Group #65866 — subgroups

Subgroups — #65866

Order 320.

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

normal subgroups only

This group has 236 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×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#4076	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	#4041	C₂	1	1	yes	maximal, characteristic, complemented
5	#4039	C₂	1	1	yes	maximal, characteristic, complemented
6	#4038	C₂	1	1	yes	maximal, characteristic, complemented
7	#4034	C₂	1	1	yes	maximal, characteristic, complemented
8	#4031	C₂	1	2	yes	maximal, complemented
9	#4030	C₂	1	2	yes	maximal, complemented
10	#4021	C₂	1	2	yes	maximal, complemented
11	#4018	C₂	1	2	yes	maximal, complemented
12	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₅	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×D₂₀	C₂²	1	2	yes	—
17	C₂×C₄×D₅	C₂²	1	2	yes	—
18	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₀	C₂²	1	2	yes	—
20	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₄²×C₅	C₂²	1	2	yes	—
24	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
25	#685	C₂²	1	2	yes	—
26	#685	C₂²	1	2	yes	—
27	#685	C₂²	1	1	yes	characteristic
28	#685	C₂²	1	2	yes	—
29	#685	C₂²	1	2	yes	—
30	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
31	#683	C₂²	1	1	yes	characteristic
32	#683	C₂²	1	2	yes	—
33	C₄×Dic₅	C₂²	1	2	yes	—
34	C₄×Dic₅	C₂²	1	1	yes	characteristic
35	#555	—	5	1	no	maximal, Sylow
36	C₂³×C₅	C₂³	1	1	yes	characteristic
37	C₂²×D₅	C₂³	1	2	yes	—
38	C₂²×D₅	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅	—	2	2	no	—
40	C₂×C₄×C₅	C₂³	1	2	yes	—
41	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅	C₂³	1	2	yes	—
43	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
44	C₂²⋊D₅	—	2	2	no	—
45	C₂²⋊D₅	—	4	2	no	—
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	D₂₀	—	2	2	no	—
51	D₂₀	—	4	2	no	—
52	C₄×D₅	—	2	2	no	—
53	C₄×D₅	—	2	4	no	—
54	Dic₁₀	—	4	2	no	—
55	C₄⋊Q₈	—	5	1	no	—
56	#124	—	5	2	no	—
57	#124	—	5	2	no	—
58	#121	—	5	1	no	—
59	#121	—	5	1	no	—
60	#121	—	5	2	no	—
61	C₄×D₄	—	5	1	no	—
62	C₄×D₄	—	5	1	no	—
63	#117	—	5	1	no	—
64	#117	—	5	2	no	—
65	#117	D₅	1	1	yes	characteristic, complemented
66	C₂²×C₅	—	2	1	no	—
67	C₂²×C₅	—	2	1	no	—
68	C₂²×C₅	—	2	1	no	—
69	C₂²×C₅	C₂⁴	1	1	yes	characteristic
70	C₂×D₅	—	2	1	no	—
71	C₂×D₅	—	2	2	no	—
72	C₂×D₅	—	2	2	no	—
73	C₂×D₅	—	2	1	no	—
74	C₂×D₅	—	2	1	no	—
75	C₂×D₅	—	2	2	no	—
76	C₄×C₅	—	2	2	no	—
77	C₄×C₅	—	2	1	no	—
78	C₄×C₅	—	2	2	no	—
79	C₄×C₅	#41	1	2	yes	—
80	Dic₅	—	2	1	no	—
81	Dic₅	—	2	1	no	—
82	Dic₅	—	2	2	no	—
83	Dic₅	#41	1	2	yes	—
84	C₂×Q₈	—	5	2	no	—
85	C₂×D₄	—	5	1	no	—
86	C₂×D₄	—	5	2	no	—
87	C₂×D₄	—	5	1	no	—
88	C₂×D₄	—	5	2	no	—
89	C₂²×C₄	—	5	2	no	—
90	C₂²×C₄	—	5	1	no	—
91	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
92	C₄⋊C₄	—	5	1	no	—
93	C₄⋊C₄	—	5	2	no	—
94	C₄⋊C₄	—	5	1	no	—
95	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
97	C₂²⋊C₄	—	5	1	no	—
98	C₂²⋊C₄	—	5	2	no	—
99	C₂²⋊C₄	—	5	2	no	—
100	C₂²⋊C₄	—	5	2	no	—
101	C₂²⋊C₄	—	5	2	no	—
102	C₂²⋊C₄	—	5	1	no	—
103	C₂²⋊C₄	C₂×D₅	1	2	yes	—
104	C₄²	—	5	2	no	—
105	C₄²	—	5	1	no	—
106	C₄²	C₂×D₅	1	2	yes	—
107	C₂×C₅	—	4	1	no	—
108	C₂×C₅	C₂×#41	1	1	yes	characteristic
109	C₂×C₅	C₂×#41	1	1	yes	characteristic
110	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
111	D₅	—	4	2	no	—
112	D₅	—	4	1	no	—
113	C₂³	—	5	1	no	—
114	C₂³	—	5	2	no	—
115	C₂³	C₂²×D₅	1	1	yes	characteristic
116	Q₈	—	20	2	no	—
117	D₄	—	10	2	no	—
118	D₄	—	10	2	no	—
119	D₄	—	20	2	no	—
120	D₄	—	20	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	5	1	no	—
123	C₂×C₄	—	5	1	no	—
124	C₂×C₄	—	5	1	no	—
125	C₂×C₄	—	5	2	no	—
126	C₂×C₄	—	10	4	no	—
127	C₂×C₄	—	10	2	no	—
128	C₂×C₄	C₂²×D₅	1	2	yes	—
129	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
130	C₂×C₄	C₂²×D₅	1	2	yes	—
131	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
132	C₅	#555	1	1	yes	characteristic, complemented, Sylow
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	10	1	no	—
137	C₂²	—	10	2	no	—
138	C₂²	—	10	2	no	—
139	C₂²	—	10	1	no	—
140	C₂²	—	10	1	no	—
141	C₂²	—	10	2	no	—
142	C₂²	C₂³×D₅	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	5	2	no	—
147	C₄	—	10	1	no	—
148	C₄	—	10	2	no	—
149	C₄	—	10	1	no	—
150	C₄	#709	1	2	yes	—
151	C₂	—	4	1	no	—
152	C₂	—	20	1	no	—
153	C₂	—	20	2	no	—
154	C₂	C₂×#709	1	1	yes	characteristic
155	C₂	D₅×#41	1	1	yes	characteristic
156	C₂	#4149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#117

