Group #12014040 — subgroups

Subgroups — #12014040

Order 1280.

238 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 87 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×#2254	C₂	1	1	yes	maximal, characteristic
2	#10602177	C₂	1	2	yes	maximal
3	#10602154	C₂	1	2	yes	maximal
4	#10599839	C₂	1	1	yes	maximal, characteristic
5	C₁₆×#168	C₂	1	1	yes	maximal, characteristic
6	C₅×#500	C₂²	1	2	yes	—
7	#65382	—	2	2	no	—
8	#65370	—	2	2	no	—
9	#64971	C₂²	1	2	yes	—
10	C₅×#367	—	2	2	no	—
11	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	#64725	—	2	2	no	—
13	#64709	—	2	4	no	—
14	C₄×#672	C₂²	1	1	yes	characteristic
15	C₈×#168	C₂²	1	1	yes	characteristic
16	#12522	—	5	1	no	maximal, Sylow
17	C₅×C₄⋊Q₈	—	2	2	no	—
18	#4015	—	2	2	no	—
19	C₂×C₅×C₁₆	D₄	1	2	yes	—
20	C₅×#107	—	2	2	no	—
21	C₅×#107	—	2	2	no	—
22	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
23	#3967	—	4	4	no	—
24	#3950	—	2	2	no	—
25	#3950	—	2	2	no	—
26	C₂×#672	D₄	1	2	yes	—
27	#3942	—	4	4	no	—
28	C₄×#168	D₄	1	2	yes	—
29	#2254	—	5	1	no	—
30	#2254	D₅	1	1	yes	characteristic
31	#2252	—	5	2	no	—
32	#1710	—	5	2	no	—
33	C₈×C₁₆	—	5	1	no	—
34	C₂×C₅×Q₈	—	4	2	no	—
35	C₂×Dic₁₀	—	4	2	no	—
36	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
38	C₅×C₄⋊C₄	—	4	2	no	—
39	C₅×C₄⋊C₄	—	4	2	no	—
40	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₅	—	4	2	no	—
42	C₄⋊Dic₅	—	4	2	no	—
43	C₂×#168	D₈	1	4	yes	—
44	C₅×C₁₆	Q₁₆	1	4	yes	—
45	#672	Q₁₆	1	4	yes	—
46	#500	—	5	2	no	—
47	#500	C₂×D₅	1	2	yes	—
48	#487	—	10	2	no	—
49	#487	—	10	2	no	—
50	#367	—	2	2	no	—
51	#367	—	10	2	no	—
52	#366	—	10	4	no	—
53	C₄×C₁₆	—	5	1	no	—
54	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
55	C₈²	—	5	1	no	—
56	C₅×Q₈	—	8	4	no	—
57	C₂×C₄×C₅	—	8	2	no	—
58	C₂×C₄×C₅	#127	1	1	yes	characteristic
59	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
60	C₂×Dic₅	—	8	2	no	—
61	Dic₁₀	—	8	4	no	—
62	C₅×C₈	C₂×Q₁₆	1	2	yes	—
63	C₅×C₈	C₂×Q₁₆	1	2	yes	—
64	#168	—	2	4	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	10	2	no	—
67	C₂×C₁₆	—	5	2	no	—
68	C₂×C₁₆	C₂²⋊D₅	1	2	yes	—
69	#107	—	2	2	no	—
70	#107	—	2	2	no	—
71	#107	—	10	2	no	—
72	#107	—	10	2	no	—
73	#103	—	20	4	no	—
74	#103	—	20	4	no	—
75	C₄×C₈	—	5	2	no	—
76	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
77	C₂²×C₅	#493	1	1	yes	characteristic
78	C₄×C₅	—	16	2	no	—
79	C₄×C₅	#494	1	1	yes	characteristic
80	C₄×C₅	#494	1	1	yes	characteristic
81	C₄×C₅	#508	1	4	yes	—
82	Dic₅	—	16	2	no	—
83	C₂×Q₈	—	4	2	no	—
84	C₂×Q₈	—	20	2	no	—
85	C₂×C₈	—	5	4	no	—
86	C₂×C₈	D₄×D₅	1	1	yes	characteristic
87	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	20	2	no	—
91	C₄⋊C₄	—	20	2	no	—
92	C₄²	D₄×D₅	1	1	yes	characteristic
93	C₁₆	—	5	4	no	—
94	C₁₆	#689	1	4	yes	—
95	C₂×C₅	#1713	1	1	yes	characteristic
96	C₂×C₅	#2249	1	2	yes	—
97	Q₈	—	8	4	no	—
98	Q₈	—	40	4	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	—	40	2	no	—
101	C₂×C₄	D₅×D₈	1	2	yes	—
102	C₂×C₄	#4086	1	1	yes	characteristic
103	C₈	—	10	4	no	—
104	C₈	D₅×Q₁₆	1	2	yes	—
105	C₈	C₂×#689	1	2	yes	—
106	C₅	#12522	1	1	yes	characteristic, complemented, Sylow
107	C₂²	#65440	1	1	yes	characteristic
108	C₄	—	16	2	no	—
109	C₄	—	80	2	no	—
110	C₄	#65205	1	4	yes	—
111	C₄	#65381	1	1	yes	characteristic
112	C₄	#65470	1	1	yes	characteristic
113	C₂	#10602155	1	1	yes	characteristic
114	C₂	#10602273	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2254

