Group #11746936 — subgroups

Subgroups — #11746936

Order 960.

180 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 64 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	#91758	C₂	1	1	yes	maximal, characteristic
2	C₃×#4016	C₂	1	1	yes	maximal, characteristic
3	C₅×#803	C₂	1	1	yes	maximal, characteristic
4	#90974	C₂	1	4	yes	maximal
5	#64992	—	3	1	no	maximal, Hall
6	C₂×Dic₃₀	C₂²	1	2	yes	—
7	C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₅	—	2	2	no	—
9	C₂×C₅×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
11	C₃×#683	—	2	2	no	—
12	#5069	—	5	1	no	maximal, Hall
13	#4016	D₃	1	1	yes	characteristic
14	#4015	—	3	1	no	—
15	C₅×#97	—	3	1	no	—
16	#3948	—	3	4	no	—
17	C₂×C₃×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×Dic₁₅	—	2	2	no	—
20	Dic₃₀	—	2	4	no	—
21	C₂×C₃×Dic₅	—	2	2	no	—
22	C₅×C₃⋊C₈	—	2	2	no	—
23	C₃×#125	—	5	1	no	—
24	#869	—	5	1	no	—
25	#810	—	5	4	no	—
26	#803	D₅	1	1	yes	characteristic
27	C₂×Dic₁₀	—	3	2	no	—
28	C₂×C₅×C₈	—	3	2	no	—
29	C₄²×C₅	C₂×D₃	1	1	yes	characteristic
30	C₄⋊Dic₅	—	6	2	no	—
31	C₄⋊Dic₅	C₂×D₃	1	2	yes	—
32	#683	—	2	2	no	—
33	#491	—	15	1	no	Sylow
34	C₂²×C₃×C₅	C₂×D₄	1	1	yes	characteristic
35	C₃×C₄×C₅	—	2	2	no	—
36	C₃×C₄×C₅	#41	1	2	yes	—
37	Dic₁₅	—	4	2	no	—
38	C₃×Dic₅	—	4	2	no	—
39	C₂×Dic₆	—	5	2	no	—
40	C₃×C₄⋊C₄	—	5	2	no	—
41	C₃×C₄⋊C₄	—	10	2	no	—
42	C₃×C₄²	C₂×D₅	1	1	yes	characteristic
43	C₄⋊Dic₃	—	10	2	no	—
44	C₂×C₃⋊C₈	C₂×D₅	1	2	yes	—
45	C₂×C₄×C₅	#67	1	2	yes	—
46	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
47	C₂×Dic₅	—	2	2	no	—
48	C₂×Dic₅	—	6	2	no	—
49	Dic₁₀	—	6	4	no	—
50	C₅×C₈	—	6	2	no	—
51	C₄⋊Q₈	—	15	1	no	—
52	#125	—	5	1	no	—
53	#103	—	15	4	no	—
54	#97	—	3	1	no	—
55	C₂×C₃×C₅	#124	1	1	yes	characteristic
56	C₂×C₃×C₅	#137	1	2	yes	—
57	C₂×C₃×C₄	—	10	2	no	—
58	C₂×C₃×C₄	D₂₀	1	2	yes	—
59	C₂×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
60	C₂×Dic₃	—	10	2	no	—
61	Dic₆	—	10	4	no	—
62	C₃⋊C₈	—	2	2	no	—
63	C₂²×C₅	C₂×#67	1	1	yes	characteristic
64	C₄×C₅	—	2	2	no	—
65	C₄×C₅	#239	1	2	yes	—
66	Dic₅	—	4	2	no	—
67	Dic₅	—	12	2	no	—
68	C₂×Q₈	—	15	2	no	—
69	C₂×C₈	—	3	2	no	—
70	C₄⋊C₄	—	5	2	no	—
71	C₄⋊C₄	—	10	2	no	—
72	C₄⋊C₄	—	30	2	no	—
73	C₄²	D₃×D₅	1	1	yes	characteristic
74	C₃×C₅	#491	1	1	yes	characteristic, complemented, Hall
75	C₂²×C₃	C₂×D₂₀	1	1	yes	characteristic
76	C₃×C₄	—	2	2	no	—
77	C₃×C₄	—	20	2	no	—
78	C₃×C₄	#709	1	2	yes	—
79	Dic₃	—	20	2	no	—
80	C₂×C₅	#947	1	1	yes	characteristic
81	C₂×C₅	#951	1	2	yes	—
82	Q₈	—	30	4	no	—
83	C₂×C₄	—	10	2	no	—
84	C₂×C₄	—	30	2	no	—
85	C₂×C₄	#1202	1	2	yes	—
86	C₂×C₄	C₂×D₃×D₅	1	1	yes	characteristic
87	C₈	—	6	2	no	—
88	C₂×C₃	#4021	1	1	yes	characteristic
89	C₂×C₃	#4055	1	2	yes	—
90	C₅	#5069	1	1	yes	characteristic, complemented, Sylow
91	C₂²	C₂×#1202	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	20	2	no	—
94	C₄	—	60	2	no	—
95	C₄	#6769	1	2	yes	—
96	C₃	#64992	1	1	yes	characteristic, complemented, Sylow
97	C₂	#91313	1	2	yes	—
98	C₂	#91374	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#91758

