Group #10604679 — subgroups

Subgroups — #10604679

Order 640.

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

normal subgroups only

This group has 260 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	C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	C₄×#700	C₂	1	1	yes	maximal, characteristic
4	#64875	C₂	1	4	yes	maximal
5	C₂×C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
6	C₂×C₄×AGL(1,𝔽₅)	C₂²	1	2	yes	—
7	C₂×#700	C₂²	1	2	yes	—
8	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#4092	—	2	1	no	—
10	Q₈×Dic₅	—	2	1	no	—
11	#4043	—	2	1	no	—
12	#4038	C₂²	1	2	yes	—
13	#4036	C₄	1	2	yes	complemented
14	#4033	—	2	1	no	—
15	C₄²×D₅	C₂²	1	1	yes	characteristic
16	#4015	C₄	1	1	yes	characteristic, complemented
17	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	1	no	—
18	#4002	—	2	1	no	—
19	C₄×#170	—	2	1	no	—
20	#1986	—	5	1	no	maximal, Sylow
21	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
22	C₂×C₅×Q₈	—	2	1	no	—
23	C₂×C₄×D₅	D₄	1	1	yes	characteristic
24	C₂×C₄×D₅	D₄	1	1	yes	characteristic
25	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₀	—	2	1	no	—
27	C₂×#170	—	2	2	no	—
28	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
29	C₄×AGL(1,𝔽₅)	—	2	4	no	—
30	#700	—	2	2	no	—
31	#700	Q₈	1	4	yes	complemented
32	C₅×C₄⋊C₄	—	2	1	no	—
33	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	#685	—	2	2	no	—
36	C₄⋊Dic₅	—	2	1	no	—
37	C₄⋊Dic₅	—	2	2	no	—
38	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
39	#683	—	2	2	no	—
40	C₄×Dic₅	—	2	2	no	—
41	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₅	D₄	1	1	yes	characteristic
43	C₄×Dic₅	D₄	1	1	yes	characteristic
44	#533	—	5	1	no	—
45	C₄×M₁₆	—	5	1	no	—
46	C₄×C₄⋊C₄	—	5	1	no	—
47	#339	—	5	4	no	—
48	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
49	C₂×AGL(1,𝔽₅)	—	2	4	no	—
50	C₂×AGL(1,𝔽₅)	—	4	2	no	—
51	C₂×AGL(1,𝔽₅)	—	4	2	no	—
52	C₅×Q₈	—	4	2	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₅	—	2	2	no	—
58	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
61	C₄×D₅	—	2	2	no	—
62	C₄×D₅	—	4	1	no	—
63	C₄×D₅	C₂×Q₈	1	2	yes	—
64	C₄×D₅	#41	1	2	yes	—
65	Dic₁₀	—	4	2	no	—
66	#170	—	2	4	no	—
67	C₂×M₁₆	—	5	2	no	—
68	C₄⋊Q₈	—	5	1	no	—
69	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
70	#125	—	5	2	no	—
71	C₂²⋊Q₈	—	10	1	no	—
72	C₂²⋊Q₈	—	10	1	no	—
73	C₄×Q₈	—	10	1	no	—
74	C₄×Q₈	—	10	1	no	—
75	#117	—	5	2	no	—
76	C₂×C₄⋊C₄	—	10	1	no	—
77	C₂×C₄²	—	5	1	no	—
78	C₂×C₄²	—	5	2	no	—
79	#97	—	10	1	no	—
80	C₄×C₈	—	10	1	no	—
81	#95	—	10	1	no	—
82	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×D₅	—	2	1	no	—
84	C₂×D₅	—	2	1	no	—
85	C₂×D₅	#124	1	1	yes	characteristic
86	C₂×D₅	C₄⋊Q₈	1	1	yes	characteristic
87	AGL(1,𝔽₅)	—	4	4	no	—
88	C₄×C₅	—	2	1	no	—
89	C₄×C₅	—	4	2	no	—
90	C₄×C₅	#104	1	2	yes	—
91	C₄×C₅	C₄×Q₈	1	2	yes	—
92	Dic₅	—	2	1	no	—
93	Dic₅	—	2	1	no	—
94	Dic₅	—	4	2	no	—
95	Dic₅	C₂²⋊Q₈	1	2	yes	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	10	1	no	—
98	C₂²×C₄	—	5	1	no	—
99	C₂²×C₄	—	5	1	no	—
100	C₂²×C₄	—	5	1	no	—
101	C₂²×C₄	—	10	2	no	—
102	M₁₆	—	5	4	no	—
103	M₁₆	—	10	2	no	—
104	C₂×C₈	—	10	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	5	2	no	—
107	C₄⋊C₄	—	10	2	no	—
108	C₄⋊C₄	—	10	2	no	—
109	C₄⋊C₄	—	10	4	no	—
110	C₄⋊C₄	—	10	1	no	—
111	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
112	C₂²⋊C₄	—	10	2	no	—
113	C₄²	—	5	1	no	—
114	C₄²	—	5	1	no	—
115	C₄²	—	5	1	no	—
116	C₄²	—	10	2	no	—
117	C₄²	—	10	4	no	—
118	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
119	C₂×C₅	#391	1	1	yes	characteristic
120	C₂×C₅	C₂×#104	1	1	yes	characteristic
121	C₂×C₅	#421	1	1	yes	characteristic
122	D₅	—	2	1	no	—
123	D₅	—	2	1	no	—
124	C₂³	—	5	1	no	—
125	Q₈	—	4	2	no	—
126	Q₈	—	20	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	5	1	no	—
129	C₂×C₄	—	5	2	no	—
130	C₂×C₄	—	5	2	no	—
131	C₂×C₄	—	5	1	no	—
132	C₂×C₄	—	5	1	no	—
133	C₂×C₄	—	10	2	no	—
134	C₂×C₄	—	10	4	no	—
135	C₂×C₄	—	10	2	no	—
136	C₂×C₄	—	20	1	no	—
137	C₂×C₄	—	20	2	no	—
138	C₂×C₄	—	20	2	no	—
139	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₈	—	10	4	no	—
143	C₅	#1986	1	1	yes	characteristic, complemented, Sylow
144	C₂²	—	5	1	no	—
145	C₂²	—	5	1	no	—
146	C₂²	—	10	1	no	—
147	C₂²	—	10	1	no	—
148	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	—	5	2	no	—
152	C₄	—	10	1	no	—
153	C₄	—	10	1	no	—
154	C₄	—	20	4	no	—
155	C₄	—	20	2	no	—
156	C₄	#4008	1	2	yes	—
157	C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
158	C₂	—	10	1	no	—
159	C₂	—	10	1	no	—
160	C₂	C₂×#4008	1	1	yes	characteristic
161	C₂	#65784	1	1	yes	characteristic
162	C₂	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#66062

