Group #35319140 — subgroups

Subgroups — #35319140

Order 21248.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#24909738	C₂	1	1	yes	maximal, characteristic
2	#24909736	C₂	1	2	yes	maximal
3	#24907080	C₂	1	2	yes	maximal
4	C₁₆×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#19128106	C₂²	1	2	yes	—
7	#19128104	—	2	2	no	—
8	#19128080	—	2	2	no	—
9	#19127527	—	2	2	no	—
10	#19127526	—	2	4	no	—
11	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₈₃	C₂²	1	2	yes	—
14	#366×C₈₃	—	2	2	no	—
15	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×#13050580	—	4	4	no	—
17	#14581792	—	2	2	no	—
18	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
19	#14581726	—	2	2	no	—
20	#14581726	—	2	2	no	—
21	#14581725	—	4	4	no	—
22	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₈₃	—	2	2	no	—
25	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
26	#107×C₈₃	—	2	2	no	—
27	#107×C₈₃	—	2	2	no	—
28	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₆₆	—	4	2	no	—
30	#13050580	—	8	8	no	—
31	C₂×C₈₃⋊C₈	D₈	1	2	yes	—
32	C₂×C₈₃⋊C₈	D₈	1	2	yes	—
33	C₄⋊Dic₈₃	—	4	2	no	—
34	C₄⋊Dic₈₃	—	4	2	no	—
35	C₈₃⋊C₁₆	Q₁₆	1	4	yes	complemented
36	C₂×Q₈×C₈₃	—	4	2	no	—
37	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₈₃	—	4	2	no	—
40	C₄⋊C₄×C₈₃	—	4	2	no	—
41	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₃	Q₁₆	1	4	yes	complemented
43	Q₈×C₈₃	—	8	4	no	—
44	C₂×C₄×C₈₃	—	8	2	no	—
45	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₈₃	—	8	2	no	—
49	Dic₁₆₆	—	8	4	no	—
50	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
51	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
52	C₈₃⋊C₈	D₁₆	1	4	yes	—
53	C₈₃⋊C₈	Q₃₂	1	4	yes	—
54	C₂²×C₈₃	#493	1	1	yes	characteristic
55	C₄×C₈₃	—	16	2	no	—
56	C₄×C₈₃	#494	1	1	yes	characteristic
57	C₄×C₈₃	#494	1	1	yes	characteristic
58	C₄×C₈₃	C₂×D₁₆	1	2	yes	—
59	C₄×C₈₃	C₂×Q₃₂	1	2	yes	—
60	Dic₈₃	—	16	2	no	—
61	#12521	—	83	1	no	maximal, Sylow
62	C₂×C₈₃	#1713	1	1	yes	characteristic
63	C₂×C₈₃	#2246	1	1	yes	characteristic
64	C₂×C₈₃	#2247	1	1	yes	characteristic
65	#2252	—	83	2	no	—
66	#2252	—	83	1	no	—
67	#2252	D₈₃	1	1	yes	characteristic
68	#1713	—	83	2	no	—
69	C₈×C₁₆	—	83	1	no	—
70	C₈₃	#12521	1	1	yes	characteristic, complemented, Sylow
71	#500	—	83	2	no	—
72	#500	—	166	2	no	—
73	#500	C₂×D₈₃	1	2	yes	—
74	#494	—	166	2	no	—
75	#366	—	2	2	no	—
76	#366	—	166	4	no	—
77	#366	—	166	2	no	—
78	C₄×C₁₆	—	83	1	no	—
79	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
80	C₈²	—	83	1	no	—
81	C₂×Q₁₆	—	332	4	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	166	2	no	—
84	C₂×C₁₆	—	83	2	no	—
85	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	332	4	no	—
89	#107	—	166	2	no	—
90	#107	—	166	2	no	—
91	C₄×C₈	—	83	1	no	—
92	C₄×C₈	—	83	1	no	—
93	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	332	2	no	—
96	Q₁₆	—	664	8	no	—
97	C₂×C₈	—	83	2	no	—
98	C₂×C₈	—	83	2	no	—
99	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	332	2	no	—
104	C₄⋊C₄	—	332	2	no	—
105	C₄²	D₄×D₈₃	1	1	yes	characteristic
106	C₁₆	—	83	4	no	—
107	C₁₆	#13050580	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	664	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	664	2	no	—
112	C₂×C₄	#14581788	1	1	yes	characteristic
113	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
114	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
115	C₈	—	83	4	no	—
116	C₈	—	83	4	no	—
117	C₈	Q₁₆×D₈₃	1	2	yes	—
118	C₈	C₂×#13050580	1	2	yes	—
119	C₂²	#19128076	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	1328	2	no	—
122	C₄	#19128079	1	1	yes	characteristic
123	C₄	#19128080	1	1	yes	characteristic
124	C₄	D₁₆×D₈₃	1	2	yes	—
125	C₄	Q₃₂×D₈₃	1	2	yes	—
126	C₂	#24907080	1	1	yes	characteristic
127	C₂	#24909711	1	1	yes	characteristic
128	C₂	#24909714	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24909738

