Group #35319784 — subgroups

Subgroups — #35319784

Order 21248.

308 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 131 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	#24909237	C₂	1	2	yes	maximal
2	M₃₂×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24907345	C₂	1	2	yes	maximal
4	#24905565	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	1	yes	characteristic
7	#19127739	C₂²	1	2	yes	—
8	#19127659	—	2	2	no	—
9	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
10	#19127525	C₄	1	8	yes	complemented
11	#19127476	C₄	1	2	yes	complemented
12	C₁₆×Dic₈₃	C₄	1	2	yes	complemented
13	#19127462	—	2	2	no	—
14	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#108×C₈₃	C₂²	1	2	yes	—
16	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
17	#14581751	—	2	2	no	—
18	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
19	#14581732	C₂×C₄	1	4	yes	—
20	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
21	#14581726	—	2	2	no	—
22	#14581724	C₂×C₄	1	4	yes	—
23	#14581702	—	4	2	no	—
24	C₈×Dic₈₃	—	4	2	no	—
25	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
27	C₈×Dic₈₃	Q₈	1	1	yes	characteristic
28	C₈×Dic₈₃	Q₈	1	1	yes	characteristic
29	C₂×M₁₆×C₈₃	—	2	2	no	—
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	M₃₂×C₈₃	—	2	2	no	—
32	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
33	#108×C₈₃	—	2	2	no	—
34	#108×C₈₃	C₂×C₄	1	4	yes	—
35	C₂²×Dic₈₃	—	2	1	no	—
36	C₂×C₈₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₈₃⋊C₈	SD₁₆	1	2	yes	—
39	C₂×C₈₃⋊C₈	SD₁₆	1	2	yes	—
40	C₄⋊Dic₈₃	—	2	4	no	—
41	C₂²⋊Dic₈₃	—	4	2	no	—
42	C₄×Dic₈₃	—	2	1	no	—
43	C₄×Dic₈₃	—	4	2	no	—
44	C₄×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₄×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₈₃⋊C₁₆	C₄²	1	4	yes	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₈₃	—	2	4	no	—
49	M₁₆×C₈₃	—	4	2	no	—
50	C₂×C₈×C₈₃	—	4	2	no	—
51	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₃	#41	1	2	yes	—
55	C₁₆×C₈₃	—	2	2	no	—
56	C₂³×C₈₃	#135	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
60	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
61	C₂×Dic₈₃	—	2	1	no	—
62	C₂×Dic₈₃	—	2	1	no	—
63	C₂×Dic₈₃	—	4	2	no	—
64	C₂×Dic₈₃	—	4	4	no	—
65	C₈×C₈₃	—	4	4	no	—
66	C₈×C₈₃	C₂×C₄²	1	1	yes	characteristic
67	C₈×C₈₃	#117	1	1	yes	characteristic
68	C₈×C₈₃	C₄×Q₈	1	2	yes	—
69	C₈₃⋊C₈	#106	1	1	yes	characteristic
70	C₈₃⋊C₈	#106	1	2	yes	—
71	C₈₃⋊C₈	#106	1	1	yes	characteristic
72	C₂²×C₈₃	—	4	1	no	—
73	C₂²×C₈₃	#427	1	1	yes	characteristic
74	C₂²×C₈₃	C₄×SD₁₆	1	2	yes	—
75	C₄×C₈₃	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
77	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
78	Dic₈₃	—	2	2	no	—
