Group #28298115 — subgroups

Subgroups — #28298115

Order 13568.

156 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 44 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	#20289651	C₂	1	2	yes	maximal
2	#20274834	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11719910	—	2	2	no	—
4	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
5	#16810469	C₄	1	1	yes	characteristic
6	#16810377	—	2	2	no	—
7	C₂×C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×#11719910	—	4	2	no	—
9	C₂²×#88322	D₄	1	2	yes	—
10	C₂×C₄×Dic₅₃	—	2	1	no	—
11	C₄⋊#88322	—	4	1	no	—
12	#13133124	—	2	1	no	—
13	#13133124	C₈	1	2	yes	—
14	C₂²×C₈×C₅₃	—	2	1	no	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₄⋊C₈×C₅₃	—	4	1	no	—
17	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
20	C₂²×Dic₅₃	—	2	2	no	—
21	C₂×C₈×C₅₃	—	4	2	no	—
22	C₂×C₈×C₅₃	—	4	1	no	—
23	C₄²×C₅₃	—	2	1	no	—
24	C₄²×C₅₃	—	2	1	no	—
25	C₄×Dic₅₃	—	4	1	no	—
26	C₄×Dic₅₃	—	4	1	no	—
27	C₂×#88322	—	2	2	no	—
28	C₂×#88322	—	4	2	no	—
29	#11719910	—	8	2	no	—
30	C₂³×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	—	2	1	no	—
32	C₂×C₄×C₅₃	—	2	1	no	—
33	C₂×C₄×C₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	1	no	—
35	C₂×C₄×C₅₃	—	2	2	no	—
36	C₂×C₄×C₅₃	—	4	1	no	—
37	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
39	C₂×Dic₅₃	—	4	2	no	—
40	C₂×Dic₅₃	—	4	2	no	—
41	C₂×Dic₅₃	—	4	2	no	—
42	C₈×C₅₃	—	8	1	no	—
43	#88322	—	4	2	no	—
44	#7065	—	53	1	no	maximal, Sylow
45	C₂²×C₅₃	—	2	1	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	#324	1	1	yes	characteristic
48	C₂²×C₅₃	#350	1	2	yes	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	4	1	no	—
51	C₄×C₅₃	—	4	1	no	—
52	C₄×C₅₃	#354	1	1	yes	characteristic
53	C₄×C₅₃	#355	1	1	yes	characteristic
54	Dic₅₃	—	8	2	no	—
55	#1832	—	53	1	no	—
56	#1322	—	53	2	no	—
57	C₂×C₅₃	—	2	2	no	—
58	C₂×C₅₃	#1323	1	1	yes	characteristic
59	C₂×C₅₃	#1324	1	1	yes	characteristic
60	C₂×C₅₃	#1328	1	1	yes	characteristic
61	C₂×C₄⋊C₈	—	53	1	no	—
62	C₂×C₄⋊C₈	#6157	1	1	yes	characteristic
63	#376	—	53	1	no	—
64	C₂²⋊C₁₆	—	106	2	no	—
65	#336	—	106	2	no	—
66	C₅₃	#7065	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₈	—	2	1	no	—
68	C₂²×C₈	—	53	2	no	—
69	C₂×C₄²	—	106	1	no	—
70	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
71	C₂×C₁₆	—	212	2	no	—
72	C₄⋊C₈	—	4	1	no	—
73	C₄⋊C₈	—	212	1	no	—
74	#95	—	53	2	no	—
75	#95	—	106	1	no	—
76	C₂²×C₄	—	106	2	no	—
77	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
78	C₂²×C₄	#11719936	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	106	2	no	—
83	C₂×C₈	—	212	2	no	—
84	C₄²	—	2	1	no	—
85	C₄²	—	2	1	no	—
86	C₄²	—	212	1	no	—
87	C₄²	—	212	1	no	—
88	C₁₆	—	424	2	no	—
89	C₂³	#13133274	1	1	yes	characteristic
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	212	2	no	—
97	C₂×C₄	—	212	2	no	—
98	C₂×C₄	—	212	2	no	—
99	C₂×C₄	#13133275	1	1	yes	characteristic
100	C₂×C₄	#13133278	1	1	yes	characteristic
101	C₈	—	8	1	no	—
102	C₈	—	212	2	no	—
103	C₂²	—	2	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	#16811469	1	1	yes	characteristic
106	C₂²	#16811522	1	2	yes	—
107	C₄	—	2	1	no	—
108	C₄	—	4	1	no	—
109	C₄	—	4	1	no	—
110	C₄	—	424	2	no	—
111	C₄	#16811529	1	1	yes	characteristic
112	C₄	#16811531	1	1	yes	characteristic
113	C₂	—	2	2	no	—
114	C₂	#20289654	1	1	yes	characteristic
115	C₂	#20289656	1	1	yes	characteristic
116	C₂	#20289664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20289651

C₂

yes

maximal

#20274834

C₂

yes

maximal, characteristic

C₂²⋊#11719910

—

C₂×C₄⋊#88322

C₂²

yes

characteristic

#16810469

C₄

yes

characteristic

#16810377

—

C₂×C₄⋊C₈×C₅₃

C₄

yes

characteristic

C₂×#11719910

—

C₂²×#88322

D₄

yes

—

C₂×C₄×Dic₅₃

—

C₄⋊#88322

—

#13133124

—

#13133124

C₈

yes

—

C₂²×C₈×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

#11719910

—

C₂³×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#7065

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#324

yes

characteristic

C₂²×C₅₃

#350

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#1832

—

#1322

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

#6157

yes

characteristic

#376

—

C₂²⋊C₁₆

—

106

—

#336

—

106

—

C₅₃

#7065

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

C₂×C₁₆

—

212

—

C₄⋊C₈

—

C₄⋊C₈

—

212

—

#95

—

#95

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

C₂²×C₄

#11719936

yes

characteristic

C₂²×C₄

C₂²⋊#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₄²

—

C₄²

—

C₄²

—

212

—

C₄²

—

212

—

C₁₆

—

424

—

C₂³

#13133274

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

—

212

—

C₂×C₄

#13133275

yes

characteristic

100

C₂×C₄

#13133278

yes

characteristic

101

C₈

—

102

C₈

—

212

—

103

C₂²

—

104

C₂²

—

105

C₂²

#16811469

yes

characteristic

106

C₂²

#16811522

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

424

—

111

C₄

#16811529

yes

characteristic

112

C₄

#16811531

yes

characteristic

113

C₂

—

114

C₂

#20289654

yes

characteristic

115

C₂

#20289656

yes

characteristic

116

C₂

#20289664

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133275
commutator	C₂²×C₄×C₅₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×C₄⋊C₈×C₅₃	C₄
socle	C₂²×C₅₃	#324
prime radical	C₂³×C₅₃	C₂²⋊C₈
cosocle	C₂×C₄⋊#88322	C₂²
hypercenter	C₂×C₄⋊C₈	#6157
hypocenter	C₅₃	#7065
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

#13133275

commutator

C₂²×C₄×C₅₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₂×C₄⋊C₈×C₅₃

C₄

socle

C₂²×C₅₃

#324

prime radical

C₂³×C₅₃

C₂²⋊C₈

cosocle

C₂×C₄⋊#88322

C₂²

C₂×C₄⋊C₈

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
66. C₅₃	#7065	semidirect

H ≅ G/N

type

66. C₅₃

#7065

semidirect

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

subgroup	index
#20289651	2
#20274834	2
#7065	53

subgroup

index

#20289651

#20274834

#7065

Subgroups — #28298115

Conjugacy classes of subgroups

Complements

Maximal subgroups