Group #20289607 — subgroups

Subgroups — #20289607

Order 6784.

148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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	#16811642	C₂	1	1	yes	maximal, characteristic
2	#16811627	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719936	—	2	1	no	—
5	C₄⋊#88324	—	2	1	no	—
6	C₄⋊#88324	—	2	1	no	—
7	C₄⋊#88324	C₄	1	2	yes	—
8	#13133274	C₄	1	2	yes	—
9	C₄×#88324	—	2	1	no	—
10	C₂×C₈×D₅₃	—	2	1	no	—
11	C₄²×D₅₃	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	C₄	1	1	yes	characteristic
13	#13133146	—	2	1	no	—
14	C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
17	C₂×C₄×D₅₃	Q₈	1	1	yes	characteristic
18	C₂×#88324	—	2	2	no	—
19	C₂×#88324	—	2	2	no	—
20	#11719936	—	4	1	no	—
21	#11719936	—	4	1	no	—
22	C₂×C₈×C₅₃	—	2	1	no	—
23	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄×Dic₅₃	D₄	1	1	yes	characteristic
26	C₄×Dic₅₃	D₄	1	1	yes	characteristic
27	C₂×#88322	—	2	1	no	—
28	C₈×D₅₃	—	4	1	no	—
29	C₈×D₅₃	—	4	1	no	—
30	C₂²×D₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄×D₅₃	—	2	1	no	—
38	C₄×D₅₃	—	2	1	no	—
39	C₄×D₅₃	—	4	1	no	—
40	C₄×D₅₃	D₈	1	1	yes	characteristic
41	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
42	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
43	C₄×D₅₃	Q₁₆	1	1	yes	characteristic
44	#88324	—	4	2	no	—
45	#88324	—	4	2	no	—
46	C₈×C₅₃	—	4	1	no	—
47	#88322	—	4	1	no	—
48	C₂²×C₅₃	#95	1	1	yes	characteristic
49	C₂×D₅₃	—	2	1	no	—
50	C₂×D₅₃	—	2	1	no	—
51	C₂×D₅₃	#106	1	1	yes	characteristic
52	C₂×D₅₃	#107	1	1	yes	characteristic
53	C₄×C₅₃	—	2	1	no	—
54	C₄×C₅₃	#102	1	1	yes	characteristic
55	C₄×C₅₃	#103	1	1	yes	characteristic
56	C₄×C₅₃	#108	1	1	yes	characteristic
57	C₄×C₅₃	#108	1	1	yes	characteristic
58	Dic₅₃	—	2	1	no	—
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	#102	1	1	yes	characteristic
61	Dic₅₃	#103	1	1	yes	characteristic
62	#1301	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	#340	1	1	yes	characteristic
64	C₂×C₅₃	#341	1	1	yes	characteristic
65	C₂×C₅₃	#344	1	1	yes	characteristic
66	D₅₃	—	2	1	no	—
67	D₅₃	—	2	1	no	—
68	#431	—	53	1	no	—
69	#423	—	53	1	no	—
70	C₂×C₄⋊C₈	—	53	1	no	—
71	C₅₃	#1301	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	106	1	no	—
73	C₂²×C₈	—	106	1	no	—
74	C₂×C₄²	—	53	1	no	—
75	C₄⋊C₈	—	53	2	no	—
76	C₄⋊C₈	—	53	1	no	—
77	C₄⋊C₈	—	106	1	no	—
78	C₄⋊C₈	—	106	1	no	—
79	C₄⋊C₈	—	106	1	no	—
80	C₄⋊C₈	#6157	1	1	yes	characteristic
81	C₂²⋊C₈	—	53	2	no	—
82	C₄×C₈	—	106	1	no	—
83	C₂²×C₄	—	53	1	no	—
84	C₂²×C₄	—	53	1	no	—
85	C₂²×C₄	—	53	1	no	—
86	M₁₆	—	212	1	no	—
87	M₁₆	—	212	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	106	2	no	—
90	C₂×C₈	—	106	2	no	—
91	C₂×C₈	—	106	1	no	—
92	C₂×C₈	—	212	1	no	—
93	C₂×C₈	—	212	1	no	—
94	C₄²	—	53	1	no	—
95	C₄²	—	53	1	no	—
96	C₄²	—	53	1	no	—
97	C₄²	C₂×#6157	1	1	yes	characteristic
98	C₂³	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	53	1	no	—
106	C₂×C₄	—	106	1	no	—
107	C₂×C₄	—	106	1	no	—
108	C₂×C₄	—	212	1	no	—
109	C₂×C₄	C₄×#6157	1	1	yes	characteristic
110	C₂×C₄	C₄⋊#6157	1	1	yes	characteristic
111	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	212	2	no	—
114	C₈	—	212	2	no	—
115	C₈	—	212	1	no	—
116	C₂²	—	53	1	no	—
117	C₂²	—	53	1	no	—
118	C₂²	—	106	1	no	—
119	C₂²	—	106	1	no	—
120	C₂²	(C₂×#6157)⋊C₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	53	1	no	—
123	C₄	—	53	1	no	—
124	C₄	—	106	1	no	—
125	C₄	—	106	1	no	—
126	C₄	#13133279	1	1	yes	characteristic
127	C₄	#13133280	1	1	yes	characteristic
128	C₄	#13133287	1	1	yes	characteristic
129	C₄	#13133288	1	1	yes	characteristic
130	C₂	—	106	1	no	—
131	C₂	—	106	1	no	—
132	C₂	#16811496	1	1	yes	characteristic
133	C₂	#16811499	1	1	yes	characteristic
134	C₂	#16811506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811642