C₂

yes

maximal, characteristic

C₃×#4016

C₂

yes

maximal, characteristic

C₅×#803

C₂

yes

maximal, characteristic

#90974

C₂

yes

maximal

#64992

—

maximal, Hall

C₂×Dic₃₀

C₂²

yes

—

C₃×C₄²×C₅

C₂²

yes

characteristic

C₄⋊Dic₁₅

—

C₂×C₅×C₃⋊C₈

C₂²

yes

—

C₃×C₄⋊Dic₅

C₂²

yes

—

—

—

D₃

yes

characteristic

#4015

—

C₅×#97

—

#3948

—

C₂×C₃×C₄×C₅

D₄

yes

—

C₂×C₃×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₁₅

—

Dic₃₀

—

C₂×C₃×Dic₅

—

C₅×C₃⋊C₈

—

—

—

—

D₅

yes

characteristic

C₂×Dic₁₀

—

C₂×C₅×C₈

—

C₄²×C₅

C₂×D₃

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×D₃

yes

—

#683

—

#491

—

Sylow

C₂²×C₃×C₅

C₂×D₄

yes

characteristic

C₃×C₄×C₅

—

C₃×C₄×C₅

#41

yes

—

Dic₁₅

—

C₃×Dic₅

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₅

yes

characteristic

C₄⋊Dic₃

—

C₂×C₃⋊C₈

C₂×D₅

yes

—

C₂×C₄×C₅

#67

yes

—

C₂×C₄×C₅

C₂²×D₃

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₄⋊Q₈

—

#125

—

#103

—

#97

—

C₂×C₃×C₅

#124

yes

characteristic

C₂×C₃×C₅

#137

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

D₂₀

yes

—

C₂×C₃×C₄

C₂²×D₅

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

C₂²×C₅

C₂×#67

yes

characteristic

C₄×C₅

—

C₄×C₅

#239

yes

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₃×D₅

yes

characteristic

C₃×C₅

#491

yes

characteristic, complemented, Hall

C₂²×C₃

C₂×D₂₀

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#709

yes

—

Dic₃

—

C₂×C₅

#947

yes

characteristic

C₂×C₅

#951

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#1202

yes

—

C₂×C₄

C₂×D₃×D₅

yes

characteristic

C₈

—

C₂×C₃

#4021

yes

characteristic

C₂×C₃

#4055

yes

—

C₅

#5069

yes

characteristic, complemented, Sylow

C₂²

C₂×#1202

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#6769

yes

—

C₃

#64992

yes

characteristic, complemented, Sylow

C₂

#91313

yes

—

C₂

#91374

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#1202
commutator, cosocle	C₂×C₃×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂×D₃×D₅
Fitting	C₃×C₄²×C₅	C₂²
socle, prime radical	C₂²×C₃×C₅	C₂×D₄
hypercenter	C₄²	D₃×D₅
hypocenter	C₃×C₅	#491
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center

C₂²

C₂×#1202

commutator, cosocle

C₂×C₃×C₄×C₅

C₂³

Frattini

C₂×C₄

C₂×D₃×D₅

Fitting

C₃×C₄²×C₅

C₂²

socle, prime radical

C₂²×C₃×C₅

C₂×D₄

hypercenter

C₄²

D₃×D₅

hypocenter

C₃×C₅

#491

LCS

G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅

UCS

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
74. C₃×C₅	#491	semidirect
90. C₅	#5069	semidirect
96. C₃	#64992	semidirect

H ≅ G/N

type

74. C₃×C₅

#491

semidirect

90. C₅

#5069

semidirect

96. C₃

#64992

semidirect

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

subgroup	index
#91758	2
C₃×#4016	2
C₅×#803	2
#90974	2
#64992	3
#5069	5

subgroup

index

Subgroups — #11746936

Conjugacy classes of subgroups

Complements

Maximal subgroups