Group #13056342 — subgroups

Subgroups — #13056342

Order 1344.

180 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 67 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	#10621353	C₂	1	1	yes	maximal, characteristic
2	#10621352	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×#97	C₂	1	1	yes	maximal, characteristic
4	#10620655	C₂	1	4	yes	maximal
5	#89475	—	3	1	no	maximal, Hall
6	C₂×Dic₄₂	C₂²	1	2	yes	—
7	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
8	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₁	—	2	2	no	—
10	C₄⋊Dic₂₁	C₂²	1	2	yes	—
11	#66629	—	2	2	no	—
12	#6433	—	3	1	no	—
13	#6432	—	3	1	no	—
14	C₇×#97	D₃	1	1	yes	characteristic
15	#6390	—	3	4	no	—
16	#4703	—	7	1	no	maximal, Hall
17	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
18	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂×Dic₂₁	—	2	2	no	—
20	C₂×Dic₂₁	—	2	2	no	—
21	Dic₄₂	—	2	4	no	—
22	C₃×C₇×C₈	—	2	2	no	—
23	C₂×Dic₁₄	—	3	2	no	—
24	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
25	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
26	C₄⋊Dic₇	—	3	2	no	—
27	C₄⋊Dic₇	—	6	2	no	—
28	Dic₇⋊C₄	—	6	2	no	—
29	#870	—	7	1	no	—
30	#869	—	7	1	no	—
31	C₃×#97	D₇	1	1	yes	characteristic
32	#816	—	7	4	no	—
33	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
34	C₃×C₄×C₇	—	2	2	no	—
35	C₃×C₄×C₇	#41	1	2	yes	—
36	Dic₂₁	—	4	2	no	—
37	Dic₂₁	—	4	2	no	—
38	#491	—	21	1	no	Sylow
39	C₂×C₄×C₇	D₁₂	1	2	yes	—
40	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
41	C₂×Dic₇	—	6	2	no	—
42	C₂×Dic₇	—	6	2	no	—
43	Dic₁₄	—	6	4	no	—
44	C₇×C₈	—	2	2	no	—
45	C₂×Dic₆	—	7	2	no	—
46	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
47	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
48	C₄⋊Dic₃	—	7	2	no	—
49	C₄⋊Dic₃	—	14	2	no	—
50	Dic₃⋊C₄	—	14	2	no	—
51	C₂×C₃×C₇	#124	1	1	yes	characteristic
52	C₂×C₃×C₇	#137	1	2	yes	—
53	C₄⋊Q₈	—	21	1	no	—
54	#125	—	21	1	no	—
55	#103	—	21	4	no	—
56	#97	D₂₁	1	1	yes	characteristic
57	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
58	C₄×C₇	—	2	2	no	—
59	C₄×C₇	#235	1	2	yes	—
60	Dic₇	—	12	2	no	—
61	Dic₇	—	12	2	no	—
62	C₂×C₃×C₄	D₂₈	1	2	yes	—
63	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
64	C₂×Dic₃	—	14	2	no	—
65	C₂×Dic₃	—	14	2	no	—
66	Dic₆	—	14	4	no	—
67	C₃×C₈	—	2	2	no	—
68	C₃×C₇	#491	1	1	yes	characteristic, complemented, Hall
69	C₂×Q₈	—	21	2	no	—
70	C₂×C₈	C₂×D₂₁	1	2	yes	—
71	C₄⋊C₄	—	21	2	no	—
72	C₄⋊C₄	—	42	2	no	—
73	C₄⋊C₄	—	42	2	no	—
74	C₄²	C₂×D₂₁	1	1	yes	characteristic
75	C₂×C₇	#875	1	1	yes	characteristic
76	C₂×C₇	#909	1	2	yes	—
77	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
78	C₃×C₄	—	2	2	no	—
79	C₃×C₄	#1157	1	2	yes	—
80	Dic₃	—	28	2	no	—
81	Dic₃	—	28	2	no	—
82	Q₈	—	42	4	no	—
83	C₂×C₄	—	42	2	no	—
84	C₂×C₄	—	42	2	no	—
85	C₂×C₄	D₈₄	1	2	yes	—
86	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
87	C₈	—	2	2	no	—
88	C₇	#4703	1	1	yes	characteristic, complemented, Sylow
89	C₂×C₃	#6438	1	1	yes	characteristic
90	C₂×C₃	#6472	1	2	yes	—
91	C₂²	C₂×D₈₄	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	84	2	no	—
94	C₄	—	84	2	no	—
95	C₄	#66728	1	2	yes	—
96	C₃	#89475	1	1	yes	characteristic, complemented, Sylow
97	C₂	#10621358	1	1	yes	characteristic
98	C₂	#10621392	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10621353