C₂

yes

maximal, characteristic

#10602177

C₂

yes

maximal

#10602154

C₂

yes

maximal

#10599839

C₂

yes

maximal, characteristic

C₁₆×#168

C₂

yes

maximal, characteristic

C₅×#500

C₂²

yes

—

#65382

—

#65370

—

#64971

C₂²

yes

—

C₅×#367

—

C₄×C₅×C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

C₈×#168

C₂²

yes

characteristic

#12522

—

maximal, Sylow

C₅×C₄⋊Q₈

—

#4015

—

C₂×C₅×C₁₆

D₄

yes

—

C₅×#107

—

C₅×#107

—

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

#3942

—

C₄×#168

D₄

yes

—

#2254

—

#2254

D₅

yes

characteristic

#2252

—

#1710

—

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₂×C₅×C₈

C₂×D₄

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₂×#168

D₈

yes

—

C₅×C₁₆

Q₁₆

yes

—

#672

Q₁₆

yes

—

#500

—

#500

C₂×D₅

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅

yes

characteristic

C₈²

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

C₂×Q₁₆

yes

—

C₅×C₈

C₂×Q₁₆

yes

—

#168

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

D₄×D₅

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×D₅

yes

characteristic

C₁₆

—

C₁₆

#689

yes

—

C₂×C₅

#1713

yes

characteristic

C₂×C₅

#2249

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

D₅×D₈

yes

—

102

C₂×C₄

#4086

yes

characteristic

103

C₈

—

104

C₈

D₅×Q₁₆

yes

—

105

C₈

C₂×#689

yes

—

106

C₅

#12522

yes

characteristic, complemented, Sylow

107

C₂²

#65440

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

#65205

yes

—

111

C₄

#65381

yes

characteristic

112

C₄

#65470

yes

characteristic

113

C₂

#10602155

yes

characteristic

114

C₂

#10602273

yes

—

	subgroup	G/H
center	C₂²	#65440
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2254	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#2254	D₅
hypocenter	C₅	#12522
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#65440

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
106. C₅	#12522	semidirect

H ≅ G/N

type

106. C₅

#12522

semidirect

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

subgroup	index
C₅×#2254	2
#10602177	2
#10602154	2
#10599839	2
C₁₆×#168	2
#12522	5

subgroup

index

C₁₆×#168

Subgroups — #12014040

Conjugacy classes of subgroups

Complements

Maximal subgroups