Group #35298080 — subgroups

Subgroups — #35298080

Order 21248.

342 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 147 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	#24907615	C₂	1	1	yes	maximal, characteristic
2	#24907209	C₂	1	1	yes	maximal, characteristic
3	#24905297	C₂	1	4	yes	maximal
4	C₈₃×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
8	#19127456	C₄	1	2	yes	—
9	#19127456	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
11	C₄³×C₈₃	C₂²	1	1	yes	characteristic
12	#336×C₈₃	—	2	2	no	—
13	C₂²×C₈₃⋊C₈	—	2	2	no	—
14	C₂²×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
15	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
16	C₄⋊(C₈₃⋊C₈)	—	4	4	no	—
17	C₄×C₈₃⋊C₈	—	2	2	no	—
18	C₄×C₈₃⋊C₈	—	2	2	no	—
19	C₂²×C₈×C₈₃	—	2	2	no	—
20	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₈₃	D₄	1	2	yes	—
23	C₂×C₄²×C₈₃	Q₈	1	2	yes	—
24	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
25	C₄⋊C₈×C₈₃	—	2	4	no	—
26	C₂×C₈₃⋊C₈	—	2	4	no	—
27	C₂×C₈₃⋊C₈	—	2	4	no	—
28	C₂×C₈₃⋊C₈	—	2	4	no	—
29	C₂×C₈₃⋊C₈	—	4	2	no	—
30	C₂×C₈₃⋊C₈	—	4	4	no	—
31	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
32	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
33	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₈×C₈₃	—	4	2	no	—
37	C₂×C₈×C₈₃	—	4	2	no	—
38	C₂×C₈×C₈₃	—	4	2	no	—
39	C₄²×C₈₃	—	2	4	no	—
40	C₄²×C₈₃	—	2	2	no	—
41	C₄²×C₈₃	—	2	2	no	—
42	C₄²×C₈₃	—	2	2	no	—
43	C₄²×C₈₃	—	2	2	no	—
44	C₄²×C₈₃	#41	1	2	yes	—
45	C₄²×C₈₃	#41	1	2	yes	—
46	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	—	2	2	no	—
48	C₂×C₄×C₈₃	—	2	2	no	—
49	C₂×C₄×C₈₃	—	2	2	no	—
50	C₂×C₄×C₈₃	—	2	2	no	—
51	C₂×C₄×C₈₃	—	2	2	no	—
52	C₂×C₄×C₈₃	—	2	2	no	—
53	C₂×C₄×C₈₃	#108	1	4	yes	—
54	C₂×C₄×C₈₃	#117	1	2	yes	—
55	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
57	C₂×C₄×C₈₃	#125	1	2	yes	—
58	C₂×C₄×C₈₃	#131	1	2	yes	—
59	C₂×C₄×C₈₃	#131	1	2	yes	—
60	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₄×C₈₃	#137	1	2	yes	—
62	C₈×C₈₃	—	8	2	no	—
63	C₈₃⋊C₈	—	4	4	no	—
64	C₈₃⋊C₈	—	8	2	no	—
65	C₂²×C₈₃	#382	1	1	yes	characteristic
66	C₂²×C₈₃	#424	1	1	yes	characteristic
67	C₂²×C₈₃	#424	1	1	yes	characteristic
68	C₂²×C₈₃	#428	1	2	yes	—
69	C₂²×C₈₃	C₂×#108	1	2	yes	—
70	C₄×C₈₃	—	2	4	no	—
71	C₄×C₈₃	—	2	2	no	—
72	C₄×C₈₃	—	4	2	no	—
73	C₄×C₈₃	#433	1	4	yes	—
74	C₄×C₈₃	#489	1	2	yes	—
75	C₄×C₈₃	#490	1	1	yes	characteristic
76	C₄×C₈₃	#491	1	1	yes	characteristic
77	#8789	—	83	1	no	maximal, Sylow
78	C₂×C₈₃	#1573	1	2	yes	—
79	C₂×C₈₃	#1574	1	2	yes	—
80	C₂×C₈₃	#1832	1	1	yes	characteristic
81	C₂×C₈₃	#1836	1	1	yes	characteristic
82	C₂×C₈₃	#1837	1	1	yes	characteristic
83	#1842	—	83	1	no	—
84	#1831	D₈₃	1	1	yes	characteristic
85	#1744	—	83	1	no	—
86	#1277	—	83	4	no	—
87	C₈₃	#8789	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₄⋊C₈	—	83	2	no	—
89	C₂×C₄⋊C₈	—	166	2	no	—
90	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
91	C₂×C₄×C₈	—	83	2	no	—
92	C₄³	C₂×D₈₃	1	1	yes	characteristic
93	#336	—	2	2	no	—
94	#336	—	83	2	no	—
95	#336	—	83	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	83	4	no	—
98	C₂²×C₈	—	166	2	no	—
99	C₂×C₄²	Dic₁₆₆	1	2	yes	—
100	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
101	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₈₃	1	2	yes	—
103	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
104	C₄⋊C₈	—	2	4	no	—
105	C₄⋊C₈	—	332	4	no	—
106	C₄⋊C₈	—	166	4	no	—
107	C₄×C₈	—	166	2	no	—
108	C₄×C₈	—	166	2	no	—
109	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
110	C₂²×C₄	Dic₈₃⋊C₄	1	2	yes	—
111	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
112	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₆₆	1	1	yes	characteristic
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	4	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	332	2	no	—
118	C₂×C₈	—	332	4	no	—
119	C₂×C₈	—	166	4	no	—
120	C₂×C₈	—	166	4	no	—
121	C₂×C₈	—	166	4	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	2	4	no	—
127	C₄²	#13050589	1	2	yes	—
128	C₄²	#13050591	1	2	yes	—
129	C₂³	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#14581727	1	4	yes	—
137	C₂×C₄	#14581750	1	2	yes	—
138	C₂×C₄	#14581779	1	1	yes	characteristic
139	C₂×C₄	#14581781	1	1	yes	characteristic
140	C₂×C₄	#14581783	1	2	yes	—
141	C₂×C₄	#14581798	1	2	yes	—
142	C₂×C₄	#14581800	1	2	yes	—
143	C₂×C₄	#14581817	1	2	yes	—
144	C₂×C₄	#14581822	1	2	yes	—
145	C₈	—	8	2	no	—
146	C₈	—	332	4	no	—
147	C₈	—	664	2	no	—
148	C₂²	#19127566	1	1	yes	characteristic
149	C₂²	#19127727	1	1	yes	characteristic
150	C₂²	#19127729	1	1	yes	characteristic
151	C₂²	#19127744	1	2	yes	—
152	C₂²	C₂×#14581727	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	#19127761	1	4	yes	—
157	C₄	#19128060	1	2	yes	—
158	C₄	#19128067	1	1	yes	characteristic
159	C₄	#19128071	1	1	yes	characteristic
160	C₂	#24906249	1	2	yes	—
161	C₂	#24906251	1	2	yes	—
162	C₂	#24907582	1	1	yes	characteristic
163	C₂	#24907596	1	1	yes	characteristic
164	C₂	#24907600	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907615

