Group #20283479 — subgroups

Subgroups — C₂×#16810717 #20283479

Order 6784.

488 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 205 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	#16810717	C₂	1	8	yes	maximal, direct product factor
4	C₂×#13133140	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×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	C₈×Dic₅₃	C₂²	1	4	yes	—
18	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	#107×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	1	yes	characteristic
25	C₂²×Dic₅₃	D₄	1	1	yes	characteristic
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	4	no	—
28	C₂×Dic₁₀₆	C₂×C₄	1	8	yes	—
29	C₂×Dic₁₀₆	C₂³	1	4	yes	—
30	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
31	C₂×C₈×C₅₃	C₂³	1	2	yes	—
32	C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
34	#11719919	—	2	8	no	—
35	C₄×Dic₅₃	—	2	8	no	—
36	C₄×Dic₅₃	D₄	1	2	yes	—
37	C₄×Dic₅₃	C₂³	1	2	yes	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	C₂³	1	2	yes	—
40	Dic₂₁₂	C₂×C₄	1	16	yes	complemented
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	4	4	no	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	8	no	—
49	C₂×Dic₅₃	—	2	2	no	—
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	Q₁₆	1	4	yes	complemented
52	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
53	C₂×Dic₅₃	C₂×D₄	1	2	yes	—
54	Dic₁₀₆	—	2	8	no	—
55	Dic₁₀₆	—	2	4	no	—
56	Dic₁₀₆	C₂²×C₄	1	8	yes	—
57	C₈×C₅₃	C₂²×C₄	1	4	yes	—
58	#88322	—	2	2	no	—
59	C₂²×C₅₃	C₄×D₄	1	2	yes	—
60	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
61	C₂²×C₅₃	#135	1	2	yes	—
62	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₅₃	—	4	4	no	—
64	C₄×C₅₃	C₄×D₄	1	2	yes	—
65	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
67	Dic₅₃	—	2	8	no	—
68	Dic₅₃	—	2	2	no	—
69	Dic₅₃	C₂×Q₁₆	1	4	yes	complemented
70	C₂×C₄×Q₁₆	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	C₄×Q₁₆	1	4	yes	complemented
72	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
73	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
75	C₂²×Q₁₆	—	53	1	no	—
76	C₂×C₄×Q₈	—	53	2	no	—
77	C₄×Q₁₆	—	53	8	no	—
78	C₂×#107	D₅₃	1	1	yes	characteristic
79	C₂×#103	—	53	2	no	—
80	C₂×C₄×C₈	—	53	1	no	—
81	C₅₃	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	53	2	no	—
83	C₂×Q₁₆	—	53	8	no	—
84	C₂×Q₁₆	—	53	4	no	—
85	C₂²×C₈	—	53	1	no	—
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₄×Q₈	—	53	8	no	—
88	C₄×Q₈	—	106	4	no	—
89	C₂×C₄⋊C₄	—	106	2	no	—
90	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
91	C₂×C₄²	—	53	1	no	—
92	C₂×C₄²	—	106	2	no	—
93	#107	C₂×D₅₃	1	4	yes	—
94	#103	—	53	8	no	—
95	C₄×C₈	—	53	4	no	—
96	C₂×Q₈	—	53	8	no	—
97	C₂×Q₈	—	53	4	no	—
98	C₂×Q₈	—	106	2	no	—
99	C₂×Q₈	—	106	4	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	53	1	no	—
102	C₂²×C₄	—	53	1	no	—
103	C₂²×C₄	—	106	2	no	—
104	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
105	Q₁₆	—	53	16	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	106	2	no	—
108	C₂×C₈	C₄×D₅₃	1	4	yes	—
109	C₂×C₈	C₂²×D₅₃	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	106	8	no	—
112	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
113	C₄²	—	53	2	no	—
114	C₄²	—	53	2	no	—
115	C₄²	—	106	8	no	—
116	C₂³	D₄×D₅₃	1	1	yes	characteristic
117	Q₈	—	53	8	no	—
118	Q₈	—	106	8	no	—
119	Q₈	—	106	4	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	—	53	4	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	106	8	no	—
126	C₂×C₄	—	106	2	no	—
127	C₂×C₄	—	106	4	no	—
128	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
129	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
130	C₂×C₄	#11719949	1	2	yes	—
131	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
132	C₈	—	106	2	no	—
133	C₈	C₂×C₄×D₅₃	1	4	yes	—
134	C₂²	#13133179	1	2	yes	—
135	C₂²	Q₁₆×D₅₃	1	2	yes	—
136	C₂²	#13133236	1	2	yes	—
137	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
138	C₄	—	4	4	no	—
139	C₄	—	53	4	no	—
140	C₄	—	106	8	no	—
141	C₄	—	106	2	no	—
142	C₄	#13133179	1	2	yes	—
143	C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
144	C₄	C₂×#11719949	1	1	yes	characteristic
145	C₂	#16810717	1	4	yes	direct product factor
146	C₂	C₂×#13133179	1	1	yes	characteristic
147	C₂	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
148	C₂	C₂×#13133236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133182

