Group #35298101 — subgroups

Subgroups — #35298101

Order 21248.

270 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 107 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	#24905755	C₂	1	1	yes	maximal, characteristic
3	#24905321	C₂	1	4	yes	maximal
4	C₈₃×#1835	C₂	1	1	yes	maximal, characteristic
5	#19127754	C₂²	1	2	yes	—
6	C₂×#14581728	—	2	2	no	—
7	#19127725	C₂²	1	2	yes	—
8	#19127423	C₄	1	4	yes	complemented
9	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
10	#382×C₈₃	C₂²	1	2	yes	—
11	#340×C₈₃	—	2	2	no	—
12	C₂×#13050574	—	2	2	no	—
13	#14581728	—	4	4	no	—
14	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
15	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
16	C₂²⋊(C₈₃⋊C₈)	—	2	2	no	—
17	C₄×C₈₃⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₈₃	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
20	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₈₃	Q₈	1	2	yes	—
23	#95×C₈₃	—	2	4	no	—
24	#13050574	—	4	4	no	—
25	C₂×C₈₃⋊C₈	—	2	4	no	—
26	C₂×C₈₃⋊C₈	—	4	2	no	—
27	C₂²×C₄×C₈₃	—	4	2	no	—
28	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	#41	1	2	yes	—
30	C₂×C₈×C₈₃	—	2	2	no	—
31	C₂×C₈×C₈₃	—	2	2	no	—
32	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
33	C₄⋊C₄×C₈₃	—	4	2	no	—
34	C₄⋊C₄×C₈₃	—	4	2	no	—
35	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₈₃	#41	1	2	yes	—
38	C₂³×C₈₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₈₃	—	4	2	no	—
40	C₂×C₄×C₈₃	—	8	4	no	—
41	C₂×C₄×C₈₃	—	8	2	no	—
42	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#117	1	2	yes	—
44	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
45	C₂×C₄×C₈₃	#125	1	2	yes	—
46	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×C₄×C₈₃	#137	1	2	yes	—
48	C₈×C₈₃	—	2	4	no	—
49	C₈₃⋊C₈	—	4	4	no	—
50	C₈₃⋊C₈	—	8	2	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	#382	1	1	yes	characteristic
53	C₂²×C₈₃	#489	1	2	yes	—
54	C₄×C₈₃	—	2	2	no	—
55	C₄×C₈₃	—	2	2	no	—
56	C₄×C₈₃	—	16	2	no	—
57	C₄×C₈₃	#428	1	2	yes	—
58	C₄×C₈₃	#490	1	1	yes	characteristic
59	C₄×C₈₃	#491	1	1	yes	characteristic
60	#8794	—	83	1	no	maximal, Sylow
61	C₂×C₈₃	—	2	2	no	—
62	C₂×C₈₃	#1833	1	2	yes	—
63	C₂×C₈₃	#1836	1	1	yes	characteristic
64	#1848	—	83	1	no	—
65	#1835	D₈₃	1	1	yes	characteristic
66	#1452	—	83	1	no	—
67	#1286	—	83	4	no	—
68	C₈₃	#8794	1	1	yes	characteristic, complemented, Sylow
69	#431	—	83	2	no	—
70	C₂×#108	—	166	2	no	—
71	#423	—	83	2	no	—
72	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
73	#382	C₂×D₈₃	1	2	yes	—
74	#340	—	2	2	no	—
75	#322	—	83	4	no	—
76	C₂×M₁₆	—	166	2	no	—
77	C₂²×C₈	C₂²⋊D₈₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
80	#108	—	332	4	no	—
81	C₄⋊C₈	—	166	4	no	—
82	C₄⋊C₈	—	166	2	no	—
83	C₂²⋊C₈	—	166	2	no	—
84	C₄×C₈	—	83	4	no	—
85	C₄×C₈	Dic₁₆₆	1	2	yes	—
86	C₄×C₈	C₄×D₈₃	1	2	yes	—
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
90	C₂²×C₄	#13050589	1	2	yes	—
91	M₁₆	—	332	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	332	2	no	—
95	C₂×C₈	—	166	4	no	—
96	C₂×C₈	Dic₈₃⋊C₄	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
100	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
101	C₄²	#13050591	1	2	yes	—
102	C₂³	#14581779	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
107	C₂×C₄	#14581750	1	2	yes	—
108	C₂×C₄	#14581781	1	1	yes	characteristic
109	C₂×C₄	#14581783	1	2	yes	—
110	C₂×C₄	#14581817	1	2	yes	—
111	C₂×C₄	#14581822	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	332	4	no	—
114	C₈	—	664	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#19127566	1	1	yes	characteristic
117	C₂²	#19128060	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	#19127744	1	2	yes	—
122	C₄	#19128067	1	1	yes	characteristic
123	C₄	#19128071	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#24907586	1	2	yes	—
126	C₂	#24907596	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal, characteristic

#24905755

C₂

yes

maximal, characteristic

#24905321

C₂

yes

maximal

C₈₃×#1835

C₂

yes

maximal, characteristic

#19127754

C₂²

yes

—

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127423

C₄

yes

complemented

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#382×C₈₃

C₂²

yes

—

#340×C₈₃

—

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₄⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₄×C₈₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

#95×C₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

#125

yes

—

C₂×C₄×C₈₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₈₃

#137

yes

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#382

yes

characteristic

C₂²×C₈₃

#489

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#428

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1833

yes

—

C₂×C₈₃

yes

—

D₈₃

yes

characteristic

#1452

—

#1286

—

C₈₃

#8794

yes

characteristic, complemented, Sylow

#431

—

C₂×#108

—

166

—

#423

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#382

C₂×D₈₃

yes

—

#340

—

#322

—

C₂×M₁₆

—

166

—

C₂²×C₈

C₂²⋊D₈₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#108

—

332

—

C₄⋊C₈

—

166

—

C₄⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

C₄×C₈

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

C₄×D₈₃

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

#13050589

yes

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

—

166

—

C₂×C₈

Dic₈₃⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×D₈₃

yes

characteristic

100

C₄²

C₂×Dic₁₆₆

yes

characteristic

101

C₄²

#13050591

yes

—

102

C₂³

#14581779

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×Dic₈₃⋊C₄

yes

characteristic

107

C₂×C₄

#14581750

yes

—

108

C₂×C₄

#14581781

yes

characteristic

109

C₂×C₄

#14581783

yes

—

110

C₂×C₄

#14581817

yes

—

111

C₂×C₄

#14581822

yes

—

112

C₈

—

113

C₈

—

332

—

114

C₈

—

664

—

115

C₂²

—

116

C₂²

#19127566

yes

characteristic

117

C₂²

#19128060

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#19127744

yes

—

122

C₄

#19128067

yes

characteristic

123

C₄

#19128071

yes

characteristic

124

C₂

—

125

C₂

#24907586

yes

—

126

C₂

#24907596

yes

characteristic

	subgroup	G/H
center	C₂²	#19127566
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1835	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1835	D₈₃
hypocenter	C₈₃	#8794
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#19127566

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1835

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #19127423	C₄	semidirect
68. C₈₃	#8794	semidirect

H ≅ G/N

type

8. #19127423

C₄

semidirect

68. C₈₃

#8794

semidirect

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

subgroup	index
#24907635	2
#24905755	2
#24905321	2
C₈₃×#1835	2
#8794	83

subgroup

index

C₈₃×#1835

Subgroups — #35298101

Conjugacy classes of subgroups

Complements

Maximal subgroups