Group #35325199 — subgroups

Subgroups — #35325199

Order 21248.

200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂×#19127541	C₂	1	2	yes	maximal
2	#24909716	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₈₃⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
4	#24905643	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₄	—	2	1	no	—
7	C₂×Dic₆₆₄	C₂²	1	2	yes	—
8	#19128081	—	2	1	no	—
9	#19127541	—	2	4	no	—
10	C₂×C₈₃⋊C₃₂	C₂²	1	2	yes	—
11	C₂×Q₃₂×C₈₃	C₂²	1	1	yes	characteristic
12	C₄×Q₁₆×C₈₃	—	2	1	no	—
13	#366×C₈₃	C₂²	1	1	yes	characteristic
14	#358×C₈₃	—	2	1	no	—
15	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₃₂	—	2	2	no	—
17	C₂×Dic₃₃₂	—	4	1	no	—
18	#14581792	—	4	1	no	—
19	Dic₆₆₄	—	2	2	no	—
20	Dic₆₆₄	—	2	2	no	—
21	Dic₆₆₄	—	4	2	no	—
22	C₈₃⋊C₃₂	—	2	2	no	—
23	C₂×Q₁₆×C₈₃	—	2	1	no	—
24	C₄×Q₈×C₈₃	—	4	1	no	—
25	Q₃₂×C₈₃	—	2	1	no	—
26	Q₃₂×C₈₃	D₄	1	2	yes	—
27	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
28	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
29	#107×C₈₃	—	2	1	no	—
30	#103×C₈₃	—	4	1	no	—
31	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂×Dic₁₆₆	—	4	2	no	—
33	Dic₃₃₂	—	4	2	no	—
34	Dic₃₃₂	—	4	2	no	—
35	Dic₃₃₂	—	8	2	no	—
36	C₄⋊Dic₈₃	—	8	1	no	—
37	C₄⋊Dic₈₃	—	8	1	no	—
38	C₂×Q₈×C₈₃	—	4	1	no	—
39	Q₁₆×C₈₃	—	2	2	no	—
40	Q₁₆×C₈₃	—	4	1	no	—
41	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
42	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊C₄×C₈₃	—	4	1	no	—
44	C₄⋊C₄×C₈₃	—	8	1	no	—
45	C₄²×C₈₃	—	8	1	no	—
46	C₄²×C₈₃	D₈	1	1	yes	characteristic
47	C₁₆×C₈₃	—	2	1	no	—
48	C₁₆×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₃	#41	1	1	yes	characteristic
50	Q₈×C₈₃	—	4	2	no	—
51	Q₈×C₈₃	—	8	1	no	—
52	C₂×C₄×C₈₃	—	8	1	no	—
53	C₂×C₄×C₈₃	—	8	1	no	—
54	C₂×C₄×C₈₃	D₁₆	1	1	yes	characteristic
55	C₂×C₄×C₈₃	D₁₆	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
57	C₂×Dic₈₃	—	8	2	no	—
58	Dic₁₆₆	—	8	2	no	—
59	Dic₁₆₆	—	8	2	no	—
60	C₈×C₈₃	—	2	1	no	—
61	C₈×C₈₃	#121	1	1	yes	characteristic
62	C₈×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	—	8	2	no	—
66	C₄×C₈₃	—	16	1	no	—
67	C₄×C₈₃	Q₆₄	1	2	yes	complemented
68	C₄×C₈₃	#459	1	1	yes	characteristic
69	C₄×C₈₃	#509	1	1	yes	characteristic
70	Dic₈₃	—	16	2	no	—
71	#13688	—	83	1	no	maximal, Sylow
72	C₂×C₈₃	#2206	1	1	yes	characteristic
73	C₂×C₈₃	C₂×Q₆₄	1	1	yes	characteristic
74	C₂×C₈₃	#2264	1	1	yes	characteristic
75	C₂×Q₆₄	—	83	2	no	—
76	#2247	—	83	1	no	—
77	C₄×Q₃₂	D₈₃	1	1	yes	characteristic
78	C₄⋊C₃₂	—	83	1	no	—
79	#1416	—	83	2	no	—
80	C₈₃	#13688	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₃₂	—	83	2	no	—
82	C₂×Q₃₂	—	166	1	no	—
83	C₂×Q₃₂	C₂×D₈₃	1	1	yes	characteristic
84	#494	—	166	1	no	—
85	C₄×Q₁₆	—	2	1	no	—
86	Q₆₄	—	166	4	no	—
87	C₂×C₃₂	—	83	2	no	—
88	#366	C₂×D₈₃	1	1	yes	characteristic
89	#358	—	2	1	no	—
90	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	1	no	—
92	C₂×Q₁₆	—	332	1	no	—
93	C₂×Q₁₆	—	166	2	no	—
94	C₄⋊Q₈	—	332	1	no	—
95	C₄×Q₈	—	4	1	no	—
96	Q₃₂	—	2	1	no	—
97	Q₃₂	—	332	2	no	—
98	Q₃₂	—	166	2	no	—
99	Q₃₂	—	166	2	no	—
100	Q₃₂	D₃₃₂	1	2	yes	—
101	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
103	#107	—	2	1	no	—
104	#103	—	4	1	no	—
105	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
106	C₃₂	—	166	2	no	—
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	332	2	no	—
109	Q₁₆	—	2	2	no	—
110	Q₁₆	—	4	1	no	—
111	Q₁₆	—	332	2	no	—
112	Q₁₆	—	332	2	no	—
113	Q₁₆	—	664	2	no	—
114	C₂×C₈	#13050575	1	1	yes	characteristic
115	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	664	1	no	—
119	C₄⋊C₄	—	664	1	no	—
120	C₄²	—	8	1	no	—
121	C₄²	#13050575	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	C₂×D₃₃₂	1	1	yes	characteristic
124	C₁₆	#13050590	1	1	yes	characteristic
125	Q₈	—	4	2	no	—
126	Q₈	—	8	1	no	—
127	Q₈	—	664	2	no	—
128	Q₈	—	664	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	664	2	no	—
132	C₂×C₄	#14581733	1	1	yes	characteristic
133	C₂×C₄	#14581733	1	1	yes	characteristic
134	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#14581766	1	1	yes	characteristic
137	C₈	#14581817	1	1	yes	characteristic
138	C₂²	C₂×#14581733	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	1328	2	no	—
143	C₄	#19127541	1	2	yes	complemented
144	C₄	#19127896	1	1	yes	characteristic
145	C₄	#19128136	1	1	yes	characteristic
146	C₂	#24909498	1	1	yes	characteristic
147	C₂	C₂×#19127541	1	1	yes	characteristic
148	C₂	#24909780	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19127541

