Group #35316259 — subgroups

Subgroups — #35316259

Order 21248.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#24909282	C₂	1	2	yes	maximal, complemented
2	C₈×D₆₆₄	C₂	1	1	yes	maximal, characteristic
3	#24905417	C₂	1	2	yes	maximal, complemented
4	#24905381	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₈₃	—	2	2	no	—
7	C₄×D₆₆₄	C₄	1	1	yes	characteristic
8	C₈×D₃₃₂	C₂²	1	2	yes	—
9	(C₂×D₈₃)⋊C₁₆	—	2	2	no	—
10	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#19127454	C₄	1	1	yes	characteristic
13	#19127430	C₄	1	2	yes	—
14	C₄⋊C₁₆×C₈₃	C₂²	1	2	yes	—
15	C₈²×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×D₆₆₄	C₈	1	1	yes	characteristic
17	C₂×C₈×D₈₃	—	2	2	no	—
18	C₄×D₃₃₂	C₂×C₄	1	2	yes	—
19	C₂×C₈₃⋊C₁₆	—	2	1	no	—
20	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
21	C₁₆×D₈₃	—	4	4	no	—
22	#14581726	C₈	1	1	yes	characteristic
23	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
24	#14581714	C₈	1	2	yes	—
25	(C₂×D₈₃)⋊C₈	—	2	2	no	—
26	C₂×C₁₆×C₈₃	—	2	2	no	—
27	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×D₃₃₂	C₂×C₈	1	2	yes	—
31	C₂×C₄×D₈₃	—	2	2	no	—
32	D₆₆₄	C₁₆	1	4	yes	complemented
33	C₂×C₈₃⋊C₈	—	2	2	no	—
34	C₈×D₈₃	—	4	2	no	—
35	C₈×D₈₃	—	4	2	no	—
36	C₄⋊Dic₈₃	C₂×C₈	1	2	yes	—
37	(C₂×D₈₃)⋊C₄	—	2	2	no	—
38	C₈₃⋊C₁₆	—	2	2	no	—
39	C₈₃⋊C₁₆	—	4	1	no	—
40	C₂×C₈×C₈₃	—	2	1	no	—
41	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
45	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
46	C₁₆×C₈₃	—	4	2	no	—
47	C₂²×D₈₃	—	2	2	no	—
48	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
51	C₂×Dic₈₃	—	2	2	no	—
52	D₃₃₂	—	2	2	no	—
53	D₃₃₂	C₂×C₁₆	1	4	yes	complemented
54	C₄×D₈₃	—	4	2	no	—
55	C₄×D₈₃	—	4	2	no	—
56	C₈×C₈₃	—	2	1	no	—
57	C₈×C₈₃	—	2	1	no	—
58	C₈×C₈₃	—	4	1	no	—
59	C₈×C₈₃	C₂×C₁₆	1	2	yes	complemented
60	C₈×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₈₃⋊C₈	—	4	2	no	—
62	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
63	C₂×D₈₃	—	2	4	no	—
64	C₂×D₈₃	—	4	2	no	—
65	C₂×D₈₃	—	4	2	no	—
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	#442	1	1	yes	characteristic
68	C₄×C₈₃	#444	1	1	yes	characteristic
69	C₄×C₈₃	C₂²×C₁₆	1	1	yes	characteristic
70	C₄×C₈₃	#504	1	1	yes	characteristic
71	Dic₈₃	—	4	2	no	—
72	#12062	—	83	1	no	maximal, Sylow
73	C₂×C₈₃	#1580	1	1	yes	characteristic
74	C₂×C₈₃	D₄×C₁₆	1	1	yes	characteristic
75	C₂×C₈₃	#2171	1	1	yes	characteristic
76	D₈₃	—	4	4	no	—
77	D₄×C₁₆	—	83	2	no	—
78	C₈×D₈	—	83	1	no	—
79	#1367	D₈₃	1	1	yes	characteristic, complemented
80	D₄⋊C₁₆	—	83	2	no	—
81	#1312	—	83	1	no	—
82	C₈₃	#12062	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₁₆	—	166	2	no	—
84	C₄×D₈	—	83	1	no	—
85	C₈×D₄	—	83	2	no	—
86	C₄⋊C₁₆	C₂×D₈₃	1	2	yes	—
87	C₂²⋊C₁₆	—	166	2	no	—
88	C₄×C₁₆	—	83	1	no	—
89	C₄×C₁₆	—	83	1	no	—
90	#335	—	83	1	no	—
91	D₄⋊C₈	—	83	2	no	—
92	C₈²	C₂×D₈₃	1	1	yes	characteristic
93	C₂×D₈	—	83	1	no	—
94	C₂²×C₈	—	166	2	no	—
95	C₄×D₄	—	83	2	no	—
96	C₂×C₁₆	—	2	2	no	—
97	C₂×C₁₆	—	332	4	no	—
98	C₂×C₁₆	—	83	2	no	—
99	C₂×C₁₆	—	166	1	no	—
100	#107	—	83	1	no	—
101	C₄⋊C₈	—	83	2	no	—
102	#102	—	83	2	no	—
103	C₂²⋊C₈	—	166	2	no	—
104	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
105	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
107	C₂×D₄	—	83	2	no	—
108	C₂²×C₄	—	166	2	no	—
109	D₈	—	83	4	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	332	2	no	—
112	C₂×C₈	—	332	2	no	—
113	C₂×C₈	—	166	2	no	—
114	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
115	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
116	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
117	C₂×C₈	#13050591	1	1	yes	characteristic
118	C₄⋊C₄	—	83	2	no	—
119	C₂²⋊C₄	—	166	2	no	—
120	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
121	C₁₆	—	4	2	no	—
122	C₁₆	—	332	1	no	—
123	C₁₆	—	166	2	no	—
124	C₂³	—	166	2	no	—
125	D₄	—	83	4	no	—
126	D₄	—	166	2	no	—
127	C₂×C₄	—	332	2	no	—
128	C₂×C₄	—	332	2	no	—
129	C₂×C₄	—	166	2	no	—
130	C₂×C₄	#14581754	1	1	yes	characteristic
131	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
132	C₂×C₄	#14581800	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	332	2	no	—
137	C₈	C₁₆×D₈₃	1	2	yes	complemented
138	C₈	#14581819	1	2	yes	—
139	C₂²	—	332	2	no	—
140	C₂²	—	332	2	no	—
141	C₂²	—	166	4	no	—
142	C₂²	#19127766	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	332	2	no	—
145	C₄	#19127812	1	1	yes	characteristic
146	C₄	#19127822	1	1	yes	characteristic
147	C₄	C₂×C₁₆×D₈₃	1	1	yes	characteristic
148	C₄	#19128119	1	1	yes	characteristic
149	C₂	—	332	4	no	—
150	C₂	#24906286	1	1	yes	characteristic
151	C₂	#24909282	1	1	yes	characteristic
152	C₂	#24909304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909282

