Group #10604652 — subgroups

Subgroups — #10604652

Order 640.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 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	#66062	C₂	1	1	yes	maximal, characteristic
2	#65681	C₂	1	1	yes	maximal, characteristic
3	C₄×D₅⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#64872	C₂	1	4	yes	maximal, complemented
5	C₂×#700	—	2	1	no	—
6	C₂×D₅⋊C₈	—	2	1	no	—
7	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	#4092	—	2	1	no	—
9	Q₈×Dic₅	—	2	1	no	—
10	#4043	C₄	1	2	yes	—
11	#4038	—	2	1	no	—
12	#4036	—	2	1	no	—
13	#4033	C₂²	1	2	yes	—
14	C₄²×D₅	C₂²	1	1	yes	characteristic
15	#4015	C₄	1	1	yes	characteristic
16	C₄⋊#170	—	2	1	no	—
17	C₄⋊#170	C₂²	1	2	yes	—
18	C₄×#170	—	2	1	no	—
19	C₄×#170	C₂²	1	2	yes	—
20	#1493	—	5	1	no	maximal, Sylow
21	C₂×C₅×Q₈	—	2	1	no	—
22	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₅	D₄	1	1	yes	characteristic
24	C₂×C₄×D₅	D₄	1	1	yes	characteristic
25	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
26	C₂×#170	—	2	2	no	—
27	C₂×#170	—	2	2	no	—
28	#700	—	4	2	no	—
29	D₅⋊C₈	—	2	4	no	—
30	C₅×C₄⋊C₄	—	2	1	no	—
31	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	#685	—	2	2	no	—
34	C₄⋊Dic₅	—	2	1	no	—
35	C₄⋊Dic₅	—	2	1	no	—
36	C₄⋊Dic₅	—	2	2	no	—
37	#683	—	2	2	no	—
38	C₄×Dic₅	—	2	2	no	—
39	C₄×Dic₅	D₄	1	1	yes	characteristic
40	C₄×Dic₅	D₄	1	1	yes	characteristic
41	C₄×Dic₅	C₂³	1	1	yes	characteristic
42	#533	—	5	1	no	—
43	#423	—	5	1	no	—
44	C₂×C₄×C₈	—	5	1	no	—
45	Q₈⋊C₈	—	5	4	no	—
46	C₂²×D₅	C₂²⋊C₄	1	1	yes	characteristic
47	C₅×Q₈	—	4	2	no	—
48	C₂×C₄×C₅	—	2	2	no	—
49	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₅	—	2	2	no	—
53	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
56	C₄×D₅	—	2	2	no	—
57	C₄×D₅	—	2	2	no	—
58	C₄×D₅	SD₁₆	1	2	yes	—
59	C₄×D₅	Q₁₆	1	2	yes	—
60	Dic₁₀	—	2	4	no	—
61	#170	—	2	4	no	—
62	#170	—	4	2	no	—
63	C₂×M₁₆	—	10	1	no	—
64	C₂²×C₈	—	10	1	no	—
65	C₄⋊Q₈	—	5	1	no	—
66	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
67	#125	—	10	1	no	—
68	C₂²⋊Q₈	—	5	2	no	—
69	C₂²⋊Q₈	—	10	1	no	—
70	C₄×Q₈	—	5	2	no	—
71	C₄×Q₈	—	10	1	no	—
72	#117	—	10	1	no	—
73	C₂×C₄²	—	5	1	no	—
74	C₄⋊C₈	—	5	2	no	—
75	C₄⋊C₈	—	10	1	no	—
76	C₄×C₈	—	5	2	no	—
77	C₄×C₈	—	10	1	no	—
78	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×D₅	—	2	1	no	—
80	C₂×D₅	—	2	1	no	—
81	C₂×D₅	#103	1	2	yes	—
82	C₄×C₅	—	4	2	no	—
83	C₄×C₅	#103	1	2	yes	—
84	C₄×C₅	#104	1	2	yes	—
85	C₄×C₅	#131	1	2	yes	—
86	Dic₅	—	2	1	no	—
87	Dic₅	—	2	1	no	—
88	Dic₅	—	4	2	no	—
89	Dic₅	C₂×SD₁₆	1	1	yes	characteristic
90	Dic₅	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	5	2	no	—
93	C₂²×C₄	—	5	1	no	—
94	C₂²×C₄	—	5	1	no	—
95	C₂²×C₄	—	5	1	no	—
96	M₁₆	—	20	2	no	—
97	C₂×C₈	—	10	2	no	—
98	C₂×C₈	—	10	4	no	—
99	C₂×C₈	—	10	2	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	10	2	no	—
102	C₄⋊C₄	—	10	2	no	—
103	C₄⋊C₄	—	10	1	no	—
104	C₄⋊C₄	—	10	1	no	—
105	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
106	C₂²⋊C₄	—	10	2	no	—
107	C₄²	—	5	1	no	—
108	C₄²	—	5	1	no	—
109	C₄²	—	5	1	no	—
110	C₄²	—	10	2	no	—
111	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₂×C₅	#408	1	1	yes	characteristic
113	C₂×C₅	C₂×#103	1	1	yes	characteristic
114	C₂×C₅	C₂×#104	1	1	yes	characteristic
115	D₅	—	2	2	no	—
116	C₂³	—	5	1	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	10	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	5	1	no	—
121	C₂×C₄	—	5	2	no	—
122	C₂×C₄	—	5	2	no	—
123	C₂×C₄	—	5	1	no	—
124	C₂×C₄	—	5	1	no	—
125	C₂×C₄	—	10	2	no	—
126	C₂×C₄	—	10	2	no	—
127	C₂×C₄	—	10	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	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
131	C₈	—	10	4	no	—
132	C₈	—	20	2	no	—
133	C₅	#1493	1	1	yes	characteristic, complemented, Sylow
134	C₂²	—	5	2	no	—
135	C₂²	—	10	1	no	—
136	C₂²	—	10	1	no	—
137	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	5	1	no	—
140	C₄	—	5	1	no	—
141	C₄	—	10	1	no	—
142	C₄	—	10	1	no	—
143	C₄	—	20	2	no	—
144	C₄	#4008	1	2	yes	—
145	C₄	#4009	1	2	yes	—
146	C₄	#4133	1	2	yes	—
147	C₂	—	10	2	no	—
148	C₂	C₂×#4008	1	1	yes	characteristic
149	C₂	C₂×#4009	1	1	yes	characteristic
150	C₂	#65787	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#66062

