Group #35291069 — subgroups

Subgroups — #35291069

Order 21248.

132 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 55 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	#24906361	C₂	1	1	yes	maximal, characteristic
2	#24905533	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#1312	C₂	1	1	yes	maximal, characteristic
4	#19127831	—	2	1	no	—
5	C₄×Dic₃₃₂	C₄	1	1	yes	characteristic
6	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
7	#19127454	C₄	1	1	yes	characteristic
8	#19127433	—	2	1	no	—
9	C₄×C₁₆×C₈₃	—	2	1	no	—
10	C₈²×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₃₂	C₈	1	1	yes	characteristic
12	C₄×Dic₁₆₆	—	2	1	no	—
13	C₂×C₈₃⋊C₁₆	—	4	1	no	—
14	#14581726	C₈	1	1	yes	characteristic
15	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
16	Dic₈₃⋊C₈	—	4	1	no	—
17	#14581717	—	2	1	no	—
18	#14581702	—	4	1	no	—
19	C₂×C₁₆×C₈₃	—	2	2	no	—
20	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₆₆	—	2	1	no	—
24	Dic₃₃₂	—	4	1	no	—
25	C₂×C₈₃⋊C₈	—	4	1	no	—
26	C₄⋊Dic₈₃	—	2	1	no	—
27	Dic₈₃⋊C₄	—	4	1	no	—
28	C₄×Dic₈₃	—	4	1	no	—
29	C₈₃⋊C₁₆	—	8	1	no	—
30	C₂×C₈×C₈₃	—	2	1	no	—
31	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₈₃	M₁₆	1	1	yes	characteristic
33	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
34	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
35	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₁₆×C₈₃	—	4	2	no	—
37	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
38	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
39	C₂×C₄×C₈₃	#104	1	1	yes	characteristic
40	C₂×Dic₈₃	—	4	1	no	—
41	C₂×Dic₈₃	—	4	1	no	—
42	Dic₁₆₆	—	4	1	no	—
43	Dic₁₆₆	—	4	1	no	—
44	C₈×C₈₃	—	2	2	no	—
45	C₈×C₈₃	—	2	1	no	—
46	C₈×C₈₃	—	2	1	no	—
47	C₈×C₈₃	—	2	1	no	—
48	C₈×C₈₃	SD₃₂	1	1	yes	characteristic
49	C₈×C₈₃	Q₃₂	1	1	yes	characteristic
50	C₈₃⋊C₈	—	8	1	no	—
51	C₂²×C₈₃	D₄⋊C₈	1	1	yes	characteristic
52	C₄×C₈₃	—	2	1	no	—
53	C₄×C₈₃	#349	1	1	yes	characteristic
54	C₄×C₈₃	#350	1	1	yes	characteristic
55	C₄×C₈₃	#358	1	1	yes	characteristic
56	C₄×C₈₃	#359	1	1	yes	characteristic
57	Dic₈₃	—	8	1	no	—
58	Dic₈₃	—	8	1	no	—
59	#7073	—	83	1	no	maximal, Sylow
60	C₂×C₈₃	#1330	1	1	yes	characteristic
61	C₂×C₈₃	#1332	1	1	yes	characteristic
62	C₂×C₈₃	#1336	1	1	yes	characteristic
63	#1591	—	83	1	no	—
64	#1370	—	83	1	no	—
65	#1312	D₈₃	1	1	yes	characteristic
66	C₈₃	#7073	1	1	yes	characteristic, complemented, Sylow
67	#446	—	166	1	no	—
68	C₄×Q₁₆	—	83	1	no	—
69	C₄⋊C₁₆	—	166	1	no	—
70	C₄×C₁₆	—	2	1	no	—
71	#335	—	83	1	no	—
72	Q₈⋊C₈	—	166	1	no	—
73	C₈²	C₂×D₈₃	1	1	yes	characteristic
74	C₂×Q₁₆	—	83	1	no	—
75	C₄×Q₈	—	166	1	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	332	1	no	—
78	#107	—	83	1	no	—
79	C₄⋊C₈	—	332	1	no	—
80	C₄⋊C₈	—	166	1	no	—
81	#103	—	166	1	no	—
82	#97	—	332	1	no	—
83	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
84	C₄×C₈	D₃₃₂	1	1	yes	characteristic
85	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
86	C₂×Q₈	—	166	1	no	—
87	Q₁₆	—	332	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	332	1	no	—
90	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
91	C₂×C₈	#13050573	1	1	yes	characteristic
92	C₂×C₈	D₆₆₄	1	1	yes	characteristic
93	C₂×C₈	#13050579	1	1	yes	characteristic
94	C₄⋊C₄	—	332	1	no	—
95	C₄⋊C₄	—	166	1	no	—
96	C₄²	—	332	1	no	—
97	C₄²	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
98	C₁₆	—	4	2	no	—
99	C₁₆	—	664	1	no	—
100	Q₈	—	332	1	no	—
101	Q₈	—	332	1	no	—
102	C₂×C₄	—	332	1	no	—
103	C₂×C₄	—	332	1	no	—
104	C₂×C₄	(C₂×D₈₃)⋊C₈	1	1	yes	characteristic
105	C₂×C₄	#14581714	1	1	yes	characteristic
106	C₂×C₄	#14581720	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	664	1	no	—
112	C₈	#14581737	1	1	yes	characteristic
113	C₈	Dic₆₆₄	1	1	yes	characteristic
114	C₂²	#19127430	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	664	1	no	—
117	C₄	—	664	1	no	—
118	C₄	#19127482	1	1	yes	characteristic
119	C₄	#19127486	1	1	yes	characteristic
120	C₄	#19127506	1	1	yes	characteristic
121	C₄	#19127509	1	1	yes	characteristic
122	C₂	#24905422	1	1	yes	characteristic
123	C₂	#24905428	1	1	yes	characteristic
124	C₂	#24905440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24906361

