Group #10598867 — subgroups

Subgroups — #10598867

Order 640.

178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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	#64974	C₂	1	1	yes	maximal, characteristic
2	#64974	C₂	1	1	yes	maximal, characteristic
3	#64974	C₂	1	1	yes	maximal, characteristic
4	C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	#64675	C₂	1	1	yes	maximal, characteristic, complemented
6	#64674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄²×D₅	C₂²	1	1	yes	characteristic
9	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	#3952	C₄	1	2	yes	—
13	#3952	C₄	1	2	yes	—
14	#3952	C₄	1	2	yes	—
15	#3946	C₄	1	2	yes	—
16	#3946	C₄	1	2	yes	—
17	#3946	C₄	1	2	yes	—
18	C₄×#168	C₂²	1	1	yes	characteristic
19	C₄×#168	C₂²	1	1	yes	characteristic
20	C₄×#168	C₂²	1	1	yes	characteristic
21	#1454	—	5	1	no	maximal, Sylow
22	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₂×#168	C₂×C₄	1	2	yes	—
33	C₂×#168	C₂×C₄	1	2	yes	—
34	C₂×#168	C₂×C₄	1	2	yes	—
35	#431	—	5	1	no	—
36	#431	—	5	1	no	—
37	#431	—	5	1	no	—
38	#322	—	5	1	no	—
39	#322	—	5	1	no	—
40	#322	D₅	1	1	yes	characteristic, complemented
41	C₈²	—	5	1	no	—
42	C₂²×D₅	C₄²	1	1	yes	characteristic
43	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₅	C₄²	1	1	yes	characteristic
47	C₂×Dic₅	C₄²	1	1	yes	characteristic
48	C₂×Dic₅	C₄²	1	1	yes	characteristic
49	C₄×D₅	—	2	2	no	—
50	C₄×D₅	—	2	2	no	—
51	C₄×D₅	—	2	2	no	—
52	C₅×C₈	—	2	2	no	—
53	C₅×C₈	—	2	2	no	—
54	C₅×C₈	—	2	2	no	—
55	#168	—	2	2	no	—
56	#168	—	2	2	no	—
57	#168	—	2	2	no	—
58	C₂×C₄²	—	5	1	no	—
59	C₄⋊C₈	—	5	2	no	—
60	C₄⋊C₈	—	5	2	no	—
61	C₄⋊C₈	—	5	2	no	—
62	C₂²⋊C₈	—	5	2	no	—
63	C₂²⋊C₈	—	5	2	no	—
64	C₂²⋊C₈	—	5	2	no	—
65	C₄×C₈	—	5	1	no	—
66	C₄×C₈	—	5	1	no	—
67	C₄×C₈	—	5	1	no	—
68	C₄×C₈	C₂×D₅	1	1	yes	characteristic
69	C₄×C₈	C₂×D₅	1	1	yes	characteristic
70	C₄×C₈	C₂×D₅	1	1	yes	characteristic
71	C₂²×C₅	C₂×C₄²	1	1	yes	characteristic
72	C₂×D₅	—	2	1	no	—
73	C₂×D₅	—	2	1	no	—
74	C₂×D₅	—	2	1	no	—
75	C₄×C₅	#131	1	2	yes	—
76	C₄×C₅	#131	1	2	yes	—
77	C₄×C₅	#131	1	2	yes	—
78	Dic₅	—	2	1	no	—
79	Dic₅	—	2	1	no	—
80	Dic₅	—	2	1	no	—
81	C₂²×C₄	—	5	1	no	—
82	C₂²×C₄	—	5	1	no	—
83	C₂²×C₄	—	5	1	no	—
84	C₂×C₈	—	5	2	no	—
85	C₂×C₈	—	5	2	no	—
86	C₂×C₈	—	5	2	no	—
87	C₂×C₈	C₄×D₅	1	2	yes	—
88	C₂×C₈	C₄×D₅	1	2	yes	—
89	C₂×C₈	C₄×D₅	1	2	yes	—
90	C₄²	—	5	1	no	—
91	C₄²	—	5	1	no	—
92	C₄²	—	5	1	no	—
93	C₄²	C₂²×D₅	1	1	yes	characteristic
94	C₂×C₅	#405	1	1	yes	characteristic
95	C₂×C₅	#405	1	1	yes	characteristic
96	C₂×C₅	#405	1	1	yes	characteristic
97	D₅	—	4	1	no	—
98	C₂³	—	5	1	no	—
99	C₂×C₄	—	5	1	no	—
100	C₂×C₄	—	5	1	no	—
101	C₂×C₄	—	5	1	no	—
102	C₂×C₄	—	10	2	no	—
103	C₂×C₄	—	10	2	no	—
104	C₂×C₄	—	10	2	no	—
105	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
106	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
107	C₂×C₄	C₂×C₄×D₅	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	2	2	no	—
110	C₈	—	2	2	no	—
111	C₈	—	10	2	no	—
112	C₈	—	10	2	no	—
113	C₈	—	10	2	no	—
114	C₅	#1454	1	1	yes	characteristic, complemented, Sylow
115	C₂²	—	10	1	no	—
116	C₂²	—	10	1	no	—
117	C₂²	—	10	1	no	—
118	C₂²	C₄²×D₅	1	1	yes	characteristic
119	C₄	—	10	1	no	—
120	C₄	—	10	1	no	—
121	C₄	—	10	1	no	—
122	C₄	#4047	1	2	yes	—
123	C₄	#4053	1	2	yes	—
124	C₄	#4053	1	2	yes	—
125	C₂	—	20	1	no	—
126	C₂	#64978	1	1	yes	characteristic
127	C₂	#64997	1	1	yes	characteristic
128	C₂	#64997	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#64974

