Group #20274934 — subgroups

Subgroups — #20274934

Order 6784.

252 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 85 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	#16811390	C₂	1	1	yes	maximal, characteristic
2	#16810671	C₂	1	2	yes	maximal, complemented
3	#16810607	C₂	1	1	yes	maximal, characteristic
4	#16810388	C₂	1	2	yes	maximal, complemented
5	C₅₃×#408	C₂	1	1	yes	maximal, characteristic
6	#13133224	—	2	4	no	—
7	C₂×#11719917	—	2	2	no	—
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂×#11719917	C₂²	1	2	yes	—
10	#13133152	—	2	4	no	—
11	#13133136	C₄	1	2	yes	—
12	#13133134	C₄	1	2	yes	—
13	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²⋊C₈×C₅₃	—	2	2	no	—
17	#41×C₅₃	—	2	2	no	—
18	#41×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₅₃	Q₈	1	2	yes	—
22	C₂²×C₄×C₅₃	D₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	M₁₆×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	1	no	—
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	#11719917	—	2	4	no	—
29	#11719917	—	2	2	no	—
30	#11719917	—	2	2	no	—
31	#11719917	C₂×C₄	1	4	yes	—
32	#11719917	D₄	1	4	yes	—
33	C₂×#88322	—	2	4	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₅₃	—	2	2	no	—
39	D₄×C₅₃	—	2	4	no	—
40	D₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	2	yes	—
48	C₈×C₅₃	—	4	1	no	—
49	C₈×C₅₃	—	4	1	no	—
50	#88322	—	2	4	no	—
51	#88322	—	2	4	no	—
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	4	1	no	—
54	C₂²×C₅₃	—	4	2	no	—
55	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	C₄×D₄	1	2	yes	—
59	C₄×C₅₃	#120	1	1	yes	characteristic
60	C₄×C₅₃	#123	1	1	yes	characteristic
61	#1858	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	—	4	2	no	—
65	C₂×C₅₃	#385	1	1	yes	characteristic
66	#568	—	53	1	no	—
67	#430	—	53	2	no	—
68	#413	—	53	1	no	—
69	#408	D₅₃	1	1	yes	characteristic
70	#341	—	53	2	no	—
71	C₅₃	#1858	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
73	#131	—	106	4	no	—
74	C₂×M₁₆	—	53	2	no	—
75	C₂×M₁₆	—	53	2	no	—
76	C₂×M₁₆	—	106	2	no	—
77	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
79	#108	—	106	4	no	—
80	#101	—	53	2	no	—
81	#100	—	53	2	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
86	C₂×D₄	Dic₁₀₆	1	2	yes	—
87	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	53	4	no	—
92	M₁₆	—	53	4	no	—
93	M₁₆	—	106	4	no	—
94	M₁₆	—	106	2	no	—
95	M₁₆	—	106	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	106	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂³	#11719919	1	2	yes	—
103	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#11719919	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	D₄×D₅₃	1	2	yes	—
112	C₂×C₄	#11719947	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	106	4	no	—
117	C₈	—	106	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×#11719919	1	1	yes	characteristic
122	C₂²	#13133194	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	#13133177	1	2	yes	—
125	C₄	#13133185	1	1	yes	characteristic
126	C₄	#13133198	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#16810495	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811390

C₂

yes

maximal, characteristic

#16810671

C₂

yes

maximal, complemented

#16810607

C₂

yes

maximal, characteristic

#16810388

C₂

yes

maximal, complemented

C₅₃×#408

C₂

yes

maximal, characteristic

#13133224

—

C₂×#11719917

—

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

#13133152

—

#13133136

C₄

yes

—

#13133134

C₄

yes

—

C₂×#41×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂²⋊C₈×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

—

—

—

C₂×C₄

yes

—

#11719917

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

#341

—

C₅₃

#1858

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₅₃

yes

characteristic

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

#108

—

106

—

#101

—

#100

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

C₄×D₅₃

yes

characteristic

C₂×D₄

Dic₁₀₆

yes

—

C₂×D₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂³

#11719919

yes

—

103

C₂³

C₂×Dic₁₀₆

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#11719919

yes

—

110

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

111

C₂×C₄

D₄×D₅₃

yes

—

112

C₂×C₄

#11719947

yes

—

113

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

106

—

117

C₈

—

106

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#11719919

yes

characteristic

122

C₂²

#13133194

yes

—

123

C₄

—

124

C₄

#13133177

yes

—

125

C₄

#13133185

yes

characteristic

126

C₄

#13133198

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#16810495

yes

characteristic

	subgroup	G/H
center	C₄	#13133185
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#408	C₂
socle	C₂×C₅₃	#385
hypercenter	#408	D₅₃
hypocenter	C₅₃	#1858
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#13133185

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#408

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #16810671	C₂	semidirect
4. #16810388	C₂	semidirect
71. C₅₃	#1858	semidirect

H ≅ G/N

type

2. #16810671

C₂

semidirect

4. #16810388

C₂

semidirect

71. C₅₃

#1858

semidirect

Subgroups — #20274934

Conjugacy classes of subgroups

Complements