C₂

yes

maximal, characteristic

#24905533

C₂

yes

maximal, characteristic

C₈₃×#1312

C₂

yes

maximal, characteristic

#19127831

—

C₄×Dic₃₃₂

C₄

yes

characteristic

C₄⋊(C₈₃⋊C₁₆)

—

#19127454

C₄

yes

characteristic

#19127433

—

C₄×C₁₆×C₈₃

—

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

C₈

yes

characteristic

C₄×Dic₁₆₆

—

C₂×C₈₃⋊C₁₆

—

#14581726

C₈

yes

characteristic

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

#14581717

—

#14581702

—

C₂×C₁₆×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₈₃⋊C₁₆

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₂×C₈

yes

characteristic

C₂×C₈×C₈₃

M₁₆

yes

characteristic

C₂×C₈×C₈₃

D₈

yes

characteristic

C₂×C₈×C₈₃

SD₁₆

yes

characteristic

C₄²×C₈₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₈

yes

C₂×C₄×C₈₃

yes

C₂×C₄×C₈₃

yes

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

SD₃₂

yes

characteristic

C₈×C₈₃

Q₃₂

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

D₄⋊C₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

Dic₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

C₈₃

#7073

yes

characteristic, complemented, Sylow

#446

—

166

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

166

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

166

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

166

—

C₂×C₁₆

—

C₂×C₁₆

—

332

—

#107

—

C₄⋊C₈

—

332

—

C₄⋊C₈

—

166

—

#103

—

166

—

#97

—

332

—

C₄×C₈

C₄×D₈₃

yes

characteristic

C₄×C₈

D₃₃₂

yes

characteristic

C₄×C₈

C₂²⋊D₈₃

yes

characteristic

C₂×Q₈

—

166

—

Q₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

C₈×D₈₃

yes

characteristic

C₂×C₈

#13050573

yes

characteristic

C₂×C₈

D₆₆₄

yes

characteristic

C₂×C₈

#13050579

yes

characteristic

C₄⋊C₄

—

332

—

C₄⋊C₄

—

166

—

C₄²

—

332

—

C₄²

(C₂×D₈₃)⋊C₄

yes

characteristic

C₁₆

—

C₁₆

—

664

—

100

Q₈

—

332

—

101

Q₈

—

332

—

102

C₂×C₄

—

332

—

103

C₂×C₄

—

332

—

104

C₂×C₄

(C₂×D₈₃)⋊C₈

yes

characteristic

105

C₂×C₄

#14581714

yes

characteristic

106

C₂×C₄

#14581720

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

664

—

112

C₈

#14581737

yes

characteristic

113

C₈

Dic₆₆₄

yes

characteristic

114

C₂²

#19127430

yes

characteristic

115

C₄

—

116

C₄

—

664

—

117

C₄

—

664

—

118

C₄

#19127482

yes

characteristic

119

C₄

#19127486

yes

characteristic

120

C₄

#19127506

yes

characteristic

121

C₄

#19127509

yes

characteristic

122

C₂

#24905422

yes

characteristic

123

C₂

#24905428

yes

characteristic

124

C₂

#24905440

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14581714
commutator	C₂×C₈×C₈₃	C₂×C₈
Frattini	C₈²	C₂×D₈₃
Fitting	C₈₃×#1312	C₂
socle, prime radical	C₂²×C₈₃	D₄⋊C₈
cosocle	C₈²×C₈₃	C₂²
hypercenter	#1312	D₈₃
hypocenter	C₈₃	#7073
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂×C₄

#14581714

commutator

C₂×C₈×C₈₃

C₂×C₈

Frattini

C₈²

C₂×D₈₃

Fitting

C₈₃×#1312

C₂

socle, prime radical

C₂²×C₈₃

D₄⋊C₈

cosocle

C₈²×C₈₃

C₂²

D₈₃

C₈₃

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

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

N	H ≅ G/N	type
66. C₈₃	#7073	semidirect

H ≅ G/N

type

66. C₈₃

#7073

semidirect

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

subgroup	index
#24906361	2
#24905533	2
C₈₃×#1312	2
#7073	83

subgroup

index

#24906361

#24905533

C₈₃×#1312

#7073

Subgroups — #35291069

Conjugacy classes of subgroups

Complements

Maximal subgroups