Group #28303544 — subgroups

Subgroups — #28303544

Order 13568.

262 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 80 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	#20290640	C₂	1	1	yes	maximal, characteristic
2	#20290637	C₂	1	1	yes	maximal, characteristic
3	#20289569	C₂	1	4	yes	maximal
4	#20273000	C₂	1	1	yes	maximal, characteristic
5	#16811628	C₂²	1	2	yes	—
6	#16811567	C₂²	1	2	yes	—
7	#16811499	—	2	2	no	—
8	#16811497	—	2	2	no	—
9	#16810674	C₂²	1	2	yes	—
10	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
11	#16810344	C₄	1	1	yes	characteristic, complemented
12	C₈×#88322	C₄	1	2	yes	complemented
13	C₅₃×#322	C₄	1	1	yes	characteristic, complemented
14	C₂×#11719936	—	2	2	no	—
15	C₂×C₄⋊#6157	—	2	2	no	—
16	#13133284	—	2	4	no	—
17	(C₂×#6157)⋊C₄	—	2	4	no	—
18	C₂×C₈×D₅₃	D₄	1	2	yes	—
19	C₄²×D₅₃	C₂³	1	1	yes	characteristic
20	#13133146	—	2	2	no	—
21	#13133129	—	2	2	no	—
22	C₄×#88322	C₂×C₄	1	2	yes	—
23	C₄×#88322	C₂×C₄	1	1	yes	characteristic
24	C₈×Dic₅₃	Q₈	1	2	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×#6157	—	4	2	no	—
28	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	#41	1	2	yes	—
30	C₂×#88324	—	4	2	no	—
31	C₄⋊#6157	—	4	4	no	—
32	#11719936	—	4	4	no	—
33	C₂×C₈×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
37	C₄×Dic₅₃	#41	1	2	yes	—
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	C₄⋊C₄	1	2	yes	—
40	C₈×D₅₃	—	2	4	no	—
41	C₂²×D₅₃	#124	1	1	yes	characteristic
42	C₂×#6157	—	8	4	no	—
43	C₂×#6157	—	8	2	no	—
44	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#117	1	2	yes	—
46	C₂×Dic₅₃	#124	1	1	yes	characteristic
47	C₂×Dic₅₃	#125	1	2	yes	—
48	C₄×D₅₃	—	4	2	no	—
49	C₄×D₅₃	#135	1	4	yes	—
50	#88324	—	8	2	no	—
51	C₈×C₅₃	—	2	2	no	—
52	C₈×C₅₃	—	4	2	no	—
53	#88322	—	2	2	no	—
54	#88322	—	4	2	no	—
55	#8791	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	#382	1	1	yes	characteristic
57	C₂×D₅₃	—	2	2	no	—
58	C₂×D₅₃	#488	1	2	yes	—
59	#6157	—	16	2	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#427	1	2	yes	—
62	Dic₅₃	—	2	2	no	—
63	Dic₅₃	#488	1	2	yes	—
64	#1837	—	53	1	no	—
65	#1835	—	53	1	no	—
66	#1450	—	53	1	no	—
67	#1286	—	53	4	no	—
68	C₂×C₅₃	#1833	1	2	yes	—
69	C₂×C₅₃	#1835	1	1	yes	characteristic
70	D₅₃	—	2	2	no	—
71	#431	—	53	2	no	—
72	#423	—	53	2	no	—
73	C₂×C₄×C₈	—	53	1	no	—
74	#382	—	53	2	no	—
75	#341	—	106	2	no	—
76	#340	—	106	2	no	—
77	#322	—	53	1	no	—
78	#322	#6157	1	1	yes	characteristic, complemented
79	C₈²	—	53	2	no	—
80	C₅₃	#8791	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	106	2	no	—
82	C₂²×C₈	—	53	2	no	—
83	C₂×C₄⋊C₄	—	106	2	no	—
84	C₂×C₄²	—	53	1	no	—
85	C₄⋊C₈	—	106	2	no	—
86	C₄⋊C₈	—	106	4	no	—
87	C₂²⋊C₈	—	106	2	no	—
88	C₄×C₈	—	53	2	no	—
89	C₄×C₈	—	53	1	no	—
90	C₄×C₈	—	53	2	no	—
91	C₄×C₈	C₂×#6157	1	2	yes	—
92	C₄×C₈	C₂×#6157	1	1	yes	characteristic
93	#95	—	106	4	no	—
94	C₂²×C₄	—	53	1	no	—
95	C₂²×C₄	—	53	2	no	—
96	C₂²×C₄	—	212	2	no	—
97	M₁₆	—	212	4	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	53	2	no	—
100	C₂×C₈	—	106	4	no	—
101	C₂×C₈	—	106	2	no	—
102	C₂×C₈	—	212	2	no	—
103	C₂×C₈	C₄⋊#6157	1	2	yes	—
104	C₄⋊C₄	—	212	4	no	—
105	C₄²	—	53	1	no	—
106	C₄²	—	53	2	no	—
107	C₄²	C₂²×#6157	1	1	yes	characteristic
108	C₂³	—	53	1	no	—
109	C₂×C₄	—	53	1	no	—
110	C₂×C₄	—	53	4	no	—
111	C₂×C₄	—	53	2	no	—
112	C₂×C₄	—	424	2	no	—
113	C₂×C₄	—	424	4	no	—
114	C₂×C₄	—	212	2	no	—
115	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
116	C₂×C₄	#13133296	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	106	2	no	—
120	C₈	—	424	2	no	—
121	C₈	—	212	2	no	—
122	C₂²	—	53	2	no	—
123	C₂²	—	106	2	no	—
124	C₂²	#16811567	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	53	2	no	—
127	C₄	—	848	2	no	—
128	C₄	—	106	2	no	—
129	C₄	#16811633	1	2	yes	—
130	C₂	—	106	2	no	—
131	C₂	#20290635	1	2	yes	—
132	C₂	#20290637	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290640

