Group #13268514 — subgroups

Subgroups — #13268514

Order 1792.

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	#11726815	C₂	1	4	yes	maximal, complemented
2	#11725819	C₂	1	1	yes	maximal, characteristic, complemented
3	#11725816	C₂	1	1	yes	maximal, characteristic
4	C₇×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#6374	—	2	8	no	—
6	#89471	C₂²	1	2	yes	—
7	#89467	—	2	2	no	—
8	#89467	C₂²	1	2	yes	—
9	#89454	—	2	2	no	—
10	C₄×C₇×C₁₆	C₂²	1	2	yes	—
11	C₇×C₈²	C₂²	1	1	yes	characteristic
12	#12517	—	7	1	no	maximal, Sylow
13	C₂×Dic₂₈	—	2	4	no	—
14	C₂×D₅₆	—	2	4	no	—
15	C₂×D₅₆	—	4	4	no	—
16	#6437	—	2	2	no	—
17	#6432	—	2	2	no	—
18	C₂×C₇×C₁₆	D₄	1	4	yes	—
19	C₄×C₇×C₈	D₄	1	2	yes	—
20	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
21	#6392	—	4	4	no	—
22	#6374	—	4	16	no	—
23	#2248	—	7	4	no	—
24	#1713	—	7	1	no	—
25	#1712	—	7	1	no	—
26	C₈×C₁₆	D₇	1	1	yes	characteristic, complemented
27	C₂×D₂₈	—	4	2	no	—
28	C₂×D₂₈	—	4	4	no	—
29	C₂×Dic₁₄	—	4	2	no	—
30	C₂×C₇×C₈	D₈	1	4	yes	—
31	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
32	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
33	C₄⋊Dic₇	—	4	4	no	—
34	Dic₂₈	—	4	8	no	—
35	D₅₆	—	4	8	no	—
36	D₅₆	—	8	8	no	—
37	C₇×C₁₆	D₈	1	8	yes	complemented
38	C₂×SD₃₂	—	14	8	no	—
39	#500	—	14	2	no	—
40	#494	—	7	2	no	—
41	#493	—	7	2	no	—
42	#493	—	14	2	no	—
43	C₄×C₁₆	C₂×D₇	1	2	yes	—
44	C₈²	C₂×D₇	1	1	yes	characteristic
45	C₂²×D₇	—	8	2	no	—
46	C₂×C₄×C₇	#127	1	1	yes	characteristic
47	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
48	C₂×Dic₇	—	8	2	no	—
49	D₂₈	—	8	4	no	—
50	D₂₈	—	8	8	no	—
51	Dic₁₄	—	8	4	no	—
52	C₇×C₈	SD₃₂	1	8	yes	complemented
53	C₇×C₈	C₂×D₈	1	4	yes	—
54	C₂×Q₁₆	—	14	4	no	—
55	C₂×D₈	—	14	4	no	—
56	C₂×D₈	—	28	4	no	—
57	C₄⋊Q₈	—	14	2	no	—
58	#127	—	14	2	no	—
59	SD₃₂	—	28	16	no	—
60	C₂×C₁₆	D₂₈	1	4	yes	—
61	#107	—	28	4	no	—
62	C₄×C₈	D₂₈	1	2	yes	—
63	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
64	C₂²×C₇	#493	1	1	yes	characteristic
65	C₂×D₇	—	16	2	no	—
66	C₂×D₇	—	16	4	no	—
67	C₄×C₇	#493	1	2	yes	—
68	C₄×C₇	C₂×SD₃₂	1	4	yes	—
69	Dic₇	—	16	2	no	—
70	C₂×Q₈	—	28	2	no	—
71	C₂×D₄	—	28	2	no	—
72	C₂×D₄	—	28	4	no	—
73	Q₁₆	—	28	8	no	—
74	D₈	—	28	8	no	—
75	D₈	—	56	8	no	—
76	C₂×C₈	D₅₆	1	4	yes	—
77	C₂×C₈	C₂×D₂₈	1	2	yes	—
78	C₄⋊C₄	—	28	4	no	—
79	C₄²	C₂×D₂₈	1	1	yes	characteristic
80	C₁₆	D₅₆	1	8	yes	complemented
81	C₂×C₇	#1712	1	1	yes	characteristic
82	C₂×C₇	#2248	1	2	yes	—
83	D₇	—	32	2	no	—
84	C₂³	—	56	2	no	—
85	Q₈	—	56	4	no	—
86	D₄	—	56	8	no	—
87	D₄	—	56	4	no	—
88	C₂×C₄	—	56	2	no	—
89	C₂×C₄	#6437	1	1	yes	characteristic
90	C₂×C₄	C₂×D₅₆	1	2	yes	—
91	C₈	#6374	1	8	yes	complemented
92	C₈	C₂×D₅₆	1	4	yes	—
93	C₇	#12517	1	1	yes	characteristic, complemented, Sylow
94	C₂²	—	112	4	no	—
95	C₂²	—	112	2	no	—
96	C₂²	#89467	1	1	yes	characteristic
97	C₄	—	112	2	no	—
98	C₄	#89467	1	2	yes	—
99	C₄	C₂×#6374	1	4	yes	—
100	C₂	—	224	2	no	—
101	C₂	#11725816	1	1	yes	characteristic
102	C₂	#11726815	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11726815

