Group #20274565 — subgroups

Subgroups — C₄×#13133146 #20274565

Order 6784.

316 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133146	C₂	1	2	yes	maximal
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
11	#13133146	C₄	1	8	yes	direct product factor
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	2	yes	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₄²×C₅₃	D₄	1	2	yes	—
25	C₄²×C₅₃	Q₈	1	2	yes	—
26	C₄×Dic₅₃	—	2	4	no	—
27	C₄×Dic₅₃	—	2	2	no	—
28	C₄×Dic₅₃	C₈	1	8	yes	complemented
29	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂×#88322	C₂×C₄	1	4	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	2	yes	—
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	C₄²	1	2	yes	—
45	C₂×Dic₅₃	C₄²	1	2	yes	—
46	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
47	C₂×Dic₅₃	C₂×C₈	1	4	yes	—
48	C₈×C₅₃	—	2	4	no	—
49	#88322	—	2	4	no	—
50	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
51	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₅₃	#117	1	1	yes	characteristic
53	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
54	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
55	C₄×C₅₃	C₄⋊C₈	1	8	yes	complemented
56	C₄×C₅₃	C₄×D₄	1	2	yes	—
57	C₄×C₅₃	C₄×Q₈	1	2	yes	—
58	Dic₅₃	—	2	2	no	—
59	Dic₅₃	—	2	4	no	—
60	Dic₅₃	C₄×C₈	1	4	yes	complemented
61	C₄×C₄⋊C₈	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
64	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
65	C₂×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
66	C₂×C₅₃	#431	1	2	yes	—
67	C₂×C₄⋊C₈	—	53	2	no	—
68	C₂×C₄×C₈	—	53	1	no	—
69	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
70	C₄³	—	53	1	no	—
71	#336	—	53	2	no	—
72	C₅₃	C₄×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	53	2	no	—
74	C₂²×C₈	C₂×D₅₃	1	2	yes	—
75	C₂×C₄²	—	53	2	no	—
76	C₂×C₄²	—	53	1	no	—
77	C₂×C₄²	—	53	2	no	—
78	C₂×C₄²	—	53	1	no	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	—	53	8	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	106	2	no	—
83	C₂²×C₄	—	53	1	no	—
84	C₂²×C₄	—	53	1	no	—
85	C₂²×C₄	—	53	2	no	—
86	C₂²×C₄	C₄×D₅₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	53	4	no	—
91	C₂×C₈	—	106	2	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	C₄×D₅₃	1	4	yes	—
94	C₄²	—	53	4	no	—
95	C₄²	—	53	8	no	—
96	C₄²	—	106	2	no	—
97	C₄²	—	106	4	no	—
98	C₄²	Dic₁₀₆	1	2	yes	—
99	C₄²	C₂²⋊D₅₃	1	2	yes	—
100	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	53	4	no	—
103	C₂×C₄	—	53	2	no	—
104	C₂×C₄	—	53	4	no	—
105	C₂×C₄	—	106	2	no	—
106	C₂×C₄	—	106	2	no	—
107	C₂×C₄	—	106	2	no	—
108	C₂×C₄	C₈×D₅₃	1	4	yes	—
109	C₂×C₄	#11719912	1	4	yes	—
110	C₂×C₄	#11719919	1	4	yes	—
111	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
112	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
113	C₂×C₄	#11719945	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₈	—	106	4	no	—
117	C₂²	C₄²×D₅₃	1	1	yes	characteristic
118	C₂²	C₂×#11719919	1	1	yes	characteristic
119	C₂²	#13133174	1	1	yes	characteristic
120	C₂²	C₂×C₈×D₅₃	1	2	yes	—
121	C₂²	C₂×#11719912	1	2	yes	—
122	C₄	—	53	4	no	—
123	C₄	—	106	2	no	—
124	C₄	—	106	4	no	—
125	C₄	#13133146	1	8	yes	direct product factor
126	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
127	C₄	C₄×Dic₁₀₆	1	2	yes	—
128	C₂	C₄×#11719919	1	1	yes	characteristic
129	C₂	C₄×C₈×D₅₃	1	1	yes	characteristic
130	C₂	C₄×#11719912	1	1	yes	characteristic
131	C₂	C₂×#13133146	1	2	yes	—
132	C₂	#16810674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133146

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₄²×Dic₅₃

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

C₄

yes

direct product factor

C₄×#88322

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₈

yes

complemented

C₄×Dic₅₃

C₂×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₄²

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

C₄⋊C₈

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₄×C₈

yes

complemented

C₄×C₄⋊C₈

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

characteristic

C₂×C₅₃

C₂×C₄⋊C₈

yes

—

C₂×C₅₃

#431

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₄³

—

#336

—

C₅₃

C₄×C₄⋊C₈

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₄²

—

106

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₂²⋊D₅₃

yes

—

100

C₂³

C₂×C₄×D₅₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

C₈×D₅₃

yes

—

109

C₂×C₄

#11719912

yes

—

110

C₂×C₄

#11719919

yes

—

111

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

112

C₂×C₄

C₂×C₄×D₅₃

yes

—

113

C₂×C₄

#11719945

yes

—

114

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

115

C₈

—

116

C₈

—

106

—

117

C₂²

C₄²×D₅₃

yes

characteristic

118

C₂²

C₂×#11719919

yes

characteristic

119

C₂²

#13133174

yes

characteristic

120

C₂²

C₂×C₈×D₅₃

yes

—

121

C₂²

C₂×#11719912

yes

—

122

C₄

—

123

C₄

—

106

—

124

C₄

—

106

—

125

C₄

#13133146

yes

direct product factor

126

C₄

C₄×C₂²⋊D₅₃

yes

—

127

C₄

C₄×Dic₁₀₆

yes

—

128

C₂

C₄×#11719919

yes

characteristic

129

C₂

C₄×C₈×D₅₃

yes

characteristic

130

C₂

C₄×#11719912

yes

characteristic

131

C₂

C₂×#13133146

yes

—

132

C₂

#16810674

yes

—

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

subgroup

G/H

center

C₂×C₄²

C₂×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₄×C₄⋊C₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
11. #13133146	C₄	direct (N × H)
28. C₄×Dic₅₃	C₈	semidirect
55. C₄×C₅₃	C₄⋊C₈	semidirect
60. Dic₅₃	C₄×C₈	semidirect
72. C₅₃	C₄×C₄⋊C₈	semidirect
125. C₄	#13133146	direct (N × H)

H ≅ G/N

type

11. #13133146

C₄

direct (N × H)

28. C₄×Dic₅₃

C₈

semidirect

55. C₄×C₅₃

C₄⋊C₈

semidirect

60. Dic₅₃

C₄×C₈

semidirect

72. C₅₃

C₄×C₄⋊C₈

semidirect

125. C₄

#13133146

direct (N × H)

Subgroups — C₄×#13133146 #20274565

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13133146 #20274565

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133146 #20274565