Group #20277068 — subgroups

Subgroups — #20277068

Order 6784.

912 conjugacy classes of subgroups; only the 389 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 389 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	C₂²×C₂²⋊Dic₅₃	C₂	1	2	yes	maximal
2	C₂²⋊C₄×Dic₅₃	C₂	1	8	yes	maximal, complemented
3	C₄×C₂²⋊Dic₅₃	C₂	1	4	yes	maximal, complemented
4	C₂²×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₅₃	C₂²	1	4	yes	—
6	C₂×C₂²⋊Dic₅₃	C₄	1	16	yes	complemented
7	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
8	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	8	yes	complemented
10	#13133124	C₂²	1	8	yes	complemented
11	C₂⁵×C₅₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₅₃	C₄	1	8	yes	complemented
14	C₂×C₂²⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂⁴×C₅₃	—	4	4	no	—
16	C₂⁴×C₅₃	C₂×C₄	1	8	yes	—
17	C₂⁴×C₅₃	C₂×C₄	1	4	yes	—
18	C₂⁴×C₅₃	C₂³	1	2	yes	—
19	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	—	4	4	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	complemented
22	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
23	C₂²×Dic₅₃	—	4	8	no	—
24	C₂²×Dic₅₃	C₂×C₄	1	16	yes	complemented
25	C₂²×Dic₅₃	C₂³	1	8	yes	—
26	C₂²⋊C₄×C₅₃	C₂×C₄	1	16	yes	complemented
27	C₂²⋊Dic₅₃	C₂×C₄	1	16	yes	complemented
28	C₂²⋊Dic₅₃	C₂×C₄	1	16	yes	complemented
29	C₄×Dic₅₃	—	4	8	no	—
30	C₂³×C₅₃	—	2	8	no	—
31	C₂³×C₅₃	—	2	4	no	—
32	C₂³×C₅₃	—	4	16	no	—
33	C₂³×C₅₃	—	4	8	no	—
34	C₂³×C₅₃	C₄²	1	16	yes	complemented
35	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
36	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
37	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
40	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	8	no	—
42	C₂×C₄×C₅₃	—	4	8	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
44	C₂×Dic₅₃	—	2	16	no	—
45	C₂×Dic₅₃	—	4	16	no	—
46	C₂×Dic₅₃	C₂²×C₄	1	16	yes	—
47	C₂²×C₅₃	—	2	8	no	—
48	C₂²×C₅₃	—	2	4	no	—
49	C₂²×C₅₃	—	2	8	no	—
50	C₂²×C₅₃	—	2	8	no	—
51	C₂²×C₅₃	—	2	8	no	—
52	C₂²×C₅₃	—	4	16	no	—
53	C₂²×C₅₃	C₂×C₄²	1	8	yes	—
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	H(5,𝔽₂)	1	4	yes	—
58	C₄×C₅₃	—	2	8	no	—
59	Dic₅₃	—	2	16	no	—
