Group #20277333 — subgroups

Subgroups — #20277333

Order 6784.

600 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 309 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	#16810522	C₂	1	8	yes	maximal, complemented
3	#16810483	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#13133176	—	2	8	no	—
6	C₂×C₄⋊Dic₅₃	C₂²	1	4	yes	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	8	yes	—
8	C₂×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	4	yes	—
10	#13133124	C₂²	1	8	yes	—
11	C₂³×C₄×C₅₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₅₃	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	4	yes	—
15	C₂⁴×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	4	4	no	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	D₄	1	4	yes	—
20	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
23	C₂²×Dic₅₃	C₂³	1	8	yes	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	16	yes	—
25	C₂²⋊Dic₅₃	—	2	8	no	—
26	C₄⋊Dic₅₃	—	2	8	no	—
27	C₄⋊Dic₅₃	—	4	8	no	—
28	C₄×Dic₅₃	—	2	8	no	—
29	C₂³×C₅₃	—	2	2	no	—
30	C₂³×C₅₃	—	2	2	no	—
31	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
32	C₂³×C₅₃	C₂×D₄	1	2	yes	—
33	C₂³×C₅₃	C₂×Q₈	1	2	yes	—
34	C₂³×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	8	no	—
36	C₂×C₄×C₅₃	—	4	8	no	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	16	yes	complemented
38	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
42	C₂×Dic₅₃	—	2	8	no	—
43	C₂×Dic₅₃	—	2	8	no	—
44	C₂×Dic₅₃	—	2	8	no	—
45	C₂²×C₅₃	—	2	2	no	—
46	C₂²×C₅₃	—	2	2	no	—
47	C₂²×C₅₃	—	2	4	no	—
48	C₂²×C₅₃	—	2	4	no	—
49	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
50	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
51	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
52	C₂²×C₅₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂²×C₅₃	H(5,𝔽₂)	1	2	yes	—
54	C₂²×C₅₃	2₋¹⁺⁴	1	2	yes	—
55	C₄×C₅₃	—	2	8	no	—
56	C₄×C₅₃	C₂×C₄⋊C₄	1	8	yes	—
57	Dic₅₃	—	4	8	no	—
58	#2316	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	4	no	—
60	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₅₃	#518	1	1	yes	characteristic
62	C₂×C₅₃	#519	1	1	yes	characteristic
63	C₂×C₅₃	#537	1	2	yes	—
64	C₂×C₅₃	#544	1	2	yes	—
65	C₂×#117	—	53	2	no	—
66	C₂²×C₄⋊C₄	D₅₃	1	1	yes	characteristic
67	#389	—	53	8	no	—
68	#380	—	53	4	no	—
69	C₅₃	#2316	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₄	C₂×D₅₃	1	2	yes	—
71	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
72	#117	—	106	8	no	—
73	C₂×C₄⋊C₄	—	53	4	no	—
74	C₂×C₄⋊C₄	—	53	8	no	—
75	C₂×C₄⋊C₄	Dic₅₃	1	8	yes	complemented
76	C₂×C₄⋊C₄	C₂×D₅₃	1	4	yes	—
77	C₂×C₂²⋊C₄	—	53	4	no	—
78	C₂×C₄²	—	53	4	no	—
79	#95	—	53	8	no	—
80	C₂⁴	C₂²×D₅₃	1	1	yes	characteristic
81	C₂²×C₄	—	4	4	no	—
82	C₂²×C₄	—	53	8	no	—
83	C₂²×C₄	Dic₁₀₆	1	4	yes	—
84	C₂²×C₄	D₂₁₂	1	4	yes	—
85	C₂²×C₄	C₂×Dic₅₃	1	8	yes	—
86	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
87	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
89	C₄⋊C₄	—	106	8	no	—
90	C₄⋊C₄	—	212	8	no	—
91	C₄⋊C₄	C₂×Dic₅₃	1	16	yes	—
92	C₂²⋊C₄	—	106	8	no	—
93	C₄²	—	106	8	no	—
94	C₂³	—	2	2	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₂×Dic₁₀₆	1	2	yes	—
97	C₂³	C₂×D₂₁₂	1	2	yes	—
98	C₂³	C₂²×Dic₅₃	1	2	yes	—
99	C₂³	C₂³×D₅₃	1	1	yes	characteristic
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	4	8	no	—
102	C₂×C₄	—	106	8	no	—
103	C₂×C₄	—	106	8	no	—
104	C₂×C₄	—	106	8	no	—
105	C₂×C₄	C₄⋊Dic₅₃	1	16	yes	complemented
106	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
107	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
108	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
109	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	2	4	no	—
114	C₂²	C₂×C₄⋊Dic₅₃	1	4	yes	—
115	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
116	C₂²	C₂²×D₂₁₂	1	1	yes	characteristic
117	C₂²	C₂²×Dic₁₀₆	1	1	yes	characteristic
118	C₂²	#13133262	1	2	yes	—
119	C₂²	#13133265	1	2	yes	—
120	C₄	—	2	8	no	—
121	C₄	—	212	8	no	—
122	C₄	C₂×C₄⋊Dic₅₃	1	8	yes	—
123	C₂	—	2	4	no	—
124	C₂	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
125	C₂	#16811098	1	1	yes	characteristic
126	C₂	#16811101	1	1	yes	characteristic
127	C₂	#16811204	1	2	yes	—
128	C₂	#16811254	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133176

