Group #35309510 — subgroups

Subgroups — #35309510

Order 21248.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#24908269	C₂	1	3	yes	maximal
2	#24907620	C₂	1	3	yes	maximal, complemented
3	C₈₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14581789	C₂²	1	1	yes	characteristic
5	C₂×#14581724	—	2	6	no	—
6	C₂×#14581714	—	2	6	no	—
7	#19127571	C₂²	1	3	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	3	yes	—
9	C₂²×D₃₃₂	—	2	3	no	—
10	#14581789	—	2	3	no	—
11	#14581789	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₈₃	—	2	3	no	—
13	C₂×C₄⋊Dic₈₃	—	2	6	no	—
14	#14581724	—	4	12	no	—
15	#14581714	—	4	12	no	—
16	C₂²×C₈×C₈₃	D₄	1	6	yes	—
17	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₈₃	—	2	6	no	—
19	C₂³×D₈₃	—	4	1	no	—
20	C₂×D₃₃₂	—	2	6	no	—
21	C₂×D₃₃₂	—	4	6	no	—
22	C₂×D₃₃₂	—	4	6	no	—
23	C₂×D₃₃₂	—	4	3	no	—
24	C₂²×Dic₈₃	—	4	3	no	—
25	C₄⋊Dic₈₃	—	4	6	no	—
26	C₄⋊Dic₈₃	—	4	12	no	—
27	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₈₃	—	2	6	no	—
29	C₂×C₈×C₈₃	—	2	6	no	—
30	C₂×C₈×C₈₃	SD₁₆	1	12	yes	—
31	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₈₃	#41	1	3	yes	—
33	C₂³×C₈₃	#127	1	1	yes	characteristic
34	C₂²×D₈₃	—	4	2	no	—
35	C₂²×D₈₃	—	8	3	no	—
36	C₂²×D₈₃	—	8	3	no	—
37	C₂×C₄×C₈₃	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₈₃	#123	1	3	yes	—
39	C₂×C₄×C₈₃	C₂×SD₁₆	1	6	yes	—
40	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	6	yes	—
41	C₂×Dic₈₃	—	8	3	no	—
42	C₂×Dic₈₃	—	8	6	no	—
43	D₃₃₂	—	4	12	no	—
44	D₃₃₂	—	8	6	no	—
45	C₈×C₈₃	—	2	12	no	—
46	C₂²×C₈₃	#399	1	1	yes	characteristic
47	C₂²×C₈₃	#492	1	3	yes	—
48	C₂²×C₈₃	#496	1	3	yes	—
49	C₂×D₈₃	—	8	6	no	—
50	C₂×D₈₃	—	16	1	no	—
51	C₂×D₈₃	—	16	3	no	—
52	C₄×C₈₃	#476	1	6	yes	—
53	C₄×C₈₃	#481	1	6	yes	—
54	Dic₈₃	—	16	3	no	—
55	#10683	—	83	1	no	maximal, Sylow
56	C₂×C₈₃	#1705	1	1	yes	characteristic
57	C₂×C₈₃	#1715	1	3	yes	—
58	C₂×C₈₃	#2094	1	3	yes	—
59	D₈₃	—	16	2	no	—
60	#1957	—	83	3	no	—
61	#1844	—	83	3	no	—
62	#1273	D₈₃	1	1	yes	characteristic, complemented
63	C₈₃	#10683	1	1	yes	characteristic, complemented, Sylow
64	C₂×#127	—	83	1	no	—
65	C₂×#106	—	166	6	no	—
66	C₂×#102	—	166	6	no	—
67	C₂×C₄×C₈	C₂×D₈₃	1	3	yes	—
68	#384	—	83	3	no	—
69	C₂²×D₄	—	166	3	no	—
70	C₂²×C₈	D₃₃₂	1	6	yes	—
71	#127	—	83	2	no	—
72	#127	—	166	3	no	—
73	C₂×C₄⋊C₄	—	166	3	no	—
74	C₂×C₄⋊C₄	—	166	6	no	—
75	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
76	#106	—	332	12	no	—
77	#102	—	332	12	no	—
78	C₄×C₈	—	2	6	no	—
79	C₂⁴	—	332	1	no	—
80	C₂×D₄	—	332	3	no	—
81	C₂×D₄	—	332	6	no	—
82	C₂×D₄	—	332	6	no	—
83	C₂×D₄	—	166	6	no	—
84	C₂²×C₄	—	332	3	no	—
85	C₂²×C₄	C₂×D₃₃₂	1	3	yes	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	#13050579	1	12	yes	—
89	C₄⋊C₄	—	332	6	no	—
90	C₄⋊C₄	—	332	12	no	—
91	C₄²	Q₈×D₈₃	1	1	yes	characteristic
92	C₄²	#13050589	1	3	yes	—
93	C₂³	—	332	2	no	—
94	C₂³	—	664	3	no	—
95	C₂³	—	664	3	no	—
96	C₂³	#14581789	1	1	yes	characteristic
97	D₄	—	332	12	no	—
98	D₄	—	664	6	no	—
99	C₂×C₄	—	664	3	no	—
100	C₂×C₄	—	664	6	no	—
101	C₂×C₄	#14581772	1	3	yes	—
102	C₂×C₄	#14581776	1	3	yes	—
103	C₂×C₄	C₂×#13050579	1	6	yes	—
104	C₂×C₄	#14581820	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	1328	1	no	—
107	C₂²	—	1328	3	no	—
108	C₂²	—	664	6	no	—
109	C₂²	#19127638	1	1	yes	characteristic
110	C₂²	#19128074	1	3	yes	—
111	C₂²	#19128090	1	3	yes	—
112	C₄	—	1328	3	no	—
113	C₄	#19127992	1	6	yes	—
114	C₄	#19128020	1	6	yes	—
115	C₂	—	1328	2	no	—
116	C₂	#24907050	1	1	yes	characteristic
117	C₂	#24907091	1	3	yes	—
118	C₂	#24908995	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24908269

