Group #20274833 — subgroups

Subgroups — #20274833

Order 6784.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133146	C₂	1	2	yes	maximal
2	#16810469	C₂	1	1	yes	maximal, characteristic
3	#16810378	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal
5	C₅₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	2	yes	—
7	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
9	#13133146	—	2	4	no	—
10	#13133124	C₄	1	2	yes	—
11	#13133124	C₄	1	2	yes	—
12	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₅₃	—	2	2	no	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₄²×C₅₃	—	2	2	no	—
23	C₄×Dic₅₃	—	2	2	no	—
24	C₄×Dic₅₃	D₄	1	2	yes	—
25	C₄×Dic₅₃	Q₈	1	2	yes	—
26	C₂×#88322	—	2	2	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₅₃	#41	1	2	yes	—
36	C₂×C₄×C₅₃	#41	1	2	yes	—
37	C₂×Dic₅₃	—	2	2	no	—
38	C₂×Dic₅₃	—	2	2	no	—
39	C₂×Dic₅₃	—	2	1	no	—
40	C₂×Dic₅₃	—	2	1	no	—
41	C₂×Dic₅₃	—	2	2	no	—
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
44	C₈×C₅₃	—	4	2	no	—
45	#88322	—	4	2	no	—
46	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₅₃	#117	1	1	yes	characteristic
48	C₂²×C₅₃	#117	1	1	yes	characteristic
49	C₂²×C₅₃	#131	1	2	yes	—
50	C₂²×C₅₃	#131	1	2	yes	—
51	C₄×C₅₃	—	4	2	no	—
52	C₄×C₅₃	#124	1	2	yes	—
53	C₄×C₅₃	#125	1	2	yes	—
54	Dic₅₃	—	2	4	no	—
55	Dic₅₃	—	4	1	no	—
56	Dic₅₃	—	4	1	no	—
57	#1832	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#382	1	1	yes	characteristic
59	C₂×C₅₃	#424	1	2	yes	—
60	C₂×C₅₃	#433	1	2	yes	—
61	C₂×C₅₃	#433	1	1	yes	characteristic
62	C₂×C₅₃	#433	1	1	yes	characteristic
63	C₂×C₄⋊C₈	—	53	2	no	—
64	#376	—	53	1	no	—
65	#336	—	53	2	no	—
66	#336	—	53	1	no	—
67	#336	D₅₃	1	1	yes	characteristic
68	C₅₃	#1832	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	53	2	no	—
70	C₂²×C₈	C₂×D₅₃	1	2	yes	—
71	C₂×C₄²	—	53	1	no	—
72	C₂×C₄²	—	53	1	no	—
73	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
74	C₄⋊C₈	—	106	4	no	—
75	#95	—	53	2	no	—
76	#95	—	53	2	no	—
77	C₂²×C₄	—	53	2	no	—
78	C₂²×C₄	—	53	1	no	—
79	C₂²×C₄	—	53	1	no	—
80	C₂²×C₄	C₄×D₅₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	106	2	no	—
86	C₂×C₈	—	106	2	no	—
87	C₂×C₈	—	106	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	53	2	no	—
90	C₄²	—	53	2	no	—
91	C₄²	—	106	2	no	—
92	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	53	4	no	—
97	C₂×C₄	—	106	2	no	—
98	C₂×C₄	—	106	2	no	—
99	C₂×C₄	—	106	1	no	—
100	C₂×C₄	—	106	1	no	—
101	C₂×C₄	—	106	2	no	—
102	C₂×C₄	—	106	2	no	—
103	C₂×C₄	#11719945	1	2	yes	—
104	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
105	C₂×C₄	#11719947	1	2	yes	—
106	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	212	2	no	—
109	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
110	C₂²	#13133173	1	1	yes	characteristic
111	C₂²	#13133174	1	1	yes	characteristic
112	C₂²	#13133223	1	2	yes	—
113	C₂²	#13133225	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	106	4	no	—
116	C₄	—	212	1	no	—
117	C₄	—	212	1	no	—
118	C₄	#13133203	1	2	yes	—
119	C₄	#13133207	1	2	yes	—
120	C₂	#16810487	1	1	yes	characteristic
121	C₂	#16810648	1	2	yes	—
122	C₂	#16810681	1	2	yes	—
123	C₂	#16810682	1	1	yes	characteristic
124	C₂	#16810685	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133146

C₂

yes

maximal

#16810469

C₂

yes

maximal, characteristic

#16810378

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₅₃×#336

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

—

#13133124

C₄

yes

—

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

#125

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#433

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#1832

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

106

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

100

C₂×C₄

—

106

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

106

—

103

C₂×C₄

#11719945

yes

—

104

C₂×C₄

D₄×D₅₃

yes

characteristic

105

C₂×C₄

#11719947

yes

—

106

C₂×C₄

Q₈×D₅₃

yes

characteristic

107

C₈

—

108

C₈

—

212

—

109

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

110

C₂²

#13133173

yes

characteristic

111

C₂²

#13133174

yes

characteristic

112

C₂²

#13133223

yes

—

113

C₂²

#13133225

yes

—

114

C₄

—

115

C₄

—

106

—

116

C₄

—

212

—

117

C₄

—

212

—

118

C₄

#13133203

yes

—

119

C₄

#13133207

yes

—

120

C₂

#16810487

yes

characteristic

121

C₂

#16810648

yes

—

122

C₂

#16810681

yes

—

123

C₂

#16810682

yes

characteristic

124

C₂

#16810685

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₄
Fitting	C₅₃×#336	C₂
hypercenter	#336	D₅₃
hypocenter	C₅₃	#1832
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₄

Fitting

C₅₃×#336

C₂

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
68. C₅₃	#1832	semidirect

H ≅ G/N

type

68. C₅₃

#1832

semidirect

Subgroups — #20274833

Conjugacy classes of subgroups

Complements