C₂

yes

maximal

#16810522

C₂

yes

maximal, complemented

#16810483

C₂

yes

maximal

C₂²×C₄⋊C₄×C₅₃

C₂

yes

maximal, characteristic

#13133176

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133124

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×Dic₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂⁴

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂²×C₅₃

C₂²×Q₈

yes

characteristic

C₂²×C₅₃

H(5,𝔽₂)

yes

—

C₂²×C₅₃

2₋¹⁺⁴

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

Dic₅₃

—

#2316

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#544

yes

—

C₂×#117

—

C₂²×C₄⋊C₄

D₅₃

yes

characteristic

#389

—

#380

—

C₅₃

#2316

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

—

C₂³×C₄

C₂×D₅₃

yes

characteristic

#117

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

C₂⁴

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₀₆

yes

—

C₂³

C₂×D₂₁₂

yes

—

C₂³

C₂²×Dic₅₃

yes

—

C₂³

C₂³×D₅₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

106

—

103

C₂×C₄

—

106

—

104

C₂×C₄

—

106

—

105

C₂×C₄

C₄⋊Dic₅₃

yes

complemented

106

C₂×C₄

C₂×Dic₁₀₆

yes

—

107

C₂×C₄

C₂×D₂₁₂

yes

—

108

C₂×C₄

C₂²×Dic₅₃

yes

—

109

C₂×C₄

C₂²×Dic₅₃

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×C₄⋊Dic₅₃

yes

—

115

C₂²

C₂³×Dic₅₃

yes

characteristic

116

C₂²

C₂²×D₂₁₂

yes

characteristic

117

C₂²

C₂²×Dic₁₀₆

yes

characteristic

118

C₂²

#13133262

yes

—

119

C₂²

#13133265

yes

—

120

C₄

—

121

C₄

—

212

—

122

C₄

C₂×C₄⋊Dic₅₃

yes

—

123

C₂

—

124

C₂

C₂²×C₄⋊Dic₅₃

yes

characteristic

125

C₂

#16811098

yes

characteristic

126

C₂

#16811101

yes

characteristic

127

C₂

#16811204

yes

—

128

C₂

#16811254

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₅₃
hypocenter	C₅₃	#2316
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₅₃

C₂

socle

C₂³×C₅₃

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #16810522	C₂	semidirect
13. C₂×C₄⋊C₄×C₅₃	C₄	semidirect
37. C₂×C₄×C₅₃	C₄⋊C₄	semidirect
69. C₅₃	#2316	semidirect
75. C₂×C₄⋊C₄	Dic₅₃	semidirect
105. C₂×C₄	C₄⋊Dic₅₃	semidirect

H ≅ G/N

type

2. #16810522

C₂

semidirect

13. C₂×C₄⋊C₄×C₅₃

C₄

semidirect

37. C₂×C₄×C₅₃

C₄⋊C₄

semidirect

69. C₅₃

#2316

semidirect

75. C₂×C₄⋊C₄

Dic₅₃

semidirect

105. C₂×C₄

C₄⋊Dic₅₃

semidirect

Subgroups — #20277333

Conjugacy classes of subgroups

Complements