Group #20274347 — subgroups

Subgroups — #20274347

Order 6784.

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

normal subgroups only

This group has 176 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	#16811019	C₂	1	1	yes	maximal, characteristic
2	#16811019	C₂	1	1	yes	maximal, characteristic
3	#16811002	C₂	1	1	yes	maximal, characteristic
4	#16810974	C₂	1	1	yes	maximal, characteristic
5	#16810973	C₂	1	1	yes	maximal, characteristic
6	#16810345	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#498	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719925	—	2	2	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133148	—	2	2	no	—
12	#13133148	—	2	2	no	—
13	#13133142	—	2	2	no	—
14	#13133141	—	2	2	no	—
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	#106×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	—	2	1	no	—
23	C₂×Dic₁₀₆	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×C₈×C₅₃	D₄	1	2	yes	—
26	C₄⋊C₄×C₅₃	—	2	1	no	—
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	#11719925	—	4	4	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	C₂×#88322	D₄	1	2	yes	—
37	C₂×#88322	D₄	1	2	yes	—
38	Q₈×C₅₃	—	4	2	no	—
39	Q₈×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	1	no	—
47	Dic₁₀₆	—	4	2	no	—
48	Dic₁₀₆	—	4	2	no	—
49	C₈×C₅₃	—	2	2	no	—
50	#88322	—	2	2	no	—
51	#88322	SD₁₆	1	4	yes	—
52	C₂²×C₅₃	#127	1	1	yes	characteristic
53	C₄×C₅₃	—	8	1	no	—
54	C₄×C₅₃	—	8	1	no	—
55	C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
56	C₄×C₅₃	#137	1	2	yes	—
57	C₄×C₅₃	#137	1	2	yes	—
58	Dic₅₃	—	8	1	no	—
59	Dic₅₃	—	8	1	no	—
60	#1716	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	#492	1	1	yes	characteristic
62	C₂×C₅₃	#497	1	1	yes	characteristic
63	C₂×C₅₃	#497	1	1	yes	characteristic
64	#498	—	53	1	no	—
65	#498	—	53	1	no	—
66	#498	D₅₃	1	1	yes	characteristic
67	#494	—	53	1	no	—
68	#487	—	53	1	no	—
69	#487	—	53	1	no	—
70	#322	—	53	1	no	—
71	C₅₃	#1716	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₁₆	—	106	2	no	—
73	C₄⋊Q₈	—	53	1	no	—
74	C₄⋊Q₈	—	53	1	no	—
75	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
77	#106	—	2	2	no	—
78	#106	—	106	2	no	—
79	#106	—	106	2	no	—
80	#103	—	106	2	no	—
81	#103	—	106	2	no	—
82	C₄×C₈	—	53	1	no	—
83	C₄×C₈	—	53	1	no	—
84	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	106	1	no	—
88	C₂×Q₈	—	106	1	no	—
89	Q₁₆	—	212	4	no	—
90	C₂×C₈	—	53	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	106	1	no	—
98	C₄⋊C₄	—	106	1	no	—
99	C₄⋊C₄	—	106	1	no	—
100	C₄⋊C₄	—	106	1	no	—
101	C₄²	C₂²×D₅₃	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	212	2	no	—
105	Q₈	—	212	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	212	1	no	—
109	C₂×C₄	—	212	1	no	—
110	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	—	53	4	no	—
115	C₈	—	106	2	no	—
116	C₂²	#13133213	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	424	1	no	—
120	C₄	—	424	1	no	—
121	C₄	SD₁₆×D₅₃	1	2	yes	—
122	C₄	#13133246	1	2	yes	—
123	C₄	#13133248	1	2	yes	—
124	C₂	#16810994	1	1	yes	characteristic
125	C₂	#16811014	1	1	yes	characteristic
126	C₂	#16811015	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811019

C₂

yes

maximal, characteristic

#16811019

C₂

yes

maximal, characteristic

#16811002

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic

#16810973

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic

C₅₃×#498

C₂

yes

maximal, characteristic

C₂×#11719925

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#106×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

SD₁₆

yes

—

C₂²×C₅₃

#127

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

#137

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

D₅₃

yes

—

—

—

—

C₅₃

#1716

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#106

—

#106

—

106

—

#106

—

106

—

#103

—

106

—

#103

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄²

C₂²×D₅₃

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

212

—

105

Q₈

—

212

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

212

—

109

C₂×C₄

—

212

—

110

C₂×C₄

D₄×D₅₃

yes

characteristic

111

C₂×C₄

D₄×D₅₃

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

—

106

—

116

C₂²

#13133213

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

424

—

120

C₄

—

424

—

121

C₄

SD₁₆×D₅₃

yes

—

122

C₄

#13133246

yes

—

123

C₄

#13133248

yes

—

124

C₂

#16810994

yes

characteristic

125

C₂

#16811014

yes

characteristic

126

C₂

#16811015

yes

characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#498	C₂
socle	C₂²×C₅₃	#127
hypercenter	#498	D₅₃
hypocenter	C₅₃	#1716
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#13133213

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#498

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
71. C₅₃	#1716	semidirect

H ≅ G/N

type

71. C₅₃

#1716

semidirect

Subgroups — #20274347

Conjugacy classes of subgroups

Complements