Group #13247524 — subgroups

Subgroups — #13247524

Order 1792.

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

normal subgroups only

This group has 258 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×#1745	C₂	1	1	yes	maximal, characteristic
2	#11725467	C₂	1	2	yes	maximal
3	#11725466	C₂	1	2	yes	maximal
4	#11725417	C₂	1	2	yes	maximal
5	C₂×C₇×#97	C₂²	1	1	yes	characteristic
6	C₂×C₇×#97	C₂²	1	1	yes	characteristic
7	C₄³×C₇	C₂²	1	1	yes	characteristic
8	#89697	C₂²	1	4	yes	—
9	C₇×#336	C₄	1	4	yes	—
10	#8802	—	7	1	no	maximal, Sylow
11	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
12	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₇	—	2	2	no	—
14	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
16	C₇×#97	—	2	2	no	—
17	C₇×#97	—	2	2	no	—
18	C₂²⋊(C₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₇⋊C₈)	—	2	4	no	—
20	C₄×C₇⋊C₈	—	2	4	no	—
21	#1745	D₇	1	1	yes	characteristic
22	#1281	—	7	2	no	—
23	#1276	—	7	2	no	—
24	#1275	—	7	2	no	—
25	C₂²×C₄×C₇	—	2	2	no	—
26	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇	#41	1	2	yes	—
28	C₂×C₇×C₈	—	2	2	no	—
29	C₂×C₇×C₈	—	2	2	no	—
30	C₂×C₇×C₈	—	2	4	no	—
31	C₂×C₇×C₈	—	2	4	no	—
32	C₄²×C₇	—	2	4	no	—
33	C₄²×C₇	—	4	4	no	—
34	C₄²×C₇	#41	1	2	yes	—
35	C₄²×C₇	#41	1	2	yes	—
36	C₂×C₇⋊C₈	—	4	4	no	—
37	#431	—	7	4	no	—
38	C₂×#97	C₂×D₇	1	1	yes	characteristic
39	C₂×#97	C₂×D₇	1	1	yes	characteristic
40	C₄³	C₂×D₇	1	1	yes	characteristic
41	#336	Dic₇	1	4	yes	—
42	C₂³×C₇	#117	1	1	yes	characteristic
43	C₂×C₄×C₇	—	2	4	no	—
44	C₂×C₄×C₇	—	4	2	no	—
45	C₂×C₄×C₇	—	4	2	no	—
46	C₂×C₄×C₇	—	4	2	no	—
47	C₂×C₄×C₇	#117	1	1	yes	characteristic
48	C₂×C₄×C₇	#117	1	1	yes	characteristic
49	C₂×C₄×C₇	#126	1	4	yes	—
50	C₂×C₄×C₇	#131	1	4	yes	—
51	C₇×C₈	—	4	2	no	—
52	C₇×C₈	—	4	2	no	—
53	C₇⋊C₈	—	8	4	no	—
54	C₂²×C₈	C₂×Dic₇	1	2	yes	—
55	C₂²×C₈	C₂×Dic₇	1	2	yes	—
56	C₂×C₄²	—	2	2	no	—
57	C₂×C₄²	C₂×Dic₇	1	2	yes	—
58	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
59	C₄⋊C₈	—	14	4	no	—
60	C₂²⋊C₈	—	14	4	no	—
61	#97	—	2	2	no	—
62	#97	—	2	2	no	—
63	C₄×C₈	—	14	4	no	—
64	C₂²×C₇	—	2	2	no	—
65	C₂²×C₇	#383	1	1	yes	characteristic
66	C₂²×C₇	#433	1	2	yes	—
67	C₄×C₇	—	2	4	no	—
68	C₄×C₇	—	4	4	no	—
69	C₄×C₇	#433	1	2	yes	—
70	C₄×C₇	#433	1	2	yes	—
71	C₂²×C₄	—	2	2	no	—
72	C₂²×C₄	#1157	1	2	yes	—
73	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	4	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	28	4	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	4	4	no	—
81	C₄²	#1157	1	2	yes	—
82	C₄²	#1157	1	2	yes	—
83	C₂×C₇	—	2	2	no	—
84	C₂×C₇	#1840	1	1	yes	characteristic
85	C₂×C₇	#1841	1	2	yes	—
86	C₂³	#6489	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	#6439	1	4	yes	—
92	C₂×C₄	#6489	1	1	yes	characteristic
93	C₂×C₄	#6489	1	1	yes	characteristic
94	C₂×C₄	#6511	1	4	yes	—
95	C₈	—	4	2	no	—
96	C₈	—	4	2	no	—
97	C₈	—	56	4	no	—
98	C₇	#8802	1	1	yes	characteristic, complemented, Sylow
99	C₂²	—	2	2	no	—
100	C₂²	#89709	1	1	yes	characteristic
101	C₂²	#89772	1	2	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	4	4	no	—
104	C₄	#89772	1	2	yes	—
105	C₄	#89772	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	#11727192	1	1	yes	characteristic
108	C₂	#11727625	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1745

