Group #35318680 — subgroups

Subgroups — #35318680

Order 21248.

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

normal subgroups only

This group has 212 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	#24909371	C₂	1	1	yes	maximal, characteristic
2	#24906830	C₂	1	2	yes	maximal
3	#24905537	C₂	1	1	yes	maximal, characteristic
4	#24905512	C₂	1	2	yes	maximal
5	C₈₃×#2247	C₂	1	1	yes	maximal, characteristic
6	#19127983	—	2	2	no	—
7	C₈×Dic₁₆₆	C₂²	1	1	yes	characteristic
8	C₄⋊(C₈₃⋊C₁₆)	C₂²	1	1	yes	characteristic
9	Dic₈₃⋊C₁₆	—	2	1	no	—
10	#19127476	—	2	1	no	—
11	#19127454	C₂²	1	2	yes	—
12	#19127447	—	2	2	no	—
13	C₂×Q₃₂×C₈₃	—	2	1	no	—
14	C₂×Q₃₂×C₈₃	—	2	1	no	—
15	#494×C₈₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
17	C₄×Dic₁₆₆	D₄	1	1	yes	characteristic
18	C₂×C₈₃⋊C₁₆	—	2	1	no	—
19	#14581723	—	4	2	no	—
20	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
21	C₄⋊(C₈₃⋊C₈)	D₄	1	1	yes	characteristic
22	Dic₈₃⋊C₈	—	2	1	no	—
23	#14581715	—	4	2	no	—
24	C₈×Dic₈₃	—	2	1	no	—
25	C₂×Q₁₆×C₈₃	—	2	2	no	—
26	C₂×Q₁₆×C₈₃	—	2	2	no	—
27	C₄⋊Q₈×C₈₃	—	2	2	no	—
28	Q₃₂×C₈₃	—	4	2	no	—
29	Q₃₂×C₈₃	—	4	2	no	—
30	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₆₆	D₈	1	1	yes	characteristic
34	C₂×C₈₃⋊C₈	—	2	1	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₄⋊Dic₈₃	D₈	1	1	yes	characteristic
37	Dic₈₃⋊C₄	—	2	1	no	—
38	C₄×Dic₈₃	—	2	1	no	—
39	C₈₃⋊C₁₆	—	4	1	no	—
40	C₂×Q₈×C₈₃	—	4	2	no	—
41	Q₁₆×C₈₃	—	4	2	no	—
42	Q₁₆×C₈₃	—	4	2	no	—
43	Q₁₆×C₈₃	—	8	2	no	—
44	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₈₃	—	4	2	no	—
47	C₄⋊C₄×C₈₃	—	4	2	no	—
48	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₃	—	2	1	no	—
50	C₁₆×C₈₃	SD₁₆	1	2	yes	—
51	Q₈×C₈₃	—	8	2	no	—
52	Q₈×C₈₃	—	8	2	no	—
53	C₂×C₄×C₈₃	—	8	2	no	—
54	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
55	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
57	C₂×Dic₈₃	—	2	1	no	—
58	C₂×Dic₈₃	—	2	1	no	—
59	Dic₁₆₆	—	2	2	no	—
60	C₈×C₈₃	—	2	1	no	—
61	C₈×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₈×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₈₃⋊C₈	—	4	1	no	—
64	C₈₃⋊C₈	—	8	2	no	—
65	C₂²×C₈₃	#466	1	1	yes	characteristic
66	C₄×C₈₃	—	2	1	no	—
67	C₄×C₈₃	—	16	2	no	—
68	C₄×C₈₃	#460	1	1	yes	characteristic
69	C₄×C₈₃	#472	1	1	yes	characteristic
70	C₄×C₈₃	#510	1	2	yes	—
71	Dic₈₃	—	4	1	no	—
72	Dic₈₃	—	4	1	no	—
73	#12443	—	83	1	no	maximal, Sylow
74	C₂×C₈₃	#1668	1	1	yes	characteristic
75	C₂×C₈₃	#2219	1	1	yes	characteristic
76	C₂×C₈₃	#2222	1	1	yes	characteristic
77	#2247	D₈₃	1	1	yes	characteristic
78	#2183	—	83	1	no	—
79	#1670	—	83	2	no	—
80	#1372	—	83	1	no	—
81	#1362	—	83	2	no	—
82	C₈₃	#12443	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₃₂	—	2	1	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	#494	C₂×D₈₃	1	2	yes	—
86	#475	—	166	2	no	—
87	C₈×Q₈	—	83	1	no	—
88	C₄⋊C₁₆	—	83	1	no	—
89	C₄⋊C₁₆	—	166	1	no	—
90	#346	—	166	1	no	—
91	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
92	#335	—	83	2	no	—
93	#332	—	166	2	no	—
94	C₂×Q₁₆	—	2	2	no	—
95	C₂×Q₁₆	—	2	2	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	C₄×Q₈	—	83	1	no	—
98	Q₃₂	—	4	2	no	—
99	Q₃₂	—	4	2	no	—
100	C₂×C₁₆	—	166	1	no	—
101	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
102	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
103	#106	—	332	2	no	—
104	C₄⋊C₈	—	83	1	no	—
105	C₄⋊C₈	—	166	1	no	—
106	C₄⋊C₈	—	166	2	no	—
107	#103	—	332	2	no	—
108	C₄×C₈	—	166	1	no	—
109	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
110	C₂×Q₈	—	4	2	no	—
111	C₂×Q₈	—	83	1	no	—
112	Q₁₆	—	4	2	no	—
113	Q₁₆	—	4	2	no	—
114	Q₁₆	—	8	2	no	—
115	C₂×C₈	—	332	2	no	—
116	C₂×C₈	—	166	1	no	—
117	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
118	C₂×C₈	#13050589	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	83	1	no	—
122	C₄⋊C₄	—	166	1	no	—
123	C₄²	—	166	1	no	—
124	C₄²	D₄×D₈₃	1	1	yes	characteristic
125	C₁₆	—	2	1	no	—
126	C₁₆	—	332	1	no	—
127	C₁₆	#13050578	1	2	yes	—
128	Q₈	—	8	2	no	—
129	Q₈	—	8	2	no	—
130	Q₈	—	166	2	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	166	1	no	—
133	C₂×C₄	—	166	1	no	—
134	C₂×C₄	#14581763	1	1	yes	characteristic
135	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
136	C₂×C₄	#14581811	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	332	1	no	—
139	C₈	—	664	2	no	—
140	C₈	C₂×#13050578	1	1	yes	characteristic
141	C₈	#14581818	1	1	yes	characteristic
142	C₂²	#19127934	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	16	2	no	—
145	C₄	—	332	1	no	—
146	C₄	—	332	1	no	—
147	C₄	#19127898	1	1	yes	characteristic
148	C₄	#19127968	1	1	yes	characteristic
149	C₄	#19128143	1	2	yes	—
150	C₂	#24906820	1	1	yes	characteristic
151	C₂	#24909568	1	1	yes	characteristic
152	C₂	#24909585	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909371

