Group #13018823 — subgroups

Subgroups — #13018823

Order 1280.

338 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 128 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	#10603859	C₂	1	1	yes	maximal, characteristic
2	#10603858	C₂	1	1	yes	maximal, characteristic
3	#10603817	C₂	1	2	yes	maximal
4	#10603816	C₂	1	2	yes	maximal
5	D₅×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#3996	—	2	2	no	—
7	C₂×#3995	—	2	2	no	—
8	#65681	C₂²	1	1	yes	characteristic
9	#65681	C₂²	1	2	yes	—
10	#65681	C₂²	1	1	yes	characteristic
11	C₄×C₈×D₅	C₂²	1	2	yes	—
12	C₄×C₈×D₅	C₂²	1	1	yes	characteristic
13	#64883	—	2	4	no	—
14	#64881	—	2	4	no	—
15	C₅×#322	C₄	1	1	yes	characteristic
16	#64675	C₄	1	1	yes	characteristic
17	#64674	C₄	1	2	yes	—
18	#8825	—	5	1	no	maximal, Sylow
19	C₂×#700	—	2	2	no	—
20	C₂×#700	—	2	1	no	—
21	C₂×#700	—	2	1	no	—
22	C₂×C₈×D₅	D₄	1	2	yes	—
23	C₂×C₈×D₅	Q₈	1	4	yes	—
24	C₄²×D₅	C₂³	1	1	yes	characteristic
25	C₄⋊#170	—	2	2	no	—
26	C₄⋊#170	—	2	2	no	—
27	C₄⋊#170	—	2	2	no	—
28	C₄⋊#170	—	2	2	no	—
29	#3996	—	4	4	no	—
30	#3995	—	4	4	no	—
31	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₈×Dic₅	D₄	1	4	yes	—
34	C₈×Dic₅	Q₈	1	2	yes	—
35	C₄×#168	C₂×C₄	1	1	yes	characteristic
36	C₄×#168	C₂×C₄	1	2	yes	—
37	#1848	—	5	1	no	—
38	#1848	—	5	1	no	—
39	#1566	—	5	2	no	—
40	#1566	—	5	2	no	—
41	C₂×#322	—	5	1	no	—
42	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×D₅	C₂×Q₈	1	2	yes	—
44	C₂×#170	—	4	2	no	—
45	C₂×#170	—	4	1	no	—
46	C₂×#170	—	4	1	no	—
47	#700	—	4	4	no	—
48	#700	—	4	2	no	—
49	#700	—	4	2	no	—
50	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
51	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
52	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
53	C₄×Dic₅	C₂×D₄	1	2	yes	—
54	C₄×Dic₅	C₂×Q₈	1	1	yes	characteristic
55	C₂×#168	C₄⋊C₄	1	4	yes	—
56	C₂×#168	C₄⋊C₄	1	2	yes	—
57	C₈×D₅	—	4	4	no	—
58	C₈×D₅	SD₁₆	1	4	yes	—
59	C₈×D₅	SD₁₆	1	4	yes	—
60	C₂×#108	—	10	2	no	—
61	C₂×#108	—	10	2	no	—
62	#423	—	5	1	no	—
63	#423	—	5	2	no	—
64	#423	—	5	1	no	—
65	C₂×C₄×C₈	—	5	2	no	—
66	C₂×C₄×C₈	—	5	1	no	—
67	#334	—	10	4	no	—
68	#334	—	10	4	no	—
69	#322	—	5	2	no	—
70	#322	—	5	1	no	—
71	#322	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂²×D₅	C₄⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
