Group #35319098 — subgroups

Subgroups — #35319098

Order 21248.

260 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 83 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	#24909691	C₂	1	2	yes	maximal, complemented
2	#24909687	C₂	1	2	yes	maximal
3	#24907073	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907069	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128106	C₂²	1	2	yes	—
7	#19128081	C₂²	1	1	yes	characteristic
8	#19128078	C₂²	1	1	yes	characteristic
9	#19128052	—	2	2	no	—
10	#19128048	—	2	2	no	—
11	#19127506	—	2	4	no	—
12	#19127503	—	2	4	no	—
13	C₄×C₁₆×C₈₃	C₂²	1	2	yes	—
14	C₈²×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₃₂	—	2	2	no	—
16	C₂×D₆₆₄	—	2	2	no	—
17	#14581792	—	2	1	no	—
18	#14581792	—	2	2	no	—
19	#14581789	—	2	1	no	—
20	#14581726	—	2	2	no	—
21	#14581726	—	2	2	no	—
22	#14581717	—	4	4	no	—
23	#14581714	—	4	4	no	—
24	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
25	C₄×C₈×C₈₃	D₄	1	2	yes	—
26	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₆₆	—	4	1	no	—
28	C₂×Dic₁₆₆	—	4	2	no	—
29	C₂×D₃₃₂	—	4	1	no	—
30	C₂×D₃₃₂	—	4	2	no	—
31	Dic₃₃₂	—	4	4	no	—
32	D₆₆₄	—	4	4	no	—
33	C₄⋊Dic₈₃	—	4	2	no	—
34	C₄⋊Dic₈₃	—	4	2	no	—
35	C₄⋊Dic₈₃	—	4	2	no	—
36	C₂×C₈×C₈₃	D₈	1	4	yes	—
37	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
38	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₈₃	—	2	4	no	—
40	C₂²×D₈₃	—	8	1	no	—
41	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
42	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
43	C₂×Dic₈₃	—	8	1	no	—
44	C₂×Dic₈₃	—	8	2	no	—
45	D₃₃₂	—	8	2	no	—
46	D₃₃₂	—	8	4	no	—
47	Dic₁₆₆	—	8	2	no	—
48	Dic₁₆₆	—	8	4	no	—
49	C₈×C₈₃	—	2	4	no	—
50	C₈×C₈₃	#135	1	4	yes	—
51	C₂²×C₈₃	#493	1	1	yes	characteristic
52	C₂×D₈₃	—	16	1	no	—
53	C₂×D₈₃	—	16	2	no	—
54	C₄×C₈₃	#495	1	2	yes	—
55	C₄×C₈₃	#508	1	4	yes	—
56	Dic₈₃	—	16	1	no	—
57	Dic₈₃	—	16	2	no	—
58	#12511	—	83	1	no	maximal, Sylow
59	C₂×C₈₃	#1709	1	1	yes	characteristic
60	C₂×C₈₃	#2249	1	2	yes	—
61	D₈₃	—	32	1	no	—
62	#2241	—	83	2	no	—
63	#2240	—	83	2	no	—
64	#1710	—	83	1	no	—
65	#1709	—	83	1	no	—
66	C₈×C₁₆	D₈₃	1	1	yes	characteristic, complemented
67	C₈₃	#12511	1	1	yes	characteristic, complemented, Sylow
68	#500	—	83	2	no	—
69	#494	—	83	1	no	—
70	#493	—	83	1	no	—
71	#487	—	166	2	no	—
72	#486	—	166	2	no	—
73	#358	—	166	4	no	—
74	#357	—	166	4	no	—
75	C₄×C₁₆	C₂×D₈₃	1	2	yes	—
76	C₈²	C₂×D₈₃	1	1	yes	characteristic
77	C₂×Q₁₆	—	166	2	no	—
78	C₂×D₈	—	166	2	no	—
79	C₄⋊Q₈	—	166	1	no	—
80	C₄⋊Q₈	—	166	2	no	—
81	#127	—	166	1	no	—
82	C₂×C₁₆	D₃₃₂	1	4	yes	—
83	#107	—	166	2	no	—
84	#107	—	166	2	no	—
85	#103	—	332	4	no	—
86	#102	—	332	4	no	—
87	C₄×C₈	D₃₃₂	1	2	yes	—
88	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
89	C₂×Q₈	—	332	1	no	—
90	C₂×Q₈	—	332	2	no	—
91	C₂×D₄	—	332	1	no	—
92	C₂×D₄	—	332	2	no	—
93	Q₁₆	—	332	4	no	—
94	D₈	—	332	4	no	—
95	C₂×C₈	D₆₆₄	1	4	yes	—
96	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
97	C₄⋊C₄	—	332	2	no	—
98	C₄⋊C₄	—	332	2	no	—
99	C₄⋊C₄	—	332	2	no	—
100	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
101	C₁₆	—	2	4	no	—
102	C₂³	—	664	1	no	—
103	Q₈	—	664	2	no	—
104	Q₈	—	664	4	no	—
105	D₄	—	664	2	no	—
106	D₄	—	664	4	no	—
107	C₂×C₄	—	664	1	no	—
108	C₂×C₄	—	664	2	no	—
109	C₂×C₄	#14581789	1	1	yes	characteristic
110	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
111	C₈	—	2	4	no	—
112	C₈	#14581814	1	4	yes	—
113	C₂²	—	1328	1	no	—
114	C₂²	—	1328	2	no	—
115	C₂²	#19128078	1	1	yes	characteristic
116	C₄	—	1328	1	no	—
117	C₄	—	1328	2	no	—
118	C₄	#19128085	1	2	yes	—
119	C₄	#19128133	1	4	yes	—
120	C₂	—	2656	1	no	—
121	C₂	#24907069	1	1	yes	characteristic
122	C₂	#24909724	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24909691

