Group #20276428 — subgroups

Subgroups — #20276428

Order 6784.

232 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 85 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×#13133239	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810428	C₂	1	2	yes	maximal, complemented
4	#16810425	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	—	2	1	no	—
7	#13133239	—	2	2	no	—
8	#13133239	C₄	1	4	yes	complemented
9	C₂×Dic₂₁₂	C₂²	1	1	yes	characteristic
10	C₂×#11719913	—	2	1	no	—
11	C₂×D₄₂₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₃	—	2	1	no	—
13	#13133151	—	2	1	no	—
14	#13133151	C₂²	1	2	yes	—
15	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	M₃₂×C₅₃	—	2	2	no	—
17	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
18	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₃	—	4	1	no	—
20	C₂²×Dic₅₃	—	4	1	no	—
21	#11719945	—	2	4	no	—
22	#11719945	—	4	2	no	—
23	C₂×D₂₁₂	—	2	1	no	—
24	C₂×C₄×D₅₃	—	4	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₈×C₅₃	D₄	1	2	yes	—
29	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄⋊Dic₅₃	—	2	2	no	—
31	C₄⋊Dic₅₃	—	4	1	no	—
32	Dic₂₁₂	—	2	1	no	—
33	Dic₂₁₂	C₂×C₄	1	2	yes	complemented
34	D₄₂₄	—	2	1	no	—
35	D₄₂₄	C₂×C₄	1	2	yes	complemented
36	#11719913	—	2	4	no	—
37	C₁₆×C₅₃	—	2	2	no	—
38	C₂³×C₅₃	D₈	1	1	yes	characteristic
39	C₂²×D₅₃	—	4	1	no	—
40	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²⋊D₅₃	—	4	4	no	—
45	C₂×Dic₅₃	—	4	1	no	—
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	—	8	2	no	—
48	D₂₁₂	—	2	2	no	—
49	D₂₁₂	—	4	1	no	—
50	C₄×D₅₃	—	4	4	no	—
51	Dic₁₀₆	—	2	2	no	—
52	Dic₁₀₆	—	4	1	no	—
53	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
54	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	#102	1	2	yes	—
59	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×D₅₃	—	4	2	no	—
61	C₂×D₅₃	—	8	1	no	—
62	C₂×D₅₃	—	8	1	no	—
63	C₄×C₅₃	#102	1	2	yes	—
64	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	Dic₅₃	—	4	2	no	—
67	Dic₅₃	—	8	2	no	—
68	#2139	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
71	C₂×C₅₃	#509	1	1	yes	characteristic
72	C₂×C₅₃	#510	1	1	yes	characteristic
73	D₅₃	—	8	2	no	—
74	C₂×#135	—	53	1	no	—
75	C₂×M₃₂	D₅₃	1	1	yes	characteristic, complemented
76	C₂×#107	—	53	1	no	—
77	#358	—	53	2	no	—
78	#357	—	53	2	no	—
79	C₅₃	#2139	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	106	1	no	—
81	#135	—	53	4	no	—
82	#135	—	106	2	no	—
83	C₂×Q₁₆	—	53	1	no	—
84	C₂×SD₁₆	—	106	1	no	—
85	C₂×D₈	—	53	1	no	—
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	106	1	no	—
88	M₃₂	—	2	2	no	—
89	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
90	#107	—	53	2	no	—
91	#107	—	106	1	no	—
92	#41	—	106	4	no	—
93	#41	—	212	2	no	—
94	C₂×Q₈	—	106	1	no	—
95	C₂×D₄	—	106	1	no	—
96	C₂×D₄	—	212	1	no	—
97	C₂²×C₄	—	212	1	no	—
98	C₂²×C₄	—	212	1	no	—
99	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
100	Q₁₆	—	53	2	no	—
101	Q₁₆	—	106	1	no	—
102	SD₁₆	—	106	4	no	—
103	D₈	—	53	2	no	—
104	D₈	—	106	1	no	—
105	C₂×C₈	C₄×D₅₃	1	2	yes	—
106	C₂×C₈	D₂₁₂	1	1	yes	characteristic
107	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₄⋊C₄	—	106	2	no	—
110	C₄⋊C₄	—	212	1	no	—
111	C₁₆	—	2	2	no	—
112	C₂³	—	212	1	no	—
113	C₂³	D₄₂₄	1	1	yes	characteristic
114	Q₈	—	106	2	no	—
115	Q₈	—	212	1	no	—
116	D₄	—	106	2	no	—
117	D₄	—	212	4	no	—
118	D₄	—	212	1	no	—
119	C₂×C₄	—	424	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	—	212	1	no	—
122	C₂×C₄	—	212	4	no	—
123	C₂×C₄	#11719913	1	2	yes	—
124	C₂×C₄	D₄₂₄	1	1	yes	characteristic
125	C₂×C₄	#11719921	1	2	yes	—
126	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
127	C₈	#11719921	1	2	yes	—
128	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
129	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	424	1	no	—
133	C₂²	—	424	1	no	—
134	C₂²	—	212	2	no	—
135	C₂²	#13133139	1	2	yes	—
136	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
137	C₄	—	424	2	no	—
138	C₄	—	212	2	no	—
139	C₄	#13133139	1	2	yes	—
140	C₄	C₂×#11719921	1	1	yes	characteristic
141	C₄	C₂×#11719913	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	424	2	no	—
144	C₂	C₂×#13133139	1	1	yes	characteristic
145	C₂	#16811061	1	1	yes	characteristic
146	C₂	#16811066	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133239

