Group #10599733 — subgroups

Subgroups — #10599733

Order 640.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×#676	C₂	1	1	yes	maximal, characteristic
2	#65415	C₂	1	2	yes	maximal
3	C₂×#3954	C₂	1	2	yes	maximal, complemented
4	C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
5	#64772	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₅	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#681	C₂²	1	2	yes	—
10	C₂²×#168	C₂²	1	1	yes	characteristic
11	C₂×#676	—	2	4	no	—
12	C₂×#676	C₄	1	4	yes	—
13	#3954	—	2	4	no	—
14	#3952	—	2	4	no	—
15	#1811	—	5	1	no	maximal, Sylow
16	C₂³×D₅	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂²×Dic₅	Q₈	1	1	yes	characteristic
19	C₂×C₄×D₅	—	2	2	no	—
20	C₂×C₄×D₅	—	2	2	no	—
21	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
22	C₂×C₄×D₅	D₄	1	2	yes	—
23	C₅×M₁₆	—	2	2	no	—
24	C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×C₈	—	2	2	no	—
26	C₂×C₅×C₈	—	2	2	no	—
27	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
28	#681	—	2	2	no	—
29	#681	—	2	2	no	—
30	C₂×#168	—	2	2	no	—
31	C₂×#168	C₂×C₄	1	2	yes	—
32	C₂×#168	D₄	1	4	yes	—
33	#676	—	2	8	no	—
34	C₂²×M₁₆	—	5	1	no	—
35	C₂×#108	—	5	2	no	—
36	#407	—	5	2	no	—
37	#341	—	5	1	no	—
38	#341	D₅	1	1	yes	characteristic, complemented
39	C₂³×C₅	C₂×Q₈	1	1	yes	characteristic
40	C₂²×D₅	—	2	2	no	—
41	C₂²×D₅	—	4	2	no	—
42	C₂²×D₅	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅	#41	1	2	yes	—
47	C₂×Dic₅	—	2	2	no	—
48	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
49	C₄×D₅	—	2	2	no	—
50	C₄×D₅	—	2	2	no	—
51	C₄×D₅	—	2	4	no	—
52	C₅×C₈	—	2	2	no	—
53	C₅×C₈	—	4	2	no	—
54	#168	—	4	2	no	—
55	#168	C₂²⋊C₄	1	4	yes	—
56	C₂³×C₄	—	5	1	no	—
57	C₂×M₁₆	—	5	4	no	—
58	C₂×M₁₆	—	5	2	no	—
59	C₂×M₁₆	—	10	4	no	—
60	C₂×M₁₆	C₂×D₅	1	2	yes	—
61	C₂²×C₈	—	5	1	no	—
62	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
63	#108	—	10	4	no	—
64	C₂²⋊C₈	—	10	4	no	—
65	C₂²×C₅	—	2	2	no	—
66	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₅	C₂²⋊Q₈	1	2	yes	—
68	C₂×D₅	—	2	2	no	—
69	C₂×D₅	—	2	4	no	—
70	C₂×D₅	—	4	4	no	—
71	C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₅	#117	1	1	yes	characteristic
73	C₄×C₅	#121	1	2	yes	—
74	Dic₅	—	2	2	no	—
75	C₂⁴	—	5	1	no	—
76	C₂²×C₄	—	5	1	no	—
77	C₂²×C₄	—	5	2	no	—
78	C₂²×C₄	—	5	2	no	—
79	C₂²×C₄	—	10	2	no	—
80	C₂²×C₄	—	10	2	no	—
81	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	2	no	—
84	M₁₆	—	10	8	no	—
85	M₁₆	—	10	2	no	—
86	M₁₆	—	10	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	5	2	no	—
90	C₂×C₈	—	5	4	no	—
91	C₂×C₈	—	10	2	no	—
92	C₂×C₈	C₄×D₅	1	2	yes	—
93	C₂×C₅	—	2	2	no	—
94	C₂×C₅	#380	1	1	yes	characteristic
95	C₂×C₅	#430	1	2	yes	—
96	D₅	—	4	2	no	—
97	C₂³	—	5	2	no	—
98	C₂³	—	10	2	no	—
99	C₂³	—	20	2	no	—
100	C₂³	Q₈×D₅	1	1	yes	characteristic
101	C₂×C₄	—	5	2	no	—
102	C₂×C₄	—	10	2	no	—
103	C₂×C₄	—	10	2	no	—
104	C₂×C₄	—	10	2	no	—
105	C₂×C₄	—	10	4	no	—
106	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
107	C₂×C₄	#709	1	2	yes	—
108	C₂×C₄	D₄×D₅	1	2	yes	—
109	C₂×C₄	D₄×D₅	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	—	4	2	no	—
112	C₈	—	5	4	no	—
113	C₈	—	20	2	no	—
114	C₅	#1811	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	2	2	no	—
116	C₂²	—	10	4	no	—
117	C₂²	—	10	2	no	—
118	C₂²	—	20	4	no	—
119	C₂²	D₅×C₄⋊C₄	1	1	yes	characteristic
120	C₂²	#4042	1	2	yes	—
121	C₄	—	10	2	no	—
122	C₄	#4018	1	1	yes	characteristic
123	C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
124	C₄	#4030	1	2	yes	—
125	C₂	—	2	2	no	—
126	C₂	—	20	2	no	—
127	C₂	#64956	1	1	yes	characteristic
128	C₂	#65182	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#676

