Group #15777637 — subgroups

Subgroups — #15777637

Order 3328.

354 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 115 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	#13117092	C₂	1	4	yes	maximal, complemented
2	#13114454	C₂	1	1	yes	maximal, characteristic, complemented
3	#13114451	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#88040	—	2	8	no	—
6	#11718360	C₂²	1	2	yes	—
7	#11718356	—	2	2	no	—
8	#11718356	C₂²	1	2	yes	—
9	#11718343	—	2	2	no	—
10	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₈²×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₂	—	2	4	no	—
13	C₂×D₁₀₄	—	2	4	no	—
14	C₂×D₁₀₄	—	4	4	no	—
15	#88128	—	2	2	no	—
16	#88123	—	2	2	no	—
17	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
18	C₄×C₈×C₁₃	D₄	1	2	yes	—
19	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	#88058	—	4	4	no	—
21	#88040	—	4	16	no	—
22	#12517	—	13	1	no	maximal, Sylow
23	C₂×D₅₂	—	4	2	no	—
24	C₂×D₅₂	—	4	4	no	—
25	C₂×Dic₂₆	—	4	2	no	—
26	C₂×C₈×C₁₃	D₈	1	4	yes	—
27	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
28	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₃	—	4	4	no	—
30	Dic₅₂	—	4	8	no	—
31	D₁₀₄	—	4	8	no	—
32	D₁₀₄	—	8	8	no	—
33	C₁₃×C₁₆	D₈	1	8	yes	complemented
34	#2248	—	13	4	no	—
35	#1713	—	13	1	no	—
36	#1712	—	13	1	no	—
37	C₈×C₁₆	D₁₃	1	1	yes	characteristic, complemented
38	C₂²×D₁₃	—	8	2	no	—
39	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
41	C₂×Dic₁₃	—	8	2	no	—
42	D₅₂	—	8	4	no	—
43	D₅₂	—	8	8	no	—
44	Dic₂₆	—	8	4	no	—
45	C₈×C₁₃	SD₃₂	1	8	yes	complemented
46	C₈×C₁₃	C₂×D₈	1	4	yes	—
47	C₂×SD₃₂	—	26	8	no	—
48	#500	—	26	2	no	—
49	#494	—	13	2	no	—
50	#493	—	13	2	no	—
51	#493	—	26	2	no	—
52	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
53	C₈²	C₂×D₁₃	1	1	yes	characteristic
54	C₂²×C₁₃	#493	1	1	yes	characteristic
55	C₂×D₁₃	—	16	2	no	—
56	C₂×D₁₃	—	16	4	no	—
57	C₄×C₁₃	#493	1	2	yes	—
58	C₄×C₁₃	C₂×SD₃₂	1	4	yes	—
59	Dic₁₃	—	16	2	no	—
60	C₂×Q₁₆	—	26	4	no	—
61	C₂×D₈	—	26	4	no	—
62	C₂×D₈	—	52	4	no	—
63	C₄⋊Q₈	—	26	2	no	—
64	#127	—	26	2	no	—
65	SD₃₂	—	52	16	no	—
66	C₂×C₁₆	D₅₂	1	4	yes	—
67	#107	—	52	4	no	—
68	C₄×C₈	D₅₂	1	2	yes	—
69	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
70	C₂×C₁₃	#1712	1	1	yes	characteristic
71	C₂×C₁₃	#2248	1	2	yes	—
72	D₁₃	—	32	2	no	—
73	C₂×Q₈	—	52	2	no	—
74	C₂×D₄	—	52	2	no	—
75	C₂×D₄	—	52	4	no	—
76	Q₁₆	—	52	8	no	—
77	D₈	—	52	8	no	—
78	D₈	—	104	8	no	—
79	C₂×C₈	D₁₀₄	1	4	yes	—
80	C₂×C₈	C₂×D₅₂	1	2	yes	—
81	C₄⋊C₄	—	52	4	no	—
82	C₄²	C₂×D₅₂	1	1	yes	characteristic
83	C₁₆	D₁₀₄	1	8	yes	complemented
84	C₁₃	#12517	1	1	yes	characteristic, complemented, Sylow
85	C₂³	—	104	2	no	—
86	Q₈	—	104	4	no	—
87	D₄	—	104	8	no	—
88	D₄	—	104	4	no	—
89	C₂×C₄	—	104	2	no	—
90	C₂×C₄	#88128	1	1	yes	characteristic
91	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
92	C₈	#88040	1	8	yes	complemented
93	C₈	C₂×D₁₀₄	1	4	yes	—
94	C₂²	—	208	2	no	—
95	C₂²	—	208	4	no	—
96	C₂²	#11718356	1	1	yes	characteristic
97	C₄	—	208	2	no	—
98	C₄	#11718356	1	2	yes	—
99	C₄	C₂×#88040	1	4	yes	—
100	C₂	—	416	2	no	—
101	C₂	#13114451	1	1	yes	characteristic
102	C₂	#13117092	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117092

