Group #20283786 — subgroups

Subgroups — #20283786

Order 6784.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#16810695	C₂	1	8	yes	maximal, complemented
2	#16810680	C₂	1	2	yes	maximal, complemented
3	#16810590	C₂	1	4	yes	maximal, complemented
4	C₅₃×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	4	yes	—
6	C₂²×#88322	C₂²	1	4	yes	—
7	C₄⋊#88322	C₂²	1	4	yes	—
8	C₂²⋊#88322	C₂²	1	4	yes	—
9	C₂²⋊#88322	C₂²	1	8	yes	—
10	#13133128	C₂²	1	4	yes	—
11	C₂×#41×C₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
13	#127×C₅₃	C₄	1	1	yes	characteristic
14	#124×C₅₃	C₄	1	2	yes	—
15	#121×C₅₃	C₄	1	4	yes	—
16	C₄×D₄×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	#41×C₅₃	—	4	2	no	—
19	#41×C₅₃	—	4	2	no	—
20	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
22	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	4	yes	—
27	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
28	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₂³	1	2	yes	—
32	#11719917	—	4	4	no	—
33	C₂×#88322	—	4	4	no	—
34	C₂×#88322	C₂³	1	8	yes	—
35	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂³×C₅₃	C₂²×C₄	1	4	yes	—
37	Q₈×C₅₃	—	4	2	no	—
38	D₄×C₅₃	—	4	4	no	—
39	D₄×C₅₃	—	4	2	no	—
40	D₄×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	#88322	—	2	8	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	4	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	8	no	—
55	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	4	no	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	H(5,𝔽₂)	1	2	yes	—
61	#2980	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	4	1	no	—
63	C₂×C₅₃	—	4	4	no	—
64	C₂×C₅₃	#518	1	1	yes	characteristic
65	C₂×C₅₃	#568	1	2	yes	—
66	#532	D₅₃	1	1	yes	characteristic
67	#435	—	53	8	no	—
68	#432	—	53	2	no	—
69	#408	—	53	4	no	—
70	C₅₃	#2980	1	1	yes	characteristic, complemented, Sylow
71	C₂×#41	C₂×D₅₃	1	2	yes	—
72	C₂×M₁₆	—	53	4	no	—
73	C₂²×C₈	—	53	4	no	—
74	C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
75	#127	Dic₅₃	1	1	yes	characteristic
76	#124	Dic₅₃	1	2	yes	—
77	#121	Dic₅₃	1	4	yes	—
78	C₄×D₄	C₂×D₅₃	1	4	yes	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	—	53	4	no	—
81	C₂²⋊C₈	—	53	4	no	—
82	C₂²⋊C₈	—	53	8	no	—
83	#97	—	53	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₅₃	1	2	yes	—
87	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
88	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
89	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
90	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₅₃	1	4	yes	—
93	M₁₆	—	212	4	no	—
94	C₂×C₈	—	53	8	no	—
95	C₂×C₈	—	212	4	no	—
96	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₅₃	1	8	yes	—
98	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
100	C₄²	C₂²×D₅₃	1	2	yes	—
101	C₂³	C₂²×Dic₅₃	1	4	yes	—
102	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	4	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
113	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
114	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
115	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
116	C₈	—	106	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	#13133262	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#16811098	1	1	yes	characteristic
130	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810695

C₂

yes

maximal, complemented

#16810680

C₂

yes

maximal, complemented

#16810590

C₂

yes

maximal, complemented

C₅₃×#532

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

C₂×#41×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

#127×C₅₃

C₄

yes

characteristic

#124×C₅₃

C₄

yes

—

#121×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂³×C₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

H(5,𝔽₂)

yes

—

#2980

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#518

yes

characteristic

C₂×C₅₃

#568

yes

—

#532

D₅₃

yes

—

—

—

C₅₃

#2980

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₅₃

yes

characteristic

#127

Dic₅₃

yes

characteristic

#124

Dic₅₃

yes

—

#121

Dic₅₃

yes

—

C₄×D₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

100

C₄²

C₂²×D₅₃

yes

—

101

C₂³

C₂²×Dic₅₃

yes

—

102

C₂³

C₂²×Dic₅₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

112

C₂×C₄

C₂²×Dic₅₃

yes

—

113

C₂×C₄

C₂²×Dic₅₃

yes

—

114

C₂×C₄

C₂²×Dic₅₃

yes

—

115

C₂×C₄

C₂³×D₅₃

yes

characteristic

116

C₈

—

106

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#13133262

yes

—

127

C₂

—

128

C₂

—

129

C₂

#16811098

yes

characteristic

130

C₂

#16811390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#532	C₂
hypercenter	#532	D₅₃
hypocenter	C₅₃	#2980
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#532

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #16810695	C₂	semidirect
2. #16810680	C₂	semidirect
3. #16810590	C₂	semidirect
70. C₅₃	#2980	semidirect

H ≅ G/N

type

1. #16810695

C₂

semidirect

2. #16810680

C₂

semidirect

3. #16810590

C₂

semidirect

70. C₅₃

#2980

semidirect

Subgroups — #20283786

Conjugacy classes of subgroups

Complements