C₂

yes

maximal, characteristic

#11725467

C₂

yes

maximal

#11725466

C₂

yes

maximal

#11725417

C₂

yes

maximal

C₂×C₇×#97

C₂²

yes

characteristic

C₂×C₇×#97

C₂²

yes

characteristic

C₄³×C₇

C₂²

yes

characteristic

#89697

C₂²

yes

—

C₇×#336

C₄

yes

—

#8802

—

maximal, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂×C₄²×C₇

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#97

—

C₇×#97

—

C₂²⋊(C₇⋊C₈)

—

C₄⋊(C₇⋊C₈)

—

C₄×C₇⋊C₈

—

#1745

D₇

yes

—

—

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

#41

yes

—

C₄²×C₇

#41

yes

—

C₂×C₇⋊C₈

—

#431

—

C₂×#97

C₂×D₇

yes

characteristic

C₂×#97

C₂×D₇

yes

characteristic

C₄³

C₂×D₇

yes

characteristic

#336

Dic₇

yes

—

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

C₂×C₄×C₇

yes

—

C₂×C₄×C₇

#131

yes

—

C₇×C₈

—

C₇×C₈

—

C₇⋊C₈

—

C₂²×C₈

C₂×Dic₇

yes

—

C₂²×C₈

C₂×Dic₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₇

yes

—

C₂×C₄²

C₂²×D₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₇

—

C₂²×C₇

#383

yes

characteristic

C₂²×C₇

#433

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#433

yes

—

C₄×C₇

#433

yes

—

C₂²×C₄

—

C₂²×C₄

#1157

yes

—

C₂²×C₄

C₂²×Dic₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

#1157

yes

—

C₄²

#1157

yes

—

C₂×C₇

—

C₂×C₇

#1840

yes

characteristic

C₂×C₇

#1841

yes

—

C₂³

#6489

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6439

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₇

#8802

yes

characteristic, complemented, Sylow

C₂²

—

100

C₂²

#89709

yes

characteristic

101

C₂²

#89772

yes

—

102

C₄

—

103

C₄

—

104

C₄

#89772

yes

—

105

C₄

#89772

yes

—

106

C₂

—

107

C₂

#11727192

yes

characteristic

108

C₂

#11727625

yes

—

	subgroup	G/H
center	C₂×C₄	#6489
commutator	C₂²×C₄×C₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#1745	C₂
socle	C₂²×C₇	#383
prime radical	C₂³×C₇	#117
cosocle	C₂×C₄²×C₇	C₂³
hypercenter	#1745	D₇
hypocenter	C₇	#8802
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1745

subgroup

G/H

center

C₂×C₄

#6489

commutator

C₂²×C₄×C₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂²×C₇

C₂³×C₇

C₂×C₄²×C₇

C₂³

D₇

C₇

G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1745

N	H ≅ G/N	type
98. C₇	#8802	semidirect

H ≅ G/N

type

98. C₇

#8802

semidirect

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

subgroup	index
C₇×#1745	2
#11725467	2
#11725466	2
#11725417	2
#8802	7

subgroup

index

Subgroups — #13247524

Conjugacy classes of subgroups

Complements

Maximal subgroups