Group #20283645 — subgroups

Subgroups — #20283645

Order 6784.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16810750	C₂	1	4	yes	maximal, complemented
3	#16810747	C₂	1	4	yes	maximal, complemented
4	#16810651	C₂	1	2	yes	maximal
5	#16810581	C₂	1	2	yes	maximal
6	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#423	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	#13133217	C₄	1	2	yes	—
11	#13133209	C₄	1	2	yes	—
12	#13133196	C₄	1	4	yes	—
13	C₄×Dic₁₀₆	C₂²	1	4	yes	—
14	#13133176	C₂²	1	2	yes	—
15	#13133146	C₂²	1	8	yes	—
16	#13133127	C₂²	1	4	yes	—
17	C₄×#88322	C₂²	1	2	yes	—
18	C₄×#88322	C₂²	1	2	yes	—
19	C₈×Dic₅₃	C₂²	1	4	yes	—
20	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₅₃	C₂²	1	4	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
26	M₁₆×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	C₂³	1	4	yes	—
28	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₂³	1	2	yes	—
31	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
32	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
33	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
34	#11719919	C₂×C₄	1	8	yes	—
35	C₄×Dic₅₃	C₂³	1	4	yes	—
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	Q₈	1	4	yes	complemented
38	C₂×#88322	C₂³	1	2	yes	—
39	C₂×#88322	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂³	1	1	yes	characteristic
41	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₅₃	#41	1	2	yes	—
48	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
50	Dic₁₀₆	—	2	8	no	—
51	C₈×C₅₃	—	2	4	no	—
52	#88322	—	2	2	no	—
53	#88322	C₂×Q₈	1	2	yes	—
54	#88322	C₂×Q₈	1	2	yes	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	C₄×Q₈	1	2	yes	—
60	C₄×C₅₃	#131	1	4	yes	—
61	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
63	Dic₅₃	—	2	8	no	—
64	#2960	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#131	1	2	yes	—
68	#533	—	53	1	no	—
69	#446	—	53	4	no	—
70	C₈×Q₈	—	53	4	no	—
71	#424	—	53	2	no	—
72	#423	D₅₃	1	1	yes	characteristic
73	#405	—	53	2	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₅₃	#2960	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
77	C₂²×C₈	—	53	1	no	—
78	C₂²×C₈	—	53	1	no	—
79	C₄⋊Q₈	—	53	2	no	—
80	#125	—	53	2	no	—
81	C₂²⋊Q₈	—	53	4	no	—
82	C₄×Q₈	—	53	4	no	—
83	#117	—	53	2	no	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₄⋊C₈	—	53	8	no	—
86	C₄⋊C₈	C₂×D₅₃	1	4	yes	—
87	#97	—	53	4	no	—
88	C₄×C₈	—	53	4	no	—
89	C₄×C₈	—	53	2	no	—
90	C₄×C₈	—	53	2	no	—
91	C₂×Q₈	—	53	4	no	—
92	C₂²×C₄	C₄×D₅₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	C₂×C₈	—	53	4	no	—
96	C₂×C₈	—	53	1	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	—	106	2	no	—
100	C₂×C₈	C₂²×D₅₃	1	4	yes	—
101	C₄⋊C₄	—	53	4	no	—
102	C₄⋊C₄	—	53	8	no	—
103	C₄⋊C₄	—	53	4	no	—
104	C₂²⋊C₄	—	53	4	no	—
105	C₄²	—	53	4	no	—
106	C₄²	C₄×D₅₃	1	1	yes	characteristic
107	C₄²	C₄×D₅₃	1	1	yes	characteristic
108	C₄²	C₂²×D₅₃	1	2	yes	—
109	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
110	Q₈	—	106	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	53	8	no	—
113	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
116	C₂×C₄	#11719947	1	2	yes	—
117	C₂×C₄	Q₈×D₅₃	1	2	yes	—
118	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	53	2	no	—
121	C₈	—	53	2	no	—
122	C₈	—	106	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#13133182	1	2	yes	—
125	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	106	8	no	—
128	C₄	#13133182	1	2	yes	—
129	C₄	#13133225	1	4	yes	—
130	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
131	C₄	C₂×#11719947	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#13133182	1	1	yes	characteristic
134	C₂	C₂×#13133225	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16810750

C₂

yes

maximal, complemented

#16810747

C₂

yes

maximal, complemented

#16810651

C₂

yes

maximal

#16810581

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₅₃×#423

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133217

C₄

yes

—

#13133209

C₄

yes

—

#13133196

C₄

yes

—

C₄×Dic₁₀₆

C₂²

yes

—

#13133176

C₂²

yes

—

#13133146

C₂²

yes

—

#13133127

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

#2960

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

C₂×#131

yes

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

D₅₃

yes

characteristic

#405

—

C₂×C₄×C₈

—

C₅₃

#2960

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₅₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₅₃

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

100

C₂×C₈

C₂²×D₅₃

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₅₃

yes

characteristic

107

C₄²

C₄×D₅₃

yes

characteristic

108

C₄²

C₂²×D₅₃

yes

—

109

C₂³

C₂×C₄×D₅₃

yes

characteristic

110

Q₈

—

106

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄×D₅₃

yes

—

114

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

115

C₂×C₄

C₂×C₄×D₅₃

yes

—

116

C₂×C₄

#11719947

yes

—

117

C₂×C₄

Q₈×D₅₃

yes

—

118

C₂×C₄

C₂³×D₅₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

106

—

123

C₂²

—

124

C₂²

#13133182

yes

—

125

C₂²

C₂²×C₄×D₅₃

yes

characteristic

126

C₄

—

127

C₄

—

106

—

128

C₄

#13133182

yes

—

129

C₄

#13133225

yes

—

130

C₄

C₂×Q₈×D₅₃

yes

characteristic

131

C₄

C₂×#11719947

yes

characteristic

132

C₂

—

133

C₂

C₂×#13133182

yes

characteristic

134

C₂

C₂×#13133225

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#423	C₂
hypercenter	#423	D₅₃
hypocenter	C₅₃	#2960
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#423

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #16810750	C₂	semidirect
3. #16810747	C₂	semidirect
37. C₂×#88322	Q₈	semidirect
75. C₅₃	#2960	semidirect

H ≅ G/N

type

2. #16810750

C₂

semidirect

3. #16810747

C₂

semidirect

37. C₂×#88322

Q₈

semidirect

75. C₅₃

#2960

semidirect

Subgroups — #20283645

Conjugacy classes of subgroups

Complements