Group #15756693 — subgroups

Subgroups — #15756693

Order 3328.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#13114993	C₂	1	2	yes	maximal
2	#13114964	C₂	1	4	yes	maximal
3	C₂×C₈²×C₁₃	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
5	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×#88057	—	2	4	no	—
7	#11718596	C₂²	1	4	yes	—
8	C₈²×C₁₃	C₄	1	4	yes	complemented
9	#11718143	—	2	8	no	—
10	C₂²×C₈×C₁₃	D₄	1	4	yes	—
11	C₂²×C₈×C₁₃	D₄	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃	—	2	4	no	—
14	C₂×C₄⋊Dic₁₃	—	2	8	no	—
15	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
16	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
17	C₄×C₈×C₁₃	Q₈	1	4	yes	—
18	C₄×C₈×C₁₃	Q₈	1	2	yes	—
19	#88057	—	4	8	no	—
20	#8815	—	13	1	no	maximal, Sylow
21	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
23	C₂²×Dic₁₃	—	4	4	no	—
24	C₂×C₈×C₁₃	—	2	8	no	—
25	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	—
26	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
27	C₂×C₈×C₁₃	SD₁₆	1	8	yes	—
28	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
31	C₄⋊Dic₁₃	—	4	8	no	—
32	C₄⋊Dic₁₃	—	4	16	no	—
33	#1844	—	13	2	no	—
34	#1834	—	13	4	no	—
35	C₂×C₈²	D₁₃	1	1	yes	characteristic
36	C₂³×C₁₃	#127	1	1	yes	characteristic
37	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
41	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
42	C₂×C₄×C₁₃	#135	1	8	yes	—
43	C₂×Dic₁₃	—	8	4	no	—
44	C₂×Dic₁₃	—	8	8	no	—
45	C₈×C₁₃	—	2	8	no	—
46	C₈×C₁₃	#106	1	8	yes	—
47	C₂×#106	—	26	4	no	—
48	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
49	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
50	#384	—	13	4	no	—
51	#340	—	26	8	no	—
52	C₈²	Dic₁₃	1	4	yes	complemented
53	C₂²×C₁₃	#384	1	1	yes	characteristic
54	C₂²×C₁₃	#492	1	2	yes	—
55	C₂²×C₁₃	#495	1	4	yes	—
56	C₄×C₁₃	C₂×#106	1	2	yes	—
57	C₄×C₁₃	#427	1	4	yes	—
58	C₄×C₁₃	#498	1	2	yes	—
59	C₄×C₁₃	#499	1	4	yes	—
60	Dic₁₃	—	16	4	no	—
61	C₂²×C₈	D₅₂	1	4	yes	—
62	C₂²×C₈	D₅₂	1	2	yes	—
63	C₂×C₄⋊C₄	—	26	8	no	—
64	C₂×C₄⋊C₄	—	26	4	no	—
65	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
66	#106	—	52	8	no	—
67	C₄×C₈	Dic₂₆	1	4	yes	—
68	C₄×C₈	Dic₂₆	1	2	yes	—
69	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
70	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
71	C₂×C₁₃	#1708	1	4	yes	—
72	C₂×C₁₃	#1844	1	1	yes	characteristic
73	C₂×C₁₃	#1846	1	2	yes	—
74	C₂²×C₄	—	52	4	no	—
75	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₅₂	1	2	yes	—
77	C₂×C₈	—	2	8	no	—
78	C₂×C₈	#6095	1	8	yes	—
79	C₂×C₈	C₄⋊Dic₁₃	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₁₃	1	4	yes	—
81	C₄⋊C₄	—	52	8	no	—
82	C₄⋊C₄	—	52	16	no	—
83	C₄²	C₂×Dic₂₆	1	2	yes	—
84	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
85	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
86	C₁₃	#8815	1	1	yes	characteristic, complemented, Sylow
87	C₂³	#88128	1	1	yes	characteristic
88	C₂×C₄	—	104	4	no	—
89	C₂×C₄	—	104	8	no	—
90	C₂×C₄	#88123	1	1	yes	characteristic
91	C₂×C₄	#88123	1	2	yes	—
92	C₂×C₄	C₂×#6095	1	4	yes	—
93	C₂×C₄	#88158	1	8	yes	—
94	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
96	C₈	—	2	8	no	—
97	C₈	#88057	1	8	yes	—
98	C₂²	#11718354	1	2	yes	—
99	C₂²	#11718357	1	4	yes	—
100	C₂²	#11718596	1	1	yes	characteristic
101	C₄	—	208	4	no	—
102	C₄	#11718341	1	2	yes	—
103	C₄	#11718344	1	4	yes	—
104	C₄	C₂×#88057	1	2	yes	—
105	C₄	#11718767	1	4	yes	—
106	C₂	#13114437	1	4	yes	—
107	C₂	#13114993	1	1	yes	characteristic
108	C₂	#13115000	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13114993