C₂

yes

maximal, characteristic

C₂×#13133151

C₂

yes

maximal, characteristic, complemented

#16810428

C₂

yes

maximal, complemented

#16810425

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

—

#13133239

—

#13133239

C₄

yes

complemented

C₂×Dic₂₁₂

C₂²

yes

characteristic

C₂×#11719913

—

C₂×D₄₂₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

—

#13133151

—

#13133151

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

C₂×C₄

yes

complemented

D₄₂₄

—

D₄₂₄

C₂×C₄

yes

complemented

#11719913

—

C₁₆×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2139

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

C₂×#135

—

C₂×M₃₂

D₅₃

yes

characteristic, complemented

C₂×#107

—

#358

—

#357

—

C₅₃

#2139

yes

characteristic, complemented, Sylow

C₂×#41

—

106

—

#135

—

#135

—

106

—

C₂×Q₁₆

—

C₂×SD₁₆

—

106

—

C₂×D₈

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

106

—

M₃₂

—

C₂×C₁₆

C₂×D₅₃

yes

—

#107

—

#107

—

106

—

#41

—

106

—

#41

—

212

—

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

106

—

102

SD₁₆

—

106

—

103

D₈

—

104

D₈

—

106

—

105

C₂×C₈

C₄×D₅₃

yes

—

106

C₂×C₈

D₂₁₂

yes

characteristic

107

C₂×C₈

C₂²⋊D₅₃

yes

—

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

212

—

111

C₁₆

—

112

C₂³

—

212

—

113

C₂³

D₄₂₄

yes

characteristic

114

Q₈

—

106

—

115

Q₈

—

212

—

116

D₄

—

106

—

117

D₄

—

212

—

118

D₄

—

212

—

119

C₂×C₄

—

424

—

120

C₂×C₄

—

212

—

121

C₂×C₄

—

212

—

122

C₂×C₄

—

212

—

123

C₂×C₄

#11719913

yes

—

124

C₂×C₄

D₄₂₄

yes

characteristic

125

C₂×C₄

#11719921

yes

—

126

C₂×C₄

C₂×D₂₁₂

yes

characteristic

127

C₈

#11719921

yes

—

128

C₈

C₂×C₄×D₅₃

yes

characteristic

129

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

424

—

133

C₂²

—

424

—

134

C₂²

—

212

—

135

C₂²

#13133139

yes

—

136

C₂²

C₂×D₄₂₄

yes

characteristic

137

C₄

—

424

—

138

C₄

—

212

—

139

C₄

#13133139

yes

—

140

C₄

C₂×#11719921

yes

characteristic

141

C₄

C₂×#11719913

yes

characteristic

142

C₂

—

143

C₂

—

424

—

144

C₂

C₂×#13133139

yes

characteristic

145

C₂

#16811061

yes

characteristic

146

C₂

#16811066

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×M₃₂	D₅₃
hypocenter	C₅₃	#2139
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₄₂₄

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×M₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

hypercenter

C₂×M₃₂

D₅₃

hypocenter

C₅₃

#2139

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂×#13133151	C₂	semidirect
3. #16810428	C₂	semidirect
4. #16810425	C₂	semidirect
5. C₂×M₃₂×C₅₃	C₂	semidirect
8. #13133239	C₄	semidirect
33. Dic₂₁₂	C₂×C₄	semidirect
35. D₄₂₄	C₂×C₄	semidirect
75. C₂×M₃₂	D₅₃	semidirect
79. C₅₃	#2139	semidirect

H ≅ G/N

type

2. C₂×#13133151

C₂

semidirect

3. #16810428

C₂

semidirect

4. #16810425

C₂

semidirect

5. C₂×M₃₂×C₅₃

C₂

semidirect

8. #13133239

C₄

semidirect

33. Dic₂₁₂

C₂×C₄

semidirect

35. D₄₂₄

C₂×C₄

semidirect

75. C₂×M₃₂

D₅₃

semidirect

79. C₅₃

#2139

semidirect

Subgroups — #20276428

Conjugacy classes of subgroups

Complements