C₂

yes

maximal, complemented

#11725819

C₂

yes

maximal, characteristic, complemented

#11725816

C₂

yes

maximal, characteristic

C₇×C₈×C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×#6374

—

#89471

C₂²

yes

—

#89467

—

#89467

C₂²

yes

—

#89454

—

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

characteristic

#12517

—

maximal, Sylow

C₂×Dic₂₈

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

—

—

—

—

—

C₈×C₁₆

D₇

yes

characteristic, complemented

C₂×D₂₈

—

C₂×D₂₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₄²×C₇

C₂×D₄

yes

characteristic

C₄⋊Dic₇

—

Dic₂₈

—

D₅₆

—

D₅₆

—

C₇×C₁₆

D₈

yes

complemented

C₂×SD₃₂

—

—

—

—

—

C₄×C₁₆

C₂×D₇

yes

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

Dic₁₄

—

C₇×C₈

SD₃₂

yes

complemented

C₇×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

D₂₈

yes

—

#107

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

#493

yes

—

C₄×C₇

C₂×SD₃₂

yes

—

Dic₇

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

D₅₆

yes

—

C₂×C₈

C₂×D₂₈

yes

—

C₄⋊C₄

—

C₄²

C₂×D₂₈

yes

characteristic

C₁₆

D₅₆

yes

C₂×C₇

yes

C₂×C₇

yes

—

D₇

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#6437

yes

characteristic

C₂×C₄

C₂×D₅₆

yes

—

C₈

#6374

yes

complemented

C₈

C₂×D₅₆

yes

—

C₇

#12517

yes

characteristic, complemented, Sylow

C₂²

—

112

—

C₂²

—

112

—

C₂²

#89467

yes

characteristic

C₄

—

112

—

C₄

#89467

yes

—

C₄

C₂×#6374

yes

—

100

C₂

—

224

—

101

C₂

#11725816

yes

characteristic

102

C₂

#11726815

yes

—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×C₈×C₁₆	C₂
socle	C₂²×C₇	#493
prime radical	#11725816	C₂
hypercenter	C₈×C₁₆	D₇
hypocenter	C₇	#12517
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

Fitting

C₇×C₈×C₁₆

C₂

C₂²×C₇

C₂

C₈×C₁₆

D₇

hypocenter

C₇

#12517

LCS

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
1. #11726815	C₂	semidirect
2. #11725819	C₂	semidirect
4. C₇×C₈×C₁₆	C₂	semidirect
26. C₈×C₁₆	D₇	semidirect
37. C₇×C₁₆	D₈	semidirect
52. C₇×C₈	SD₃₂	semidirect
80. C₁₆	D₅₆	semidirect
91. C₈	#6374	semidirect
93. C₇	#12517	semidirect

H ≅ G/N

type

1. #11726815

C₂

semidirect

2. #11725819

C₂

semidirect

4. C₇×C₈×C₁₆

C₂

semidirect

26. C₈×C₁₆

D₇

semidirect

37. C₇×C₁₆

D₈

semidirect

52. C₇×C₈

SD₃₂

semidirect

80. C₁₆

D₅₆

semidirect

91. C₈

#6374

semidirect

93. C₇

#12517

semidirect

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

subgroup	index
#11726815	2
#11725819	2
#11725816	2
C₇×C₈×C₁₆	2
#12517	7

subgroup

index

C₇×C₈×C₁₆

Subgroups — #13268514

Conjugacy classes of subgroups

Complements

Maximal subgroups