Group #35325378 — subgroups

Subgroups — #35325378

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	#24909716	C₂	1	1	yes	maximal, characteristic
2	#24905643	C₂	1	4	yes	maximal
3	C₄×C₈₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#2252	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₆₄	—	2	1	no	—
6	C₂×Dic₆₆₄	C₂²	1	2	yes	—
7	#19128081	—	2	1	no	—
8	C₂×C₈₃⋊C₃₂	C₂²	1	2	yes	—
9	#500×C₈₃	—	2	1	no	—
10	#366×C₈₃	—	2	1	no	—
11	#366×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	Dic₆₆₄	—	2	4	no	—
17	Dic₆₆₄	—	4	2	no	—
18	C₈₃⋊C₃₂	—	2	2	no	—
19	C₄⋊Q₈×C₈₃	—	4	1	no	—
20	C₂×C₁₆×C₈₃	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
22	#107×C₈₃	—	2	2	no	—
23	#107×C₈₃	—	4	1	no	—
24	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂×Dic₁₆₆	—	4	2	no	—
26	Dic₃₃₂	—	4	2	no	—
27	Dic₃₃₂	—	4	2	no	—
28	Dic₃₃₂	—	8	2	no	—
29	C₄⋊Dic₈₃	—	8	1	no	—
30	C₄⋊Dic₈₃	—	8	1	no	—
31	C₂×Q₈×C₈₃	—	8	1	no	—
32	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
33	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₈₃	—	4	2	no	—
35	C₄⋊C₄×C₈₃	—	8	1	no	—
36	C₄²×C₈₃	D₈	1	1	yes	characteristic
37	C₁₆×C₈₃	SD₁₆	1	2	yes	—
38	C₁₆×C₈₃	#41	1	2	yes	—
39	Q₈×C₈₃	—	16	2	no	—
40	C₂×C₄×C₈₃	—	8	2	no	—
41	C₂×C₄×C₈₃	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₈₃	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₈₃	—	8	2	no	—
45	Dic₁₆₆	—	8	2	no	—
46	Dic₁₆₆	—	8	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	—	16	2	no	—
52	C₄×C₈₃	SD₆₄	1	2	yes	—
53	C₄×C₈₃	Q₆₄	1	2	yes	—
54	C₄×C₈₃	#486	1	1	yes	characteristic
55	C₄×C₈₃	C₂×SD₃₂	1	1	yes	characteristic
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	D₈₃	1	1	yes	characteristic
62	#2247	—	83	1	no	—
63	#1416	—	83	4	no	—
64	C₄×C₃₂	—	83	1	no	—
65	C₈₃	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	83	2	no	—
67	C₂×Q₃₂	—	166	1	no	—
68	#500	—	2	1	no	—
69	#494	—	166	1	no	—
70	C₂×C₃₂	—	83	2	no	—
71	#366	—	2	1	no	—
72	#366	C₂×D₈₃	1	2	yes	—
73	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
74	C₂×Q₁₆	—	332	1	no	—
75	C₂×Q₁₆	—	166	2	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	332	1	no	—
78	Q₃₂	—	332	2	no	—
79	Q₃₂	—	166	4	no	—
80	C₂×C₁₆	C₂²⋊D₈₃	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
82	#107	—	2	2	no	—
83	#107	—	4	1	no	—
84	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
85	C₃₂	—	166	2	no	—
86	C₂×Q₈	—	8	1	no	—
87	C₂×Q₈	—	332	2	no	—
88	Q₁₆	—	332	2	no	—
89	Q₁₆	—	332	2	no	—
90	Q₁₆	—	664	2	no	—
91	C₂×C₈	#13050575	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	664	1	no	—
96	C₄⋊C₄	—	664	1	no	—
97	C₄²	#13050575	1	1	yes	characteristic
98	C₁₆	#13050577	1	2	yes	—
99	C₁₆	#13050591	1	2	yes	—
100	Q₈	—	16	2	no	—
101	Q₈	—	664	2	no	—
102	Q₈	—	664	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	664	2	no	—
105	C₂×C₄	#14581733	1	1	yes	characteristic
106	C₂×C₄	#14581733	1	1	yes	characteristic
107	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
108	C₈	#14581735	1	2	yes	—
109	C₈	#14581781	1	1	yes	characteristic
110	C₈	C₂×#13050577	1	1	yes	characteristic
111	C₂²	C₂×#14581733	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	1328	2	no	—
114	C₄	#19127539	1	2	yes	—
115	C₄	#19127541	1	2	yes	—
116	C₄	#19128049	1	1	yes	characteristic
117	C₄	C₂×#14581735	1	1	yes	characteristic
118	C₂	#24909688	1	1	yes	characteristic
119	C₂	C₂×#19127539	1	1	yes	characteristic
120	C₂	C₂×#19127541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909716

