Group #20277056 — subgroups

Subgroups — C₄×#13133176 #20277056

Order 6784.

752 conjugacy classes of subgroups; only the 421 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 421 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×#13133176	C₂	1	2	yes	maximal
2	C₄×C₄⋊Dic₅₃	C₂	1	4	yes	maximal, complemented
3	C₄×C₂²⋊Dic₅₃	C₂	1	4	yes	maximal
4	#16810469	C₂	1	2	yes	maximal, complemented
5	C₄²×Dic₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic
7	#13133176	C₄	1	16	yes	direct product factor
8	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
10	C₂×C₄×Dic₅₃	C₂²	1	8	yes	—
11	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
12	#13133124	C₂²	1	8	yes	—
13	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₅₃	C₄	1	8	yes	complemented
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	—	2	8	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
28	C₂²×Dic₅₃	C₂³	1	8	yes	—
29	C₄²×C₅₃	C₂×C₄	1	4	yes	—
30	C₄²×C₅₃	C₂×C₄	1	8	yes	—
31	C₄²×C₅₃	C₂×C₄	1	4	yes	—
32	C₂²⋊Dic₅₃	C₂×C₄	1	16	yes	—
33	C₄⋊Dic₅₃	C₂×C₄	1	16	yes	complemented
34	C₄×Dic₅₃	—	2	16	no	—
35	C₄×Dic₅₃	C₂×C₄	1	16	yes	complemented
36	C₂³×C₅₃	—	2	4	no	—
37	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	2	16	no	—
42	C₂×C₄×C₅₃	C₄²	1	16	yes	complemented
43	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₅₃	#41	1	8	yes	—
50	C₂×Dic₅₃	—	2	16	no	—
51	C₂×Dic₅₃	C₂²×C₄	1	16	yes	—
52	C₂²×C₅₃	—	2	4	no	—
53	C₂²×C₅₃	—	2	8	no	—
54	C₂²×C₅₃	C₂×C₄²	1	4	yes	—
55	C₂²×C₅₃	C₂³×C₄	1	2	yes	—
56	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×#41	1	4	yes	—
58	C₄×C₅₃	—	2	4	no	—
59	C₄×C₅₃	C₂×C₄²	1	4	yes	—
60	C₄×C₅₃	C₂×C₄²	1	4	yes	—
61	C₄×C₅₃	#117	1	8	yes	complemented
62	Dic₅₃	—	2	16	no	—
63	C₄×#117	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	2	4	no	—
65	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#117	1	2	yes	—
67	C₂×C₅₃	C₄×#41	1	4	yes	—
68	C₂×#117	—	53	2	no	—
69	C₂²×C₄²	D₅₃	1	1	yes	characteristic
70	C₄×C₄⋊C₄	—	53	4	no	—
71	C₄×C₂²⋊C₄	—	53	4	no	—
72	#376	—	53	2	no	—
73	C₄³	—	53	2	no	—
74	C₅₃	C₄×#117	1	1	yes	characteristic, complemented, Sylow
75	C₂³×C₄	C₂×D₅₃	1	2	yes	—
76	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
77	#117	—	53	16	no	—
78	C₂×C₄⋊C₄	—	53	4	no	—
79	C₂×C₂²⋊C₄	—	53	4	no	—
80	C₂×C₄²	—	53	4	no	—
81	C₂×C₄²	—	53	8	no	—
82	C₂×C₄²	Dic₅₃	1	8	yes	complemented
83	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
84	C₂×C₄²	C₂×D₅₃	1	2	yes	—
85	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
86	#95	—	53	8	no	—
87	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
88	C₂²×C₄	—	2	8	no	—
89	C₂²×C₄	—	53	8	no	—
90	C₂²×C₄	C₄×D₅₃	1	8	yes	—
91	C₂²×C₄	C₂×Dic₅₃	1	8	yes	—
92	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
93	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
94	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
97	C₄⋊C₄	—	53	16	no	—
98	C₂²⋊C₄	—	53	16	no	—
99	C₄²	—	53	16	no	—
100	C₄²	—	106	16	no	—
101	C₄²	C₂×Dic₅₃	1	4	yes	—
102	C₄²	C₂×Dic₅₃	1	8	yes	—
103	C₄²	C₂×Dic₅₃	1	4	yes	—
104	C₂³	—	2	4	no	—
105	C₂³	C₂×C₄×D₅₃	1	4	yes	—
106	C₂³	C₂²×Dic₅₃	1	2	yes	—
107	C₂³	C₂³×D₅₃	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	16	no	—
110	C₂×C₄	—	53	16	no	—
111	C₂×C₄	—	106	16	no	—
112	C₂×C₄	C₄×Dic₅₃	1	16	yes	complemented
113	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
114	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
115	C₂×C₄	#11719945	1	8	yes	—
116	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
117	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
118	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
119	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
120	C₂²	—	2	4	no	—
121	C₂²	—	2	8	no	—
122	C₂²	C₂×C₄×Dic₅₃	1	4	yes	—
123	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
124	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
125	C₂²	C₂×#11719945	1	4	yes	—
126	C₄	—	2	4	no	—
127	C₄	—	106	16	no	—
128	C₄	C₂×C₄×Dic₅₃	1	4	yes	—
129	C₄	C₂×C₄×Dic₅₃	1	4	yes	—
130	C₄	#13133176	1	8	yes	direct product factor
131	C₂	—	2	4	no	—
132	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
133	C₂	C₂×#13133176	1	2	yes	—
134	C₂	C₄×#11719945	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133176

