Group #20274361 — subgroups

Subgroups — #20274361

Order 6784.

164 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16811028	C₂	1	1	yes	maximal, characteristic
2	#16811021	C₂	1	1	yes	maximal, characteristic
3	#16810974	C₂	1	1	yes	maximal, characteristic
4	#16810974	C₂	1	1	yes	maximal, characteristic
5	#16810974	C₂	1	1	yes	maximal, characteristic
6	#16810974	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#322	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	1	yes	characteristic
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133151	—	2	2	no	—
13	#13133149	—	2	2	no	—
14	#13133142	—	2	2	no	—
15	#13133142	—	2	2	no	—
16	#13133142	—	2	2	no	—
17	#13133142	—	2	2	no	—
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₀₆	—	2	1	no	—
22	C₂×Dic₁₀₆	—	2	1	no	—
23	C₂×Dic₁₀₆	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×C₈×C₅₃	D₄	1	2	yes	—
26	C₂×C₈×C₅₃	D₄	1	2	yes	—
27	C₂×C₈×C₅₃	D₄	1	2	yes	—
28	C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₅₃	—	2	1	no	—
30	C₄⋊Dic₅₃	—	2	1	no	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₅₃	—	4	1	no	—
41	C₂×Dic₅₃	—	4	1	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	Dic₁₀₆	—	4	2	no	—
45	Dic₁₀₆	—	4	2	no	—
46	Dic₁₀₆	—	4	2	no	—
47	Dic₁₀₆	—	4	2	no	—
48	C₈×C₅₃	—	2	2	no	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	#127	1	1	yes	characteristic
52	C₄×C₅₃	#135	1	2	yes	—
53	C₄×C₅₃	#137	1	2	yes	—
54	C₄×C₅₃	#137	1	2	yes	—
55	Dic₅₃	—	8	1	no	—
56	Dic₅₃	—	8	1	no	—
57	Dic₅₃	—	8	1	no	—
58	Dic₅₃	—	8	1	no	—
59	#1718	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#495	1	1	yes	characteristic
61	C₂×C₅₃	#497	1	1	yes	characteristic
62	C₂×C₅₃	#497	1	1	yes	characteristic
63	#500	—	53	1	no	—
64	#498	—	53	1	no	—
65	#487	—	53	1	no	—
66	#487	—	53	1	no	—
67	#487	—	53	1	no	—
68	#487	—	53	1	no	—
69	#322	D₅₃	1	1	yes	characteristic
70	C₅₃	#1718	1	1	yes	characteristic, complemented, Sylow
71	C₄⋊Q₈	—	53	1	no	—
72	C₄⋊Q₈	—	53	1	no	—
73	C₄⋊Q₈	—	53	1	no	—
74	C₄⋊Q₈	—	53	1	no	—
75	#107	—	106	2	no	—
76	#106	—	106	2	no	—
77	#103	—	106	2	no	—
78	#103	—	106	2	no	—
79	#103	—	106	2	no	—
80	#103	—	106	2	no	—
81	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
82	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
83	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
84	C₂×Q₈	—	106	1	no	—
85	C₂×Q₈	—	106	1	no	—
86	C₂×Q₈	—	106	1	no	—
87	C₂×Q₈	—	106	1	no	—
88	C₂×C₈	D₂₁₂	1	2	yes	—
89	C₂×C₈	D₂₁₂	1	2	yes	—
90	C₂×C₈	D₂₁₂	1	2	yes	—
91	C₄⋊C₄	—	106	1	no	—
92	C₄⋊C₄	—	106	1	no	—
93	C₄⋊C₄	—	106	1	no	—
94	C₄⋊C₄	—	106	1	no	—
95	C₄⋊C₄	—	106	1	no	—
96	C₄⋊C₄	—	106	1	no	—
97	C₄⋊C₄	—	106	1	no	—
98	C₄⋊C₄	—	106	1	no	—
99	C₄²	C₂²×D₅₃	1	1	yes	characteristic
100	Q₈	—	212	2	no	—
101	Q₈	—	212	2	no	—
102	Q₈	—	212	2	no	—
103	Q₈	—	212	2	no	—
104	C₂×C₄	—	212	1	no	—
105	C₂×C₄	—	212	1	no	—
106	C₂×C₄	—	212	1	no	—
107	C₂×C₄	—	212	1	no	—
108	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
109	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
110	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	2	2	no	—
114	C₂²	#13133214	1	1	yes	characteristic
115	C₄	—	424	1	no	—
116	C₄	—	424	1	no	—
117	C₄	—	424	1	no	—
118	C₄	—	424	1	no	—
119	C₄	#13133239	1	2	yes	—
120	C₄	#13133250	1	2	yes	—
121	C₄	#13133250	1	2	yes	—
122	C₂	#16811007	1	1	yes	characteristic
123	C₂	#16811017	1	1	yes	characteristic
124	C₂	#16811017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic

#16811021

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic

C₅₃×#322

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

—

—

—

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

#127

yes

characteristic

C₄×C₅₃

#135

yes

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

—

D₅₃

yes

characteristic

C₅₃

#1718

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#107

—

106

—

#106

—

106

—

#103

—

106

—

#103

—

106

—

#103

—

106

—

#103

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄²

C₂²×D₅₃

yes

characteristic

100

Q₈

—

212

—

101

Q₈

—

212

—

102

Q₈

—

212

—

103

Q₈

—

212

—

104

C₂×C₄

—

212

—

105

C₂×C₄

—

212

—

106

C₂×C₄

—

212

—

107

C₂×C₄

—

212

—

108

C₂×C₄

C₂×D₂₁₂

yes

characteristic

109

C₂×C₄

C₂×D₂₁₂

yes

characteristic

110

C₂×C₄

C₂×D₂₁₂

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

#13133214

yes

characteristic

115

C₄

—

424

—

116

C₄

—

424

—

117

C₄

—

424

—

118

C₄

—

424

—

119

C₄

#13133239

yes

—

120

C₄

#13133250

yes

—

121

C₄

#13133250

yes

—

122

C₂

#16811007

yes

characteristic

123

C₂

#16811017

yes

characteristic

124

C₂

#16811017

yes

characteristic

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#322	C₂
socle	C₂²×C₅₃	#127
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1718
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#13133214

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#322

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
70. C₅₃	#1718	semidirect

H ≅ G/N

type

70. C₅₃

#1718

semidirect

Subgroups — #20274361

Conjugacy classes of subgroups

Complements