Group #66025 — subgroups

Subgroups — #66025

Order 320.

234 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 99 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×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#4044	C₂	1	2	yes	maximal, complemented
3	#4043	C₂	1	1	yes	maximal, characteristic, complemented
4	#4041	C₂	1	1	yes	maximal, characteristic
5	#4040	C₂	1	2	yes	maximal, complemented
6	#4039	C₂	1	1	yes	maximal, characteristic, complemented
7	#4039	C₂	1	2	yes	maximal, complemented
8	#4038	C₂	1	2	yes	maximal, complemented
9	#4033	C₂	1	1	yes	maximal, characteristic, complemented
10	#4021	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄²×D₅	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₂₀	C₂²	1	1	yes	characteristic
13	C₂×D₂₀	C₂²	1	2	yes	—
14	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₅	C₂²	1	2	yes	—
18	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
19	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊C₄	C₂²	1	2	yes	—
23	C₄²×C₅	C₂²	1	1	yes	characteristic
24	#685	C₂²	1	2	yes	—
25	#685	C₂²	1	2	yes	—
26	#685	C₂²	1	2	yes	—
27	#685	C₂²	1	2	yes	—
28	#685	C₂²	1	2	yes	—
29	C₄⋊Dic₅	C₂²	1	2	yes	—
30	#683	C₂²	1	2	yes	—
31	C₄×Dic₅	C₂²	1	1	yes	characteristic
32	C₄×Dic₅	C₂²	1	1	yes	characteristic
33	C₄×Dic₅	C₂²	1	1	yes	characteristic
34	C₄×Dic₅	C₂²	1	2	yes	—
35	#529	—	5	1	no	maximal, Sylow
36	C₂²×D₅	C₂³	1	2	yes	—
37	C₂²×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅	C₂³	1	2	yes	—
39	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅	C₂³	1	2	yes	—
42	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂×Dic₅	C₂³	1	1	yes	characteristic
44	C₂×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×Dic₅	C₂³	1	2	yes	—
46	C₂×Dic₅	C₂³	1	1	yes	characteristic
47	D₂₀	—	2	4	no	—
48	D₂₀	—	2	2	no	—
49	C₄×D₅	—	2	2	no	—
50	C₄×D₅	—	2	2	no	—
51	C₄×D₅	—	2	2	no	—
52	C₄×D₅	—	2	4	no	—
53	Dic₁₀	—	2	2	no	—
54	#126	—	5	2	no	—
55	#125	D₅	1	1	yes	characteristic, complemented
56	#124	—	5	1	no	—
57	#123	—	5	2	no	—
58	C₂²⋊Q₈	—	5	1	no	—
59	#121	—	5	1	no	—
60	C₄×Q₈	—	5	1	no	—
61	C₄×D₄	—	5	1	no	—
62	C₄×D₄	—	5	2	no	—
63	#117	—	5	2	no	—
64	C₂×C₄²	—	5	1	no	—
65	C₂²×C₅	C₂⁴	1	1	yes	characteristic
66	C₂×D₅	—	2	1	no	—
67	C₂×D₅	—	2	1	no	—
68	C₂×D₅	—	2	2	no	—
69	C₂×D₅	—	2	2	no	—
70	C₂×D₅	—	2	2	no	—
71	C₂×D₅	#41	1	2	yes	—
72	C₄×C₅	—	2	1	no	—
73	C₄×C₅	—	2	1	no	—
74	C₄×C₅	—	2	2	no	—
75	C₄×C₅	—	2	2	no	—
76	C₄×C₅	#41	1	2	yes	—
77	Dic₅	—	2	2	no	—
78	Dic₅	—	2	1	no	—
79	Dic₅	—	2	1	no	—
80	Dic₅	#41	1	2	yes	—
81	C₂×Q₈	—	5	1	no	—
82	C₂×D₄	—	5	2	no	—
83	C₂×D₄	—	5	1	no	—
84	C₂²×C₄	—	5	1	no	—
85	C₂²×C₄	—	5	1	no	—
86	C₂²×C₄	—	5	1	no	—
87	C₂²×C₄	—	5	2	no	—
88	C₄⋊C₄	—	5	2	no	—
89	C₄⋊C₄	—	5	2	no	—
90	C₄⋊C₄	C₂×D₅	1	2	yes	—
91	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
92	C₄⋊C₄	C₂×D₅	1	2	yes	—
93	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
94	C₂²⋊C₄	—	5	2	no	—
95	C₂²⋊C₄	—	5	2	no	—
96	C₂²⋊C₄	—	5	2	no	—
97	C₂²⋊C₄	—	5	2	no	—
98	C₂²⋊C₄	—	5	2	no	—
99	C₄²	—	5	1	no	—
100	C₄²	—	5	1	no	—
101	C₄²	—	5	1	no	—
102	C₄²	—	5	2	no	—
103	C₄²	C₂×D₅	1	1	yes	characteristic
104	C₂×C₅	C₂×#41	1	1	yes	characteristic
105	C₂×C₅	C₂×#41	1	1	yes	characteristic
106	C₂×C₅	C₂×#41	1	1	yes	characteristic
107	D₅	—	2	2	no	—
108	D₅	—	4	2	no	—
109	C₂³	—	5	1	no	—
110	C₂³	—	5	2	no	—
111	Q₈	—	10	2	no	—
112	D₄	—	10	4	no	—
113	D₄	—	10	2	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	5	1	no	—
116	C₂×C₄	—	5	1	no	—
117	C₂×C₄	—	5	2	no	—
118	C₂×C₄	—	10	2	no	—
119	C₂×C₄	—	10	4	no	—
120	C₂×C₄	—	10	2	no	—
121	C₂×C₄	—	10	2	no	—
122	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₅	1	2	yes	—
125	C₂×C₄	C₂²×D₅	1	2	yes	—
126	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₅	#529	1	1	yes	characteristic, complemented, Sylow
128	C₂²	—	5	2	no	—
129	C₂²	—	10	1	no	—
130	C₂²	—	10	1	no	—
131	C₂²	—	10	2	no	—
132	C₂²	—	10	2	no	—
133	C₂²	—	10	2	no	—
134	C₂²	C₂³×D₅	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	2	no	—
139	C₄	—	5	2	no	—
140	C₄	—	10	1	no	—
141	C₄	—	10	2	no	—
142	C₄	—	10	1	no	—
143	C₄	#713	1	2	yes	—
144	C₂	—	10	2	no	—
145	C₂	—	20	2	no	—
146	C₂	C₂×#713	1	1	yes	characteristic
147	C₂	D₅×#41	1	1	yes	characteristic
148	C₂	D₅×#41	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#125

