Group #20283652 — subgroups

Subgroups — #20283652

Order 6784.

348 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 181 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16810752	C₂	1	8	yes	maximal, complemented
3	#16810651	C₂	1	2	yes	maximal
4	#16810581	C₂	1	2	yes	maximal
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
6	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	#13133217	C₄	1	2	yes	—
10	#13133209	C₄	1	2	yes	—
11	#13133196	C₄	1	4	yes	—
12	C₄×Dic₁₀₆	C₂²	1	4	yes	—
13	#13133176	C₂²	1	2	yes	—
14	#13133146	C₂²	1	8	yes	—
15	#13133128	C₂²	1	4	yes	—
16	#13133127	C₂²	1	4	yes	—
17	C₈×Dic₅₃	C₂²	1	4	yes	—
18	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
24	M₁₆×C₅₃	—	2	4	no	—
25	C₂×C₈×C₅₃	C₂³	1	4	yes	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂³	1	2	yes	—
29	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
31	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
32	#11719919	C₂×C₄	1	8	yes	—
33	C₄×Dic₅₃	C₂³	1	4	yes	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	Q₈	1	4	yes	complemented
36	C₂×#88322	C₂³	1	2	yes	—
37	C₂×#88322	C₂³	1	2	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₅₃	#41	1	2	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
47	Dic₁₀₆	—	4	4	no	—
48	C₈×C₅₃	—	2	4	no	—
49	#88322	—	2	2	no	—
50	#88322	C₂×Q₈	1	4	yes	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	C₄×Q₈	1	2	yes	—
57	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
59	Dic₅₃	—	2	8	no	—
60	#2961	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	—	2	2	no	—
62	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
63	C₂×C₅₃	#568	1	2	yes	—
64	#533	—	53	1	no	—
65	#446	—	53	8	no	—
66	#424	—	53	2	no	—
67	#423	D₅₃	1	1	yes	characteristic
68	#405	—	53	2	no	—
69	C₂×#97	—	53	1	no	—
70	C₅₃	#2961	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
72	C₂²×C₈	—	53	1	no	—
73	C₂²×C₈	—	53	1	no	—
74	C₄⋊Q₈	—	53	2	no	—
75	#125	—	53	2	no	—
76	C₂²⋊Q₈	—	53	4	no	—
77	C₄×Q₈	—	53	4	no	—
78	#117	—	53	2	no	—
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	—	53	8	no	—
81	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
82	#97	—	53	4	no	—
83	#97	—	53	4	no	—
84	C₄×C₈	—	53	4	no	—
85	C₂×Q₈	—	53	4	no	—
86	C₂²×C₄	C₄×D₅₃	1	2	yes	—
87	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	53	4	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	—	106	2	no	—
93	C₂×C₈	C₂²×D₅₃	1	4	yes	—
94	C₄⋊C₄	—	53	4	no	—
95	C₄⋊C₄	—	53	8	no	—
96	C₄⋊C₄	—	53	4	no	—
97	C₂²⋊C₄	—	53	4	no	—
98	C₄²	—	53	4	no	—
99	C₄²	C₄×D₅₃	1	1	yes	characteristic
100	C₄²	C₄×D₅₃	1	1	yes	characteristic
101	C₄²	C₂²×D₅₃	1	2	yes	—
102	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
103	Q₈	—	212	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	53	8	no	—
106	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₄	#11719947	1	2	yes	—
110	C₂×C₄	Q₈×D₅₃	1	2	yes	—
111	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₈	—	53	4	no	—
114	C₈	—	106	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#13133182	1	2	yes	—
117	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	106	8	no	—
121	C₄	#13133182	1	2	yes	—
122	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
123	C₄	C₂×#11719947	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×#13133182	1	1	yes	characteristic
126	C₂	#16811389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810752

C₂

yes

maximal, complemented

#16810651

C₂

yes

maximal

#16810581

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic

C₅₃×#423

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133217

C₄

yes

—

#13133209

C₄

yes

—

#13133196

C₄

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

C₂²

yes

—

#13133146

C₂²

yes

—

#13133128

C₂²

yes

—

#13133127

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

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

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

#2961

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

#568

yes

—

—

—

—

D₅₃

yes

characteristic

#405

—

C₂×#97

—

C₅₃

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₄×D₅₃

yes

characteristic

100

C₄²

C₄×D₅₃

yes

characteristic

101

C₄²

C₂²×D₅₃

yes

—

102

C₂³

C₂×C₄×D₅₃

yes

characteristic

103

Q₈

—

212

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₅₃

yes

—

107

C₂×C₄

C₂×C₄×D₅₃

yes

—

108

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

109

C₂×C₄

#11719947

yes

—

110

C₂×C₄

Q₈×D₅₃

yes

—

111

C₂×C₄

C₂³×D₅₃

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

106

—

115

C₂²

—

116

C₂²

#13133182

yes

—

117

C₂²

C₂²×C₄×D₅₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

106

—

121

C₄

#13133182

yes

—

122

C₄

C₂×Q₈×D₅₃

yes

characteristic

123

C₄

C₂×#11719947

yes

characteristic

124

C₂

—

125

C₂

C₂×#13133182

yes

characteristic

126

C₂

#16811389

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#423	C₂
hypercenter	#423	D₅₃
hypocenter	C₅₃	#2961
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#423

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #16810752	C₂	semidirect
35. C₂×#88322	Q₈	semidirect
70. C₅₃	#2961	semidirect

H ≅ G/N

type

2. #16810752

C₂

semidirect

35. C₂×#88322

Q₈

semidirect

70. C₅₃

#2961

semidirect

Subgroups — #20283652

Conjugacy classes of subgroups

Complements