Group #65867 — subgroups

Subgroups — #65867

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×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	#4070	C₂	1	1	yes	maximal, characteristic, complemented
4	#4043	C₂	1	1	yes	maximal, characteristic, complemented
5	#4038	C₂	1	1	yes	maximal, characteristic, complemented
6	#4034	C₂	1	1	yes	maximal, characteristic, complemented
7	#4033	C₂	1	1	yes	maximal, characteristic, complemented
8	#4031	C₂	1	2	yes	maximal
9	#4025	C₂	1	2	yes	maximal, complemented
10	#4022	C₂	1	2	yes	maximal, complemented
11	C₄×Dic₁₀	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×C₄×D₅	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₀	C₂²	1	2	yes	—
16	C₂×Dic₁₀	C₂²	1	1	yes	characteristic
17	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₄²×C₅	C₂²	1	2	yes	—
21	C₂²⋊Dic₅	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₅	C₂²	1	2	yes	—
23	#685	C₂²	1	1	yes	characteristic
24	#685	C₂²	1	2	yes	—
25	#685	C₂²	1	2	yes	—
26	C₄⋊Dic₅	C₂²	1	2	yes	—
27	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
28	#683	C₂²	1	1	yes	characteristic
29	#683	C₂²	1	2	yes	—
30	#683	C₂²	1	2	yes	—
31	#683	C₂²	1	2	yes	—
32	C₄×Dic₅	C₂²	1	2	yes	—
33	C₄×Dic₅	C₂²	1	2	yes	—
34	C₄×Dic₅	C₂²	1	1	yes	characteristic
35	#556	—	5	1	no	maximal, Sylow
36	C₂³×C₅	C₂³	1	1	yes	characteristic
37	C₂²×D₅	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅	—	2	2	no	—
39	C₂×C₄×C₅	C₂³	1	2	yes	—
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	C₂²⋊D₅	—	2	2	no	—
44	C₂×Dic₅	C₂³	1	1	yes	characteristic
45	C₂×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×Dic₅	C₂³	1	2	yes	—
47	C₂×Dic₅	C₂³	1	2	yes	—
48	C₂×Dic₅	C₂³	1	1	yes	characteristic
49	C₄×D₅	—	2	2	no	—
50	Dic₁₀	—	2	2	no	—
51	Dic₁₀	—	2	4	no	—
52	C₄⋊Q₈	—	5	1	no	—
53	#126	—	5	2	no	—
54	#126	—	5	2	no	—
55	#124	—	5	2	no	—
56	C₂²⋊Q₈	—	5	1	no	—
57	C₂²⋊Q₈	—	5	1	no	—
58	C₄×Q₈	—	5	1	no	—
59	C₄×Q₈	—	5	2	no	—
60	C₄×D₄	—	5	1	no	—
61	#117	—	5	1	no	—
62	#117	D₅	1	1	yes	characteristic, complemented
63	C₂²×C₅	—	2	1	no	—
64	C₂²×C₅	—	2	1	no	—
65	C₂²×C₅	—	2	1	no	—
66	C₂²×C₅	C₂⁴	1	1	yes	characteristic
67	C₂×D₅	—	2	1	no	—
68	C₂×D₅	—	2	1	no	—
69	C₂×D₅	—	2	1	no	—
70	C₄×C₅	—	2	2	no	—
71	C₄×C₅	—	2	1	no	—
72	C₄×C₅	—	2	2	no	—
73	C₄×C₅	#41	1	2	yes	—
74	Dic₅	—	2	2	no	—
75	Dic₅	—	2	2	no	—
76	Dic₅	—	2	1	no	—
77	Dic₅	—	2	1	no	—
78	Dic₅	#41	1	2	yes	—
79	C₂×Q₈	—	5	1	no	—
80	C₂×Q₈	—	5	2	no	—
81	C₂×D₄	—	5	1	no	—
82	C₂²×C₄	—	5	1	no	—
83	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
84	C₄⋊C₄	—	5	1	no	—
85	C₄⋊C₄	—	5	1	no	—
86	C₄⋊C₄	—	5	2	no	—
87	C₄⋊C₄	—	5	2	no	—
88	C₄⋊C₄	—	5	2	no	—
89	C₄⋊C₄	—	5	2	no	—
90	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
91	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
92	C₂²⋊C₄	—	5	1	no	—
93	C₂²⋊C₄	—	5	2	no	—
94	C₂²⋊C₄	—	5	2	no	—
95	C₂²⋊C₄	—	5	1	no	—
96	C₂²⋊C₄	—	5	2	no	—
97	C₂²⋊C₄	C₂×D₅	1	2	yes	—
98	C₄²	—	5	2	no	—
99	C₄²	—	5	1	no	—
100	C₄²	—	5	2	no	—
101	C₄²	C₂×D₅	1	2	yes	—
102	C₂×C₅	—	4	1	no	—
103	C₂×C₅	C₂×#41	1	1	yes	characteristic
104	C₂×C₅	C₂×#41	1	1	yes	characteristic
105	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
106	D₅	—	4	1	no	—
107	C₂³	—	5	1	no	—
108	C₂³	C₂²×D₅	1	1	yes	characteristic
109	Q₈	—	10	4	no	—
110	Q₈	—	10	2	no	—
111	D₄	—	10	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	5	1	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	5	1	no	—
116	C₂×C₄	—	5	2	no	—
117	C₂×C₄	—	5	2	no	—
118	C₂×C₄	—	10	2	no	—
119	C₂×C₄	C₂²×D₅	1	2	yes	—
120	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₅	1	2	yes	—
122	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
123	C₅	#556	1	1	yes	characteristic, complemented, Sylow
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	10	1	no	—
128	C₂²	—	10	1	no	—
129	C₂²	—	10	1	no	—
130	C₂²	C₂³×D₅	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	5	2	no	—
135	C₄	—	10	1	no	—
136	C₄	—	10	2	no	—
137	C₄	—	10	2	no	—
138	C₄	—	10	1	no	—
139	C₄	#709	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	20	1	no	—
142	C₂	C₂×#709	1	1	yes	characteristic
143	C₂	D₅×#41	1	1	yes	characteristic
144	C₂	#4150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#117

