Group #20283339 — subgroups

Subgroups — C₂×#16810682 #20283339

Order 6784.

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

normal subgroups only

This group has 412 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	C₂×#13133174	C₂	1	1	yes	maximal, characteristic
2	#16810682	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133146	C₂	1	2	yes	maximal, complemented
4	C₂×#13133129	C₂	1	2	yes	maximal
5	C₂×#13133128	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
8	C₂²×#88322	C₂²	1	2	yes	—
9	#13133174	C₄	1	4	yes	—
10	#13133174	C₂²	1	4	yes	—
11	C₂×#11719919	C₄	1	2	yes	—
12	C₂×#11719921	C₄	1	2	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133146	C₂²	1	8	yes	complemented
15	#13133129	C₂²	1	8	yes	—
16	#13133128	C₂²	1	4	yes	complemented
17	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₂²	1	4	yes	complemented
20	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
24	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	4	no	—
26	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₅₃	C₂³	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₂³	1	4	yes	—
30	C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₄²×C₅₃	C₂³	1	2	yes	—
32	#11719921	C₂×C₄	1	8	yes	—
33	#11719919	C₂×C₄	1	8	yes	—
34	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
35	C₄×Dic₅₃	C₂³	1	2	yes	—
36	C₂×#88322	—	2	4	no	—
37	C₂×#88322	C₂³	1	4	yes	—
38	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×D₅₃	—	2	2	no	—
40	C₂²×D₅₃	—	2	4	no	—
41	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	4	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	2	no	—
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
50	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
51	C₄×D₅₃	—	2	4	no	—
52	C₄×D₅₃	—	2	4	no	—
53	C₈×C₅₃	—	2	4	no	—
54	#88322	—	2	4	no	—
55	C₂²×C₅₃	#117	1	2	yes	—
56	C₂²×C₅₃	#131	1	4	yes	—
57	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
58	C₂×D₅₃	—	2	2	no	—
59	C₂×D₅₃	—	2	4	no	—
60	C₂×D₅₃	—	4	4	no	—
61	C₄×C₅₃	—	2	4	no	—
62	C₄×C₅₃	#117	1	2	yes	—
63	C₄×C₅₃	C₂×#41	1	2	yes	—
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	—	2	4	no	—
66	C₂×#433	—	53	1	no	maximal, Sylow
67	C₂×C₅₃	#433	1	4	yes	complemented
68	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#131	1	2	yes	—
70	D₅₃	—	4	2	no	—
71	C₂×#117	—	53	1	no	—
72	#433	—	53	8	no	—
73	C₂×C₄⋊C₈	—	53	2	no	—
74	C₂×C₂²⋊C₈	—	53	2	no	—
75	C₂×#97	—	53	1	no	—
76	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic, complemented
77	C₅₃	C₂×#433	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₄	—	53	1	no	—
79	C₂²×C₈	—	53	2	no	—
80	C₂²×C₈	C₂×D₅₃	1	2	yes	—
81	#117	—	53	4	no	—
82	#117	—	53	4	no	—
83	C₂×C₄⋊C₄	—	53	2	no	—
84	C₂×C₂²⋊C₄	—	53	2	no	—
85	C₂×C₄²	—	53	1	no	—
86	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
87	C₄⋊C₈	—	53	8	no	—
88	C₂²⋊C₈	—	53	8	no	—
89	#97	—	53	4	no	—
90	C₄×C₈	C₂×D₅₃	1	4	yes	complemented
91	C₂⁴	—	53	1	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	—	53	1	no	—
94	C₂²×C₄	—	53	2	no	—
95	C₂²×C₄	—	53	2	no	—
96	C₂²×C₄	—	106	4	no	—
97	C₂²×C₄	C₄×D₅₃	1	2	yes	—
98	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₈	—	2	4	no	—
100	C₂×C₈	—	53	4	no	—
101	C₂×C₈	—	106	4	no	—
102	C₂×C₈	C₂²×D₅₃	1	4	yes	—
103	C₄⋊C₄	—	53	8	no	—
104	C₂²⋊C₄	—	53	8	no	—
105	C₄²	—	53	2	no	—
106	C₄²	—	53	2	no	—
107	C₄²	C₄×D₅₃	1	2	yes	—
108	C₄²	C₂²×D₅₃	1	2	yes	—
109	C₂³	—	53	2	no	—
110	C₂³	—	106	2	no	—
111	C₂³	—	106	4	no	—
112	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	53	4	no	—
115	C₂×C₄	—	53	2	no	—
116	C₂×C₄	—	106	2	no	—
117	C₂×C₄	—	106	4	no	—
118	C₂×C₄	—	106	4	no	—
119	C₂×C₄	—	106	4	no	—
120	C₂×C₄	C₂×C₄×D₅₃	1	4	yes	—
121	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
122	C₂×C₄	#11719945	1	4	yes	—
123	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₈	—	106	4	no	—
126	C₂²	—	106	2	no	—
127	C₂²	—	106	4	no	—
128	C₂²	—	212	4	no	—
129	C₂²	#13133174	1	2	yes	—
130	C₂²	#13133223	1	4	yes	—
131	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
132	C₄	—	2	4	no	—
133	C₄	—	106	2	no	—
134	C₄	—	106	4	no	—
135	C₄	#13133174	1	2	yes	—
136	C₄	C₂×#11719945	1	2	yes	—
137	C₂	—	212	2	no	—
138	C₂	#16810682	1	4	yes	direct product factor
139	C₂	C₂×#13133174	1	1	yes	characteristic
140	C₂	C₂×#13133223	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133174

