Group #20283178 — subgroups

Subgroups — #20283178

Order 6784.

716 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 249 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×#13133265	C₂	1	1	yes	maximal, characteristic
2	#16811101	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810596	C₂	1	12	yes	maximal, complemented
4	C₅₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133265	—	2	4	no	—
6	#13133265	—	2	4	no	—
7	C₂×#11719949	C₄	1	4	yes	complemented
8	C₂×#11719945	C₂²	1	6	yes	—
9	C₂×Q₈×D₅₃	C₄	1	4	yes	complemented
10	Q₈×Dic₅₃	—	2	4	no	—
11	#13133176	C₂²	1	6	yes	—
12	#13133131	C₂²	1	16	yes	—
13	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₅₃	—	2	4	no	—
15	#117×C₅₃	C₂²	1	6	yes	—
16	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
17	C₂×Q₈×C₅₃	D₄	1	4	yes	—
18	C₂×Q₈×C₅₃	D₄	1	4	yes	—
19	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
20	C₂×C₂²⋊D₅₃	C₂³	1	4	yes	—
21	#11719949	—	2	16	no	—
22	Q₈×D₅₃	—	2	16	no	—
23	#11719945	—	2	12	no	—
24	#11719945	—	2	12	no	—
25	C₂×D₂₁₂	C₂×C₄	1	6	yes	—
26	C₂×C₄×D₅₃	C₂×C₄	1	12	yes	—
27	C₂×Dic₁₀₆	C₂×C₄	1	6	yes	—
28	C₄⋊C₄×C₅₃	—	2	6	no	—
29	C₂²⋊C₄×C₅₃	C₂³	1	4	yes	—
30	C₄²×C₅₃	—	2	6	no	—
31	C₂²⋊Dic₅₃	C₂³	1	4	yes	—
32	C₄⋊Dic₅₃	—	2	6	no	—
33	C₄×Dic₅₃	—	2	6	no	—
34	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
36	Q₈×C₅₃	C₂²⋊C₄	1	16	yes	—
37	C₂×C₄×C₅₃	—	2	4	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
39	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
40	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
41	C₂²⋊D₅₃	—	2	4	no	—
42	C₂²⋊D₅₃	—	2	4	no	—
43	C₂×Dic₅₃	—	2	4	no	—
44	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
45	D₂₁₂	—	2	12	no	—
46	C₄×D₅₃	—	2	24	no	—
47	Dic₁₀₆	—	2	12	no	—
48	C₂²×C₅₃	—	4	1	no	—
49	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₅₃	—	2	4	no	—
53	C₂×D₅₃	—	4	4	no	—
54	C₄×C₅₃	—	4	4	no	—
55	C₄×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
56	Dic₅₃	—	2	4	no	—
57	Dic₅₃	—	4	4	no	—
58	#2884	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	—	2	1	no	—
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
63	D₅₃	—	4	4	no	—
64	C₂×2₋¹⁺⁴	—	53	1	no	—
65	#519	—	53	1	no	—
66	#519	D₅₃	1	1	yes	characteristic, complemented
67	#410	—	53	12	no	—
68	C₅₃	#2884	1	1	yes	characteristic, complemented, Sylow
69	2₋¹⁺⁴	—	106	4	no	—
70	2₋¹⁺⁴	—	106	4	no	—
71	C₂×#41	—	53	4	no	—
72	C₂×#41	—	53	6	no	—
73	C₂²×Q₈	—	53	4	no	—
74	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
75	C₄×Q₈	—	2	4	no	—
76	C₄×Q₈	—	106	4	no	—
77	#117	—	53	6	no	—
78	#117	C₂×D₅₃	1	6	yes	—
79	#99	—	53	16	no	—
80	#41	—	106	16	no	—
81	#41	—	106	12	no	—
82	#41	—	106	12	no	—
83	C₂×Q₈	—	53	6	no	—
84	C₂×Q₈	—	106	16	no	—
85	C₂×Q₈	C₄×D₅₃	1	4	yes	—
86	C₂×Q₈	D₂₁₂	1	4	yes	—
87	C₂×Q₈	C₂²⋊D₅₃	1	4	yes	—
88	C₂×D₄	—	53	4	no	—
89	C₂×D₄	—	53	6	no	—
90	C₂²×C₄	—	53	12	no	—
91	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
92	C₄⋊C₄	—	2	6	no	—
93	C₄⋊C₄	—	106	6	no	—
94	C₂²⋊C₄	—	53	4	no	—
95	C₂²⋊C₄	C₂²×D₅₃	1	4	yes	—
96	C₄²	—	2	6	no	—
97	C₄²	—	106	6	no	—
98	C₂³	—	53	4	no	—
99	C₂³	C₂³×D₅₃	1	1	yes	characteristic
100	Q₈	—	106	12	no	—
101	Q₈	#11719921	1	16	yes	—
102	D₄	—	106	4	no	—
103	D₄	—	106	4	no	—
104	D₄	—	106	12	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	53	4	no	—
107	C₂×C₄	—	106	24	no	—
108	C₂×C₄	—	106	4	no	—
109	C₂×C₄	C₂×C₄×D₅₃	1	6	yes	—
110	C₂×C₄	C₂×D₂₁₂	1	6	yes	—
111	C₂×C₄	C₂×C₂²⋊D₅₃	1	6	yes	—
112	C₂²	—	4	1	no	—
113	C₂²	—	106	4	no	—
114	C₂²	—	212	4	no	—
115	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
116	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	106	4	no	—
120	C₄	—	212	4	no	—
121	C₄	C₂×#11719921	1	12	yes	—
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	212	4	no	—
126	C₂	C₂²×#11719921	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133265

