Group #35325377 — subgroups

Subgroups — #35325377

Order 21248.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#24909738	C₂	1	1	yes	maximal, characteristic
2	#24909716	C₂	1	1	yes	maximal, characteristic
3	#24905645	C₂	1	4	yes	maximal
4	C₄×C₃₂×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₆₄	—	2	1	no	—
6	C₂×Dic₆₆₄	C₂²	1	2	yes	—
7	#19128106	—	2	1	no	—
8	#19128081	—	2	1	no	—
9	#19127527	—	2	1	no	—
10	#19127527	C₂²	1	2	yes	—
11	C₂×C₃₂×C₈₃	C₂²	1	2	yes	—
12	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₃₂	—	2	2	no	—
14	C₂×Dic₃₃₂	—	4	1	no	—
15	#14581792	—	4	1	no	—
16	#14581792	—	4	1	no	—
17	Dic₆₆₄	—	2	4	no	—
18	Dic₆₆₄	—	4	2	no	—
19	#14581726	—	2	2	no	—
20	#14581726	—	4	1	no	—
21	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
22	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₃₂×C₈₃	—	2	2	no	—
25	C₂×Dic₁₆₆	—	4	2	no	—
26	C₂×Dic₁₆₆	—	8	1	no	—
27	Dic₃₃₂	—	4	2	no	—
28	Dic₃₃₂	—	4	2	no	—
29	Dic₃₃₂	—	8	2	no	—
30	C₄⋊Dic₈₃	—	4	2	no	—
31	C₄⋊Dic₈₃	—	8	1	no	—
32	C₄⋊Dic₈₃	—	8	1	no	—
33	C₄⋊Dic₈₃	—	8	1	no	—
34	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
35	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₈₃	D₈	1	1	yes	characteristic
37	C₁₆×C₈₃	SD₁₆	1	2	yes	—
38	C₁₆×C₈₃	#41	1	2	yes	—
39	C₂×C₄×C₈₃	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₈₃	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₈₃	—	8	2	no	—
43	C₂×Dic₈₃	—	8	2	no	—
44	Dic₁₆₆	—	8	2	no	—
45	Dic₁₆₆	—	8	2	no	—
46	Dic₁₆₆	—	16	2	no	—
47	C₈×C₈₃	SD₃₂	1	2	yes	—
48	C₈×C₈₃	#124	1	1	yes	characteristic
49	C₈×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₈₃	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₈₃	SD₆₄	1	2	yes	—
52	C₄×C₈₃	Q₆₄	1	2	yes	—
53	C₄×C₈₃	#486	1	1	yes	characteristic
54	C₄×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
55	Dic₈₃	—	16	2	no	—
56	Dic₈₃	—	16	2	no	—
57	#13720	—	83	1	no	maximal, Sylow
58	C₂×C₈₃	#2240	1	1	yes	characteristic
59	C₂×C₈₃	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₈₃	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	—	83	1	no	—
62	#2247	—	83	1	no	—
63	#1416	—	83	4	no	—
64	C₄×C₃₂	D₈₃	1	1	yes	characteristic
65	C₈₃	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	83	2	no	—
67	C₂×Q₃₂	—	166	1	no	—
68	#500	—	166	1	no	—
69	#494	—	166	1	no	—
70	C₂×C₃₂	C₂×D₈₃	1	2	yes	—
71	#366	—	83	2	no	—
72	#366	—	166	1	no	—
73	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
74	C₂×Q₁₆	—	332	1	no	—
75	C₂×Q₁₆	—	166	2	no	—
76	C₄⋊Q₈	—	332	1	no	—
77	C₄⋊Q₈	—	332	1	no	—
78	Q₃₂	—	332	2	no	—
79	Q₃₂	—	166	4	no	—
80	C₂×C₁₆	D₃₃₂	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
82	#107	—	332	1	no	—
83	#107	—	166	2	no	—
84	C₄×C₈	D₃₃₂	1	1	yes	characteristic
85	C₃₂	—	2	2	no	—
86	C₂×Q₈	—	332	2	no	—
87	C₂×Q₈	—	664	1	no	—
88	Q₁₆	—	332	2	no	—
89	Q₁₆	—	332	2	no	—
90	Q₁₆	—	664	2	no	—
91	C₂×C₈	D₆₆₄	1	1	yes	characteristic
92	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
93	C₄⋊C₄	—	332	2	no	—
94	C₄⋊C₄	—	664	1	no	—
95	C₄⋊C₄	—	664	1	no	—
96	C₄⋊C₄	—	664	1	no	—
97	C₄²	D₆₆₄	1	1	yes	characteristic
98	C₁₆	#13050579	1	2	yes	—
99	C₁₆	#13050590	1	2	yes	—
100	Q₈	—	1328	2	no	—
101	Q₈	—	664	2	no	—
102	Q₈	—	664	2	no	—
103	C₂×C₄	—	664	2	no	—
104	C₂×C₄	—	664	2	no	—
105	C₂×C₄	D₁₃₂₈	1	1	yes	characteristic
106	C₂×C₄	D₁₃₂₈	1	1	yes	characteristic
107	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
108	C₈	#14581737	1	2	yes	—
109	C₈	#14581780	1	1	yes	characteristic
110	C₈	C₂×#13050579	1	1	yes	characteristic
111	C₂²	C₂×D₁₃₂₈	1	1	yes	characteristic
112	C₄	—	1328	2	no	—
113	C₄	—	1328	2	no	—
114	C₄	#19127540	1	2	yes	—
115	C₄	Dic₁₃₂₈	1	2	yes	—
116	C₄	#19128048	1	1	yes	characteristic
117	C₄	C₂×#14581737	1	1	yes	characteristic
118	C₂	#24909687	1	1	yes	characteristic
119	C₂	C₂×#19127540	1	1	yes	characteristic
120	C₂	C₂×Dic₁₃₂₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909738

