Group #20285030 — subgroups

Subgroups — C₂×#16810977 #20285030

Order 6784.

412 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133209	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810977	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133141	C₂	1	2	yes	maximal, complemented
4	C₂×#13133138	C₂	1	2	yes	maximal
5	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#88322	C₂²	1	2	yes	—
8	#13133209	—	2	2	no	—
9	#13133209	C₂²	1	4	yes	complemented
10	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	C₂×#11719919	—	2	2	no	—
12	#13133141	C₂²	1	8	yes	complemented
13	#13133138	C₂²	1	8	yes	—
14	C₄×#88322	C₂²	1	4	yes	complemented
15	C₂²×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
17	#124×C₅₃	—	2	2	no	—
18	#124×C₅₃	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂⁴×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	—	2	1	no	—
23	C₂×Q₈×C₅₃	—	2	4	no	—
24	C₂×Q₈×C₅₃	C₂³	1	2	yes	—
25	C₂×D₄×C₅₃	—	2	1	no	—
26	C₂×D₄×C₅₃	—	2	4	no	—
27	C₂×D₄×C₅₃	C₂³	1	2	yes	—
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂²×C₄×C₅₃	D₄	1	2	yes	—
30	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
31	C₂²×Dic₅₃	—	2	2	no	—
32	C₂²⋊C₄×C₅₃	—	2	8	no	—
33	C₄²×C₅₃	D₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	2	yes	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	C₂³	1	4	yes	—
37	#11719919	—	2	8	no	—
38	C₂×#88322	—	2	4	no	—
39	C₂×#88322	C₂³	1	4	yes	—
40	C₂³×C₅₃	—	2	2	no	—
41	C₂³×C₅₃	—	4	2	no	—
42	C₂³×C₅₃	—	4	4	no	—
43	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₅₃	—	2	4	no	—
45	Q₈×C₅₃	—	4	2	no	—
46	D₄×C₅₃	—	2	4	no	—
47	D₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	2	4	no	—
49	C₂×C₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
52	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#41	1	4	yes	—
54	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₅₃	—	2	4	no	—
56	C₂×Dic₅₃	—	4	4	no	—
57	#88322	—	2	4	no	—
58	C₂²×C₅₃	—	4	2	no	—
59	C₂²×C₅₃	—	4	4	no	—
60	C₂²×C₅₃	—	8	4	no	—
61	C₂²×C₅₃	#124	1	2	yes	—
62	C₂²×C₅₃	#135	1	4	yes	—
63	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₅₃	—	2	4	no	—
65	C₄×C₅₃	—	4	2	no	—
66	C₄×C₅₃	#124	1	2	yes	—
67	C₄×C₅₃	C₂×#41	1	2	yes	—
68	Dic₅₃	—	4	4	no	—
69	C₂×#488	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	8	2	no	—
71	C₂×C₅₃	#488	1	4	yes	complemented
72	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
73	C₂×C₅₃	C₂×#135	1	2	yes	—
74	C₂×#125	—	53	1	no	—
75	C₂×#124	D₅₃	1	1	yes	characteristic
76	#488	—	53	8	no	—
77	C₂×#103	—	53	2	no	—
78	C₂×#102	—	53	2	no	—
79	C₂×C₄×C₈	—	53	1	no	—
80	C₅₃	C₂×#488	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
83	C₂²×C₈	—	53	2	no	—
84	#125	—	53	4	no	—
85	#125	—	106	2	no	—
86	#124	—	2	2	no	—
87	#124	C₂×D₅₃	1	4	yes	—
88	C₂×C₄⋊C₄	—	53	2	no	—
89	C₂×C₄⋊C₄	—	106	2	no	—
90	C₂×C₂²⋊C₄	—	2	2	no	—
91	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
92	#103	—	53	8	no	—
93	#102	—	53	8	no	—
94	C₄×C₈	—	53	4	no	—
95	C₂⁴	—	2	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	C₂²×D₅₃	1	2	yes	—
99	C₂×D₄	—	2	4	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	C₂²×D₅₃	1	2	yes	—
102	C₂²×C₄	—	2	1	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
105	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
106	C₂×C₈	—	53	4	no	—
107	C₂×C₈	—	106	4	no	—
108	C₄⋊C₄	—	53	4	no	—
109	C₄⋊C₄	—	106	2	no	—
110	C₄⋊C₄	—	106	8	no	—
111	C₂²⋊C₄	—	2	8	no	—
112	C₄²	C₂²⋊D₅₃	1	2	yes	—
113	C₄²	C₂²×D₅₃	1	2	yes	—
114	C₂³	—	2	2	no	—
115	C₂³	—	4	2	no	—
116	C₂³	—	4	4	no	—
117	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	106	4	no	—
126	C₂×C₄	—	212	4	no	—
127	C₂×C₄	#11719947	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
131	C₈	—	106	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	8	4	no	—
135	C₂²	#13133204	1	2	yes	—
136	C₂²	#13133237	1	4	yes	—
137	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	—	212	4	no	—
141	C₄	#13133204	1	2	yes	—
142	C₄	C₂×#11719947	1	2	yes	—
143	C₂	—	8	2	no	—
144	C₂	#16810977	1	4	yes	direct product factor
145	C₂	C₂×#13133204	1	1	yes	characteristic
146	C₂	C₂×#13133237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133209

