Group #20274580 — subgroups

Subgroups — #20274580

Order 6784.

296 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₅₃	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810377	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#88322	C₂²	1	1	yes	characteristic
7	C₂²×#88322	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
9	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
10	C₄×#88322	C₄	1	4	yes	complemented
11	C₈×Dic₅₃	—	2	4	no	—
12	#13133124	C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	C₄	1	4	yes	complemented
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
19	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
20	C₂×C₈×C₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₂×C₈×C₅₃	C₂×C₄	1	4	yes	—
23	C₄²×C₅₃	C₂×C₄	1	2	yes	—
24	C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₅₃	C₈	1	8	yes	complemented
26	C₄×Dic₅₃	—	2	2	no	—
27	C₄×Dic₅₃	—	2	2	no	—
28	C₂×#88322	—	2	1	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂×#88322	C₂×C₄	1	2	yes	—
31	C₂×#88322	C₂×C₄	1	2	yes	—
32	C₂×#88322	D₄	1	2	yes	—
33	C₂×#88322	Q₈	1	2	yes	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	C₄²	1	2	yes	—
37	C₂×C₄×C₅₃	C₄²	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	4	no	—
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	C₂×C₈	1	8	yes	—
48	C₈×C₅₃	—	2	4	no	—
49	#88322	—	2	2	no	—
50	#88322	C₄⋊C₄	1	4	yes	complemented
51	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₅₃	C₄×Q₈	1	1	yes	characteristic
54	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
55	C₂²×C₅₃	#131	1	2	yes	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	C₄×C₈	1	4	yes	complemented
58	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₅₃	#117	1	1	yes	characteristic
60	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
61	C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
62	Dic₅₃	—	2	8	no	—
63	C₈×C₄⋊C₄	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×C₅₃	#405	1	1	yes	characteristic
67	C₂×C₅₃	C₈×D₄	1	2	yes	—
68	C₂×C₅₃	C₈×Q₈	1	2	yes	—
69	C₂×C₄⋊C₈	D₅₃	1	1	yes	characteristic
70	C₂×C₄×C₈	—	53	1	no	—
71	C₂×C₄×C₈	—	53	2	no	—
72	C₄×C₄⋊C₄	—	53	1	no	—
73	#336	—	53	2	no	—
74	C₅₃	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	53	1	no	—
76	C₂²×C₈	—	53	1	no	—
77	C₂²×C₈	C₂×D₅₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	53	2	no	—
79	C₂×C₄²	—	53	2	no	—
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	C₄⋊C₈	Dic₅₃	1	4	yes	complemented
82	C₄×C₈	—	53	4	no	—
83	C₄×C₈	—	106	4	no	—
84	#95	—	53	2	no	—
85	C₂²×C₄	—	53	4	no	—
86	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	53	2	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	53	2	no	—
95	C₂×C₈	—	106	1	no	—
96	C₂×C₈	—	106	1	no	—
97	C₂×C₈	C₂×Dic₅₃	1	4	yes	—
98	C₄⋊C₄	—	53	8	no	—
99	C₄²	—	106	2	no	—
100	C₄²	—	106	2	no	—
101	C₄²	C₄×D₅₃	1	2	yes	—
102	C₄²	C₂×Dic₅₃	1	2	yes	—
103	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	53	8	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	106	4	no	—
108	C₂×C₄	C₈×D₅₃	1	4	yes	—
109	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
110	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
111	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
113	C₂×C₄	#11719947	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
115	C₂×C₄	#11719949	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₈	—	53	4	no	—
119	C₈	—	106	2	no	—
120	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
121	C₂²	#13133179	1	1	yes	characteristic
122	C₂²	#13133182	1	1	yes	characteristic
123	C₂²	C₂×C₈×D₅₃	1	2	yes	—
124	C₂²	#13133225	1	2	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	106	8	no	—
127	C₄	C₈×Dic₅₃	1	4	yes	complemented
128	C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
129	C₄	#13133175	1	1	yes	characteristic
130	C₄	D₄×Dic₅₃	1	1	yes	characteristic
131	C₄	Q₈×Dic₅₃	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
133	C₂	C₂×C₈×Dic₅₃	1	1	yes	characteristic
134	C₂	#16810581	1	1	yes	characteristic
135	C₂	#16810688	1	2	yes	—
136	C₂	#16810747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×C₈×Dic₅₃

