Group #15777659 — subgroups

Subgroups — #15777659

Order 3328.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117118	C₂	1	1	yes	maximal, characteristic
2	#13117109	C₂	1	2	yes	maximal
3	#13114445	C₂	1	2	yes	maximal
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	C₂²	1	2	yes	—
7	#11718754	—	2	2	no	—
8	#11718742	—	2	2	no	—
9	#11718343	C₂²	1	2	yes	—
10	C₁₃×#367	—	2	2	no	—
11	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	#11718097	—	2	2	no	—
13	#11718081	—	2	4	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	—	2	2	no	—
17	#88123	—	2	2	no	—
18	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
19	C₁₃×#107	—	2	2	no	—
20	C₁₃×#107	—	2	2	no	—
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88075	—	4	4	no	—
23	#88058	—	2	2	no	—
24	#88058	—	2	2	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	#88050	—	4	4	no	—
27	C₄×#1055	D₄	1	2	yes	—
28	#12522	—	13	1	no	maximal, Sylow
29	C₂×Q₈×C₁₃	—	4	2	no	—
30	C₂×Dic₂₆	—	4	2	no	—
31	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₁₃×C₄⋊C₄	—	4	2	no	—
34	C₁₃×C₄⋊C₄	—	4	2	no	—
35	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊Dic₁₃	—	4	2	no	—
37	C₄⋊Dic₁₃	—	4	2	no	—
38	C₂×#1055	D₈	1	4	yes	—
39	C₁₃×C₁₆	Q₁₆	1	4	yes	—
40	#6090	Q₁₆	1	4	yes	—
41	#2254	—	13	1	no	—
42	#2254	D₁₃	1	1	yes	characteristic
43	#2252	—	13	2	no	—
44	#1710	—	13	2	no	—
45	C₈×C₁₆	—	13	1	no	—
46	Q₈×C₁₃	—	8	4	no	—
47	C₂×C₄×C₁₃	—	8	2	no	—
48	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
50	C₂×Dic₁₃	—	8	2	no	—
51	Dic₂₆	—	8	4	no	—
52	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
53	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
54	#1055	—	2	4	no	—
55	#500	—	13	2	no	—
56	#500	C₂×D₁₃	1	2	yes	—
57	#487	—	26	2	no	—
58	#487	—	26	2	no	—
59	#367	—	2	2	no	—
60	#367	—	26	2	no	—
61	#366	—	26	4	no	—
62	C₄×C₁₆	—	13	1	no	—
63	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
64	C₈²	—	13	1	no	—
65	C₂²×C₁₃	#493	1	1	yes	characteristic
66	C₄×C₁₃	—	16	2	no	—
67	C₄×C₁₃	#494	1	1	yes	characteristic
68	C₄×C₁₃	#494	1	1	yes	characteristic
69	C₄×C₁₃	#508	1	4	yes	—
70	Dic₁₃	—	16	2	no	—
71	C₄⋊Q₈	—	2	2	no	—
72	C₄⋊Q₈	—	26	2	no	—
73	C₂×C₁₆	—	13	2	no	—
74	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
75	#107	—	2	2	no	—
76	#107	—	2	2	no	—
77	#107	—	26	2	no	—
78	#107	—	26	2	no	—
79	#103	—	52	4	no	—
80	#103	—	52	4	no	—
81	C₄×C₈	—	13	2	no	—
82	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
83	C₂×C₁₃	#1713	1	1	yes	characteristic
84	C₂×C₁₃	#2249	1	2	yes	—
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	52	2	no	—
87	C₂×C₈	—	13	4	no	—
88	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
89	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	52	2	no	—
93	C₄⋊C₄	—	52	2	no	—
94	C₄²	D₄×D₁₃	1	1	yes	characteristic
95	C₁₆	—	13	4	no	—
96	C₁₆	#6107	1	4	yes	—
97	C₁₃	#12522	1	1	yes	characteristic, complemented, Sylow
98	Q₈	—	8	4	no	—
99	Q₈	—	104	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	104	2	no	—
102	C₂×C₄	D₈×D₁₃	1	2	yes	—
103	C₂×C₄	#88194	1	1	yes	characteristic
104	C₈	—	26	4	no	—
105	C₈	Q₁₆×D₁₃	1	2	yes	—
106	C₈	C₂×#6107	1	2	yes	—
107	C₂²	#11718812	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	208	2	no	—
110	C₄	#11718577	1	4	yes	—
111	C₄	#11718753	1	1	yes	characteristic
112	C₄	#11718842	1	1	yes	characteristic
113	C₂	#13114453	1	1	yes	characteristic
114	C₂	#13117096	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117118

C₂

yes

maximal, characteristic

#13117109

C₂

yes

maximal

#13114445

C₂

yes

maximal

C₁₆×#1055

C₂

yes

maximal, characteristic

C₁₃×#2254

C₂

yes

maximal, characteristic

C₁₃×#500

C₂²

yes

—

#11718754

—

#11718742

—

#11718343

C₂²

yes

—

C₁₃×#367

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

#88050

—

C₄×#1055

D₄

yes

—

#12522

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

D₈

yes

—

C₁₃×C₁₆

Q₁₆

yes

—

#6090

Q₁₆

yes

—

#2254

—

#2254

D₁₃

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

—

#1055

—

#500

—

#500

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

Dic₁₃

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

D₄×D₁₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×D₁₃

yes

characteristic

C₁₆

—

C₁₆

#6107

yes

—

C₁₃

#12522

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

104

—

100

C₂×C₄

—

101

C₂×C₄

—

104

—

102

C₂×C₄

D₈×D₁₃

yes

—

103

C₂×C₄

#88194

yes

characteristic

104

C₈

—

105

C₈

Q₁₆×D₁₃

yes

—

106

C₈

C₂×#6107

yes

—

107

C₂²

#11718812

yes

characteristic

108

C₄

—

109

C₄

—

208

—

110

C₄

#11718577

yes

—

111

C₄

#11718753

yes

characteristic

112

C₄

#11718842

yes

characteristic

113

C₂

#13114453

yes

characteristic

114

C₂

#13117096

yes

—

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2254	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#2254	D₁₃
hypocenter	C₁₃	#12522
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#11718812

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2254

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
97. C₁₃	#12522	semidirect

H ≅ G/N

type

97. C₁₃

#12522

semidirect

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

subgroup	index
#13117118	2
#13117109	2
#13114445	2
C₁₆×#1055	2
C₁₃×#2254	2
#12522	13

subgroup

index

#13117118

#13117109

#13114445

C₁₆×#1055

C₁₃×#2254

#12522

Subgroups — #15777659

Conjugacy classes of subgroups

Complements

Maximal subgroups