C₂

yes

maximal

#13114964

C₂

yes

maximal

C₂×C₈²×C₁₃

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×C₄×C₈×C₁₃

C₂²

yes

characteristic

C₂×#88057

—

#11718596

C₂²

yes

—

C₈²×C₁₃

C₄

yes

complemented

#11718143

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

C₂×C₄

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

C₄×C₈×C₁₃

Q₈

yes

—

#88057

—

#8815

—

maximal, Sylow

C₂²×C₄×C₁₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

SD₁₆

yes

—

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

C₂×Q₈

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#1844

—

#1834

—

C₂×C₈²

D₁₃

yes

characteristic

C₂³×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃

#135

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₈×C₁₃

—

C₈×C₁₃

#106

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

C₂×C₄×C₈

C₂×D₁₃

yes

characteristic

#384

—

#340

—

C₈²

Dic₁₃

yes

C₂²×C₁₃

yes

C₂²×C₁₃

yes

—

C₂²×C₁₃

#495

yes

—

C₄×C₁₃

C₂×#106

yes

—

C₄×C₁₃

#427

yes

—

C₄×C₁₃

#498

yes

—

C₄×C₁₃

#499

yes

—

Dic₁₃

—

C₂²×C₈

D₅₂

yes

—

C₂²×C₈

D₅₂

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#106

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

Dic₂₆

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₄×C₈

C₂×Dic₁₃

yes

—

C₂×C₁₃

#1708

yes

—

C₂×C₁₃

#1844

yes

characteristic

C₂×C₁₃

#1846

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₂

yes

characteristic

C₂²×C₄

C₂×D₅₂

yes

—

C₂×C₈

—

C₂×C₈

#6095

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₂×C₈

C₄⋊Dic₁₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₂₆

yes

—

C₄²

C₂×Dic₂₆

yes

characteristic

C₄²

C₂²×Dic₁₃

yes

characteristic

C₁₃

#8815

yes

characteristic, complemented, Sylow

C₂³

#88128

yes

characteristic

C₂×C₄

—

104

—

C₂×C₄

—

104

—

C₂×C₄

#88123

yes

characteristic

C₂×C₄

#88123

yes

—

C₂×C₄

C₂×#6095

yes

—

C₂×C₄

#88158

yes

—

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₁₃

yes

—

C₈

—

C₈

#88057

yes

—

C₂²

#11718354

yes

—

C₂²

#11718357

yes

—

100

C₂²

#11718596

yes

characteristic

101

C₄

—

208

—

102

C₄

#11718341

yes

—

103

C₄

#11718344

yes

—

104

C₄

C₂×#88057

yes

—

105

C₄

#11718767

yes

—

106

C₂

#13114437

yes

—

107

C₂

#13114993

yes

characteristic

108

C₂

#13115000

yes

—

	subgroup	G/H
center	C₂³	#88128
commutator	C₄²×C₁₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₂×C₈²×C₁₃	C₂
socle, prime radical	C₂³×C₁₃	#127
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₂×C₈²	D₁₃
hypocenter	C₁₃	#8815
LCS	G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#88128

commutator

C₄²×C₁₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₂×C₈²×C₁₃

C₂

socle, prime radical

C₂³×C₁₃

#127

cosocle

C₂×C₄²×C₁₃

C₂³

hypercenter

C₂×C₈²

D₁₃

hypocenter

C₁₃

#8815

LCS

G ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
8. C₈²×C₁₃	C₄	semidirect
52. C₈²	Dic₁₃	semidirect
86. C₁₃	#8815	semidirect

H ≅ G/N

type

8. C₈²×C₁₃

C₄

semidirect

52. C₈²

Dic₁₃

semidirect

86. C₁₃

#8815

semidirect

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

subgroup	index
#13114993	2
#13114964	2
C₂×C₈²×C₁₃	2
#8815	13

subgroup

index

#13114993

#13114964

C₂×C₈²×C₁₃

#8815

Subgroups — #15756693

Conjugacy classes of subgroups

Complements

Maximal subgroups