Group #20285051 — subgroups

Subgroups — C₂×#16810989 #20285051

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×#13133214	C₂	1	1	yes	maximal, characteristic
2	#16810989	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133137	C₂	1	4	yes	maximal, complemented
4	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₁₂	—	2	2	no	—
7	C₂²×D₂₁₂	C₂²	1	2	yes	—
8	C₂²×#88322	C₂²	1	2	yes	—
9	#13133214	—	2	2	no	—
10	#13133214	C₂²	1	4	yes	—
11	#13133137	C₂²	1	16	yes	complemented
12	#13133128	C₂²	1	4	yes	complemented
13	#125×C₅₃	—	2	2	no	—
14	#125×C₅₃	C₂²	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂³×D₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂×D₂₁₂	—	2	8	no	—
23	C₂×D₂₁₂	—	2	2	no	—
24	C₂×D₂₁₂	—	2	8	no	—
25	C₂×D₂₁₂	—	4	8	no	—
26	C₂×D₂₁₂	C₂³	1	4	yes	—
27	C₄⋊C₄×C₅₃	—	2	8	no	—
28	C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
30	C₄²×C₅₃	D₄	1	2	yes	—
31	C₄²×C₅₃	C₂³	1	2	yes	—
32	C₂×#88322	—	2	4	no	—
33	C₂×#88322	C₂³	1	4	yes	—
34	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×D₅₃	—	2	4	no	—
36	C₂²×D₅₃	—	4	4	no	—
37	C₂²×D₅₃	—	4	8	no	—
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	4	yes	—
44	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
45	D₂₁₂	—	2	8	no	—
46	D₂₁₂	—	4	16	no	—
47	D₂₁₂	—	4	4	no	—
48	#88322	—	2	4	no	—
49	C₂²×C₅₃	#124	1	2	yes	—
50	C₂²×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
51	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
52	C₂×D₅₃	—	4	4	no	—
53	C₂×D₅₃	—	4	8	no	—
54	C₂×D₅₃	—	8	8	no	—
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	—	4	4	no	—
57	C₄×C₅₃	#124	1	2	yes	—
58	C₄×C₅₃	C₂×#41	1	2	yes	—
59	C₂×#490	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#490	1	4	yes	complemented
61	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
62	C₂×C₅₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
63	D₅₃	—	8	4	no	—
64	C₂×#127	—	53	1	no	—
65	C₂×#125	D₅₃	1	1	yes	characteristic, complemented
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	—	53	2	no	—
71	C₂²×D₄	—	106	2	no	—
72	C₂²×C₈	—	53	2	no	—
73	#127	—	53	4	no	—
74	#127	—	106	2	no	—
75	#125	—	2	2	no	—
76	#125	C₂×D₅₃	1	4	yes	complemented
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	#102	—	53	16	no	—
81	#97	—	53	4	no	—
82	C₂⁴	—	106	2	no	—
83	C₂×D₄	—	53	4	no	—
84	C₂×D₄	—	106	8	no	—
85	C₂×D₄	—	106	8	no	—
86	C₂×D₄	—	106	2	no	—
87	C₂×D₄	—	212	8	no	—
88	C₂²×C₄	—	2	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	—	2	8	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
96	C₄²	C₂²⋊D₅₃	1	2	yes	—
97	C₄²	C₂²×D₅₃	1	2	yes	—
98	C₂³	—	106	4	no	—
99	C₂³	—	212	4	no	—
100	C₂³	—	212	8	no	—
101	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
102	D₄	—	106	8	no	—
103	D₄	—	212	16	no	—
104	D₄	—	212	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	#11719949	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	—	424	8	no	—
114	C₂²	—	212	4	no	—
115	C₂²	—	212	8	no	—
116	C₂²	#13133206	1	2	yes	—
117	C₂²	#13133242	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	4	4	no	—
121	C₄	#13133206	1	2	yes	—
122	C₄	C₂×#11719949	1	2	yes	—
123	C₂	—	424	4	no	—
124	C₂	#16810989	1	4	yes	direct product factor
125	C₂	C₂×#13133206	1	1	yes	characteristic
126	C₂	C₂×#13133242	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133214

C₂

yes

maximal, characteristic

#16810989

C₂

yes

maximal, direct product factor

C₂×#13133137

C₂

yes

maximal, complemented

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₁₂

—

C₂²×D₂₁₂

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133214

—

#13133214

C₂²

yes

—

#13133137

C₂²

yes

complemented

#13133128

C₂²

yes

complemented

#125×C₅₃

—

#125×C₅₃

C₂²

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂³×D₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×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

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

#88322

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

AGL(1,ℤ/8)

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×#41

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂×#127

—

C₂×#125

D₅₃

yes

characteristic, complemented

#490

—

C₂×#102

—

C₂×#97

—

C₅₃

C₂×#490

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

106

—

C₂²×C₈

—

#127

—

#127

—

106

—

#125

—

#125

C₂×D₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#102

—

#97

—

C₂⁴

—

106

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

—

106

—

C₂³

—

212

—

100

C₂³

—

212

—

101

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

102

D₄

—

106

—

103

D₄

—

212

—

104

D₄

—

212

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

#11719949

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

—

424

—

114

C₂²

—

212

—

115

C₂²

—

212

—

116

C₂²

#13133206

yes

—

117

C₂²

#13133242

yes

—

118

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

#13133206

yes

—

122

C₄

C₂×#11719949

yes

—

123

C₂

—

424

—

124

C₂

#16810989

yes

direct product factor

125

C₂

C₂×#13133206

yes

characteristic

126

C₂

C₂×#13133242

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	C₂×#490
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #16810989	C₂	direct (N × H)
3. C₂×#13133137	C₂	semidirect
4. C₂×#13133128	C₂	semidirect
5. C₂×#125×C₅₃	C₂	semidirect
11. #13133137	C₂²	semidirect
12. #13133128	C₂²	semidirect
14. #125×C₅₃	C₂²	semidirect
60. C₂×C₅₃	#490	semidirect
65. C₂×#125	D₅₃	semidirect
69. C₅₃	C₂×#490	semidirect
76. #125	C₂×D₅₃	semidirect
124. C₂	#16810989	direct (N × H)

H ≅ G/N

type

2. #16810989

C₂

direct (N × H)

3. C₂×#13133137

C₂

semidirect

4. C₂×#13133128

C₂

semidirect

5. C₂×#125×C₅₃

C₂

semidirect

11. #13133137

C₂²

semidirect

12. #13133128

C₂²

semidirect

14. #125×C₅₃

C₂²

semidirect

60. C₂×C₅₃

#490

semidirect

65. C₂×#125

D₅₃

semidirect

69. C₅₃

C₂×#490

semidirect

76. #125

C₂×D₅₃

semidirect

124. C₂

#16810989

direct (N × H)

Subgroups — C₂×#16810989 #20285051

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810989 #20285051

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810989 #20285051