Group #10604987 — subgroups

Subgroups — #10604987

Order 640.

260 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 76 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×#4090	C₂	1	1	yes	maximal, characteristic
2	#65758	C₂	1	1	yes	maximal, characteristic
3	C₄×#704	C₂	1	1	yes	maximal, characteristic
4	#64919	C₂	1	4	yes	maximal
5	C₂²×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂²×Dic₁₀	C₄	1	1	yes	characteristic
7	C₂×#704	C₂²	1	2	yes	—
8	C₂×#704	C₂²	1	2	yes	—
9	#4090	—	2	1	no	—
10	#4090	—	2	1	no	—
11	#4090	—	2	1	no	—
12	#4090	—	2	1	no	—
13	C₂×#683	C₄	1	2	yes	—
14	C₂×#683	C₂²	1	2	yes	—
15	C₂×C₄×Dic₅	C₂²	1	1	yes	characteristic
16	#4004	—	2	1	no	—
17	#4002	—	2	1	no	—
18	C₄⋊#170	—	2	1	no	—
19	C₄×#170	—	2	1	no	—
20	#1997	—	5	1	no	maximal, Sylow
21	C₂×C₅×Q₈	—	2	1	no	—
22	C₂×C₅×Q₈	—	2	1	no	—
23	C₂×C₅×Q₈	—	4	2	no	—
24	C₂²×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₅	C₂×C₄	1	2	yes	—
28	C₂²×Dic₅	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀	—	2	1	no	—
30	C₂×Dic₁₀	—	2	1	no	—
31	C₂×Dic₁₀	—	4	2	no	—
32	#704	—	2	2	no	—
33	#704	—	2	4	no	—
34	#704	Q₈	1	4	yes	complemented
35	C₂×#170	—	2	2	no	—
36	C₂×#170	—	2	2	no	—
37	#683	—	2	2	no	—
38	#683	—	2	2	no	—
39	#683	—	2	2	no	—
40	#683	—	2	2	no	—
41	C₄×Dic₅	D₄	1	1	yes	characteristic
42	C₄×Dic₅	D₄	1	1	yes	characteristic
43	C₄×Dic₅	D₄	1	1	yes	characteristic
44	C₄×Dic₅	D₄	1	1	yes	characteristic
45	C₂×C₄⋊Q₈	—	5	1	no	—
46	#423	—	5	1	no	—
47	C₄×M₁₆	—	5	1	no	—
48	#343	—	5	4	no	—
49	C₂³×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₅×Q₈	—	4	2	no	—
51	C₅×Q₈	—	4	2	no	—
52	C₂×C₄×C₅	—	2	2	no	—
53	C₂×C₄×C₅	—	2	2	no	—
54	C₂×C₄×C₅	—	4	2	no	—
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	—	2	2	no	—
59	C₂×Dic₅	—	4	2	no	—
60	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
63	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
64	C₂×Dic₅	C₂×Q₈	1	2	yes	—
65	C₂×Dic₅	#41	1	2	yes	—
66	Dic₁₀	—	4	2	no	—
67	Dic₁₀	—	4	2	no	—
68	#170	—	2	4	no	—
69	#170	—	4	2	no	—
70	C₂²×Q₈	—	5	1	no	—
71	C₂²×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₂×M₁₆	—	5	2	no	—
73	C₂×M₁₆	—	5	2	no	—
74	C₄⋊Q₈	—	10	1	no	—
75	C₄⋊Q₈	—	10	1	no	—
76	C₄⋊Q₈	—	10	1	no	—
77	C₄⋊Q₈	—	10	1	no	—
78	C₂×C₄⋊C₄	—	5	2	no	—
79	C₂×C₄⋊C₄	—	5	2	no	—
80	C₂×C₄²	—	5	1	no	—
81	C₄⋊C₈	—	10	1	no	—
82	C₄⋊C₈	—	10	1	no	—
83	#97	—	10	1	no	—
84	C₄×C₈	—	10	1	no	—
85	C₂²×C₅	—	2	1	no	—
86	C₂²×C₅	—	2	1	no	—
87	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂²×C₅	C₄×Q₈	1	2	yes	—
89	C₄×C₅	—	2	1	no	—
90	C₄×C₅	—	4	2	no	—
91	C₄×C₅	#100	1	2	yes	—
92	Dic₅	—	2	1	no	—
93	Dic₅	—	4	2	no	—
94	Dic₅	#101	1	2	yes	—
95	Dic₅	C₂²⋊Q₈	1	2	yes	—
96	Dic₅	#124	1	1	yes	characteristic
97	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	1	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂×Q₈	—	10	1	no	—
102	C₂×Q₈	—	10	1	no	—
103	C₂×Q₈	—	20	2	no	—
104	C₂²×C₄	—	5	1	no	—
105	C₂²×C₄	—	5	1	no	—
106	C₂²×C₄	—	5	2	no	—
107	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
108	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
109	M₁₆	—	5	4	no	—
110	M₁₆	—	10	2	no	—
111	M₁₆	—	10	4	no	—
112	C₂×C₈	—	10	2	no	—
113	C₂×C₈	—	10	2	no	—
114	C₄⋊C₄	—	10	2	no	—
115	C₄⋊C₄	—	10	2	no	—
116	C₄⋊C₄	—	10	2	no	—
117	C₄⋊C₄	—	10	2	no	—
118	C₄²	—	5	1	no	—
119	C₄²	—	5	1	no	—
120	C₄²	—	5	1	no	—
121	C₄²	—	5	1	no	—
122	C₂×C₅	—	2	2	no	—
123	C₂×C₅	#391	1	1	yes	characteristic
124	C₂×C₅	C₂×#100	1	1	yes	characteristic
125	C₂×C₅	C₂×#101	1	1	yes	characteristic
126	C₂³	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	20	2	no	—
130	Q₈	—	20	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	5	2	no	—
135	C₂×C₄	—	5	2	no	—
136	C₂×C₄	—	5	1	no	—
137	C₂×C₄	—	5	1	no	—
138	C₂×C₄	—	5	1	no	—
139	C₂×C₄	—	5	1	no	—
140	C₂×C₄	—	10	2	no	—
141	C₂×C₄	—	10	2	no	—
142	C₂×C₄	—	20	2	no	—
143	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
145	C₈	—	10	4	no	—
146	C₈	—	20	2	no	—
147	C₅	#1997	1	1	yes	characteristic, complemented, Sylow
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	Q₈×AGL(1,𝔽₅)	1	2	yes	—
151	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	5	2	no	—
155	C₄	—	5	2	no	—
156	C₄	—	5	1	no	—
157	C₄	—	5	1	no	—
158	C₄	—	10	1	no	—
159	C₄	—	20	2	no	—
160	C₄	#4013	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	D₅⋊#101	1	1	yes	characteristic
163	C₂	(C₂×AGL(1,𝔽₅))⋊Q₈	1	1	yes	characteristic
164	C₂	C₂×#4013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#4090

