Group #35319142 — subgroups

Subgroups — #35319142

Order 21248.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#24909743	C₂	1	2	yes	maximal
2	#24909736	C₂	1	2	yes	maximal
3	#24907073	C₂	1	1	yes	maximal, characteristic
4	C₈×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1710	C₂	1	1	yes	maximal, characteristic
6	#19128106	C₂²	1	2	yes	—
7	#19128052	—	2	2	no	—
8	#19127528	—	2	4	no	—
9	#19127526	—	2	4	no	—
10	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	#500×C₈₃	C₂²	1	2	yes	—
13	#487×C₈₃	—	2	2	no	—
14	C₈²×C₈₃	C₂²	1	1	yes	characteristic
15	#14581792	—	2	2	no	—
16	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
17	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
18	#14581726	—	2	2	no	—
19	#14581726	—	2	2	no	—
20	#14581717	—	4	4	no	—
21	C₄⋊Q₈×C₈₃	—	2	2	no	—
22	#107×C₈₃	—	2	2	no	—
23	#107×C₈₃	—	2	2	no	—
24	#103×C₈₃	—	4	4	no	—
25	C₄×C₈×C₈₃	D₄	1	2	yes	—
26	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₆₆	—	4	2	no	—
28	C₄⋊Dic₈₃	—	4	2	no	—
29	C₄⋊Dic₈₃	—	4	2	no	—
30	C₈₃⋊C₁₆	Q₁₆	1	4	yes	—
31	C₈₃⋊C₁₆	Q₁₆	1	4	yes	—
32	C₂×Q₈×C₈₃	—	4	2	no	—
33	C₂×C₈×C₈₃	D₈	1	4	yes	—
34	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₈₃	—	4	2	no	—
37	C₄⋊C₄×C₈₃	—	4	2	no	—
38	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₈₃	—	8	4	no	—
40	C₂×C₄×C₈₃	—	8	2	no	—
41	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
43	C₂×Dic₈₃	—	8	2	no	—
44	Dic₁₆₆	—	8	4	no	—
45	C₈×C₈₃	—	2	4	no	—
46	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
47	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
48	C₂²×C₈₃	#493	1	1	yes	characteristic
49	C₄×C₈₃	—	16	2	no	—
50	C₄×C₈₃	#494	1	1	yes	characteristic
51	C₄×C₈₃	#494	1	1	yes	characteristic
52	C₄×C₈₃	#508	1	4	yes	—
53	Dic₈₃	—	16	2	no	—
54	#12522	—	83	1	no	maximal, Sylow
55	C₂×C₈₃	#1713	1	1	yes	characteristic
56	C₂×C₈₃	#2249	1	2	yes	—
57	#2254	—	83	2	no	—
58	#2252	—	83	2	no	—
59	#1710	—	83	1	no	—
60	#1710	D₈₃	1	1	yes	characteristic
61	C₈×C₁₆	—	83	1	no	—
62	C₈₃	#12522	1	1	yes	characteristic, complemented, Sylow
63	#500	—	83	2	no	—
64	#500	C₂×D₈₃	1	2	yes	—
65	#487	—	2	2	no	—
66	#487	—	166	2	no	—
67	#367	—	166	4	no	—
68	#366	—	166	4	no	—
69	C₄×C₁₆	—	83	1	no	—
70	C₄×C₁₆	—	83	1	no	—
71	C₈²	C₂×D₈₃	1	1	yes	characteristic
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	—	166	2	no	—
74	C₂×C₁₆	—	83	2	no	—
75	C₂×C₁₆	—	83	2	no	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	166	2	no	—
79	#107	—	166	2	no	—
80	#103	—	4	4	no	—
81	#103	—	332	4	no	—
82	C₄×C₈	C₂²⋊D₈₃	1	2	yes	—
83	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	—	332	2	no	—
86	C₂×C₈	#13050575	1	4	yes	—
87	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
88	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	332	2	no	—
92	C₄⋊C₄	—	332	2	no	—
93	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
94	C₁₆	—	83	4	no	—
95	C₁₆	—	83	4	no	—
96	Q₈	—	8	4	no	—
97	Q₈	—	664	4	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	664	2	no	—
100	C₂×C₄	#14581788	1	1	yes	characteristic
101	C₂×C₄	C₂×#13050575	1	2	yes	—
102	C₈	—	2	4	no	—
103	C₈	Q₁₆×D₈₃	1	2	yes	—
104	C₈	Q₁₆×D₈₃	1	2	yes	—
105	C₂²	#19128077	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	1328	2	no	—
108	C₄	#19128079	1	1	yes	characteristic
109	C₄	#19128079	1	1	yes	characteristic
110	C₄	#19128131	1	4	yes	—
111	C₂	#24907079	1	1	yes	characteristic
112	C₂	#24909722	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909743

