Group #66005 — subgroups

Subgroups — #66005

Order 320.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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
2	Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	#4090	C₂	1	1	yes	maximal, characteristic, complemented
4	#4082	C₂	1	1	yes	maximal, characteristic
5	D₄×Dic₅	C₂	1	1	yes	maximal, characteristic
6	#4025	C₂	1	2	yes	maximal, complemented
7	#4025	C₂	1	2	yes	maximal, complemented
8	#4024	C₂	1	2	yes	maximal, complemented
9	#4023	C₂	1	2	yes	maximal, complemented
10	C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅	C₂²	1	2	yes	—
15	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₅	C₂²	1	2	yes	—
21	C₂²⋊Dic₅	C₂²	1	2	yes	—
22	C₂²⋊Dic₅	C₂²	1	2	yes	—
23	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₅	C₂²	1	2	yes	—
26	#683	C₂²	1	1	yes	characteristic
27	#683	C₂²	1	2	yes	—
28	#683	C₂²	1	1	yes	characteristic
29	#683	C₂²	1	2	yes	—
30	#683	C₂²	1	2	yes	—
31	C₄×Dic₅	C₂²	1	2	yes	—
32	C₄×Dic₅	C₂²	1	1	yes	characteristic
33	C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₄×Dic₅	C₂²	1	2	yes	—
35	#556	—	5	1	no	maximal, Sylow
36	C₂³×C₅	C₂³	1	2	yes	—
37	C₅×Q₈	—	2	2	no	—
38	C₅×D₄	—	2	2	no	—
39	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂³	1	2	yes	—
41	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×Dic₅	—	2	4	no	—
44	C₂×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×Dic₅	C₂³	1	2	yes	—
46	C₂×Dic₅	C₂³	1	1	yes	characteristic
47	C₂×Dic₅	C₂³	1	2	yes	—
48	C₂×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×Dic₅	C₂³	1	1	yes	characteristic
50	Dic₁₀	—	2	2	no	—
51	Dic₁₀	—	2	2	no	—
52	C₄⋊Q₈	—	5	1	no	—
53	#126	—	5	2	no	—
54	#126	—	5	2	no	—
55	#124	—	5	1	no	—
56	#124	D₅	1	1	yes	characteristic
57	C₂²⋊Q₈	—	5	2	no	—
58	C₄×Q₈	—	5	1	no	—
59	C₄×Q₈	—	5	1	no	—
60	C₄×Q₈	—	5	1	no	—
61	C₄×D₄	—	5	1	no	—
62	#117	—	5	2	no	—
63	C₂²×C₅	—	2	2	no	—
64	C₂²×C₅	—	2	2	no	—
65	C₂²×C₅	—	2	2	no	—
66	C₂²×C₅	C₂⁴	1	1	yes	characteristic
67	C₄×C₅	—	2	1	no	—
68	C₄×C₅	—	2	2	no	—
69	C₄×C₅	—	2	1	no	—
70	C₄×C₅	#41	1	2	yes	—
71	Dic₅	—	2	2	no	—
72	Dic₅	—	2	1	no	—
73	Dic₅	—	2	2	no	—
74	Dic₅	—	2	1	no	—
75	Dic₅	—	2	1	no	—
76	Dic₅	#41	1	2	yes	—
77	C₂×Q₈	—	5	1	no	—
78	C₂×Q₈	—	5	1	no	—
79	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
80	C₂×D₄	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₄	—	5	2	no	—
82	C₄⋊C₄	—	5	1	no	—
83	C₄⋊C₄	—	5	2	no	—
84	C₄⋊C₄	—	5	1	no	—
85	C₄⋊C₄	—	5	2	no	—
86	C₄⋊C₄	—	5	2	no	—
87	C₄⋊C₄	—	5	1	no	—
88	C₄⋊C₄	—	5	1	no	—
89	C₄⋊C₄	—	5	2	no	—
90	C₂²⋊C₄	—	5	2	no	—
91	C₂²⋊C₄	—	5	2	no	—
92	C₂²⋊C₄	—	5	2	no	—
93	C₂²⋊C₄	C₂×D₅	1	2	yes	—
94	C₂²⋊C₄	C₂×D₅	1	2	yes	—
95	C₄²	—	5	2	no	—
96	C₄²	—	5	2	no	—
97	C₄²	—	5	1	no	—
98	C₄²	—	5	1	no	—
99	C₄²	C₂×D₅	1	1	yes	characteristic
100	C₂×C₅	—	4	2	no	—
101	C₂×C₅	C₂×#41	1	1	yes	characteristic
102	C₂×C₅	C₂×#41	1	1	yes	characteristic
103	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
104	C₂³	C₂²×D₅	1	2	yes	—
105	Q₈	—	2	2	no	—
106	Q₈	—	10	2	no	—
107	Q₈	—	10	2	no	—
108	D₄	—	2	2	no	—
109	C₂×C₄	—	5	1	no	—
110	C₂×C₄	—	5	1	no	—
111	C₂×C₄	—	5	2	no	—
112	C₂×C₄	—	5	1	no	—
113	C₂×C₄	—	5	2	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	10	4	no	—
116	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₅	1	2	yes	—
118	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₅	#556	1	1	yes	characteristic, complemented, Sylow
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂³×D₅	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	5	2	no	—
129	C₄	—	10	2	no	—
130	C₄	—	10	2	no	—
131	C₄	—	10	1	no	—
132	C₄	—	10	1	no	—
133	C₄	—	10	1	no	—
134	C₄	#711	1	2	yes	—
135	C₂	—	4	2	no	—
136	C₂	C₂×#711	1	1	yes	characteristic
137	C₂	D₅×#41	1	1	yes	characteristic
138	C₂	#4150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#124