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₅

C₂

yes

maximal, characteristic

#4070

C₂

yes

maximal, characteristic, complemented

#4043

C₂

yes

maximal, characteristic, complemented

#4038

C₂

yes

maximal, characteristic, complemented

#4034

C₂

yes

maximal, characteristic, complemented

#4033

C₂

yes

maximal, characteristic, complemented

#4031

C₂

yes

maximal

#4025

C₂

yes

maximal, complemented

#4022

C₂

yes

maximal, complemented

C₄×Dic₁₀

C₂

yes

maximal, complemented

C₂²×C₄×C₅

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

characteristic

C₂×Dic₁₀

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

characteristic

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

C₂²⋊Dic₅

C₂²

yes

—

#685

C₂²

yes

characteristic

#685

C₂²

yes

—

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

#683

C₂²

yes

characteristic

#683

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

C₄×Dic₅

C₂²

yes

—

C₄×Dic₅

C₂²

yes

—

C₄×Dic₅

C₂²

yes

characteristic

#556

—

maximal, Sylow

C₂³×C₅

C₂³

yes

characteristic

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

—

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

characteristic

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

characteristic

C₄×D₅

—

Dic₁₀

—

Dic₁₀

—

C₄⋊Q₈

—

#126

—

#126

—

#124

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

D₅

yes

characteristic, complemented

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

#41

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

C₂×D₅

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅

yes

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×D₅

yes

—

102

C₂×C₅

—

103

C₂×C₅

C₂×#41

yes

characteristic

104

C₂×C₅

C₂×#41

yes

characteristic

105

C₂×C₅

2₋¹⁺⁴

yes

characteristic

106

D₅

—

107

C₂³

—

108

C₂³

C₂²×D₅

yes

characteristic

109

Q₈

—

110

Q₈

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×D₅

yes

—

120

C₂×C₄

C₂²×D₅

yes

characteristic

121

C₂×C₄

C₂²×D₅

yes

—

122

C₂×C₄

C₂²×D₅

yes

characteristic

123

C₅

#556

yes

characteristic, complemented, Sylow

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂³×D₅

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#709

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂×#709

yes

characteristic

143

C₂

D₅×#41

yes

characteristic

144

C₂

#4150

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#117	C₂
prime radical	C₂×C₂²⋊D₅	C₂²
hypercenter	#117	D₅
hypocenter	C₅	#556
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

Fitting

C₅×#117

C₂

prime radical

C₂×C₂²⋊D₅

C₂²

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₅×#117	C₂	semidirect
3. #4070	C₂	semidirect
4. #4043	C₂	semidirect
5. #4038	C₂	semidirect
6. #4034	C₂	semidirect
7. #4033	C₂	semidirect
9. #4025	C₂	semidirect
10. #4022	C₂	semidirect
11. C₄×Dic₁₀	C₂	semidirect
62. #117	D₅	semidirect
123. C₅	#556	semidirect

H ≅ G/N

type

1. C₅×#117

C₂

semidirect

3. #4070

C₂

semidirect

4. #4043

C₂

semidirect

5. #4038

C₂

semidirect

6. #4034

C₂

semidirect

7. #4033

C₂

semidirect

9. #4025

C₂

semidirect

10. #4022

C₂

semidirect

11. C₄×Dic₁₀

C₂

semidirect

62. #117

D₅

semidirect

123. C₅

#556

semidirect

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

subgroup	index
C₅×#117	2
C₄×C₂²⋊D₅	2
#4070	2
#4043	2
#4038	2
#4034	2
#4033	2
#4031	2
#4025	2
#4022	2
C₄×Dic₁₀	2
#556	5

subgroup

index

C₄×C₂²⋊D₅

C₄×Dic₁₀

Subgroups — #65867

Conjugacy classes of subgroups

Complements

Maximal subgroups