Group #13268501 — subgroups

Subgroups — #13268501

Order 1792.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₇×#1710	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729135	C₂	1	4	yes	maximal, complemented
3	#11725816	C₂	1	1	yes	maximal, characteristic
4	#11725243	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#500	C₂²	1	2	yes	—
6	C₇×#487	—	2	2	no	—
7	#89467	—	2	2	no	—
8	#89467	C₂²	1	2	yes	—
9	C₇×C₈²	C₂²	1	1	yes	characteristic
10	#89286	—	2	8	no	—
11	C₄×C₇⋊C₁₆	C₂²	1	2	yes	—
12	#12513	—	7	1	no	maximal, Sylow
13	C₇×C₄⋊Q₈	—	2	2	no	—
14	C₂×D₅₆	—	2	4	no	—
15	C₂×D₅₆	—	4	4	no	—
16	#6437	—	2	2	no	—
17	C₇×#107	—	2	4	no	—
18	C₇×#103	—	4	4	no	—
19	C₄×C₇×C₈	D₄	1	2	yes	—
20	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₇⋊C₁₆	D₄	1	4	yes	—
22	#2240	—	7	4	no	—
23	#1712	—	7	1	no	—
24	#1710	D₇	1	1	yes	characteristic, complemented
25	#1311	—	7	1	no	—
26	C₂×C₇×Q₈	—	4	2	no	—
27	C₂×D₂₈	—	4	2	no	—
28	C₂×D₂₈	—	4	4	no	—
29	C₂×C₇×C₈	D₈	1	4	yes	—
30	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
31	C₇×C₄⋊C₄	—	4	2	no	—
32	C₇×C₄⋊C₄	—	4	2	no	—
33	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
34	D₅₆	—	4	8	no	—
35	D₅₆	—	8	8	no	—
36	C₇⋊C₁₆	—	2	4	no	—
37	#500	C₂×D₇	1	2	yes	—
38	#493	—	7	2	no	—
39	#493	—	14	2	no	—
40	#487	—	2	2	no	—
41	#357	—	14	8	no	—
42	C₄×C₁₆	—	7	2	no	—
43	C₈²	C₂×D₇	1	1	yes	characteristic
44	C₂²×D₇	—	8	2	no	—
45	C₇×Q₈	—	8	4	no	—
46	C₂×C₄×C₇	—	8	2	no	—
47	C₂×C₄×C₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
49	D₂₈	—	8	4	no	—
50	D₂₈	—	8	8	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	—	2	2	no	—
56	#127	—	14	2	no	—
57	C₂×C₁₆	—	7	4	no	—
58	#107	—	2	4	no	—
59	#103	—	4	4	no	—
60	C₄×C₈	C₂²⋊D₇	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	—	16	2	no	—
66	C₄×C₇	#495	1	2	yes	—
67	C₄×C₇	#509	1	4	yes	—
68	C₂×Q₈	—	4	2	no	—
69	C₂×D₄	—	28	4	no	—
70	C₂×D₄	—	28	2	no	—
71	D₈	—	28	8	no	—
72	D₈	—	56	8	no	—
73	C₂×C₈	#1141	1	4	yes	—
74	C₂×C₈	D₄×D₇	1	2	yes	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
78	C₁₆	—	14	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	—	8	4	no	—
84	D₄	—	56	4	no	—
85	D₄	—	56	8	no	—
86	C₂×C₄	—	8	2	no	—
87	C₂×C₄	C₂×#1141	1	2	yes	—
88	C₂×C₄	#6503	1	1	yes	characteristic
89	C₈	—	2	4	no	—
90	C₈	#6482	1	4	yes	—
91	C₇	#12513	1	1	yes	characteristic, complemented, Sylow
92	C₂²	—	112	2	no	—
93	C₂²	—	112	4	no	—
94	C₂²	#89845	1	1	yes	characteristic
95	C₄	—	16	2	no	—
96	C₄	#89963	1	2	yes	—
97	C₄	#89965	1	4	yes	—
98	C₂	—	224	2	no	—
99	C₂	#11729051	1	1	yes	characteristic
100	C₂	#11729075	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1710

C₂

yes

maximal, characteristic, complemented

#11729135

C₂

yes

maximal, complemented

#11725816

C₂

yes

maximal, characteristic

#11725243

C₂

yes

maximal, characteristic, complemented

C₇×#500

C₂²

yes

—

C₇×#487

—

#89467

—

#89467

C₂²

yes

—

C₇×C₈²

C₂²

yes

characteristic

#89286

—

C₄×C₇⋊C₁₆

C₂²

yes

—

#12513

—

maximal, Sylow

C₇×C₄⋊Q₈

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

C₇×#107

—

C₇×#103

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

C₂×C₇⋊C₁₆

D₄

yes

—

#2240

—

#1712

—

#1710

D₇

yes

characteristic, complemented

#1311

—

C₂×C₇×Q₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×C₇×C₈

D₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×D₄

yes

characteristic

D₅₆

—

D₅₆

—

C₇⋊C₁₆

—

#500

C₂×D₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₇

yes

characteristic

C₂²×D₇

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

D₂₈

—

D₂₈

—

C₇×C₈

—

C₇×C₈

#135

yes

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

C₄×C₈

C₂²⋊D₇

yes

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

#495

yes

—

C₄×C₇

#509

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

#1141

yes

—

C₂×C₈

D₄×D₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₇

yes

characteristic

C₁₆

—

C₂×C₇

#1709

yes

characteristic

C₂×C₇

#2250

yes

—

D₇

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×#1141

yes

—

C₂×C₄

#6503

yes

characteristic

C₈

—

C₈

#6482

yes

—

C₇

#12513

yes

characteristic, complemented, Sylow

C₂²

—

112

—

C₂²

—

112

—

C₂²

#89845

yes

characteristic

C₄

—

C₄

#89963

yes

—

C₄

#89965

yes

—

C₂

—

224

—

C₂

#11729051

yes

characteristic

100

C₂

#11729075

yes

—

	subgroup	G/H
center	C₂²	#89845
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1710	C₂
socle	C₂²×C₇	#493
prime radical	#11725816	C₂
hypercenter	#1710	D₇
hypocenter	C₇	#12513
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#89845

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 ◁ #1710

N	H ≅ G/N	type
1. C₇×#1710	C₂	semidirect
2. #11729135	C₂	semidirect
4. #11725243	C₂	semidirect
24. #1710	D₇	semidirect
91. C₇	#12513	semidirect

H ≅ G/N

type

1. C₇×#1710

C₂

semidirect

2. #11729135

C₂

semidirect

4. #11725243

C₂

semidirect

24. #1710

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
C₇×#1710	2
#11729135	2
#11725816	2
#11725243	2
#12513	7

subgroup

index

Subgroups — #13268501

Conjugacy classes of subgroups

Complements

Maximal subgroups