C₂

yes

maximal, characteristic

Q₈×Dic₅

C₂

yes

maximal, characteristic, complemented

#4090

C₂

yes

maximal, characteristic, complemented

#4082

C₂

yes

maximal, characteristic

D₄×Dic₅

C₂

yes

maximal, characteristic

#4025

C₂

yes

maximal, complemented

#4025

C₂

yes

maximal, complemented

#4024

C₂

yes

maximal, complemented

#4023

C₂

yes

maximal, complemented

C₄×Dic₁₀

C₂

yes

maximal, characteristic, complemented

C₄×Dic₁₀

C₂

yes

maximal, characteristic, complemented

C₂×C₅×Q₈

C₂²

yes

characteristic

C₂×C₅×D₄

C₂²

yes

characteristic

C₂²×Dic₅

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

characteristic

C₅×C₂²⋊C₄

C₂²

yes

—

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

characteristic

C₂²⋊Dic₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

C₄⋊Dic₅

C₂²

yes

characteristic

C₄⋊Dic₅

C₂²

yes

—

#683

C₂²

yes

characteristic

#683

C₂²

yes

—

#683

C₂²

yes

characteristic

#683

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

—

#556

—

maximal, Sylow

C₂³×C₅

C₂³

yes

—

C₅×Q₈

—

C₅×D₄

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

Dic₁₀

—

Dic₁₀

—

C₄⋊Q₈

—

—

—

—

D₅

yes

characteristic

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂⁴

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#41

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅

yes

characteristic

C₂×D₄

C₂×D₅

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₅

yes

characteristic

100

C₂×C₅

—

101

C₂×C₅

C₂×#41

yes

characteristic

102

C₂×C₅

C₂×#41

yes

characteristic

103

C₂×C₅

2₋¹⁺⁴

yes

characteristic

104

C₂³

C₂²×D₅

yes

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²×D₅

yes

characteristic

117

C₂×C₄

C₂²×D₅

yes

—

118

C₂×C₄

C₂²×D₅

yes

characteristic

119

C₂×C₄

C₂²×D₅

yes

characteristic

120

C₅

#556

yes

characteristic, complemented, Sylow

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂³×D₅

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#711

yes

—

135

C₂

—

136

C₂

C₂×#711

yes

characteristic

137

C₂

D₅×#41

yes

characteristic

138

C₂

#4150

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#124	C₂
prime radical	C₂×C₅×D₄	C₂²
hypercenter	#124	D₅
hypocenter	C₅	#556
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

Fitting

C₅×#124

C₂

prime radical

C₂×C₅×D₄

C₂²

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. Q₈×Dic₅	C₂	semidirect
3. #4090	C₂	semidirect
6. #4025	C₂	semidirect
7. #4025	C₂	semidirect
8. #4024	C₂	semidirect
9. #4023	C₂	semidirect
10. C₄×Dic₁₀	C₂	semidirect
11. C₄×Dic₁₀	C₂	semidirect
120. C₅	#556	semidirect

H ≅ G/N

type

2. Q₈×Dic₅

C₂

semidirect

3. #4090

C₂

semidirect

6. #4025

C₂

semidirect

7. #4025

C₂

semidirect

8. #4024

C₂

semidirect

9. #4023

C₂

semidirect

10. C₄×Dic₁₀

C₂

semidirect

11. C₄×Dic₁₀

C₂

semidirect

120. C₅

#556

semidirect

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

subgroup	index
C₅×#124	2
Q₈×Dic₅	2
#4090	2
#4082	2
D₄×Dic₅	2
#4025	2
#4024	2
#4023	2
C₄×Dic₁₀	2
#556	5

subgroup

index

Q₈×Dic₅

D₄×Dic₅

C₄×Dic₁₀

Subgroups — #66005

Conjugacy classes of subgroups

Complements

Maximal subgroups