Group #20275496 — subgroups

Subgroups — #20275496

Order 6784.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#16810665	C₂	1	2	yes	maximal
2	C₂×#13133149	C₂	1	1	yes	maximal, characteristic
3	#16810578	C₂	1	1	yes	maximal, characteristic
4	#16810391	C₂	1	2	yes	maximal
5	C₂×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	1	yes	characteristic
7	#13133176	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	#13133151	—	2	2	no	—
10	#13133149	—	2	1	no	—
11	#13133149	—	2	1	no	—
12	#13133149	—	2	2	no	—
13	#13133127	—	2	1	no	—
14	C₈×Dic₅₃	—	2	1	no	—
15	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	#108×C₅₃	C₄	1	4	yes	complemented
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	—	2	2	no	—
20	M₁₆×C₅₃	—	2	2	no	—
21	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	Q₈	1	2	yes	—
27	C₂²⋊Dic₅₃	—	2	1	no	—
28	C₄⋊Dic₅₃	—	2	2	no	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	D₄	1	2	yes	—
31	C₄×Dic₅₃	—	2	1	no	—
32	#11719917	—	2	1	no	—
33	#11719917	Q₈	1	2	yes	—
34	C₂×#88322	—	2	1	no	—
35	C₂³×C₅₃	#41	1	1	yes	characteristic
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	#41	1	1	yes	characteristic
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	—	4	2	no	—
44	C₂×Dic₅₃	—	8	2	no	—
45	C₈×C₅₃	—	2	4	no	—
46	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
47	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	#88322	—	2	2	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₅₃	#121	1	1	yes	characteristic
52	C₂²×C₅₃	#125	1	1	yes	characteristic
53	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
55	C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₅₃	—	4	2	no	—
58	Dic₅₃	—	8	2	no	—
59	#1951	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	2	1	no	—
61	C₂×C₅₃	—	2	1	no	—
62	C₂×C₅₃	#389	1	1	yes	characteristic
63	C₂×C₅₃	#471	1	2	yes	—
64	C₂×#108	D₅₃	1	1	yes	characteristic
65	#428	—	53	2	no	—
66	C₂×#106	—	53	1	no	—
67	#405	—	53	1	no	—
68	#343	—	53	2	no	—
69	C₅₃	#1951	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₁₆	—	53	1	no	—
71	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
72	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
73	#117	—	53	1	no	—
74	C₂×C₄⋊C₄	—	53	2	no	—
75	#108	Dic₅₃	1	4	yes	complemented
76	#107	—	106	2	no	—
77	#106	—	106	2	no	—
78	#106	—	106	1	no	—
79	#106	—	106	1	no	—
80	#97	—	106	1	no	—
81	C₄×C₈	—	106	1	no	—
82	C₂²×C₄	—	106	2	no	—
83	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	53	2	no	—
86	M₁₆	—	106	1	no	—
87	M₁₆	C₂×Dic₅₃	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	106	1	no	—
90	C₂×C₈	Dic₁₀₆	1	2	yes	—
91	C₂×C₈	D₂₁₂	1	1	yes	characteristic
92	C₂×C₈	D₂₁₂	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
94	C₄⋊C₄	—	53	2	no	—
95	C₄⋊C₄	—	106	2	no	—
96	C₄⋊C₄	—	106	2	no	—
97	C₂²⋊C₄	—	106	1	no	—
98	C₄²	—	106	1	no	—
99	C₂³	#11719949	1	1	yes	characteristic
100	C₂×C₄	—	106	2	no	—
101	C₂×C₄	—	424	2	no	—
102	C₂×C₄	—	212	2	no	—
103	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
104	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
105	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
106	C₂×C₄	#11719947	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	106	2	no	—
111	C₈	C₄⋊Dic₅₃	1	2	yes	—
112	C₈	C₄⋊Dic₅₃	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₅₃	1	1	yes	characteristic
115	C₂²	#13133192	1	1	yes	characteristic
116	C₂²	#13133208	1	1	yes	characteristic
117	C₄	—	424	2	no	—
118	C₄	—	212	2	no	—
119	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
120	C₄	D₄×Dic₅₃	1	1	yes	characteristic
121	C₄	#13133197	1	1	yes	characteristic
122	C₄	#13133216	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#16810522	1	1	yes	characteristic
126	C₂	#16810887	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810665

C₂

yes

maximal

C₂×#13133149

C₂

yes

maximal, characteristic

#16810578

C₂

yes

maximal, characteristic

#16810391

C₂

yes

maximal

C₂×#108×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

characteristic

#13133176

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₂²

yes

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

D₄

yes

—

C₄×Dic₅₃

—

#11719917

—

#11719917

Q₈

yes

—

C₂×#88322

—

C₂³×C₅₃

#41

yes

characteristic

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

#41

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊C₄

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₄×C₅₃

C₄⋊Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#1951

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#389

yes

characteristic

C₂×C₅₃

#471

yes

—

C₂×#108

D₅₃

yes

—

—

—

—

C₅₃

#1951

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₅₃

yes

complemented

#107

—

106

—

#106

—

106

—

#106

—

106

—

#106

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

106

—

M₁₆

C₂×Dic₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₂³

#11719949

yes

characteristic

100

C₂×C₄

—

106

—

101

C₂×C₄

—

424

—

102

C₂×C₄

—

212

—

103

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

104

C₂×C₄

C₂×D₂₁₂

yes

characteristic

105

C₂×C₄

D₄×D₅₃

yes

characteristic

106

C₂×C₄

#11719947

yes

characteristic

107

C₂×C₄

Q₈×D₅₃

yes

characteristic

108

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

109

C₈

—

110

C₈

—

106

—

111

C₈

C₄⋊Dic₅₃

yes

—

112

C₈

C₄⋊Dic₅₃

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₅₃

yes

characteristic

115

C₂²

#13133192

yes

characteristic

116

C₂²

#13133208

yes

characteristic

117

C₄

—

424

—

118

C₄

—

212

—

119

C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

120

C₄

D₄×Dic₅₃

yes

characteristic

121

C₄

#13133197

yes

characteristic

122

C₄

#13133216

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#16810522

yes

characteristic

126

C₂

#16810887

yes

—

	subgroup	G/H
center	C₂²	#13133192
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×#108×C₅₃	C₂
socle	C₂²×C₅₃	#121
hypercenter	C₂×#108	D₅₃
hypocenter	C₅₃	#1951
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#13133192

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×#108×C₅₃

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₅₃	C₄	semidirect
69. C₅₃	#1951	semidirect
75. #108	Dic₅₃	semidirect

H ≅ G/N

type

17. #108×C₅₃

C₄

semidirect

69. C₅₃

#1951

semidirect

75. #108

Dic₅₃

semidirect

Subgroups — #20275496

Conjugacy classes of subgroups

Complements