C₂

yes

maximal, characteristic, complemented

#16810977

C₂

yes

maximal, direct product factor

C₂×#13133141

C₂

yes

maximal, complemented

C₂×#13133138

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

#13133209

—

#13133209

C₂²

yes

complemented

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#11719919

—

#13133141

C₂²

yes

complemented

#13133138

C₂²

yes

—

C₄×#88322

C₂²

yes

complemented

C₂²×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×D₄×C₅₃

C₂²

yes

characteristic

#124×C₅₃

—

#124×C₅₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

#135

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

C₂×#488

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×#125

—

C₂×#124

D₅₃

yes

—

—

—

C₂×C₄×C₈

—

C₅₃

C₂×#488

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

characteristic

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

#125

—

#125

—

106

—

#124

—

#124

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

#102

—

C₄×C₈

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

C₂²×D₅₃

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₂²⋊D₅₃

yes

—

105

C₂²×C₄

C₂²×D₅₃

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

106

—

111

C₂²⋊C₄

—

112

C₄²

C₂²⋊D₅₃

yes

—

113

C₄²

C₂²×D₅₃

yes

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

118

Q₈

—

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

212

—

127

C₂×C₄

#11719947

yes

—

128

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂×C₄

C₂³×D₅₃

yes

characteristic

131

C₈

—

106

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#13133204

yes

—

136

C₂²

#13133237

yes

—

137

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

212

—

141

C₄

#13133204

yes

—

142

C₄

C₂×#11719947

yes

—

143

C₂

—

144

C₂

#16810977

yes

direct product factor

145

C₂

C₂×#13133204

yes

characteristic

146

C₂

C₂×#13133237

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#124×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#124	D₅₃
hypocenter	C₅₃	C₂×#488
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#124×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#13133209	C₂	semidirect
2. #16810977	C₂	direct (N × H)
3. C₂×#13133141	C₂	semidirect
5. C₂×C₄×#88322	C₂	semidirect
9. #13133209	C₂²	semidirect
12. #13133141	C₂²	semidirect
14. C₄×#88322	C₂²	semidirect
71. C₂×C₅₃	#488	semidirect
80. C₅₃	C₂×#488	semidirect
144. C₂	#16810977	direct (N × H)

H ≅ G/N

type

1. C₂×#13133209

C₂

semidirect

2. #16810977

C₂

direct (N × H)

3. C₂×#13133141

C₂

semidirect

5. C₂×C₄×#88322

C₂

semidirect

9. #13133209

C₂²

semidirect

12. #13133141

C₂²

semidirect

14. C₄×#88322

C₂²

semidirect

71. C₂×C₅₃

#488

semidirect

80. C₅₃

C₂×#488

semidirect

144. C₂

#16810977

direct (N × H)

Subgroups — C₂×#16810977 #20285030

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810977 #20285030

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810977 #20285030