60	#2273	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	8	no	—
62	C₂×C₅₃	—	2	4	no	—
63	C₂×C₅₃	C₂²×C₄²	1	1	yes	characteristic
64	C₂×C₅₃	#518	1	4	yes	—
65	C₂×C₅₃	#518	1	2	yes	—
66	C₂²×C₂²⋊C₄	—	53	2	no	—
67	C₂²×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
68	C₄×C₂²⋊C₄	—	53	4	no	—
69	C₄×C₂²⋊C₄	—	53	8	no	—
70	C₅₃	#2273	1	1	yes	characteristic, complemented, Sylow
71	C₂⁵	C₂×D₅₃	1	1	yes	characteristic
72	C₂³×C₄	—	53	4	no	—
73	C₂³×C₄	C₂×D₅₃	1	2	yes	—
74	C₂×C₂²⋊C₄	—	53	16	no	—
75	C₂×C₂²⋊C₄	—	53	4	no	—
76	C₂×C₂²⋊C₄	—	53	4	no	—
77	C₂×C₂²⋊C₄	Dic₅₃	1	8	yes	complemented
78	C₂×C₂²⋊C₄	C₂×D₅₃	1	4	yes	—
79	C₂×C₄²	—	53	8	no	—
80	#95	—	53	8	no	—
81	C₂⁴	—	4	4	no	—
82	C₂⁴	C₄×D₅₃	1	8	yes	—
83	C₂⁴	C₂×Dic₅₃	1	4	yes	—
84	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
85	C₂⁴	C₂²×D₅₃	1	2	yes	—
86	C₂²×C₄	—	4	4	no	—
87	C₂²×C₄	—	53	8	no	—
88	C₂²×C₄	—	53	16	no	—
89	C₂²×C₄	—	212	8	no	—
90	C₂²×C₄	C₂×Dic₅₃	1	8	yes	complemented
91	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
92	C₂²⋊C₄	—	53	16	no	—
93	C₂²⋊C₄	—	53	16	no	—
94	C₂²⋊C₄	C₂×Dic₅₃	1	16	yes	complemented
95	C₄²	—	212	8	no	—
96	C₂³	—	2	4	no	—
97	C₂³	—	2	8	no	—
98	C₂³	—	4	16	no	—
99	C₂³	—	4	8	no	—
100	C₂³	C₄×Dic₅₃	1	16	yes	complemented
101	C₂³	C₂×C₄×D₅₃	1	4	yes	—
102	C₂³	C₂×C₄×D₅₃	1	4	yes	—
103	C₂³	C₂×C₄×D₅₃	1	4	yes	—
104	C₂³	C₂²×Dic₅₃	1	4	yes	—
105	C₂³	C₂²×Dic₅₃	1	2	yes	—
106	C₂³	C₂³×D₅₃	1	1	yes	characteristic
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	4	8	no	—
109	C₂×C₄	—	53	16	no	—
110	C₂×C₄	—	106	16	no	—
111	C₂×C₄	—	212	16	no	—
112	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
113	C₂²	—	2	8	no	—
114	C₂²	—	2	8	no	—
115	C₂²	—	2	8	no	—
116	C₂²	—	2	4	no	—
117	C₂²	—	2	8	no	—
118	C₂²	—	4	16	no	—
119	C₂²	C₂×C₄×Dic₅₃	1	4	yes	—
120	C₂²	C₂×C₄×Dic₅₃	1	8	yes	—
121	C₂²	C₂²×C₄×D₅₃	1	2	yes	—
122	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
123	C₂²	#13133262	1	4	yes	—
124	C₄	—	2	8	no	—
125	C₄	—	106	16	no	—
126	C₂	—	2	8	no	—
127	C₂	—	2	4	no	—
128	C₂	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
129	C₂	#16811096	1	4	yes	—
130	C₂	#16811098	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊Dic₅₃

