Group #65931 — subgroups

Subgroups — #65931

Order 320.

294 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 103 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	C₂×#711	C₂	1	2	yes	maximal, complemented
2	C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	#4085	C₂	1	2	yes	maximal, complemented
4	#4085	C₂	1	2	yes	maximal, complemented
5	#4076	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#683	C₂	1	1	yes	maximal, characteristic, complemented
7	#4042	C₂	1	2	yes	maximal, complemented
8	#4030	C₂	1	2	yes	maximal, complemented
9	#4024	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×D₄	C₂²	1	2	yes	—
11	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
14	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₅	C₂²	1	2	yes	—
16	C₂²×Dic₅	C₂²	1	2	yes	—
17	C₂²×Dic₅	C₂²	1	1	yes	characteristic
18	C₂²×Dic₅	C₂²	1	1	yes	characteristic
19	#711	—	2	8	no	—
20	C₂×D₂₀	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₅	C₂²	1	2	yes	—
22	C₂×Dic₁₀	C₂²	1	2	yes	—
23	C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂²⋊Dic₅	C₂²	1	2	yes	—
26	#685	C₂²	1	2	yes	—
27	#685	C₂²	1	2	yes	—
28	C₄⋊Dic₅	C₂²	1	1	yes	characteristic
29	#683	C₂²	1	2	yes	—
30	#683	C₂²	1	2	yes	—
31	#683	C₂²	1	2	yes	—
32	#537	—	5	1	no	maximal, Sylow
33	C₂³×C₅	C₂³	1	2	yes	—
34	C₂³×C₅	C₂³	1	1	yes	characteristic
35	C₂²×D₅	C₂³	1	2	yes	—
36	C₅×D₄	—	2	4	no	—
37	C₅×D₄	—	4	1	no	—
38	C₂×C₄×C₅	—	4	1	no	—
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	C₂²⋊D₅	—	2	4	no	—
43	C₂²⋊D₅	—	2	4	no	—
44	C₂²⋊D₅	—	4	2	no	—
45	C₂×Dic₅	—	2	4	no	—
46	C₂×Dic₅	—	4	1	no	—
47	C₂×Dic₅	D₄	1	4	yes	complemented
48	C₂×Dic₅	C₂³	1	2	yes	—
49	C₂×Dic₅	C₂³	1	2	yes	—
50	C₂×Dic₅	C₂³	1	2	yes	—
51	D₂₀	—	4	1	no	—
52	C₄×D₅	—	2	4	no	—
53	Dic₁₀	—	2	4	no	—
54	C₂×#41	—	5	2	no	—
55	C₂²⋊Q₈	—	5	2	no	—
56	C₂²⋊Q₈	—	5	2	no	—
57	#121	—	5	2	no	—
58	#121	—	5	2	no	—
59	#121	—	5	1	no	—
60	#121	—	5	2	no	—
61	#121	D₅	1	1	yes	characteristic, complemented
62	C₂×C₄⋊C₄	—	5	1	no	—
63	C₂²×C₅	—	2	1	no	—
64	C₂²×C₅	—	2	1	no	—
65	C₂²×C₅	—	2	2	no	—
66	C₂²×C₅	—	2	2	no	—
67	C₂²×C₅	—	2	2	no	—
68	C₂²×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₅	C₂⁴	1	1	yes	characteristic
70	C₂×D₅	—	2	2	no	—
71	C₂×D₅	—	2	2	no	—
72	C₂×D₅	—	2	2	no	—
73	C₄×C₅	—	2	2	no	—
74	C₄×C₅	—	2	1	no	—
75	C₄×C₅	—	2	1	no	—
76	Dic₅	—	2	2	no	—
77	Dic₅	—	2	2	no	—
78	Dic₅	C₂×D₄	1	4	yes	—
79	#41	—	10	8	no	—
80	C₂×Q₈	—	5	2	no	—
81	C₂×D₄	—	5	2	no	—
82	C₂×D₄	—	5	2	no	—
83	C₂×D₄	—	5	2	no	—
84	C₂×D₄	—	5	1	no	—
85	C₂×D₄	C₂×D₅	1	2	yes	—
86	C₂×D₄	C₂×D₅	1	1	yes	characteristic
87	C₂²×C₄	—	5	1	no	—
88	C₂²×C₄	—	5	2	no	—
89	C₂²×C₄	—	5	1	no	—
90	C₂²×C₄	—	5	2	no	—
91	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
92	C₄⋊C₄	—	5	2	no	—
93	C₄⋊C₄	—	5	2	no	—
94	C₄⋊C₄	—	5	2	no	—
95	C₄⋊C₄	—	5	1	no	—
96	C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
97	C₂²⋊C₄	—	5	2	no	—
98	C₂²⋊C₄	—	5	2	no	—
99	C₂²⋊C₄	—	5	2	no	—
100	C₂²⋊C₄	C₂×D₅	1	2	yes	—
101	C₂×C₅	—	2	2	no	—
102	C₂×C₅	—	4	2	no	—
103	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₅	H(5,𝔽₂)	1	1	yes	characteristic
105	C₂×C₅	2₋¹⁺⁴	1	1	yes	characteristic
106	D₅	—	4	2	no	—
107	C₂³	—	5	2	no	—
108	C₂³	C₂²×D₅	1	2	yes	—
109	C₂³	C₂²×D₅	1	1	yes	characteristic
110	Q₈	—	10	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	1	no	—
113	D₄	—	10	4	no	—
114	D₄	—	10	4	no	—
115	D₄	—	20	1	no	—
116	D₄	—	20	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	5	4	no	—
119	C₂×C₄	—	5	2	no	—
120	C₂×C₄	—	5	2	no	—
121	C₂×C₄	—	5	2	no	—
122	C₂×C₄	—	10	4	no	—
123	C₂×C₄	—	10	4	no	—
124	C₂×C₄	—	20	1	no	—
125	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₅	1	2	yes	—
127	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₅	#537	1	1	yes	characteristic, complemented, Sylow
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	10	2	no	—
135	C₂²	—	10	2	no	—
136	C₂²	—	10	2	no	—
137	C₂²	D₄×D₅	1	2	yes	—
138	C₂²	C₂³×D₅	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	5	4	no	—
143	C₄	—	10	2	no	—
144	C₄	—	10	2	no	—
145	C₂	—	2	2	no	—
146	C₂	—	4	2	no	—
147	C₂	—	20	2	no	—
148	C₂	C₂×D₄×D₅	1	1	yes	characteristic
149	C₂	#4144	1	1	yes	characteristic
150	C₂	#4150	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#711

