Group #28309908 — subgroups

Subgroups — #28309908

Order 13568.

216 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 84 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	#20290276	C₂	1	2	yes	maximal
2	#20289741	C₂	1	4	yes	maximal, complemented
3	C₄×#13133156	C₂	1	1	yes	maximal, characteristic
4	#16811645	—	2	2	no	—
5	#16811486	—	2	2	no	—
6	#16811484	C₂²	1	4	yes	—
7	C₂×#13133156	C₄	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	#16810398	C₂²	1	2	yes	—
10	C₄×#11719908	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₄⋊#88324	—	2	4	no	—
13	#13133274	—	4	2	no	—
14	#13133272	—	4	2	no	—
15	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₅₃	Q₈	1	1	yes	characteristic
17	C₄²×D₅₃	D₄	1	1	yes	characteristic
18	#13133156	—	2	4	no	—
19	C₂×#11719908	C₂×C₄	1	2	yes	—
20	C₄×#88322	Q₈	1	1	yes	characteristic
21	C₈×Dic₅₃	D₄	1	1	yes	characteristic
22	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×D₅₃	SD₁₆	1	2	yes	—
27	C₂×#88324	—	4	4	no	—
28	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₄×Dic₅₃	SD₁₆	1	2	yes	—
32	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
34	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
35	C₈×D₅₃	—	2	2	no	—
36	C₈×D₅₃	—	2	2	no	—
37	C₁₆×C₅₃	—	2	2	no	—
38	#11719908	M₁₆	1	4	yes	—
39	C₂²×D₅₃	#106	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#106	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₅₃	#106	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
44	C₄×D₅₃	—	2	1	no	—
45	C₄×D₅₃	—	2	1	no	—
46	C₄×D₅₃	—	4	2	no	—
47	#88324	—	8	4	no	—
48	C₈×C₅₃	#108	1	2	yes	—
49	C₈×C₅₃	C₂×M₁₆	1	2	yes	—
50	#88322	—	2	1	no	—
51	#88322	—	2	1	no	—
52	#12021	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
54	C₂×D₅₃	—	2	1	no	—
55	C₂×D₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	#423	1	1	yes	characteristic
58	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	—	2	2	no	—
61	C₄×M₃₂	—	53	1	no	—
62	#1568	—	53	2	no	—
63	#1368	—	53	4	no	—
64	C₂×C₅₃	#1566	1	1	yes	characteristic
65	C₂×C₅₃	#2155	1	2	yes	—
66	D₅₃	—	4	1	no	—
67	C₂×M₃₂	—	53	2	no	—
68	#432	—	106	2	no	—
69	C₂×C₄×C₈	—	53	1	no	—
70	#346	—	53	2	no	—
71	C₄×C₁₆	—	53	1	no	—
72	C₄×C₁₆	#6157	1	1	yes	characteristic
73	#335	—	106	2	no	—
74	#334	—	53	4	no	—
75	C₅₃	#12021	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	53	1	no	—
77	C₂²×C₈	—	53	1	no	—
78	C₂×C₄²	—	53	1	no	—
79	M₃₂	—	106	4	no	—
80	C₂×C₁₆	—	53	2	no	—
81	C₂×C₁₆	C₂×#6157	1	2	yes	—
82	C₄⋊C₈	—	106	4	no	—
83	C₂²⋊C₈	—	212	2	no	—
84	#97	—	212	2	no	—
85	C₄×C₈	—	53	1	no	—
86	C₄×C₈	—	53	1	no	—
87	C₄×C₈	—	53	1	no	—
88	C₄×C₈	C₂×#6157	1	1	yes	characteristic
89	C₂²×C₄	—	53	1	no	—
90	C₂²×C₄	—	53	2	no	—
91	C₂×C₈	—	53	1	no	—
92	C₂×C₈	—	53	1	no	—
93	C₂×C₈	—	106	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₂×C₈	—	212	4	no	—
96	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
97	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
98	C₄²	—	53	2	no	—
99	C₄²	—	53	1	no	—
100	C₄²	C₄⋊#6157	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	—	53	4	no	—
103	C₂³	—	53	1	no	—
104	C₂×C₄	—	53	1	no	—
105	C₂×C₄	—	53	2	no	—
106	C₂×C₄	—	106	1	no	—
107	C₂×C₄	—	106	1	no	—
108	C₂×C₄	—	212	2	no	—
109	C₂×C₄	#13133285	1	2	yes	—
110	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
111	C₈	—	106	1	no	—
112	C₈	—	106	1	no	—
113	C₈	—	424	4	no	—
114	C₈	#13133288	1	2	yes	—
115	C₈	D₅₃⋊M₁₆	1	2	yes	—
116	C₂²	—	106	1	no	—
117	C₂²	—	106	2	no	—
118	C₂²	C₂×#13133285	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	106	1	no	—
121	C₄	—	106	2	no	—
122	C₄	#16811628	1	1	yes	characteristic
123	C₄	C₂×#13133288	1	1	yes	characteristic
124	C₂	—	212	1	no	—
125	C₂	#20290271	1	1	yes	characteristic
126	C₂	#20291062	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290276

