Group #20283273 — subgroups

Subgroups — C₂×#16810649 #20283273

Order 6784.

512 conjugacy classes of subgroups; only the 269 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 269 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×#11719917	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
3	#16810649	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133146	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	—	2	4	no	—
7	C₂×#11719917	C₂²	1	4	yes	—
8	C₂²×#88322	C₂²	1	2	yes	—
9	#13133176	C₄	1	4	yes	—
10	#13133176	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
12	C₂×C₂²⋊Dic₅₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
14	#13133146	C₂²	1	16	yes	complemented
15	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	—	2	4	no	—
17	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₈×C₅₃	C₂²	1	4	yes	—
19	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	D₄	1	4	yes	—
23	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₅₃	—	2	4	no	—
28	C₂×C₈×C₅₃	—	2	8	no	—
29	C₂×C₈×C₅₃	C₂³	1	4	yes	—
30	C₂²⋊Dic₅₃	C₂×C₄	1	8	yes	—
31	C₄⋊Dic₅₃	C₂×C₄	1	8	yes	—
32	C₄×Dic₅₃	C₂×C₄	1	4	yes	—
33	C₄×Dic₅₃	C₂³	1	4	yes	—
34	#11719917	—	2	8	no	—
35	C₂×#88322	—	2	4	no	—
36	C₂×#88322	C₂³	1	4	yes	—
37	C₂³×C₅₃	—	2	4	no	—
38	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
39	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
40	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
47	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
48	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	8	no	—
51	C₂×Dic₅₃	C₂²×C₄	1	8	yes	—
52	C₈×C₅₃	—	2	4	no	—
53	#88322	—	2	4	no	—
54	C₂²×C₅₃	—	2	4	no	—
55	C₂²×C₅₃	—	2	8	no	—
56	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
57	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
58	C₂²×C₅₃	#131	1	4	yes	—
59	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
61	C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
62	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
64	Dic₅₃	—	2	8	no	—
65	C₂×#424	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	2	4	no	—
67	C₂×C₅₃	#424	1	4	yes	complemented
68	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#131	1	2	yes	—
70	C₂²×M₁₆	—	53	1	no	—
71	C₂³×C₈	D₅₃	1	1	yes	characteristic
72	C₂×#117	—	53	1	no	—
73	#424	—	53	8	no	—
74	C₂×C₄⋊C₈	—	53	4	no	—
75	C₅₃	C₂×#424	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
77	C₂×M₁₆	—	53	4	no	—
78	C₂×M₁₆	—	106	4	no	—
79	C₂²×C₈	—	2	4	no	—
80	C₂²×C₈	—	53	2	no	—
81	C₂²×C₈	C₂×D₅₃	1	4	yes	—
82	C₂²×C₈	C₂×D₅₃	1	2	yes	—
83	#117	—	53	4	no	—
84	#117	—	53	4	no	—
85	C₂×C₄⋊C₄	—	53	2	no	—
86	C₂×C₂²⋊C₄	—	53	2	no	—
87	C₂×C₄²	—	53	2	no	—
88	C₄⋊C₈	—	53	16	no	—
89	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
90	C₂²×C₄	—	53	4	no	—
91	C₂²×C₄	Dic₁₀₆	1	4	yes	—
92	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₅₃	1	2	yes	—
94	C₂²×C₄	C₂²⋊D₅₃	1	4	yes	—
95	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
96	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
97	M₁₆	—	106	8	no	—
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	8	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	4	no	—
106	C₄²	—	53	4	no	—
107	C₂³	—	2	4	no	—
108	C₂³	#11719919	1	4	yes	—
109	C₂³	C₂×C₄×D₅₃	1	2	yes	—
110	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
111	C₂×C₄	—	53	8	no	—
112	C₂×C₄	—	106	8	no	—
113	C₂×C₄	#11719919	1	8	yes	—
114	C₂×C₄	#11719919	1	4	yes	—
115	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
116	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
117	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
121	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	106	4	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	8	no	—
126	C₂²	C₂×#11719919	1	4	yes	—
127	C₂²	C₂×#11719919	1	2	yes	—
128	C₂²	#13133223	1	4	yes	—
129	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
130	C₄	—	106	8	no	—
131	C₄	C₂×#11719919	1	4	yes	—
132	C₄	C₂×#11719919	1	2	yes	—
133	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
134	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
135	C₂	—	2	4	no	—
136	C₂	#16810649	1	4	yes	direct product factor
137	C₂	C₂²×#11719919	1	1	yes	characteristic
138	C₂	C₂×#13133223	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal, characteristic

C₂×#13133176

C₂

yes

maximal, characteristic

#16810649

C₂

yes

maximal, direct product factor

C₂×#13133146

C₂

yes

maximal, complemented

C₂³×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133176

C₄

yes

—

#13133176

C₂²

yes

—

C₂×C₄⋊Dic₅₃

C₄

yes

—

C₂×C₂²⋊Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

#13133146

C₂²

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

—

C₂²⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

#131

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

—

C₂×#424

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#424

yes

complemented

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#131

yes

—

C₂²×M₁₆

—

C₂³×C₈

D₅₃

yes

characteristic

C₂×#117

—

#424

—

C₂×C₄⋊C₈

—

C₅₃

C₂×#424

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂²×C₈

C₂×D₅₃

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂⁴

C₄×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

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

C₂³

—

108

C₂³

#11719919

yes

—

109

C₂³

C₂×C₄×D₅₃

yes

—

110

C₂³

C₂×C₄×D₅₃

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

#11719919

yes

—

114

C₂×C₄

#11719919

yes

—

115

C₂×C₄

C₂×Dic₁₀₆

yes

—

116

C₂×C₄

C₂×Dic₁₀₆

yes

—

117

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

118

C₂×C₄

C₂×C₄×D₅₃

yes

—

119

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

121

C₂×C₄

C₂³×D₅₃

yes

characteristic

122

C₈

—

123

C₈

—

106

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×#11719919

yes

—

127

C₂²

C₂×#11719919

yes

—

128

C₂²

#13133223

yes

—

129

C₂²

C₂²×C₄×D₅₃

yes

characteristic

130

C₄

—

106

—

131

C₄

C₂×#11719919

yes

—

132

C₄

C₂×#11719919

yes

—

133

C₄

C₂²×C₂²⋊D₅₃

yes

characteristic

134

C₄

C₂²×Dic₁₀₆

yes

characteristic

135

C₂

—

136

C₂

#16810649

yes

direct product factor

137

C₂

C₂²×#11719919

yes

characteristic

138

C₂

C₂×#13133223

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂³×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂³×C₈	D₅₃
hypocenter	C₅₃	C₂×#424
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂³×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂³×C₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
3. #16810649	C₂	direct (N × H)
4. C₂×#13133146	C₂	semidirect
14. #13133146	C₂²	semidirect
67. C₂×C₅₃	#424	semidirect
75. C₅₃	C₂×#424	semidirect
136. C₂	#16810649	direct (N × H)

H ≅ G/N

type

3. #16810649

C₂

direct (N × H)

4. C₂×#13133146

C₂

semidirect

14. #13133146

C₂²

semidirect

67. C₂×C₅₃

#424

semidirect

75. C₅₃

C₂×#424

semidirect

136. C₂

#16810649

direct (N × H)

Subgroups — C₂×#16810649 #20283273

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810649 #20283273

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810649 #20283273