Group #20275666 — subgroups

Subgroups — #20275666

Order 6784.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#16810386	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	—	2	1	no	—
6	#13133217	—	2	2	no	—
7	#13133217	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₅₃	—	2	1	no	—
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₅₃	—	2	1	no	—
12	#13133124	—	2	1	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	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	—	2	2	no	—
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂×Dic₁₀₆	—	2	4	no	—
22	C₂×Dic₁₀₆	—	4	2	no	—
23	C₂×Dic₁₀₆	—	4	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄⋊Dic₅₃	—	2	4	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₅₃	Q₈	1	4	yes	complemented
33	C₄×Dic₅₃	—	4	2	no	—
34	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	D₈	1	2	yes	—
39	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅₃	#41	1	2	yes	—
44	C₂×Dic₅₃	—	2	4	no	—
45	C₂×Dic₅₃	—	2	4	no	—
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	4	2	no	—
50	Dic₁₀₆	—	4	8	no	—
51	C₈×C₅₃	—	4	2	no	—
52	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
54	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
55	C₂²×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²×C₅₃	#137	1	2	yes	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	#102	1	4	yes	—
59	C₄×C₅₃	C₄×Q₈	1	2	yes	—
60	C₄×C₅₃	#124	1	1	yes	characteristic
61	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₅₃	—	4	4	no	—
63	Dic₅₃	—	4	4	no	—
64	#1988	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	#391	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
67	C₂×C₅₃	#419	1	1	yes	characteristic
68	C₂×C₅₃	#474	1	2	yes	—
69	C₂×C₅₃	#475	1	2	yes	—
70	C₂×C₄⋊Q₈	—	53	1	no	—
71	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
72	C₄×C₄⋊C₄	—	53	1	no	—
73	#340	—	53	4	no	—
74	C₅₃	#1988	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	106	1	no	—
76	C₂²×C₈	C₂×D₅₃	1	2	yes	—
77	C₄⋊Q₈	—	53	4	no	—
78	C₄⋊Q₈	—	106	2	no	—
79	C₂×C₄⋊C₄	—	53	2	no	—
80	C₂×C₄⋊C₄	—	53	2	no	—
81	C₂×C₄⋊C₄	—	106	1	no	—
82	C₂×C₄²	—	106	1	no	—
83	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
84	C₄⋊C₈	—	2	2	no	—
85	#95	—	106	1	no	—
86	C₂×Q₈	—	106	4	no	—
87	C₂×Q₈	—	212	2	no	—
88	C₂×Q₈	—	212	2	no	—
89	C₂²×C₄	—	106	2	no	—
90	C₂²×C₄	—	106	2	no	—
91	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
92	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₄⋊C₄	—	53	4	no	—
97	C₄⋊C₄	—	53	4	no	—
98	C₄⋊C₄	—	106	2	no	—
99	C₄⋊C₄	—	106	2	no	—
100	C₄⋊C₄	—	106	4	no	—
101	C₄²	—	212	2	no	—
102	C₄²	C₄×D₅₃	1	2	yes	—
103	C₄²	C₂²⋊D₅₃	1	2	yes	—
104	C₂³	C₂×D₂₁₂	1	1	yes	characteristic
105	Q₈	—	212	8	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	106	4	no	—
108	C₂×C₄	—	106	4	no	—
109	C₂×C₄	—	212	2	no	—
110	C₂×C₄	—	212	2	no	—
111	C₂×C₄	—	212	2	no	—
112	C₂×C₄	—	212	2	no	—
113	C₂×C₄	#11719913	1	2	yes	—
114	C₂×C₄	D₄₂₄	1	2	yes	—
115	C₂×C₄	#11719921	1	2	yes	—
116	C₂×C₄	#11719921	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄	#11719947	1	2	yes	—
119	C₂×C₄	Q₈×D₅₃	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₂²	C₂×#11719921	1	1	yes	characteristic
123	C₂²	#13133197	1	2	yes	—
124	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
125	C₂²	C₂×#11719913	1	1	yes	characteristic
126	C₂²	#13133250	1	2	yes	—
127	C₄	—	2	2	no	—
128	C₄	—	212	4	no	—
129	C₄	—	212	4	no	—
130	C₄	#13133139	1	4	yes	—
131	C₄	#13133182	1	2	yes	—
132	C₄	#13133204	1	1	yes	characteristic
133	C₄	#13133215	1	1	yes	characteristic
134	C₂	#16810528	1	1	yes	characteristic
135	C₂	C₂×#13133139	1	1	yes	characteristic
136	C₂	#16810633	1	1	yes	characteristic
137	C₂	#16810904	1	2	yes	—
138	C₂	#16810909	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810386

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

#13133217

—

#13133217

C₄

yes

complemented

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

—

#13133124

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

Q₈

yes

complemented

C₄×Dic₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

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×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂²×C₅₃

C₂×SD₁₆

yes

characteristic

C₂²×C₅₃

#137

yes

—

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

#124

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#475

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

#340

—

C₅₃

#1988

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂²×C₈

C₂×D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#95

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄²

—

212

—

102

C₄²

C₄×D₅₃

yes

—

103

C₄²

C₂²⋊D₅₃

yes

—

104

C₂³

C₂×D₂₁₂

yes

characteristic

105

Q₈

—

212

—

106

C₂×C₄

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

212

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

#11719913

yes

—

114

C₂×C₄

D₄₂₄

yes

—

115

C₂×C₄

#11719921

yes

—

116

C₂×C₄

#11719921

yes

—

117

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

118

C₂×C₄

#11719947

yes

—

119

C₂×C₄

Q₈×D₅₃

yes

—

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

121

C₈

—

122

C₂²

C₂×#11719921

yes

characteristic

123

C₂²

#13133197

yes

—

124

C₂²

C₂×D₄₂₄

yes

characteristic

125

C₂²

C₂×#11719913

yes

characteristic

126

C₂²

#13133250

yes

—

127

C₄

—

128

C₄

—

212

—

129

C₄

—

212

—

130

C₄

#13133139

yes

—

131

C₄

#13133182

yes

—

132

C₄

#13133204

yes

characteristic

133

C₄

#13133215

yes

characteristic

134

C₂

#16810528

yes

characteristic

135

C₂

C₂×#13133139

yes

characteristic

136

C₂

#16810633

yes

characteristic

137

C₂

#16810904

yes

—

138

C₂

#16810909

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	#1988
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

#1988

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #13133217	C₄	semidirect
32. C₄⋊Dic₅₃	Q₈	semidirect
74. C₅₃	#1988	semidirect

H ≅ G/N

type

7. #13133217

C₄

semidirect

32. C₄⋊Dic₅₃

Q₈

semidirect

74. C₅₃

#1988

semidirect

Subgroups — #20275666

Conjugacy classes of subgroups

Complements