79	Dic₈₃	—	4	1	no	—
80	Dic₈₃	—	8	4	no	—
81	#12668	—	83	1	no	maximal, Sylow
82	C₂×C₈₃	—	2	1	no	—
83	C₂×C₈₃	—	2	2	no	—
84	C₂×C₈₃	C₄×#106	1	1	yes	characteristic
85	#2155	—	83	2	no	—
86	C₄×M₃₂	—	83	1	no	—
87	#1772	—	83	2	no	—
88	#1385	D₈₃	1	1	yes	characteristic
89	#1384	—	83	1	no	—
90	C₈₃	#12668	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₃₂	—	83	1	no	—
92	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
93	C₂×#108	C₂×D₈₃	1	2	yes	—
94	#427	—	83	2	no	—
95	#405	—	166	2	no	—
96	C₂×C₄×C₈	—	83	1	no	—
97	#365	—	83	8	no	—
98	#346	—	83	2	no	—
99	C₄×C₁₆	—	83	2	no	—
100	#339	—	166	2	no	—
101	C₂×M₁₆	—	2	2	no	—
102	C₂²×C₈	—	83	1	no	—
103	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
104	#117	—	166	2	no	—
105	C₂×C₄²	—	83	1	no	—
106	M₃₂	—	2	2	no	—
107	M₃₂	—	83	4	no	—
108	C₂×C₁₆	—	83	2	no	—
109	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
110	#108	—	2	2	no	—
111	#108	C₄×D₈₃	1	4	yes	—
112	#107	—	166	2	no	—
113	#106	—	83	4	no	—
114	#97	—	332	2	no	—
115	C₄×C₈	—	332	2	no	—
116	C₄×C₈	—	83	1	no	—
117	C₄×C₈	—	83	1	no	—
118	C₄×C₈	—	83	1	no	—
119	C₄×C₈	—	83	1	no	—
120	C₂²×C₄	—	166	1	no	—
121	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
122	M₁₆	—	2	4	no	—
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	83	2	no	—
126	C₂×C₈	—	83	2	no	—
127	C₂×C₈	—	83	1	no	—
128	C₂×C₈	—	83	1	no	—
129	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
130	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
131	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
132	C₂×C₈	#13050589	1	2	yes	—
133	C₄⋊C₄	—	166	4	no	—
134	C₂²⋊C₄	—	332	2	no	—
135	C₄²	—	332	2	no	—
136	C₄²	—	83	1	no	—
137	C₄²	—	83	1	no	—
138	C₄²	—	166	1	no	—
139	C₁₆	—	2	2	no	—
140	C₁₆	—	83	4	no	—
141	C₂³	#14581815	1	1	yes	characteristic
142	C₂×C₄	—	332	2	no	—
143	C₂×C₄	—	332	4	no	—
144	C₂×C₄	—	166	1	no	—
145	C₂×C₄	—	166	1	no	—
146	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
147	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	2	yes	—
148	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
149	C₂×C₄	#14581811	1	1	yes	characteristic
150	C₈	—	4	4	no	—
151	C₈	—	83	1	no	—
152	C₈	—	83	2	no	—
153	C₈	—	83	1	no	—
154	C₈	C₄²×D₈₃	1	1	yes	characteristic
155	C₈	#14581748	1	1	yes	characteristic
156	C₈	#14581757	1	2	yes	—
157	C₂²	—	4	1	no	—
158	C₂²	#19127738	1	1	yes	characteristic
159	C₂²	#19127786	1	2	yes	—
160	C₄	—	332	1	no	—
161	C₄	—	664	4	no	—
162	C₄	—	166	2	no	—
163	C₄	#19127552	1	1	yes	characteristic
164	C₄	#106×D₈₃	1	1	yes	characteristic
165	C₄	#19127792	1	2	yes	—
166	C₂	—	2	1	no	—
167	C₂	—	2	2	no	—
168	C₂	#24907351	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909237

