Group #13247509 — subgroups

Subgroups — #13247509

Order 1792.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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×#1452	C₂	1	1	yes	maximal, characteristic
2	#11727486	C₂	1	2	yes	maximal
3	#11725553	C₂	1	4	yes	maximal
4	C₇×#431	C₂²	1	2	yes	—
5	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
6	#89698	C₂²	1	4	yes	—
7	C₇×#322	C₄	1	4	yes	—
8	#89253	—	2	2	no	—
9	#89252	—	2	2	no	—
10	#8798	—	7	1	no	maximal, Sylow
11	C₂²×C₇×C₈	Q₈	1	2	yes	—
12	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
13	C₇×C₄⋊C₈	—	2	2	no	—
14	C₇×C₂²⋊C₈	—	2	2	no	—
15	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
16	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
17	C₄×C₇×C₈	D₄	1	1	yes	characteristic
18	C₄×C₇×C₈	D₄	1	1	yes	characteristic
19	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
20	C₄⋊(C₇⋊C₈)	—	2	4	no	—
21	#6377	—	2	4	no	—
22	#1568	—	7	2	no	—
23	#1452	D₇	1	1	yes	characteristic
24	#1287	—	7	4	no	—
25	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₇	#41	1	2	yes	—
27	C₂×C₇×C₈	—	2	4	no	—
28	C₂×C₇×C₈	—	4	2	no	—
29	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
30	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
31	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
33	C₄²×C₇	#41	1	2	yes	—
34	C₂×C₇⋊C₈	—	4	4	no	—
35	#432	—	7	4	no	—
36	#431	C₂×D₇	1	2	yes	—
37	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
38	#335	—	14	2	no	—
39	#334	—	14	2	no	—
40	#322	Dic₇	1	4	yes	—
41	C₂³×C₇	#125	1	1	yes	characteristic
42	C₂×C₄×C₇	—	2	2	no	—
43	C₂×C₄×C₇	—	4	2	no	—
44	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇	#117	1	2	yes	—
46	C₂×C₄×C₇	#124	1	2	yes	—
47	C₂×C₄×C₇	#125	1	1	yes	characteristic
48	C₇×C₈	—	2	2	no	—
49	C₇×C₈	—	2	2	no	—
50	C₇×C₈	—	4	4	no	—
51	C₇⋊C₈	—	8	4	no	—
52	C₂²×C₈	Dic₁₄	1	2	yes	—
53	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
54	C₄⋊C₈	—	2	2	no	—
55	C₄⋊C₈	—	14	4	no	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	14	4	no	—
58	#97	—	14	4	no	—
59	C₄×C₈	D₂₈	1	1	yes	characteristic
60	C₄×C₈	D₂₈	1	1	yes	characteristic
61	C₄×C₈	C₂×Dic₇	1	4	yes	—
62	C₄×C₈	C₂×Dic₇	1	2	yes	—
63	C₂²×C₇	—	2	2	no	—
64	C₂²×C₇	—	2	1	no	—
65	C₂²×C₇	#382	1	1	yes	characteristic
66	C₄×C₇	—	2	2	no	—
67	C₄×C₇	—	2	2	no	—
68	C₄×C₇	—	2	1	no	—
69	C₄×C₇	#430	1	2	yes	—
70	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
71	C₂²×C₄	#1157	1	2	yes	—
72	C₂×C₈	—	2	4	no	—
73	C₂×C₈	—	4	2	no	—
74	C₂×C₈	—	28	4	no	—
75	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
76	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
77	C₄²	C₂×D₂₈	1	1	yes	characteristic
78	C₄²	#1157	1	2	yes	—
79	C₄²	C₂²×Dic₇	1	1	yes	characteristic
80	C₂×C₇	—	4	1	no	—
81	C₂×C₇	#1833	1	2	yes	—
82	C₂×C₇	#1838	1	1	yes	characteristic
83	C₂³	#6433	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	#6433	1	1	yes	characteristic
87	C₂×C₄	#6438	1	2	yes	—
88	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
89	C₂×C₄	#6489	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	2	2	no	—
92	C₈	—	4	4	no	—
93	C₈	—	56	4	no	—
94	C₇	#8798	1	1	yes	characteristic, complemented, Sylow
95	C₂²	—	2	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	#89708	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	2	no	—
100	C₄	—	2	1	no	—
101	C₄	#89897	1	2	yes	—
102	C₂	—	4	1	no	—
103	C₂	#11727595	1	1	yes	characteristic
104	C₂	#11727650	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1452

C₂

yes

maximal, characteristic

#11727486

C₂

yes

maximal

#11725553

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

#89698

C₂²

yes

—

C₇×#322

C₄

yes

—

—

—

—

C₂²×C₇×C₈

Q₈

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

#6377

—

#1568

—

#1452

D₇

yes

characteristic

#1287

—

C₂²×C₄×C₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

#432

—

#431

C₂×D₇

yes

—

C₂×C₄×C₈

C₂×D₇

yes

—

—

Dic₇

yes

—

C₂³×C₇

#125

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

—

C₂×C₄×C₇

#125

yes

characteristic

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

Dic₁₄

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

D₂₈

yes

characteristic

C₄×C₈

C₂×Dic₇

yes

—

C₄×C₈

C₂×Dic₇

yes

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#430

yes

—

C₂²×C₄

C₂×Dic₁₄

yes

characteristic

C₂²×C₄

#1157

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₇

yes

—

C₂×C₈

C₄⋊Dic₇

yes

—

C₄²

C₂×D₂₈

yes

characteristic

C₄²

#1157

yes

—

C₄²

C₂²×Dic₇

yes

characteristic

C₂×C₇

—

C₂×C₇

#1833

yes

—

C₂×C₇

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6433

yes

characteristic

C₂×C₄

#6438

yes

—

C₂×C₄

C₂×C₄⋊Dic₇

yes

characteristic

C₂×C₄

#6489

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₇

#8798

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#89708

yes

characteristic

C₄

—

C₄

—

100

C₄

—

101

C₄

#89897

yes

—

102

C₂

—

103

C₂

#11727595

yes

characteristic

104

C₂

#11727650

yes

—

	subgroup	G/H
center	C₂²	#89708
commutator	C₄²×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1452	C₂
socle	C₂²×C₇	#382
prime radical	C₂³×C₇	#125
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1452	D₇
hypocenter	C₇	#8798
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#89708

commutator

C₄²×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
94. C₇	#8798	semidirect

H ≅ G/N

type

94. C₇

#8798

semidirect

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

subgroup	index
C₇×#1452	2
#11727486	2
#11725553	2
#8798	7

subgroup

index

Subgroups — #13247509

Conjugacy classes of subgroups

Complements

Maximal subgroups