Group #35319139 — subgroups

Subgroups — #35319139

Order 21248.

262 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 103 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	#24909736	C₂	1	4	yes	maximal
2	#24907081	C₂	1	1	yes	maximal, characteristic
3	C₈×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1713	C₂	1	1	yes	maximal, characteristic
5	#19128106	—	2	1	no	—
6	#19128106	C₂²	1	2	yes	—
7	#19128081	—	2	1	no	—
8	#19127526	—	2	8	no	—
9	C₄×C₈₃⋊C₁₆	C₂²	1	2	yes	—
10	#500×C₈₃	—	2	1	no	—
11	#500×C₈₃	C₂²	1	2	yes	—
12	#494×C₈₃	—	2	1	no	—
13	C₈²×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₃₂	—	4	2	no	—
15	#14581792	—	2	2	no	—
16	C₂×C₈₃⋊C₁₆	D₄	1	4	yes	—
17	#14581726	—	2	4	no	—
18	#14581726	—	4	2	no	—
19	C₂×Q₁₆×C₈₃	—	4	2	no	—
20	C₄⋊Q₈×C₈₃	—	2	2	no	—
21	#107×C₈₃	—	2	4	no	—
22	#107×C₈₃	—	4	2	no	—
23	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₆₆	—	4	2	no	—
27	Dic₃₃₂	—	8	4	no	—
28	C₄⋊Dic₈₃	—	4	2	no	—
29	C₄⋊Dic₈₃	—	4	2	no	—
30	C₈₃⋊C₁₆	Q₁₆	1	8	yes	complemented
31	C₂×Q₈×C₈₃	—	4	2	no	—
32	Q₁₆×C₈₃	—	8	4	no	—
33	C₂×C₈×C₈₃	D₈	1	2	yes	—
34	C₂×C₈×C₈₃	D₈	1	2	yes	—
35	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
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	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₈₃	—	8	2	no	—
45	Dic₁₆₆	—	8	4	no	—
46	C₈×C₈₃	D₁₆	1	4	yes	—
47	C₈×C₈₃	Q₃₂	1	4	yes	—
48	C₈×C₈₃	C₂×Q₁₆	1	4	yes	—
49	C₂²×C₈₃	#493	1	1	yes	characteristic
50	C₄×C₈₃	—	16	2	no	—
51	C₄×C₈₃	#494	1	2	yes	—
52	C₄×C₈₃	C₂×D₁₆	1	2	yes	—
53	C₄×C₈₃	C₂×Q₃₂	1	2	yes	—
54	Dic₈₃	—	16	2	no	—
55	#12521	—	83	1	no	maximal, Sylow
56	C₂×C₈₃	#1713	1	1	yes	characteristic
57	C₂×C₈₃	#2246	1	1	yes	characteristic
58	C₂×C₈₃	#2247	1	1	yes	characteristic
59	#2252	—	83	4	no	—
60	#1713	—	83	1	no	—
61	#1713	D₈₃	1	1	yes	characteristic
62	C₈×C₁₆	—	83	1	no	—
63	C₈₃	#12521	1	1	yes	characteristic, complemented, Sylow
64	#500	—	2	1	no	—
65	#500	—	83	2	no	—
66	#500	—	166	1	no	—
67	#500	C₂×D₈₃	1	2	yes	—
68	#494	—	2	1	no	—
69	#494	—	166	1	no	—
70	#366	—	166	8	no	—
71	C₄×C₁₆	—	83	2	no	—
72	C₈²	C₂×D₈₃	1	1	yes	characteristic
73	C₂×Q₁₆	—	4	2	no	—
74	C₂×Q₁₆	—	332	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	166	2	no	—
77	C₂×C₁₆	—	83	4	no	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	332	2	no	—
81	#107	—	166	4	no	—
82	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	332	2	no	—
87	Q₁₆	—	8	4	no	—
88	Q₁₆	—	664	4	no	—
89	C₂×C₈	#13050575	1	2	yes	—
90	C₂×C₈	#13050575	1	2	yes	—
91	C₂×C₈	D₄×D₈₃	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	332	2	no	—
95	C₄⋊C₄	—	332	2	no	—
96	C₄²	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
97	C₁₆	—	83	8	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	664	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	664	2	no	—
102	C₂×C₄	#14581788	1	1	yes	characteristic
103	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
104	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
105	C₈	#14581733	1	4	yes	—
106	C₈	#14581738	1	4	yes	—
107	C₈	Q₁₆×D₈₃	1	4	yes	—
108	C₂²	#19128077	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	1328	2	no	—
111	C₄	#19128079	1	2	yes	—
112	C₄	C₂×#14581733	1	2	yes	—
113	C₄	C₂×#14581738	1	2	yes	—
114	C₂	#24907079	1	1	yes	characteristic
115	C₂	#24909712	1	1	yes	characteristic
116	C₂	#24909715	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909736