C₂

yes

maximal

M₃₂×Dic₈₃

C₂

yes

maximal, characteristic

#24907345

C₂

yes

maximal

#24905565

C₂

yes

maximal, characteristic

C₈₃×#1385

C₂

yes

maximal, characteristic

C₂×#14581732

C₂²

yes

characteristic

#19127739

C₂²

yes

—

#19127659

—

C₂×C₈×Dic₈₃

C₂²

yes

characteristic

#19127525

C₄

yes

complemented

#19127476

C₄

yes

complemented

C₁₆×Dic₈₃

C₄

yes

complemented

#19127462

—

C₂×M₃₂×C₈₃

C₂²

yes

characteristic

C₂×#108×C₈₃

C₂²

yes

—

C₂²×C₈₃⋊C₈

D₄

yes

characteristic

#14581751

—

C₂×C₄×Dic₈₃

D₄

yes

characteristic

#14581732

C₂×C₄

yes

—

C₂×C₈₃⋊C₁₆

C₂×C₄

yes

—

#14581726

—

#14581724

C₂×C₄

yes

—

#14581702

—

C₈×Dic₈₃

—

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈×Dic₈₃

C₂×C₄

yes

characteristic

C₈×Dic₈₃

Q₈

yes

characteristic

C₈×Dic₈₃

Q₈

yes

characteristic

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

C₂³

yes

characteristic

M₃₂×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

—

#108×C₈₃

—

#108×C₈₃

C₂×C₄

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₈₃⋊C₈

SD₁₆

yes

—

C₂×C₈₃⋊C₈

SD₁₆

yes

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

C₄⋊C₄

yes

characteristic

C₄×Dic₈₃

C₄⋊C₄

yes

characteristic

C₈₃⋊C₁₆

C₄²

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

C₂²×C₄

yes

—

C₂×C₈×C₈₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₃

#41

yes

—

C₁₆×C₈₃

—

C₂³×C₈₃

#135

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₄²

yes

characteristic

C₈×C₈₃

#117

yes

characteristic

C₈×C₈₃

C₄×Q₈

yes

—

C₈₃⋊C₈

#106

yes

characteristic

C₈₃⋊C₈

#106

yes

—

C₈₃⋊C₈

#106

yes

characteristic

C₂²×C₈₃

—

C₂²×C₈₃

#427

yes

characteristic

C₂²×C₈₃

C₄×SD₁₆

yes

—

C₄×C₈₃

C₄×C₄⋊C₄

yes

characteristic

C₄×C₈₃

C₂×#106

yes

characteristic

C₄×C₈₃

C₄×SD₁₆

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

#12668

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₄×#106

yes

characteristic

#2155

—

C₄×M₃₂

—

#1772

—

#1385

D₈₃

yes

characteristic

#1384

—

C₈₃

#12668

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

C₂×D₈₃

yes

characteristic

C₂×#108

C₂×D₈₃

yes

—

#427

—

#405

—

166

—

C₂×C₄×C₈

—

#365

—

#346

—

C₄×C₁₆

—

100

#339

—

166

—

101

C₂×M₁₆

—

102

C₂²×C₈

—

103

C₂²×C₈

C₂²×D₈₃

yes

characteristic

104

#117

—

166

—

105

C₂×C₄²

—

106

M₃₂

—

107

M₃₂

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₄×D₈₃

yes

—

110

#108

—

111

#108

C₄×D₈₃

yes

—

112

#107

—

166

—

113

#106

—

114

#97

—

332

—

115

C₄×C₈

—

332

—

116

C₄×C₈

—

117

C₄×C₈

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₂²×C₄

—

166

—

121

C₂²×C₄

D₄×D₈₃

yes

characteristic

122

M₁₆

—

123

M₁₆

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

130

C₂×C₈

C₂×C₄×D₈₃

yes

—

131

C₂×C₈

Q₈×D₈₃

yes

characteristic

132

C₂×C₈

#13050589

yes

—

133

C₄⋊C₄

—

166

—

134

C₂²⋊C₄

—

332

—

135

C₄²

—

332

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

166

—

139

C₁₆

—

140

C₁₆

—

141

C₂³

#14581815

yes

characteristic

142

C₂×C₄

—

332

—

143

C₂×C₄

—

332

—

144

C₂×C₄

—

166

—

145

C₂×C₄

—

166

—

146

C₂×C₄

C₄⋊C₄×D₈₃

yes

characteristic

147

C₂×C₄

(C₂²⋊D₈₃)⋊C₄

yes

—

148

C₂×C₄

SD₁₆×D₈₃

yes

—

149

C₂×C₄

#14581811

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

C₄²×D₈₃

yes

characteristic

155

C₈

#14581748

yes

characteristic

156

C₈

#14581757

yes

—

157

C₂²

—

158

C₂²

#19127738

yes

characteristic

159

C₂²

#19127786

yes

—

160

C₄

—

332

—

161

C₄

—

664

—

162

C₄

—

166

—

163

C₄

#19127552

yes

characteristic

164

C₄

#106×D₈₃

yes

characteristic

165

C₄

#19127792

yes

—

166

C₂

—

167

C₂

—

168

C₂

#24907351

yes

characteristic

	subgroup	G/H
center	C₄	#106×D₈₃
commutator	C₈×C₈₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1385	C₂
socle	C₂×C₈₃	C₄×#106
prime radical	C₂³×C₈₃	#135
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1385	D₈₃
hypocenter	C₈₃	#12668
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

subgroup

G/H

center

C₄

#106×D₈₃

commutator

C₈×C₈₃

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₈₃

Fitting

C₈₃×#1385

C₂

C₂×C₈₃

C₂³×C₈₃

C₂²×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1385

N	H ≅ G/N	type
10. #19127525	C₄	semidirect
11. #19127476	C₄	semidirect
12. C₁₆×Dic₈₃	C₄	semidirect
90. C₈₃	#12668	semidirect

H ≅ G/N

type

10. #19127525

C₄

semidirect

11. #19127476

C₄

semidirect

12. C₁₆×Dic₈₃

C₄

semidirect

90. C₈₃

#12668

semidirect

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

subgroup	index
#24909237	2
M₃₂×Dic₈₃	2
#24907345	2
#24905565	2
C₈₃×#1385	2
#12668	83

subgroup

index

M₃₂×Dic₈₃

C₈₃×#1385

Subgroups — #35319784

Conjugacy classes of subgroups

Complements

Maximal subgroups