C₂

yes

maximal, characteristic

#24909716

C₂

yes

maximal, characteristic

#24905645

C₂

yes

maximal

C₄×C₃₂×C₈₃

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

—

C₂×Dic₆₆₄

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×C₃₂×C₈₃

C₂²

yes

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

#14581792

—

Dic₆₆₄

—

Dic₆₆₄

—

#14581726

—

#14581726

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₃₂×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄²×C₈₃

D₈

yes

characteristic

C₁₆×C₈₃

SD₁₆

yes

—

C₁₆×C₈₃

#41

yes

—

C₂×C₄×C₈₃

D₁₆

yes

characteristic

C₂×C₄×C₈₃

D₁₆

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

SD₃₂

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈×C₈₃

C₂×SD₁₆

yes

characteristic

C₂²×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

SD₆₄

yes

—

C₄×C₈₃

Q₆₄

yes

—

C₄×C₈₃

#486

yes

characteristic

C₄×C₈₃

C₂×SD₃₂

yes

characteristic

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×SD₆₄

yes

characteristic

C₂×C₈₃

C₂×Q₆₄

yes

—

—

—

C₄×C₃₂

D₈₃

yes

characteristic

C₈₃

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

166

—

#500

—

166

—

#494

—

166

—

C₂×C₃₂

C₂×D₈₃

yes

—

#366

—

#366

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

332

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

332

—

C₄⋊Q₈

—

332

—

Q₃₂

—

332

—

Q₃₂

—

166

—

C₂×C₁₆

D₃₃₂

yes

characteristic

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

#107

—

332

—

#107

—

166

—

C₄×C₈

D₃₃₂

yes

characteristic

C₃₂

—

C₂×Q₈

—

332

—

C₂×Q₈

—

664

—

Q₁₆

—

332

—

Q₁₆

—

332

—

Q₁₆

—

664

—

C₂×C₈

D₆₆₄

yes

characteristic

C₂×C₈

C₂×D₃₃₂

yes

characteristic

C₄⋊C₄

—

332

—

C₄⋊C₄

—

664

—

C₄⋊C₄

—

664

—

C₄⋊C₄

—

664

—

C₄²

D₆₆₄

yes

characteristic

C₁₆

#13050579

yes

—

C₁₆

#13050590

yes

—

100

Q₈

—

1328

—

101

Q₈

—

664

—

102

Q₈

—

664

—

103

C₂×C₄

—

664

—

104

C₂×C₄

—

664

—

105

C₂×C₄

D₁₃₂₈

yes

characteristic

106

C₂×C₄

D₁₃₂₈

yes

characteristic

107

C₂×C₄

C₂×D₆₆₄

yes

characteristic

108

C₈

#14581737

yes

—

109

C₈

#14581780

yes

characteristic

110

C₈

C₂×#13050579

yes

characteristic

111

C₂²

C₂×D₁₃₂₈

yes

characteristic

112

C₄

—

1328

—

113

C₄

—

1328

—

114

C₄

#19127540

yes

—

115

C₄

Dic₁₃₂₈

yes

—

116

C₄

#19128048

yes

characteristic

117

C₄

C₂×#14581737

yes

characteristic

118

C₂

#24909687

yes

characteristic

119

C₂

C₂×#19127540

yes

characteristic

120

C₂

C₂×Dic₁₃₂₈

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₃₂₈
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₄×C₃₂×C₈₃	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₈₃
hypocenter	C₈₃	#13720
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₃₂₈

commutator, cosocle

C₂×C₁₆×C₈₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₄×C₃₂×C₈₃

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₁₆

hypercenter

C₄×C₃₂

D₈₃

hypocenter

C₈₃

#13720

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

N	H ≅ G/N	type
65. C₈₃	#13720	semidirect

H ≅ G/N

type

65. C₈₃

#13720

semidirect

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

subgroup	index
#24909738	2
#24909716	2
#24905645	2
C₄×C₃₂×C₈₃	2
#13720	83

subgroup

index

C₄×C₃₂×C₈₃

Subgroups — #35325377

Conjugacy classes of subgroups

Complements

Maximal subgroups