Group #20276760 — subgroups

Subgroups — #20276760

Order 6784.

166 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₂×#13133161	C₂	1	2	yes	maximal
2	#16811003	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810426	C₂	1	2	yes	maximal, complemented
5	C₄×D₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₂	—	2	1	no	—
7	C₂×Dic₂₁₂	C₂²	1	2	yes	—
8	#13133217	—	2	1	no	—
9	#13133161	—	2	4	no	—
10	C₂×#11719908	C₂²	1	2	yes	—
11	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₄×D₄×C₅₃	—	2	1	no	—
13	#107×C₅₃	C₂²	1	1	yes	characteristic
14	#102×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₅₃	—	2	1	no	—
17	C₂²×C₄×C₅₃	—	4	1	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	D₈×C₅₃	—	2	1	no	—
20	D₈×C₅₃	D₄	1	2	yes	—
21	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₅₃	—	2	1	no	—
24	C₂²⋊C₄×C₅₃	—	4	1	no	—
25	C₄²×C₅₃	D₄	1	1	yes	characteristic
26	C₄⋊Dic₅₃	—	4	1	no	—
27	C₄⋊Dic₅₃	—	4	1	no	—
28	Dic₂₁₂	—	2	2	no	—
29	Dic₂₁₂	—	2	2	no	—
30	Dic₂₁₂	—	4	2	no	—
31	#11719908	—	2	2	no	—
32	C₂³×C₅₃	—	4	1	no	—
33	D₄×C₅₃	—	2	2	no	—
34	D₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	—	8	2	no	—
37	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₃	—	4	2	no	—
41	Dic₁₀₆	—	4	2	no	—
42	Dic₁₀₆	—	4	2	no	—
43	C₈×C₅₃	—	2	1	no	—
44	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₈×C₅₃	#41	1	1	yes	characteristic
46	C₂²×C₅₃	—	4	2	no	—
47	C₂²×C₅₃	—	8	1	no	—
48	C₂²×C₅₃	—	8	1	no	—
49	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	8	1	no	—
52	C₄×C₅₃	SD₃₂	1	2	yes	complemented
53	C₄×C₅₃	#121	1	1	yes	characteristic
54	C₄×C₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	Dic₅₃	—	8	2	no	—
56	#2208	—	53	1	no	maximal, Sylow
57	C₂×C₅₃	—	8	2	no	—
58	C₂×C₅₃	#459	1	1	yes	characteristic
59	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
60	C₂×C₅₃	#510	1	1	yes	characteristic
61	C₂×SD₃₂	—	53	2	no	—
62	#494	—	53	1	no	—
63	C₄×D₈	D₅₃	1	1	yes	characteristic
64	C₄⋊C₁₆	—	53	1	no	—
65	#358	—	53	2	no	—
66	C₅₃	#2208	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₁₆	—	53	2	no	—
68	C₂×Q₁₆	—	106	1	no	—
69	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
70	C₄⋊Q₈	—	106	1	no	—
71	C₄×D₄	—	2	1	no	—
72	SD₃₂	—	106	4	no	—
73	C₂×C₁₆	—	53	2	no	—
74	#107	C₂×D₅₃	1	1	yes	characteristic
75	#102	—	2	1	no	—
76	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
77	C₂×Q₈	—	106	2	no	—
78	C₂×D₄	—	2	1	no	—
79	C₂²×C₄	—	4	1	no	—
80	Q₁₆	—	106	2	no	—
81	Q₁₆	—	106	2	no	—
82	Q₁₆	—	212	2	no	—
83	D₈	—	2	1	no	—
84	D₈	D₂₁₂	1	2	yes	—
85	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
86	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	212	1	no	—
89	C₄⋊C₄	—	212	1	no	—
90	C₂²⋊C₄	—	4	1	no	—
91	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
92	C₁₆	—	106	2	no	—
93	C₂³	—	4	1	no	—
94	Q₈	—	212	2	no	—
95	Q₈	—	212	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	212	2	no	—
101	C₂×C₄	#11719922	1	1	yes	characteristic
102	C₂×C₄	#11719922	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	C₂×D₂₁₂	1	1	yes	characteristic
106	C₈	#11719945	1	1	yes	characteristic
107	C₂²	—	4	2	no	—
108	C₂²	—	8	1	no	—
109	C₂²	—	8	1	no	—
110	C₂²	C₂×#11719922	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	8	1	no	—
113	C₄	—	424	2	no	—
114	C₄	#13133161	1	2	yes	complemented
115	C₄	#13133191	1	1	yes	characteristic
116	C₄	#13133242	1	1	yes	characteristic
117	C₂	—	8	2	no	—
118	C₂	#16810818	1	1	yes	characteristic
119	C₂	C₂×#13133161	1	1	yes	characteristic
120	C₂	#16811063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133161

C₂

yes

maximal

#16811003

C₂

yes

maximal, characteristic, complemented

C₄⋊#11719908

C₂

yes

maximal, characteristic, complemented

#16810426

C₂

yes

maximal, complemented

C₄×D₈×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₂²

yes

—

#13133217

—

#13133161

—

C₂×#11719908

C₂²

yes

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

—

#107×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

—

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

#41

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

SD₃₂

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2208

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#459

yes

characteristic

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

#510

yes

characteristic

C₂×SD₃₂

—

#494

—

C₄×D₈

D₅₃

yes

characteristic

C₄⋊C₁₆

—

#358

—

C₅₃

#2208

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×D₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

—

106

—

C₄×D₄

—

SD₃₂

—

106

—

C₂×C₁₆

—

#107

C₂×D₅₃

yes

characteristic

#102

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×D₄

—

C₂²×C₄

—

Q₁₆

—

106

—

Q₁₆

—

106

—

Q₁₆

—

212

—

D₈

—

D₈

D₂₁₂

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₁₆

—

106

—

C₂³

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

212

—

101

C₂×C₄

#11719922

yes

characteristic

102

C₂×C₄

#11719922

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

104

C₈

—

105

C₈

C₂×D₂₁₂

yes

characteristic

106

C₈

#11719945

yes

characteristic

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×#11719922

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

424

—

114

C₄

#13133161

yes

complemented

115

C₄

#13133191

yes

characteristic

116

C₄

#13133242

yes

characteristic

117

C₂

—

118

C₂

#16810818

yes

characteristic

119

C₂

C₂×#13133161

yes

characteristic

120

C₂

#16811063

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₂×C₈×C₅₃	C₂³
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₄×D₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₄×D₈	D₅₃
hypocenter	C₅₃	#2208
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₂×C₈×C₅₃

C₂³

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₄×D₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₄×D₈

D₅₃

hypocenter

C₅₃

#2208

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

N	H ≅ G/N	type
2. #16811003	C₂	semidirect
3. C₄⋊#11719908	C₂	semidirect
4. #16810426	C₂	semidirect
52. C₄×C₅₃	SD₃₂	semidirect
66. C₅₃	#2208	semidirect
114. C₄	#13133161	semidirect

H ≅ G/N

type

2. #16811003

C₂

semidirect

3. C₄⋊#11719908

C₂

semidirect

4. #16810426

C₂

semidirect

52. C₄×C₅₃

SD₃₂

semidirect

66. C₅₃

#2208

semidirect

114. C₄

#13133161

semidirect

Subgroups — #20276760

Conjugacy classes of subgroups

Complements