C₂

yes

maximal

#24909736

C₂

yes

maximal

#24907073

C₂

yes

maximal, characteristic

C₈×C₈₃⋊C₁₆

C₂

yes

maximal, characteristic

C₈₃×#1710

C₂

yes

maximal, characteristic

#19128106

C₂²

yes

—

#19128052

—

#19127528

—

#19127526

—

C₄×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₄×C₈₃⋊C₁₆

C₂²

yes

characteristic

#500×C₈₃

C₂²

yes

—

#487×C₈₃

—

C₈²×C₈₃

C₂²

yes

characteristic

#14581792

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581726

—

#14581726

—

#14581717

—

C₄⋊Q₈×C₈₃

—

#107×C₈₃

—

#107×C₈₃

—

#103×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

Q₁₆

yes

—

C₈₃⋊C₁₆

Q₁₆

yes

—

C₂×Q₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₂×C₈×C₈₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

C₂×Q₁₆

yes

—

C₈×C₈₃

C₂×Q₁₆

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

—

—

D₈₃

yes

characteristic

C₈×C₁₆

—

C₈₃

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₈₃

yes

—

#487

—

#487

—

166

—

#367

—

166

—

#366

—

166

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₈₃

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

166

—

#107

—

166

—

#103

—

#103

—

332

—

C₄×C₈

C₂²⋊D₈₃

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

332

—

C₂×C₈

#13050575

yes

—

C₂×C₈

D₄×D₈₃

yes

characteristic

C₂×C₈

D₄×D₈₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

C₁₆

—

C₁₆

—

Q₈

—

Q₈

—

664

—

C₂×C₄

—

C₂×C₄

—

664

—

100

C₂×C₄

#14581788

yes

characteristic

101

C₂×C₄

C₂×#13050575

yes

—

102

C₈

—

103

C₈

Q₁₆×D₈₃

yes

—

104

C₈

Q₁₆×D₈₃

yes

—

105

C₂²

#19128077

yes

characteristic

106

C₄

—

107

C₄

—

1328

—

108

C₄

#19128079

yes

characteristic

109

C₄

#19128079

yes

characteristic

110

C₄

#19128131

yes

—

111

C₂

#24907079

yes

characteristic

112

C₂

#24909722

yes

—

	subgroup	G/H
center	C₂²	#19128077
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1710	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#1710	D₈₃
hypocenter	C₈₃	#12522
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#19128077

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1710

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
62. C₈₃	#12522	semidirect

H ≅ G/N

type

62. C₈₃

#12522

semidirect

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

subgroup	index
#24909743	2
#24909736	2
#24907073	2
C₈×C₈₃⋊C₁₆	2
C₈₃×#1710	2
#12522	83

subgroup

index

#24909743

#24909736

#24907073

C₈×C₈₃⋊C₁₆

C₈₃×#1710

#12522

Subgroups — #35319142

Conjugacy classes of subgroups

Complements

Maximal subgroups