C₂

yes

maximal, characteristic

#10621352

C₂

yes

maximal, characteristic

C₃×C₇×#97

C₂

yes

maximal, characteristic

#10620655

C₂

yes

maximal

#89475

—

maximal, Hall

C₂×Dic₄₂

C₂²

yes

—

C₂×C₃×C₇×C₈

C₂²

yes

—

C₃×C₄²×C₇

C₂²

yes

characteristic

C₄⋊Dic₂₁

—

C₄⋊Dic₂₁

C₂²

yes

—

#66629

—

#6433

—

#6432

—

C₇×#97

D₃

yes

—

—

C₂×C₃×C₄×C₇

D₄

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₂₁

—

C₂×Dic₂₁

—

Dic₄₂

—

C₃×C₇×C₈

—

C₂×Dic₁₄

—

C₂×C₇×C₈

C₂×D₃

yes

—

C₄²×C₇

C₂×D₃

yes

characteristic

C₄⋊Dic₇

—

C₄⋊Dic₇

—

Dic₇⋊C₄

—

#870

—

#869

—

C₃×#97

D₇

yes

characteristic

#816

—

C₂²×C₃×C₇

C₂×D₄

yes

characteristic

C₃×C₄×C₇

—

C₃×C₄×C₇

#41

yes

—

Dic₂₁

—

Dic₂₁

—

#491

—

Sylow

C₂×C₄×C₇

D₁₂

yes

—

C₂×C₄×C₇

C₂²×D₃

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

—

Dic₁₄

—

C₇×C₈

—

C₂×Dic₆

—

C₂×C₃×C₈

C₂×D₇

yes

—

C₃×C₄²

C₂×D₇

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃×C₇

#124

yes

characteristic

C₂×C₃×C₇

#137

yes

—

C₄⋊Q₈

—

#125

—

#103

—

#97

D₂₁

yes

characteristic

C₂²×C₇

C₂×D₁₂

yes

characteristic

C₄×C₇

—

C₄×C₇

#235

yes

—

Dic₇

—

Dic₇

—

C₂×C₃×C₄

D₂₈

yes

—

C₂×C₃×C₄

C₂²×D₇

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

—

C₃×C₇

#491

yes

characteristic, complemented, Hall

C₂×Q₈

—

C₂×C₈

C₂×D₂₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₁

yes

C₂×C₇

yes

C₂×C₇

yes

—

C₂²×C₃

C₂×D₂₈

yes

characteristic

C₃×C₄

—

C₃×C₄

#1157

yes

—

Dic₃

—

Dic₃

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈₄

yes

—

C₂×C₄

C₂²×D₂₁

yes

characteristic

C₈

—

C₇

#4703

yes

characteristic, complemented, Sylow

C₂×C₃

#6438

yes

characteristic

C₂×C₃

#6472

yes

—

C₂²

C₂×D₈₄

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#66728

yes

—

C₃

#89475

yes

characteristic, complemented, Sylow

C₂

#10621358

yes

characteristic

C₂

#10621392

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈₄
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₁
Fitting	C₃×C₇×#97	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	#97	D₂₁
hypocenter	C₃×C₇	#491
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×D₈₄

commutator, cosocle

C₂×C₃×C₄×C₇

C₂³

Frattini

C₂×C₄

C₂²×D₂₁

Fitting

C₃×C₇×#97

C₂

socle, prime radical

C₂²×C₃×C₇

C₂×D₄

D₂₁

C₃×C₇

G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
68. C₃×C₇	#491	semidirect
88. C₇	#4703	semidirect
96. C₃	#89475	semidirect

H ≅ G/N

type

68. C₃×C₇

#491

semidirect

88. C₇

#4703

semidirect

96. C₃

#89475

semidirect

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

subgroup	index
#10621353	2
#10621352	2
C₃×C₇×#97	2
#10620655	2
#89475	3
#4703	7

subgroup

index

C₃×C₇×#97

Subgroups — #13056342

Conjugacy classes of subgroups

Complements

Maximal subgroups