C₂

yes

maximal, characteristic

#24909736

C₂

yes

maximal

#24907080

C₂

yes

maximal

C₁₆×C₈₃⋊C₈

C₂

yes

maximal, characteristic

C₈₃×#2252

C₂

yes

maximal, characteristic

#19128106

C₂²

yes

—

—

—

—

—

C₄×C₈₃⋊C₁₆

C₂²

yes

characteristic

C₈×C₈₃⋊C₈

C₂²

yes

characteristic

#500×C₈₃

C₂²

yes

—

#366×C₈₃

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₂×#13050580

—

#14581792

—

C₂×C₈₃⋊C₁₆

D₄

yes

—

#14581726

—

#14581726

—

#14581725

—

C₄×C₈₃⋊C₈

D₄

yes

characteristic

C₄×C₈₃⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₈₃

—

C₂×C₁₆×C₈₃

D₄

yes

—

#107×C₈₃

—

#107×C₈₃

—

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

#13050580

—

C₂×C₈₃⋊C₈

D₈

yes

—

C₂×C₈₃⋊C₈

D₈

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₈₃⋊C₁₆

Q₁₆

yes

complemented

C₂×Q₈×C₈₃

—

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

C₁₆×C₈₃

Q₁₆

yes

complemented

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

C₂×Q₁₆

yes

—

C₈×C₈₃

C₂×Q₁₆

yes

—

C₈₃⋊C₈

D₁₆

yes

—

C₈₃⋊C₈

Q₃₂

yes

—

C₂²×C₈₃

#493

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

C₂×D₁₆

yes

—

C₄×C₈₃

C₂×Q₃₂

yes

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

#1713

—

C₈×C₁₆

—

C₈₃

#12521

yes

characteristic, complemented, Sylow

#500

—

#500

—

166

—

#500

C₂×D₈₃

yes

—

#494

—

166

—

#366

—

#366

—

166

—

#366

—

166

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₈²

—

C₂×Q₁₆

—

332

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₈₃

yes

—

#107

—

#107

—

#107

—

332

—

#107

—

166

—

#107

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

332

—

Q₁₆

—

664

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₈₃

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₈₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

332

—

104

C₄⋊C₄

—

332

—

105

C₄²

D₄×D₈₃

yes

characteristic

106

C₁₆

—

107

C₁₆

#13050580

yes

complemented

108

Q₈

—

109

Q₈

—

664

—

110

C₂×C₄

—

111

C₂×C₄

—

664

—

112

C₂×C₄

#14581788

yes

characteristic

113

C₂×C₄

D₈×D₈₃

yes

characteristic

114

C₂×C₄

D₈×D₈₃

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₈₃

yes

—

118

C₈

C₂×#13050580

yes

—

119

C₂²

#19128076

yes

characteristic

120

C₄

—

121

C₄

—

1328

—

122

C₄

#19128079

yes

characteristic

123

C₄

#19128080

yes

characteristic

124

C₄

D₁₆×D₈₃

yes

—

125

C₄

Q₃₂×D₈₃

yes

—

126

C₂

#24907080

yes

characteristic

127

C₂

#24909711

yes

characteristic

128

C₂

#24909714

yes

characteristic

	subgroup	G/H
center	C₂²	#19128076
commutator, cosocle	C₄×C₈×C₈₃	C₂³
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#2252	C₂
socle, prime radical	C₂²×C₈₃	#493
hypercenter	#2252	D₈₃
hypocenter	C₈₃	#12521
LCS	G ▷ C₄×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#19128076

commutator, cosocle

C₄×C₈×C₈₃

C₂³

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#2252

C₂

C₂²×C₈₃

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
35. C₈₃⋊C₁₆	Q₁₆	semidirect
42. C₁₆×C₈₃	Q₁₆	semidirect
70. C₈₃	#12521	semidirect
107. C₁₆	#13050580	semidirect

H ≅ G/N

type

35. C₈₃⋊C₁₆

Q₁₆

semidirect

42. C₁₆×C₈₃

Q₁₆

semidirect

70. C₈₃

#12521

semidirect

107. C₁₆

#13050580

semidirect

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

subgroup	index
#24909738	2
#24909736	2
#24907080	2
C₁₆×C₈₃⋊C₈	2
C₈₃×#2252	2
#12521	83

subgroup

index

#24909738

#24909736

#24907080

C₁₆×C₈₃⋊C₈

C₈₃×#2252

#12521

Subgroups — #35319140

Conjugacy classes of subgroups

Complements

Maximal subgroups