Group #12013989 — subgroups

Subgroups — #12013989

Order 1280.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#10602103	C₂	1	4	yes	maximal
3	#10598858	C₂	1	1	yes	maximal, characteristic
4	C₈×#672	C₂	1	1	yes	maximal, characteristic
5	C₅×#500	—	2	1	no	—
6	C₅×#500	—	2	1	no	—
7	C₅×#494	C₂²	1	2	yes	—
8	#64988	—	2	1	no	—
9	#64971	—	2	1	no	—
10	#64971	C₂²	1	2	yes	—
11	C₅×C₈²	C₂²	1	1	yes	characteristic
12	#64784	—	2	8	no	—
13	C₄×#672	C₂²	1	2	yes	—
14	#12510	—	5	1	no	maximal, Sylow
15	C₂×C₅×Q₁₆	—	2	4	no	—
16	C₅×C₄⋊Q₈	—	2	2	no	—
17	C₂×Dic₂₀	—	4	2	no	—
18	#4015	—	2	2	no	—
19	C₅×#107	—	4	2	no	—
20	C₅×#107	—	4	2	no	—
21	C₄×C₅×C₈	D₄	1	1	yes	characteristic
22	C₄×C₅×C₈	D₄	1	1	yes	characteristic
23	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
24	#3950	—	2	4	no	—
25	#3950	—	4	2	no	—
26	C₂×#672	D₄	1	4	yes	—
27	#2241	—	5	4	no	—
28	#1713	—	5	1	no	—
29	#1713	D₅	1	1	yes	characteristic
30	C₈×C₁₆	—	5	1	no	—
31	C₂×C₅×Q₈	—	4	2	no	—
32	C₂×Dic₁₀	—	4	2	no	—
33	C₅×Q₁₆	—	4	8	no	—
34	C₂×C₅×C₈	D₈	1	2	yes	—
35	C₂×C₅×C₈	D₈	1	2	yes	—
36	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
37	C₅×C₄⋊C₄	—	4	2	no	—
38	C₅×C₄⋊C₄	—	4	2	no	—
39	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₅	—	4	2	no	—
41	C₄⋊Dic₅	—	4	2	no	—
42	Dic₂₀	—	8	4	no	—
43	#672	—	2	4	no	—
44	#500	—	2	1	no	—
45	#500	—	2	1	no	—
46	#500	—	5	2	no	—
47	#500	—	10	1	no	—
48	#494	—	10	1	no	—
49	#494	C₂×D₅	1	2	yes	—
50	#358	—	10	8	no	—
51	C₄×C₁₆	—	5	2	no	—
52	C₈²	C₂×D₅	1	1	yes	characteristic
53	C₅×Q₈	—	8	4	no	—
54	C₂×C₄×C₅	—	8	2	no	—
55	C₂×C₄×C₅	#127	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₅	—	8	2	no	—
59	Dic₁₀	—	8	4	no	—
60	C₅×C₈	SD₃₂	1	4	yes	—
61	C₅×C₈	Q₃₂	1	4	yes	—
62	C₅×C₈	#135	1	4	yes	—
63	C₂×Q₁₆	—	2	4	no	—
64	C₂×Q₁₆	—	20	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	10	2	no	—
67	C₂×C₁₆	—	5	4	no	—
68	#107	—	4	2	no	—
69	#107	—	4	2	no	—
70	#107	—	10	4	no	—
71	#107	—	20	2	no	—
72	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
73	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
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	#495	1	2	yes	—
78	C₄×C₅	C₂×SD₃₂	1	2	yes	—
79	C₄×C₅	C₂×Q₃₂	1	2	yes	—
80	Dic₅	—	16	2	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	20	2	no	—
83	Q₁₆	—	4	8	no	—
84	Q₁₆	—	40	4	no	—
85	C₂×C₈	#686	1	2	yes	—
86	C₂×C₈	#686	1	2	yes	—
87	C₂×C₈	D₄×D₅	1	2	yes	—
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	C₂×C₂²⋊D₅	1	1	yes	characteristic
93	C₁₆	—	10	4	no	—
94	C₂×C₅	#1709	1	1	yes	characteristic
95	C₂×C₅	#2247	1	1	yes	characteristic
96	C₂×C₅	#2248	1	1	yes	characteristic
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	C₂×#686	1	1	yes	characteristic
102	C₂×C₄	C₂×#686	1	1	yes	characteristic
103	C₂×C₄	#4086	1	1	yes	characteristic
104	C₈	#3960	1	4	yes	—
105	C₈	#3961	1	4	yes	—
106	C₈	#4058	1	4	yes	—
107	C₅	#12510	1	1	yes	characteristic, complemented, Sylow
108	C₂²	#65362	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	80	2	no	—
111	C₄	#65444	1	2	yes	—
112	C₄	C₂×#3960	1	2	yes	—
113	C₄	C₂×#3961	1	2	yes	—
114	C₂	#10602068	1	1	yes	characteristic
115	C₂	#10602105	1	1	yes	characteristic
116	C₂	#10602106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1713

C₂

yes

maximal, characteristic

#10602103

C₂

yes

maximal

#10598858

C₂

yes

maximal, characteristic

C₈×#672

C₂

yes

maximal, characteristic

C₅×#500

—

C₅×#500

—

C₅×#494

C₂²

yes

—

#64988

—

#64971

—

#64971

C₂²

yes

—

C₅×C₈²

C₂²

yes

characteristic

#64784

—

C₄×#672

C₂²

yes

—

#12510

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₅×C₄⋊Q₈

—

C₂×Dic₂₀

—

#4015

—

C₅×#107

—

C₅×#107

—

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

D₄

yes

characteristic

C₄×C₅×C₈

C₂³

yes

—

—

D₄

yes

—

#2241

—

#1713

—

#1713

D₅

yes

characteristic

C₈×C₁₆

—

C₂×C₅×Q₈

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

Dic₂₀

—

—

—

—

—

—

—

C₂×D₅

yes

—

#358

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

characteristic

C₂×Dic₅

—

Dic₁₀

—

C₅×C₈

SD₃₂

yes

—

C₅×C₈

Q₃₂

yes

—

C₅×C₈

#135

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²⋊D₅

yes

characteristic

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

#495

yes

—

C₄×C₅

C₂×SD₃₂

yes

—

C₄×C₅

C₂×Q₃₂

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#686

yes

—

C₂×C₈

#686

yes

—

C₂×C₈

D₄×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₂²⋊D₅

yes

characteristic

C₁₆

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×#686

yes

characteristic

102

C₂×C₄

C₂×#686

yes

characteristic

103

C₂×C₄

#4086

yes

characteristic

104

C₈

#3960

yes

—

105

C₈

#3961

yes

—

106

C₈

#4058

yes

—

107

C₅

#12510

yes

characteristic, complemented, Sylow

108

C₂²

#65362

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#65444

yes

—

112

C₄

C₂×#3960

yes

—

113

C₄

C₂×#3961

yes

—

114

C₂

#10602068

yes

characteristic

115

C₂

#10602105

yes

characteristic

116

C₂

#10602106

yes

characteristic

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1713	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#1713	D₅
hypocenter	C₅	#12510
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

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 ◁ #1713

N	H ≅ G/N	type
107. C₅	#12510	semidirect

H ≅ G/N

type

107. C₅

#12510

semidirect

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

subgroup	index
C₅×#1713	2
#10602103	2
#10598858	2
C₈×#672	2
#12510	5

subgroup

index

Subgroups — #12013989

Conjugacy classes of subgroups

Complements

Maximal subgroups