Group #14510765 — subgroups

Subgroups — C₅×#73041 #14510765

Order 1920.

170 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 32 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×#5407	C₂	1	1	yes	maximal, characteristic
2	C₅×#5403	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#504⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2223	—	3	1	no	maximal, Hall
5	C₅×#986	—	2	1	no	—
6	C₅×#984	—	2	1	no	—
7	C₅×#131⋊C₃	C₂²	1	1	yes	characteristic
8	C₅×#800	—	4	1	no	maximal
9	#73041	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₅×SD₃₂	—	3	1	no	—
11	C₅×#504	D₃	1	1	yes	characteristic, complemented
12	C₅×#473	—	3	1	no	—
13	C₅×#472	—	3	1	no	—
14	C₅×#368	—	3	1	no	—
15	C₅×#362	—	3	1	no	—
16	C₅×#361	—	3	1	no	—
17	C₅×#41⋊C₃	D₄	1	1	yes	characteristic
18	C₅×GL(2,𝔽₃)	—	4	1	no	—
19	C₅×#226	—	4	1	no	—
20	C₅×Dic₁₂	—	4	1	no	—
21	C₅×D₂₄	—	4	1	no	—
22	C₃×C₅×C₁₆	—	4	1	no	—
23	#5407	C₂×C₅	1	1	yes	characteristic
24	#5403	C₂×C₅	1	1	yes	characteristic, complemented
25	#504⋊C₃	C₂×C₅	1	1	yes	characteristic, complemented
26	C₅×#137	—	6	1	no	—
27	C₅×AGL(1,ℤ/8)	—	6	1	no	—
28	C₂×C₅×Q₁₆	—	3	1	no	—
29	C₂×C₅×D₈	—	3	1	no	—
30	C₅×#131	C₂×D₃	1	1	yes	characteristic
31	C₅×SD₃₂	—	6	1	no	—
32	C₅×SD₃₂	—	6	1	no	—
33	C₅×M₃₂	—	3	1	no	—
34	C₂×C₅×C₁₆	—	3	1	no	—
35	C₅×#108	—	3	1	no	—
36	C₅×#108	—	3	1	no	—
37	C₅×#101	—	6	1	no	—
38	C₅×#100	—	6	1	no	—
39	#2223	—	3	1	no	Sylow
40	C₅×D₁₂	—	8	1	no	—
41	C₅×Dic₆	—	8	1	no	—
42	C₅×SL(2,𝔽₃)	D₈	1	1	yes	characteristic
43	C₃×C₅×C₈	—	4	1	no	—
44	#986	—	2	1	no	—
45	#984	—	2	1	no	—
46	#131⋊C₃	C₂²×C₅	1	1	yes	characteristic
47	#800	—	4	1	no	—
48	C₅×#41	D₁₂	1	1	yes	characteristic
49	C₂×C₅×Q₈	—	6	1	no	—
50	C₂×C₅×D₄	—	6	1	no	—
51	C₅×Q₁₆	—	6	1	no	—
52	C₅×Q₁₆	—	6	1	no	—
53	C₅×Q₁₆	—	12	1	no	—
54	C₅×SD₁₆	—	12	1	no	—
55	C₅×SD₁₆	—	12	1	no	—
56	C₅×D₈	—	6	1	no	—
57	C₅×D₈	—	6	1	no	—
58	C₅×D₈	—	12	1	no	—
59	C₅×M₁₆	—	3	1	no	—
60	C₅×M₁₆	—	6	1	no	—
61	C₅×M₁₆	—	6	1	no	—
62	C₂×C₅×C₈	—	3	1	no	—
63	C₅×C₁₆	—	3	1	no	—
64	C₅×C₁₆	S₄	1	1	yes	characteristic
65	C₂×SD₃₂	—	3	1	no	—
66	#504	C₅×D₃	1	1	yes	characteristic, complemented
67	#473	—	3	1	no	—
68	#472	—	3	1	no	—
69	#368	—	3	1	no	—
70	#362	—	3	1	no	—
71	#361	—	3	1	no	—
72	C₂×C₅×D₃	—	16	1	no	—
73	C₃×C₄×C₅	—	4	1	no	—
74	C₅×Dic₃	—	16	1	no	—
