Group #35319173 — subgroups

Subgroups — #35319173

Order 21248.

274 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 91 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	#24909716	C₂	1	2	yes	maximal
2	#24909691	C₂	1	2	yes	maximal
3	#24907081	C₂	1	1	yes	maximal, characteristic
4	#24907060	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₄	—	2	4	no	—
7	#19128106	—	2	1	no	—
8	#19128106	—	2	1	no	—
9	#19128106	C₂²	1	1	yes	characteristic
10	#19128099	—	2	1	no	—
11	#19128081	C₂²	1	1	yes	characteristic
12	#19128081	C₂²	1	2	yes	—
13	#19128052	—	2	1	no	—
14	#19127506	—	2	4	no	—
15	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₈²×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₃₂	—	2	2	no	—
18	C₂×Dic₃₃₂	—	2	2	no	—
19	C₂×Dic₃₃₂	—	2	2	no	—
20	#14581792	—	2	1	no	—
21	#14581792	—	2	2	no	—
22	#14581792	—	2	1	no	—
23	Dic₆₆₄	—	4	8	no	—
24	#14581726	—	2	2	no	—
25	#14581726	—	4	2	no	—
26	#14581726	—	4	2	no	—
27	#14581724	—	4	2	no	—
28	#14581717	—	4	2	no	—
29	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
30	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₆₆	—	4	1	no	—
34	C₂×Dic₁₆₆	—	4	2	no	—
35	C₂×Dic₁₆₆	—	4	1	no	—
36	Dic₃₃₂	—	4	4	no	—
37	Dic₃₃₂	—	4	4	no	—
38	Dic₃₃₂	—	4	4	no	—
39	C₄⋊Dic₈₃	—	4	1	no	—
40	C₄⋊Dic₈₃	—	4	1	no	—
41	C₄⋊Dic₈₃	—	4	1	no	—
42	C₄⋊Dic₈₃	—	4	1	no	—
43	C₄⋊Dic₈₃	—	4	2	no	—
44	C₄⋊Dic₈₃	—	4	2	no	—
45	C₂×C₈×C₈₃	D₈	1	2	yes	—
46	C₂×C₈×C₈₃	D₈	1	2	yes	—
47	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
48	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₈₃	—	2	4	no	—
50	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₈₃	—	8	1	no	—
54	C₂×Dic₈₃	—	8	2	no	—
55	C₂×Dic₈₃	—	8	1	no	—
56	Dic₁₆₆	—	8	2	no	—
57	Dic₁₆₆	—	8	2	no	—
58	Dic₁₆₆	—	8	2	no	—
59	Dic₁₆₆	—	8	2	no	—
60	C₈×C₈₃	—	2	2	no	—
61	C₈×C₈₃	Q₃₂	1	4	yes	complemented
62	C₈×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₈₃	#493	1	1	yes	characteristic
64	C₄×C₈₃	#496	1	2	yes	—
65	C₄×C₈₃	C₂×Q₃₂	1	2	yes	—
66	C₄×C₈₃	#510	1	2	yes	—
67	Dic₈₃	—	16	2	no	—
68	Dic₈₃	—	16	1	no	—
69	Dic₈₃	—	16	1	no	—
70	#12528	—	83	1	no	maximal, Sylow
71	C₂×C₈₃	#1721	1	1	yes	characteristic
72	C₂×C₈₃	#2247	1	1	yes	characteristic
73	C₂×C₈₃	#2251	1	1	yes	characteristic
74	#2247	—	83	2	no	—
75	#2241	—	83	2	no	—
76	#1713	—	83	1	no	—
77	#1707	—	83	1	no	—
78	#1312	D₈₃	1	1	yes	characteristic
79	C₈₃	#12528	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₃₂	—	166	4	no	—
81	#500	—	83	1	no	—
82	#500	—	166	1	no	—
83	#500	—	166	1	no	—
84	#498	—	166	1	no	—
85	#494	—	83	1	no	—
86	#494	—	83	2	no	—
87	#487	—	166	1	no	—
88	#358	—	166	4	no	—
89	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
90	C₈²	C₂×D₈₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	166	2	no	—
92	C₂×Q₁₆	—	166	2	no	—
93	C₂×Q₁₆	—	166	2	no	—
94	C₄⋊Q₈	—	166	1	no	—
95	C₄⋊Q₈	—	166	1	no	—
96	C₄⋊Q₈	—	166	2	no	—
97	Q₃₂	—	332	8	no	—
98	C₂×C₁₆	D₃₃₂	1	4	yes	—
99	#107	—	332	2	no	—
100	#107	—	332	2	no	—
101	#107	—	166	2	no	—
102	#106	—	332	2	no	—
103	#103	—	332	2	no	—
104	C₄×C₈	D₃₃₂	1	1	yes	characteristic
105	C₄×C₈	D₃₃₂	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
107	C₂×Q₈	—	332	1	no	—
108	C₂×Q₈	—	332	2	no	—
109	C₂×Q₈	—	332	1	no	—
110	Q₁₆	—	332	4	no	—
111	Q₁₆	—	332	4	no	—
112	Q₁₆	—	332	4	no	—
113	C₂×C₈	D₆₆₄	1	2	yes	—
114	C₂×C₈	D₆₆₄	1	2	yes	—
115	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
116	C₄⋊C₄	—	332	1	no	—
117	C₄⋊C₄	—	332	1	no	—
118	C₄⋊C₄	—	332	1	no	—
119	C₄⋊C₄	—	332	1	no	—
120	C₄⋊C₄	—	332	2	no	—
121	C₄⋊C₄	—	332	2	no	—
122	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	664	2	no	—
125	Q₈	—	664	2	no	—
126	Q₈	—	664	2	no	—
127	Q₈	—	664	2	no	—
128	C₂×C₄	—	664	1	no	—
129	C₂×C₄	—	664	2	no	—
130	C₂×C₄	—	664	1	no	—
131	C₂×C₄	#14581789	1	1	yes	characteristic
132	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₆₆₄	1	4	yes	complemented
136	C₈	#14581820	1	4	yes	—
137	C₂²	#19128078	1	1	yes	characteristic
138	C₄	—	1328	1	no	—
139	C₄	—	1328	2	no	—
140	C₄	—	1328	1	no	—
141	C₄	#19128090	1	2	yes	—
142	C₄	C₂×Dic₆₆₄	1	2	yes	—
143	C₄	#19128144	1	2	yes	—
144	C₂	#24907129	1	1	yes	characteristic
145	C₂	#24909716	1	1	yes	characteristic
146	C₂	#24909734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909716

