Group #66018 — subgroups

Subgroups — #66018

Order 320.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	#4092	C₂	1	2	yes	maximal, complemented
3	#4085	C₂	1	2	yes	maximal, complemented
4	#4032	C₂	1	2	yes	maximal, complemented
5	#4030	C₂	1	2	yes	maximal, complemented
6	#4029	C₂	1	2	yes	maximal, complemented
7	#4024	C₂	1	2	yes	maximal, complemented
8	#4021	C₂	1	1	yes	maximal, characteristic, complemented
9	#4016	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	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂²×Dic₅	C₂²	1	2	yes	—
15	C₂×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅	C₂²	1	2	yes	—
17	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₅	C₂²	1	2	yes	—
22	#685	C₂²	1	2	yes	—
23	#685	C₂²	1	2	yes	—
24	#685	C₂²	1	2	yes	—
25	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₅	C₂²	1	2	yes	—
28	#683	C₂²	1	2	yes	—
29	#683	C₂²	1	2	yes	—
30	#683	C₂²	1	2	yes	—
31	#562	—	5	1	no	maximal, Sylow
32	C₂³×C₅	C₂³	1	2	yes	—
33	C₂²×D₅	C₂³	1	2	yes	—
34	C₅×Q₈	—	4	1	no	—
35	C₅×D₄	—	4	1	no	—
36	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²⋊D₅	—	4	2	no	—
41	C₂²⋊D₅	—	4	2	no	—
42	C₂×Dic₅	—	4	2	no	—
43	C₂×Dic₅	C₂³	1	2	yes	—
44	C₂×Dic₅	C₂³	1	2	yes	—
45	C₂×Dic₅	C₂³	1	2	yes	—
46	D₂₀	—	4	1	no	—
47	C₄×D₅	—	4	2	no	—
48	Dic₁₀	—	4	1	no	—
49	#125	—	5	1	no	—
50	#124	—	5	1	no	—
51	#124	D₅	1	1	yes	characteristic, complemented
52	#123	—	5	2	no	—
53	#123	—	5	2	no	—
54	C₂²⋊Q₈	—	5	2	no	—
55	C₂²⋊Q₈	—	5	2	no	—
56	#121	—	5	2	no	—
57	#121	—	5	2	no	—
58	C₂²×C₅	—	2	2	no	—
59	C₂²×C₅	—	2	2	no	—
60	C₂²×C₅	—	2	2	no	—
61	C₂²×C₅	C₂⁴	1	1	yes	characteristic
62	C₂×D₅	—	2	2	no	—
63	C₂×D₅	—	2	2	no	—
64	C₂×D₅	—	2	2	no	—
65	C₄×C₅	—	2	1	no	—
66	C₄×C₅	—	2	2	no	—
67	C₄×C₅	—	2	1	no	—
68	C₄×C₅	—	2	1	no	—
69	Dic₅	—	2	2	no	—
70	Dic₅	—	2	2	no	—
71	Dic₅	—	2	2	no	—
72	C₂×Q₈	—	5	1	no	—
73	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
74	C₂×D₄	—	5	2	no	—
75	C₂×D₄	—	5	2	no	—
76	C₂×D₄	—	5	1	no	—
77	C₂×D₄	C₂×D₅	1	1	yes	characteristic
78	C₂²×C₄	—	5	2	no	—
79	C₂²×C₄	—	5	2	no	—
80	C₄⋊C₄	—	5	2	no	—
81	C₄⋊C₄	—	5	2	no	—
82	C₄⋊C₄	—	5	2	no	—
83	C₄⋊C₄	—	5	2	no	—
84	C₄⋊C₄	—	5	1	no	—
85	C₄⋊C₄	—	5	1	no	—
86	C₂²⋊C₄	—	5	2	no	—
87	C₂²⋊C₄	—	5	2	no	—
88	C₂²⋊C₄	—	5	2	no	—
89	C₂²⋊C₄	—	5	2	no	—
90	C₂²⋊C₄	C₂×D₅	1	2	yes	—
91	C₂²⋊C₄	C₂×D₅	1	2	yes	—
92	C₄²	C₂×D₅	1	1	yes	characteristic
93	C₂×C₅	—	4	2	no	—
94	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
95	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
96	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
97	D₅	—	4	2	no	—
98	C₂³	—	5	2	no	—
99	C₂³	C₂²×D₅	1	2	yes	—
100	Q₈	—	4	1	no	—
101	Q₈	—	20	1	no	—
102	D₄	—	4	1	no	—
103	D₄	—	20	1	no	—
104	D₄	—	20	2	no	—
105	D₄	—	20	2	no	—
106	C₂×C₄	—	5	2	no	—
107	C₂×C₄	—	5	2	no	—
108	C₂×C₄	—	5	2	no	—
109	C₂×C₄	—	20	2	no	—
110	C₂×C₄	—	20	2	no	—
111	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
112	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₅	1	2	yes	—
115	C₅	#562	1	1	yes	characteristic, complemented, Sylow
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	10	2	no	—
120	C₂²	—	10	2	no	—
121	C₂²	—	10	2	no	—
122	C₂²	C₂³×D₅	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	10	2	no	—
128	C₄	—	10	2	no	—
129	C₄	—	10	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	20	2	no	—
132	C₂	#4144	1	1	yes	characteristic
133	C₂	#4149	1	1	yes	characteristic
134	C₂	#4150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#124