C₂

yes

maximal, characteristic, complemented

#4076

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₅

C₂

yes

maximal, characteristic, complemented

#4041

C₂

yes

maximal, characteristic, complemented

#4039

C₂

yes

maximal, characteristic, complemented

#4038

C₂

yes

maximal, characteristic, complemented

#4034

C₂

yes

maximal, characteristic, complemented

#4031

C₂

yes

maximal, complemented

#4030

C₂

yes

maximal, complemented

#4021

C₂

yes

maximal, complemented

#4018

C₂

yes

maximal, complemented

C₂²×C₄×C₅

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

—

C₂×D₂₀

C₂²

yes

characteristic

C₂×D₂₀

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

—

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

—

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₅×C₂²⋊C₄

C₂²

yes

—

C₄²×C₅

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

characteristic

#685

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

characteristic

#685

C₂²

yes

—

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

#683

C₂²

yes

characteristic

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

#555

—

maximal, Sylow

C₂³×C₅

C₂³

yes

characteristic

C₂²×D₅

C₂³

yes

—

C₂²×D₅

C₂³

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

D₂₀

—

D₂₀

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

C₄⋊Q₈

—

—

—

—

—

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

#117

D₅

yes

characteristic, complemented

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#41

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

#41

yes

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

C₂×D₅

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₅

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×D₅

yes

—

107

C₂×C₅

—

108

C₂×C₅

C₂×#41

yes

characteristic

109

C₂×C₅

C₂×#41

yes

characteristic

110

C₂×C₅

H(5,𝔽₂)

yes

characteristic

111

D₅

—

112

D₅

—

113

C₂³

—

114

C₂³

—

115

C₂³

C₂²×D₅

yes

characteristic

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²×D₅

yes

—

129

C₂×C₄

C₂²×D₅

yes

characteristic

130

C₂×C₄

C₂²×D₅

yes

—

131

C₂×C₄

C₂²×D₅

yes

characteristic

132

C₅

#555

yes

characteristic, complemented, Sylow

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₅

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#709

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

154

C₂

C₂×#709

yes

characteristic

155

C₂

D₅×#41

yes

characteristic

156

C₂

#4149

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#117	C₂
hypercenter	#117	D₅
hypocenter	C₅	#555
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₅×#117	C₂	semidirect
2. #4076	C₂	semidirect
3. C₄×C₂²⋊D₅	C₂	semidirect
4. #4041	C₂	semidirect
5. #4039	C₂	semidirect
6. #4038	C₂	semidirect
7. #4034	C₂	semidirect
8. #4031	C₂	semidirect
9. #4030	C₂	semidirect
10. #4021	C₂	semidirect
11. #4018	C₂	semidirect
65. #117	D₅	semidirect
132. C₅	#555	semidirect

H ≅ G/N

type

1. C₅×#117

C₂

semidirect

2. #4076

C₂

semidirect

3. C₄×C₂²⋊D₅

C₂

semidirect

4. #4041

C₂

semidirect

5. #4039

C₂

semidirect

6. #4038

C₂

semidirect

7. #4034

C₂

semidirect

8. #4031

C₂

semidirect

9. #4030

C₂

semidirect

10. #4021

C₂

semidirect

11. #4018

C₂

semidirect

65. #117

D₅

semidirect

132. C₅

#555

semidirect

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

subgroup	index
C₅×#117	2
#4076	2
C₄×C₂²⋊D₅	2
#4041	2
#4039	2
#4038	2
#4034	2
#4031	2
#4030	2
#4021	2
#4018	2
#555	5

subgroup

index

C₄×C₂²⋊D₅

Subgroups — #65866

Conjugacy classes of subgroups

Complements

Maximal subgroups