Group #13268500 — subgroups

Subgroups — #13268500

Order 1792.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#11726812	C₂	1	4	yes	maximal, complemented
2	#11725818	C₂	1	1	yes	maximal, characteristic, complemented
3	#11725816	C₂	1	1	yes	maximal, characteristic
4	C₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#89467	—	2	2	no	—
6	#89467	C₂²	1	2	yes	—
7	#89454	C₂²	1	2	yes	—
8	#89453	—	2	2	no	—
9	C₄×C₇×C₁₆	C₂²	1	2	yes	—
10	C₇×C₈²	C₂²	1	1	yes	characteristic
11	#89304	—	2	8	no	—
12	#12513	—	7	1	no	maximal, Sylow
13	C₂×D₅₆	—	2	4	no	—
14	C₂×D₅₆	—	4	4	no	—
15	#6437	—	2	2	no	—
16	#6432	—	2	2	no	—
17	C₂×C₇×C₁₆	D₄	1	4	yes	—
18	C₄×C₇×C₈	D₄	1	2	yes	—
19	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
20	#6392	—	2	4	no	—
21	#6390	—	4	4	no	—
22	#2240	—	7	4	no	—
23	#1712	—	7	1	no	—
24	#1710	—	7	1	no	—
25	#1311	D₇	1	1	yes	characteristic, complemented
26	C₂×D₂₈	—	4	2	no	—
27	C₂×D₂₈	—	4	4	no	—
28	C₂×Dic₁₄	—	4	2	no	—
29	C₂×C₇×C₈	D₈	1	4	yes	—
30	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
31	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
32	C₄⋊Dic₇	—	4	2	no	—
33	C₄⋊Dic₇	—	4	2	no	—
34	D₅₆	—	4	8	no	—
35	D₅₆	—	8	8	no	—
36	C₇×C₁₆	—	2	4	no	—
37	#500	—	7	2	no	—
38	#493	—	7	2	no	—
39	#493	—	14	2	no	—
40	#487	—	14	2	no	—
41	#357	—	14	8	no	—
42	C₄×C₁₆	C₂×D₇	1	2	yes	—
43	C₈²	C₂×D₇	1	1	yes	characteristic
44	C₂²×D₇	—	8	2	no	—
45	C₂×C₄×C₇	#127	1	1	yes	characteristic
46	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
47	C₂×Dic₇	—	8	2	no	—
48	D₂₈	—	8	4	no	—
49	D₂₈	—	8	8	no	—
50	Dic₁₄	—	8	4	no	—
51	C₇×C₈	—	2	4	no	—
52	C₇×C₈	#135	1	4	yes	—
53	C₂×D₈	—	14	4	no	—
54	C₂×D₈	—	28	4	no	—
55	C₄⋊Q₈	—	14	2	no	—
56	#127	—	14	2	no	—
57	C₂×C₁₆	D₂₈	1	4	yes	—
58	#107	—	14	4	no	—
59	#103	—	28	4	no	—
60	C₄×C₈	D₂₈	1	2	yes	—
61	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
62	C₂²×C₇	#493	1	1	yes	characteristic
63	C₂×D₇	—	16	2	no	—
64	C₂×D₇	—	16	4	no	—
65	C₄×C₇	#495	1	2	yes	—
66	C₄×C₇	#509	1	4	yes	—
67	Dic₇	—	16	2	no	—
68	C₂×Q₈	—	28	2	no	—
69	C₂×D₄	—	28	2	no	—
70	C₂×D₄	—	28	4	no	—
71	D₈	—	28	8	no	—
72	D₈	—	56	8	no	—
73	C₂×C₈	D₅₆	1	4	yes	—
74	C₂×C₈	C₂×D₂₈	1	2	yes	—
75	C₄⋊C₄	—	28	2	no	—
76	C₄⋊C₄	—	28	2	no	—
77	C₄²	C₂×D₂₈	1	1	yes	characteristic
78	C₁₆	—	2	4	no	—
79	C₂×C₇	#1709	1	1	yes	characteristic
80	C₂×C₇	#2250	1	2	yes	—
81	D₇	—	32	2	no	—
82	C₂³	—	56	2	no	—
83	Q₈	—	56	4	no	—
84	D₄	—	56	8	no	—
85	D₄	—	56	4	no	—
86	C₂×C₄	—	56	2	no	—
87	C₂×C₄	#6437	1	1	yes	characteristic
88	C₂×C₄	C₂×D₅₆	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#6467	1	4	yes	—
91	C₇	#12513	1	1	yes	characteristic, complemented, Sylow
92	C₂²	—	112	4	no	—
93	C₂²	—	112	2	no	—
94	C₂²	#89467	1	1	yes	characteristic
95	C₄	—	112	2	no	—
96	C₄	#89468	1	2	yes	—
97	C₄	#89680	1	4	yes	—
98	C₂	—	224	2	no	—
99	C₂	#11725814	1	1	yes	characteristic
100	C₂	#11726835	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11726812

C₂

yes

maximal, complemented

#11725818

C₂

yes

maximal, characteristic, complemented

#11725816

C₂

yes

maximal, characteristic

C₇×#1311

C₂

yes

maximal, characteristic, complemented

#89467

—

#89467

C₂²

yes

—

#89454

C₂²

yes

—

#89453

—

C₄×C₇×C₁₆

C₂²

yes

—

C₇×C₈²

C₂²

yes

—

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

#6432

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

—

—

—

—

—

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

—

C₄⋊Dic₇

—

D₅₆

—

D₅₆

—

C₇×C₁₆

—

—

—

—

—

—

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

—

C₇×C₈

#135

yes

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₂₈

yes

—

#107

—

#103

—

C₄×C₈

D₂₈

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

#495

yes

—

C₄×C₇

#509

yes

—

Dic₇

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

D₅₆

yes

—

C₂×C₈

C₂×D₂₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₈

yes

characteristic

C₁₆

—

C₂×C₇

#1709

yes

characteristic

C₂×C₇

#2250

yes

—

D₇

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

#6437

yes

characteristic

C₂×C₄

C₂×D₅₆

yes

—

C₈

—

C₈

#6467

yes

—

C₇

#12513

yes

characteristic, complemented, Sylow

C₂²

—

112

—

C₂²

—

112

—

C₂²

#89467

yes

characteristic

C₄

—

112

—

C₄

#89468

yes

—

C₄

#89680

yes

—

C₂

—

224

—

C₂

#11725814

yes

characteristic

100

C₂

#11726835

yes

—

	subgroup	G/H
center	C₂²	#89467
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1311	C₂
socle	C₂²×C₇	#493
prime radical	#11725816	C₂
hypercenter	#1311	D₇
hypocenter	C₇	#12513
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#89467

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

C₂

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #11726812	C₂	semidirect
2. #11725818	C₂	semidirect
4. C₇×#1311	C₂	semidirect
25. #1311	D₇	semidirect
91. C₇	#12513	semidirect

H ≅ G/N

type

1. #11726812

C₂

semidirect

2. #11725818

C₂

semidirect

4. C₇×#1311

C₂

semidirect

25. #1311

D₇

semidirect

91. C₇

#12513

semidirect

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

subgroup	index
#11726812	2
#11725818	2
#11725816	2
C₇×#1311	2
#12513	7

subgroup

index

Subgroups — #13268500

Conjugacy classes of subgroups

Complements

Maximal subgroups