Group #20274873 — subgroups

Subgroups — #20274873

Order 6784.

280 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133151	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133150	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810492	C₂	1	2	yes	maximal
5	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	C₂×#11719919	—	2	4	no	—
9	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#13133151	—	2	2	no	—
11	#13133150	—	2	4	no	—
12	C₈×Dic₅₃	C₄	1	4	yes	complemented
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	#107×C₅₃	—	2	2	no	—
16	C₂²×C₄×C₅₃	—	2	2	no	—
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅₃	—	2	2	no	—
19	C₂²×Dic₅₃	D₄	1	2	yes	—
20	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₅₃	D₄	1	2	yes	—
22	C₂×C₈×C₅₃	Q₈	1	2	yes	—
23	C₄⋊C₄×C₅₃	—	2	2	no	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	#11719919	—	4	8	no	—
28	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
29	C₄×Dic₅₃	Q₈	1	2	yes	—
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	C₂×#88322	D₄	1	2	yes	—
32	C₂×#88322	Q₈	1	2	yes	—
33	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	4	4	no	—
36	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
42	C₂×Dic₅₃	—	4	2	no	—
43	C₂×Dic₅₃	—	4	4	no	—
44	C₂×Dic₅₃	C₄⋊C₄	1	4	yes	—
45	C₂×Dic₅₃	D₈	1	4	yes	—
46	C₂×Dic₅₃	Q₁₆	1	4	yes	—
47	C₈×C₅₃	C₄⋊C₄	1	4	yes	complemented
48	#88322	C₄⋊C₄	1	4	yes	complemented
49	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₅₃	#127	1	1	yes	characteristic
51	C₂²×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
52	C₂²×C₅₃	C₂×D₈	1	2	yes	—
53	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
54	C₄×C₅₃	—	8	2	no	—
55	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
58	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₅₃	—	8	2	no	—
60	Dic₅₃	#107	1	8	yes	complemented
61	#1845	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	#384	1	1	yes	characteristic
63	C₂×C₅₃	C₂×#107	1	2	yes	—
64	C₂×C₅₃	#493	1	1	yes	characteristic
65	C₂×C₅₃	#494	1	1	yes	characteristic
66	C₂×C₅₃	#500	1	2	yes	—
67	C₂×#107	—	53	1	no	—
68	C₂×#107	—	53	2	no	—
69	C₂×#107	D₅₃	1	1	yes	characteristic
70	C₂×C₄×C₈	—	53	1	no	—
71	#384	—	53	2	no	—
72	C₅₃	#1845	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	53	1	no	—
74	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	53	2	no	—
76	C₂×C₄⋊C₄	—	106	4	no	—
77	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
78	C₂×C₄²	—	53	1	no	—
79	#107	—	2	2	no	—
80	#107	—	106	4	no	—
81	#107	—	106	2	no	—
82	C₄×C₈	—	53	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	53	2	no	—
85	C₂²×C₄	—	106	2	no	—
86	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	C₂×C₈	—	53	2	no	—
88	C₂×C₈	—	53	2	no	—
89	C₂×C₈	—	53	2	no	—
90	C₂×C₈	Dic₁₀₆	1	2	yes	—
91	C₂×C₈	C₄×D₅₃	1	2	yes	—
92	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	106	2	no	—
96	C₄⋊C₄	—	106	2	no	—
97	C₄⋊C₄	—	212	8	no	—
98	C₄²	—	53	2	no	—
99	C₄²	—	53	2	no	—
100	C₂³	D₄×D₅₃	1	1	yes	characteristic
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	53	4	no	—
104	C₂×C₄	—	53	4	no	—
105	C₂×C₄	—	53	4	no	—
106	C₂×C₄	—	212	2	no	—
107	C₂×C₄	—	212	4	no	—
108	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
112	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₈	—	53	4	no	—
115	C₈	#11719919	1	4	yes	complemented
116	C₂²	C₄⋊C₄×D₅₃	1	1	yes	characteristic
117	C₂²	#13133213	1	1	yes	characteristic
118	C₂²	#13133216	1	1	yes	characteristic
119	C₂²	D₈×D₅₃	1	2	yes	—
120	C₂²	Q₁₆×D₅₃	1	2	yes	—
121	C₄	—	8	2	no	—
122	C₄	—	53	8	no	—
123	C₄	—	424	2	no	—
124	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
125	C₄	C₂×#11719919	1	1	yes	characteristic
126	C₄	#13133215	1	1	yes	characteristic
127	C₄	#13133216	1	1	yes	characteristic
128	C₂	#16810492	1	1	yes	characteristic
129	C₂	#107×D₅₃	1	2	yes	—
130	C₂	#16810998	1	1	yes	characteristic
131	C₂	#16811001	1	1	yes	characteristic
132	C₂	#16811027	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133150

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

#16810492

C₂

yes

maximal

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133151

—

#13133150

—

C₈×Dic₅₃

C₄

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

Q₈

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₄⋊C₄

yes

—

C₂×Dic₅₃

D₈

yes

—

C₂×Dic₅₃

Q₁₆

yes

—

C₈×C₅₃

C₄⋊C₄

yes

complemented

#88322

C₄⋊C₄

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#127

yes

characteristic

C₂²×C₅₃

C₄⋊Q₈

yes

characteristic

C₂²×C₅₃

C₂×D₈

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#107

—

C₂×#107

—

C₂×#107

D₅₃

yes

characteristic

C₂×C₄×C₈

—

#384

—

C₅₃

#1845

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

#107

—

#107

—

106

—

#107

—

106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

C₄²

—

100

C₂³

D₄×D₅₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

212

—

107

C₂×C₄

—

212

—

108

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

109

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

110

C₂×C₄

D₄×D₅₃

yes

characteristic

111

C₂×C₄

Q₈×D₅₃

yes

characteristic

112

C₂×C₄

Q₈×D₅₃

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

114

C₈

—

115

C₈

#11719919

yes

complemented

116

C₂²

C₄⋊C₄×D₅₃

yes

characteristic

117

C₂²

#13133213

yes

characteristic

118

C₂²

#13133216

yes

characteristic

119

C₂²

D₈×D₅₃

yes

—

120

C₂²

Q₁₆×D₅₃

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

424

—

124

C₄

C₄⋊C₄×D₅₃

yes

characteristic

125

C₄

C₂×#11719919

yes

characteristic

126

C₄

#13133215

yes

characteristic

127

C₄

#13133216

yes

characteristic

128

C₂

#16810492

yes

characteristic

129

C₂

#107×D₅₃

yes

—

130

C₂

#16810998

yes

characteristic

131

C₂

#16811001

yes

characteristic

132

C₂

#16811027

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	#1845
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
12. C₈×Dic₅₃	C₄	semidirect
47. C₈×C₅₃	C₄⋊C₄	semidirect
48. #88322	C₄⋊C₄	semidirect
60. Dic₅₃	#107	semidirect
72. C₅₃	#1845	semidirect
115. C₈	#11719919	semidirect

H ≅ G/N

type

12. C₈×Dic₅₃

C₄

semidirect

47. C₈×C₅₃

C₄⋊C₄

semidirect

48. #88322

C₄⋊C₄

semidirect

60. Dic₅₃

#107

semidirect

72. C₅₃

#1845

semidirect

115. C₈

#11719919

semidirect

Subgroups — #20274873

Conjugacy classes of subgroups

Complements