Group #35298208 — subgroups

Subgroups — #35298208

Order 21248.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	2	yes	maximal
2	#24906222	C₂	1	4	yes	maximal
3	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581728	—	2	4	no	—
5	#19127725	C₂²	1	4	yes	—
6	#19127454	—	2	8	no	—
7	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	C₈²×C₈₃	C₄	1	1	yes	characteristic
10	C₈²×C₈₃	C₄	1	2	yes	—
11	C₈²×C₈₃	C₄	1	1	yes	characteristic
12	C₂×#13050574	—	2	4	no	—
13	#14581728	—	4	8	no	—
14	C₄⋊(C₈₃⋊C₈)	—	2	8	no	—
15	C₂²×C₈×C₈₃	D₄	1	2	yes	—
16	C₂²×C₈×C₈₃	Q₈	1	4	yes	—
17	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₈₃	D₄	1	4	yes	—
23	C₄×C₈×C₈₃	Q₈	1	2	yes	—
24	#13050574	—	4	8	no	—
25	C₂×C₈₃⋊C₈	—	4	4	no	—
26	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₈₃	C₂×Q₈	1	2	yes	—
28	C₂×C₈×C₈₃	—	2	8	no	—
29	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₈₃	C₄⋊C₄	1	4	yes	—
33	C₂×C₈×C₈₃	D₈	1	4	yes	—
34	C₂×C₈×C₈₃	Q₁₆	1	4	yes	—
35	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₈₃	C₂×D₄	1	2	yes	—
37	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₈₃	—	2	4	no	—
40	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₈₃	C₂×Q₁₆	1	2	yes	—
46	C₈×C₈₃	—	2	4	no	—
47	C₈×C₈₃	#107	1	4	yes	—
48	C₈×C₈₃	#107	1	4	yes	—
49	C₈×C₈₃	#108	1	8	yes	complemented
50	C₈₃⋊C₈	—	8	4	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	#384	1	1	yes	characteristic
53	C₂²×C₈₃	#500	1	2	yes	—
54	C₄×C₈₃	—	2	2	no	—
55	C₄×C₈₃	C₂×#107	1	2	yes	—
56	C₄×C₈₃	C₂×#108	1	4	yes	—
57	C₄×C₈₃	#493	1	1	yes	characteristic
58	C₄×C₈₃	#494	1	1	yes	characteristic
59	#8823	—	83	1	no	maximal, Sylow
60	C₂×C₈₃	—	2	2	no	—
61	C₂×C₈₃	#1845	1	1	yes	characteristic
62	C₂×C₈₃	#1848	1	2	yes	—
63	#1848	—	83	2	no	—
64	#1567	—	83	4	no	—
65	C₂×C₈²	D₈₃	1	1	yes	characteristic
66	C₈₃	#8823	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	166	4	no	—
68	#423	—	83	4	no	—
69	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
70	C₂×C₄×C₈	C₂×D₈₃	1	2	yes	—
71	#335	—	166	8	no	—
72	C₈²	Dic₈₃	1	1	yes	characteristic
73	C₈²	Dic₈₃	1	2	yes	—
74	C₈²	Dic₈₃	1	1	yes	characteristic
75	C₂×M₁₆	—	166	4	no	—
76	C₂²×C₈	Dic₁₆₆	1	4	yes	—
77	C₂²×C₈	D₃₃₂	1	2	yes	—
78	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
79	#108	—	332	8	no	—
80	C₄⋊C₈	—	166	8	no	—
81	C₄×C₈	Dic₁₆₆	1	2	yes	—
82	C₄×C₈	D₃₃₂	1	4	yes	—
83	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
84	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
87	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₁₆₆	1	2	yes	—
89	M₁₆	—	332	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	332	4	no	—
92	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
93	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
94	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₈₃	1	4	yes	—
96	C₂×C₈	D₆₆₄	1	4	yes	—
97	C₂×C₈	Dic₃₃₂	1	4	yes	—
98	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
99	C₄²	C₂×D₃₃₂	1	2	yes	—
100	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
101	C₂³	#14581792	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₈₃	1	2	yes	—
105	C₂×C₄	#14581789	1	1	yes	characteristic
106	C₂×C₄	#14581792	1	2	yes	—
107	C₂×C₄	C₂×D₆₆₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₃₃₂	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	664	4	no	—
111	C₈	#14581726	1	4	yes	—
112	C₈	#14581726	1	4	yes	—
113	C₈	#14581728	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#19127571	1	1	yes	characteristic
116	C₂²	#19128106	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#14581726	1	2	yes	—
119	C₄	C₂×#14581728	1	4	yes	—
120	C₄	#19128078	1	1	yes	characteristic
121	C₄	#19128081	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#24907623	1	1	yes	characteristic
124	C₂	#24907635	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907635

