Group #20283480 — subgroups

Subgroups — C₂×Q₁₆×Dic₅₃ #20283480

Order 6784.

488 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 245 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×Q₈×Dic₅₃	C₂	1	2	yes	maximal
2	Q₁₆×Dic₅₃	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133141	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	Q₈×Dic₅₃	—	2	4	no	—
9	Q₈×Dic₅₃	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	2	no	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×C₄×Dic₅₃	—	2	2	no	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133151	C₂²	1	4	yes	—
15	#13133141	C₂²	1	8	yes	—
16	C₈×Dic₅₃	C₂²	1	4	yes	—
17	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
18	C₂×Q₁₆×C₅₃	C₄	1	8	yes	complemented
19	C₂×Q₁₆×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	2	no	—
22	C₂×Q₈×C₅₃	—	2	4	no	—
23	C₂×Q₈×C₅₃	C₂×C₄	1	8	yes	—
24	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
25	C₂²×C₄×C₅₃	—	2	2	no	—
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×Dic₅₃	—	2	2	no	—
28	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
29	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
30	Q₁₆×C₅₃	C₂×C₄	1	16	yes	complemented
31	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×C₅₃	C₂³	1	2	yes	—
33	C₄⋊Dic₅₃	—	2	2	no	—
34	C₄⋊Dic₅₃	—	2	8	no	—
35	C₄⋊Dic₅₃	C₂³	1	4	yes	—
36	C₄×Dic₅₃	—	2	8	no	—
37	C₄×Dic₅₃	D₄	1	2	yes	—
38	C₄×Dic₅₃	C₂³	1	2	yes	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	C₂³	1	2	yes	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	2	8	no	—
43	Q₈×C₅₃	—	2	4	no	—
44	Q₈×C₅₃	C₂²×C₄	1	8	yes	—
45	C₂×C₄×C₅₃	—	2	8	no	—
46	C₂×C₄×C₅₃	—	2	4	no	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₅₃	—	2	2	no	—
52	C₂×Dic₅₃	—	2	4	no	—
53	C₂×Dic₅₃	—	4	4	no	—
54	C₂×Dic₅₃	Q₁₆	1	4	yes	direct product factor
55	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
56	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
57	C₈×C₅₃	C₂²×C₄	1	4	yes	—
58	#88322	—	2	2	no	—
59	C₂²×C₅₃	C₄×D₄	1	2	yes	—
60	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
61	C₂²×C₅₃	#135	1	2	yes	—
62	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₅₃	—	2	8	no	—
64	C₄×C₅₃	C₄×D₄	1	2	yes	—
65	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
67	Dic₅₃	—	2	2	no	—
68	Dic₅₃	—	4	4	no	—
69	Dic₅₃	C₂×Q₁₆	1	4	yes	direct product factor
70	C₂×C₄×Q₁₆	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	C₄×Q₁₆	1	4	yes	complemented
72	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
75	C₂²×Q₁₆	D₅₃	1	1	yes	characteristic
76	C₂×C₄×Q₈	—	53	2	no	—
77	C₄×Q₁₆	—	53	8	no	—
78	C₂×#107	—	53	1	no	—
79	C₂×#103	—	53	2	no	—
80	C₂×C₄×C₈	—	53	1	no	—
81	C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
83	C₂×Q₁₆	Dic₅₃	1	8	yes	direct product factor
84	C₂×Q₁₆	C₂×D₅₃	1	4	yes	—
85	C₂²×C₈	—	53	1	no	—
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×Q₈	—	53	8	no	—
88	C₄×Q₈	—	106	4	no	—
89	C₂×C₄⋊C₄	—	53	2	no	—
90	C₂×C₄⋊C₄	—	106	2	no	—
91	C₂×C₄²	—	53	1	no	—
92	C₂×C₄²	—	106	2	no	—
93	#107	—	53	4	no	—
94	#103	—	53	8	no	—
95	C₄×C₈	—	53	4	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	C₂×Dic₅₃	1	8	yes	—
99	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	Q₁₆	C₂×Dic₅₃	1	16	yes	direct product factor
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
109	C₂×C₈	C₂²×D₅₃	1	2	yes	—
110	C₄⋊C₄	—	53	4	no	—
111	C₄⋊C₄	—	106	8	no	—
112	C₄⋊C₄	—	106	2	no	—
113	C₄²	—	53	2	no	—
114	C₄²	—	53	2	no	—
115	C₄²	—	106	8	no	—
116	C₂³	D₄×D₅₃	1	1	yes	characteristic
117	Q₈	—	2	8	no	—
118	Q₈	—	2	4	no	—
119	Q₈	C₂²×Dic₅₃	1	8	yes	—
120	C₂×C₄	—	2	8	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	53	4	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	106	2	no	—
126	C₂×C₄	—	106	4	no	—
127	C₂×C₄	—	212	4	no	—
128	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
129	C₂×C₄	#11719947	1	2	yes	—
130	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
131	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
132	C₈	—	106	2	no	—
133	C₈	C₂²×Dic₅₃	1	4	yes	—
134	C₂²	D₄×Dic₅₃	1	2	yes	—
135	C₂²	Q₁₆×D₅₃	1	2	yes	—
136	C₂²	#13133240	1	2	yes	—
137	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	53	4	no	—
140	C₄	—	106	2	no	—
141	C₄	—	212	4	no	—
142	C₄	D₄×Dic₅₃	1	2	yes	—
143	C₄	C₂³×Dic₅₃	1	1	yes	characteristic
144	C₄	C₂×#11719947	1	1	yes	characteristic
145	C₂	Q₁₆×Dic₅₃	1	4	yes	direct product factor
146	C₂	C₂×D₄×Dic₅₃	1	1	yes	characteristic
147	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
148	C₂	C₂×#13133240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₅₃

