Group #35313087 — subgroups

Subgroups — #35313087

Order 21248.

686 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 235 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	#24916053	C₂	1	3	yes	maximal
2	#24907161	C₂	1	3	yes	maximal
3	C₂×#338×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
5	C₂×#14581751	C₂²	1	3	yes	—
6	#19127743	—	2	12	no	—
7	#19127739	—	2	6	no	—
8	#19127455	—	2	6	no	—
9	C₂²×M₁₆×C₈₃	C₂²	1	3	yes	—
10	#338×C₈₃	C₄	1	4	yes	complemented
11	C₂²×C₈₃⋊C₈	—	2	3	no	—
12	C₂²×C₈₃⋊C₈	—	2	3	no	—
13	C₂²×C₈₃⋊C₈	Q₈	1	2	yes	—
14	#14581751	—	2	6	no	—
15	#14581751	—	2	6	no	—
16	C₂×C₄⋊Dic₈₃	D₄	1	6	yes	—
17	C₂×C₂²⋊Dic₈₃	—	2	3	no	—
18	C₂×C₄×Dic₈₃	—	2	3	no	—
19	#14581726	—	2	12	no	—
20	#14581724	—	2	12	no	—
21	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₈₃	—	2	6	no	—
23	C₂×M₁₆×C₈₃	—	2	6	no	—
24	C₂×M₁₆×C₈₃	C₂×C₄	1	6	yes	—
25	C₂×M₁₆×C₈₃	Q₈	1	6	yes	—
26	C₂²×C₈×C₈₃	D₄	1	6	yes	—
27	C₂²×Dic₈₃	—	2	6	no	—
28	C₂×C₈₃⋊C₈	—	2	6	no	—
29	C₂×C₈₃⋊C₈	—	4	3	no	—
30	C₂×C₈₃⋊C₈	—	4	1	no	—
31	C₄⋊Dic₈₃	—	2	12	no	—
32	C₂²⋊Dic₈₃	—	4	6	no	—
33	C₄×Dic₈₃	—	4	3	no	—
34	C₄×Dic₈₃	—	4	3	no	—
35	C₂⁴×C₈₃	#41	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
38	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
39	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	C₂×Q₈	1	3	yes	—
41	C₂²×C₄×C₈₃	#41	1	3	yes	—
42	M₁₆×C₈₃	—	2	12	no	—
43	M₁₆×C₈₃	—	2	12	no	—
44	C₂×C₈×C₈₃	—	2	6	no	—
45	C₂×C₈×C₈₃	—	2	6	no	—
46	C₂×C₈×C₈₃	C₄⋊C₄	1	12	yes	—
47	C₂³×C₈₃	—	4	2	no	—
48	C₂³×C₈₃	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₈₃	#121	1	3	yes	—
50	C₂³×C₈₃	#125	1	3	yes	—
51	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	3	yes	—
52	C₂×C₄×C₈₃	C₄×D₄	1	3	yes	—
53	C₂×C₄×C₈₃	#120	1	3	yes	—
54	C₂×C₄×C₈₃	C₂²⋊Q₈	1	6	yes	—
55	C₂×C₄×C₈₃	C₂²⋊Q₈	1	3	yes	—
56	C₂×C₄×C₈₃	#123	1	3	yes	—
57	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₄⋊Q₈	1	3	yes	—
59	C₂×C₄×C₈₃	C₄⋊Q₈	1	3	yes	—
60	C₂×Dic₈₃	—	4	12	no	—
61	C₂×Dic₈₃	—	4	6	no	—
62	C₈×C₈₃	—	2	12	no	—
63	C₈₃⋊C₈	—	4	2	no	—
64	C₂²×C₈₃	—	2	6	no	—
65	C₂²×C₈₃	—	4	1	no	—
66	C₂²×C₈₃	—	4	3	no	—
67	C₂²×C₈₃	#389	1	3	yes	—
68	C₂²×C₈₃	#392	1	1	yes	characteristic
69	C₂²×C₈₃	#398	1	3	yes	—
70	C₄×C₈₃	#384	1	1	yes	characteristic
71	C₄×C₈₃	#385	1	3	yes	—
72	C₄×C₈₃	#395	1	3	yes	—
73	C₄×C₈₃	#399	1	1	yes	characteristic
74	Dic₈₃	—	8	6	no	—
75	#11337	—	83	1	no	maximal, Sylow
76	C₂×C₈₃	—	2	3	no	—