C₂

yes

maximal, characteristic

#16811627

C₂

yes

maximal, characteristic

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#13133274

C₄

yes

—

C₄×#88324

—

C₂×C₈×D₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊#88322

C₄

yes

characteristic

#13133146

—

C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

Q₈

yes

—

—

—

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₂×#88322

—

C₈×D₅₃

—

C₈×D₅₃

—

C₂²×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

D₈

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

SD₁₆

yes

characteristic

C₄×D₅₃

Q₁₆

yes

characteristic

#88324

—

#88324

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#95

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

yes

C₂×D₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

Dic₅₃

yes

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#431

—

#423

—

C₂×C₄⋊C₈

—

C₅₃

#1301

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₄⋊C₈

#6157

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

212

—

109

C₂×C₄

C₄×#6157

yes

characteristic

110

C₂×C₄

C₄⋊#6157

yes

characteristic

111

C₂×C₄

C₂²⋊#6157

yes

characteristic

112

C₈

—

113

C₈

—

212

—

114

C₈

—

212

—

115

C₈

—

212

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

106

—

119

C₂²

—

106

—

120

C₂²

(C₂×#6157)⋊C₄

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

#13133279

yes

characteristic

127

C₄

#13133280

yes

characteristic

128

C₄

#13133287

yes

characteristic

129

C₄

#13133288

yes

characteristic

130

C₂

—

106

—

131

C₂

—

106

—

132

C₂

#16811496

yes

characteristic

133

C₂

#16811499

yes

characteristic

134

C₂

#16811506

yes

characteristic

	subgroup	G/H
center	C₂²	(C₂×#6157)⋊C₄
commutator	C₂×C₄×C₅₃	C₄²
Frattini	C₄²	C₂×#6157
Fitting	C₄⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	#95
hypercenter	C₄⋊C₈	#6157
hypocenter	C₅₃	#1301
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

(C₂×#6157)⋊C₄

commutator

C₂×C₄×C₅₃

C₄²

Frattini

C₄²

C₂×#6157

Fitting

C₄⋊C₈×C₅₃

C₄

C₂²×C₅₃

C₄⋊C₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
71. C₅₃	#1301	semidirect

H ≅ G/N

type

71. C₅₃

#1301

semidirect

Subgroups — #20289607

Conjugacy classes of subgroups

Complements