Group #20285216 — subgroups

Subgroups — C₂×#16811022 #20285216

Order 6784.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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
2	#16811022	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133150	C₂	1	2	yes	maximal
4	C₂×#13133148	C₂	1	2	yes	maximal
5	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	#13133209	—	2	2	no	—
10	#13133209	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×#11719919	—	2	2	no	—
13	#13133150	C₂²	1	8	yes	—
14	#13133148	C₂²	1	8	yes	—
15	C₄×#88322	C₂²	1	4	yes	—
16	#125×C₅₃	—	2	2	no	—
17	#125×C₅₃	C₂²	1	4	yes	—
18	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	D₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	2	2	no	—
25	C₄⋊C₄×C₅₃	—	2	8	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₄²×C₅₃	C₂³	1	2	yes	—
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	C₂³	1	4	yes	—
32	#11719919	—	2	8	no	—
33	C₂×#88322	Q₈	1	4	yes	—
34	C₂×#88322	Q₈	1	4	yes	—
35	C₂×#88322	C₂³	1	2	yes	—
36	C₂×#88322	C₂³	1	2	yes	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
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	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	4	4	no	—
48	#88322	C₂×Q₈	1	4	yes	—
49	#88322	C₂×Q₈	1	4	yes	—
50	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
51	C₂²×C₅₃	#135	1	4	yes	—
52	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
53	C₄×C₅₃	—	2	4	no	—
54	C₄×C₅₃	—	4	4	no	—
55	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
56	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
57	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
58	Dic₅₃	—	4	4	no	—
59	C₂×#499	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#499	1	4	yes	complemented
61	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₅₃	C₂×#135	1	2	yes	—
63	C₂×#125	—	53	1	no	—
64	C₂×#125	D₅₃	1	1	yes	characteristic
65	#499	—	53	8	no	—
66	C₂×#107	—	53	2	no	—
67	C₂×#106	—	53	2	no	—
68	C₂×C₄×C₈	—	53	1	no	—
69	C₅₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₈	—	53	1	no	—
71	C₂²×C₈	—	53	1	no	—
72	#125	—	2	2	no	—
73	#125	—	53	4	no	—
74	#125	—	106	2	no	—
75	#125	C₂×D₅₃	1	4	yes	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	53	2	no	—
78	C₂×C₄⋊C₄	—	106	2	no	—
79	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	#107	—	53	8	no	—
82	#106	—	53	8	no	—
83	C₄×C₈	—	53	4	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	106	2	no	—
86	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	C₂×C₈	—	53	4	no	—
89	C₂×C₈	—	53	4	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₄⋊C₄	—	2	8	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	53	4	no	—
95	C₄⋊C₄	—	106	2	no	—
96	C₄⋊C₄	—	106	8	no	—
97	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
98	C₄²	C₂²⋊D₅₃	1	2	yes	—
99	C₄²	C₂²×D₅₃	1	2	yes	—
100	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	106	4	no	—
105	C₂×C₄	—	212	4	no	—
106	C₂×C₄	Q₈×D₅₃	1	2	yes	—
107	C₂×C₄	Q₈×D₅₃	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
111	C₈	—	53	4	no	—
112	C₈	—	53	4	no	—
113	C₂²	#13133215	1	2	yes	—
114	C₂²	#13133237	1	4	yes	—
115	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
116	C₄	—	2	4	no	—
117	C₄	—	4	4	no	—
118	C₄	—	212	4	no	—
119	C₄	#13133215	1	2	yes	—
120	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
121	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
122	C₂	#16811022	1	4	yes	direct product factor
123	C₂	C₂×#13133215	1	1	yes	characteristic
124	C₂	C₂×#13133237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal, characteristic

#16811022

C₂

yes

maximal, direct product factor

C₂×#13133150

C₂

yes

maximal

C₂×#13133148

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133209

—

#13133209

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

C₄×#88322

C₂²

yes

—

#125×C₅₃

—

#125×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₂×#88322

Q₈

yes

—

C₂×#88322

Q₈

yes

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

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

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

C₂×#135

yes

—

C₂×#125

—

C₂×#125

D₅₃

yes

—

—

—

C₂×C₄×C₈

—

C₅₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

#125

—

#125

—

#125

—

106

—

#125

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#107

—

#106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

100

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

106

—

105

C₂×C₄

—

212

—

106

C₂×C₄

Q₈×D₅₃

yes

—

107

C₂×C₄

Q₈×D₅₃

yes

—

108

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

109

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

110

C₂×C₄

C₂³×D₅₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₂²

#13133215

yes

—

114

C₂²

#13133237

yes

—

115

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

212

—

119

C₄

#13133215

yes

—

120

C₄

C₂×Q₈×D₅₃

yes

characteristic

121

C₄

C₂×Q₈×D₅₃

yes

characteristic

122

C₂

#16811022

yes

direct product factor

123

C₂

C₂×#13133215

yes

characteristic

124

C₂

C₂×#13133237

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×#499
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. #16811022	C₂	direct (N × H)
60. C₂×C₅₃	#499	semidirect
69. C₅₃	C₂×#499	semidirect
122. C₂	#16811022	direct (N × H)

H ≅ G/N

type

2. #16811022

C₂

direct (N × H)

60. C₂×C₅₃

#499

semidirect

69. C₅₃

C₂×#499

semidirect

122. C₂

#16811022

direct (N × H)

Subgroups — C₂×#16811022 #20285216

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811022 #20285216

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811022 #20285216