C₂

yes

maximal

#24907620

C₂

yes

maximal, complemented

C₈₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#14581789

C₂²

yes

—

—

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂²×D₃₃₂

—

#14581789

—

#14581789

Q₈

yes

—

C₂×C₄⋊Dic₈₃

—

C₂×C₄⋊Dic₈₃

—

#14581724

—

#14581714

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

—

C₂³×D₈₃

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

C₂²×Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#127

yes

characteristic

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

#123

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

C₈×C₈₃

—

C₂²×C₈₃

#399

yes

characteristic

C₂²×C₈₃

#492

yes

—

C₂²×C₈₃

#496

yes

—

C₂×D₈₃

—

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#476

yes

—

C₄×C₈₃

#481

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#2094

yes

—

D₈₃

—

#1957

—

#1844

—

#1273

D₈₃

yes

characteristic, complemented

C₈₃

#10683

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#106

—

166

—

C₂×#102

—

166

—

C₂×C₄×C₈

C₂×D₈₃

yes

—

#384

—

C₂²×D₄

—

166

—

C₂²×C₈

D₃₃₂

yes

—

#127

—

#127

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#106

—

332

—

#102

—

332

—

C₄×C₈

—

C₂⁴

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

166

—

C₂²×C₄

—

332

—

C₂²×C₄

C₂×D₃₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#13050579

yes

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄²

Q₈×D₈₃

yes

characteristic

C₄²

#13050589

yes

—

C₂³

—

332

—

C₂³

—

664

—

C₂³

—

664

—

C₂³

#14581789

yes

characteristic

D₄

—

332

—

D₄

—

664

—

C₂×C₄

—

664

—

100

C₂×C₄

—

664

—

101

C₂×C₄

#14581772

yes

—

102

C₂×C₄

#14581776

yes

—

103

C₂×C₄

C₂×#13050579

yes

—

104

C₂×C₄

#14581820

yes

—

105

C₈

—

106

C₂²

—

1328

—

107

C₂²

—

1328

—

108

C₂²

—

664

—

109

C₂²

#19127638

yes

characteristic

110

C₂²

#19128074

yes

—

111

C₂²

#19128090

yes

—

112

C₄

—

1328

—

113

C₄

#19127992

yes

—

114

C₄

#19128020

yes

—

115

C₂

—

1328

—

116

C₂

#24907050

yes

characteristic

117

C₂

#24907091

yes

—

118

C₂

#24908995

yes

—

	subgroup	G/H
center	C₂³	#14581789
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1273	C₂
socle	C₂³×C₈₃	#127
prime radical	C₂×#14581789	C₂²
hypercenter	#1273	D₈₃
hypocenter	C₈₃	#10683
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#14581789

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1273

C₂

C₂³×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #24907620	C₂	semidirect
3. C₈₃×#1273	C₂	semidirect
62. #1273	D₈₃	semidirect
63. C₈₃	#10683	semidirect

H ≅ G/N

type

2. #24907620

C₂

semidirect

3. C₈₃×#1273

C₂

semidirect

62. #1273

D₈₃

semidirect

63. C₈₃

#10683

semidirect

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

subgroup	index
#24908269	2
#24907620	2
C₈₃×#1273	2
#10683	83

subgroup

index

#24908269

#24907620

C₈₃×#1273

#10683

Subgroups — #35309510

Conjugacy classes of subgroups

Complements

Maximal subgroups