Group #15777657 — subgroups

Subgroups — #15777657

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	#13117116	C₂	1	2	yes	maximal
2	#13117109	C₂	1	2	yes	maximal
3	#13114446	C₂	1	1	yes	maximal, characteristic
4	C₈×#6090	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1710	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	C₂²	1	2	yes	—
7	C₁₃×#487	—	2	2	no	—
8	#11718343	C₂²	1	2	yes	—
9	#11718342	—	2	2	no	—
10	C₈²×C₁₃	C₂²	1	1	yes	characteristic
11	#11718082	—	2	4	no	—
12	#11718081	—	2	4	no	—
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	—	2	2	no	—
16	#88123	—	2	2	no	—
17	C₁₃×#107	—	2	2	no	—
18	C₁₃×#107	—	2	2	no	—
19	C₁₃×#103	—	4	4	no	—
20	C₄×C₈×C₁₃	D₄	1	2	yes	—
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88058	—	2	2	no	—
23	#88058	—	2	2	no	—
24	#88056	—	4	4	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	C₂×#6090	D₄	1	2	yes	—
27	#12522	—	13	1	no	maximal, Sylow
28	C₂×Q₈×C₁₃	—	4	2	no	—
29	C₂×Dic₂₆	—	4	2	no	—
30	C₂×C₈×C₁₃	D₈	1	4	yes	—
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	#6090	Q₁₆	1	4	yes	—
39	#6090	Q₁₆	1	4	yes	—
40	#2254	—	13	2	no	—
41	#2252	—	13	2	no	—
42	#1710	—	13	1	no	—
43	#1710	D₁₃	1	1	yes	characteristic
44	C₈×C₁₆	—	13	1	no	—
45	Q₈×C₁₃	—	8	4	no	—
46	C₂×C₄×C₁₃	—	8	2	no	—
47	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
49	C₂×Dic₁₃	—	8	2	no	—
50	Dic₂₆	—	8	4	no	—
51	C₈×C₁₃	—	2	4	no	—
52	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
53	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
54	#500	—	13	2	no	—
55	#500	C₂×D₁₃	1	2	yes	—
56	#487	—	2	2	no	—
57	#487	—	26	2	no	—
58	#367	—	26	4	no	—
59	#366	—	26	4	no	—
60	C₄×C₁₆	—	13	1	no	—
61	C₄×C₁₆	—	13	1	no	—
62	C₈²	C₂×D₁₃	1	1	yes	characteristic
63	C₂²×C₁₃	#493	1	1	yes	characteristic
64	C₄×C₁₃	—	16	2	no	—
65	C₄×C₁₃	#494	1	1	yes	characteristic
66	C₄×C₁₃	#494	1	1	yes	characteristic
67	C₄×C₁₃	#508	1	4	yes	—
68	Dic₁₃	—	16	2	no	—
69	C₄⋊Q₈	—	2	2	no	—
70	C₄⋊Q₈	—	26	2	no	—
71	C₂×C₁₆	—	13	2	no	—
72	C₂×C₁₆	—	13	2	no	—
73	#107	—	2	2	no	—
74	#107	—	2	2	no	—
75	#107	—	26	2	no	—
76	#107	—	26	2	no	—
77	#103	—	4	4	no	—
78	#103	—	52	4	no	—
79	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
80	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
81	C₂×C₁₃	#1713	1	1	yes	characteristic
82	C₂×C₁₃	#2249	1	2	yes	—
83	C₂×Q₈	—	4	2	no	—
84	C₂×Q₈	—	52	2	no	—
85	C₂×C₈	#6104	1	4	yes	—
86	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
87	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	52	2	no	—
91	C₄⋊C₄	—	52	2	no	—
92	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
93	C₁₆	—	13	4	no	—
94	C₁₆	—	13	4	no	—
95	C₁₃	#12522	1	1	yes	characteristic, complemented, Sylow
96	Q₈	—	8	4	no	—
97	Q₈	—	104	4	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	—	104	2	no	—
100	C₂×C₄	C₂×#6104	1	2	yes	—
101	C₂×C₄	#88194	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	Q₁₆×D₁₃	1	2	yes	—
104	C₈	Q₁₆×D₁₃	1	2	yes	—
105	C₂²	#11718734	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	208	2	no	—
108	C₄	#11718842	1	1	yes	characteristic
109	C₄	#11718842	1	1	yes	characteristic
110	C₄	#11718855	1	4	yes	—
111	C₂	#13114452	1	1	yes	characteristic
112	C₂	#13117095	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117116

C₂

yes

maximal

#13117109

C₂

yes

maximal

#13114446

C₂

yes

maximal, characteristic

C₈×#6090

C₂

yes

maximal, characteristic

C₁₃×#1710

C₂

yes

maximal, characteristic

C₁₃×#500

C₂²

yes

—

C₁₃×#487

—

#11718343

C₂²

yes

—

#11718342

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

#88123

—

C₁₃×#107

—

C₁₃×#107

—

C₁₃×#103

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

C₂×#6090

D₄

yes

—

#12522

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

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

—

#6090

Q₁₆

yes

—

#6090

Q₁₆

yes

—

—

—

—

D₁₃

yes

characteristic

C₈×C₁₆

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

—

#500

—

#500

C₂×D₁₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

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

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₁₃

yes

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

#6104

yes

—

C₂×C₈

D₄×D₁₃

yes

characteristic

C₂×C₈

D₄×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₃

#12522

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

104

—

C₂×C₄

—

C₂×C₄

—

104

—

100

C₂×C₄

C₂×#6104

yes

—

101

C₂×C₄

#88194

yes

characteristic

102

C₈

—

103

C₈

Q₁₆×D₁₃

yes

—

104

C₈

Q₁₆×D₁₃

yes

—

105

C₂²

#11718734

yes

characteristic

106

C₄

—

107

C₄

—

208

—

108

C₄

#11718842

yes

characteristic

109

C₄

#11718842

yes

characteristic

110

C₄

#11718855

yes

—

111

C₂

#13114452

yes

characteristic

112

C₂

#13117095

yes

—

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1710	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#1710	D₁₃
hypocenter	C₁₃	#12522
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

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

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

H ≅ G/N

type

95. C₁₃

#12522

semidirect

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

subgroup	index
#13117116	2
#13117109	2
#13114446	2
C₈×#6090	2
C₁₃×#1710	2
#12522	13

subgroup

index

C₁₃×#1710

Subgroups — #15777657

Conjugacy classes of subgroups

Complements

Maximal subgroups