Group #20275650 — subgroups

Subgroups — #20275650

Order 6784.

292 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810377	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	C₄	1	1	yes	characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₄×Dic₁₀₆	—	2	2	no	—
8	C₄×Dic₁₀₆	—	2	2	no	—
9	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719919	C₄	1	2	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
12	C₄×#88322	—	2	2	no	—
13	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₀₆	—	2	2	no	—
22	C₂×Dic₁₀₆	—	2	2	no	—
23	C₂×Dic₁₀₆	C₈	1	4	yes	complemented
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	D₄	1	2	yes	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	#11719919	—	2	4	no	—
30	C₄×Dic₅₃	—	2	2	no	—
31	C₄×Dic₅₃	—	2	2	no	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	Q₈	1	4	yes	complemented
35	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₅₃	#41	1	2	yes	—
45	C₂×Dic₅₃	—	2	2	no	—
46	C₂×Dic₅₃	—	2	2	no	—
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
50	Dic₁₀₆	—	2	8	no	—
51	C₈×C₅₃	—	4	2	no	—
52	#88322	—	2	4	no	—
53	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
55	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	#131	1	2	yes	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
60	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₅₃	#124	1	1	yes	characteristic
62	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₅₃	—	2	4	no	—
64	Dic₅₃	—	4	2	no	—
65	#1984	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	#391	1	1	yes	characteristic
67	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₅₃	#408	1	1	yes	characteristic
69	C₂×C₅₃	C₈×Q₈	1	2	yes	—
70	C₂×C₅₃	#446	1	2	yes	—
71	C₂×C₄×Q₈	—	53	1	no	—
72	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
73	C₂×C₄×C₈	—	53	1	no	—
74	#336	—	53	4	no	—
75	C₅₃	#1984	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	53	1	no	—
77	C₂²×C₈	—	53	2	no	—
78	C₂²×C₈	C₂×D₅₃	1	2	yes	—
79	C₄×Q₈	—	106	2	no	—
80	C₄×Q₈	—	106	2	no	—
81	C₂×C₄⋊C₄	—	53	2	no	—
82	C₂×C₄⋊C₄	—	53	1	no	—
83	C₂×C₄²	—	53	2	no	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	C₄×C₈	—	106	2	no	—
87	C₂×Q₈	—	53	4	no	—
88	C₂×Q₈	—	106	2	no	—
89	C₂×Q₈	—	106	2	no	—
90	C₂²×C₄	—	53	2	no	—
91	C₂²×C₄	—	53	2	no	—
92	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	53	4	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₄⋊C₄	—	106	4	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₄²	—	106	2	no	—
103	C₄²	—	106	2	no	—
104	C₄²	D₂₁₂	1	2	yes	—
105	C₄²	C₂²⋊D₅₃	1	2	yes	—
106	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
107	Q₈	—	106	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	53	4	no	—
110	C₂×C₄	—	106	2	no	—
111	C₂×C₄	—	106	2	no	—
112	C₂×C₄	—	106	2	no	—
113	C₂×C₄	—	106	4	no	—
114	C₂×C₄	C₈×D₅₃	1	2	yes	—
115	C₂×C₄	#11719912	1	2	yes	—
116	C₂×C₄	#11719921	1	2	yes	—
117	C₂×C₄	#11719921	1	2	yes	—
118	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
119	C₂×C₄	#11719947	1	2	yes	—
120	C₂×C₄	Q₈×D₅₃	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₈	—	4	2	no	—
123	C₈	—	106	4	no	—
124	C₂²	C₂×#11719921	1	1	yes	characteristic
125	C₂²	#13133182	1	2	yes	—
126	C₂²	C₂×C₈×D₅₃	1	1	yes	characteristic
127	C₂²	C₂×#11719912	1	1	yes	characteristic
128	C₂²	#13133225	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	106	4	no	—
131	C₄	—	212	2	no	—
132	C₄	#13133129	1	4	yes	—
133	C₄	#13133197	1	2	yes	—
134	C₄	#13133204	1	1	yes	characteristic
135	C₄	#13133215	1	1	yes	characteristic
136	C₂	#16810528	1	1	yes	characteristic
137	C₂	C₂×#13133129	1	1	yes	characteristic
138	C₂	#16810591	1	1	yes	characteristic
139	C₂	#16810747	1	2	yes	—
140	C₂	#16810750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₀₆

C₂

yes

maximal, characteristic

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₄

yes

characteristic

C₂²×#88322

C₂²

yes

—

C₄×Dic₁₀₆

—

C₄×Dic₁₀₆

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×#11719919

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄×#88322

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₈

yes

complemented

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₅₃

#408

yes

characteristic

C₂×C₅₃

C₈×Q₈

yes

—

C₂×C₅₃

#446

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#336

—

C₅₃

#1984

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄²

—

106

—

103

C₄²

—

106

—

104

C₄²

D₂₁₂

yes

—

105

C₄²

C₂²⋊D₅₃

yes

—

106

C₂³

C₂×C₄×D₅₃

yes

characteristic

107

Q₈

—

106

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

106

—

114

C₂×C₄

C₈×D₅₃

yes

—

115

C₂×C₄

#11719912

yes

—

116

C₂×C₄

#11719921

yes

—

117

C₂×C₄

#11719921

yes

—

118

C₂×C₄

C₂×D₂₁₂

yes

characteristic

119

C₂×C₄

#11719947

yes

—

120

C₂×C₄

Q₈×D₅₃

yes

—

121

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

122

C₈

—

123

C₈

—

106

—

124

C₂²

C₂×#11719921

yes

characteristic

125

C₂²

#13133182

yes

—

126

C₂²

C₂×C₈×D₅₃

yes

characteristic

127

C₂²

C₂×#11719912

yes

characteristic

128

C₂²

#13133225

yes

—

129

C₄

—

130

C₄

—

106

—

131

C₄

—

212

—

132

C₄

#13133129

yes

—

133

C₄

#13133197

yes

—

134

C₄

#13133204

yes

characteristic

135

C₄

#13133215

yes

characteristic

136

C₂

#16810528

yes

characteristic

137

C₂

C₂×#13133129

yes

characteristic

138

C₂

#16810591

yes

characteristic

139

C₂

#16810747

yes

—

140

C₂

#16810750

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1984
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
23. C₂×Dic₁₀₆	C₈	semidirect
34. C₂×#88322	Q₈	semidirect
75. C₅₃	#1984	semidirect

H ≅ G/N

type

23. C₂×Dic₁₀₆

C₈

semidirect

34. C₂×#88322

Q₈

semidirect

75. C₅₃

#1984

semidirect

Subgroups — #20275650

Conjugacy classes of subgroups

Complements