Group #13117056 — subgroups

Subgroups — #13117056

Order 1664.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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	#11718798	C₂	1	1	yes	maximal, characteristic
2	#11718786	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#349	C₂	1	1	yes	maximal, characteristic
4	#11718109	C₂	1	2	yes	maximal, complemented
5	#11718105	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	#88178	—	2	1	no	—
8	C₂×#6099	—	2	1	no	—
9	C₂×Dic₅₂	C₂²	1	1	yes	characteristic
10	C₁₃×M₃₂	C₂²	1	2	yes	—
11	#88059	—	2	1	no	—
12	#88059	C₂²	1	2	yes	—
13	#88057	C₂²	1	1	yes	characteristic
14	#88056	—	2	1	no	—
15	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
16	C₂×Dic₂₆	—	2	1	no	—
17	C₁₃×M₁₆	—	2	2	no	—
18	C₂×C₈×C₁₃	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂²⋊Dic₁₃	—	4	1	no	—
21	C₄⋊Dic₁₃	—	2	1	no	—
22	#6101	—	4	1	no	—
23	#6099	—	2	2	no	—
24	#6099	—	4	1	no	—
25	C₂×#1055	—	4	1	no	—
26	Dic₅₂	—	2	2	no	—
27	C₁₃×C₁₆	—	2	2	no	—
28	#2239	—	13	1	no	maximal, Sylow
29	C₂³×C₁₃	SD₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₃	—	2	1	no	—
31	C₂×C₄×C₁₃	—	2	1	no	—
32	C₂×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₁₃	—	4	1	no	—
35	C₂×Dic₁₃	—	4	1	no	—
36	Dic₂₆	—	4	1	no	—
37	Dic₂₆	—	4	1	no	—
38	C₈×C₁₃	—	2	1	no	—
39	C₈×C₁₃	#41	1	2	yes	—
40	#1055	—	4	2	no	—
41	#470	—	13	1	no	—
42	#430	—	13	1	no	—
43	#365	—	13	2	no	—
44	#362	—	13	2	no	—
45	#349	D₁₃	1	1	yes	characteristic
46	C₂²×C₁₃	—	2	1	no	—
47	C₂²×C₁₃	—	4	1	no	—
48	C₂²×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
49	C₄×C₁₃	—	2	1	no	—
50	C₄×C₁₃	#123	1	1	yes	characteristic
51	C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
52	Dic₁₃	—	8	1	no	—
53	Dic₁₃	—	8	1	no	—
54	C₂×Q₁₆	—	13	1	no	—
55	C₂×M₁₆	—	26	1	no	—
56	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic
57	C₂²⋊Q₈	—	26	1	no	—
58	M₃₂	C₂×D₁₃	1	2	yes	—
59	#108	—	13	2	no	—
60	#108	—	26	1	no	—
61	#106	—	13	1	no	—
62	#103	—	26	1	no	—
63	C₂×C₁₃	—	2	1	no	—
64	C₂×C₁₃	—	4	1	no	—
65	C₂×C₁₃	#481	1	1	yes	characteristic
66	C₂×Q₈	—	26	1	no	—
67	C₂²×C₄	D₅₂	1	1	yes	characteristic
68	Q₁₆	—	26	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	26	2	no	—
71	M₁₆	—	52	1	no	—
72	C₂×C₈	—	52	1	no	—
73	C₂×C₈	D₅₂	1	1	yes	characteristic
74	C₂×C₈	C₂²×D₁₃	1	1	yes	characteristic
75	C₄⋊C₄	—	26	1	no	—
76	C₄⋊C₄	—	52	1	no	—
77	C₂²⋊C₄	—	52	1	no	—
78	C₁₆	—	2	2	no	—
79	C₁₃	#2239	1	1	yes	characteristic, complemented, Sylow
80	C₂³	#6095	1	1	yes	characteristic
81	Q₈	—	52	1	no	—
82	Q₈	—	52	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	52	1	no	—
86	C₂×C₄	—	52	1	no	—
87	C₂×C₄	#6095	1	1	yes	characteristic
88	C₂×C₄	C₂×D₅₂	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	52	2	no	—
91	C₈	#6129	1	2	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	C₂×#6095	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	104	1	no	—
97	C₄	—	104	1	no	—
98	C₄	#88140	1	1	yes	characteristic
99	C₄	#88162	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	4	1	no	—
102	C₂	#11718396	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11718798

C₂

yes

maximal, characteristic

#11718786

C₂

yes

maximal, characteristic

C₁₃×#349

C₂

yes

maximal, characteristic

#11718109

C₂

yes

maximal, complemented

#11718105

C₂

yes

maximal, complemented

C₂×C₁₃×M₁₆

C₂²

yes

characteristic

#88178

—

C₂×#6099

—

C₂×Dic₅₂

C₂²

yes

characteristic

C₁₃×M₃₂

C₂²

yes

—

#88059

—

#88059

C₂²

yes

—

#88057

C₂²

yes

characteristic

#88056

—

C₂²×C₄×C₁₃

D₄

yes

characteristic

C₂×Dic₂₆

—

C₁₃×M₁₆

—

C₂×C₈×C₁₃

D₄

yes

characteristic

C₂×C₈×C₁₃

C₂³

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

—

—

—

—

Dic₅₂

—

C₁₃×C₁₆

—

#2239

—

maximal, Sylow

C₂³×C₁₃

SD₁₆

yes

characteristic

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

SD₁₆

yes

characteristic

C₂×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

#41

yes

—

—

—

—

—

—

D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×SD₁₆

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₃

yes

characteristic

C₂²⋊Q₈

—

M₃₂

C₂×D₁₃

yes

—

—

—

—

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

#481

yes

characteristic

C₂×Q₈

—

C₂²×C₄

D₅₂

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

D₅₂

yes

characteristic

C₂×C₈

C₂²×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₁₃

#2239

yes

characteristic, complemented, Sylow

C₂³

#6095

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6095

yes

characteristic

C₂×C₄

C₂×D₅₂

yes

characteristic

C₈

—

C₈

—

C₈

#6129

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#6095

yes

characteristic

C₄

—

C₄

—

104

—

C₄

—

104

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#11718396

yes

characteristic

	subgroup	G/H
center	C₂	#11718396
commutator, cosocle	C₂×C₈×C₁₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₃
Fitting	C₁₃×#349	C₂
socle	C₂×C₁₃	#481
prime radical	C₂³×C₁₃	SD₁₆
hypercenter	#349	D₁₃
hypocenter	C₁₃	#2239
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

subgroup

G/H

center

C₂

#11718396

commutator, cosocle

C₂×C₈×C₁₃

C₂³

Frattini

C₂×C₈

C₂²×D₁₃

Fitting

C₁₃×#349

C₂

socle

C₂×C₁₃

#481

prime radical

C₂³×C₁₃

SD₁₆

D₁₃

C₁₃

G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

N	H ≅ G/N	type
4. #11718109	C₂	semidirect
5. #11718105	C₂	semidirect
79. C₁₃	#2239	semidirect

H ≅ G/N

type

4. #11718109

C₂

semidirect

5. #11718105

C₂

semidirect

79. C₁₃

#2239

semidirect

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

subgroup	index
#11718798	2
#11718786	2
C₁₃×#349	2
#11718109	2
#11718105	2
#2239	13

subgroup

index

C₁₃×#349

Subgroups — #13117056

Conjugacy classes of subgroups

Complements

Maximal subgroups