C₂

yes

maximal, characteristic

#24907209

C₂

yes

maximal, characteristic

#24905297

C₂

yes

maximal

C₈₃×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₈₃⋊C₈)

—

C₂×C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127456

C₄

yes

—

#19127456

C₄

yes

—

C₂×C₄⋊C₈×C₈₃

C₂²

yes

—

C₄³×C₈₃

C₂²

yes

characteristic

#336×C₈₃

—

C₂²×C₈₃⋊C₈

—

C₂²×C₈₃⋊C₈

C₂×C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

—

C₄×C₈₃⋊C₈

—

C₂²×C₈×C₈₃

—

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

Q₈

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₄⋊C₄

yes

—

C₂²×C₄×C₈₃

C₄⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

#41

yes

—

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#108

yes

—

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

—

C₂×C₄×C₈₃

#131

yes

—

C₂×C₄×C₈₃

#131

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

C₂²×C₈₃

yes

—

C₂²×C₈₃

C₂×#108

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#433

yes

—

C₄×C₈₃

#489

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

yes

—

C₂×C₈₃

#1574

yes

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

yes

characteristic

#1744

—

#1277

—

C₈₃

#8789

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

166

—

C₂×C₄⋊C₈

C₂×D₈₃

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₈₃

yes

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

166

—

C₂×C₄²

Dic₁₆₆

yes

—

100

C₂×C₄²

C₄×D₈₃

yes

characteristic

101

C₂×C₄²

C₄×D₈₃

yes

characteristic

102

C₂×C₄²

C₂²⋊D₈₃

yes

—

103

C₂×C₄²

C₂²×D₈₃

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

332

—

106

C₄⋊C₈

—

166

—

107

C₄×C₈

—

166

—

108

C₄×C₈

—

166

—

109

C₂²×C₄

Dic₈₃⋊C₄

yes

—

110

C₂²×C₄

Dic₈₃⋊C₄

yes

—

111

C₂²×C₄

C₂×C₄×D₈₃

yes

characteristic

112

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₆₆

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

332

—

118

C₂×C₈

—

332

—

119

C₂×C₈

—

166

—

120

C₂×C₈

—

166

—

121

C₂×C₈

—

166

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#13050589

yes

—

128

C₄²

#13050591

yes

—

129

C₂³

C₂×Dic₈₃⋊C₄

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#14581727

yes

—

137

C₂×C₄

#14581750

yes

—

138

C₂×C₄

#14581779

yes

characteristic

139

C₂×C₄

#14581781

yes

characteristic

140

C₂×C₄

#14581783

yes

—

141

C₂×C₄

#14581798

yes

—

142

C₂×C₄

#14581800

yes

—

143

C₂×C₄

#14581817

yes

—

144

C₂×C₄

#14581822

yes

—

145

C₈

—

146

C₈

—

332

—

147

C₈

—

664

—

148

C₂²

#19127566

yes

characteristic

149

C₂²

#19127727

yes

characteristic

150

C₂²

#19127729

yes

characteristic

151

C₂²

#19127744

yes

—

152

C₂²

C₂×#14581727

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#19127761

yes

—

157

C₄

#19128060

yes

—

158

C₄

#19128067

yes

characteristic

159

C₄

#19128071

yes

characteristic

160

C₂

#24906249

yes

—

161

C₂

#24906251

yes

—

162

C₂

#24907582

yes

characteristic

163

C₂

#24907596

yes

characteristic

164

C₂

#24907600

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₃
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1831	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1831	D₈₃
hypocenter	C₈₃	#8789
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₈₃

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1831

C₂

socle, prime radical

C₂³×C₈₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
87. C₈₃	#8789	semidirect

H ≅ G/N

type

87. C₈₃

#8789

semidirect

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

subgroup	index
#24907615	2
#24907209	2
#24905297	2
C₈₃×#1831	2
#8789	83

subgroup

index

C₈₃×#1831

Subgroups — #35298080

Conjugacy classes of subgroups

Complements

Maximal subgroups