Group #12014017 — subgroups

Subgroups — #12014017

Order 1280.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₅×#1712	C₂	1	1	yes	maximal, characteristic
2	#10602084	C₂	1	4	yes	maximal, complemented
3	#10598858	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#672	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#493	—	2	2	no	—
6	C₅×#493	C₂²	1	2	yes	—
7	C₂×#3959	—	2	8	no	—
8	#64988	C₂²	1	2	yes	—
9	#64971	—	2	2	no	—
10	C₅×C₈²	C₂²	1	1	yes	characteristic
11	C₄×#672	C₂²	1	2	yes	—
12	#12517	—	5	1	no	maximal, Sylow
13	C₂×C₅×D₈	—	2	4	no	—
14	C₂×C₅×D₈	—	4	4	no	—
15	C₅×#127	—	2	2	no	—
16	C₂×Dic₂₀	—	2	4	no	—
17	#4015	—	2	2	no	—
18	C₄×C₅×C₈	D₄	1	2	yes	—
19	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
20	#3959	—	4	16	no	—
21	#3950	—	4	4	no	—
22	C₂×#672	D₄	1	4	yes	—
23	#2248	—	5	4	no	—
24	#1713	—	5	1	no	—
25	#1712	D₅	1	1	yes	characteristic
26	C₈×C₁₆	—	5	1	no	—
27	C₂×C₅×D₄	—	4	2	no	—
28	C₂×C₅×D₄	—	4	4	no	—
29	C₂×Dic₁₀	—	4	2	no	—
30	C₅×D₈	—	4	8	no	—
31	C₅×D₈	—	8	8	no	—
32	C₂×C₅×C₈	D₈	1	4	yes	—
33	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
34	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₅	—	4	4	no	—
36	Dic₂₀	—	4	8	no	—
37	#672	D₈	1	8	yes	complemented
38	C₂×SD₃₂	—	10	8	no	—
39	#500	—	10	2	no	—
40	#494	—	5	2	no	—
41	#493	—	2	2	no	—
42	#493	C₂×D₅	1	2	yes	—
43	C₄×C₁₆	—	5	2	no	—
44	C₈²	C₂×D₅	1	1	yes	characteristic
45	C₂³×C₅	—	8	2	no	—
46	C₅×D₄	—	8	4	no	—
47	C₅×D₄	—	8	8	no	—
48	C₂×C₄×C₅	#127	1	1	yes	characteristic
49	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
50	C₂×Dic₅	—	8	2	no	—
51	Dic₁₀	—	8	4	no	—
52	C₅×C₈	SD₃₂	1	8	yes	complemented
53	C₅×C₈	C₂×D₈	1	4	yes	—
54	C₂×Q₁₆	—	10	4	no	—
55	C₂×D₈	—	2	4	no	—
56	C₂×D₈	—	4	4	no	—
57	C₄⋊Q₈	—	10	2	no	—
58	#127	—	2	2	no	—
59	SD₃₂	—	20	16	no	—
60	C₂×C₁₆	—	5	4	no	—
61	#107	—	20	4	no	—
62	C₄×C₈	C₂²⋊D₅	1	2	yes	—
63	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
64	C₂²×C₅	—	16	2	no	—
65	C₂²×C₅	—	16	4	no	—
66	C₂²×C₅	#493	1	1	yes	characteristic
67	C₄×C₅	#493	1	2	yes	—
68	C₄×C₅	C₂×SD₃₂	1	4	yes	—
69	Dic₅	—	16	2	no	—
70	C₂×Q₈	—	20	2	no	—
71	C₂×D₄	—	4	4	no	—
72	C₂×D₄	—	4	2	no	—
73	Q₁₆	—	20	8	no	—
74	D₈	—	4	8	no	—
75	D₈	—	8	8	no	—
76	C₂×C₈	#686	1	4	yes	—
77	C₂×C₈	D₄×D₅	1	2	yes	—
78	C₄⋊C₄	—	20	4	no	—
79	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
80	C₁₆	—	5	8	no	—
81	C₂×C₅	—	32	2	no	—
82	C₂×C₅	#1712	1	1	yes	characteristic
83	C₂×C₅	#2248	1	2	yes	—
84	C₂³	—	8	2	no	—
85	Q₈	—	40	4	no	—
86	D₄	—	8	8	no	—
87	D₄	—	8	4	no	—
88	C₂×C₄	—	40	2	no	—
89	C₂×C₄	C₂×#686	1	2	yes	—
90	C₂×C₄	#4086	1	1	yes	characteristic
91	C₈	#3959	1	8	yes	complemented
92	C₈	D₅×D₈	1	4	yes	—
93	C₅	#12517	1	1	yes	characteristic, complemented, Sylow
94	C₂²	—	16	2	no	—
95	C₂²	—	16	4	no	—
96	C₂²	#65362	1	1	yes	characteristic
97	C₄	—	80	2	no	—
98	C₄	C₂×#3959	1	4	yes	—
99	C₄	#65440	1	2	yes	—
100	C₂	—	32	2	no	—
101	C₂	#10602069	1	1	yes	characteristic
102	C₂	#10602084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1712

