Group #20287826 — subgroups

Subgroups — C₂×Q₃₂×D₅₃ #20287826

Order 6784.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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×Q₁₆×D₅₃	C₂	1	2	yes	maximal
2	C₂×Dic₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13133163	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₅₃	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₅₃	—	2	2	no	—
8	C₂×Dic₂₁₂	C₂²	1	2	yes	—
9	C₂×#11719925	—	2	2	no	—
10	Q₁₆×D₅₃	—	2	4	no	—
11	Q₁₆×D₅₃	C₂²	1	8	yes	—
12	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
13	Dic₄₂₄	C₂²	1	4	yes	complemented
14	#13133163	C₂²	1	8	yes	complemented
15	C₂×#11719908	C₂²	1	1	yes	characteristic
16	C₁₆×D₅₃	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₅₃	C₂²	1	2	yes	—
18	Q₃₂×C₅₃	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	2	no	—
21	Q₈×D₅₃	—	2	8	no	—
22	Q₈×D₅₃	—	4	4	no	—
23	C₂×C₄×D₅₃	—	4	2	no	—
24	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₁₀₆	—	4	2	no	—
27	Q₁₆×C₅₃	—	2	2	no	—
28	Q₁₆×C₅₃	C₂³	1	4	yes	—
29	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	#11719925	—	2	8	no	—
31	C₂×#88322	D₄	1	1	yes	characteristic
32	Dic₂₁₂	—	2	2	no	—
33	Dic₂₁₂	C₂³	1	4	yes	—
34	C₈×D₅₃	D₄	1	2	yes	—
35	C₈×D₅₃	C₂³	1	2	yes	—
36	C₁₆×C₅₃	C₂³	1	2	yes	—
37	#11719908	C₂³	1	2	yes	—
38	C₂²×D₅₃	D₈	1	1	yes	characteristic
39	Q₈×C₅₃	—	2	4	no	—
40	Q₈×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₅₃	—	4	2	no	—
44	C₂×Dic₅₃	D₈	1	1	yes	characteristic
45	C₄×D₅₃	—	4	8	no	—
46	C₄×D₅₃	D₈	1	2	yes	—
47	C₄×D₅₃	C₂×D₄	1	2	yes	—
48	Dic₁₀₆	—	2	4	no	—
49	Dic₁₀₆	—	4	2	no	—
50	Dic₁₀₆	—	4	8	no	—
51	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₈×C₅₃	C₂⁴	1	1	yes	characteristic
53	#88322	C₂×D₄	1	2	yes	—
54	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂×D₅₃	Q₃₂	1	4	yes	direct product factor
56	C₂×D₅₃	C₂×D₈	1	2	yes	—
57	C₄×C₅₃	—	4	4	no	—
58	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
59	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
60	Dic₅₃	—	4	4	no	—
61	Dic₅₃	C₂×D₈	1	2	yes	—
62	C₂²×Q₃₂	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	C₂×Q₃₂	1	2	yes	complemented
64	C₂×C₅₃	C₂²×D₈	1	1	yes	characteristic
65	D₅₃	C₂×Q₃₂	1	4	yes	direct product factor
66	C₂²×Q₁₆	—	53	2	no	—
67	C₂×Q₃₂	—	53	2	no	—
68	C₂×Q₃₂	—	53	8	no	—
69	C₂×Q₃₂	—	53	1	no	—
70	C₂×Q₃₂	D₅₃	1	1	yes	characteristic, direct product factor
71	C₂²×C₁₆	—	53	1	no	—
72	C₅₃	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	106	2	no	—
74	C₂×Q₁₆	—	53	8	no	—
75	C₂×Q₁₆	—	53	2	no	—
76	C₂×Q₁₆	—	106	4	no	—
77	C₂×Q₁₆	—	106	2	no	—
78	C₂×Q₁₆	C₂×D₅₃	1	2	yes	—
79	C₂²×C₈	—	53	1	no	—
80	Q₃₂	—	53	8	no	—
81	Q₃₂	—	53	4	no	—
82	Q₃₂	C₂×D₅₃	1	4	yes	direct product factor
83	C₂×C₁₆	—	53	1	no	—
84	C₂×C₁₆	—	53	4	no	—
85	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	106	8	no	—
88	C₂×Q₈	—	106	2	no	—
89	C₂×Q₈	—	212	2	no	—
90	C₂×Q₈	—	212	4	no	—
91	C₂²×C₄	—	53	1	no	—
92	C₂²×C₄	—	212	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	53	4	no	—
95	Q₁₆	—	106	8	no	—
96	Q₁₆	—	106	2	no	—
97	Q₁₆	C₂²×D₅₃	1	4	yes	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
102	C₁₆	—	53	2	no	—
103	C₁₆	C₂²×D₅₃	1	2	yes	—
104	C₂³	—	53	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	106	4	no	—
108	Q₈	—	212	8	no	—
109	Q₈	—	212	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	2	no	—
113	C₂×C₄	—	53	2	no	—
114	C₂×C₄	—	212	8	no	—
115	C₂×C₄	—	212	2	no	—
116	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
117	C₈	—	53	2	no	—
118	C₈	D₄×D₅₃	1	1	yes	characteristic
119	C₈	C₂³×D₅₃	1	1	yes	characteristic
120	C₂²	—	53	4	no	—
121	C₂²	—	53	2	no	—
122	C₂²	D₈×D₅₃	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	53	2	no	—
125	C₄	—	212	4	no	—
126	C₄	D₈×D₅₃	1	1	yes	characteristic
127	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
128	C₂	—	53	4	no	—
129	C₂	Q₃₂×D₅₃	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₅₃