C₂

yes

maximal, complemented

#13114454

C₂

yes

maximal, characteristic, complemented

#13114451

C₂

yes

maximal, characteristic

C₈×C₁₃×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#88040

—

#11718360

C₂²

yes

—

#11718356

—

#11718356

C₂²

yes

—

#11718343

—

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₂×Dic₅₂

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

Dic₅₂

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

D₈

yes

—

—

—

C₈×C₁₆

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

Dic₂₆

—

C₈×C₁₃

SD₃₂

yes

complemented

C₈×C₁₃

C₂×D₈

yes

—

C₂×SD₃₂

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#493

yes

—

C₄×C₁₃

C₂×SD₃₂

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₅₂

yes

—

#107

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

104

—

C₂×C₈

D₁₀₄

yes

—

C₂×C₈

C₂×D₅₂

yes

—

C₄⋊C₄

—

C₄²

C₂×D₅₂

yes

characteristic

C₁₆

D₁₀₄

yes

complemented

C₁₃

#12517

yes

characteristic, complemented, Sylow

C₂³

—

104

—

Q₈

—

104

—

D₄

—

104

—

D₄

—

104

—

C₂×C₄

—

104

—

C₂×C₄

#88128

yes

characteristic

C₂×C₄

C₂×D₁₀₄

yes

—

C₈

#88040

yes

complemented

C₈

C₂×D₁₀₄

yes

—

C₂²

—

208

—

C₂²

—

208

—

C₂²

#11718356

yes

characteristic

C₄

—

208

—

C₄

#11718356

yes

—

C₄

C₂×#88040

yes

—

100

C₂

—

416

—

101

C₂

#13114451

yes

characteristic

102

C₂

#13117092

yes

—

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₈×C₁₃×C₁₆	C₂
socle	C₂²×C₁₃	#493
prime radical	#13114451	C₂
hypercenter	C₈×C₁₆	D₁₃
hypocenter	C₁₃	#12517
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₈×C₁₃×C₁₆

C₂

C₂²×C₁₃

C₂

C₈×C₁₆

D₁₃

hypocenter

C₁₃

#12517

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #13117092	C₂	semidirect
2. #13114454	C₂	semidirect
4. C₈×C₁₃×C₁₆	C₂	semidirect
33. C₁₃×C₁₆	D₈	semidirect
37. C₈×C₁₆	D₁₃	semidirect
45. C₈×C₁₃	SD₃₂	semidirect
83. C₁₆	D₁₀₄	semidirect
84. C₁₃	#12517	semidirect
92. C₈	#88040	semidirect

H ≅ G/N

type

1. #13117092

C₂

semidirect

2. #13114454

C₂

semidirect

4. C₈×C₁₃×C₁₆

C₂

semidirect

33. C₁₃×C₁₆

D₈

semidirect

37. C₈×C₁₆

D₁₃

semidirect

45. C₈×C₁₃

SD₃₂

semidirect

83. C₁₆

D₁₀₄

semidirect

84. C₁₃

#12517

semidirect

92. C₈

#88040

semidirect

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

subgroup	index
#13117092	2
#13114454	2
#13114451	2
C₈×C₁₃×C₁₆	2
#12517	13

subgroup

index

C₈×C₁₃×C₁₆

Subgroups — #15777637

Conjugacy classes of subgroups

Complements

Maximal subgroups