C₂

yes

maximal

C₄×C₄⋊Dic₅₃

C₂

yes

maximal, characteristic

#16810377

C₂

yes

maximal

C₂×C₄⋊C₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₂×C₄⋊Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₄×#88322

C₄

yes

complemented

C₈×Dic₅₃

—

#13133124

C₄

yes

—

C₂²×C₈×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₄⋊C₈×C₅₃

C₄

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₈

yes

complemented

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

C₂×C₄

yes

—

C₂×#88322

D₄

yes

—

C₂×#88322

Q₈

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

—

C₈×C₅₃

—

#88322

—

#88322

C₄⋊C₄

yes

complemented

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

characteristic

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×C₈

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

#117

yes

characteristic

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₄×Q₈

yes

characteristic

Dic₅₃

—

C₈×C₄⋊C₄

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×C₅₃

#405

yes

characteristic

C₂×C₅₃

C₈×D₄

yes

—

C₂×C₅₃

C₈×Q₈

yes

—

C₂×C₄⋊C₈

D₅₃

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

—

#336

—

C₅₃

C₈×C₄⋊C₄

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

Dic₅₃

yes

complemented

C₄×C₈

—

C₄×C₈

—

106

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₄⋊C₄

—

C₄²

—

106

—

100

C₄²

—

106

—

101

C₄²

C₄×D₅₃

yes

—

102

C₄²

C₂×Dic₅₃

yes

—

103

C₂³

C₂×C₄×D₅₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

C₈×D₅₃

yes

—

109

C₂×C₄

C₄×Dic₅₃

yes

—

110

C₂×C₄

C₄×Dic₅₃

yes

—

111

C₂×C₄

C₂×C₄×D₅₃

yes

characteristic

112

C₂×C₄

D₄×D₅₃

yes

characteristic

113

C₂×C₄

#11719947

yes

characteristic

114

C₂×C₄

Q₈×D₅₃

yes

characteristic

115

C₂×C₄

#11719949

yes

characteristic

116

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

—

106

—

120

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

121

C₂²

#13133179

yes

characteristic

122

C₂²

#13133182

yes

characteristic

123

C₂²

C₂×C₈×D₅₃

yes

—

124

C₂²

#13133225

yes

—

125

C₄

—

126

C₄

—

106

—

127

C₄

C₈×Dic₅₃

yes

complemented

128

C₄

C₄⋊C₄×D₅₃

yes

characteristic

129

C₄

#13133175

yes

characteristic

130

C₄

D₄×Dic₅₃

yes

characteristic

131

C₄

Q₈×Dic₅₃

yes

characteristic

132

C₂

C₄⋊C₄×Dic₅₃

yes

characteristic

133

C₂

C₂×C₈×Dic₅₃

yes

characteristic

134

C₂

#16810581

yes

characteristic

135

C₂

#16810688

yes

—

136

C₂

#16810747

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂×C₅₃	C₂×C₄×C₈
Fitting	C₂×C₄⋊C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₈×C₄⋊C₄
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂×C₅₃

C₂×C₄×C₈

Fitting

C₂×C₄⋊C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₅₃

hypocenter

C₅₃

C₈×C₄⋊C₄

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
10. C₄×#88322	C₄	semidirect
15. C₄⋊C₈×C₅₃	C₄	semidirect
25. C₄⋊Dic₅₃	C₈	semidirect
50. #88322	C₄⋊C₄	semidirect
57. C₄×C₅₃	C₄×C₈	semidirect
74. C₅₃	C₈×C₄⋊C₄	semidirect
81. C₄⋊C₈	Dic₅₃	semidirect
127. C₄	C₈×Dic₅₃	semidirect

H ≅ G/N

type

10. C₄×#88322

C₄

semidirect

15. C₄⋊C₈×C₅₃

C₄

semidirect

25. C₄⋊Dic₅₃

C₈

semidirect

50. #88322

C₄⋊C₄

semidirect

57. C₄×C₅₃

C₄×C₈

semidirect

74. C₅₃

C₈×C₄⋊C₄

semidirect

81. C₄⋊C₈

Dic₅₃

semidirect

127. C₄

C₈×Dic₅₃

semidirect

Subgroups — #20274580

Conjugacy classes of subgroups

Complements