75	C₂×Dic₅	#127	1	1	yes	characteristic
76	C₂×Dic₅	C₄⋊Q₈	1	2	yes	—
77	C₄×D₅	—	2	4	no	—
78	C₄×D₅	C₂×SD₁₆	1	2	yes	—
79	C₄×D₅	C₂×SD₁₆	1	2	yes	—
80	#170	—	8	2	no	—
81	#170	—	8	1	no	—
82	#170	—	8	1	no	—
83	C₅×C₈	—	2	4	no	—
84	C₅×C₈	#106	1	4	yes	—
85	#168	—	2	4	no	—
86	#168	#106	1	4	yes	—
87	C₂×M₁₆	—	10	2	no	—
88	C₂×M₁₆	—	10	1	no	—
89	C₂×M₁₆	—	10	1	no	—
90	C₂²×C₈	—	5	2	no	—
91	C₂²×C₈	—	5	4	no	—
92	C₂×C₄²	—	5	1	no	—
93	#108	—	20	4	no	—
94	#108	—	20	4	no	—
95	C₄⋊C₈	—	10	2	no	—
96	C₄⋊C₈	—	10	2	no	—
97	C₄⋊C₈	—	10	2	no	—
98	C₄⋊C₈	—	10	2	no	—
99	C₄×C₈	—	5	4	no	—
100	C₄×C₈	—	5	2	no	—
101	C₄×C₈	—	5	1	no	—
102	C₄×C₈	—	5	2	no	—
103	C₄×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
104	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₂²×C₅	#384	1	1	yes	characteristic
106	C₂×D₅	—	2	2	no	—
107	C₂×D₅	#498	1	2	yes	—
108	C₄×C₅	C₂×#106	1	2	yes	—
109	C₄×C₅	#430	1	4	yes	—
110	Dic₅	—	2	2	no	—
111	Dic₅	#492	1	1	yes	characteristic
112	Dic₅	#492	1	1	yes	characteristic
113	C₂²×C₄	—	5	1	no	—
114	C₂²×C₄	—	5	2	no	—
115	M₁₆	—	20	4	no	—
116	M₁₆	—	20	2	no	—
117	M₁₆	—	20	2	no	—
118	C₂×C₈	—	5	2	no	—
119	C₂×C₈	—	5	4	no	—
120	C₂×C₈	—	5	4	no	—
121	C₂×C₈	—	5	4	no	—
122	C₂×C₈	—	20	4	no	—
123	C₂×C₈	—	20	1	no	—
124	C₂×C₈	—	20	2	no	—
125	C₂×C₈	—	20	1	no	—
126	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
127	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
128	C₄²	—	5	2	no	—
129	C₄²	—	5	1	no	—
130	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₂×C₅	#1844	1	1	yes	characteristic
132	C₂×C₅	#1849	1	2	yes	—
133	D₅	—	2	2	no	—
134	C₂³	—	5	1	no	—
135	C₂×C₄	—	5	1	no	—
136	C₂×C₄	—	5	2	no	—
137	C₂×C₄	—	5	2	no	—
138	C₂×C₄	—	5	2	no	—
139	C₂×C₄	—	10	4	no	—
140	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
142	C₈	—	2	4	no	—
143	C₈	—	5	4	no	—
144	C₈	—	10	4	no	—
145	C₈	—	40	2	no	—
146	C₈	—	40	1	no	—
147	C₈	—	40	1	no	—
148	C₈	#3993	1	4	yes	—
149	C₅	#8825	1	1	yes	characteristic, complemented, Sylow
150	C₂²	—	5	2	no	—
151	C₂²	—	10	2	no	—
152	C₂²	#65688	1	1	yes	characteristic
153	C₄	—	5	1	no	—
154	C₄	—	5	1	no	—
155	C₄	—	10	2	no	—
156	C₄	C₂×#3993	1	2	yes	—
157	C₄	#65729	1	4	yes	—
158	C₂	—	10	2	no	—
159	C₂	#10603847	1	1	yes	characteristic
160	C₂	#10603890	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10603859

