Group #20271208 — subgroups

Subgroups — C₄²×D₄×C₅₃ #20271208

Order 6784.

924 conjugacy classes of subgroups; only the 576 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 576 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×D₄×C₅₃	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₅₃	C₂	1	2	yes	maximal, complemented
3	C₄×C₄⋊C₄×C₅₃	C₂	1	3	yes	maximal, complemented
4	C₄×C₂²⋊C₄×C₅₃	C₂	1	6	yes	maximal, complemented
5	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
7	C₂³×C₄×C₅₃	C₂²	1	6	yes	—
8	C₄×D₄×C₅₃	C₄	1	24	yes	direct product factor
9	C₂×C₄⋊C₄×C₅₃	C₂²	1	3	yes	—
10	C₂×C₂²⋊C₄×C₅₃	C₂²	1	6	yes	—
11	C₂×C₄²×C₅₃	—	2	8	no	—
12	C₂×C₄²×C₅₃	C₂²	1	6	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	#95×C₅₃	C₂²	1	6	yes	—
17	C₂⁴×C₅₃	C₂³	1	2	yes	—
18	C₂×D₄×C₅₃	C₂×C₄	1	12	yes	—
19	C₂²×C₄×C₅₃	—	2	12	no	—
20	C₂²×C₄×C₅₃	—	2	12	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	24	yes	complemented
22	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	6	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
26	C₄⋊C₄×C₅₃	C₂×C₄	1	12	yes	complemented
27	C₂²⋊C₄×C₅₃	C₂×C₄	1	24	yes	complemented
28	C₄²×C₅₃	—	2	12	no	—
29	C₄²×C₅₃	—	2	6	no	—
30	C₄²×C₅₃	C₂×C₄	1	12	yes	complemented
31	C₄²×C₅₃	D₄	1	4	yes	direct product factor
32	C₂³×C₅₃	—	2	2	no	—
33	C₂³×C₅₃	—	2	6	no	—
34	C₂³×C₅₃	C₂²×C₄	1	12	yes	—
35	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
36	D₄×C₅₃	C₄²	1	16	yes	direct product factor
37	C₂×C₄×C₅₃	—	2	6	no	—
38	C₂×C₄×C₅₃	—	2	6	no	—
39	C₂×C₄×C₅₃	—	2	6	no	—
40	C₂×C₄×C₅₃	—	2	48	no	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	12	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	6	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	6	yes	—
46	C₂×C₄×C₅₃	#41	1	6	yes	—
47	C₂²×C₅₃	—	2	12	no	—
48	C₂²×C₅₃	—	2	12	no	—
49	C₂²×C₅₃	C₂×C₄²	1	8	yes	complemented
50	C₂²×C₅₃	C₂³×C₄	1	3	yes	—
51	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
52	C₂²×C₅₃	C₂×#41	1	3	yes	—
53	C₄×C₅₃	—	2	12	no	—
54	C₄×C₅₃	—	2	6	no	—
55	C₄×C₅₃	C₂×C₄²	1	4	yes	complemented
56	C₄×C₅₃	C₄×D₄	1	12	yes	direct product factor
57	C₄²×D₄	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₅₃	—	2	8	no	—
