Group #20273010 — subgroups

Subgroups — #20273010

Order 6784.

190 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16810676	C₂	1	1	yes	maximal, characteristic
2	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
3	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810344	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#431	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₄	1	2	yes	—
9	C₄⋊#88322	C₄	1	2	yes	—
10	C₂²⋊#88322	C₄	1	2	yes	—
11	#13133128	C₄	1	2	yes	—
12	C₄×#88322	C₂²	1	1	yes	characteristic
13	C₄×#88322	C₂²	1	1	yes	characteristic
14	C₄×#88322	C₂²	1	1	yes	characteristic
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₅₃	C₄	1	2	yes	—
18	C₂²⋊C₈×C₅₃	C₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂³	1	1	yes	characteristic
30	#11719917	—	2	4	no	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	C₂×C₄	1	2	yes	—
33	C₂×#88322	C₂×C₄	1	2	yes	—
34	C₂×#88322	C₂×C₄	1	2	yes	—
35	C₂³×C₅₃	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	2	no	—
47	#88322	—	2	2	no	—
48	#88322	—	2	2	no	—
49	#88322	—	2	2	no	—
50	#88322	M₁₆	1	4	yes	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
59	C₄×C₅₃	#131	1	2	yes	—
60	C₄×C₅₃	#131	1	2	yes	—
61	#1453	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	4	1	no	—
63	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
64	C₂×C₅₃	#405	1	1	yes	characteristic
65	C₂×C₅₃	#405	1	1	yes	characteristic
66	#431	—	53	1	no	—
67	#431	D₅₃	1	1	yes	characteristic
68	C₄×M₁₆	—	53	1	no	—
69	#322	—	53	1	no	—
70	#322	—	53	1	no	—
71	#322	—	53	1	no	—
72	C₈²	—	53	1	no	—
73	C₅₃	#1453	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₁₆	—	53	2	no	—
75	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
76	C₄⋊C₈	—	53	2	no	—
77	C₄⋊C₈	Dic₅₃	1	2	yes	—
78	C₂²⋊C₈	—	53	2	no	—
79	C₂²⋊C₈	Dic₅₃	1	2	yes	—
80	#97	—	53	2	no	—
81	C₄×C₈	—	53	1	no	—
82	C₄×C₈	—	53	1	no	—
83	C₄×C₈	—	53	1	no	—
84	C₄×C₈	—	53	1	no	—
85	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
89	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
90	M₁₆	—	106	4	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
96	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
97	C₄²	C₄×D₅₃	1	1	yes	characteristic
98	C₄²	C₄×D₅₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
100	C₄²	C₂²×D₅₃	1	1	yes	characteristic
101	C₂³	C₄×Dic₅₃	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
106	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
107	C₂×C₄	C₄×Dic₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
110	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	2	2	no	—
113	C₈	—	53	4	no	—
114	C₈	—	106	2	no	—
115	C₈	—	106	2	no	—
116	C₈	—	106	2	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	M₁₆×D₅₃	1	2	yes	—
125	C₄	#13133223	1	2	yes	—
126	C₄	#13133224	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	M₁₆×Dic₅₃	1	1	yes	characteristic
129	C₂	#16810578	1	1	yes	characteristic
130	C₂	#16810579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810676

C₂

yes

maximal, characteristic

C₄×#11719917

C₂

yes

maximal, characteristic

#16810345

C₂

yes

maximal, characteristic, complemented

#16810345

C₂

yes

maximal, characteristic, complemented

#16810344

C₂

yes

maximal, characteristic, complemented

C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#431

C₂

yes

maximal, characteristic

C₂×#11719917

C₄

yes

—

C₄⋊#88322

C₄

yes

—

C₂²⋊#88322

C₄

yes

—

#13133128

C₄

yes

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

—

C₂²⋊C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

#11719917

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

M₁₆

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

#131

yes

—

#1453

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₄×M₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

yes

characteristic

C₄×M₁₆

—

#322

—

#322

—

#322

—

C₈²

—

C₅₃

#1453

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₅₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₅₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₄×D₅₃

yes

characteristic

C₄²

C₂×Dic₅₃

yes

characteristic

100

C₄²

C₂²×D₅₃

yes

characteristic

101

C₂³

C₄×Dic₅₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×Dic₅₃

yes

characteristic

106

C₂×C₄

C₄×Dic₅₃

yes

characteristic

107

C₂×C₄

C₄×Dic₅₃

yes

characteristic

108

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

109

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

110

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

106

—

115

C₈

—

106

—

116

C₈

—

106

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

M₁₆×D₅₃

yes

—

125

C₄

#13133223

yes

—

126

C₄

#13133224

yes

—

127

C₂

—

128

C₂

M₁₆×Dic₅₃

yes

characteristic

129

C₂

#16810578

yes

characteristic

130

C₂

#16810579

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂×C₄²
Fitting	C₅₃×#431	C₂
hypercenter	#431	D₅₃
hypocenter	C₅₃	#1453
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #431

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅₃

commutator, socle

C₂²×C₅₃

C₂×C₄²

Fitting

C₅₃×#431

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #431

N	H ≅ G/N	type
3. #16810345	C₂	semidirect
4. #16810345	C₂	semidirect
5. #16810344	C₂	semidirect
6. C₈×#88322	C₂	semidirect
73. C₅₃	#1453	semidirect

H ≅ G/N

type

3. #16810345

C₂

semidirect

4. #16810345

C₂

semidirect

5. #16810344

C₂

semidirect

6. C₈×#88322

C₂

semidirect

73. C₅₃

#1453

semidirect

Subgroups — #20273010

Conjugacy classes of subgroups

Complements