C₂

yes

maximal, characteristic

#10603858

C₂

yes

maximal, characteristic

#10603817

C₂

yes

maximal

#10603816

C₂

yes

maximal

D₅×#322

C₂

yes

maximal, characteristic

C₂×#3996

—

C₂×#3995

—

#65681

C₂²

yes

characteristic

#65681

C₂²

yes

—

#65681

C₂²

yes

characteristic

C₄×C₈×D₅

C₂²

yes

—

C₄×C₈×D₅

C₂²

yes

—

—

C₄

yes

characteristic

#64675

C₄

yes

characteristic

#64674

C₄

yes

—

—

—

—

—

C₂×C₈×D₅

D₄

yes

—

C₂×C₈×D₅

Q₈

yes

—

C₄²×D₅

C₂³

yes

—

—

—

—

—

—

C₄×C₅×C₈

C₂×C₄

yes

—

C₄×C₅×C₈

C₂×C₄

yes

characteristic

C₈×Dic₅

D₄

yes

—

C₈×Dic₅

Q₈

yes

—

C₄×#168

C₂×C₄

yes

characteristic

C₄×#168

C₂×C₄

yes

—

—

—

—

—

—

C₂×C₄×D₅

C₂×D₄

yes

characteristic

C₂×C₄×D₅

C₂×Q₈

yes

—

—

—

—

—

—

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₂×C₅×C₈

C₄⋊C₄

yes

—

C₄²×C₅

C₂²×C₄

yes

characteristic

C₄×Dic₅

C₂×D₄

yes

—

C₄×Dic₅

C₂×Q₈

yes

characteristic

C₂×#168

C₄⋊C₄

yes

—

C₂×#168

C₄⋊C₄

yes

—

C₈×D₅

—

C₈×D₅

SD₁₆

yes

—

C₈×D₅

SD₁₆

yes

—

—

—

—

—

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

—

—

—

—

yes

C₂²×D₅

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂×C₄⋊C₄

yes

—

C₂×Dic₅

#127

yes

characteristic

C₂×Dic₅

C₄⋊Q₈

yes

—

C₄×D₅

—

C₄×D₅

C₂×SD₁₆

yes

—

C₄×D₅

C₂×SD₁₆

yes

—

#170

—

#170

—

#170

—

C₅×C₈

—

C₅×C₈

#106

yes

—

#168

—

#168

#106

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

—

104

C₄×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

105

C₂²×C₅

#384

yes

characteristic

106

C₂×D₅

—

107

C₂×D₅

#498

yes

—

108

C₄×C₅

C₂×#106

yes

—

109

C₄×C₅

#430

yes

—

110

Dic₅

—

111

Dic₅

#492

yes

characteristic

112

Dic₅

#492

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

127

C₂×C₈

C₄⋊AGL(1,𝔽₅)

yes

—

128

C₄²

—

129

C₄²

—

130

C₄²

C₂²×AGL(1,𝔽₅)

yes

characteristic

131

C₂×C₅

#1844

yes

characteristic

132

C₂×C₅

#1849

yes

—

133

D₅

—

134

C₂³

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

141

C₂×C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#3993

yes

—

149

C₅

#8825

yes

characteristic, complemented, Sylow

150

C₂²

—

151

C₂²

—

152

C₂²

#65688

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

C₂×#3993

yes

—

157

C₄

#65729

yes

—

158

C₂

—

159

C₂

#10603847

yes

characteristic

160

C₂

#10603890

yes

—

	subgroup	G/H
center	C₂²	#65688
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₄²	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#322	C₄
socle	C₂²×C₅	#384
prime radical	C₂²×D₅	C₄⋊Q₈
cosocle	C₄²×D₅	C₂³
hypercenter	#322	AGL(1,𝔽₅)
hypocenter	C₅	#8825
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#65688

commutator

C₄²×C₅

C₂²×C₄

Frattini

C₄²

C₂²×AGL(1,𝔽₅)

C₄

C₂²×C₅

C₂²×D₅

C₄⋊Q₈

cosocle

C₄²×D₅

C₂³

C₅

G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
149. C₅	#8825	semidirect

H ≅ G/N

type

149. C₅

#8825

semidirect

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

subgroup	index
#10603859	2
#10603858	2
#10603817	2
#10603816	2
D₅×#322	2
#8825	5

subgroup

index

Subgroups — #13018823

Conjugacy classes of subgroups

Complements

Maximal subgroups