Group #66009 — subgroups

Subgroups — #66009

Order 320.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×Q₈×D₅	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#711	C₂	1	1	yes	maximal, characteristic
3	C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	#4090	C₂	1	1	yes	maximal, characteristic, complemented
5	#4082	C₂	1	1	yes	maximal, characteristic, complemented
6	#4029	C₂	1	4	yes	maximal, complemented
7	#4024	C₂	1	4	yes	maximal, complemented
8	#4018	C₂	1	1	yes	maximal, characteristic, complemented
9	#4015	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
13	C₂²×Dic₅	C₂²	1	2	yes	—
14	Q₈×D₅	—	2	4	no	—
15	#711	—	2	4	no	—
16	C₂×C₄×D₅	C₂²	1	2	yes	—
17	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₀	C₂²	1	2	yes	—
21	C₅×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₂²⋊Dic₅	C₂²	1	4	yes	—
24	#685	C₂²	1	2	yes	—
25	C₄⋊Dic₅	C₂²	1	4	yes	—
26	#683	C₂²	1	2	yes	—
27	#683	C₂²	1	4	yes	—
28	C₄×Dic₅	C₂²	1	1	yes	characteristic
29	#536	—	5	1	no	maximal, Sylow
30	C₂³×C₅	C₂³	1	2	yes	—
31	C₂²×D₅	C₂³	1	1	yes	characteristic
32	C₅×Q₈	—	2	2	no	—
33	C₅×D₄	—	2	2	no	—
34	C₂×C₄×C₅	C₂³	1	2	yes	—
35	C₂×C₄×C₅	C₂³	1	2	yes	—
36	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²⋊D₅	—	2	4	no	—
38	C₂×Dic₅	—	2	4	no	—
39	C₂×Dic₅	C₂³	1	2	yes	—
40	C₂×Dic₅	C₂³	1	2	yes	—
41	C₂×Dic₅	C₂³	1	2	yes	—
42	C₂×Dic₅	C₂³	1	1	yes	characteristic
43	C₄×D₅	—	2	4	no	—
44	C₄×D₅	D₄	1	4	yes	—
45	Dic₁₀	—	2	2	no	—
46	Dic₁₀	—	2	2	no	—
47	Dic₁₀	—	2	4	no	—
48	C₂×#41	—	5	1	no	—
49	C₂²×Q₈	—	5	1	no	—
50	C₄⋊Q₈	—	5	1	no	—
51	C₄⋊Q₈	—	5	1	no	—
52	#124	—	5	1	no	—
53	#124	D₅	1	1	yes	characteristic, complemented
54	#123	—	5	4	no	—
55	C₂²⋊Q₈	—	5	4	no	—
56	#117	—	5	1	no	—
57	C₂²×C₅	—	2	2	no	—
58	C₂²×C₅	—	2	4	no	—
59	C₂²×C₅	C₂⁴	1	1	yes	characteristic
60	C₂×D₅	—	2	2	no	—
61	C₂×D₅	C₂×D₄	1	2	yes	—
62	C₄×C₅	—	2	2	no	—
63	C₄×C₅	—	2	2	no	—
64	C₄×C₅	C₂×D₄	1	2	yes	—
65	Dic₅	—	2	2	no	—
66	Dic₅	—	2	2	no	—
67	Dic₅	—	2	2	no	—
68	Dic₅	C₂×D₄	1	2	yes	—
69	#41	—	10	4	no	—
70	C₂×Q₈	—	5	2	no	—
71	C₂×Q₈	—	5	1	no	—
72	C₂×Q₈	—	5	1	no	—
73	C₂×Q₈	—	10	4	no	—
74	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
75	C₂×D₄	—	5	2	no	—
76	C₂×D₄	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄	—	5	1	no	—
78	C₂²×C₄	—	5	2	no	—
79	C₂²×C₄	—	5	2	no	—
80	C₄⋊C₄	—	5	2	no	—
81	C₄⋊C₄	—	5	4	no	—
82	C₄⋊C₄	—	5	4	no	—
83	C₂²⋊C₄	—	5	4	no	—
84	C₂²⋊C₄	—	5	2	no	—
85	C₂²⋊C₄	C₂×D₅	1	4	yes	—
86	C₄²	—	5	1	no	—
87	C₄²	C₂×D₅	1	1	yes	characteristic
88	C₂×C₅	—	4	2	no	—
89	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₅	2₋¹⁺⁴	1	2	yes	—
91	D₅	—	2	2	no	—
92	C₂³	—	5	1	no	—
93	C₂³	C₂²×D₅	1	2	yes	—
94	Q₈	—	2	2	no	—
95	Q₈	—	10	4	no	—
96	Q₈	—	10	2	no	—
97	Q₈	—	10	2	no	—
98	D₄	—	2	2	no	—
99	D₄	—	10	4	no	—
100	C₂×C₄	—	5	1	no	—
101	C₂×C₄	—	5	4	no	—
102	C₂×C₄	—	5	2	no	—
103	C₂×C₄	—	5	2	no	—
104	C₂×C₄	—	5	2	no	—
105	C₂×C₄	—	10	4	no	—
106	C₂×C₄	—	10	4	no	—
107	C₂×C₄	C₂²×D₅	1	2	yes	—
108	C₂×C₄	C₂²×D₅	1	2	yes	—
109	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
110	C₅	#536	1	1	yes	characteristic, complemented, Sylow
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	5	2	no	—
114	C₂²	—	10	2	no	—
115	C₂²	C₂³×D₅	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	5	2	no	—
119	C₄	—	10	2	no	—
120	C₄	—	10	2	no	—
121	C₄	—	10	2	no	—
122	C₄	D₄×D₅	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	10	2	no	—
125	C₂	C₂×D₄×D₅	1	1	yes	characteristic
126	C₂	#4150	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₅

