃	#8795
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#19127567

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
15. #322×C₈₃	C₄	semidirect
91. C₈₃	#8795	semidirect
101. #322	Dic₈₃	semidirect

H ≅ G/N

type

15. #322×C₈₃

C₄

semidirect

91. C₈₃

#8795

semidirect

101. #322

Dic₈₃

semidirect

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

subgroup	index
#24907627	2
#24907601	2
#24905322	2
C₈₃×#1452	2
#8795	83

subgroup

index

C₈₃×#1452

Subgroups — #35298107

Conjugacy classes of subgroups

Complements

Maximal subgroups