Group #20274507 — subgroups

Subgroups — C₄×#13133134 #20274507

Order 6784.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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×#13133134	C₂	1	2	yes	maximal
2	C₄×#11719917	C₂	1	2	yes	maximal, complemented
3	#16810392	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	4	yes	—
6	#13133134	C₄	1	8	yes	direct product factor
7	#13133128	—	2	2	no	—
8	C₄×#88322	—	2	2	no	—
9	C₂²×D₄×C₅₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅₃	C₄	1	2	yes	—
11	C₄×D₄×C₅₃	—	2	2	no	—
12	C₄×D₄×C₅₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₅₃	—	2	2	no	—
18	C₂×D₄×C₅₃	—	2	2	no	—
19	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	—	2	2	no	—
22	C₂²×C₄×C₅₃	—	4	4	no	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂²⋊C₄×C₅₃	—	2	4	no	—
29	C₄²×C₅₃	D₄	1	2	yes	—
30	C₄²×C₅₃	D₄	1	2	yes	—
31	#11719917	—	2	4	no	—
32	#11719917	C₂×C₄	1	8	yes	complemented
33	C₂×#88322	—	2	4	no	—
34	C₂³×C₅₃	—	2	2	no	—
35	C₂³×C₅₃	—	2	2	no	—
36	C₂³×C₅₃	—	4	2	no	—
37	C₂³×C₅₃	—	4	2	no	—
38	C₂³×C₅₃	C₄²	1	4	yes	—
39	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	D₄×C₅₃	—	2	8	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	—	2	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	4	no	—
45	C₂×C₄×C₅₃	—	4	4	no	—
46	C₂×C₄×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₅₃	#41	1	2	yes	—
53	#88322	—	2	8	no	—
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	4	no	—
57	C₂²×C₅₃	—	4	4	no	—
58	C₂²×C₅₃	—	4	4	no	—
59	C₂²×C₅₃	—	4	2	no	—
60	C₂²×C₅₃	—	4	2	no	—
61	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₅₃	#117	1	1	yes	characteristic
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	4	2	no	—
67	C₄×C₅₃	#100	1	2	yes	complemented
68	C₄×C₅₃	C₄×D₄	1	4	yes	—
69	C₄×#100	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	—	4	4	no	—
73	C₂×C₅₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₅₃	C₂×#100	1	1	yes	characteristic
75	C₂×C₅₃	#413	1	1	yes	characteristic
76	C₂×C₄×D₄	D₅₃	1	1	yes	characteristic
77	C₂×#100	—	53	2	no	—
78	C₄×M₁₆	—	53	2	no	—
79	#343	—	53	2	no	—
80	C₅₃	C₄×#100	1	1	yes	characteristic, complemented, Sylow
81	C₂²×D₄	C₂×D₅₃	1	1	yes	characteristic
82	C₂³×C₄	Dic₅₃	1	2	yes	—
83	C₂×M₁₆	—	53	4	no	—
84	C₄×D₄	—	2	2	no	—
85	C₄×D₄	—	2	2	no	—
86	C₂×C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
88	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
89	#100	—	53	8	no	—
90	#97	—	106	2	no	—
91	C₄×C₈	—	106	2	no	—
92	C₂⁴	C₂×Dic₅₃	1	2	yes	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	C₄×D₅₃	1	4	yes	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	4	4	no	—
99	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
100	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
101	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
103	M₁₆	—	53	8	no	—
104	M₁₆	—	106	4	no	—
105	C₂×C₈	—	106	4	no	—
106	C₄⋊C₄	—	2	2	no	—
107	C₂²⋊C₄	—	2	4	no	—
108	C₄²	C₂²⋊D₅₃	1	2	yes	—
109	C₄²	C₂²⋊D₅₃	1	2	yes	—
110	C₂³	—	2	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	—	4	2	no	—
114	C₂³	C₄×Dic₅₃	1	4	yes	—
115	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
116	D₄	—	2	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	4	4	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
124	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
127	C₂×C₄	#11719945	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
129	C₈	—	106	8	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	4	no	—
133	C₂²	—	4	4	no	—
134	C₂²	—	4	4	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	4	2	no	—
137	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
138	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
139	C₂²	#13133176	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	2	no	—
143	C₄	#13133134	1	2	yes	direct product factor
144	C₄	C₄×C₂²⋊D₅₃	1	4	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	4	no	—
148	C₂	C₄×C₂²⋊Dic₅₃	1	1	yes	characteristic
149	C₂	C₂×#13133134	1	1	yes	characteristic
150	C₂	#16810607	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133134

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, complemented

#16810392

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133134

C₄

yes

direct product factor

#13133128

—

C₄×#88322

—

C₂²×D₄×C₅₃

C₂²

yes

characteristic

C₂³×C₄×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

D₄

yes

—

#11719917

—

#11719917

C₂×C₄

yes

complemented

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄²

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#100

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×#100

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×C₂²⋊C₄

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄×D₄

D₅₃

yes

characteristic

C₂×#100

—

C₄×M₁₆

—

#343

—

C₅₃

C₄×#100

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₅₃

yes

characteristic

C₂³×C₄

Dic₅₃

yes

—

C₂×M₁₆

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#100

—

#97

—

106

—

C₄×C₈

—

106

—

C₂⁴

C₂×Dic₅₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

100

C₂²×C₄

C₂×Dic₅₃

yes

—

101

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

102

C₂²×C₄

C₂²×D₅₃

yes

characteristic

103

M₁₆

—

104

M₁₆

—

106

—

105

C₂×C₈

—

106

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

C₂²⋊D₅₃

yes

—

109

C₄²

C₂²⋊D₅₃

yes

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

C₄×Dic₅₃

yes

—

115

C₂³

C₂²×Dic₅₃

yes

characteristic

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²⋊Dic₅₃

yes

—

124

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

125

C₂×C₄

C₂²⋊Dic₅₃

yes

characteristic

126

C₂×C₄

C₂×C₄×D₅₃

yes

—

127

C₂×C₄

#11719945

yes

—

128

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

129

C₈

—

106

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

138

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

139

C₂²

#13133176

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#13133134

yes

direct product factor

144

C₄

C₄×C₂²⋊D₅₃

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₄×C₂²⋊Dic₅₃

yes

characteristic

149

C₂

C₂×#13133134

yes

characteristic

150

C₂

#16810607

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄×D₄×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×D₄	D₅₃
hypocenter	C₅₃	C₄×#100
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄×D₄×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×D₄

D₅₃

hypocenter

C₅₃

C₄×#100

LCS

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
2. C₄×#11719917	C₂	semidirect
3. #16810392	C₂	semidirect
6. #13133134	C₄	direct (N × H)
32. #11719917	C₂×C₄	semidirect
67. C₄×C₅₃	#100	semidirect
80. C₅₃	C₄×#100	semidirect
143. C₄	#13133134	direct (N × H)

H ≅ G/N

type

2. C₄×#11719917

C₂

semidirect

3. #16810392

C₂

semidirect

6. #13133134

C₄

direct (N × H)

32. #11719917

C₂×C₄

semidirect

67. C₄×C₅₃

#100

semidirect

80. C₅₃

C₄×#100

semidirect

143. C₄

#13133134

direct (N × H)

Subgroups — C₄×#13133134 #20274507

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13133134 #20274507

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13133134 #20274507