Group #20283501 — subgroups

Subgroups — C₂×#16810729 #20283501

Order 6784.

480 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133182	C₂	1	2	yes	maximal
3	#16810729	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133140	C₂	1	2	yes	maximal
5	C₂×#13133127	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
8	C₂×Dic₂₁₂	C₄	1	8	yes	complemented
9	C₂×Dic₂₁₂	C₂²	1	4	yes	—
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	#13133182	—	2	4	no	—
12	#13133182	C₂²	1	8	yes	—
13	C₂×#11719919	—	2	2	no	—
14	C₂×C₄×Dic₅₃	—	2	2	no	—
15	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
16	#13133140	C₂²	1	8	yes	—
17	#13133127	C₂²	1	4	yes	—
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	#106×C₅₃	C₂²	1	4	yes	—
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₅₃	—	2	2	no	—
24	C₂²×Dic₅₃	D₄	1	2	yes	—
25	C₂×Dic₁₀₆	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	4	no	—
27	C₂×Dic₁₀₆	C₂×C₄	1	8	yes	—
28	C₂×Dic₁₀₆	C₂³	1	4	yes	—
29	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₅₃	C₂³	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
33	#11719919	—	2	8	no	—
34	C₄×Dic₅₃	—	2	8	no	—
35	C₄×Dic₅₃	D₄	1	2	yes	—
36	C₄×Dic₅₃	C₂³	1	2	yes	—
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	C₂³	1	2	yes	—
39	Dic₂₁₂	C₂×C₄	1	16	yes	complemented
40	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	4	4	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#41	1	2	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	8	no	—
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	—	2	4	no	—
50	C₂×Dic₅₃	C₂×D₄	1	4	yes	—
51	Dic₁₀₆	—	2	8	no	—
52	Dic₁₀₆	—	2	4	no	—
53	Dic₁₀₆	C₂²×C₄	1	8	yes	—
54	C₈×C₅₃	C₂²×C₄	1	4	yes	—
55	#88322	—	2	2	no	—
56	C₂²×C₅₃	C₄×D₄	1	2	yes	—
57	C₂²×C₅₃	#137	1	4	yes	—
58	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	4	4	no	—
60	C₄×C₅₃	C₄×D₄	1	2	yes	—
61	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
63	Dic₅₃	—	2	8	no	—
64	Dic₅₃	—	2	4	no	—
65	C₂×#441	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	#441	1	4	yes	complemented
67	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
68	C₂×C₅₃	C₂×#137	1	2	yes	—
69	C₂²×Q₁₆	—	53	1	no	—
70	C₂×C₄×Q₈	—	53	2	no	—
71	#441	—	53	8	no	—
72	C₂×#106	D₅₃	1	1	yes	characteristic
73	C₂×#103	—	53	2	no	—
74	C₂×#97	—	53	1	no	—
75	C₅₃	C₂×#441	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	53	2	no	—
77	C₂×Q₁₆	—	53	8	no	—
78	C₂×Q₁₆	—	53	4	no	—
79	C₂²×C₈	—	53	1	no	—
80	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
81	C₄×Q₈	—	53	8	no	—
82	C₄×Q₈	—	106	4	no	—
83	C₂×C₄⋊C₄	—	106	2	no	—
84	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
85	C₂×C₄²	—	53	1	no	—
86	C₂×C₄²	—	106	2	no	—
87	#106	C₂×D₅₃	1	4	yes	—
88	#103	—	53	8	no	—
89	#97	—	53	4	no	—
90	C₂×Q₈	—	53	8	no	—
91	C₂×Q₈	—	53	4	no	—
92	C₂×Q₈	—	106	2	no	—
93	C₂×Q₈	—	106	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	53	2	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
98	Q₁₆	—	53	16	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	106	2	no	—
101	C₂×C₈	C₄×D₅₃	1	4	yes	—
102	C₂×C₈	C₂²×D₅₃	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	106	8	no	—
105	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
106	C₄²	—	53	2	no	—
107	C₄²	—	53	2	no	—
108	C₄²	—	106	8	no	—
109	C₂³	D₄×D₅₃	1	1	yes	characteristic
110	Q₈	—	53	8	no	—
111	Q₈	—	106	8	no	—
112	Q₈	—	106	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	53	4	no	—
116	C₂×C₄	—	106	8	no	—
117	C₂×C₄	—	106	4	no	—
118	C₂×C₄	—	106	4	no	—
119	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
120	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
121	C₂×C₄	#11719949	1	2	yes	—
122	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
123	C₈	—	106	2	no	—
124	C₈	C₂×C₄×D₅₃	1	4	yes	—
125	C₂²	#13133179	1	2	yes	—
126	C₂²	#13133246	1	4	yes	—
127	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	106	8	no	—
130	C₄	—	106	4	no	—
131	C₄	#13133179	1	2	yes	—
132	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
133	C₄	C₂×#11719949	1	1	yes	characteristic
134	C₂	#16810729	1	4	yes	direct product factor
135	C₂	C₂×#13133179	1	1	yes	characteristic
136	C₂	C₂×#13133246	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133182