C₂

yes

maximal, characteristic, complemented

C₂×#711

C₂

yes

maximal, characteristic

C₅×#124

C₂

yes

maximal, characteristic, complemented

#4090

C₂

yes

maximal, characteristic, complemented

#4082

C₂

yes

maximal, characteristic, complemented

#4029

C₂

yes

maximal, complemented

#4024

C₂

yes

maximal, complemented

#4018

C₂

yes

maximal, characteristic, complemented

#4015

C₂

yes

maximal, characteristic, complemented

C₂×C₅×Q₈

C₂²

yes

characteristic

C₂×C₅×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

—

Q₈×D₅

—

#711

—

C₂×C₄×D₅

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

—

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

characteristic

C₂²⋊Dic₅

C₂²

yes

—

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

#536

—

maximal, Sylow

C₂³×C₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

characteristic

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₄×D₅

—

C₄×D₅

D₄

yes

—

Dic₁₀

—

Dic₁₀

—

Dic₁₀

—

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

D₅

yes

characteristic, complemented

#123

—

C₂²⋊Q₈

—

#117

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

C₂×D₄

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂×D₄

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂×D₄

yes

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

—

C₄²

C₂×D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

C₂²×D₄

yes

characteristic

C₂×C₅

2₋¹⁺⁴

yes

—

D₅

—

C₂³

—

C₂³

C₂²×D₅

yes

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²×D₅

yes

—

108

C₂×C₄

C₂²×D₅

yes

—

109

C₂×C₄

C₂²×D₅

yes

characteristic

110

C₅

#536

yes

characteristic, complemented, Sylow

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂³×D₅

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₄×D₅

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₂×D₄×D₅

yes

characteristic

126

C₂

#4150

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#124	C₂
prime radical	C₂×#711	C₂
hypercenter	#124	D₅
hypocenter	C₅	#536
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₂×Q₈×D₅	C₂	semidirect
3. C₅×#124	C₂	semidirect
4. #4090	C₂	semidirect
5. #4082	C₂	semidirect
6. #4029	C₂	semidirect
7. #4024	C₂	semidirect
8. #4018	C₂	semidirect
9. #4015	C₂	semidirect
53. #124	D₅	semidirect
110. C₅	#536	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₅

C₂

semidirect

3. C₅×#124

C₂

semidirect

4. #4090

C₂

semidirect

5. #4082

C₂

semidirect

6. #4029

C₂

semidirect

7. #4024

C₂

semidirect

8. #4018

C₂

semidirect

9. #4015

C₂

semidirect

53. #124

D₅

semidirect

110. C₅

#536

semidirect

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

subgroup	index
C₂×Q₈×D₅	2
C₂×#711	2
C₅×#124	2
#4090	2
#4082	2
#4029	2
#4024	2
#4018	2
#4015	2
#536	5

subgroup

index

C₂×Q₈×D₅

Subgroups — #66009

Conjugacy classes of subgroups

Complements

Maximal subgroups