Group #10617384 — subgroups

Subgroups — #10617384

Order 640.

314 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 144 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	#65788	C₂	1	4	yes	maximal, complemented
3	#65707	C₂	1	4	yes	maximal, complemented
4	#65706	C₂	1	4	yes	maximal
5	#65684	C₂	1	1	yes	maximal, characteristic
6	#65684	C₂	1	1	yes	maximal, characteristic
7	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	#4092	C₄	1	2	yes	—
9	Q₈×Dic₅	C₂²	1	2	yes	—
10	#4043	C₄	1	2	yes	—
11	#4038	C₂²	1	2	yes	—
12	#4036	C₄	1	2	yes	—
13	#4033	C₂²	1	2	yes	—
14	C₄²×D₅	C₂²	1	1	yes	characteristic
15	#4015	C₄	1	1	yes	characteristic
16	#4004	C₂²	1	2	yes	—
17	#4004	C₂²	1	2	yes	—
18	#4004	C₂²	1	4	yes	—
19	#4003	C₂²	1	2	yes	—
20	#4003	C₂²	1	2	yes	—
21	#4002	C₂²	1	2	yes	—
22	#4002	C₂²	1	2	yes	—
23	#4002	C₂²	1	4	yes	—
24	C₄⋊#170	C₂²	1	4	yes	—
25	C₄×#170	C₂²	1	4	yes	—
26	#2987	—	5	1	no	maximal, Sylow
27	C₂×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
32	C₂×#170	C₂³	1	4	yes	—
33	C₂×#170	C₂³	1	4	yes	—
34	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
35	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	#685	C₂×C₄	1	4	yes	—
38	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₅	C₂×C₄	1	4	yes	—
41	#683	C₂×C₄	1	4	yes	—
42	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₅	C₂³	1	4	yes	—
44	C₄×Dic₅	C₂³	1	1	yes	characteristic
45	C₄×Dic₅	C₂³	1	1	yes	characteristic
46	#533	—	5	1	no	—
47	#446	—	5	4	no	—
48	#446	—	5	4	no	—
49	#433	—	5	4	no	—
50	#432	—	5	1	no	—
51	#432	—	5	1	no	—
52	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
53	C₅×Q₈	—	4	2	no	—
54	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×Dic₅	C₂²×C₄	1	4	yes	—
61	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
62	C₄×D₅	—	2	2	no	—
63	C₄×D₅	—	4	1	no	—
64	C₄×D₅	—	4	1	no	—
65	Dic₁₀	—	4	2	no	—
66	#170	—	2	4	no	—
67	#170	—	2	4	no	—
68	C₄⋊Q₈	—	5	1	no	—
69	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
70	#125	—	5	2	no	—
71	C₂²⋊Q₈	—	5	2	no	—
72	C₂²⋊Q₈	—	5	2	no	—
73	C₄×Q₈	—	5	2	no	—
74	C₄×Q₈	—	5	2	no	—
75	#117	—	5	2	no	—
76	C₂×C₄²	—	5	1	no	—
77	C₄⋊C₈	—	5	4	no	—
78	C₄⋊C₈	—	5	2	no	—
79	C₄⋊C₈	—	5	2	no	—
80	C₄⋊C₈	—	5	4	no	—
81	C₂²⋊C₈	—	5	2	no	—
82	C₂²⋊C₈	—	5	2	no	—
83	#97	—	5	2	no	—
84	#97	—	5	2	no	—
85	#97	—	5	4	no	—
86	C₄×C₈	—	5	4	no	—
87	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
88	C₂×D₅	—	2	1	no	—
89	C₂×D₅	—	2	1	no	—
90	C₂×D₅	—	2	1	no	—
91	C₄×C₅	—	2	1	no	—
92	C₄×C₅	—	2	1	no	—
93	C₄×C₅	—	2	1	no	—
94	C₄×C₅	—	2	4	no	—
95	Dic₅	—	2	4	no	—
96	Dic₅	—	2	1	no	—
97	Dic₅	—	2	1	no	—
98	Dic₅	2₋¹⁺⁴	1	2	yes	—
99	C₂×Q₈	—	5	2	no	—
100	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
101	C₂²×C₄	—	5	1	no	—
102	C₂²×C₄	—	5	1	no	—
103	C₂²×C₄	—	5	1	no	—
104	C₂×C₈	—	5	4	no	—
105	C₂×C₈	—	5	4	no	—
106	C₄⋊C₄	—	5	2	no	—
107	C₄⋊C₄	—	5	4	no	—
108	C₄⋊C₄	—	5	4	no	—
109	C₄⋊C₄	—	5	2	no	—
110	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
111	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
112	C₂²⋊C₄	—	5	4	no	—
113	C₄²	—	5	4	no	—
114	C₄²	—	5	1	no	—
115	C₄²	—	5	1	no	—
116	C₄²	—	5	1	no	—
117	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
118	C₂×C₅	#519	1	1	yes	characteristic
119	C₂×C₅	#568	1	1	yes	characteristic
120	C₂×C₅	#568	1	1	yes	characteristic
121	D₅	—	4	1	no	—
122	C₂³	—	5	1	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	20	2	no	—
125	C₂×C₄	—	5	1	no	—
126	C₂×C₄	—	5	1	no	—
127	C₂×C₄	—	5	1	no	—
128	C₂×C₄	—	5	4	no	—
129	C₂×C₄	—	10	2	no	—
130	C₂×C₄	—	20	1	no	—
131	C₂×C₄	—	20	1	no	—
132	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
136	C₈	—	10	4	no	—
137	C₈	—	10	4	no	—
138	C₅	#2987	1	1	yes	characteristic, complemented, Sylow
139	C₂²	—	10	1	no	—
140	C₂²	—	10	1	no	—
141	C₂²	—	10	1	no	—
142	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	4	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	5	2	no	—
148	C₄	—	10	1	no	—
149	C₄	—	10	1	no	—
150	C₄	—	10	4	no	—
151	C₂	—	20	1	no	—
152	C₂	#66256	1	1	yes	characteristic
153	C₂	#66260	1	1	yes	characteristic
154	C₂	#66262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#66062