59	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
60	C₂×C₅₃	C₂×C₄×D₄	1	3	yes	—
61	C₂×C₅₃	C₄×#41	1	3	yes	—
62	C₂×C₄×D₄	C₂×C₅₃	1	3	yes	—
63	C₂²×C₄²	C₂×C₅₃	1	2	yes	complemented
64	C₄×C₄⋊C₄	C₂×C₅₃	1	3	yes	complemented
65	C₄×C₂²⋊C₄	C₂×C₅₃	1	6	yes	complemented
66	C₄³	C₂×C₅₃	1	1	yes	characteristic, complemented
67	C₅₃	C₄²×D₄	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×D₄	C₂²×C₅₃	1	1	yes	characteristic
69	C₂³×C₄	C₂²×C₅₃	1	6	yes	—
70	C₄×D₄	C₄×C₅₃	1	24	yes	direct product factor
71	C₂×C₄⋊C₄	C₂²×C₅₃	1	3	yes	—
72	C₂×C₂²⋊C₄	C₂²×C₅₃	1	6	yes	—
73	C₂×C₄²	—	2	8	no	—
74	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
75	C₂×C₄²	C₂²×C₅₃	1	6	yes	—
76	C₂×C₄²	C₂²×C₅₃	1	3	yes	—
77	C₂×C₄²	C₂²×C₅₃	1	1	yes	characteristic
78	#95	C₂²×C₅₃	1	6	yes	—
79	C₂⁴	C₂³×C₅₃	1	2	yes	—
80	C₂×D₄	C₂×C₄×C₅₃	1	12	yes	—
81	C₂²×C₄	—	2	12	no	—
82	C₂²×C₄	—	2	12	no	—
83	C₂²×C₄	C₂×C₄×C₅₃	1	24	yes	complemented
84	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
85	C₂²×C₄	C₂³×C₅₃	1	6	yes	—
86	C₂²×C₄	C₂³×C₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂³×C₅₃	1	3	yes	—
88	C₄⋊C₄	C₂×C₄×C₅₃	1	12	yes	complemented
89	C₂²⋊C₄	C₂×C₄×C₅₃	1	24	yes	complemented
90	C₄²	—	2	12	no	—
91	C₄²	—	2	6	no	—
92	C₄²	C₂×C₄×C₅₃	1	12	yes	complemented
93	C₄²	D₄×C₅₃	1	4	yes	direct product factor
94	C₂³	—	2	2	no	—
95	C₂³	—	2	6	no	—
96	C₂³	C₂²×C₄×C₅₃	1	12	yes	—
97	C₂³	C₂⁴×C₅₃	1	1	yes	characteristic
98	D₄	C₄²×C₅₃	1	16	yes	direct product factor
99	C₂×C₄	—	2	6	no	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	6	no	—
102	C₂×C₄	—	2	48	no	—
103	C₂×C₄	C₂²×C₄×C₅₃	1	6	yes	—
104	C₂×C₄	C₂²×C₄×C₅₃	1	12	yes	—
105	C₂×C₄	C₂²×C₄×C₅₃	1	6	yes	—
106	C₂×C₄	C₂²×C₄×C₅₃	1	6	yes	—
107	C₂×C₄	C₂×D₄×C₅₃	1	6	yes	—
108	C₂×C₄	#41×C₅₃	1	6	yes	—
109	C₂²	—	2	12	no	—
110	C₂²	—	2	12	no	—
111	C₂²	C₂×C₄²×C₅₃	1	8	yes	complemented
112	C₂²	C₂³×C₄×C₅₃	1	3	yes	—
113	C₂²	C₂²×D₄×C₅₃	1	1	yes	characteristic
114	C₂²	C₂×#41×C₅₃	1	3	yes	—
115	C₄	—	2	12	no	—
116	C₄	—	2	6	no	—
117	C₄	C₂×C₄²×C₅₃	1	4	yes	complemented
118	C₄	C₄×D₄×C₅₃	1	12	yes	direct product factor
119	C₂	—	2	8	no	—
120	C₂	C₂²×C₄²×C₅₃	1	1	yes	characteristic
121	C₂	C₂×C₄×D₄×C₅₃	1	3	yes	—
122	C₂	C₄×#41×C₅₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×D₄×C₅₃