C₂

yes

maximal

#24909691

C₂

yes

maximal

#24907081

C₂

yes

maximal, characteristic

#24907060

C₂

yes

maximal, characteristic

C₈₃×#1312

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

—

#19128106

—

#19128106

—

#19128106

C₂²

yes

characteristic

#19128099

—

#19128081

C₂²

yes

characteristic

#19128081

C₂²

yes

—

#19128052

—

#19127506

—

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

#14581792

—

#14581792

—

Dic₆₆₄

—

—

—

—

—

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

Q₃₂

yes

complemented

C₈×C₈₃

AGL(1,ℤ/8)

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₄×C₈₃

#496

yes

—

C₄×C₈₃

C₂×Q₃₂

yes

—

C₄×C₈₃

#510

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

—

—

D₈₃

yes

characteristic

C₈₃

#12528

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

166

—

#500

—

#500

—

166

—

#500

—

166

—

#498

—

166

—

#494

—

#494

—

#487

—

166

—

#358

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

Q₃₂

—

332

—

C₂×C₁₆

D₃₃₂

yes

—

#107

—

332

—

100

#107

—

332

—

101

#107

—

166

—

102

#106

—

332

—

103

#103

—

332

—

104

C₄×C₈

D₃₃₂

yes

characteristic

105

C₄×C₈

D₃₃₂

yes

characteristic

106

C₄×C₈

C₂²×D₈₃

yes

characteristic

107

C₂×Q₈

—

332

—

108

C₂×Q₈

—

332

—

109

C₂×Q₈

—

332

—

110

Q₁₆

—

332

—

111

Q₁₆

—

332

—

112

Q₁₆

—

332

—

113

C₂×C₈

D₆₆₄

yes

—

114

C₂×C₈

D₆₆₄

yes

—

115

C₂×C₈

C₂×D₃₃₂

yes

—

116

C₄⋊C₄

—

332

—

117

C₄⋊C₄

—

332

—

118

C₄⋊C₄

—

332

—

119

C₄⋊C₄

—

332

—

120

C₄⋊C₄

—

332

—

121

C₄⋊C₄

—

332

—

122

C₄²

C₂×D₃₃₂

yes

characteristic

123

C₁₆

—

124

Q₈

—

664

—

125

Q₈

—

664

—

126

Q₈

—

664

—

127

Q₈

—

664

—

128

C₂×C₄

—

664

—

129

C₂×C₄

—

664

—

130

C₂×C₄

—

664

—

131

C₂×C₄

#14581789

yes

characteristic

132

C₂×C₄

C₂×D₆₆₄

yes

characteristic

133

C₂×C₄

C₂×D₆₆₄

yes

characteristic

134

C₈

—

135

C₈

Dic₆₆₄

yes

complemented

136

C₈

#14581820

yes

—

137

C₂²

#19128078

yes

characteristic

138

C₄

—

1328

—

139

C₄

—

1328

—

140

C₄

—

1328

—

141

C₄

#19128090

yes

—

142

C₄

C₂×Dic₆₆₄

yes

—

143

C₄

#19128144

yes

—

144

C₂

#24907129

yes

characteristic

145

C₂

#24909716

yes

characteristic

146

C₂

#24909734

yes

characteristic

	subgroup	G/H
center	C₂²	#19128078
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1312	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#1312	D₈₃
hypocenter	C₈₃	#12528
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#19128078

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1312

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
61. C₈×C₈₃	Q₃₂	semidirect
79. C₈₃	#12528	semidirect
135. C₈	Dic₆₆₄	semidirect

H ≅ G/N

type

61. C₈×C₈₃

Q₃₂

semidirect

79. C₈₃

#12528

semidirect

135. C₈

Dic₆₆₄

semidirect

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

subgroup	index
#24909716	2
#24909691	2
#24907081	2
#24907060	2
C₈₃×#1312	2
#12528	83

subgroup

index

C₈₃×#1312

Subgroups — #35319173

Conjugacy classes of subgroups

Complements

Maximal subgroups