Group #10604279 — subgroups

Subgroups — #10604279

Order 640.

228 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 62 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×#4034	C₂	1	1	yes	maximal, characteristic
2	#65757	C₂	1	1	yes	maximal, characteristic
3	#65757	C₂	1	1	yes	maximal, characteristic
4	#64873	C₂	1	4	yes	maximal, complemented
5	C₂²×Dic₁₀	—	2	1	no	—
6	C₂×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
7	C₂×C₄⋊Dic₅	C₄	1	1	yes	characteristic
8	C₂×#683	—	2	1	no	—
9	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
10	#4034	—	2	1	no	—
11	#4034	—	2	1	no	—
12	#4034	C₄	1	2	yes	—
13	#4034	C₂²	1	2	yes	—
14	C₂²⋊#170	—	2	1	no	—
15	C₂²⋊#170	—	2	1	no	—
16	#4004	C₂²	1	2	yes	—
17	#4004	C₂²	1	2	yes	—
18	#4002	—	2	1	no	—
19	#4002	—	2	1	no	—
20	#1556	—	5	1	no	maximal, Sylow
21	C₂²×C₄×C₅	—	2	1	no	—
22	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₅	—	2	1	no	—
24	C₂²×Dic₅	D₄	1	1	yes	characteristic
25	C₂²×Dic₅	D₄	1	1	yes	characteristic
26	C₂×Dic₁₀	—	2	2	no	—
27	C₂×Dic₁₀	—	2	2	no	—
28	C₂×Dic₁₀	—	4	4	no	—
29	C₂×#170	—	2	2	no	—
30	C₂×#170	—	2	2	no	—
31	C₅×C₄⋊C₄	—	2	1	no	—
32	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₅	—	2	1	no	—
34	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
35	#683	—	2	2	no	—
36	#683	—	2	2	no	—
37	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₅	D₄	1	1	yes	characteristic
39	C₄×Dic₅	D₄	1	1	yes	characteristic
40	C₄×Dic₅	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊Q₈	—	5	1	no	—
42	#432	—	5	1	no	—
43	#432	—	5	1	no	—
44	#332	—	5	4	no	—
45	C₂³×C₅	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅	—	2	2	no	—
47	C₂×C₄×C₅	—	2	2	no	—
48	C₂×C₄×C₅	—	4	2	no	—
49	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₅	—	2	2	no	—
52	C₂×Dic₅	—	2	2	no	—
53	C₂×Dic₅	—	2	2	no	—
54	C₂×Dic₅	—	4	2	no	—
55	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
58	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
59	Dic₁₀	—	4	4	no	—
60	Dic₁₀	—	4	4	no	—
61	#170	—	4	2	no	—
62	#170	—	4	2	no	—
63	C₂²×Q₈	—	10	1	no	—
64	C₄⋊Q₈	—	5	2	no	—
65	C₄⋊Q₈	—	5	2	no	—
66	C₄⋊Q₈	—	10	1	no	—
67	C₄⋊Q₈	—	10	1	no	—
68	C₂×C₄⋊C₄	—	5	1	no	—
69	C₂×C₄⋊C₄	—	10	1	no	—
70	C₂×C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
71	C₂×C₄²	—	5	1	no	—
72	C₄⋊C₈	—	5	2	no	—
73	C₄⋊C₈	—	5	2	no	—
74	C₂²⋊C₈	—	10	1	no	—
75	C₂²⋊C₈	—	10	1	no	—
76	#97	—	10	1	no	—
77	#97	—	10	1	no	—
78	C₂²×C₅	—	2	1	no	—
79	C₂²×C₅	—	2	1	no	—
80	C₂²×C₅	#101	1	2	yes	—
81	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₄×C₅	—	2	1	no	—
83	C₄×C₅	—	2	1	no	—
84	C₄×C₅	—	4	2	no	—
85	Dic₅	—	2	1	no	—
86	Dic₅	—	2	1	no	—
87	Dic₅	—	4	2	no	—
88	Dic₅	AGL(1,ℤ/8)	1	2	yes	—
89	Dic₅	#137	1	2	yes	—
90	C₂×Q₈	—	10	2	no	—
91	C₂×Q₈	—	10	2	no	—
92	C₂×Q₈	—	20	4	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	5	1	no	—
95	C₂²×C₄	—	5	1	no	—
96	C₂²×C₄	—	10	1	no	—
97	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₂×C₈	—	10	2	no	—
99	C₂×C₈	—	10	2	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	5	2	no	—
102	C₄⋊C₄	—	10	2	no	—
103	C₄⋊C₄	—	10	2	no	—
104	C₄⋊C₄	—	10	1	no	—
105	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
106	C₄²	—	5	1	no	—
107	C₄²	—	5	1	no	—
108	C₄²	—	5	1	no	—
109	C₄²	—	5	1	no	—
110	C₂×C₅	—	2	2	no	—
111	C₂×C₅	C₂×#101	1	1	yes	characteristic
112	C₂×C₅	#418	1	1	yes	characteristic
113	C₂×C₅	#419	1	1	yes	characteristic
114	C₂³	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
115	Q₈	—	20	4	no	—
116	Q₈	—	20	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	5	1	no	—
121	C₂×C₄	—	5	1	no	—
122	C₂×C₄	—	5	1	no	—
123	C₂×C₄	—	5	1	no	—
124	C₂×C₄	—	10	2	no	—
125	C₂×C₄	—	10	2	no	—
126	C₂×C₄	—	10	2	no	—
127	C₂×C₄	—	20	2	no	—
128	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₈	—	20	2	no	—
131	C₈	—	20	2	no	—
132	C₅	#1556	1	1	yes	characteristic, complemented, Sylow
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#4005	1	2	yes	—
136	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	5	2	no	—
141	C₄	—	5	2	no	—
142	C₄	—	10	1	no	—
143	C₄	—	10	1	no	—
144	C₄	—	20	2	no	—
145	C₂	—	2	2	no	—
146	C₂	C₂×#4005	1	1	yes	characteristic
147	C₂	#65767	1	1	yes	characteristic
148	C₂	#65782	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4034

