Group #12014038 — subgroups

Subgroups — #12014038

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×#1710	C₂	1	1	yes	maximal, characteristic
2	#10602178	C₂	1	2	yes	maximal
3	#10602177	C₂	1	2	yes	maximal
4	#10598857	C₂	1	1	yes	maximal, characteristic
5	C₈×#672	C₂	1	1	yes	maximal, characteristic
6	C₅×#500	C₂²	1	2	yes	—
7	C₅×#487	—	2	2	no	—
8	#64971	C₂²	1	2	yes	—
9	#64970	—	2	2	no	—
10	C₅×C₈²	C₂²	1	1	yes	characteristic
11	#64710	—	2	4	no	—
12	#64709	—	2	4	no	—
13	C₄×#672	C₂²	1	1	yes	characteristic
14	C₄×#672	C₂²	1	1	yes	characteristic
15	#12522	—	5	1	no	maximal, Sylow
16	C₅×C₄⋊Q₈	—	2	2	no	—
17	#4015	—	2	2	no	—
18	C₅×#107	—	2	2	no	—
19	C₅×#107	—	2	2	no	—
20	C₅×#103	—	4	4	no	—
21	C₄×C₅×C₈	D₄	1	2	yes	—
22	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
23	#3950	—	2	2	no	—
24	#3950	—	2	2	no	—
25	#3948	—	4	4	no	—
26	C₂×#672	D₄	1	2	yes	—
27	C₂×#672	D₄	1	2	yes	—
28	#2254	—	5	2	no	—
29	#2252	—	5	2	no	—
30	#1710	—	5	1	no	—
31	#1710	D₅	1	1	yes	characteristic
32	C₈×C₁₆	—	5	1	no	—
33	C₂×C₅×Q₈	—	4	2	no	—
34	C₂×Dic₁₀	—	4	2	no	—
35	C₂×C₅×C₈	D₈	1	4	yes	—
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	#672	Q₁₆	1	4	yes	—
44	#672	Q₁₆	1	4	yes	—
45	#500	—	5	2	no	—
46	#500	C₂×D₅	1	2	yes	—
47	#487	—	2	2	no	—
48	#487	—	10	2	no	—
49	#367	—	10	4	no	—
50	#366	—	10	4	no	—
51	C₄×C₁₆	—	5	1	no	—
52	C₄×C₁₆	—	5	1	no	—
53	C₈²	C₂×D₅	1	1	yes	characteristic
54	C₅×Q₈	—	8	4	no	—
55	C₂×C₄×C₅	—	8	2	no	—
56	C₂×C₄×C₅	#127	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
58	C₂×Dic₅	—	8	2	no	—
59	Dic₁₀	—	8	4	no	—
60	C₅×C₈	—	2	4	no	—
61	C₅×C₈	C₂×Q₁₆	1	2	yes	—
62	C₅×C₈	C₂×Q₁₆	1	2	yes	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	10	2	no	—
65	C₂×C₁₆	—	5	2	no	—
66	C₂×C₁₆	—	5	2	no	—
67	#107	—	2	2	no	—
68	#107	—	2	2	no	—
69	#107	—	10	2	no	—
70	#107	—	10	2	no	—
71	#103	—	4	4	no	—
72	#103	—	20	4	no	—
73	C₄×C₈	C₂²⋊D₅	1	2	yes	—
74	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
75	C₂²×C₅	#493	1	1	yes	characteristic
76	C₄×C₅	—	16	2	no	—
77	C₄×C₅	#494	1	1	yes	characteristic
78	C₄×C₅	#494	1	1	yes	characteristic
79	C₄×C₅	#508	1	4	yes	—
80	Dic₅	—	16	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	20	2	no	—
83	C₂×C₈	#686	1	4	yes	—
84	C₂×C₈	D₄×D₅	1	1	yes	characteristic
85	C₂×C₈	D₄×D₅	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	4	2	no	—
88	C₄⋊C₄	—	20	2	no	—
89	C₄⋊C₄	—	20	2	no	—
90	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
91	C₁₆	—	5	4	no	—
92	C₁₆	—	5	4	no	—
93	C₂×C₅	#1713	1	1	yes	characteristic
94	C₂×C₅	#2249	1	2	yes	—
95	Q₈	—	8	4	no	—
96	Q₈	—	40	4	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	40	2	no	—
99	C₂×C₄	C₂×#686	1	2	yes	—
100	C₂×C₄	#4086	1	1	yes	characteristic
101	C₈	—	2	4	no	—
102	C₈	D₅×Q₁₆	1	2	yes	—
103	C₈	D₅×Q₁₆	1	2	yes	—
104	C₅	#12522	1	1	yes	characteristic, complemented, Sylow
105	C₂²	#65362	1	1	yes	characteristic
106	C₄	—	16	2	no	—
107	C₄	—	80	2	no	—
108	C₄	#65470	1	1	yes	characteristic
109	C₄	#65470	1	1	yes	characteristic
110	C₄	#65483	1	4	yes	—
111	C₂	#10602091	1	1	yes	characteristic
112	C₂	#10602113	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1710

C₂

yes

maximal, characteristic

#10602178

C₂

yes

maximal

#10602177

C₂

yes

maximal

#10598857

C₂

yes

maximal, characteristic

C₈×#672

C₂

yes

maximal, characteristic

C₅×#500

C₂²

yes

—

C₅×#487

—

#64971

C₂²

yes

—

#64970

—

C₅×C₈²

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#672

C₂²

yes

characteristic

#12522

—

maximal, Sylow

C₅×C₄⋊Q₈

—

—

—

—

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

C₂×#672

D₄

yes

—

—

—

—

D₅

yes

characteristic

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₂×C₅×C₈

D₈

yes

—

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

—

#672

Q₁₆

yes

—

#672

Q₁₆

yes

—

#500

—

#500

C₂×D₅

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

C₂×Q₁₆

yes

—

C₅×C₈

C₂×Q₁₆

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₅

yes

—

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

#686

yes

—

C₂×C₈

D₄×D₅

yes

characteristic

C₂×C₈

D₄×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₅

yes

characteristic

C₁₆

—

C₁₆

—

C₂×C₅

#1713

yes

characteristic

C₂×C₅

#2249

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#686

yes

—

100

C₂×C₄

#4086

yes

characteristic

101

C₈

—

102

C₈

D₅×Q₁₆

yes

—

103

C₈

D₅×Q₁₆

yes

—

104

C₅

#12522

yes

characteristic, complemented, Sylow

105

C₂²

#65362

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

#65470

yes

characteristic

109

C₄

#65470

yes

characteristic

110

C₄

#65483

yes

—

111

C₂

#10602091

yes

characteristic

112

C₂

#10602113

yes

—

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1710	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#1710	D₅
hypocenter	C₅	#12522
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#65362

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 ◁ C₈² ◁ #1710

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

H ≅ G/N

type

104. C₅

#12522

semidirect

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

subgroup	index
C₅×#1710	2
#10602178	2
#10602177	2
#10598857	2
C₈×#672	2
#12522	5

subgroup

index

Subgroups — #12014038

Conjugacy classes of subgroups

Complements

Maximal subgroups