Group #20276427 — subgroups

Subgroups — #20276427

Order 6784.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133157	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13133151	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810427	C₂	1	2	yes	maximal, complemented
4	#16810424	C₂	1	2	yes	maximal, complemented
5	C₂×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₅₃	—	2	1	no	—
7	#13133157	—	2	2	no	—
8	C₂×#11719908	C₂²	1	2	yes	—
9	#13133151	—	2	1	no	—
10	#13133151	C₂²	1	2	yes	—
11	C₂×#41×C₅₃	—	2	1	no	—
12	#135×C₅₃	—	2	2	no	—
13	#135×C₅₃	C₄	1	4	yes	complemented
14	C₂×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₅₃	—	2	1	no	—
16	C₂×D₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	#41×C₅₃	—	2	4	no	—
19	#41×C₅₃	—	4	2	no	—
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	C₂×D₄×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	4	1	no	—
26	Q₁₆×C₅₃	—	2	1	no	—
27	Q₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
28	SD₁₆×C₅₃	—	2	4	no	—
29	D₈×C₅₃	—	2	1	no	—
30	D₈×C₅₃	C₂×C₄	1	2	yes	complemented
31	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	—	4	1	no	—
37	#11719908	—	2	2	no	—
38	C₂³×C₅₃	—	4	1	no	—
39	C₂³×C₅₃	D₈	1	1	yes	characteristic
40	Q₈×C₅₃	—	2	2	no	—
41	Q₈×C₅₃	—	4	1	no	—
42	D₄×C₅₃	—	2	2	no	—
43	D₄×C₅₃	—	4	4	no	—
44	D₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	4	no	—
47	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
49	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₅₃	—	4	2	no	—
52	C₂×Dic₅₃	—	8	2	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	—	4	2	no	—
59	C₂²×C₅₃	—	8	1	no	—
60	C₂²×C₅₃	—	8	1	no	—
61	C₂²×C₅₃	#102	1	2	yes	—
62	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₄×C₅₃	—	4	2	no	—
64	C₄×C₅₃	#102	1	2	yes	—
65	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
67	Dic₅₃	—	8	2	no	—
68	#2139	—	53	1	no	maximal, Sylow
69	C₂×C₅₃	—	2	2	no	—
70	C₂×C₅₃	—	8	2	no	—
71	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
72	C₂×C₅₃	#509	1	1	yes	characteristic
73	C₂×C₅₃	#510	1	1	yes	characteristic
74	C₂×#135	D₅₃	1	1	yes	characteristic
75	C₂×M₃₂	—	53	1	no	—
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	—	2	1	no	—
81	#135	—	2	2	no	—
82	#135	Dic₅₃	1	4	yes	complemented
83	C₂×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
84	C₂×SD₁₆	—	2	1	no	—
85	C₂×D₈	C₂×D₅₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	106	1	no	—
88	M₃₂	—	106	2	no	—
89	C₂×C₁₆	—	53	2	no	—
90	#107	—	53	2	no	—
91	#107	—	106	1	no	—
92	#41	—	2	4	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	4	1	no	—
97	C₂²×C₄	—	4	1	no	—
98	C₂²×C₄	—	212	1	no	—
99	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
100	Q₁₆	—	2	1	no	—
101	Q₁₆	C₂×Dic₅₃	1	2	yes	complemented
102	SD₁₆	—	2	4	no	—
103	D₈	—	2	1	no	—
104	D₈	C₂×Dic₅₃	1	2	yes	complemented
105	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
106	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
107	C₂×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₄⋊C₄	—	106	2	no	—
110	C₄⋊C₄	—	212	1	no	—
111	C₁₆	—	106	2	no	—
112	C₂³	—	4	1	no	—
113	C₂³	#11719922	1	1	yes	characteristic
114	Q₈	—	2	2	no	—
115	Q₈	—	4	1	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	4	no	—
118	D₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	4	no	—
121	C₂×C₄	—	424	2	no	—
122	C₂×C₄	—	212	2	no	—
123	C₂×C₄	#11719922	1	1	yes	characteristic
124	C₂×C₄	#11719923	1	2	yes	—
125	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
126	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
127	C₈	C₂²⋊Dic₅₃	1	2	yes	—
128	C₈	C₂²×Dic₅₃	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	—	4	2	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	#13133138	1	2	yes	—
136	C₂²	C₂×#11719922	1	1	yes	characteristic
137	C₄	—	4	2	no	—
138	C₄	—	424	2	no	—
139	C₄	#13133138	1	2	yes	—
140	C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
141	C₄	C₂×#11719923	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	—	8	2	no	—
144	C₂	C₂×#13133138	1	1	yes	characteristic
145	C₂	#16811058	1	1	yes	characteristic
146	C₂	#16811063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133157