C₂

yes

maximal, characteristic

#24905643

C₂

yes

maximal

C₄×C₈₃⋊C₃₂

C₂

yes

maximal, characteristic

C₈₃×#2252

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

—

C₂×Dic₆₆₄

C₂²

yes

—

#19128081

—

C₂×C₈₃⋊C₃₂

C₂²

yes

—

#500×C₈₃

—

#366×C₈₃

—

#366×C₈₃

C₂²

yes

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×Dic₃₃₂

—

#14581792

—

Dic₆₆₄

—

Dic₆₆₄

—

C₈₃⋊C₃₂

—

C₄⋊Q₈×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂³

yes

characteristic

#107×C₈₃

—

#107×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

D₈

yes

characteristic

C₁₆×C₈₃

SD₁₆

yes

—

C₁₆×C₈₃

#41

yes

—

Q₈×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

SD₃₂

yes

—

C₈×C₈₃

#124

yes

characteristic

C₈×C₈₃

C₂×SD₁₆

yes

characteristic

C₂²×C₈₃

C₂×D₁₆

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

SD₆₄

yes

—

C₄×C₈₃

Q₆₄

yes

—

C₄×C₈₃

#486

yes

characteristic

C₄×C₈₃

C₂×SD₃₂

yes

characteristic

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

C₂×SD₆₄

yes

characteristic

C₂×C₈₃

C₂×Q₆₄

yes

characteristic

#2252

D₈₃

yes

characteristic

#2247

—

#1416

—

C₄×C₃₂

—

C₈₃

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

166

—

#500

—

#494

—

166

—

C₂×C₃₂

—

#366

—

#366

C₂×D₈₃

yes

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

332

—

C₂×Q₁₆

—

166

—

C₄⋊Q₈

—

C₄⋊Q₈

—

332

—

Q₃₂

—

332

—

Q₃₂

—

166

—

C₂×C₁₆

C₂²⋊D₈₃

yes

characteristic

C₂×C₁₆

C₂²×D₈₃

yes

characteristic

#107

—

#107

—

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

C₃₂

—

166

—

C₂×Q₈

—

C₂×Q₈

—

332

—

Q₁₆

—

332

—

Q₁₆

—

332

—

Q₁₆

—

664

—

C₂×C₈

#13050575

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

664

—

C₄⋊C₄

—

664

—

C₄²

#13050575

yes

characteristic

C₁₆

#13050577

yes

—

C₁₆

#13050591

yes

—

100

Q₈

—

101

Q₈

—

664

—

102

Q₈

—

664

—

103

C₂×C₄

—

104

C₂×C₄

—

664

—

105

C₂×C₄

#14581733

yes

characteristic

106

C₂×C₄

#14581733

yes

characteristic

107

C₂×C₄

C₂×#13050575

yes

characteristic

108

C₈

#14581735

yes

—

109

C₈

#14581781

yes

characteristic

110

C₈

C₂×#13050577

yes

characteristic

111

C₂²

C₂×#14581733

yes

characteristic

112

C₄

—

113

C₄

—

1328

—

114

C₄

#19127539

yes

—

115

C₄

#19127541

yes

—

116

C₄

#19128049

yes

characteristic

117

C₄

C₂×#14581735

yes

characteristic

118

C₂

#24909688

yes

characteristic

119

C₂

C₂×#19127539

yes

characteristic

120

C₂

C₂×#19127541

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14581733
commutator, cosocle	C₂×C₁₆×C₈₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₃
Fitting	C₈₃×#2252	C₂
socle, prime radical	C₂²×C₈₃	C₂×D₁₆
hypercenter	#2252	D₈₃
hypocenter	C₈₃	#13720
LCS	G ▷ C₂×C₁₆×C₈₃ ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

subgroup

G/H

center

C₂²

C₂×#14581733

commutator, cosocle

C₂×C₁₆×C₈₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₈₃

Fitting

C₈₃×#2252

C₂

socle, prime radical

C₂²×C₈₃

C₂×D₁₆

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

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
#24909716	2
#24905643	2
C₄×C₈₃⋊C₃₂	2
C₈₃×#2252	2
#13720	83

subgroup

index

#24909716

#24905643

C₄×C₈₃⋊C₃₂

C₈₃×#2252

#13720

Subgroups — #35325378

Conjugacy classes of subgroups

Complements

Maximal subgroups