C₂

yes

maximal, characteristic

#64974

C₂

yes

maximal, characteristic

#64974

C₂

yes

maximal, characteristic

C₅×#322

C₂

yes

maximal, characteristic, complemented

#64675

C₂

yes

maximal, characteristic, complemented

#64674

C₂

yes

maximal, characteristic, complemented

C₈×#168

C₂

yes

maximal, characteristic, complemented

C₄²×D₅

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

characteristic

C₄×C₅×C₈

C₂²

yes

characteristic

#3952

C₄

yes

—

#3952

C₄

yes

—

#3952

C₄

yes

—

#3946

C₄

yes

—

#3946

C₄

yes

—

#3946

C₄

yes

—

C₄×#168

C₂²

yes

characteristic

C₄×#168

C₂²

yes

characteristic

C₄×#168

C₂²

yes

characteristic

#1454

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₅×C₈

C₂×C₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

—

C₄²×C₅

C₂³

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₂×#168

C₂×C₄

yes

—

C₂×#168

C₂×C₄

yes

—

C₂×#168

C₂×C₄

yes

—

—

—

—

—

—

D₅

yes

characteristic, complemented

C₈²

—

C₂²×D₅

C₄²

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄²

yes

characteristic

C₂×Dic₅

C₄²

yes

characteristic

C₂×Dic₅

C₄²

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

#168

—

#168

—

#168

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅

yes

characteristic

C₄×C₈

C₂×D₅

yes

characteristic

C₄×C₈

C₂×D₅

yes

characteristic

C₂²×C₅

C₂×C₄²

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

#131

yes

—

C₄×C₅

#131

yes

—

C₄×C₅

#131

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₅

yes

—

C₂×C₈

C₄×D₅

yes

—

C₂×C₈

C₄×D₅

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₅

yes

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

yes

D₅

—

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×C₄×D₅

yes

characteristic

106

C₂×C₄

C₂×C₄×D₅

yes

characteristic

107

C₂×C₄

C₂×C₄×D₅

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₅

#1454

yes

characteristic, complemented, Sylow

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₄²×D₅

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

#4047

yes

—

123

C₄

#4053

yes

—

124

C₄

#4053

yes

—

125

C₂

—

126

C₂

#64978

yes

characteristic

127

C₂

#64997

yes

characteristic

128

C₂

#64997

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₅
commutator, socle	C₂²×C₅	C₂×C₄²
Fitting	C₅×#322	C₂
prime radical	C₂²×D₅	C₄²
cosocle	C₄²×C₅	C₂³
hypercenter	#322	D₅
hypocenter	C₅	#1454
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₅

commutator, socle

C₂²×C₅

C₂×C₄²

Fitting

C₅×#322

C₂

prime radical

C₂²×D₅

C₄²

cosocle

C₄²×C₅

C₂³

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. C₅×#322	C₂	semidirect
5. #64675	C₂	semidirect
6. #64674	C₂	semidirect
7. C₈×#168	C₂	semidirect
40. #322	D₅	semidirect
114. C₅	#1454	semidirect

H ≅ G/N

type

4. C₅×#322

C₂

semidirect

5. #64675

C₂

semidirect

6. #64674

C₂

semidirect

7. C₈×#168

C₂

semidirect

40. #322

D₅

semidirect

114. C₅

#1454

semidirect

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

subgroup	index
#64974	2
C₅×#322	2
#64675	2
#64674	2
C₈×#168	2
#1454	5

subgroup

index

Subgroups — #10598867

Conjugacy classes of subgroups

Complements

Maximal subgroups