C₂

yes

maximal, characteristic, complemented

#4044

C₂

yes

maximal, complemented

#4043

C₂

yes

maximal, characteristic, complemented

#4041

C₂

yes

maximal, characteristic

#4040

C₂

yes

maximal, complemented

#4039

C₂

yes

maximal, characteristic, complemented

#4039

C₂

yes

maximal, complemented

#4038

C₂

yes

maximal, complemented

#4033

C₂

yes

maximal, characteristic, complemented

#4021

C₂

yes

maximal, characteristic, complemented

C₄²×D₅

C₂

yes

maximal, characteristic, complemented

C₂×D₂₀

C₂²

yes

characteristic

C₂×D₂₀

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

characteristic

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₅×C₄⋊C₄

C₂²

yes

—

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₅×C₄⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

characteristic

#685

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

characteristic

C₄×Dic₅

C₂²

yes

—

#529

—

maximal, Sylow

C₂²×D₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

#126

—

#125

D₅

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

#41

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#41

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

#41

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅

yes

—

C₄⋊C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

C₂×D₅

yes

—

C₄⋊C₄

C₂×D₅

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×D₅

yes

characteristic

104

C₂×C₅

C₂×#41

yes

characteristic

105

C₂×C₅

C₂×#41

yes

characteristic

106

C₂×C₅

C₂×#41

yes

characteristic

107

D₅

—

108

D₅

—

109

C₂³

—

110

C₂³

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂²×D₅

yes

characteristic

123

C₂×C₄

C₂²×D₅

yes

characteristic

124

C₂×C₄

C₂²×D₅

yes

—

125

C₂×C₄

C₂²×D₅

yes

—

126

C₂×C₄

C₂²×D₅

yes

characteristic

127

C₅

#529

yes

characteristic, complemented, Sylow

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂³×D₅

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#713

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#713

yes

characteristic

147

C₂

D₅×#41

yes

characteristic

148

C₂

D₅×#41

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#125	C₂
prime radical	#4041	C₂
hypercenter	#125	D₅
hypocenter	C₅	#529
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₅×#125	C₂	semidirect
2. #4044	C₂	semidirect
3. #4043	C₂	semidirect
5. #4040	C₂	semidirect
6. #4039	C₂	semidirect
7. #4039	C₂	semidirect
8. #4038	C₂	semidirect
9. #4033	C₂	semidirect
10. #4021	C₂	semidirect
11. C₄²×D₅	C₂	semidirect
55. #125	D₅	semidirect
127. C₅	#529	semidirect

H ≅ G/N

type

1. C₅×#125

C₂

semidirect

2. #4044

C₂

semidirect

3. #4043

C₂

semidirect

5. #4040

C₂

semidirect

6. #4039

C₂

semidirect

7. #4039

C₂

semidirect

8. #4038

C₂

semidirect

9. #4033

C₂

semidirect

10. #4021

C₂

semidirect

11. C₄²×D₅

C₂

semidirect

55. #125

D₅

semidirect

127. C₅

#529

semidirect

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

subgroup	index
C₅×#125	2
#4044	2
#4043	2
#4041	2
#4040	2
#4039	2
#4038	2
#4033	2
#4021	2
C₄²×D₅	2
#529	5

subgroup

index

C₄²×D₅

Subgroups — #66025

Conjugacy classes of subgroups

Complements

Maximal subgroups