Group #10604318 — subgroups

Subgroups — #10604318

Order 640.

292 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 82 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	C₂×#4038	C₂	1	1	yes	maximal, characteristic
2	C₂×#4003	C₂	1	2	yes	maximal
3	#64918	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
5	C₂²×#170	C₂²	1	4	yes	—
6	C₂×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₂×#685	C₄	1	2	yes	—
8	C₂×C₄⋊Dic₅	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₅	C₂²	1	2	yes	—
10	#4038	—	2	4	no	—
11	#4003	—	2	4	no	—
12	#1818	—	5	1	no	maximal, Sylow
13	C₂³×D₅	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₅	C₂×C₄	1	2	yes	—
17	C₂²×Dic₅	C₂³	1	1	yes	characteristic
18	C₂×C₄×D₅	—	2	4	no	—
19	C₂×C₄×D₅	D₄	1	4	yes	—
20	C₂×#170	—	2	4	no	—
21	C₂×#170	—	2	8	no	—
22	C₅×C₄⋊C₄	—	2	2	no	—
23	#685	—	2	4	no	—
24	C₄⋊Dic₅	—	2	2	no	—
25	C₄×Dic₅	—	2	4	no	—
26	C₂×#117	—	5	1	no	—
27	C₂×C₂²⋊C₈	—	5	2	no	—
28	#336	—	5	4	no	—
29	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂²×D₅	—	2	2	no	—
31	C₂²×D₅	—	2	2	no	—
32	C₂²×D₅	—	2	2	no	—
33	C₂²×D₅	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₅	—	2	2	no	—
35	C₂×C₄×C₅	—	2	2	no	—
36	C₂×C₄×C₅	—	2	4	no	—
37	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
38	C₂×Dic₅	—	2	2	no	—
39	C₂×Dic₅	—	2	4	no	—
40	C₂×Dic₅	C₂×D₄	1	2	yes	—
41	C₂×Dic₅	#41	1	4	yes	—
42	C₄×D₅	—	2	8	no	—
43	#170	—	4	4	no	—
44	C₂³×C₄	—	5	1	no	—
45	C₂²×C₈	—	5	4	no	—
46	#117	—	10	4	no	—
47	C₂×C₄⋊C₄	—	5	1	no	—
48	C₂×C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂×C₂²⋊C₄	—	5	2	no	—
50	C₂×C₄²	—	5	2	no	—
51	C₂²⋊C₈	—	10	4	no	—
52	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₅	#117	1	2	yes	—
54	C₂²×C₅	#131	1	2	yes	—
55	C₂²×C₅	#131	1	2	yes	—
56	C₂×D₅	—	2	2	no	—
57	C₂×D₅	—	2	2	no	—
58	C₂×D₅	—	2	2	no	—
59	C₂×D₅	—	4	4	no	—
60	C₄×C₅	—	2	2	no	—
61	C₄×C₅	—	4	2	no	—
62	Dic₅	—	4	2	no	—
63	Dic₅	#123	1	4	yes	—
64	C₂⁴	—	5	1	no	—
65	C₂²×C₄	—	5	1	no	—
66	C₂²×C₄	—	5	4	no	—
67	C₂²×C₄	—	5	2	no	—
68	C₂²×C₄	—	10	4	no	—
69	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
70	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₂×C₈	—	10	4	no	—
72	C₂×C₈	—	10	8	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	10	2	no	—
75	C₂²⋊C₄	—	10	4	no	—
76	C₄²	—	10	4	no	—
77	C₂×C₅	#381	1	1	yes	characteristic
78	C₂×C₅	#408	1	1	yes	characteristic
79	C₂×C₅	#408	1	1	yes	characteristic
80	C₂×C₅	#433	1	4	yes	—
81	D₅	—	4	2	no	—
82	C₂³	—	5	2	no	—
83	C₂³	—	10	2	no	—
84	C₂³	—	10	2	no	—
85	C₂³	—	10	2	no	—
86	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	5	4	no	—
91	C₂×C₄	—	5	2	no	—
92	C₂×C₄	—	10	8	no	—
93	C₂×C₄	—	10	4	no	—
94	C₂×C₄	—	10	2	no	—
95	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
96	C₈	—	20	4	no	—
97	C₅	#1818	1	1	yes	characteristic, complemented, Sylow
98	C₂²	—	10	2	no	—
99	C₂²	—	10	2	no	—
100	C₂²	—	10	2	no	—
101	C₂²	—	20	4	no	—
102	C₂²	#4130	1	2	yes	—
103	C₂²	#4131	1	2	yes	—
104	C₂²	#4133	1	2	yes	—
105	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	4	2	no	—
108	C₄	—	5	4	no	—
109	C₄	—	20	2	no	—
110	C₂	—	20	2	no	—
111	C₂	#65706	1	4	yes	—
112	C₂	#65764	1	1	yes	characteristic
113	C₂	#65776	1	1	yes	characteristic
114	C₂	#65787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4038

C₂

yes

maximal, characteristic

C₂×#4003

C₂

yes

maximal

#64918

C₂

yes

maximal, complemented

C₂²×C₄×D₅

C₂²

yes

characteristic

C₂²×#170

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₄

yes

characteristic

C₂×#685

C₄

yes

—

C₂×C₄⋊Dic₅

C₄

yes

characteristic

C₂×C₄×Dic₅

C₂²

yes

—

—

—

—

C₂³×D₅

C₂×C₄

yes

characteristic

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

D₄

yes

—

C₂×#170

—

C₂×#170

—

C₅×C₄⋊C₄

—

#685

—

C₄⋊Dic₅

—

C₄×Dic₅

—

C₂×#117

—

C₂×C₂²⋊C₈

—

#336

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂×D₄

yes

—

C₂×Dic₅

#41

yes

—

C₄×D₅

—

#170

—

C₂³×C₄

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

AGL(1,𝔽₅)

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

#117

yes

—

C₂²×C₅

#131

yes

—

C₂²×C₅

#131

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

#123

yes

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

—

D₅

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²×AGL(1,𝔽₅)

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

—

C₈

—

C₅

#1818

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

#4130

yes

—

103

C₂²

#4131

yes

—

104

C₂²

#4133

yes

—

105

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

—

110

C₂

—

111

C₂

#65706

yes

—

112

C₂

#65764

yes

characteristic

113

C₂

#65776

yes

characteristic

114

C₂

#65787

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂³×C₅	C₂²×C₄
Fitting	C₂×C₅×C₄⋊C₄	C₄
prime radical	C₂³×D₅	C₂×C₄
cosocle	C₂²×Dic₅	C₂³
hypercenter	C₂×C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#1818
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×AGL(1,𝔽₅)

commutator, socle

C₂³×C₅

C₂²×C₄

Fitting

C₂×C₅×C₄⋊C₄

C₄

prime radical

C₂³×D₅

C₂×C₄

cosocle

C₂²×Dic₅

C₂³

C₂×C₄⋊C₄

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. #64918	C₂	semidirect
97. C₅	#1818	semidirect

H ≅ G/N

type

3. #64918

C₂

semidirect

97. C₅

#1818

semidirect

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

subgroup	index
C₂×#4038	2
C₂×#4003	2
#64918	2
#1818	5

subgroup

index

Subgroups — #10604318

Conjugacy classes of subgroups

Complements

Maximal subgroups