Group #20291681 — subgroups

Subgroups — C₂×#16811682 #20291681

Order 6784.

368 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 202 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	#16811682	C₂	1	8	yes	maximal, direct product factor
2	C₂×#13133292	C₂	1	2	yes	maximal, complemented
3	C₂×#13133291	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₅₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₂²×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13133225	C₂	1	1	yes	maximal, characteristic
7	#13133292	C₂²	1	8	yes	complemented
8	#13133291	C₂²	1	4	yes	complemented
9	D₅₃⋊C₁₆	C₂²	1	8	yes	complemented
10	C₂×#11719910	C₂²	1	1	yes	characteristic
11	C₂×#11719910	C₂²	1	1	yes	characteristic
12	C₂×#11719910	C₂²	1	2	yes	—
13	C₂×#11719910	C₂²	1	4	yes	complemented
14	C₂×#11719945	C₄	1	1	yes	characteristic
15	#13133225	C₄	1	4	yes	—
16	#13133225	C₂²	1	4	yes	—
17	C₂×#11719912	C₄	1	2	yes	—
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₃	C₂²	1	2	yes	—
20	C₂×M₁₆×C₅₃	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₅₃	C₈	1	2	yes	—
23	#11719945	C₈	1	4	yes	—
24	#11719945	C₂×C₄	1	4	yes	—
25	C₂×D₂₁₂	C₈	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
27	C₂×Dic₁₀₆	C₈	1	1	yes	characteristic
28	M₁₆×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	C₂³	1	1	yes	characteristic
33	C₂×#88322	C₂³	1	2	yes	—
34	#11719912	C₂×C₄	1	8	yes	—
35	C₈×D₅₃	C₂×C₄	1	4	yes	—
36	C₈×D₅₃	C₂³	1	4	yes	—
37	#11719910	C₂³	1	2	yes	—
38	#11719910	C₂³	1	2	yes	—
39	#11719910	C₂³	1	4	yes	—
40	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₅₃	C₂×C₈	1	8	yes	—
47	C₂×Dic₅₃	C₂×C₈	1	2	yes	—
48	D₂₁₂	C₂×C₈	1	4	yes	—
49	C₄×D₅₃	C₂×C₈	1	4	yes	—
50	C₄×D₅₃	C₂²×C₄	1	4	yes	—
51	Dic₁₀₆	C₂×C₈	1	4	yes	—
52	C₈×C₅₃	C₂²×C₄	1	4	yes	—
53	#88322	C₂²×C₄	1	1	yes	characteristic
54	#88322	C₂²×C₄	1	2	yes	—
55	#88322	C₂⁴	1	1	yes	characteristic
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
58	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×D₅₃	—	2	4	no	—
60	C₂×D₅₃	C₂²×C₈	1	4	yes	—
61	C₄×C₅₃	C₂²×C₈	1	2	yes	—
62	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
64	Dic₅₃	C₂²×C₈	1	4	yes	—
65	C₂×#504	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	—	2	2	no	—
67	C₂×C₅₃	#504	1	2	yes	complemented
68	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
69	D₅₃	—	2	4	no	—
70	C₂×#131	—	53	1	no	—
71	#504	—	53	8	no	—
72	C₂×M₃₂	—	53	1	no	—
73	C₂×M₃₂	—	53	2	no	—
74	C₂²×C₁₆	—	53	1	no	—
75	C₂²×C₁₆	—	53	2	no	—
76	C₅₃	C₂×#504	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	53	1	no	—
78	#131	—	53	4	no	—
79	#131	—	53	4	no	—
80	C₂×M₁₆	—	53	2	no	—
81	C₂×M₁₆	#6157	1	1	yes	characteristic
82	C₂²×C₈	—	53	1	no	—
83	C₂²×C₈	—	53	2	no	—
84	M₃₂	—	53	4	no	—
85	M₃₂	—	53	8	no	—
86	C₂×C₁₆	—	53	1	no	—
87	C₂×C₁₆	—	53	1	no	—
88	C₂×C₁₆	—	53	2	no	—
89	C₂×C₁₆	—	53	4	no	—
90	C₂×C₁₆	—	53	8	no	—
91	#41	—	53	4	no	—
92	#41	—	53	4	no	—
93	C₂×Q₈	—	53	1	no	—
94	C₂×D₄	—	53	2	no	—
95	C₂×D₄	—	53	1	no	—
96	C₂²×C₄	—	53	2	no	—
97	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
98	M₁₆	—	53	8	no	—
99	M₁₆	C₂×#6157	1	4	yes	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	—	53	2	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	53	4	no	—
105	C₂×C₈	—	53	4	no	—
106	C₂×C₈	C₂×#6157	1	2	yes	—
107	C₁₆	—	53	2	no	—
108	C₁₆	—	53	2	no	—
109	C₁₆	—	53	4	no	—
110	C₂³	—	53	2	no	—
111	C₂³	D₅₃⋊C₈	1	1	yes	characteristic
112	Q₈	—	53	4	no	—
113	D₄	—	53	8	no	—
114	D₄	—	53	4	no	—
115	C₂×C₄	—	53	2	no	—
116	C₂×C₄	—	53	4	no	—
117	C₂×C₄	—	53	4	no	—
118	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
119	C₂×C₄	D₅₃⋊C₈	1	2	yes	—
120	C₂×C₄	C₂²×#6157	1	2	yes	—
121	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
122	C₈	—	53	1	no	—
123	C₈	—	53	2	no	—
124	C₈	—	53	1	no	—
125	C₈	C₂²×#6157	1	4	yes	—
126	C₂²	—	2	2	no	—
127	C₂²	—	53	4	no	—
128	C₂²	—	106	4	no	—
129	C₂²	C₂×D₅₃⋊C₈	1	2	yes	—
130	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
131	C₄	—	53	4	no	—
132	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
133	C₄	C₂×D₅₃⋊C₈	1	2	yes	—
134	C₄	C₂³×#6157	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	106	4	no	—
137	C₂	#16811682	1	2	yes	direct product factor
138	C₂	C₂²×D₅₃⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811682

