Group #28300048 — subgroups

Subgroups — #28300048

Order 13568.

526 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 228 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	#20290429	C₂	1	2	yes	maximal
2	C₂×#88324⋊C₈	C₂	1	4	yes	maximal
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88324	C₂²	1	4	yes	—
5	#16811489	C₄	1	8	yes	—
6	#88324⋊C₈	C₄	1	16	yes	complemented
7	C₂×C₄×#88322	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
9	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
10	C₄²×Dic₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic
12	C₂²×#88324	C₂×C₄	1	8	yes	—
13	C₄×#88324	C₂×C₄	1	16	yes	—
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
17	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
18	C₄×#88322	—	2	2	no	—
19	C₈×Dic₅₃	—	2	4	no	—
20	C₈×Dic₅₃	C₂×C₄	1	8	yes	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
26	C₂²×Dic₅₃	C₄²	1	2	yes	—
27	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
28	C₂×#88324	—	2	16	no	—
29	C₂×#88324	C₄²	1	16	yes	—
30	C₂×C₈×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	C₄²	1	4	yes	—
33	C₄²×C₅₃	M₁₆	1	2	yes	—
34	C₄²×C₅₃	M₁₆	1	2	yes	—
35	C₄×Dic₅₃	—	2	8	no	—
36	C₄×Dic₅₃	M₁₆	1	4	yes	—
37	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	—	2	2	no	—
40	C₂×#88322	C₄²	1	4	yes	—
41	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	#97	1	4	yes	—
44	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	C₂×Dic₅₃	—	2	4	no	—
49	C₂×Dic₅₃	#97	1	4	yes	—
50	C₂×Dic₅₃	C₂×C₄²	1	4	yes	—
51	C₂×Dic₅₃	#131	1	8	yes	—
52	#88324	—	2	16	no	—
53	C₈×C₅₃	—	2	4	no	—
54	#88322	—	2	4	no	—
55	#7843	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	C₄³	1	1	yes	characteristic
57	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
59	C₂²×C₅₃	#405	1	4	yes	—
60	C₄×C₅₃	—	2	4	no	—
61	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
62	C₄×C₅₃	C₄×M₁₆	1	2	yes	—
63	Dic₅₃	—	2	4	no	—
64	Dic₅₃	#405	1	8	yes	—
65	#1744	—	53	2	no	—
66	C₄²×C₈	—	53	1	no	—
67	C₂×#322	—	53	4	no	—
68	C₂×C₅₃	#1452	1	2	yes	—
69	C₂×C₅₃	#1452	1	2	yes	—
70	C₂×C₅₃	C₄×#97	1	1	yes	characteristic
71	C₂×C₅₃	#1726	1	2	yes	—
72	C₂×C₄×C₈	—	53	4	no	—
73	C₂×C₄×C₈	—	53	2	no	—
74	C₂×C₄×C₈	—	53	2	no	—
75	C₂×C₄×C₈	—	53	1	no	—
76	C₂×C₄×C₈	#6157	1	1	yes	characteristic
77	C₄³	—	53	1	no	—
78	#336	—	53	8	no	—
79	#322	—	53	16	no	—
80	C₅₃	#7843	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	53	8	no	—
82	C₂²×C₈	—	53	2	no	—
83	C₂²×C₈	C₂×#6157	1	2	yes	—
84	C₂×C₄²	—	53	1	no	—
85	C₂×C₄²	—	53	1	no	—
86	C₂×C₄²	—	53	4	no	—
87	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
88	C₄×C₈	—	2	2	no	—
89	C₄×C₈	—	53	16	no	—
90	C₄×C₈	—	53	8	no	—
91	C₄×C₈	—	106	4	no	—
92	C₄×C₈	—	106	2	no	—
93	C₂²×C₄	—	53	2	no	—
94	C₂²×C₄	—	53	2	no	—
95	C₂²×C₄	C₄×#6157	1	2	yes	—
96	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	53	16	no	—
100	C₂×C₈	—	53	4	no	—
101	C₂×C₈	—	106	16	no	—
102	C₂×C₈	—	106	2	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	C₄×#6157	1	4	yes	—
105	C₄²	—	53	4	no	—
106	C₄²	—	53	4	no	—
107	C₄²	—	106	8	no	—
108	C₄²	#11719936	1	2	yes	—
109	C₄²	#11719940	1	2	yes	—
110	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	53	4	no	—
113	C₂×C₄	—	53	8	no	—
114	C₂×C₄	—	53	4	no	—
115	C₂×C₄	—	106	4	no	—
116	C₂×C₄	#13133272	1	4	yes	—
117	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
118	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
119	C₂×C₄	D₅₃⋊M₁₆	1	2	yes	—
120	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₈	—	106	16	no	—
123	C₈	—	106	4	no	—
124	C₂²	C₄²×#6157	1	1	yes	characteristic
125	C₂²	C₂×#13133272	1	1	yes	characteristic
126	C₂²	#16811582	1	1	yes	characteristic
127	C₂²	#16811586	1	4	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	53	8	no	—
130	C₄	—	106	4	no	—
131	C₄	C₄×#11719940	1	2	yes	—
132	C₄	C₄×#11719936	1	2	yes	—
133	C₂	#20289928	1	2	yes	—
134	C₂	#20289929	1	2	yes	—
135	C₂	C₄×#13133272	1	1	yes	characteristic
136	C₂	#20290371	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290429

