Group #35298218 — subgroups

Subgroups — #35298218

Order 21248.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#24907635	C₂	1	2	yes	maximal
2	#24906222	C₂	1	4	yes	maximal
3	C₂×#322×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581728	—	2	4	no	—
5	#19127725	C₂²	1	4	yes	—
6	#19127454	—	2	8	no	—
7	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	#322×C₈₃	C₄	1	4	yes	—
10	C₂×#13050574	—	2	4	no	—
11	#14581728	—	4	8	no	—
12	C₄⋊(C₈₃⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₈₃	D₄	1	2	yes	—
14	C₂²×C₈×C₈₃	Q₈	1	4	yes	—
15	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₈₃	D₄	1	4	yes	—
19	C₄×C₈×C₈₃	Q₈	1	2	yes	—
20	#13050574	—	4	8	no	—
21	C₂×C₈₃⋊C₈	—	4	4	no	—
22	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₈₃	C₂×Q₈	1	2	yes	—
24	C₂×C₈×C₈₃	—	4	4	no	—
25	C₂×C₈×C₈₃	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₈₃	D₈	1	4	yes	—
28	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
29	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₈₃	C₂×D₄	1	2	yes	—
31	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
32	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₈₃	—	2	4	no	—
34	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
37	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
40	C₈×C₈₃	—	2	8	no	—
41	C₈×C₈₃	#107	1	8	yes	—
42	C₈₃⋊C₈	—	8	4	no	—
43	C₂²×C₈₃	—	2	2	no	—
44	C₂²×C₈₃	#384	1	1	yes	characteristic
45	C₂²×C₈₃	#500	1	2	yes	—
46	C₄×C₈₃	—	2	2	no	—
47	C₄×C₈₃	C₂×#107	1	2	yes	—
48	C₄×C₈₃	#430	1	4	yes	—
49	C₄×C₈₃	#493	1	1	yes	characteristic
50	C₄×C₈₃	#494	1	1	yes	characteristic
51	#8826	—	83	1	no	maximal, Sylow
52	C₂×C₈₃	—	2	2	no	—
53	C₂×C₈₃	#1845	1	1	yes	characteristic
54	C₂×C₈₃	#1849	1	2	yes	—
55	#1848	—	83	2	no	—
56	#1567	—	83	4	no	—
57	C₂×#322	D₈₃	1	1	yes	characteristic
58	C₈₃	#8826	1	1	yes	characteristic, complemented, Sylow
59	C₂×#108	—	166	4	no	—
60	#423	—	83	4	no	—
61	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
62	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
63	#335	—	166	8	no	—
64	#322	Dic₈₃	1	4	yes	—
65	C₂×M₁₆	—	166	4	no	—
66	C₂²×C₈	Dic₁₆₆	1	4	yes	—
67	C₂²×C₈	D₃₃₂	1	2	yes	—
68	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
69	#108	—	332	8	no	—
70	C₄⋊C₈	—	166	8	no	—
71	C₄×C₈	Dic₁₆₆	1	2	yes	—
72	C₄×C₈	D₃₃₂	1	4	yes	—
73	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
74	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
75	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₁₆₆	1	2	yes	—
77	M₁₆	—	332	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	332	4	no	—
80	C₂×C₈	C₄⋊Dic₈₃	1	8	yes	—
81	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
82	C₂×C₈	D₆₆₄	1	4	yes	—
83	C₂×C₈	Dic₃₃₂	1	4	yes	—
84	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
85	C₄²	C₂×D₃₃₂	1	2	yes	—
86	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
87	C₂³	#14581792	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
91	C₂×C₄	#14581789	1	1	yes	characteristic
92	C₂×C₄	#14581792	1	2	yes	—
93	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
94	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	664	4	no	—
97	C₈	#14581726	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#19127571	1	1	yes	characteristic
100	C₂²	#19128106	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#14581726	1	2	yes	—
103	C₄	#19127750	1	4	yes	—
104	C₄	#19128078	1	1	yes	characteristic
105	C₄	#19128081	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	#24907623	1	1	yes	characteristic
108	C₂	#24907638	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal

#24906222

C₂

yes

maximal

C₂×#322×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127454

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₄

yes

—

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

Q₈

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

Q₁₆

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₂³×C₈₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

#107

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#384

yes

characteristic

C₂²×C₈₃

#500

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#107

yes

—

C₄×C₈₃

#430

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1845

yes

characteristic

C₂×C₈₃

#1849

yes

—

#1848

—

#1567

—

C₂×#322

D₈₃

yes

characteristic

C₈₃

#8826

yes

characteristic, complemented, Sylow

C₂×#108

—

166

—

#423

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄×C₈

C₂×D₈₃

yes

—

#335

—

166

—

#322

Dic₈₃

yes

—

C₂×M₁₆

—

166

—

C₂²×C₈

Dic₁₆₆

yes

—

C₂²×C₈

D₃₃₂

yes

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#108

—

332

—

C₄⋊C₈

—

166

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₆

yes

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

D₆₆₄

yes

—

C₂×C₈

Dic₃₃₂

yes

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×D₃₃₂

yes

—

C₄²

C₂×Dic₁₆₆

yes

characteristic

C₂³

#14581792

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

C₂×C₄

#14581789

yes

characteristic

C₂×C₄

#14581792

yes

—

C₂×C₄

C₂×D₆₆₄

yes

—

C₂×C₄

C₂×Dic₃₃₂

yes

—

C₈

—

C₈

—

664

—

C₈

#14581726

yes

—

C₂²

—

C₂²

#19127571

yes

characteristic

100

C₂²

#19128106

yes

—

101

C₄

—

102

C₄

C₂×#14581726

yes

—

103

C₄

#19127750

yes

—

104

C₄

#19128078

yes

characteristic

105

C₄

#19128081

yes

characteristic

106

C₂

—

107

C₂

#24907623

yes

characteristic

108

C₂

#24907638

yes

—

	subgroup	G/H
center	C₂×C₄	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#322×C₈₃	C₂
socle	C₂²×C₈₃	#384
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×#322	D₈₃
hypocenter	C₈₃	#8826
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14581789

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×#322×C₈₃

C₂

socle

C₂²×C₈₃

#384

prime radical

C₂³×C₈₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
58. C₈₃	#8826	semidirect

H ≅ G/N

type

58. C₈₃

#8826

semidirect

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

subgroup	index
#24907635	2
#24906222	2
C₂×#322×C₈₃	2
#8826	83

subgroup

index

#24907635

#24906222

C₂×#322×C₈₃

#8826

Subgroups — #35298218

Conjugacy classes of subgroups

Complements

Maximal subgroups