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₅₃

C₂

yes

maximal

#16810469

C₂

yes

maximal, complemented

C₄²×Dic₅₃

C₂

yes

maximal, complemented

C₂²×C₄²×C₅₃

C₂

yes

maximal, characteristic

#13133176

C₄

yes

direct product factor

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

complemented

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₄

yes

complemented

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

C₂×C₄²

yes

—

C₄×C₅₃

#117

yes

complemented

Dic₅₃

—

C₄×#117

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

C₂×#117

yes

—

C₂×C₅₃

C₄×#41

yes

—

C₂×#117

—

C₂²×C₄²

D₅₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

#376

—

C₄³

—

C₅₃

C₄×#117

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

Dic₅₃

yes

complemented

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#95

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

106

—

101

C₄²

C₂×Dic₅₃

yes

—

102

C₄²

C₂×Dic₅₃

yes

—

103

C₄²

C₂×Dic₅₃

yes

—

104

C₂³

—

105

C₂³

C₂×C₄×D₅₃

yes

—

106

C₂³

C₂²×Dic₅₃

yes

—

107

C₂³

C₂³×D₅₃

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

C₄×Dic₅₃

yes

complemented

113

C₂×C₄

C₂×C₄×D₅₃

yes

—

114

C₂×C₄

C₂×C₄×D₅₃

yes

—

115

C₂×C₄

#11719945

yes

—

116

C₂×C₄

C₂²×Dic₅₃

yes

—

117

C₂×C₄

C₂²×Dic₅₃

yes

—

118

C₂×C₄

C₂²×Dic₅₃

yes

—

119

C₂×C₄

C₂²×Dic₅₃

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄×Dic₅₃

yes

—

123

C₂²

C₂²×C₄×D₅₃

yes

—

124

C₂²

C₂³×Dic₅₃

yes

characteristic

125

C₂²

C₂×#11719945

yes

—

126

C₄

—

127

C₄

—

106

—

128

C₄

C₂×C₄×Dic₅₃

yes

—

129

C₄

C₂×C₄×Dic₅₃

yes

—

130

C₄

#13133176

yes

direct product factor

131

C₂

—

132

C₂

C₂²×C₄×Dic₅₃

yes

characteristic

133

C₂

C₂×#13133176

yes

—

134

C₂

C₄×#11719945

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Frattini	C₂³	C₂³×D₅₃
Fitting	C₂²×C₄²×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₄²	D₅₃
hypocenter	C₅₃	C₄×#117
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

subgroup

G/H

center

C₂×C₄²

C₂×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Frattini

C₂³

C₂³×D₅₃

Fitting

C₂²×C₄²×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂²×C₄²

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄² ◁ C₂²×C₄²

N	H ≅ G/N	type
2. C₄×C₄⋊Dic₅₃	C₂	semidirect
4. #16810469	C₂	semidirect
5. C₄²×Dic₅₃	C₂	semidirect
7. #13133176	C₄	direct (N × H)
15. C₂×C₄²×C₅₃	C₄	semidirect
33. C₄⋊Dic₅₃	C₂×C₄	semidirect
35. C₄×Dic₅₃	C₂×C₄	semidirect
42. C₂×C₄×C₅₃	C₄²	semidirect
61. C₄×C₅₃	#117	semidirect
74. C₅₃	C₄×#117	semidirect
82. C₂×C₄²	Dic₅₃	semidirect
112. C₂×C₄	C₄×Dic₅₃	semidirect
130. C₄	#13133176	direct (N × H)

H ≅ G/N

type

2. C₄×C₄⋊Dic₅₃

C₂

semidirect

4. #16810469

C₂

semidirect

5. C₄²×Dic₅₃

C₂

semidirect

7. #13133176

C₄

direct (N × H)

15. C₂×C₄²×C₅₃

C₄

semidirect

33. C₄⋊Dic₅₃

C₂×C₄

semidirect

35. C₄×Dic₅₃

C₂×C₄

semidirect

42. C₂×C₄×C₅₃

C₄²

semidirect

61. C₄×C₅₃

#117

semidirect

74. C₅₃

C₄×#117

semidirect

82. C₂×C₄²

Dic₅₃

semidirect

112. C₂×C₄

C₄×Dic₅₃

semidirect

130. C₄

#13133176

direct (N × H)

Subgroups — C₄×#13133176 #20277056

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13133176 #20277056

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133176 #20277056