Group #28299006 — subgroups

Subgroups — #28299006

Order 13568.

110 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 48 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	C₄⋊#13133267	C₂	1	1	yes	maximal, characteristic
2	#20289822	C₂	1	1	yes	maximal, characteristic
3	#20276620	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133267	—	2	1	no	—
5	C₂×#13133267	—	2	1	no	—
6	C₈×Dic₁₀₆	C₄	1	1	yes	characteristic
7	#16810441	—	2	1	no	—
8	#16810398	—	2	1	no	—
9	C₄×#11719908	C₂²	1	1	yes	characteristic
10	C₅₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
11	#13133267	—	2	2	no	—
12	#13133267	—	4	1	no	—
13	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
14	C₂×#11719908	D₄	1	1	yes	characteristic
15	C₂×#11719908	D₄	1	1	yes	characteristic
16	C₄⋊#88322	C₈	1	1	yes	characteristic
17	#13133146	—	2	1	no	—
18	C₈×Dic₅₃	—	2	1	no	—
19	C₂×C₁₆×C₅₃	—	2	1	no	—
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
24	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
25	C₄⋊Dic₅₃	C₁₆	1	1	yes	characteristic
26	#11719919	—	2	1	no	—
27	C₄×Dic₅₃	—	2	1	no	—
28	C₂×#88322	—	2	1	no	—
29	C₁₆×C₅₃	—	4	1	no	—
30	#11719908	—	2	1	no	—
31	#11719908	SD₁₆	1	1	yes	characteristic
32	#11719908	Q₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
35	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
36	C₂×Dic₅₃	—	2	1	no	—
37	C₂×Dic₅₃	—	2	1	no	—
38	Dic₁₀₆	—	4	1	no	—
39	C₈×C₅₃	—	2	1	no	—
40	C₈×C₅₃	#103	1	1	yes	characteristic
41	C₈×C₅₃	#104	1	1	yes	characteristic
42	#88322	—	4	1	no	—
43	#7393	—	53	1	no	maximal, Sylow
44	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
45	C₄×C₅₃	—	2	1	no	—
46	C₄×C₅₃	—	2	1	no	—
47	C₄×C₅₃	Q₈⋊C₈	1	1	yes	characteristic
48	C₄×C₅₃	#350	1	1	yes	characteristic
49	Dic₅₃	—	4	1	no	—
50	Dic₅₃	—	4	1	no	—
51	#2183	—	53	1	no	—
52	C₄⋊C₃₂	—	53	1	no	—
53	#1397	—	53	1	no	—
54	C₂×C₅₃	Q₈⋊C₁₆	1	1	yes	characteristic
55	C₂×C₅₃	#1400	1	1	yes	characteristic
56	C₂×C₅₃	#1401	1	1	yes	characteristic
57	C₈×Q₈	—	53	1	no	—
58	C₂×C₃₂	—	106	1	no	—
59	C₂×C₃₂	—	106	1	no	—
60	C₄⋊C₁₆	—	106	1	no	—
61	C₄⋊C₁₆	#6157	1	1	yes	characteristic
62	#346	—	106	1	no	—
63	C₄×C₁₆	—	53	1	no	—
64	C₅₃	#7393	1	1	yes	characteristic, complemented, Sylow
65	C₄×Q₈	—	53	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	53	1	no	—
68	C₂×C₁₆	—	53	1	no	—
69	C₄⋊C₈	—	53	1	no	—
70	C₄⋊C₈	—	106	1	no	—
71	C₄×C₈	—	106	1	no	—
72	C₄×C₈	C₂×#6157	1	1	yes	characteristic
73	C₃₂	—	106	2	no	—
74	C₃₂	—	212	1	no	—
75	C₂×Q₈	—	53	1	no	—
76	C₂×C₈	—	106	1	no	—
77	C₂×C₈	#11719936	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
79	C₄⋊C₄	—	53	1	no	—
80	C₄⋊C₄	—	106	1	no	—
81	C₄²	—	106	1	no	—
82	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
83	C₁₆	—	4	1	no	—
84	C₁₆	—	53	1	no	—
85	C₁₆	—	53	1	no	—
86	C₁₆	—	106	1	no	—
87	Q₈	—	212	1	no	—
88	C₂×C₄	—	106	1	no	—
89	C₂×C₄	—	106	1	no	—
90	C₂×C₄	#13133274	1	1	yes	characteristic
91	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
92	C₂×C₄	#13133292	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	212	1	no	—
95	C₈	#13133280	1	1	yes	characteristic
96	C₈	#13133281	1	1	yes	characteristic
97	C₂²	#16811518	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	—	212	1	no	—
101	C₄	—	212	1	no	—
102	C₄	#16811473	1	1	yes	characteristic
103	C₄	#16811522	1	1	yes	characteristic
104	C₂	#20289682	1	1	yes	characteristic
105	C₂	#20289831	1	1	yes	characteristic
106	C₂	#20289833	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#13133267