C₂

yes

maximal, characteristic

#10602084

C₂

yes

maximal, complemented

#10598858

C₂

yes

maximal, characteristic, complemented

C₈×#672

C₂

yes

maximal, characteristic, complemented

C₅×#493

—

C₅×#493

C₂²

yes

—

C₂×#3959

—

#64988

C₂²

yes

—

#64971

—

C₅×C₈²

C₂²

yes

characteristic

C₄×#672

C₂²

yes

—

#12517

—

maximal, Sylow

C₂×C₅×D₈

—

C₂×C₅×D₈

—

C₅×#127

—

C₂×Dic₂₀

—

#4015

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

D₄

yes

—

#2248

—

#1713

—

#1712

D₅

yes

characteristic

C₈×C₁₆

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×Dic₁₀

—

C₅×D₈

—

C₅×D₈

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

Dic₂₀

—

#672

D₈

yes

complemented

C₂×SD₃₂

—

—

—

—

C₂×D₅

yes

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂³×C₅

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

SD₃₂

yes

complemented

C₅×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

C₂×C₁₆

—

#107

—

C₄×C₈

C₂²⋊D₅

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#493

yes

characteristic

C₄×C₅

#493

yes

—

C₄×C₅

C₂×SD₃₂

yes

—

Dic₅

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

C₂×C₈

#686

yes

—

C₂×C₈

D₄×D₅

yes

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₅

yes

characteristic

C₁₆

—

C₂×C₅

—

C₂×C₅

#1712

yes

characteristic

C₂×C₅

#2248

yes

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×#686

yes

—

C₂×C₄

yes

C₈

yes

C₈

D₅×D₈

yes

—

C₅

#12517

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

C₂²

#65362

yes

characteristic

C₄

—

C₄

C₂×#3959

yes

—

C₄

#65440

yes

—

100

C₂

—

101

C₂

#10602069

yes

characteristic

102

C₂

#10602084

yes

—

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting, prime radical	C₅×#1712	C₂
socle	C₂²×C₅	#493
hypercenter	#1712	D₅
hypocenter	C₅	#12517
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#65362

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

Fitting, prime radical

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
2. #10602084	C₂	semidirect
3. #10598858	C₂	semidirect
4. C₈×#672	C₂	semidirect
37. #672	D₈	semidirect
52. C₅×C₈	SD₃₂	semidirect
91. C₈	#3959	semidirect
93. C₅	#12517	semidirect

H ≅ G/N

type

2. #10602084

C₂

semidirect

3. #10598858

C₂

semidirect

4. C₈×#672

C₂

semidirect

37. #672

D₈

semidirect

52. C₅×C₈

SD₃₂

semidirect

91. C₈

#3959

semidirect

93. C₅

#12517

semidirect

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

subgroup	index
C₅×#1712	2
#10602084	2
#10598858	2
C₈×#672	2
#12517	5

subgroup

index

Subgroups — #12014017

Conjugacy classes of subgroups

Complements

Maximal subgroups