C₂

yes

maximal

#24909716

C₂

yes

maximal, characteristic

C₄⋊(C₈₃⋊C₃₂)

C₂

yes

maximal, characteristic

#24905643

C₂

yes

maximal

C₄×Q₃₂×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

—

C₂×Dic₆₆₄

C₂²

yes

—

#19128081

—

#19127541

—

C₂×C₈₃⋊C₃₂

C₂²

yes

—

C₂×Q₃₂×C₈₃

C₂²

yes

characteristic

C₄×Q₁₆×C₈₃

—

#366×C₈₃

C₂²

yes

characteristic

#358×C₈₃

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

Dic₆₆₄

—

Dic₆₆₄

—

Dic₆₆₄

—

C₈₃⋊C₃₂

—

C₂×Q₁₆×C₈₃

—

C₄×Q₈×C₈₃

—

Q₃₂×C₈₃

—

Q₃₂×C₈₃

D₄

yes

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

#107×C₈₃

—

#103×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₄²×C₈₃

D₈

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

#41

yes

characteristic

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

D₁₆

yes

characteristic

C₂×C₄×C₈₃

D₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

yes

C₈×C₈₃

yes

C₂²×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

Q₆₄

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×Q₆₄

yes

characteristic

C₂×C₈₃

#2264

yes

characteristic

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

D₈₃

yes

characteristic

C₄⋊C₃₂

—

#1416

—

C₈₃

#13688

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

166

—

C₂×Q₃₂

C₂×D₈₃

yes

characteristic

#494

—

166

—

C₄×Q₁₆

—

Q₆₄

—

166

—

C₂×C₃₂

—

#366

C₂×D₈₃

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

332

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

332

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

332

—

Q₃₂

—

166

—

Q₃₂

—

166

—

100

Q₃₂

D₃₃₂

yes

—

101

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

102

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

103

#107

—

104

#103

—

105

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

106

C₃₂

—

166

—

107

C₂×Q₈

—

108

C₂×Q₈

—

332

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

332

—

112

Q₁₆

—

332

—

113

Q₁₆

—

664

—

114

C₂×C₈

#13050575

yes

characteristic

115

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

664

—

119

C₄⋊C₄

—

664

—

120

C₄²

—

121

C₄²

#13050575

yes

characteristic

122

C₁₆

—

123

C₁₆

C₂×D₃₃₂

yes

characteristic

124

C₁₆

#13050590

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

664

—

128

Q₈

—

664

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

664

—

132

C₂×C₄

#14581733

yes

characteristic

133

C₂×C₄

#14581733

yes

characteristic

134

C₂×C₄

C₂×#13050575

yes

characteristic

135

C₈

—

136

C₈

#14581766

yes

characteristic

137

C₈

#14581817

yes

characteristic

138

C₂²

C₂×#14581733

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

1328

—

143

C₄

#19127541

yes

complemented

144

C₄

#19127896

yes

characteristic

145

C₄

#19128136

yes

characteristic

146

C₂

#24909498

yes

characteristic

147

C₂

C₂×#19127541

yes

characteristic

148

C₂

#24909780

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14581733
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₄×Q₃₂×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₈₃
hypocenter	C₈₃	#13688
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#14581733

commutator, cosocle

C₂×C₁₆×C₈₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₄×Q₃₂×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₈₃

hypocenter

C₈₃

#13688

LCS

G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
67. C₄×C₈₃	Q₆₄	semidirect
80. C₈₃	#13688	semidirect
143. C₄	#19127541	semidirect

H ≅ G/N

type

67. C₄×C₈₃

Q₆₄

semidirect

80. C₈₃

#13688

semidirect

143. C₄

#19127541

semidirect

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

subgroup	index
C₂×#19127541	2
#24909716	2
C₄⋊(C₈₃⋊C₃₂)	2
#24905643	2
C₄×Q₃₂×C₈₃	2
#13688	83

subgroup

index

C₄⋊(C₈₃⋊C₃₂)

C₄×Q₃₂×C₈₃

Subgroups — #35325199

Conjugacy classes of subgroups

Complements

Maximal subgroups