C₂

yes

maximal, complemented

C₅×#121

C₂

yes

maximal, characteristic, complemented

#4085

C₂

yes

maximal, complemented

#4085

C₂

yes

maximal, complemented

#4076

C₂

yes

maximal, characteristic, complemented

C₂×#683

C₂

yes

maximal, characteristic, complemented

#4042

C₂

yes

maximal, complemented

#4030

C₂

yes

maximal, complemented

#4024

C₂

yes

maximal, complemented

C₂×C₅×D₄

C₂²

yes

—

C₂×C₅×D₄

C₂²

yes

characteristic

C₂²×C₄×C₅

C₂²

yes

characteristic

C₂×C₂²⋊D₅

C₂²

yes

—

C₂×C₂²⋊D₅

C₂²

yes

—

C₂×C₂²⋊D₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

—

C₂²×Dic₅

C₂²

yes

characteristic

C₂²×Dic₅

C₂²

yes

characteristic

#711

—

C₂×D₂₀

C₂²

yes

characteristic

C₂×C₄×D₅

C₂²

yes

—

C₂×Dic₁₀

C₂²

yes

—

C₅×C₄⋊C₄

C₂²

yes

characteristic

C₅×C₂²⋊C₄

C₂²

yes

—

C₂²⋊Dic₅

C₂²

yes

—

#685

C₂²

yes

—

#685

C₂²

yes

—

C₄⋊Dic₅

C₂²

yes

characteristic

#683

C₂²

yes

—

#683

C₂²

yes

—

#683

C₂²

yes

—

#537

—

maximal, Sylow

C₂³×C₅

C₂³

yes

—

C₂³×C₅

C₂³

yes

characteristic

C₂²×D₅

C₂³

yes

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

characteristic

C₂×C₄×C₅

C₂³

yes

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂²⋊D₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

D₄

yes

complemented

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

C₂×Dic₅

C₂³

yes

—

D₂₀

—

C₄×D₅

—

Dic₁₀

—

C₂×#41

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

—

—

—

—

D₅

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×D₄

yes

—

C₂²×C₅

C₂⁴

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂×D₄

yes

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅

yes

—

C₂×D₄

C₂×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

C₂×D₅

yes

—

101

C₂×C₅

—

102

C₂×C₅

—

103

C₂×C₅

C₂²×D₄

yes

characteristic

104

C₂×C₅

H(5,𝔽₂)

yes

characteristic

105

C₂×C₅

2₋¹⁺⁴

yes

characteristic

106

D₅

—

107

C₂³

—

108

C₂³

C₂²×D₅

yes

—

109

C₂³

C₂²×D₅

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

115

D₄

—

116

D₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂²×D₅

yes

characteristic

126

C₂×C₄

C₂²×D₅

yes

—

127

C₂×C₄

C₂²×D₅

yes

characteristic

128

C₅

#537

yes

characteristic, complemented, Sylow

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

D₄×D₅

yes

—

138

C₂²

C₂³×D₅

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂×D₄×D₅

yes

characteristic

149

C₂

#4144

yes

characteristic

150

C₂

#4150

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁴
Fitting	C₅×#121	C₂
hypercenter	#121	D₅
hypocenter	C₅	#537
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅

commutator, socle, cosocle

C₂²×C₅

C₂⁴

C₂

D₅

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. C₂×#711	C₂	semidirect
2. C₅×#121	C₂	semidirect
3. #4085	C₂	semidirect
4. #4085	C₂	semidirect
5. #4076	C₂	semidirect
6. C₂×#683	C₂	semidirect
7. #4042	C₂	semidirect
8. #4030	C₂	semidirect
9. #4024	C₂	semidirect
47. C₂×Dic₅	D₄	semidirect
61. #121	D₅	semidirect
128. C₅	#537	semidirect

H ≅ G/N

type

1. C₂×#711

C₂

semidirect

2. C₅×#121

C₂

semidirect

3. #4085

C₂

semidirect

4. #4085

C₂

semidirect

5. #4076

C₂

semidirect

6. C₂×#683

C₂

semidirect

7. #4042

C₂

semidirect

8. #4030

C₂

semidirect

9. #4024

C₂

semidirect

47. C₂×Dic₅

D₄

semidirect

61. #121

D₅

semidirect

128. C₅

#537

semidirect

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

subgroup	index
C₂×#711	2
C₅×#121	2
#4085	2
#4076	2
C₂×#683	2
#4042	2
#4030	2
#4024	2
#537	5

subgroup

index

Subgroups — #65931

Conjugacy classes of subgroups

Complements

Maximal subgroups