C₂

yes

maximal, characteristic

#65757

C₂

yes

maximal, characteristic

#65757

C₂

yes

maximal, characteristic

#64873

C₂

yes

maximal, complemented

C₂²×Dic₁₀

—

C₂×C₅×C₄⋊C₄

C₄

yes

characteristic

C₂×C₄⋊Dic₅

C₄

yes

characteristic

C₂×#683

—

C₂×C₄×Dic₅

C₂²

yes

—

—

C₄

yes

—

#4034

C₂²

yes

—

C₂²⋊#170

—

C₂²⋊#170

—

#4004

C₂²

yes

—

#4004

C₂²

yes

—

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

—

C₂²×Dic₅

D₄

yes

characteristic

C₂²×Dic₅

D₄

yes

characteristic

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×#170

—

C₂×#170

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

—

#683

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

C₂³

yes

characteristic

C₂×C₄⋊Q₈

—

#432

—

#432

—

#332

—

C₂³×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

AGL(1,𝔽₅)

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#101

yes

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

AGL(1,ℤ/8)

yes

—

Dic₅

#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₂×C₅

—

111

C₂×C₅

C₂×#101

yes

characteristic

112

C₂×C₅

#418

yes

characteristic

113

C₂×C₅

#419

yes

characteristic

114

C₂³

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

115

Q₈

—

116

Q₈

—

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

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

129

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

130

C₈

—

131

C₈

—

132

C₅

#1556

yes

characteristic, complemented, Sylow

133

C₂²

—

134

C₂²

—

135

C₂²

#4005

yes

—

136

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₂

—

146

C₂

C₂×#4005

yes

characteristic

147

C₂

#65767

yes

characteristic

148

C₂

#65782

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₄⋊C₄	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₅	C₂²⋊C₄
cosocle	C₄×Dic₅	C₂³
hypercenter	C₂×C₄⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#1556
LCS	G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₂×C₅×C₄⋊C₄

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂³×C₅

C₂²⋊C₄

cosocle

C₄×Dic₅

C₂³

C₂×C₄⋊C₄

C₅

G ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
4. #64873	C₂	semidirect
132. C₅	#1556	semidirect

H ≅ G/N

type

4. #64873

C₂

semidirect

132. C₅

#1556

semidirect

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

subgroup	index
C₂×#4034	2
#65757	2
#64873	2
#1556	5

subgroup

index

Subgroups — #10604279

Conjugacy classes of subgroups

Complements

Maximal subgroups