C₂

yes

maximal

#16810729

C₂

yes

maximal, direct product factor

C₂×#13133140

C₂

yes

maximal

C₂×#13133127

C₂

yes

maximal, characteristic

C₂×#106×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

#13133182

—

#13133182

C₂²

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133140

C₂²

yes

—

#13133127

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#106×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂³

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

Dic₂₁₂

C₂×C₄

yes

complemented

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂²×C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

C₂×#137

yes

—

C₂²×Q₁₆

—

C₂×C₄×Q₈

—

#441

—

C₂×#106

D₅₃

yes

characteristic

C₂×#103

—

C₂×#97

—

C₅₃

C₂×#441

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

106

—

#106

C₂×D₅₃

yes

—

#103

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₂×C₈

C₄×D₅₃

yes

—

102

C₂×C₈

C₂²×D₅₃

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

C₂²×D₅₃

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

106

—

109

C₂³

D₄×D₅₃

yes

characteristic

110

Q₈

—

111

Q₈

—

106

—

112

Q₈

—

106

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

—

106

—

119

C₂×C₄

C₂×C₄×D₅₃

yes

—

120

C₂×C₄

D₄×D₅₃

yes

characteristic

121

C₂×C₄

#11719949

yes

—

122

C₂×C₄

C₂³×D₅₃

yes

characteristic

123

C₈

—

106

—

124

C₈

C₂×C₄×D₅₃

yes

—

125

C₂²

#13133179

yes

—

126

C₂²

#13133246

yes

—

127

C₂²

C₂×D₄×D₅₃

yes

characteristic

128

C₄

—

129

C₄

—

106

—

130

C₄

—

106

—

131

C₄

#13133179

yes

—

132

C₄

C₂²×C₄×D₅₃

yes

characteristic

133

C₄

C₂×#11719949

yes

characteristic

134

C₂

#16810729

yes

direct product factor

135

C₂

C₂×#13133179

yes

characteristic

136

C₂

C₂×#13133246

yes

—

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#106×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#106	D₅₃
hypocenter	C₅₃	C₂×#441
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#106×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #16810729	C₂	direct (N × H)
8. C₂×Dic₂₁₂	C₄	semidirect
39. Dic₂₁₂	C₂×C₄	semidirect
66. C₂×C₅₃	#441	semidirect
75. C₅₃	C₂×#441	semidirect
134. C₂	#16810729	direct (N × H)

H ≅ G/N

type

3. #16810729

C₂

direct (N × H)

8. C₂×Dic₂₁₂

C₄

semidirect

39. Dic₂₁₂

C₂×C₄

semidirect

66. C₂×C₅₃

#441

semidirect

75. C₅₃

C₂×#441

semidirect

134. C₂

#16810729

direct (N × H)

Subgroups — C₂×#16810729 #20283501

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810729 #20283501

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810729 #20283501