C₂

yes

maximal

C₂×Dic₄₂₄

C₂

yes

maximal, characteristic, complemented

C₂×#13133163

C₂

yes

maximal, complemented

Q₃₂×D₅₃

C₂

yes

maximal, direct product factor

C₂×C₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₅₃

—

C₂×Dic₂₁₂

C₂²

yes

—

C₂×#11719925

—

Q₁₆×D₅₃

—

Q₁₆×D₅₃

C₂²

yes

—

C₂×C₈×D₅₃

C₂²

yes

characteristic

Dic₄₂₄

C₂²

yes

complemented

#13133163

C₂²

yes

complemented

C₂×#11719908

C₂²

yes

characteristic

C₁₆×D₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₂²

yes

—

Q₃₂×C₅₃

C₂²

yes

complemented

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂³

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

#11719925

—

C₂×#88322

D₄

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

C₂³

yes

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

C₂³

yes

—

C₁₆×C₅₃

C₂³

yes

—

#11719908

C₂³

yes

—

C₂²×D₅₃

D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

D₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

—

C₄×D₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂⁴

yes

characteristic

#88322

C₂×D₄

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

Q₃₂

yes

direct product factor

C₂×D₅₃

C₂×D₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

C₂²×D₄

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂×D₈

yes

—

C₂²×Q₃₂

—

maximal, Sylow

C₂×C₅₃

C₂×Q₃₂

yes

complemented

C₂×C₅₃

C₂²×D₈

yes

characteristic

D₅₃

C₂×Q₃₂

yes

direct product factor

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₅₃

yes

characteristic, direct product factor

C₂²×C₁₆

—

C₅₃

C₂²×Q₃₂

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

C₂×D₅₃

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

C₂²×C₄

—

212

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

—

106

—

Q₁₆

C₂²×D₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₅₃

yes

characteristic

102

C₁₆

—

103

C₁₆

C₂²×D₅₃

yes

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

106

—

108

Q₈

—

212

—

109

Q₈

—

212

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

212

—

115

C₂×C₄

—

212

—

116

C₂×C₄

D₄×D₅₃

yes

characteristic

117

C₈

—

118

C₈

D₄×D₅₃

yes

characteristic

119

C₈

C₂³×D₅₃

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

D₈×D₅₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

212

—

126

C₄

D₈×D₅₃

yes

characteristic

127

C₄

C₂×D₄×D₅₃

yes

characteristic

128

C₂

—

129

C₂

Q₃₂×D₅₃

yes

direct product factor

130

C₂

C₂×D₈×D₅₃

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₅₃
commutator	C₈×C₅₃	C₂⁴
Frattini	C₈	C₂³×D₅₃
Fitting	C₂×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₅₃
hypocenter	C₅₃	C₂²×Q₃₂
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₅₃

commutator

C₈×C₅₃

C₂⁴

Frattini

C₈

C₂³×D₅₃

Fitting

C₂×Q₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂×Q₃₂

D₅₃

hypocenter

C₅₃

C₂²×Q₃₂

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
2. C₂×Dic₄₂₄	C₂	semidirect
3. C₂×#13133163	C₂	semidirect
4. Q₃₂×D₅₃	C₂	direct (N × H)
6. C₂×Q₃₂×C₅₃	C₂	semidirect
13. Dic₄₂₄	C₂²	semidirect
14. #13133163	C₂²	semidirect
18. Q₃₂×C₅₃	C₂²	semidirect
55. C₂×D₅₃	Q₃₂	direct (N × H)
63. C₂×C₅₃	C₂×Q₃₂	semidirect
65. D₅₃	C₂×Q₃₂	direct (N × H)
70. C₂×Q₃₂	D₅₃	direct (N × H)
72. C₅₃	C₂²×Q₃₂	semidirect
82. Q₃₂	C₂×D₅₃	direct (N × H)
129. C₂	Q₃₂×D₅₃	direct (N × H)

H ≅ G/N

type

2. C₂×Dic₄₂₄

C₂

semidirect

3. C₂×#13133163

C₂

semidirect

4. Q₃₂×D₅₃

C₂

direct (N × H)

6. C₂×Q₃₂×C₅₃

C₂

semidirect

13. Dic₄₂₄

C₂²

semidirect

14. #13133163

C₂²

semidirect

18. Q₃₂×C₅₃

C₂²

semidirect

55. C₂×D₅₃

Q₃₂

direct (N × H)

63. C₂×C₅₃

C₂×Q₃₂

semidirect

65. D₅₃

C₂×Q₃₂

direct (N × H)

70. C₂×Q₃₂

D₅₃

direct (N × H)

72. C₅₃

C₂²×Q₃₂

semidirect

82. Q₃₂

C₂×D₅₃

direct (N × H)

129. C₂

Q₃₂×D₅₃

direct (N × H)

Subgroups — C₂×Q₃₂×D₅₃ #20287826

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q32×D53 #20287826

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₃₂×D₅₃ #20287826