C₂

yes

maximal

C₂²⋊C₄×Dic₅₃

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₅₃

C₂

yes

maximal, complemented

C₂²×C₂²⋊C₄×C₅₃

C₂

yes

maximal, characteristic

C₂³×Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₄

yes

complemented

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

complemented

#13133124

C₂²

yes

complemented

C₂⁵×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

C₄

yes

complemented

C₂×C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂×C₄

yes

complemented

C₂²×Dic₅₃

C₂³

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

complemented

C₂²⋊Dic₅₃

C₂×C₄

yes

complemented

C₂²⋊Dic₅₃

C₂×C₄

yes

complemented

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

complemented

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂×C₄²

yes

—

C₂²×C₅₃

C₂³×C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₄×C₅₃

—

Dic₅₃

—

#2273

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄²

yes

characteristic

C₂×C₅₃

#518

yes

—

C₂×C₅₃

#518

yes

—

C₂²×C₂²⋊C₄

—

C₂²×C₂²⋊C₄

D₅₃

yes

characteristic

C₄×C₂²⋊C₄

—

C₄×C₂²⋊C₄

—

C₅₃

#2273

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₅₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

Dic₅₃

yes

complemented

C₂×C₂²⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

#95

—

C₂⁴

—

C₂⁴

C₄×D₅₃

yes

—

C₂⁴

C₂×Dic₅₃

yes

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂⁴

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×Dic₅₃

yes

complemented

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×Dic₅₃

yes

complemented

C₄²

—

212

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

C₄×Dic₅₃

yes

complemented

101

C₂³

C₂×C₄×D₅₃

yes

—

102

C₂³

C₂×C₄×D₅₃

yes

—

103

C₂³

C₂×C₄×D₅₃

yes

—

104

C₂³

C₂²×Dic₅₃

yes

—

105

C₂³

C₂²×Dic₅₃

yes

—

106

C₂³

C₂³×D₅₃

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

212

—

112

C₂×C₄

C₂²×Dic₅₃

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂×C₄×Dic₅₃

yes

—

120

C₂²

C₂×C₄×Dic₅₃

yes

—

121

C₂²

C₂²×C₄×D₅₃

yes

—

122

C₂²

C₂³×Dic₅₃

yes

characteristic

123

C₂²

#13133262

yes

—

124

C₄

—

125

C₄

—

106

—

126

C₂

—

127

C₂

—

128

C₂

C₂²×C₄×Dic₅₃

yes

characteristic

129

C₂

#16811096

yes

—

130

C₂

#16811098

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂×C₅₃	C₂²×C₄²
Fitting	C₂²×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#2273
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂×C₅₃

C₂²×C₄²

Fitting

C₂²×C₂²⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂²×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×C₂²⋊C₄

N	H ≅ G/N	type
2. C₂²⋊C₄×Dic₅₃	C₂	semidirect
3. C₄×C₂²⋊Dic₅₃	C₂	semidirect
6. C₂×C₂²⋊Dic₅₃	C₄	semidirect
9. C₂×C₄×Dic₅₃	C₂²	semidirect
10. #13133124	C₂²	semidirect
13. C₂×C₂²⋊C₄×C₅₃	C₄	semidirect
21. C₂²×C₄×C₅₃	C₂×C₄	semidirect
24. C₂²×Dic₅₃	C₂×C₄	semidirect
26. C₂²⋊C₄×C₅₃	C₂×C₄	semidirect
27. C₂²⋊Dic₅₃	C₂×C₄	semidirect
28. C₂²⋊Dic₅₃	C₂×C₄	semidirect
34. C₂³×C₅₃	C₄²	semidirect
70. C₅₃	#2273	semidirect
77. C₂×C₂²⋊C₄	Dic₅₃	semidirect
90. C₂²×C₄	C₂×Dic₅₃	semidirect
94. C₂²⋊C₄	C₂×Dic₅₃	semidirect
100. C₂³	C₄×Dic₅₃	semidirect

H ≅ G/N

type

2. C₂²⋊C₄×Dic₅₃

C₂

semidirect

3. C₄×C₂²⋊Dic₅₃

C₂

semidirect

6. C₂×C₂²⋊Dic₅₃

C₄

semidirect

9. C₂×C₄×Dic₅₃

C₂²

semidirect

10. #13133124

C₂²

semidirect

13. C₂×C₂²⋊C₄×C₅₃

C₄

semidirect

21. C₂²×C₄×C₅₃

C₂×C₄

semidirect

24. C₂²×Dic₅₃

C₂×C₄

semidirect

26. C₂²⋊C₄×C₅₃

C₂×C₄

semidirect

27. C₂²⋊Dic₅₃

C₂×C₄

semidirect

28. C₂²⋊Dic₅₃

C₂×C₄

semidirect

34. C₂³×C₅₃

C₄²

semidirect

70. C₅₃

#2273

semidirect

77. C₂×C₂²⋊C₄

Dic₅₃

semidirect

90. C₂²×C₄

C₂×Dic₅₃

semidirect

94. C₂²⋊C₄

C₂×Dic₅₃

semidirect

100. C₂³

C₄×Dic₅₃

semidirect

Subgroups — #20277068

Conjugacy classes of subgroups

Complements