C₂

yes

maximal, characteristic, complemented

C₂×#13133151

C₂

yes

maximal, characteristic, complemented

#16810427

C₂

yes

maximal, complemented

#16810424

C₂

yes

maximal, complemented

C₂×#135×C₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₅₃

—

#13133157

—

C₂×#11719908

C₂²

yes

—

#13133151

—

#13133151

C₂²

yes

—

C₂×#41×C₅₃

—

#135×C₅₃

—

#135×C₅₃

C₄

yes

complemented

C₂×Q₁₆×C₅₃

C₂²

yes

characteristic

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₄

yes

complemented

SD₁₆×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

C₂×C₄

yes

complemented

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

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

#2139

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

yes

characteristic

C₂×M₃₂

—

C₂×#107

—

#358

—

#357

—

C₅₃

#2139

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

#135

Dic₅₃

yes

complemented

C₂×Q₁₆

C₂×D₅₃

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

#107

—

#107

—

106

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

100

Q₁₆

—

101

Q₁₆

C₂×Dic₅₃

yes

complemented

102

SD₁₆

—

103

D₈

—

104

D₈

C₂×Dic₅₃

yes

complemented

105

C₂×C₈

C₂×Dic₅₃

yes

—

106

C₂×C₈

C₂²⋊D₅₃

yes

—

107

C₂×C₈

C₂²⋊D₅₃

yes

characteristic

108

C₂×C₈

C₂²×D₅₃

yes

characteristic

109

C₄⋊C₄

—

106

—

110

C₄⋊C₄

—

212

—

111

C₁₆

—

106

—

112

C₂³

—

113

C₂³

#11719922

yes

characteristic

114

Q₈

—

115

Q₈

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

424

—

122

C₂×C₄

—

212

—

123

C₂×C₄

#11719922

yes

characteristic

124

C₂×C₄

#11719923

yes

—

125

C₂×C₄

C₂²⋊Dic₅₃

yes

—

126

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

127

C₈

C₂²⋊Dic₅₃

yes

—

128

C₈

C₂²×Dic₅₃

yes

characteristic

129

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

#13133138

yes

—

136

C₂²

C₂×#11719922

yes

characteristic

137

C₄

—

138

C₄

—

424

—

139

C₄

#13133138

yes

—

140

C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

141

C₄

C₂×#11719923

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

C₂×#13133138

yes

characteristic

145

C₂

#16811058

yes

characteristic

146

C₂

#16811063

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11719922
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₈
hypercenter	C₂×#135	D₅₃
hypocenter	C₅₃	#2139
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

subgroup

G/H

center

C₂²

C₂×#11719922

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×#135×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂×#13133157	C₂	semidirect
2. C₂×#13133151	C₂	semidirect
3. #16810427	C₂	semidirect
4. #16810424	C₂	semidirect
13. #135×C₅₃	C₄	semidirect
27. Q₁₆×C₅₃	C₂×C₄	semidirect
30. D₈×C₅₃	C₂×C₄	semidirect
79. C₅₃	#2139	semidirect
82. #135	Dic₅₃	semidirect
101. Q₁₆	C₂×Dic₅₃	semidirect
104. D₈	C₂×Dic₅₃	semidirect

H ≅ G/N

type

1. C₂×#13133157

C₂

semidirect

2. C₂×#13133151

C₂

semidirect

3. #16810427

C₂

semidirect

4. #16810424

C₂

semidirect

13. #135×C₅₃

C₄

semidirect

27. Q₁₆×C₅₃

C₂×C₄

semidirect

30. D₈×C₅₃

C₂×C₄

semidirect

79. C₅₃

#2139

semidirect

82. #135

Dic₅₃

semidirect

101. Q₁₆

C₂×Dic₅₃

semidirect

104. D₈

C₂×Dic₅₃

semidirect

Subgroups — #20276427

Conjugacy classes of subgroups

Complements