77	C₂×C₈₃	—	2	3	no	—
78	C₂×C₈₃	#1444	1	1	yes	characteristic
79	C₂×C₈₃	#2017	1	2	yes	—
80	#2911	—	83	3	no	—
81	C₂×#338	D₈₃	1	1	yes	characteristic
82	#1729	—	83	3	no	—
83	C₈₃	#11337	1	1	yes	characteristic, complemented, Sylow
84	C₂²×M₁₆	C₂×D₈₃	1	3	yes	—
85	C₂³×C₈	—	83	1	no	—
86	C₂×#117	—	83	3	no	—
87	#428	—	166	12	no	—
88	#427	—	166	6	no	—
89	#338	Dic₈₃	1	4	yes	complemented
90	#336	—	166	6	no	—
91	C₂³×C₄	C₂²×D₈₃	1	1	yes	characteristic
92	C₂×M₁₆	—	2	6	no	—
93	C₂×M₁₆	—	2	6	no	—
94	C₂×M₁₆	Dic₁₆₆	1	6	yes	—
95	C₂×M₁₆	C₂×Dic₈₃	1	6	yes	—
96	C₂²×C₈	—	83	2	no	—
97	C₂²×C₈	—	166	3	no	—
98	C₂²×C₈	—	166	3	no	—
99	C₂²×C₈	D₃₃₂	1	6	yes	—
100	#117	—	166	6	no	—
101	#117	—	166	6	no	—
102	C₂×C₄⋊C₄	—	83	6	no	—
103	C₂×C₂²⋊C₄	—	166	3	no	—
104	C₂×C₄²	—	166	3	no	—
105	#107	—	166	12	no	—
106	#106	—	166	12	no	—
107	C₂⁴	#13050591	1	1	yes	characteristic
108	C₂²×C₄	—	166	6	no	—
109	C₂²×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
110	C₂²×C₄	D₄×D₈₃	1	3	yes	—
111	C₂²×C₄	C₂×D₃₃₂	1	3	yes	—
112	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
113	C₂²×C₄	C₂×Dic₁₆₆	1	3	yes	—
114	C₂²×C₄	#13050589	1	3	yes	—
115	M₁₆	—	2	12	no	—
116	M₁₆	—	2	12	no	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	2	6	no	—
119	C₂×C₈	—	332	3	no	—
120	C₂×C₈	—	332	1	no	—
121	C₂×C₈	—	166	6	no	—
122	C₂×C₈	C₄⋊Dic₈₃	1	12	yes	—
123	C₄⋊C₄	—	166	12	no	—
124	C₂²⋊C₄	—	332	6	no	—
125	C₄²	—	332	3	no	—
126	C₄²	—	332	3	no	—
127	C₂³	—	4	2	no	—
128	C₂³	Q₈×Dic₈₃	1	1	yes	characteristic
129	C₂³	#14581767	1	3	yes	—
130	C₂³	#14581783	1	3	yes	—
131	C₂×C₄	—	332	12	no	—
132	C₂×C₄	—	332	6	no	—
133	C₂×C₄	C₂×C₄⋊Dic₈₃	1	3	yes	—
134	C₂×C₄	D₄×Dic₈₃	1	3	yes	—
135	C₂×C₄	#14581762	1	3	yes	—
136	C₂×C₄	#14581769	1	6	yes	—
137	C₂×C₄	#14581772	1	3	yes	—
138	C₂×C₄	#14581776	1	3	yes	—
139	C₂×C₄	#14581789	1	1	yes	characteristic
140	C₂×C₄	#14581791	1	3	yes	—
141	C₂×C₄	#14581792	1	3	yes	—
142	C₈	—	2	12	no	—
143	C₈	—	332	2	no	—
144	C₂²	—	2	6	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	1	no	—
147	C₂²	#19127600	1	3	yes	—
148	C₂²	#19127611	1	1	yes	characteristic
149	C₂²	#19127633	1	3	yes	—
150	C₄	—	664	6	no	—
151	C₄	#19127571	1	1	yes	characteristic
152	C₄	#19127576	1	3	yes	—
153	C₄	#19127622	1	3	yes	—
154	C₄	#19127638	1	1	yes	characteristic
155	C₂	—	2	3	no	—
156	C₂	—	2	3	no	—
157	C₂	#24905728	1	1	yes	characteristic
158	C₂	#24908568	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24916053