C₂

yes

maximal

C₂²×C₄²×C₅₃

C₂

yes

maximal, complemented

C₄×C₄⋊C₄×C₅₃

C₂

yes

maximal, complemented

C₄×C₂²⋊C₄×C₅₃

C₂

yes

maximal, complemented

C₄³×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₂²

yes

—

C₄×D₄×C₅₃

C₄

yes

direct product factor

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#95×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

complemented

C₂²⋊C₄×C₅₃

C₂×C₄

yes

complemented

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

complemented

C₄²×C₅₃

D₄

yes

direct product factor

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

D₄×C₅₃

C₄²

yes

direct product factor

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

complemented

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂×#41

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄²

yes

complemented

C₄×C₅₃

C₄×D₄

yes

direct product factor

C₄²×D₄

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

C₂×C₄×D₄

yes

—

C₂×C₅₃

C₄×#41

yes

—

C₂×C₄×D₄

C₂×C₅₃

yes

—

C₂²×C₄²

C₂×C₅₃

yes

complemented

C₄×C₄⋊C₄

C₂×C₅₃

yes

complemented

C₄×C₂²⋊C₄

C₂×C₅₃

yes

complemented

C₄³

C₂×C₅₃

yes

characteristic, complemented

C₅₃

C₄²×D₄

yes

characteristic, direct product factor, Sylow

C₂²×D₄

C₂²×C₅₃

yes

characteristic

C₂³×C₄

C₂²×C₅₃

yes

—

C₄×D₄

C₄×C₅₃

yes

direct product factor

C₂×C₄⋊C₄

C₂²×C₅₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₅₃

yes

—

C₂×C₄²

C₂²×C₅₃

yes

—

C₂×C₄²

C₂²×C₅₃

yes

—

C₂×C₄²

C₂²×C₅₃

yes

characteristic

#95

C₂²×C₅₃

yes

—

C₂⁴

C₂³×C₅₃

yes

—

C₂×D₄

C₂×C₄×C₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₅₃

yes

complemented

C₂²×C₄

C₂³×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

—

C₂²×C₄

C₂³×C₅₃

yes

characteristic

C₂²×C₄

C₂³×C₅₃

yes

—

C₄⋊C₄

C₂×C₄×C₅₃

yes

complemented

C₂²⋊C₄

C₂×C₄×C₅₃

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂×C₄×C₅₃

yes

complemented

C₄²

D₄×C₅₃

yes

direct product factor

C₂³

—

C₂³

—

C₂³

C₂²×C₄×C₅₃

yes

—

C₂³

C₂⁴×C₅₃

yes

characteristic

D₄

C₄²×C₅₃

yes

direct product factor

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×C₄×C₅₃

yes

—

104

C₂×C₄

C₂²×C₄×C₅₃

yes

—

105

C₂×C₄

C₂²×C₄×C₅₃

yes

—

106

C₂×C₄

C₂²×C₄×C₅₃

yes

—

107

C₂×C₄

C₂×D₄×C₅₃

yes

—

108

C₂×C₄

#41×C₅₃

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄²×C₅₃

yes

complemented

112

C₂²

C₂³×C₄×C₅₃

yes

—

113

C₂²

C₂²×D₄×C₅₃

yes

characteristic

114

C₂²

C₂×#41×C₅₃

yes

—

115

C₄

—

116

C₄

—

117

C₄

C₂×C₄²×C₅₃

yes

complemented

118

C₄

C₄×D₄×C₅₃

yes

direct product factor

119

C₂

—

120

C₂

C₂²×C₄²×C₅₃

yes

characteristic

121

C₂

C₂×C₄×D₄×C₅₃

yes

—

122

C₂

C₄×#41×C₅₃

yes

—

	subgroup	G/H
center	C₂×C₄²×C₅₃	C₂²
commutator	C₂	C₂²×C₄²×C₅₃
Frattini, cosocle	C₂³	C₂⁴×C₅₃
socle	C₂³×C₅₃	C₂⁴
prime radical	C₂²×D₄×C₅₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄²×C₅₃