C₂

yes

maximal

#16810717

C₂

yes

maximal, direct product factor

C₂×#13133140

C₂

yes

maximal

C₂×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#107×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

—

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×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

Q₁₆

yes

complemented

C₂×Dic₅₃

C₂×D₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

C₂²×C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

—

#88322

—

C₂²×C₅₃

C₄×D₄

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

#135

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

—

Dic₅₃

C₂×Q₁₆

yes

complemented

C₂×C₄×Q₁₆

—

maximal, Sylow

C₂×C₅₃

C₄×Q₁₆

yes

complemented

C₂×C₅₃

C₂×C₄×D₄

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₅₃

C₂×#135

yes

characteristic

C₂²×Q₁₆

—

C₂×C₄×Q₈

—

C₄×Q₁₆

—

C₂×#107

D₅₃

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

C₅₃

C₂×C₄×Q₁₆

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

—

#107

C₂×D₅₃

yes

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

106

—

104

C₂²×C₄

C₂²×D₅₃

yes

characteristic

105

Q₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

106

—

108

C₂×C₈

C₄×D₅₃

yes

—

109

C₂×C₈

C₂²×D₅₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

106

—

112

C₄⋊C₄

C₂²×D₅₃

yes

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

106

—

116

C₂³

D₄×D₅₃

yes

characteristic

117

Q₈

—

118

Q₈

—

106

—

119

Q₈

—

106

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

106

—

126

C₂×C₄

—

106

—

127

C₂×C₄

—

106

—

128

C₂×C₄

C₂×C₄×D₅₃

yes

—

129

C₂×C₄

D₄×D₅₃

yes

characteristic

130

C₂×C₄

#11719949

yes

—

131

C₂×C₄

C₂³×D₅₃

yes

characteristic

132

C₈

—

106

—

133

C₈

C₂×C₄×D₅₃

yes

—

134

C₂²

#13133179

yes

—

135

C₂²

Q₁₆×D₅₃

yes

—

136

C₂²

#13133236

yes

—

137

C₂²

C₂×D₄×D₅₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

106

—

141

C₄

—

106

—

142

C₄

#13133179

yes

—

143

C₄

C₂²×C₄×D₅₃

yes

characteristic

144

C₄

C₂×#11719949

yes

characteristic

145

C₂

#16810717

yes

direct product factor

146

C₂

C₂×#13133179

yes

characteristic

147

C₂

C₂×Q₁₆×D₅₃

yes

characteristic

148

C₂

C₂×#13133236

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#107×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#107	D₅₃
hypocenter	C₅₃	C₂×C₄×Q₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₅₃

commutator

C₄×C₅₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#107×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×#107

D₅₃

hypocenter

C₅₃

C₂×C₄×Q₁₆

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
3. #16810717	C₂	direct (N × H)
8. C₂×Dic₂₁₂	C₄	semidirect
40. Dic₂₁₂	C₂×C₄	semidirect
51. C₂×Dic₅₃	Q₁₆	semidirect
69. Dic₅₃	C₂×Q₁₆	semidirect
71. C₂×C₅₃	C₄×Q₁₆	semidirect
81. C₅₃	C₂×C₄×Q₁₆	semidirect
145. C₂	#16810717	direct (N × H)

H ≅ G/N

type

3. #16810717

C₂

direct (N × H)

8. C₂×Dic₂₁₂

C₄

semidirect

40. Dic₂₁₂

C₂×C₄

semidirect

51. C₂×Dic₅₃

Q₁₆

semidirect

69. Dic₅₃

C₂×Q₁₆

semidirect

71. C₂×C₅₃

C₄×Q₁₆

semidirect

81. C₅₃

C₂×C₄×Q₁₆

semidirect

145. C₂

#16810717

direct (N × H)

Subgroups — C₂×#16810717 #20283479

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810717 #20283479

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810717 #20283479