C₂

yes

maximal, complemented

C₈×D₆₆₄

C₂

yes

maximal, characteristic

#24905417

C₂

yes

maximal, complemented

#24905381

C₂

yes

maximal, characteristic, complemented

C₈₃×#1367

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₈₃

—

C₄×D₆₆₄

C₄

yes

characteristic

C₈×D₃₃₂

C₂²

yes

—

(C₂×D₈₃)⋊C₁₆

—

C₄×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₄×C₈₃⋊C₁₆

C₂²

yes

characteristic

#19127454

C₄

yes

characteristic

#19127430

C₄

yes

—

C₄⋊C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×D₆₆₄

C₈

yes

characteristic

C₂×C₈×D₈₃

—

C₄×D₃₃₂

C₂×C₄

yes

—

C₂×C₈₃⋊C₁₆

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₁₆×D₈₃

—

#14581726

C₈

yes

characteristic

C₄⋊(C₈₃⋊C₈)

C₂×C₄

yes

—

#14581714

C₈

yes

—

(C₂×D₈₃)⋊C₈

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×D₃₃₂

C₂×C₈

yes

—

C₂×C₄×D₈₃

—

D₆₆₄

C₁₆

yes

complemented

C₂×C₈₃⋊C₈

—

C₈×D₈₃

—

C₈×D₈₃

—

C₄⋊Dic₈₃

C₂×C₈

yes

—

(C₂×D₈₃)⋊C₄

—

C₈₃⋊C₁₆

—

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₈₃

#131

yes

characteristic

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

C₂×C₁₆

yes

complemented

C₄×D₈₃

—

C₄×D₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂×C₁₆

yes

complemented

C₈×C₈₃

AGL(1,ℤ/8)