C₂

yes

maximal

C₂×#88324⋊C₈

C₂

yes

maximal

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂×C₄×#88324

C₂²

yes

—

#16811489

C₄

yes

—

#88324⋊C₈

C₄

yes

complemented

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₄

yes

characteristic

C₂²×#88324

C₂×C₄

yes

—

C₄×#88324

C₂×C₄

yes

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

characteristic

C₄×#88322

—

C₈×Dic₅₃

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

C₄²

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₄²

yes

—

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂×#88324

—

C₂×#88324

C₄²

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

M₁₆

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₄²

yes

—

C₂³×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#97

yes

—

C₂×Dic₅₃

C₂×C₄²

yes

—

C₂×Dic₅₃

#131

yes

—

#88324

—

C₈×C₅₃

—

#88322

—

#7843

—

maximal, Sylow

C₂²×C₅₃

C₄³

yes

characteristic

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

C₂×#97

yes

characteristic

C₂²×C₅₃

#405

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₄×M₁₆

yes

—

C₄×C₅₃

C₄×M₁₆

yes

—

Dic₅₃

—

Dic₅₃

#405

yes

—

#1744

—

C₄²×C₈

—

C₂×#322

—

C₂×C₅₃

#1452

yes

—

C₂×C₅₃

#1452

yes

—

C₂×C₅₃

C₄×#97

yes

characteristic

C₂×C₅₃

#1726

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#6157

yes

characteristic

C₄³

—

#336

—

#322

—

C₅₃

#7843

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×#6157

yes

—

C₂²×C₄

C₂²×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

C₄×#6157

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

106

—

108

C₄²

#11719936

yes

—

109

C₄²

#11719940

yes

—

110

C₂³

C₂×C₄×#6157

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

#13133272

yes

—

117

C₂×C₄

C₂×C₄×#6157

yes

—

118

C₂×C₄

C₂×#11719940

yes

characteristic

119

C₂×C₄

D₅₃⋊M₁₆

yes

—

120

C₂×C₄

C₂×#11719936

yes

characteristic

121

C₈

—

122

C₈

—

106

—

123

C₈

—

106

—

124

C₂²

C₄²×#6157

yes

characteristic

125

C₂²

C₂×#13133272

yes

characteristic

126

C₂²

#16811582

yes

characteristic

127

C₂²

#16811586

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

106

—

131

C₄

C₄×#11719940

yes

—

132

C₄

C₄×#11719936

yes

—

133

C₂

#20289928

yes

—

134

C₂

#20289929

yes

—

135

C₂

C₄×#13133272

yes

characteristic

136

C₂

#20290371

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#6157
commutator	C₂²×C₅₃	C₄³
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₂×C₄²
cosocle	C₂×C₄×Dic₅₃	C₂³
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	#7843
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×#6157

commutator

C₂²×C₅₃

C₄³

Fitting

C₂×C₄×C₈×C₅₃

C₄

socle, prime radical

C₂³×C₅₃

C₂×C₄²

cosocle

C₂×C₄×Dic₅₃

C₂³

C₂×C₄×C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
6. #88324⋊C₈	C₄	semidirect
80. C₅₃	#7843	semidirect

H ≅ G/N

type

6. #88324⋊C₈

C₄

semidirect

80. C₅₃

#7843

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20290429	2
C₂×#88324⋊C₈	2
C₄×C₈×Dic₅₃	2
#7843	53

subgroup

index

#20290429

C₂×#88324⋊C₈

C₄×C₈×Dic₅₃

#7843

Subgroups — #28300048

Conjugacy classes of subgroups

Complements

Maximal subgroups