C₂

yes

maximal, characteristic

#65415

C₂

yes

maximal

C₂×#3954

C₂

yes

maximal, complemented

C₅×#341

C₂

yes

maximal, characteristic, complemented

#64772

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₅

C₂²

yes

characteristic

C₂×C₅×M₁₆

C₂²

yes

—

C₂²×C₅×C₈

C₂²

yes

characteristic

C₂×#681

C₂²

yes

—

C₂²×#168

C₂²

yes

characteristic

C₂×#676

—

C₂×#676

C₄

yes

—

—

—

—

C₂³×D₅

Q₈

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂²×Dic₅

Q₈

yes

characteristic

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

C₂×C₄

yes

—

C₂×C₄×D₅

D₄

yes

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₂×C₅×C₈

—

C₂×C₅×C₈

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#168

D₄

yes

—

#676

—

C₂²×M₁₆

—

—

—

—

D₅

yes

characteristic, complemented

C₂³×C₅

C₂×Q₈

yes

characteristic

C₂²×D₅

—

C₂²×D₅

—

C₂²×D₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₂×Dic₅

C₄⋊C₄

yes

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

C₂²⋊C₄

yes

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅

yes

characteristic

#108

—

C₂²⋊C₈

—

C₂²×C₅

—

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅

C₂²⋊Q₈

yes

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

C₂×C₂²⋊C₄

yes

C₄×C₅

yes

C₄×C₅

yes

—

Dic₅

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅

yes

—

C₂×C₅

—

C₂×C₅

#380

yes

characteristic

C₂×C₅

#430

yes

—

D₅

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

Q₈×D₅

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄×D₅

yes

characteristic

107

C₂×C₄

#709

yes

—

108

C₂×C₄

D₄×D₅

yes

—

109

C₂×C₄

D₄×D₅

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₅

#1811

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

D₅×C₄⋊C₄

yes

characteristic

120

C₂²

#4042

yes

—

121

C₄

—

122

C₄

#4018

yes

characteristic

123

C₄

D₅×C₂²⋊C₄

yes

characteristic

124

C₄

#4030

yes

—

125

C₂

—

126

C₂

—

127

C₂

#64956

yes

characteristic

128

C₂

#65182

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₅×#341	C₂
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂³×D₅	Q₈
cosocle	C₂²×C₄×C₅	C₂³
hypercenter	#341	D₅
hypocenter	C₅	#1811
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₅×#341

C₂

socle

C₂²×C₅

C₂×C₄⋊C₄

prime radical

C₂³×D₅

Q₈

cosocle

C₂²×C₄×C₅

C₂³

D₅

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
3. C₂×#3954	C₂	semidirect
4. C₅×#341	C₂	semidirect
5. #64772	C₂	semidirect
38. #341	D₅	semidirect
114. C₅	#1811	semidirect

H ≅ G/N

type

3. C₂×#3954

C₂

semidirect

4. C₅×#341

C₂

semidirect

5. #64772

C₂

semidirect

38. #341

D₅

semidirect

114. C₅

#1811

semidirect

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

subgroup	index
C₂²×#676	2
#65415	2
C₂×#3954	2
C₅×#341	2
#64772	2
#1811	5

subgroup

index

C₂²×#676

Subgroups — #10599733

Conjugacy classes of subgroups

Complements

Maximal subgroups