C₂

yes

maximal, direct product factor

C₂×#13133292

C₂

yes

maximal, complemented

C₂×#13133291

C₂

yes

maximal, characteristic, complemented

C₂×D₅₃⋊C₁₆

C₂

yes

maximal, complemented

C₂²×#11719910

C₂

yes

maximal, characteristic, complemented

C₂×#13133225

C₂

yes

maximal, characteristic

#13133292

C₂²

yes

complemented

#13133291

C₂²

yes

complemented

D₅₃⋊C₁₆

C₂²

yes

complemented

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

characteristic

C₂×#11719910

C₂²

yes

—

C₂×#11719910

C₂²

yes

complemented

C₂×#11719945

C₄

yes

characteristic

#13133225

C₄

yes

—

#13133225

C₂²

yes

—

C₂×#11719912

C₄

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

—

C₂×M₁₆×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

C₈

yes

—

#11719945

C₈

yes

—

#11719945

C₂×C₄

yes

—

C₂×D₂₁₂

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₈

yes

characteristic

M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

#11719912

C₂×C₄

yes

—

C₈×D₅₃

C₂×C₄

yes

—

C₈×D₅₃

C₂³

yes

—

#11719910

C₂³

yes

—

#11719910

C₂³

yes

—

#11719910

C₂³

yes

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

C₂×C₈

yes

—

C₂×Dic₅₃

C₂×C₈

yes

—

D₂₁₂

C₂×C₈

yes

—

C₄×D₅₃

C₂×C₈

yes

—

C₄×D₅₃

C₂²×C₄

yes

—

Dic₁₀₆

C₂×C₈

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

#88322

C₂²×C₄

yes

characteristic

#88322

C₂²×C₄

yes

—

#88322

C₂⁴

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

—

C₄×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

C₂³×C₄

yes

characteristic

Dic₅₃

C₂²×C₈

yes

—

C₂×#504

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#504

yes

complemented

C₂×C₅₃

C₂³×C₈

yes

characteristic

D₅₃

—

C₂×#131

—

#504

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₅₃

C₂×#504

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

#6157

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

characteristic

M₁₆

—

M₁₆

C₂×#6157

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#6157

yes

—

107

C₁₆

—

108

C₁₆

—

109

C₁₆

—

110

C₂³

—

111

C₂³

D₅₃⋊C₈

yes

characteristic

112

Q₈

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

D₅₃⋊C₈

yes

characteristic

119

C₂×C₄

D₅₃⋊C₈

yes

—

120

C₂×C₄

C₂²×#6157

yes

—

121

C₂×C₄

C₂²×#6157

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

C₂²×#6157

yes

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

106

—

129

C₂²

C₂×D₅₃⋊C₈

yes

—

130

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

131

C₄

—

132

C₄

C₂×D₅₃⋊C₈

yes

characteristic

133

C₄

C₂×D₅₃⋊C₈

yes

—

134

C₄

C₂³×#6157

yes

characteristic

135

C₂

—

136

C₂

—

106

—

137

C₂

#16811682

yes

direct product factor

138

C₂

C₂²×D₅₃⋊C₈

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂³×C₈
Frattini	C₄	C₂³×#6157
Fitting	C₂×M₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₈
hypercenter	C₂×M₁₆	#6157
hypocenter	C₅₃	C₂×#504
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²×#6157

commutator

C₂×C₅₃

C₂³×C₈

Frattini

C₄

C₂³×#6157

Fitting

C₂×M₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂²×C₈

C₂×M₁₆

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. #16811682	C₂	direct (N × H)
2. C₂×#13133292	C₂	semidirect
3. C₂×#13133291	C₂	semidirect
4. C₂×D₅₃⋊C₁₆	C₂	semidirect
5. C₂²×#11719910	C₂	semidirect
7. #13133292	C₂²	semidirect
8. #13133291	C₂²	semidirect
9. D₅₃⋊C₁₆	C₂²	semidirect
13. C₂×#11719910	C₂²	semidirect
67. C₂×C₅₃	#504	semidirect
76. C₅₃	C₂×#504	semidirect
137. C₂	#16811682	direct (N × H)

H ≅ G/N

type

1. #16811682

C₂

direct (N × H)

2. C₂×#13133292

C₂

semidirect

3. C₂×#13133291

C₂

semidirect

4. C₂×D₅₃⋊C₁₆

C₂

semidirect

5. C₂²×#11719910

C₂

semidirect

7. #13133292

C₂²

semidirect

8. #13133291

C₂²

semidirect

9. D₅₃⋊C₁₆

C₂²

semidirect

13. C₂×#11719910

C₂²

semidirect

67. C₂×C₅₃

#504

semidirect

76. C₅₃

C₂×#504

semidirect

137. C₂

#16811682

direct (N × H)

Subgroups — C₂×#16811682 #20291681

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811682 #20291681

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811682 #20291681