C₂

yes

maximal

Q₁₆×Dic₅₃

C₂

yes

maximal, direct product factor

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133141

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

Q₈×Dic₅₃

—

Q₈×Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133141

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₂×Q₁₆×C₅₃

C₄

yes

complemented

C₂×Q₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

Q₁₆×C₅₃

C₂×C₄

yes

complemented

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₁₆

yes

direct product factor

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₁₆

yes

direct product factor

C₂×C₄×Q₁₆

—

maximal, Sylow

C₂×C₅₃

C₄×Q₁₆

yes

complemented

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×#135

yes

characteristic

C₂²×Q₁₆

D₅₃

yes

characteristic

C₂×C₄×Q₈

—

C₄×Q₁₆

—

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

—

C₅₃

C₂×C₄×Q₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂×Q₁₆

Dic₅₃

yes

direct product factor

C₂×Q₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₂×C₄²

—

106

—

#107

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×Q₈

C₂²×D₅₃

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₂²×D₅₃

yes

characteristic

105

Q₁₆

C₂×Dic₅₃

yes

direct product factor

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

C₂×Dic₅₃

yes

—

109

C₂×C₈

C₂²×D₅₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

—

106

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

106

—

116

C₂³

D₄×D₅₃

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

C₂²×Dic₅₃

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

212

—

128

C₂×C₄

D₄×D₅₃

yes

characteristic

129

C₂×C₄

#11719947

yes

—

130

C₂×C₄

C₂²×Dic₅₃

yes

—

131

C₂×C₄

C₂³×D₅₃

yes

characteristic

132

C₈

—

106

—

133

C₈

C₂²×Dic₅₃

yes

—

134

C₂²

D₄×Dic₅₃

yes

—

135

C₂²

Q₁₆×D₅₃

yes

—

136

C₂²

#13133240

yes

—

137

C₂²

C₂×D₄×D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

—

212

—

142

C₄

D₄×Dic₅₃

yes

—

143

C₄

C₂³×Dic₅₃

yes

characteristic

144

C₄

C₂×#11719947

yes

characteristic

145

C₂

Q₁₆×Dic₅₃

yes

direct product factor

146

C₂

C₂×D₄×Dic₅₃

yes

characteristic

147

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

148

C₂

C₂×#13133240

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂²×Q₁₆×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂²×Q₁₆	D₅₃
hypocenter	C₅₃	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂²×Q₁₆×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂²×Q₁₆

D₅₃

hypocenter

C₅₃

C₂×C₄×Q₁₆

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. Q₁₆×Dic₅₃	C₂	direct (N × H)
18. C₂×Q₁₆×C₅₃	C₄	semidirect
30. Q₁₆×C₅₃	C₂×C₄	semidirect
54. C₂×Dic₅₃	Q₁₆	direct (N × H)
69. Dic₅₃	C₂×Q₁₆	direct (N × H)
71. C₂×C₅₃	C₄×Q₁₆	semidirect
81. C₅₃	C₂×C₄×Q₁₆	semidirect
83. C₂×Q₁₆	Dic₅₃	direct (N × H)
105. Q₁₆	C₂×Dic₅₃	direct (N × H)
145. C₂	Q₁₆×Dic₅₃	direct (N × H)

H ≅ G/N

type

2. Q₁₆×Dic₅₃

C₂

direct (N × H)

18. C₂×Q₁₆×C₅₃

C₄

semidirect

30. Q₁₆×C₅₃

C₂×C₄

semidirect

54. C₂×Dic₅₃

Q₁₆

direct (N × H)

69. Dic₅₃

C₂×Q₁₆

direct (N × H)

71. C₂×C₅₃

C₄×Q₁₆

semidirect

81. C₅₃

C₂×C₄×Q₁₆

semidirect

83. C₂×Q₁₆

Dic₅₃

direct (N × H)

105. Q₁₆

C₂×Dic₅₃

direct (N × H)

145. C₂

Q₁₆×Dic₅₃

direct (N × H)

Subgroups — C₂×Q₁₆×Dic₅₃ #20283480

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q16×Dic53 #20283480

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₁₆×Dic₅₃ #20283480