C₂

yes

maximal

#20289741

C₂

yes

maximal, complemented

C₄×#13133156

C₂

yes

maximal, characteristic

#16811645

—

#16811486

—

#16811484

C₂²

yes

—

C₂×#13133156

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810398

C₂²

yes

—

C₄×#11719908

C₄

yes

characteristic

C₄×C₁₆×C₅₃

C₄

yes

—

—

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

Q₈

yes

characteristic

C₄²×D₅₃

D₄

yes

characteristic

#13133156

—

C₂×#11719908

C₂×C₄

yes

—

C₄×#88322

Q₈

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₅₃

SD₁₆

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

M₁₆

yes

—

C₂²×D₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

#106

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#106

yes

characteristic

C₂×Dic₅₃

C₂×SD₁₆

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

C₂×M₁₆

yes

—

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₄×M₃₂

—

#1568

—

#1368

—

C₂×C₅₃

#1566

yes

characteristic

C₂×C₅₃

#2155

yes

—

D₅₃

—

C₂×M₃₂

—

#432

—

106

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

#6157

yes

characteristic

#335

—

106

—

#334

—

C₅₃

#12021

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6157

yes

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

#97

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊#6157

yes

characteristic

C₂×C₈

C₂²×#6157

yes

characteristic

C₄²

—

C₄²

—

100

C₄²

C₄⋊#6157

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₂³

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

212

—

109

C₂×C₄

#13133285

yes

—

110

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

111

C₈

—

106

—

112

C₈

—

106

—

113

C₈

—

424

—

114

C₈

#13133288

yes

—

115

C₈

D₅₃⋊M₁₆

yes

—

116

C₂²

—

106

—

117

C₂²

—

106

—

118

C₂²

C₂×#13133285

yes

characteristic

119

C₄

—

120

C₄

—

106

—

121

C₄

—

106

—

122

C₄

#16811628

yes

characteristic

123

C₄

C₂×#13133288

yes

characteristic

124

C₂

—

212

—

125

C₂

#20290271

yes

characteristic

126

C₂

#20291062

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133285
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂²×D₅₃	#106
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#12021
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#13133285

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₄×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂²×D₅₃

C₈×Dic₅₃

C₂³

C₄×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #20289741	C₂	semidirect
75. C₅₃	#12021	semidirect

H ≅ G/N

type

2. #20289741

C₂

semidirect

75. C₅₃

#12021

semidirect

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

subgroup	index
#20290276	2
#20289741	2
C₄×#13133156	2
#12021	53

subgroup

index

Subgroups — #28309908

Conjugacy classes of subgroups

Complements

Maximal subgroups