C₂

yes

maximal, complemented

#24909687

C₂

yes

maximal

#24907073

C₂

yes

maximal, characteristic, complemented

#24907069

C₂

yes

maximal, characteristic

C₈×C₁₆×C₈₃

C₂

yes

maximal, characteristic, complemented

#19128106

C₂²

yes

—

#19128081

C₂²

yes

characteristic

#19128078

C₂²

yes

—

—

—

—

C₄×C₁₆×C₈₃

C₂²

yes

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

C₂×D₆₆₄

—

—

—

—

—

—

—

—

C₂×C₁₆×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×D₃₃₂

—

C₂×D₃₃₂

—

Dic₃₃₂

—

D₆₆₄

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₂²×D₈₃

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

D₃₃₂

—

D₃₃₂

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

#135

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

#495

yes

—

C₄×C₈₃

#508

yes

—

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

D₈₃

—

—

—

—

—

C₈×C₁₆

D₈₃

yes

characteristic, complemented

C₈₃

#12511

yes

characteristic, complemented, Sylow

—

—

—

—

166

—

#486

—

166

—

#358

—

166

—

#357

—

166

—

C₄×C₁₆

C₂×D₈₃

yes

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

166

—

C₂×D₈

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

#127

—

166

—

C₂×C₁₆

D₃₃₂

yes

—

#107

—

166

—

#107

—

166

—

#103

—

332

—

#102

—

332

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

332

—

C₂×Q₈

—

332

—

C₂×D₄

—

332

—

C₂×D₄

—

332

—

Q₁₆

—

332

—

D₈

—

332

—

C₂×C₈

D₆₆₄

yes

—

C₂×C₈

C₂×D₃₃₂

yes

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

100

C₄²

C₂×D₃₃₂

yes

characteristic

101

C₁₆

—

102

C₂³

—

664

—

103

Q₈

—

664

—

104

Q₈

—

664

—

105

D₄

—

664

—

106

D₄

—

664

—

107

C₂×C₄

—

664

—

108

C₂×C₄

—

664

—

109

C₂×C₄

#14581789

yes

characteristic

110

C₂×C₄

C₂×D₆₆₄

yes

—

111

C₈

—

112

C₈

#14581814

yes

—

113

C₂²

—

1328

—

114

C₂²

—

1328

—

115

C₂²

#19128078

yes

characteristic

116

C₄

—

1328

—

117

C₄

—

1328

—

118

C₄

#19128085

yes

—

119

C₄

#19128133

yes

—

120

C₂

—

2656

—

121

C₂

#24907069

yes

characteristic

122

C₂

#24909724

yes

—

	subgroup	G/H
center	C₂²	#19128078
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈×C₁₆×C₈₃	C₂
socle	C₂²×C₈₃	#493
prime radical	#19128078	C₂²
hypercenter	C₈×C₁₆	D₈₃
hypocenter	C₈₃	#12511
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#19128078

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈×C₁₆×C₈₃

C₂

C₂²×C₈₃

C₂²

C₈×C₁₆

D₈₃

hypocenter

C₈₃

#12511

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #24909691	C₂	semidirect
3. #24907073	C₂	semidirect
5. C₈×C₁₆×C₈₃	C₂	semidirect
66. C₈×C₁₆	D₈₃	semidirect
67. C₈₃	#12511	semidirect

H ≅ G/N

type

1. #24909691

C₂

semidirect

3. #24907073

C₂

semidirect

5. C₈×C₁₆×C₈₃

C₂

semidirect

66. C₈×C₁₆

D₈₃

semidirect

67. C₈₃

#12511

semidirect

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

subgroup	index
#24909691	2
#24909687	2
#24907073	2
#24907069	2
C₈×C₁₆×C₈₃	2
#12511	83

subgroup

index

C₈×C₁₆×C₈₃

Subgroups — #35319098

Conjugacy classes of subgroups

Complements

Maximal subgroups