C₂

yes

maximal, characteristic

#20289822

C₂

yes

maximal, characteristic

#20276620

C₂

yes

maximal, characteristic

C₂×#13133267

—

C₂×#13133267

—

C₈×Dic₁₀₆

C₄

yes

—

—

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

C₄

yes

characteristic

#13133267

—

#13133267

—

C₄×Dic₁₀₆

C₈

yes

characteristic

C₂×#11719908

D₄

yes

characteristic

C₂×#11719908

D₄

yes

characteristic

C₄⋊#88322

C₈

yes

characteristic

#13133146

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Dic₁₀₆

C₁₆

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄⋊Dic₅₃

C₁₆

yes

characteristic

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

C₁₆×C₅₃

—

#11719908

—

#11719908

SD₁₆

yes

characteristic

#11719908

Q₁₆

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

M₃₂

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₈⋊C₈

yes

characteristic

C₄×C₅₃

#350

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2183

—

C₄⋊C₃₂

—

#1397

—

C₂×C₅₃

Q₈⋊C₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₈×Q₈

—

C₂×C₃₂

—

106

—

C₂×C₃₂

—

106

—

C₄⋊C₁₆

—

106

—

C₄⋊C₁₆

#6157

yes

characteristic

#346

—

106

—

C₄×C₁₆

—

C₅₃

#7393

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₄×C₈

—

106

—

C₄×C₈

C₂×#6157

yes

characteristic

C₃₂

—

106

—

C₃₂

—

212

—

C₂×Q₈

—

C₂×C₈

—

106

—

C₂×C₈

#11719936

yes

characteristic

C₂×C₈

C₂²⋊#6157

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

—

106

—

C₄²

D₅₃⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

106

—

Q₈

—

212

—

C₂×C₄

—

106

—

C₂×C₄

—

106

—

C₂×C₄

#13133274

yes

characteristic

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

C₂×C₄

#13133292

yes

characteristic

C₈

—

C₈

—

212

—

C₈

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

100

C₄

—

212

—

101

C₄

—

212

—

102

C₄

#16811473

yes

characteristic

103

C₄

#16811522

yes

characteristic

104

C₂

#20289682

yes

characteristic

105

C₂

#20289831

yes

characteristic

106

C₂

#20289833

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂×C₁₆
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×C₄⋊C₁₆	C₄
socle, prime radical	C₂²×C₅₃	C₂²⋊C₁₆
cosocle	C₄×#11719908	C₂²
hypercenter	C₄⋊C₁₆	#6157
hypocenter	C₅₃	#7393
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂×C₈

C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂×C₁₆

Frattini

C₄×C₈

C₂×#6157

Fitting

C₅₃×C₄⋊C₁₆

C₄

socle, prime radical

C₂²×C₅₃

C₂²⋊C₁₆

C₂²

C₄⋊C₁₆

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
64. C₅₃	#7393	semidirect

H ≅ G/N

type

64. C₅₃

#7393

semidirect

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

subgroup	index
C₄⋊#13133267	2
#20289822	2
#20276620	2
#7393	53

subgroup

index

Subgroups — #28299006

Conjugacy classes of subgroups

Complements

Maximal subgroups