C₂

yes

maximal, characteristic

#16811101

C₂

yes

maximal, characteristic, complemented

#16810596

C₂

yes

maximal, complemented

C₅₃×#519

C₂

yes

maximal, characteristic, complemented

#13133265

—

#13133265

—

C₂×#11719949

C₄

yes

complemented

C₂×#11719945

C₂²

yes

—

C₂×Q₈×D₅₃

C₄

yes

complemented

Q₈×Dic₅₃

—

#13133176

C₂²

yes

—

#13133131

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂×C₂²⋊D₅₃

C₂³

yes

—

#11719949

—

Q₈×D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂²×D₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

Dic₅₃

—

Dic₅₃

—

#2884

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₂²⋊C₄

yes

characteristic

D₅₃

—

C₂×2₋¹⁺⁴

—

#519

—

#519

D₅₃

yes

characteristic, complemented

#410

—

C₅₃

#2884

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

106

—

2₋¹⁺⁴

—

106

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

#117

C₂×D₅₃

yes

—

#99

—

#41

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

C₄×D₅₃

yes

—

C₂×Q₈

D₂₁₂

yes

—

C₂×Q₈

C₂²⋊D₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂²×D₅₃

yes

—

C₄²

—

C₄²

—

106

—

C₂³

—

C₂³

C₂³×D₅₃

yes

characteristic

100

Q₈

—

106

—

101

Q₈

#11719921

yes

—

102

D₄

—

106

—

103

D₄

—

106

—

104

D₄

—

106

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

C₂×C₄×D₅₃

yes

—

110

C₂×C₄

C₂×D₂₁₂

yes

—

111

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

112

C₂²

—

113

C₂²

—

106

—

114

C₂²

—

212

—

115

C₂²

C₂²×C₄×D₅₃

yes

characteristic

116

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

117

C₂²

C₂²×D₂₁₂

yes

characteristic

118

C₄

—

119

C₄

—

106

—

120

C₄

—

212

—

121

C₄

C₂×#11719921

yes

—

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

—

212

—

126

C₂

C₂²×#11719921

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#11719921
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂³	C₂³×D₅₃
Fitting	C₅₃×#519	C₂
socle	C₂×C₅₃	C₂²×C₂²⋊C₄
hypercenter	#519	D₅₃
hypocenter	C₅₃	#2884
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

subgroup

G/H

center

C₂

C₂²×#11719921

commutator

C₂²×C₅₃

C₂³×C₄

Frattini

C₂³

C₂³×D₅₃

Fitting

C₅₃×#519

C₂

socle

C₂×C₅₃

C₂²×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₂²×Q₈ ◁ #519

N	H ≅ G/N	type
2. #16811101	C₂	semidirect
3. #16810596	C₂	semidirect
4. C₅₃×#519	C₂	semidirect
7. C₂×#11719949	C₄	semidirect
9. C₂×Q₈×D₅₃	C₄	semidirect
66. #519	D₅₃	semidirect
68. C₅₃	#2884	semidirect

H ≅ G/N

type

2. #16811101

C₂

semidirect

3. #16810596

C₂

semidirect

4. C₅₃×#519

C₂

semidirect

7. C₂×#11719949

C₄

semidirect

9. C₂×Q₈×D₅₃

C₄

semidirect

66. #519

D₅₃

semidirect

68. C₅₃

#2884

semidirect

Subgroups — #20283178

Conjugacy classes of subgroups

Complements