Group #20285049 — subgroups

Subgroups — C₂×#16810987 #20285049

Order 6784.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133209	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810987	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133138	C₂	1	4	yes	maximal, complemented
4	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	#13133209	—	2	2	no	—
8	#13133209	C₂²	1	4	yes	complemented
9	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
10	C₂×#11719919	—	2	2	no	—
11	#13133138	C₂²	1	16	yes	complemented
12	#13133128	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₅₃	—	2	2	no	—
14	C₂²×D₄×C₅₃	C₂²	1	2	yes	—
15	#127×C₅₃	—	2	2	no	—
16	#127×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	—	2	2	no	—
20	C₂×D₄×C₅₃	—	2	8	no	—
21	C₂×D₄×C₅₃	—	2	8	no	—
22	C₂×D₄×C₅₃	—	4	8	no	—
23	C₂×D₄×C₅₃	C₂³	1	4	yes	—
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	—	2	2	no	—
27	C₄²×C₅₃	D₄	1	2	yes	—
28	C₄²×C₅₃	C₂³	1	2	yes	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	C₂³	1	4	yes	—
31	#11719919	—	2	8	no	—
32	C₂×#88322	—	2	4	no	—
33	C₂×#88322	C₂³	1	4	yes	—
34	C₂³×C₅₃	—	2	4	no	—
35	C₂³×C₅₃	—	4	4	no	—
36	C₂³×C₅₃	—	4	8	no	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	D₄×C₅₃	—	2	8	no	—
39	D₄×C₅₃	—	4	16	no	—
40	D₄×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	4	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	4	4	no	—
48	#88322	—	2	4	no	—
49	C₂²×C₅₃	—	4	4	no	—
50	C₂²×C₅₃	—	4	8	no	—
51	C₂²×C₅₃	—	8	8	no	—
52	C₂²×C₅₃	#124	1	2	yes	—
53	C₂²×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
54	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	#124	1	2	yes	—
57	C₄×C₅₃	C₂×#41	1	2	yes	—
58	Dic₅₃	—	4	4	no	—
59	C₂×#490	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	8	4	no	—
61	C₂×C₅₃	#490	1	4	yes	complemented
62	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
63	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
64	C₂×#127	D₅₃	1	1	yes	characteristic
65	C₂×#125	—	53	1	no	—
66	#490	—	53	8	no	—
67	C₂×#102	—	53	4	no	—
68	C₂×#97	—	53	1	no	—
69	C₅₃	C₂×#490	1	1	yes	characteristic, complemented, Sylow
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₅₃	1	2	yes	—
72	C₂²×C₈	—	53	2	no	—
73	#127	—	2	2	no	—
74	#127	C₂×D₅₃	1	4	yes	—
75	#125	—	53	4	no	—
76	#125	—	106	2	no	—
77	C₂×C₄⋊C₄	—	53	2	no	—
78	C₂×C₄⋊C₄	—	106	2	no	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	#102	—	53	16	no	—
81	#97	—	53	4	no	—
82	C₂⁴	—	2	2	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂²×D₅₃	1	4	yes	—
88	C₂²×C₄	—	106	2	no	—
89	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
91	C₂×C₈	—	53	4	no	—
92	C₂×C₈	—	106	4	no	—
93	C₄⋊C₄	—	53	4	no	—
94	C₄⋊C₄	—	106	2	no	—
95	C₄⋊C₄	—	106	8	no	—
96	C₄²	C₂²⋊D₅₃	1	2	yes	—
97	C₄²	C₂²×D₅₃	1	2	yes	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	4	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	212	4	no	—
108	C₂×C₄	#11719947	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
112	C₈	—	106	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	8	8	no	—
116	C₂²	#13133204	1	2	yes	—
117	C₂²	#13133243	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	212	4	no	—
121	C₄	#13133204	1	2	yes	—
122	C₄	C₂×#11719947	1	2	yes	—
123	C₂	—	8	4	no	—
124	C₂	#16810987	1	4	yes	direct product factor
125	C₂	C₂×#13133204	1	1	yes	characteristic
126	C₂	C₂×#13133243	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal, characteristic, complemented

#16810987

C₂

yes

maximal, direct product factor

C₂×#13133138

C₂

yes

maximal, complemented

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

#13133209

—

#13133209

C₂²

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

#13133138

C₂²

yes

complemented

#13133128

C₂²

yes

complemented

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

C₂²

yes

—

#127×C₅₃

—

#127×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

C₂×#490

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#127

D₅₃

yes

—

—

—

C₂×#97

—

C₅₃

C₂×#490

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

C₂×D₅₃

yes

—

C₂²×C₈

—

#127

—

#127

C₂×D₅₃

yes

—

#125

—

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

#97

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₅₃

yes

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

212

—

108

C₂×C₄

#11719947

yes

—

109

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

111

C₂×C₄

C₂³×D₅₃

yes

characteristic

112

C₈

—

106

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#13133204

yes

—

117

C₂²

#13133243

yes

—

118

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

119

C₄

—

120

C₄

—

212

—

121

C₄

#13133204

yes

—

122

C₄

C₂×#11719947

yes

—

123

C₂

—

124

C₂

#16810987

yes

direct product factor

125

C₂

C₂×#13133204

yes

characteristic

126

C₂

C₂×#13133243

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#127×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#127	D₅₃
hypocenter	C₅₃	C₂×#490
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#127×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#13133209	C₂	semidirect
2. #16810987	C₂	direct (N × H)
3. C₂×#13133138	C₂	semidirect
4. C₂×#13133128	C₂	semidirect
8. #13133209	C₂²	semidirect
11. #13133138	C₂²	semidirect
12. #13133128	C₂²	semidirect
61. C₂×C₅₃	#490	semidirect
69. C₅₃	C₂×#490	semidirect
124. C₂	#16810987	direct (N × H)

H ≅ G/N

type

1. C₂×#13133209

C₂

semidirect

2. #16810987

C₂

direct (N × H)

3. C₂×#13133138

C₂

semidirect

4. C₂×#13133128

C₂

semidirect

8. #13133209

C₂²

semidirect

11. #13133138

C₂²

semidirect

12. #13133128

C₂²

semidirect

61. C₂×C₅₃

#490

semidirect

69. C₅₃

C₂×#490

semidirect

124. C₂

#16810987

direct (N × H)

Subgroups — C₂×#16810987 #20285049

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810987 #20285049

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810987 #20285049