Group #13258305 — subgroups

Subgroups — #13258305

Order 1792.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2845	C₂	1	1	yes	maximal, characteristic
2	#11727302	C₂	1	3	yes	maximal
3	#11725436	C₂	1	3	yes	maximal
4	C₇×#398	—	2	6	no	—
5	C₇×#382	C₂²	1	3	yes	—
6	#89707	C₂²	1	1	yes	characteristic
7	C₂×#6377	C₂²	1	3	yes	—
8	#89324	—	2	6	no	—
9	#10586	—	7	1	no	maximal, Sylow
10	C₂×C₇×C₄⋊C₄	—	2	6	no	—
11	C₂×C₇×C₄⋊C₄	D₄	1	6	yes	—
12	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
13	C₂×C₄⋊Dic₇	—	2	3	no	—
14	C₂²×C₇⋊C₈	—	2	3	no	—
15	C₇×#95	—	2	2	no	—
16	C₇×#95	—	2	6	no	—
17	#6377	—	2	3	no	—
18	#6377	Q₈	1	6	yes	—
19	#2845	D₇	1	1	yes	characteristic
20	#1836	—	7	3	no	—
21	#1295	—	7	3	no	—
22	C₂²×C₄×C₇	—	2	6	no	—
23	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
24	C₂²×Dic₇	—	4	1	no	—
25	C₇×C₄⋊C₄	—	4	6	no	—
26	C₇×C₄⋊C₄	—	8	6	no	—
27	C₄²×C₇	C₂×Q₈	1	3	yes	—
28	C₄²×C₇	#41	1	1	yes	characteristic
29	C₄⋊Dic₇	—	4	6	no	—
30	C₂×C₇⋊C₈	—	2	6	no	—
31	C₂×C₇⋊C₈	—	4	3	no	—
32	C₂×C₇⋊C₈	—	4	3	no	—
33	C₂×#97	—	7	3	no	—
34	#398	—	2	6	no	—
35	#384	—	7	1	no	—
36	#382	C₂×D₇	1	3	yes	—
37	#340	—	14	6	no	—
38	C₂³×C₇	#120	1	1	yes	characteristic
39	C₂×C₄×C₇	—	2	3	no	—
40	C₂×C₄×C₇	—	2	3	no	—
41	C₂×C₄×C₇	—	4	6	no	—
42	C₂×C₄×C₇	—	4	6	no	—
43	C₂×C₄×C₇	—	4	6	no	—
44	C₂×C₄×C₇	C₂²⋊Q₈	1	3	yes	—
45	C₂×C₄×C₇	C₄⋊Q₈	1	3	yes	—
46	C₂×Dic₇	—	8	3	no	—
47	C₇⋊C₈	—	4	6	no	—
48	C₂²×C₈	—	14	3	no	—
49	C₂×C₄⋊C₄	—	2	6	no	—
50	C₂×C₄⋊C₄	—	14	3	no	—
51	C₂×C₄⋊C₄	C₂²⋊D₇	1	6	yes	—
52	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
53	#97	—	7	6	no	—
54	#97	—	14	3	no	—
55	#95	—	2	2	no	—
56	#95	—	2	6	no	—
57	C₂²×C₇	#395	1	1	yes	characteristic
58	C₂²×C₇	#454	1	3	yes	—
59	C₂²×C₇	#455	1	3	yes	—
60	C₄×C₇	—	2	3	no	—
61	C₄×C₇	—	2	3	no	—
62	C₄×C₇	—	8	6	no	—
63	Dic₇	—	16	1	no	—
64	C₂²×C₄	—	2	6	no	—
65	C₂²×C₄	—	28	1	no	—
66	C₂²×C₄	C₂×C₂²⋊D₇	1	3	yes	—
67	C₂×C₈	—	14	6	no	—
68	C₂×C₈	—	28	3	no	—
69	C₂×C₈	—	28	3	no	—
70	C₄⋊C₄	—	4	6	no	—
71	C₄⋊C₄	—	8	6	no	—
72	C₄⋊C₄	—	28	6	no	—
73	C₄²	Q₈×D₇	1	3	yes	—
74	C₄²	#1161	1	1	yes	characteristic
75	C₂×C₇	#1658	1	1	yes	characteristic
76	C₂×C₇	#1662	1	3	yes	—
77	C₂×C₇	#2061	1	3	yes	—
78	C₂³	#6516	1	1	yes	characteristic
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	4	6	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	56	3	no	—
85	C₂×C₄	#6507	1	3	yes	—
86	C₂×C₄	#6509	1	3	yes	—
87	C₈	—	28	6	no	—
88	C₇	#10586	1	1	yes	characteristic, complemented, Sylow
89	C₂²	#90001	1	1	yes	characteristic
90	C₂²	#90014	1	3	yes	—
91	C₂²	#90017	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	112	1	no	—
96	C₂	#11729655	1	3	yes	—
97	C₂	#11729708	1	3	yes	—
98	C₂	#11729714	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2845

C₂

yes

maximal, characteristic

#11727302

C₂

yes

maximal

#11725436

C₂

yes

maximal

C₇×#398

—

C₇×#382

C₂²

yes

—

#89707

C₂²

yes

characteristic

C₂×#6377

C₂²

yes

—

#89324

—

#10586

—

maximal, Sylow

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₂×C₄⋊Dic₇

—

C₂²×C₇⋊C₈

—

C₇×#95

—

C₇×#95

—

#6377

—

#6377

Q₈

yes

—

#2845

D₇

yes

characteristic

#1836

—

#1295

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂²×Dic₇

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

C₂×Q₈

yes

—

C₄²×C₇

#41

yes

characteristic

C₄⋊Dic₇

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×C₇⋊C₈

—

C₂×#97

—

#398

—

#384

—

#382

C₂×D₇

yes

—

#340

—

C₂³×C₇

#120

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₇

C₄⋊Q₈

yes

—

C₂×Dic₇

—

C₇⋊C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₇

yes

—

C₂×C₄²

C₂²×D₇

yes

—

—

—

—

C₂²×C₇

#395

yes

characteristic

C₂²×C₇

#454

yes

—

C₂²×C₇

#455

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₇

yes

—

C₄²

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂×C₇

#2061

yes

—

C₂³

#6516

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#6507

yes

—

C₂×C₄

#6509

yes

—

C₈

—

C₇

#10586

yes

characteristic, complemented, Sylow

C₂²

#90001

yes

characteristic

C₂²

#90014

yes

—

C₂²

#90017

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

112

—

C₂

#11729655

yes

—

C₂

#11729708

yes

—

C₂

#11729714

yes

characteristic

	subgroup	G/H
center	C₂³	#6516
commutator, cosocle	C₂×C₄²×C₇	C₂³
Frattini	C₂×C₄²	C₂²×D₇
Fitting	C₇×#2845	C₂
socle, prime radical	C₂³×C₇	#120
hypercenter	#2845	D₇
hypocenter	C₇	#10586
LCS	G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#6516

commutator, cosocle

C₂×C₄²×C₇

C₂³

Frattini

C₂×C₄²

C₂²×D₇

C₂

C₂³×C₇

D₇

C₇

G ▷ C₂×C₄²×C₇ ▷ C₂²×C₇ ▷ C₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
88. C₇	#10586	semidirect

H ≅ G/N

type

88. C₇

#10586

semidirect

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

subgroup	index
C₇×#2845	2
#11727302	2
#11725436	2
#10586	7

subgroup

index

Subgroups — #13258305

Conjugacy classes of subgroups

Complements

Maximal subgroups