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₈×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂²×C₁₆

yes

characteristic

C₄×C₈₃

#504

yes

characteristic

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

D₄×C₁₆

yes

characteristic

C₂×C₈₃

#2171

yes

characteristic

D₈₃

—

D₄×C₁₆

—

C₈×D₈

—

#1367

D₈₃

yes

characteristic, complemented

D₄⋊C₁₆

—

#1312

—

C₈₃

#12062

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

166

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

C₂×D₈₃

yes

—

C₂²⋊C₁₆

—

166

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×D₈

—

C₂²×C₈

—

166

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

332

—

C₂×C₁₆

—

C₂×C₁₆

—

166

—

100

#107

—

101

C₄⋊C₈

—

102

#102

—

103

C₂²⋊C₈

—

166

—

104

C₄×C₈

C₄×D₈₃

yes

characteristic

105

C₄×C₈

C₄×D₈₃

yes

characteristic

106

C₄×C₈

C₂²×D₈₃

yes

characteristic

107

C₂×D₄

—

108

C₂²×C₄

—

166

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

332

—

112

C₂×C₈

—

332

—

113

C₂×C₈

—

166

—

114

C₂×C₈

C₈×D₈₃

yes

characteristic

115

C₂×C₈

C₈×D₈₃

yes

characteristic

116

C₂×C₈

D₄×D₈₃

yes

characteristic

117

C₂×C₈

#13050591

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

166

—

120

C₄²

C₂×C₄×D₈₃

yes

characteristic

121

C₁₆

—

122

C₁₆

—

332

—

123

C₁₆

—

166

—

124

C₂³

—

166

—

125

D₄

—

126

D₄

—

166

—

127

C₂×C₄

—

332

—

128

C₂×C₄

—

332

—

129

C₂×C₄

—

166

—

130

C₂×C₄

#14581754

yes

characteristic

131

C₂×C₄

C₂×C₈×D₈₃

yes

characteristic

132

C₂×C₄

#14581800

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

332

—

137

C₈

C₁₆×D₈₃

yes

complemented

138

C₈

#14581819

yes

—

139

C₂²

—

332

—

140

C₂²

—

332

—

141

C₂²

—

166

—

142

C₂²

#19127766

yes

characteristic

143

C₄

—

144

C₄

—

332

—

145

C₄

#19127812

yes

characteristic

146

C₄

#19127822

yes

characteristic

147

C₄

C₂×C₁₆×D₈₃

yes

characteristic

148

C₄

#19128119

yes

characteristic

149

C₂

—

332

—

150

C₂

#24906286

yes

characteristic

151

C₂

#24909282

yes

characteristic

152

C₂

#24909304

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₃
commutator	C₄×C₈₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1367	C₂
socle	C₂²×C₈₃	C₈×D₄
prime radical	C₂×D₆₆₄	C₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1367	D₈₃
hypocenter	C₈₃	#12062
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₈₃

commutator

C₄×C₈₃

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1367

C₂

socle

C₂²×C₈₃

C₈×D₄

prime radical

C₂×D₆₆₄

C₈

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
1. #24909282	C₂	semidirect
3. #24905417	C₂	semidirect
4. #24905381	C₂	semidirect
5. C₈₃×#1367	C₂	semidirect
32. D₆₆₄	C₁₆	semidirect
53. D₃₃₂	C₂×C₁₆	semidirect
59. C₈×C₈₃	C₂×C₁₆	semidirect
79. #1367	D₈₃	semidirect
82. C₈₃	#12062	semidirect
137. C₈	C₁₆×D₈₃	semidirect

H ≅ G/N

type

1. #24909282

C₂

semidirect

3. #24905417

C₂

semidirect

4. #24905381

C₂

semidirect

5. C₈₃×#1367

C₂

semidirect

32. D₆₆₄

C₁₆

semidirect

53. D₃₃₂

C₂×C₁₆

semidirect

59. C₈×C₈₃

C₂×C₁₆

semidirect

79. #1367

D₈₃

semidirect

82. C₈₃

#12062

semidirect

137. C₈

C₁₆×D₈₃

semidirect

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

subgroup	index
#24909282	2
C₈×D₆₆₄	2
#24905417	2
#24905381	2
C₈₃×#1367	2
#12062	83

subgroup

index

C₈×D₆₆₄

C₈₃×#1367

Subgroups — #35316259

Conjugacy classes of subgroups

Complements

Maximal subgroups