75	#41⋊C₃	C₅×D₄	1	1	yes	characteristic
76	GL(2,𝔽₃)	—	4	1	no	—
77	#226	—	4	1	no	—
78	Dic₁₂	—	4	1	no	—
79	D₂₄	—	4	1	no	—
80	C₃×C₁₆	—	4	1	no	—
81	C₂³×C₅	—	12	1	no	—
82	C₅×Q₈	—	12	1	no	—
83	C₅×Q₈	—	12	1	no	—
84	C₅×Q₈	D₂₄	1	1	yes	characteristic
85	C₅×D₄	—	3	1	no	—
86	C₅×D₄	—	12	1	no	—
87	C₅×D₄	—	12	1	no	—
88	C₂×C₄×C₅	—	3	1	no	—
89	C₂×C₄×C₅	—	12	1	no	—
90	C₅×C₈	—	3	1	no	—
91	C₅×C₈	—	12	1	no	—
92	C₅×C₈	—	12	1	no	—
93	C₅×C₈	C₂×S₄	1	1	yes	characteristic
94	#137	—	6	1	no	—
95	AGL(1,ℤ/8)	—	6	1	no	—
96	C₂×Q₁₆	—	3	1	no	—
97	C₂×D₈	—	3	1	no	—
98	#131	C₂×C₅×D₃	1	1	yes	characteristic
99	SD₃₂	—	6	1	no	—
100	SD₃₂	—	6	1	no	—
101	M₃₂	—	3	1	no	—
102	C₂×C₁₆	—	3	1	no	—
103	#108	—	3	1	no	—
104	#108	—	3	1	no	—
105	#101	—	6	1	no	—
106	#100	—	6	1	no	—
107	C₂×C₃×C₅	—	4	1	no	—
108	C₅×D₃	—	32	1	no	—
109	D₁₂	—	8	1	no	—
110	Dic₆	—	8	1	no	—
111	SL(2,𝔽₃)	C₅×D₈	1	1	yes	characteristic
112	C₃×C₈	—	4	1	no	—
113	C₂²×C₅	—	3	1	no	—
114	C₂²×C₅	—	24	1	no	—
115	C₂²×C₅	—	48	1	no	—
116	C₄×C₅	—	3	1	no	—
117	C₄×C₅	—	24	1	no	—
118	C₄×C₅	C₄⋊S₄	1	1	yes	characteristic
119	#41	C₅×D₁₂	1	1	yes	characteristic
120	C₂×Q₈	—	6	1	no	—
121	C₂×D₄	—	6	1	no	—
122	Q₁₆	—	6	1	no	—
123	Q₁₆	—	6	1	no	—
124	Q₁₆	—	12	1	no	—
125	SD₁₆	—	12	1	no	—
126	SD₁₆	—	12	1	no	—
127	D₈	—	6	1	no	—
128	D₈	—	6	1	no	—
129	D₈	—	12	1	no	—
130	M₁₆	—	3	1	no	—
131	M₁₆	—	6	1	no	—
132	M₁₆	—	6	1	no	—
133	C₂×C₈	—	3	1	no	—
134	C₁₆	—	3	1	no	—
135	C₁₆	C₅×S₄	1	1	yes	characteristic
136	C₃×C₅	—	4	1	no	Hall
137	C₂×D₃	—	16	1	no	—
138	C₃×C₄	—	4	1	no	—
139	Dic₃	—	16	1	no	—
140	C₂×C₅	—	6	1	no	—
141	C₂×C₅	—	48	1	no	—
142	C₂×C₅	#5402	1	1	yes	characteristic
143	C₂³	—	12	1	no	—
144	Q₈	—	12	1	no	—
145	Q₈	—	12	1	no	—
146	Q₈	C₅×D₂₄	1	1	yes	characteristic
147	D₄	—	3	1	no	—
148	D₄	—	12	1	no	—
149	D₄	—	12	1	no	—
150	C₂×C₄	—	3	1	no	—
151	C₂×C₄	—	12	1	no	—
152	C₈	—	3	1	no	—
153	C₈	—	12	1	no	—
154	C₈	—	12	1	no	—
155	C₈	C₂×C₅×S₄	1	1	yes	characteristic
156	C₂×C₃	—	4	1	no	—
157	D₃	—	32	1	no	—
158	C₅	#73041	1	1	yes	characteristic, direct product factor, Sylow
159	C₂²	—	3	1	no	—
160	C₂²	—	24	1	no	—
161	C₂²	—	48	1	no	—
162	C₄	—	3	1	no	—
163	C₄	—	24	1	no	—
164	C₄	C₅×C₄⋊S₄	1	1	yes	characteristic
165	C₃	—	4	1	no	Sylow
166	C₂	—	6	1	no	—
167	C₂	—	48	1	no	—
168	C₂	C₅×#5402	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#5407