C₂

yes

maximal

#24907081

C₂

yes

maximal, characteristic

C₈×C₈₃⋊C₁₆

C₂

yes

maximal, characteristic

C₈₃×#1713

C₂

yes

maximal, characteristic

#19128106

—

#19128106

C₂²

yes

—

#19128081

—

#19127526

—

C₄×C₈₃⋊C₁₆

C₂²

yes

—

#500×C₈₃

—

#500×C₈₃

C₂²

yes

—

#494×C₈₃

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

—

#14581792

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581726

—

#14581726

—

C₂×Q₁₆×C₈₃

—

C₄⋊Q₈×C₈₃

—

#107×C₈₃

—

#107×C₈₃

—

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₈₃

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂×D₄

yes

—

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

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×Dic₈₃

—

Dic₁₆₆

—

C₈×C₈₃

D₁₆

yes

—

C₈×C₈₃

Q₃₂

yes

—

C₈×C₈₃

C₂×Q₁₆

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

#494

yes

—

C₄×C₈₃

C₂×D₁₆

yes

—

C₄×C₈₃

C₂×Q₃₂

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

C₈×C₁₆

—

C₈₃

#12521

yes

characteristic, complemented, Sylow

#500

—

#500

—

#500

—

166

—

#500

C₂×D₈₃

yes

—

#494

—

#494

—

166

—

#366

—

166

—

C₄×C₁₆

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

332

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

332

—

#107

—

166

—

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

332

—

Q₁₆

—

Q₁₆

—

664

—

C₂×C₈

#13050575

yes

—

C₂×C₈

#13050575

yes

—

C₂×C₈

D₄×D₈₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

332

—

C₄⋊C₄

—

332

—

C₄²

C₂×C₂²⋊D₈₃

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

664

—

100

C₂×C₄

—

101

C₂×C₄

—

664

—

102

C₂×C₄

#14581788

yes

characteristic

103

C₂×C₄

C₂×#13050575

yes

characteristic

104

C₂×C₄

C₂×#13050575

yes

characteristic

105

C₈

#14581733

yes

—

106

C₈

#14581738

yes

—

107

C₈

Q₁₆×D₈₃

yes

—

108

C₂²

#19128077

yes

characteristic

109

C₄

—

110

C₄

—

1328

—

111

C₄

#19128079

yes

—

112

C₄

C₂×#14581733

yes

—

113

C₄

C₂×#14581738

yes

—

114

C₂

#24907079

yes

characteristic

115

C₂

#24909712

yes

characteristic

116

C₂

#24909715

yes

characteristic

	subgroup	G/H
center	C₂²	#19128077
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1713	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#1713	D₈₃
hypocenter	C₈₃	#12521
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#19128077

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1713

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
30. C₈₃⋊C₁₆	Q₁₆	semidirect
63. C₈₃	#12521	semidirect

H ≅ G/N

type

30. C₈₃⋊C₁₆

Q₁₆

semidirect

63. C₈₃

#12521

semidirect

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

subgroup	index
#24909736	2
#24907081	2
C₈×C₈₃⋊C₁₆	2
C₈₃×#1713	2
#12521	83

subgroup

index

#24909736

#24907081

C₈×C₈₃⋊C₁₆

C₈₃×#1713

#12521

Subgroups — #35319139

Conjugacy classes of subgroups

Complements

Maximal subgroups