Group #20285218 — subgroups

Subgroups — C₂×#16811024 #20285218

Order 6784.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#13133208	C₂	1	2	yes	maximal
2	#16811024	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133148	C₂	1	2	yes	maximal
5	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	#13133208	—	2	4	no	—
9	#13133208	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	2	no	—
11	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂×#11719919	—	2	2	no	—
13	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#13133151	C₂²	1	4	yes	—
15	#13133148	C₂²	1	8	yes	—
16	C₈×Dic₅₃	C₂²	1	4	yes	—
17	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	#107×C₅₃	C₂²	1	4	yes	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	Q₈	1	4	yes	—
26	C₂×C₈×C₅₃	C₂³	1	2	yes	—
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	—	2	8	no	—
31	C₄⋊Dic₅₃	C₂³	1	4	yes	—
32	#11719919	—	2	8	no	—
33	C₄×Dic₅₃	D₄	1	2	yes	—
34	C₄×Dic₅₃	C₂³	1	2	yes	—
35	C₂×#88322	Q₈	1	4	yes	—
36	C₂×#88322	C₂³	1	2	yes	—
37	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	4	4	no	—
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	—	2	2	no	—
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	4	4	no	—
48	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
49	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
50	C₈×C₅₃	C₂×Q₈	1	4	yes	—
51	#88322	C₂×Q₈	1	4	yes	—
52	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
53	C₂²×C₅₃	#135	1	2	yes	—
54	C₂²×C₅₃	#135	1	2	yes	—
55	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₅₃	—	4	4	no	—
57	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
58	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
59	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
60	Dic₅₃	—	2	2	no	—
61	Dic₅₃	—	2	2	no	—
62	Dic₅₃	—	4	4	no	—
63	C₂×#499	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	#499	1	4	yes	complemented
65	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
68	C₂×#125	—	53	2	no	—
69	#499	—	53	8	no	—
70	C₂×#107	—	53	1	no	—
71	C₂×#107	D₅₃	1	1	yes	characteristic
72	C₂×#106	—	53	2	no	—
73	C₂×C₄×C₈	—	53	1	no	—
74	C₅₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	53	1	no	—
76	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
77	#125	—	53	8	no	—
78	#125	—	106	4	no	—
79	C₂×C₄⋊C₄	—	53	2	no	—
80	C₂×C₄⋊C₄	—	106	2	no	—
81	C₂×C₄⋊C₄	—	106	2	no	—
82	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
83	C₂×C₄²	—	53	1	no	—
84	#107	—	53	4	no	—
85	#107	C₂×D₅₃	1	4	yes	—
86	#106	—	53	8	no	—
87	C₄×C₈	—	53	4	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	53	1	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	106	2	no	—
92	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
93	C₂×C₈	—	53	4	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	Dic₁₀₆	1	4	yes	—
96	C₂×C₈	C₂²×D₅₃	1	2	yes	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	53	4	no	—
99	C₄⋊C₄	—	106	8	no	—
100	C₄⋊C₄	—	106	8	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
103	C₄²	—	53	2	no	—
104	C₄²	—	53	2	no	—
105	C₂³	D₄×D₅₃	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	—	53	2	no	—
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	106	2	no	—
111	C₂×C₄	—	106	2	no	—
112	C₂×C₄	—	106	4	no	—
113	C₂×C₄	—	212	4	no	—
114	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
115	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₅₃	1	2	yes	—
117	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
118	C₈	—	53	4	no	—
119	C₈	C₂×Dic₁₀₆	1	4	yes	—
120	C₂²	#13133216	1	2	yes	—
121	C₂²	#13133236	1	2	yes	—
122	C₂²	#13133240	1	2	yes	—
123	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
124	C₄	—	4	4	no	—
125	C₄	—	106	2	no	—
126	C₄	—	106	2	no	—
127	C₄	—	212	4	no	—
128	C₄	#13133216	1	2	yes	—
129	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
130	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
131	C₂	#16811024	1	4	yes	direct product factor
132	C₂	C₂×#13133216	1	1	yes	characteristic
133	C₂	C₂×#13133236	1	1	yes	characteristic
134	C₂	C₂×#13133240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133208

C₂

yes

maximal

#16811024

C₂

yes

maximal, direct product factor

C₂×#13133151

C₂

yes

maximal, characteristic

C₂×#13133148

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#107×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

#13133208

—

#13133208

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133148

C₂²

yes

—

C₈×Dic₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

#107×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂×#88322

Q₈

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

C₈×C₅₃

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₄⋊Q₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

D₅₃

yes

characteristic

C₂×#106

—

C₂×C₄×C₈

—

C₅₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

#125

—

#125

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

—

#107

—

#107

C₂×D₅₃

yes

—

#106

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

C₂²×D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

C₂²×D₅₃

yes

—

103

C₄²

—

104

C₄²

—

105

C₂³

D₄×D₅₃

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

106

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

106

—

113

C₂×C₄

—

212

—

114

C₂×C₄

C₂×Dic₁₀₆

yes

—

115

C₂×C₄

D₄×D₅₃

yes

characteristic

116

C₂×C₄

Q₈×D₅₃

yes

—

117

C₂×C₄

C₂³×D₅₃

yes

characteristic

118

C₈

—

119

C₈

C₂×Dic₁₀₆

yes

—

120

C₂²

#13133216

yes

—

121

C₂²

#13133236

yes

—

122

C₂²

#13133240

yes

—

123

C₂²

C₂×D₄×D₅₃

yes

characteristic

124

C₄

—

125

C₄

—

106

—

126

C₄

—

106

—

127

C₄

—

212

—

128

C₄

#13133216

yes

—

129

C₄

C₂×Q₈×D₅₃

yes

characteristic

130

C₄

C₂²×Dic₁₀₆

yes

characteristic

131

C₂

#16811024

yes

direct product factor

132

C₂

C₂×#13133216

yes

characteristic

133

C₂

C₂×#13133236

yes

characteristic

134

C₂

C₂×#13133240

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	C₂×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
2. #16811024	C₂	direct (N × H)
64. C₂×C₅₃	#499	semidirect
74. C₅₃	C₂×#499	semidirect
131. C₂	#16811024	direct (N × H)

H ≅ G/N

type

2. #16811024

C₂

direct (N × H)

64. C₂×C₅₃

#499

semidirect

74. C₅₃

C₂×#499

semidirect

131. C₂

#16811024

direct (N × H)

Subgroups — C₂×#16811024 #20285218

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811024 #20285218

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811024 #20285218