C₂

yes

maximal, characteristic

#24906830

C₂

yes

maximal

#24905537

C₂

yes

maximal, characteristic

#24905512

C₂

yes

maximal

C₈₃×#2247

C₂

yes

maximal, characteristic

#19127983

—

C₈×Dic₁₆₆

C₂²

yes

characteristic

C₄⋊(C₈₃⋊C₁₆)

C₂²

yes

characteristic

Dic₈₃⋊C₁₆

—

#19127476

—

#19127454

C₂²

yes

—

#19127447

—

C₂×Q₃₂×C₈₃

—

C₂×Q₃₂×C₈₃

—

#494×C₈₃

C₂²

yes

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₄×Dic₁₆₆

D₄

yes

characteristic

C₂×C₈₃⋊C₁₆

—

#14581723

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

D₄

yes

characteristic

Dic₈₃⋊C₈

—

#14581715

—

C₈×Dic₈₃

—

C₂×Q₁₆×C₈₃

—

C₂×Q₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

—

Q₃₂×C₈₃

—

Q₃₂×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

D₈

yes

characteristic

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

D₈

yes

characteristic

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

SD₁₆

yes

—

Q₈×C₈₃

—

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#121

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×SD₁₆

yes

characteristic

C₈×C₈₃

AGL(1,ℤ/8)

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

#466

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

D₈₃

yes

—

—

—

—

C₈₃

#12443

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

#494

C₂×D₈₃

yes

—

#475

—

166

—

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

166

—

#346

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

#335

—

#332

—

166

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

—

100

C₂×C₁₆

—

166

—

101

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

102

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

103

#106

—

332

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

166

—

106

C₄⋊C₈

—

166

—

107

#103

—

332

—

108

C₄×C₈

—

166

—

109

C₄×C₈

C₂²×D₈₃

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

C₂×C₈

—

332

—

116

C₂×C₈

—

166

—

117

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

118

C₂×C₈

#13050589

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

166

—

123

C₄²

—

166

—

124

C₄²

D₄×D₈₃

yes

characteristic

125

C₁₆

—

126

C₁₆

—

332

—

127

C₁₆

#13050578

yes

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

166

—

131

C₂×C₄

—

132

C₂×C₄

—

166

—

133

C₂×C₄

—

166

—

134

C₂×C₄

#14581763

yes

characteristic

135

C₂×C₄

D₈×D₈₃

yes

characteristic

136

C₂×C₄

#14581811

yes

characteristic

137

C₈

—

138

C₈

—

332

—

139

C₈

—

664

—

140

C₈

C₂×#13050578

yes

characteristic

141

C₈

#14581818

yes

characteristic

142

C₂²

#19127934

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

332

—

146

C₄

—

332

—

147

C₄

#19127898

yes

characteristic

148

C₄

#19127968

yes

characteristic

149

C₄

#19128143

yes

—

150

C₂

#24906820

yes

characteristic

151

C₂

#24909568

yes

characteristic

152

C₂

#24909585

yes

characteristic

	subgroup	G/H
center	C₂²	#19127934
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#2247	C₂
socle, prime radical	C₂²×C₈₃	#466
hypercenter	#2247	D₈₃
hypocenter	C₈₃	#12443
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#19127934

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#2247

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
82. C₈₃	#12443	semidirect

H ≅ G/N

type

82. C₈₃

#12443

semidirect

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

subgroup	index
#24909371	2
#24906830	2
#24905537	2
#24905512	2
C₈₃×#2247	2
#12443	83

subgroup

index

C₈₃×#2247

Subgroups — #35318680

Conjugacy classes of subgroups

Complements

Maximal subgroups