C₂

yes

maximal

#24906222

C₂

yes

maximal

C₂×C₈²×C₈₃

C₂

yes

maximal, characteristic

C₂×#14581728

—

#19127725

C₂²

yes

—

#19127454

—

C₂×C₄×C₈×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

C₈²×C₈₃

C₄

yes

characteristic

C₈²×C₈₃

C₄

yes

—

C₈²×C₈₃

C₄

yes

characteristic

C₂×#13050574

—

#14581728

—

C₄⋊(C₈₃⋊C₈)

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂²×C₈×C₈₃

Q₈

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

D₄

yes

—

C₄×C₈×C₈₃

Q₈

yes

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

Q₁₆

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

characteristic

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₃

#127

yes

characteristic

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×C₄×C₈₃

C₂×D₈

yes

—

C₂×C₄×C₈₃

C₂×Q₁₆

yes

—

C₈×C₈₃

—

C₈×C₈₃

#107

yes

—

C₈×C₈₃

#107

yes

—

C₈×C₈₃

#108

yes

complemented

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#384

yes

characteristic

C₂²×C₈₃

#500

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×#107

yes

—

C₄×C₈₃

C₂×#108

yes

—

C₄×C₈₃

yes

C₄×C₈₃

yes

—

C₂×C₈₃

—

C₂×C₈₃

#1845

yes

characteristic

C₂×C₈₃

#1848

yes

—

#1848

—

#1567

—

C₂×C₈²

D₈₃

yes

characteristic

C₈₃

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

166

—

#423

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×C₄×C₈

C₂×D₈₃

yes

—

#335

—

166

—

C₈²

Dic₈₃

yes

characteristic

C₈²

Dic₈₃

yes

—

C₈²

Dic₈₃

yes

characteristic

C₂×M₁₆

—

166

—

C₂²×C₈

Dic₁₆₆

yes

—

C₂²×C₈

D₃₃₂

yes

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#108

—

332

—

C₄⋊C₈

—

166

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

D₃₃₂

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

characteristic

C₄×C₈

C₂×Dic₈₃

yes

characteristic

C₄×C₈

C₂×Dic₈₃

yes

—

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₆₆

yes

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

D₆₆₄

yes

—

C₂×C₈

Dic₃₃₂

yes

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×D₃₃₂

yes

—

100

C₄²

C₂×Dic₁₆₆

yes

characteristic

101

C₂³

#14581792

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

—

105

C₂×C₄

#14581789

yes

characteristic

106

C₂×C₄

#14581792

yes

—

107

C₂×C₄

C₂×D₆₆₄

yes

—

108

C₂×C₄

C₂×Dic₃₃₂

yes

—

109

C₈

—

110

C₈

—

664

—

111

C₈

#14581726

yes

—

112

C₈

#14581726

yes

—

113

C₈

#14581728

yes

complemented

114

C₂²

—

115

C₂²

#19127571

yes

characteristic

116

C₂²

#19128106

yes

—

117

C₄

—

118

C₄

C₂×#14581726

yes

—

119

C₄

C₂×#14581728

yes

—

120

C₄

#19128078

yes

characteristic

121

C₄

#19128081

yes

characteristic

122

C₂

—

123

C₂

#24907623

yes

characteristic

124

C₂

#24907635

yes

—

	subgroup	G/H
center	C₂×C₄	#14581789
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×C₈²×C₈₃	C₂
socle	C₂²×C₈₃	#384
prime radical	C₂³×C₈₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	C₂×C₈²	D₈₃
hypocenter	C₈₃	#8823
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#14581789

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₂×C₈²×C₈₃

C₂

socle

C₂²×C₈₃

#384

prime radical

C₂³×C₈₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₈₃

C₂³

hypercenter

C₂×C₈²

D₈₃

hypocenter

C₈₃

#8823

LCS

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₈₃	#108	semidirect
66. C₈₃	#8823	semidirect
113. C₈	#14581728	semidirect

H ≅ G/N

type

49. C₈×C₈₃

#108

semidirect

66. C₈₃

#8823

semidirect

113. C₈

#14581728

semidirect

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

subgroup	index
#24907635	2
#24906222	2
C₂×C₈²×C₈₃	2
#8823	83

subgroup

index

#24907635

#24906222

C₂×C₈²×C₈₃

#8823

Subgroups — #35298208

Conjugacy classes of subgroups

Complements

Maximal subgroups