C₂

yes

maximal, characteristic

C₄×C₄⋊AGL(1,𝔽₅)

C₂

yes

maximal, characteristic

C₄×#700

C₂

yes

maximal, characteristic

#64875

C₂

yes

maximal

C₂×C₄⋊AGL(1,𝔽₅)

—

C₂×C₄×AGL(1,𝔽₅)

C₂²

yes

—

C₂×#700

C₂²

yes

—

C₅×C₄⋊Q₈

C₄

yes

characteristic, complemented

#4092

—

Q₈×Dic₅

—

#4043

—

#4038

C₂²

yes

—

#4036

C₄

yes

complemented

#4033

—

C₄²×D₅

C₂²

yes

characteristic

#4015

C₄

yes

characteristic, complemented

(C₂×AGL(1,𝔽₅))⋊C₄

—

—

—

—

C₂²×AGL(1,𝔽₅)

—

C₂×C₅×Q₈

—

C₂×C₄×D₅

D₄

yes

characteristic

C₂×C₄×D₅

D₄

yes

characteristic

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

—

—

—

—

—

Q₈

yes

complemented

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

#685

—

C₄⋊Dic₅

—

C₄⋊Dic₅

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

#533

—

C₄×M₁₆

—

C₄×C₄⋊C₄

—

#339

—

C₂²×D₅

C₂×D₄

yes

—

—

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×Q₈

yes

—

C₄×D₅

#41

yes

—

Dic₁₀

—

#170

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

AGL(1,𝔽₅)

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#97

—

C₄×C₈

—

#95

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

#124

yes

characteristic

C₂×D₅

C₄⋊Q₈

yes

characteristic

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#104

yes

—

C₄×C₅

C₄×Q₈

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

C₂²⋊Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

119

C₂×C₅

#391

yes

characteristic

120

C₂×C₅

C₂×#104

yes

characteristic

121

C₂×C₅

#421

yes

characteristic

122

D₅

—

123

D₅

—

124

C₂³

—

125

Q₈

—

126

Q₈

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

140

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

141

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

142

C₈

—

143

C₅

#1986

yes

characteristic, complemented, Sylow

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#4008

yes

—

157

C₄

Q₈×AGL(1,𝔽₅)

yes

—

158

C₂

—

159

C₂

—

160

C₂

C₂×#4008

yes

characteristic

161

C₂

#65784

yes

characteristic

162

C₂

(C₂×AGL(1,𝔽₅))⋊Q₈

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	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#1986
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

C₂×D₄

cosocle

C₂×C₄×D₅

C₂³

C₄⋊Q₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
8. C₅×C₄⋊Q₈	C₄	semidirect
13. #4036	C₄	semidirect
16. #4015	C₄	semidirect
31. #700	Q₈	semidirect
69. C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
143. C₅	#1986	semidirect

H ≅ G/N

type

8. C₅×C₄⋊Q₈

C₄

semidirect

13. #4036

C₄

semidirect

16. #4015

C₄

semidirect

31. #700

Q₈

semidirect

69. C₄⋊Q₈

AGL(1,𝔽₅)

semidirect

143. C₅

#1986

semidirect

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

subgroup	index
#66062	2
C₄×C₄⋊AGL(1,𝔽₅)	2
C₄×#700	2
#64875	2
#1986	5

subgroup

index

C₄×C₄⋊AGL(1,𝔽₅)

Subgroups — #10604679

Conjugacy classes of subgroups

Complements

Maximal subgroups