C₂

yes

maximal

#24907161

C₂

yes

maximal

C₂×#338×C₈₃

C₂

yes

maximal, characteristic

C₂³×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×#14581751

C₂²

yes

—

#19127743

—

#19127739

—

#19127455

—

C₂²×M₁₆×C₈₃

C₂²

yes

—

#338×C₈₃

C₄

yes

complemented

C₂²×C₈₃⋊C₈

—

C₂²×C₈₃⋊C₈

—

C₂²×C₈₃⋊C₈

Q₈

yes

—

#14581751

—

#14581751

—

C₂×C₄⋊Dic₈₃

D₄

yes

—

C₂×C₂²⋊Dic₈₃

—

C₂×C₄×Dic₈₃

—

#14581726

—

#14581724

—

C₂³×C₄×C₈₃

C₂³

yes

characteristic

C₂×M₁₆×C₈₃

—

C₂×M₁₆×C₈₃

—

C₂×M₁₆×C₈₃

C₂×C₄

yes

—

C₂×M₁₆×C₈₃

Q₈

yes

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

—

C₄×Dic₈₃

—

C₄×Dic₈₃

—

C₂⁴×C₈₃

#41

yes

characteristic

C₂²×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂²×C₄×C₈₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

—

C₂²×C₄×C₈₃

#41

yes

—

M₁₆×C₈₃

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂³×C₈₃

—

C₂³×C₈₃

C₄×Q₈

yes

characteristic

C₂³×C₈₃

#121

yes

—

C₂³×C₈₃

#125

yes

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

C₄×D₄

yes

—

C₂×C₄×C₈₃

#120

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#389

yes

—

C₂²×C₈₃

#392

yes

characteristic

C₂²×C₈₃

#398

yes

—

C₄×C₈₃

#384

yes

characteristic

C₄×C₈₃

#385

yes

—

C₄×C₈₃

#395

yes

—

C₄×C₈₃

#399

yes

characteristic

Dic₈₃

—

#11337

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

#1444

yes

characteristic

C₂×C₈₃

#2017

yes

—

#2911

—

C₂×#338

D₈₃

yes

characteristic

#1729

—

C₈₃

#11337

yes

characteristic, complemented, Sylow

C₂²×M₁₆

C₂×D₈₃

yes

—

C₂³×C₈

—

C₂×#117

—

#428

—

166

—

#427

—

166

—

#338

Dic₈₃

yes

complemented

#336

—

166

—

C₂³×C₄

C₂²×D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₁₆₆

yes

—

C₂×M₁₆

C₂×Dic₈₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

166

—

C₂²×C₈

—

166

—

C₂²×C₈

D₃₃₂

yes

—

100

#117

—

166

—

101

#117

—

166

—

102

C₂×C₄⋊C₄

—

103

C₂×C₂²⋊C₄

—

166

—

104

C₂×C₄²

—

166

—

105

#107

—

166

—

106

#106

—

166

—

107

C₂⁴

#13050591

yes

characteristic

108

C₂²×C₄

—

166

—

109

C₂²×C₄

C₂²×Dic₈₃

yes

characteristic

110

C₂²×C₄

D₄×D₈₃

yes

—

111

C₂²×C₄

C₂×D₃₃₂

yes

—

112

C₂²×C₄

Q₈×D₈₃

yes

characteristic

113

C₂²×C₄

C₂×Dic₁₆₆

yes

—

114

C₂²×C₄

#13050589

yes

—

115

M₁₆

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

332

—

120

C₂×C₈

—

332

—

121

C₂×C₈

—

166

—

122

C₂×C₈

C₄⋊Dic₈₃

yes

—

123

C₄⋊C₄

—

166

—

124

C₂²⋊C₄

—

332

—

125

C₄²

—

332

—

126

C₄²

—

332

—

127

C₂³

—

128

C₂³

Q₈×Dic₈₃

yes

characteristic

129

C₂³

#14581767

yes

—

130

C₂³

#14581783

yes

—

131

C₂×C₄

—

332

—

132

C₂×C₄

—

332

—

133

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

134

C₂×C₄

D₄×Dic₈₃

yes

—

135

C₂×C₄

#14581762

yes

—

136

C₂×C₄

#14581769

yes

—

137

C₂×C₄

#14581772

yes

—

138

C₂×C₄

#14581776

yes

—

139

C₂×C₄

#14581789

yes

characteristic

140

C₂×C₄

#14581791

yes

—

141

C₂×C₄

#14581792

yes

—

142

C₈

—

143

C₈

—

332

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

#19127600

yes

—

148

C₂²

#19127611

yes

characteristic

149

C₂²

#19127633

yes

—

150

C₄

—

664

—

151

C₄

#19127571

yes

characteristic

152

C₄

#19127576

yes

—

153

C₄

#19127622

yes

—

154

C₄

#19127638

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#24905728

yes

characteristic

158

C₂

#24908568

yes

—

	subgroup	G/H
center	C₂×C₄	#14581789
commutator	C₂²×C₄×C₈₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₃
Fitting	C₂×#338×C₈₃	C₂
socle	C₂²×C₈₃	#392
prime radical	C₂⁴×C₈₃	#41
cosocle	C₂³×C₄×C₈₃	C₂³
hypercenter	C₂×#338	D₈₃
hypocenter	C₈₃	#11337
LCS	G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#14581789

commutator

C₂²×C₄×C₈₃

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₈₃

Fitting

C₂×#338×C₈₃

C₂

C₂²×C₈₃

C₂⁴×C₈₃

C₂³×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. #338×C₈₃	C₄	semidirect
83. C₈₃	#11337	semidirect
89. #338	Dic₈₃	semidirect

H ≅ G/N

type

10. #338×C₈₃

C₄

semidirect

83. C₈₃

#11337

semidirect

89. #338

Dic₈₃

semidirect

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

subgroup	index
#24916053	2
#24907161	2
C₂×#338×C₈₃	2
#11337	83

subgroup

index

#24916053

#24907161

C₂×#338×C₈₃

#11337

Subgroups — #35313087

Conjugacy classes of subgroups

Complements

Maximal subgroups