C₂

yes

maximal, characteristic

#65758

C₂

yes

maximal, characteristic

C₄×#704

C₂

yes

maximal, characteristic

#64919

C₂

yes

maximal

C₂²×C₅×Q₈

C₄

yes

characteristic

C₂²×Dic₁₀

C₄

yes

characteristic

C₂×#704

C₂²

yes

—

C₂×#704

C₂²

yes

—

—

—

—

—

C₄

yes

—

C₂×#683

C₂²

yes

—

C₂×C₄×Dic₅

C₂²

yes

—

—

—

—

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

C₂×C₄

yes

—

C₂²×C₄×C₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₄

yes

characteristic

C₂²×Dic₅

C₂×C₄

yes

—

C₂²×Dic₅

C₂³

yes

characteristic

C₂×Dic₁₀

—

C₂×Dic₁₀

—

C₂×Dic₁₀

—

#704

—

#704

—

#704

Q₈

yes

—

—

—

—

—

—

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₄×Dic₅

D₄

yes

characteristic

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

—

#343

—

C₂³×C₅

C₂²×C₄

yes

characteristic

C₅×Q₈

—

C₅×Q₈

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂²⋊C₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×D₄

yes

characteristic

C₂×Dic₅

C₂×Q₈

yes

—

C₂×Dic₅

#41

yes

—

Dic₁₀

—

Dic₁₀

—

#170

—

#170

—

C₂²×Q₈

—

C₂²×Q₈

AGL(1,𝔽₅)

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅

C₄×Q₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#100

yes

—

Dic₅

—

Dic₅

—

Dic₅

#101

yes

—

Dic₅

C₂²⋊Q₈

yes

—

Dic₅

#124

yes

characteristic

Dic₅

C₄⋊Q₈

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

—

108

C₂²×C₄

C₂×AGL(1,𝔽₅)

yes

characteristic

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₂×C₅

—

123

C₂×C₅

#391

yes

characteristic

124

C₂×C₅

C₂×#100

yes

characteristic

125

C₂×C₅

C₂×#101

yes

characteristic

126

C₂³

C₂²×AGL(1,𝔽₅)

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

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

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

144

C₂×C₄

C₂²⋊AGL(1,𝔽₅)

yes

characteristic

145

C₈

—

146

C₈

—

147

C₅

#1997

yes

characteristic, complemented, Sylow

148

C₂²

—

149

C₂²

—

150

C₂²

Q₈×AGL(1,𝔽₅)

yes

—

151

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#4013

yes

—

161

C₂

—

162

C₂

D₅⋊#101

yes

characteristic

163

C₂

(C₂×AGL(1,𝔽₅))⋊Q₈

yes

characteristic

164

C₂

C₂×#4013

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)
commutator, prime radical	C₂³×C₅	C₂²×C₄
Frattini	C₂³	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₅×Q₈	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₅	C₂³
hypercenter	C₂²×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#1997
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

commutator, prime radical

C₂³×C₅

C₂²×C₄

Frattini

C₂³

C₂²×AGL(1,𝔽₅)

Fitting

C₂²×C₅×Q₈

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

cosocle

C₂²×Dic₅

C₂³

C₂²×Q₈

C₅

G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
34. #704	Q₈	semidirect
147. C₅	#1997	semidirect

H ≅ G/N

type

34. #704

Q₈

semidirect

147. C₅

#1997

semidirect

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

subgroup	index
C₂×#4090	2
#65758	2
C₄×#704	2
#64919	2
#1997	5

subgroup

index

Subgroups — #10604987

Conjugacy classes of subgroups

Complements

Maximal subgroups