Group #35298213 — subgroups

Subgroups — #35298213

Order 21248.

338 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 163 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	1	yes	maximal, characteristic
2	#24906218	C₂	1	4	yes	maximal
3	C₂×#19127422	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581728	—	2	2	no	—
6	#19127725	C₂²	1	2	yes	—
7	C₂×C₄×C₈₃⋊C₈	C₂²	1	2	yes	—
8	#19127451	—	2	8	no	—
9	#19127422	C₄	1	4	yes	—
10	C₂×#108×C₈₃	—	2	2	no	—
11	#423×C₈₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	C₂×#13050574	—	2	2	no	—
14	C₂²×C₈₃⋊C₈	Q₈	1	4	yes	—
15	#14581728	—	4	4	no	—
16	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
17	C₄×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₈₃⋊C₈	D₄	1	4	yes	—
19	C₂×M₁₆×C₈₃	—	2	2	no	—
20	C₂²×C₈×C₈₃	D₄	1	2	yes	—
21	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
22	#108×C₈₃	—	4	4	no	—
23	C₄⋊C₈×C₈₃	—	2	4	no	—
24	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₈₃	Q₈	1	2	yes	—
26	#13050574	—	4	4	no	—
27	C₂×C₈₃⋊C₈	—	4	4	no	—
28	C₂×C₈₃⋊C₈	—	4	2	no	—
29	C₂×C₈₃⋊C₈	C₄⋊C₄	1	8	yes	—
30	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	C₂×Q₈	1	2	yes	—
32	M₁₆×C₈₃	—	4	4	no	—
33	C₂×C₈×C₈₃	—	4	2	no	—
34	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
36	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
37	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₈₃	C₂×D₄	1	2	yes	—
39	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
40	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₄×C₈₃	—	2	4	no	—
42	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
46	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
47	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
48	C₈×C₈₃	—	8	2	no	—
49	C₈×C₈₃	#106	1	8	yes	—
50	C₈₃⋊C₈	—	2	8	no	—
51	C₈₃⋊C₈	—	8	2	no	—
52	C₂²×C₈₃	—	2	2	no	—
53	C₂²×C₈₃	#384	1	1	yes	characteristic
54	C₂²×C₈₃	#498	1	2	yes	—
55	C₄×C₈₃	—	2	2	no	—
56	C₄×C₈₃	C₂×#106	1	2	yes	—
57	C₄×C₈₃	#430	1	4	yes	—
58	C₄×C₈₃	#492	1	1	yes	characteristic
59	C₄×C₈₃	#492	1	1	yes	characteristic
60	#8825	—	83	1	no	maximal, Sylow
61	C₂×C₈₃	—	2	2	no	—
62	C₂×C₈₃	#1844	1	1	yes	characteristic
63	C₂×C₈₃	#1849	1	2	yes	—
64	#1848	—	83	1	no	—
65	#1848	D₈₃	1	1	yes	characteristic
66	#1566	—	83	4	no	—
67	C₂×#322	—	83	1	no	—
68	C₈₃	#8825	1	1	yes	characteristic, complemented, Sylow
69	C₂×#108	—	2	2	no	—
70	C₂×#108	—	166	2	no	—
71	#423	—	83	2	no	—
72	#423	C₂×D₈₃	1	2	yes	—
73	C₂×C₄×C₈	—	83	2	no	—
74	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
75	#334	—	166	8	no	—
76	#322	—	83	4	no	—
77	C₂×M₁₆	—	2	2	no	—
78	C₂×M₁₆	—	166	2	no	—
79	C₂²×C₈	—	83	4	no	—
80	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
81	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
82	#108	—	4	4	no	—
83	#108	—	332	4	no	—
84	C₄⋊C₈	—	2	4	no	—
85	C₄⋊C₈	—	166	4	no	—
86	C₄×C₈	—	83	4	no	—
87	C₄×C₈	—	83	4	no	—
88	C₄×C₈	Dic₁₆₆	1	2	yes	—
89	C₄×C₈	C₄×D₈₃	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
91	C₂²×C₄	Q₈×D₈₃	1	2	yes	—
92	M₁₆	—	4	4	no	—
93	M₁₆	—	332	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	332	2	no	—
96	C₂×C₈	—	332	4	no	—
97	C₂×C₈	—	83	8	no	—
98	C₂×C₈	Dic₈₃⋊C₄	1	4	yes	—
99	C₂×C₈	#13050577	1	4	yes	—
100	C₂×C₈	#13050578	1	4	yes	—
101	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
102	C₄²	D₄×D₈₃	1	2	yes	—
103	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
104	C₂³	#14581790	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	C₄⋊C₄×D₈₃	1	2	yes	—
107	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
108	C₂×C₄	#14581788	1	1	yes	characteristic
109	C₂×C₄	#14581791	1	2	yes	—
110	C₂×C₄	C₂×#13050577	1	2	yes	—
111	C₂×C₄	C₂×#13050578	1	2	yes	—
112	C₈	—	8	2	no	—
113	C₈	—	664	2	no	—
114	C₈	—	166	8	no	—
115	C₈	#14581723	1	8	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	#19127570	1	1	yes	characteristic
118	C₂²	#19128097	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂×#14581723	1	2	yes	—
121	C₄	#19127748	1	4	yes	—
122	C₄	#19128073	1	1	yes	characteristic
123	C₄	#19128075	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#24907619	1	1	yes	characteristic
126	C₂	#24907636	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal, characteristic