C₂

yes

maximal, characteristic

C₅×#5403

C₂

yes

maximal, characteristic, complemented

C₅×#504⋊C₃

C₂

yes

maximal, characteristic, complemented

—

—

—

C₅×#131⋊C₃

C₂²

yes

—

C₅

yes

maximal, characteristic, direct product factor, Hall

C₂×C₅×SD₃₂

—

C₅×#504

D₃

yes

characteristic, complemented

—

—

—

—

—

C₅×#41⋊C₃

D₄

yes

characteristic

C₅×GL(2,𝔽₃)

—

C₅×#226

—

C₅×Dic₁₂

—

C₅×D₂₄

—

C₃×C₅×C₁₆

—

#5407

C₂×C₅

yes

characteristic

#5403

C₂×C₅

yes

characteristic, complemented

#504⋊C₃

C₂×C₅

yes

characteristic, complemented

C₅×#137

—

C₅×AGL(1,ℤ/8)

—

C₂×C₅×Q₁₆

—

C₂×C₅×D₈

—

C₅×#131

C₂×D₃

yes

characteristic

C₅×SD₃₂

—

C₅×SD₃₂

—

C₅×M₃₂

—

C₂×C₅×C₁₆

—

—

—

—

—

—

C₅×D₁₂

—

C₅×Dic₆

—

C₅×SL(2,𝔽₃)

D₈

yes

characteristic

C₃×C₅×C₈

—

#986

—

#984

—

#131⋊C₃

C₂²×C₅

yes

characteristic

#800

—

C₅×#41

D₁₂

yes

characteristic

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×Q₁₆

—

C₅×SD₁₆

—

C₅×SD₁₆

—

C₅×D₈

—

C₅×D₈

—

C₅×D₈

—

C₅×M₁₆

—

C₅×M₁₆

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₅×C₁₆

—

C₅×C₁₆

S₄

yes

characteristic

C₂×SD₃₂

—

#504

C₅×D₃

yes

characteristic, complemented

—

—

—

—

—

C₂×C₅×D₃

—

C₃×C₄×C₅

—

C₅×Dic₃

—

#41⋊C₃

C₅×D₄

yes

characteristic

GL(2,𝔽₃)

—

#226

—

Dic₁₂

—

D₂₄

—

C₃×C₁₆

—

C₂³×C₅

—

C₅×Q₈

—

C₅×Q₈

—

C₅×Q₈

D₂₄

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

—

C₅×C₈

C₂×S₄

yes

characteristic

#137

—

AGL(1,ℤ/8)

—

C₂×Q₁₆

—

C₂×D₈

—

#131

C₂×C₅×D₃

yes

characteristic

SD₃₂

—

100

SD₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

#108

—

104

#108

—

105

#101

—

106

#100

—

107

C₂×C₃×C₅

—

108

C₅×D₃

—

109

D₁₂

—

110

Dic₆

—

111

SL(2,𝔽₃)

C₅×D₈

yes

characteristic

112

C₃×C₈

—

113

C₂²×C₅

—

114

C₂²×C₅

—

115

C₂²×C₅

—

116

C₄×C₅

—

117

C₄×C₅

—

118

C₄×C₅

C₄⋊S₄

yes

characteristic

119

#41

C₅×D₁₂

yes

characteristic

120

C₂×Q₈

—

121

C₂×D₄

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

125

SD₁₆

—

126

SD₁₆

—

127

D₈

—

128

D₈

—

129

D₈

—

130

M₁₆

—

131

M₁₆

—

132

M₁₆

—

133

C₂×C₈

—

134

C₁₆

—

135

C₁₆

C₅×S₄

yes

characteristic

136

C₃×C₅

—

Hall

137

C₂×D₃

—

138

C₃×C₄

—

139

Dic₃

—

140

C₂×C₅

—

141

C₂×C₅

—

142

C₂×C₅

#5402

yes

characteristic

143

C₂³

—

144

Q₈

—

145

Q₈

—

146

Q₈

C₅×D₂₄

yes

characteristic

147

D₄

—

148

D₄

—

149

D₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

C₂×C₅×S₄

yes

characteristic

156

C₂×C₃

—

157

D₃

—

158

C₅

#73041

yes

characteristic, direct product factor, Sylow

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₄

—

163

C₄

—

164

C₄

C₅×C₄⋊S₄

yes

characteristic

165

C₃

—

Sylow

166

C₂

—

167

C₂

—

168

C₂

C₅×#5402

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₅	#5402
commutator, cosocle	#131⋊C₃	C₂²×C₅
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#504	D₃
prime radical	C₅×#5407	C₂
hypercenter	C₅×C₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×D₈
LCS	G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆

subgroup

G/H

C₂×C₅

C₂²×C₅

C₈

C₂×C₅×S₄

D₃

C₂

C₅×C₁₆

S₄

hypocenter

SL(2,𝔽₃)

C₅×D₈

LCS

G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆

N	H ≅ G/N	type
2. C₅×#5403	C₂	semidirect
3. C₅×#504⋊C₃	C₂	semidirect
9. #73041	C₅	direct (N × H)
11. C₅×#504	D₃	semidirect
24. #5403	C₂×C₅	semidirect
25. #504⋊C₃	C₂×C₅	semidirect
66. #504	C₅×D₃	semidirect
158. C₅	#73041	direct (N × H)

H ≅ G/N

type

2. C₅×#5403

C₂

semidirect

3. C₅×#504⋊C₃

C₂

semidirect

9. #73041

C₅

direct (N × H)

11. C₅×#504

D₃

semidirect

24. #5403

C₂×C₅

semidirect

25. #504⋊C₃

C₂×C₅

semidirect

66. #504

C₅×D₃

semidirect

158. C₅

#73041

direct (N × H)

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

subgroup	index
C₅×#5407	2
C₅×#5403	2
C₅×#504⋊C₃	2
C₅×#2223	3
C₅×#800	4
#73041	5

subgroup

index

C₅×#504⋊C₃

Subgroups — C₅×#73041 #14510765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#73041 #14510765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#73041 #14510765