C₂

yes

maximal, characteristic, complemented

#4092

C₂

yes

maximal, complemented

#4085

C₂

yes

maximal, complemented

#4032

C₂

yes

maximal, complemented

#4030

C₂

yes

maximal, complemented

#4029

C₂

yes

maximal, complemented

#4024

C₂

yes

maximal, complemented

#4021

C₂

yes

maximal, characteristic, complemented

#4016

C₂

yes

maximal, characteristic, complemented

C₂×C₅×Q₈

C₂²

yes

characteristic

C₂×C₅×D₄

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

—

C₂×C₂²⋊D₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

—

C₂×D₂₀

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

characteristic

C₅×C₂²⋊C₄

C₂²

yes

—

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

characteristic

C₂²⋊Dic₅

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

C₄⋊Dic₅

C₂²

yes

characteristic

C₄⋊Dic₅

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

#562

—

maximal, Sylow

C₂³×C₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

#125

—

#124

—

#124

D₅

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

C₂×D₅

yes

characteristic

C₂×C₅

—

C₂×C₅

yes

C₂×C₅

yes

C₂×C₅

2₋¹⁺⁴

yes

characteristic

D₅

—

C₂³

—

C₂³

C₂²×D₅

yes

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×D₅

yes

characteristic

112

C₂×C₄

C₂²×D₅

yes

characteristic

113

C₂×C₄

C₂²×D₅

yes

characteristic

114

C₂×C₄

C₂²×D₅

yes

—

115

C₅

#562

yes

characteristic, complemented, Sylow

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₅

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

132

C₂

#4144

yes

characteristic

133

C₂

#4149

yes

characteristic

134

C₂

#4150

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#124	C₂
hypercenter	#124	D₅
hypocenter	C₅	#562
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. C₅×#124	C₂	semidirect
2. #4092	C₂	semidirect
3. #4085	C₂	semidirect
4. #4032	C₂	semidirect
5. #4030	C₂	semidirect
6. #4029	C₂	semidirect
7. #4024	C₂	semidirect
8. #4021	C₂	semidirect
9. #4016	C₂	semidirect
51. #124	D₅	semidirect
115. C₅	#562	semidirect

H ≅ G/N

type

1. C₅×#124

C₂

semidirect

2. #4092

C₂

semidirect

3. #4085

C₂

semidirect

4. #4032

C₂

semidirect

5. #4030

C₂

semidirect

6. #4029

C₂

semidirect

7. #4024

C₂

semidirect

8. #4021

C₂

semidirect

9. #4016

C₂

semidirect

51. #124

D₅

semidirect

115. C₅

#562

semidirect

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

subgroup	index
C₅×#124	2
#4092	2
#4085	2
#4032	2
#4030	2
#4029	2
#4024	2
#4021	2
#4016	2
#562	5

subgroup

index

Subgroups — #66018

Conjugacy classes of subgroups

Complements

Maximal subgroups