C₂

yes

maximal, characteristic

#65681

C₂

yes

maximal, characteristic

C₄×D₅⋊C₈

C₂

yes

maximal, characteristic

#64872

C₂

yes

maximal, complemented

C₂×#700

—

C₂×D₅⋊C₈

—

C₅×C₄⋊Q₈

C₄

yes

characteristic

#4092

—

Q₈×Dic₅

—

#4043

C₄

yes

—

#4038

—

#4036

—

#4033

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

#4015

C₄

yes

characteristic

C₄⋊#170

—

C₄⋊#170

C₂²

yes

—

C₄×#170

—

C₄×#170

C₂²

yes

—

#1493

—

maximal, Sylow

C₂×C₅×Q₈

—

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

characteristic

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

—

C₂×#170

—

#700

—

D₅⋊C₈

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

#685

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

C₂³

yes

characteristic

#533

—

#423

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₂²×D₅

C₂²⋊C₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

SD₁₆

yes

—

C₄×D₅

Q₁₆

yes

—

Dic₁₀

—

#170

—

#170

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

AGL(1,𝔽₅)

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

#103

yes

—

C₄×C₅

—

C₄×C₅

#103

yes

—

C₄×C₅

#104

yes

—

C₄×C₅

#131

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂×SD₁₆

yes

characteristic

Dic₅

C₂×Q₁₆

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

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

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

112

C₂×C₅

#408

yes

characteristic

113

C₂×C₅

C₂×#103

yes

characteristic

114

C₂×C₅

C₂×#104

yes

characteristic

115

D₅

—

116

C₂³

—

117

Q₈

—

118

Q₈

—

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

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

131

C₈

—

132

C₈

—

133

C₅

#1493

yes

characteristic, complemented, Sylow

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#4008

yes

—

145

C₄

#4009

yes

—

146

C₄

#4133

yes

—

147

C₂

—

148

C₂

C₂×#4008

yes

characteristic

149

C₂

C₂×#4009

yes

characteristic

150

C₂

#65787

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	C₅×C₄⋊Q₈	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₅	C₂²⋊C₄
cosocle	C₄×Dic₅	C₂³
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#1493
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₄⋊Q₈

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂²×D₅

C₂²⋊C₄

cosocle

C₄×Dic₅

C₂³

C₄⋊Q₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
4. #64872	C₂	semidirect
133. C₅	#1493	semidirect

H ≅ G/N

type

4. #64872

C₂

semidirect

133. C₅

#1493

semidirect

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

subgroup	index
#66062	2
#65681	2
C₄×D₅⋊C₈	2
#64872	2
#1493	5

subgroup

index

C₄×D₅⋊C₈

Subgroups — #10604652

Conjugacy classes of subgroups

Complements

Maximal subgroups