Group #15777624 — subgroups

Subgroups — #15777624

Order 3328.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13117061	C₂	1	4	yes	maximal, complemented
2	#13114451	C₂	1	1	yes	maximal, characteristic
3	#13112753	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#500	C₂²	1	2	yes	—
6	C₁₃×#487	—	2	2	no	—
7	#11718356	—	2	2	no	—
8	#11718356	C₂²	1	2	yes	—
9	C₈²×C₁₃	C₂²	1	1	yes	characteristic
10	#11718083	—	2	8	no	—
11	C₄×#6090	C₂²	1	2	yes	—
12	C₁₃×C₄⋊Q₈	—	2	2	no	—
13	C₂×D₁₀₄	—	2	4	no	—
14	C₂×D₁₀₄	—	4	4	no	—
15	#88128	—	2	2	no	—
16	C₁₃×#107	—	2	4	no	—
17	C₁₃×#103	—	4	4	no	—
18	C₄×C₈×C₁₃	D₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×#6090	D₄	1	4	yes	—
21	#12513	—	13	1	no	maximal, Sylow
22	C₂×Q₈×C₁₃	—	4	2	no	—
23	C₂×D₅₂	—	4	2	no	—
24	C₂×D₅₂	—	4	4	no	—
25	C₂×C₈×C₁₃	D₈	1	4	yes	—
26	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
27	C₁₃×C₄⋊C₄	—	4	2	no	—
28	C₁₃×C₄⋊C₄	—	4	2	no	—
29	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
30	D₁₀₄	—	4	8	no	—
31	D₁₀₄	—	8	8	no	—
32	#6090	—	2	4	no	—
33	#2240	—	13	4	no	—
34	#1712	—	13	1	no	—
35	#1710	D₁₃	1	1	yes	characteristic, complemented
36	#1311	—	13	1	no	—
37	C₂²×D₁₃	—	8	2	no	—
38	Q₈×C₁₃	—	8	4	no	—
39	C₂×C₄×C₁₃	—	8	2	no	—
40	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
41	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
42	D₅₂	—	8	4	no	—
43	D₅₂	—	8	8	no	—
44	C₈×C₁₃	—	2	4	no	—
45	C₈×C₁₃	#135	1	4	yes	—
46	#500	C₂×D₁₃	1	2	yes	—
47	#493	—	13	2	no	—
48	#493	—	26	2	no	—
49	#487	—	2	2	no	—
50	#357	—	26	8	no	—
51	C₄×C₁₆	—	13	2	no	—
52	C₈²	C₂×D₁₃	1	1	yes	characteristic
53	C₂²×C₁₃	#493	1	1	yes	characteristic
54	C₂×D₁₃	—	16	2	no	—
55	C₂×D₁₃	—	16	4	no	—
56	C₄×C₁₃	—	16	2	no	—
57	C₄×C₁₃	#495	1	2	yes	—
58	C₄×C₁₃	#509	1	4	yes	—
59	C₂×D₈	—	26	4	no	—
60	C₂×D₈	—	52	4	no	—
61	C₄⋊Q₈	—	2	2	no	—
62	#127	—	26	2	no	—
63	C₂×C₁₆	—	13	4	no	—
64	#107	—	2	4	no	—
65	#103	—	4	4	no	—
66	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
67	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
68	C₂×C₁₃	#1709	1	1	yes	characteristic
69	C₂×C₁₃	#2250	1	2	yes	—
70	D₁₃	—	32	2	no	—
71	C₂×Q₈	—	4	2	no	—
72	C₂×D₄	—	52	4	no	—
73	C₂×D₄	—	52	2	no	—
74	D₈	—	52	8	no	—
75	D₈	—	104	8	no	—
76	C₂×C₈	#6104	1	4	yes	—
77	C₂×C₈	D₄×D₁₃	1	2	yes	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
81	C₁₆	—	26	4	no	—
82	C₁₃	#12513	1	1	yes	characteristic, complemented, Sylow
83	C₂³	—	104	2	no	—
84	Q₈	—	8	4	no	—
85	D₄	—	104	4	no	—
86	D₄	—	104	8	no	—
87	C₂×C₄	—	8	2	no	—
88	C₂×C₄	C₂×#6104	1	2	yes	—
89	C₂×C₄	#88194	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	#88173	1	4	yes	—
92	C₂²	—	208	2	no	—
93	C₂²	—	208	4	no	—
94	C₂²	#11718734	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#11718852	1	2	yes	—
97	C₄	#11718854	1	4	yes	—
98	C₂	—	416	2	no	—
99	C₂	#13114443	1	1	yes	characteristic
100	C₂	#13117103	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117061

C₂

yes

maximal, complemented

#13114451

C₂

yes

maximal, characteristic

#13112753

C₂

yes

maximal, characteristic, complemented

C₁₃×#1710

C₂

yes

maximal, characteristic, complemented

C₁₃×#500

C₂²

yes

—

C₁₃×#487

—

#11718356

—

#11718356

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718083

—

C₄×#6090

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₁₃×#107

—

C₁₃×#103

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₂×#6090

D₄

yes

—

#12513

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

D₁₀₄

—

D₁₀₄

—

—

—

—

D₁₃

yes

characteristic, complemented

#1311

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

C₈×C₁₃

#135

yes

—

#500

C₂×D₁₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#495

yes

—

C₄×C₁₃

#509

yes

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₁₃

yes

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

104

—

C₂×C₈

#6104

yes

—

C₂×C₈

D₄×D₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

C₁₆

—

C₁₃

#12513

yes

characteristic, complemented, Sylow

C₂³

—

104

—

Q₈

—

D₄

—

104

—

D₄

—

104

—

C₂×C₄

—

C₂×C₄

C₂×#6104

yes

—

C₂×C₄

#88194

yes

characteristic

C₈

—

C₈

#88173

yes

—

C₂²

—

208

—

C₂²

—

208

—

C₂²

#11718734

yes

characteristic

C₄

—

C₄

#11718852

yes

—

C₄

#11718854

yes

—

C₂

—

416

—

C₂

#13114443

yes

characteristic

100

C₂

#13117103

yes

—

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1710	C₂
socle	C₂²×C₁₃	#493
prime radical	#13114451	C₂
hypercenter	#1710	D₁₃
hypocenter	C₁₃	#12513
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1710

C₂

C₂²×C₁₃

C₂

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
1. #13117061	C₂	semidirect
3. #13112753	C₂	semidirect
4. C₁₃×#1710	C₂	semidirect
35. #1710	D₁₃	semidirect
82. C₁₃	#12513	semidirect

H ≅ G/N

type

1. #13117061

C₂

semidirect

3. #13112753

C₂

semidirect

4. C₁₃×#1710

C₂

semidirect

35. #1710

D₁₃

semidirect

82. C₁₃

#12513

semidirect

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

subgroup	index
#13117061	2
#13114451	2
#13112753	2
C₁₃×#1710	2
#12513	13

subgroup

index

C₁₃×#1710

Subgroups — #15777624

Conjugacy classes of subgroups

Complements

Maximal subgroups