C₂

yes

maximal, characteristic

#16810682

C₂

yes

maximal, direct product factor

C₂×#13133146

C₂

yes

maximal, complemented

C₂×#13133129

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅₃

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

—

#13133174

C₄

yes

—

#13133174

C₂²

yes

—

C₂×#11719919

C₄

yes

—

C₂×#11719921

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133146

C₂²

yes

complemented

#13133129

C₂²

yes

—

#13133128

C₂²

yes

complemented

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

complemented

C₂³×D₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719921

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

#131

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

—

C₂×#117

—

#433

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

D₅₃

yes

characteristic, complemented

C₅₃

C₂×#433

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×D₅₃

yes

complemented

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

C₂²×D₅₃

yes

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₄×D₅₃

yes

—

108

C₄²

C₂²×D₅₃

yes

—

109

C₂³

—

110

C₂³

—

106

—

111

C₂³

—

106

—

112

C₂³

C₂×C₄×D₅₃

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

C₂×C₄×D₅₃

yes

—

121

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

122

C₂×C₄

#11719945

yes

—

123

C₂×C₄

C₂³×D₅₃

yes

characteristic

124

C₈

—

125

C₈

—

106

—

126

C₂²

—

106

—

127

C₂²

—

106

—

128

C₂²

—

212

—

129

C₂²

#13133174

yes

—

130

C₂²

#13133223

yes

—

131

C₂²

C₂²×C₄×D₅₃

yes

characteristic

132

C₄

—

133

C₄

—

106

—

134

C₄

—

106

—

135

C₄

#13133174

yes

—

136

C₄

C₂×#11719945

yes

—

137

C₂

—

212

—

138

C₂

#16810682

yes

direct product factor

139

C₂

C₂×#13133174

yes

characteristic

140

C₂

C₂×#13133223

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#433
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16810682	C₂	direct (N × H)
3. C₂×#13133146	C₂	semidirect
5. C₂×#13133128	C₂	semidirect
6. C₂×C₄×C₈×C₅₃	C₂	semidirect
14. #13133146	C₂²	semidirect
16. #13133128	C₂²	semidirect
19. C₄×C₈×C₅₃	C₂²	semidirect
67. C₂×C₅₃	#433	semidirect
76. C₂×C₄×C₈	D₅₃	semidirect
77. C₅₃	C₂×#433	semidirect
90. C₄×C₈	C₂×D₅₃	semidirect
138. C₂	#16810682	direct (N × H)

H ≅ G/N

type

2. #16810682

C₂

direct (N × H)

3. C₂×#13133146

C₂

semidirect

5. C₂×#13133128

C₂

semidirect

6. C₂×C₄×C₈×C₅₃

C₂

semidirect

14. #13133146

C₂²

semidirect

16. #13133128

C₂²

semidirect

19. C₄×C₈×C₅₃

C₂²

semidirect

67. C₂×C₅₃

#433

semidirect

76. C₂×C₄×C₈

D₅₃

semidirect

77. C₅₃

C₂×#433

semidirect

90. C₄×C₈

C₂×D₅₃

semidirect

138. C₂

#16810682

direct (N × H)

Subgroups — C₂×#16810682 #20283339

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810682 #20283339

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810682 #20283339