C₂²

commutator

C₂

C₂²×C₄²×C₅₃

Frattini, cosocle

C₂³

C₂⁴×C₅₃

socle

C₂³×C₅₃

C₂⁴

prime radical

C₂²×D₄×C₅₃

C₂²

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₂²×C₄²×C₅₃	C₂	semidirect
3. C₄×C₄⋊C₄×C₅₃	C₂	semidirect
4. C₄×C₂²⋊C₄×C₅₃	C₂	semidirect
5. C₄³×C₅₃	C₂	semidirect
8. C₄×D₄×C₅₃	C₄	direct (N × H)
21. C₂²×C₄×C₅₃	C₂×C₄	semidirect
26. C₄⋊C₄×C₅₃	C₂×C₄	semidirect
27. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
30. C₄²×C₅₃	C₂×C₄	semidirect
31. C₄²×C₅₃	D₄	direct (N × H)
36. D₄×C₅₃	C₄²	direct (N × H)
49. C₂²×C₅₃	C₂×C₄²	semidirect
55. C₄×C₅₃	C₂×C₄²	semidirect
56. C₄×C₅₃	C₄×D₄	direct (N × H)
57. C₄²×D₄	C₅₃	direct (N × H)
63. C₂²×C₄²	C₂×C₅₃	semidirect
64. C₄×C₄⋊C₄	C₂×C₅₃	semidirect
65. C₄×C₂²⋊C₄	C₂×C₅₃	semidirect
66. C₄³	C₂×C₅₃	semidirect
67. C₅₃	C₄²×D₄	direct (N × H)
70. C₄×D₄	C₄×C₅₃	direct (N × H)
83. C₂²×C₄	C₂×C₄×C₅₃	semidirect
88. C₄⋊C₄	C₂×C₄×C₅₃	semidirect
89. C₂²⋊C₄	C₂×C₄×C₅₃	semidirect
92. C₄²	C₂×C₄×C₅₃	semidirect
93. C₄²	D₄×C₅₃	direct (N × H)
98. D₄	C₄²×C₅₃	direct (N × H)
111. C₂²	C₂×C₄²×C₅₃	semidirect
117. C₄	C₂×C₄²×C₅₃	semidirect
118. C₄	C₄×D₄×C₅₃	direct (N × H)

H ≅ G/N

type

2. C₂²×C₄²×C₅₃

C₂

semidirect

3. C₄×C₄⋊C₄×C₅₃

C₂

semidirect

4. C₄×C₂²⋊C₄×C₅₃

C₂

semidirect

5. C₄³×C₅₃

C₂

semidirect

8. C₄×D₄×C₅₃

C₄

direct (N × H)

21. C₂²×C₄×C₅₃

C₂×C₄

semidirect

26. C₄⋊C₄×C₅₃

C₂×C₄

semidirect

27. C₂²⋊C₄×C₅₃

C₂×C₄

semidirect

30. C₄²×C₅₃

C₂×C₄

semidirect

31. C₄²×C₅₃

D₄

direct (N × H)

36. D₄×C₅₃

C₄²

direct (N × H)

49. C₂²×C₅₃

C₂×C₄²

semidirect

55. C₄×C₅₃

C₂×C₄²

semidirect

56. C₄×C₅₃

C₄×D₄

direct (N × H)

57. C₄²×D₄

C₅₃

direct (N × H)

63. C₂²×C₄²

C₂×C₅₃

semidirect

64. C₄×C₄⋊C₄

C₂×C₅₃

semidirect

65. C₄×C₂²⋊C₄

C₂×C₅₃

semidirect

66. C₄³

C₂×C₅₃

semidirect

67. C₅₃

C₄²×D₄

direct (N × H)

70. C₄×D₄

C₄×C₅₃

direct (N × H)

83. C₂²×C₄

C₂×C₄×C₅₃

semidirect

88. C₄⋊C₄

C₂×C₄×C₅₃

semidirect

89. C₂²⋊C₄

C₂×C₄×C₅₃

semidirect

92. C₄²

C₂×C₄×C₅₃

semidirect

93. C₄²

D₄×C₅₃

direct (N × H)

98. D₄

C₄²×C₅₃

direct (N × H)

111. C₂²

C₂×C₄²×C₅₃

semidirect

117. C₄

C₂×C₄²×C₅₃

semidirect

118. C₄

C₄×D₄×C₅₃

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×D₄×C₅₃	2
C₂²×C₄²×C₅₃	2
C₄×C₄⋊C₄×C₅₃	2
C₄×C₂²⋊C₄×C₅₃	2
C₄³×C₅₃	2
C₄²×D₄	53

subgroup

index

C₂×C₄×D₄×C₅₃

C₂²×C₄²×C₅₃

C₄×C₄⋊C₄×C₅₃

C₄×C₂²⋊C₄×C₅₃

C₄³×C₅₃

C₄²×D₄

Subgroups — C₄²×D₄×C₅₃ #20271208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C42×D4×C53 #20271208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×D₄×C₅₃ #20271208