C₂

yes

maximal, characteristic

#20290637

C₂

yes

maximal, characteristic

#20289569

C₂

yes

maximal

#20273000

C₂

yes

maximal, characteristic

#16811628

C₂²

yes

—

#16811567

C₂²

yes

—

#16811499

—

#16811497

—

#16810674

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810344

C₄

yes

characteristic, complemented

C₈×#88322

C₄

yes

complemented

C₅₃×#322

C₄

yes

characteristic, complemented

C₂×#11719936

—

C₂×C₄⋊#6157

—

#13133284

—

(C₂×#6157)⋊C₄

—

C₂×C₈×D₅₃

D₄

yes

—

C₄²×D₅₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₂×C₄×D₅₃

#41

yes

—

C₂×#88324

—

C₄⋊#6157

—

#11719936

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₄×Dic₅₃

#41

yes

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₈×D₅₃

—

C₂²×D₅₃

yes

—

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×Dic₅₃

#124

yes

characteristic

C₂×Dic₅₃

#125

yes

—

C₄×D₅₃

—

C₄×D₅₃

#135

yes

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

#488

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

#427

yes

—

Dic₅₃

—

Dic₅₃

#488

yes

—

—

—

—

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

#1835

yes

characteristic

D₅₃

—

#431

—

#423

—

C₂×C₄×C₈

—

#382

—

#341

—

106

—

#340

—

106

—

#322

—

#322

#6157

yes

characteristic, complemented

C₈²

—

C₅₃

#8791

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

—

C₄×C₈

C₂×#6157

yes

characteristic

#95

—

106

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₂×C₈

—

106

—

102

C₂×C₈

—

212

—

103

C₂×C₈

C₄⋊#6157

yes

—

104

C₄⋊C₄

—

212

—

105

C₄²

—

106

C₄²

—

107

C₄²

C₂²×#6157

yes

characteristic

108

C₂³

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

424

—

113

C₂×C₄

—

424

—

114

C₂×C₄

—

212

—

115

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

116

C₂×C₄

#13133296

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

106

—

120

C₈

—

424

—

121

C₈

—

212

—

122

C₂²

—

123

C₂²

—

106

—

124

C₂²

#16811567

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

848

—

128

C₄

—

106

—

129

C₄

#16811633

yes

—

130

C₂

—

106

—

131

C₂

#20290635

yes

—

132

C₂

#20290637

yes

characteristic

	subgroup	G/H
center	C₂²	#16811567
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#322	C₄
socle	C₂²×C₅₃	#382
prime radical	C₂²×D₅₃	#124
cosocle	C₄²×D₅₃	C₂³
hypercenter	#322	#6157
hypocenter	C₅₃	#8791
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#16811567

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₄²

C₂²×#6157

Fitting

C₅₃×#322

C₄

C₂²×C₅₃

C₂²×D₅₃

C₄²×D₅₃

C₂³

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
11. #16810344	C₄	semidirect
12. C₈×#88322	C₄	semidirect
13. C₅₃×#322	C₄	semidirect
78. #322	#6157	semidirect
80. C₅₃	#8791	semidirect

H ≅ G/N

type

11. #16810344

C₄

semidirect

12. C₈×#88322

C₄

semidirect

13. C₅₃×#322

C₄

semidirect

78. #322

#6157

semidirect

80. C₅₃

#8791

semidirect

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

subgroup	index
#20290640	2
#20290637	2
#20289569	2
#20273000	2
#8791	53

subgroup

index

Subgroups — #28303544

Conjugacy classes of subgroups

Complements

Maximal subgroups