#24906218

C₂

yes

maximal

C₂×#19127422

C₂

yes

maximal, characteristic

C₈₃×#1848

C₂

yes

maximal, characteristic

C₂×#14581728

—

#19127725

C₂²

yes

—

C₂×C₄×C₈₃⋊C₈

C₂²

yes

—

#19127451

—

#19127422

C₄

yes

—

C₂×#108×C₈₃

—

#423×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×#13050574

—

C₂²×C₈₃⋊C₈

Q₈

yes

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

—

C₄×C₈₃⋊C₈

D₄

yes

—

C₂×M₁₆×C₈₃

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#108×C₈₃

—

C₄⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

—

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

SD₁₆

yes

—

C₂×C₈×C₈₃

SD₁₆

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

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₂×C₄×C₈₃

C₂×SD₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

#106

yes

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#384

yes

characteristic

C₂²×C₈₃

#498

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#106

yes

—

C₄×C₈₃

#430

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1844

yes

characteristic

C₂×C₈₃

#1849

yes

—

#1848

—

#1848

D₈₃

yes

characteristic

#1566

—

C₂×#322

—

C₈₃

#8825

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

166

—

#423

—

#423

C₂×D₈₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#334

—

166

—

#322

—

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₈₃

yes

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#108

—

#108

—

332

—

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

C₄×D₈₃

yes

—

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

Q₈×D₈₃

yes

—

M₁₆

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

—

332

—

C₂×C₈

—

C₂×C₈

Dic₈₃⋊C₄

yes

—

C₂×C₈

#13050577

yes

—

100

C₂×C₈

#13050578

yes

—

101

C₄²

C₂×C₄×D₈₃

yes

characteristic

102

C₄²

D₄×D₈₃

yes

—

103

C₄²

C₂×Dic₁₆₆

yes

characteristic

104

C₂³

#14581790

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

C₄⋊C₄×D₈₃

yes

—

107

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

108

C₂×C₄

#14581788

yes

characteristic

109

C₂×C₄

#14581791

yes

—

110

C₂×C₄

C₂×#13050577

yes

—

111

C₂×C₄

C₂×#13050578

yes

—

112

C₈

—

113

C₈

—

664

—

114

C₈

—

166

—

115

C₈

#14581723

yes

—

116

C₂²

—

117

C₂²

#19127570

yes

characteristic

118

C₂²

#19128097

yes

—

119

C₄

—

120

C₄

C₂×#14581723

yes

—

121

C₄

#19127748

yes

—

122

C₄

#19128073

yes

characteristic

123

C₄

#19128075

yes

characteristic

124

C₂

—

125

C₂

#24907619

yes

characteristic

126

C₂

#24907636

yes

—

	subgroup	G/H
center	C₂×C₄	#14581788
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1848	C₂
socle	C₂²×C₈₃	#384
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1848	D₈₃
hypocenter	C₈₃	#8825
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#14581788

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1848

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 ◁ #1848

N	H ≅ G/N	type
68. C₈₃	#8825	semidirect

H ≅ G/N

type

68. C₈₃

#8825

semidirect

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

subgroup	index
#24907635	2
#24906218	2
C₂×#19127422	2
C₈₃×#1848	2
#8825	83

subgroup

index

C₈₃×#1848

Subgroups — #35298213

Conjugacy classes of subgroups

Complements

Maximal subgroups