Group #20274751 — subgroups

Subgroups — #20274751

Order 6784.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂³×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133152	C₂	1	2	yes	maximal, complemented
3	#16810587	C₂	1	1	yes	maximal, characteristic
4	#16810387	C₂	1	2	yes	maximal, complemented
5	C₅₃×#407	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₂²×#88322	—	2	4	no	—
8	C₂²×#88322	C₄	1	4	yes	—
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133152	—	2	4	no	—
12	C₂²⋊#88322	—	2	2	no	—
13	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₅₃	—	2	2	no	—
16	C₂⁴×C₅₃	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	—	2	2	no	—
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₅₃	—	2	2	no	—
24	M₁₆×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	2	no	—
26	#11719917	—	2	2	no	—
27	#11719917	—	2	2	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	—	2	8	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂×#88322	D₄	1	4	yes	—
34	C₂³×C₅₃	—	2	4	no	—
35	C₂³×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂³×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	—	2	2	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₈×C₅₃	—	4	2	no	—
48	#88322	—	2	2	no	—
49	#88322	—	4	2	no	—
50	#88322	C₂²⋊C₄	1	4	yes	—
51	C₂²×C₅₃	—	2	8	no	—
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	#108	1	4	yes	complemented
56	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₅₃	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₅₃	#117	1	1	yes	characteristic
61	C₄×C₅₃	#121	1	2	yes	—
62	#1810	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	4	no	—
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	#380	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#108	1	2	yes	—
67	C₂³×C₈	—	53	1	no	—
68	C₂×#108	—	53	2	no	—
69	#407	—	53	1	no	—
70	#407	D₅₃	1	1	yes	characteristic
71	#341	—	53	2	no	—
72	C₅₃	#1810	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
74	C₂×M₁₆	—	53	2	no	—
75	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
76	C₂²×C₈	—	53	1	no	—
77	C₂²×C₈	—	53	4	no	—
78	C₂²×C₈	—	53	1	no	—
79	C₂²×C₈	—	106	4	no	—
80	#108	—	106	4	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₂²⋊C₈	—	106	2	no	—
83	C₂⁴	Dic₁₀₆	1	1	yes	characteristic
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	Dic₁₀₆	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₅₃	1	2	yes	—
88	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
89	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	106	2	no	—
93	M₁₆	—	106	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	53	2	no	—
96	C₂×C₈	—	53	4	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	—	106	8	no	—
99	C₂×C₈	—	106	2	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	2	2	no	—
103	C₂³	#11719919	1	2	yes	—
104	C₂³	C₂×Dic₁₀₆	1	1	yes	characteristic
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	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	2	no	—
115	C₈	—	53	4	no	—
116	C₈	—	106	2	no	—
117	C₈	—	212	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#13133152	1	4	yes	complemented
123	C₂²	C₂×#11719919	1	1	yes	characteristic
124	C₂²	#13133194	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	C₂²⋊C₄×D₅₃	1	1	yes	characteristic
127	C₄	#13133173	1	1	yes	characteristic
128	C₄	#13133189	1	2	yes	—
129	C₂	—	2	4	no	—
130	C₂	—	2	2	no	—
131	C₂	#16810481	1	1	yes	characteristic
132	C₂	C₂×#13133152	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×#88322

C₂

yes

maximal, characteristic

C₂×#13133152

C₂

yes

maximal, complemented

#16810587

C₂

yes

maximal, characteristic

#16810387

C₂

yes

maximal, complemented

C₅₃×#407

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

—

C₂²×#88322

C₄

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133152

—

C₂²⋊#88322

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×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

—

#88322

—

#88322

—

#88322

C₂²⋊C₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#108

yes

complemented

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1810

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#380

yes

characteristic

C₂×C₅₃

C₂×#108

yes

—

C₂³×C₈

—

C₂×#108

—

#407

—

#407

D₅₃

yes

characteristic

#341

—

C₅₃

#1810

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

106

—

#108

—

106

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂⁴

Dic₁₀₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₀₆

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₂×C₈

—

106

—

101

C₂³

—

102

C₂³

—

103

C₂³

#11719919

yes

—

104

C₂³

C₂×Dic₁₀₆

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

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

—

212

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#13133152

yes

complemented

123

C₂²

C₂×#11719919

yes

characteristic

124

C₂²

#13133194

yes

—

125

C₄

—

126

C₄

C₂²⋊C₄×D₅₃

yes

characteristic

127

C₄

#13133173

yes

characteristic

128

C₄

#13133189

yes

—

129

C₂

—

130

C₂

—

131

C₂

#16810481

yes

characteristic

132

C₂

C₂×#13133152

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#407	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₄
hypercenter	#407	D₅₃
hypocenter	C₅₃	#1810
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#407

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#13133152	C₂	semidirect
4. #16810387	C₂	semidirect
55. C₂²×C₅₃	#108	semidirect
72. C₅₃	#1810	semidirect
122. C₂²	#13133152	semidirect

H ≅ G/N

type

2. C₂×#13133152

C₂

semidirect

4. #16810387

C₂

semidirect

55. C₂²×C₅₃

#108

semidirect

72. C₅₃

#1810

semidirect

122. C₂²

#13133152

semidirect

Subgroups — #20274751

Conjugacy classes of subgroups

Complements