C₂

yes

maximal, characteristic

#65788

C₂

yes

maximal, complemented

#65707

C₂

yes

maximal, complemented

#65706

C₂

yes

maximal

#65684

C₂

yes

maximal, characteristic

#65684

C₂

yes

maximal, characteristic

C₅×C₄⋊Q₈

C₄

yes

characteristic

#4092

C₄

yes

—

Q₈×Dic₅

C₂²

yes

—

#4043

C₄

yes

—

#4038

C₂²

yes

—

#4036

C₄

yes

—

#4033

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

#4015

C₄

yes

characteristic

#4004

C₂²

yes

—

#4004

C₂²

yes

—

#4004

C₂²

yes

—

#4003

C₂²

yes

—

#4003

C₂²

yes

—

#4002

C₂²

yes

—

#4002

C₂²

yes

—

#4002

C₂²

yes

—

C₄⋊#170

C₂²

yes

—

C₄×#170

C₂²

yes

—

#2987

—

maximal, Sylow

C₂×C₅×Q₈

C₂×C₄

yes

—

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

C₂×C₄

yes

characteristic

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

characteristic

#685

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

#683

C₂×C₄

yes

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂³

yes

—

C₄×Dic₅

C₂³

yes

characteristic

C₄×Dic₅

C₂³

yes

—

—

—

—

—

—

C₂²×D₅

C₂²×C₄

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂⁴

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

#170

—

#170

—

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⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

2₋¹⁺⁴

yes

—

C₂×Q₈

—

100

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

111

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

118

C₂×C₅

#519

yes

characteristic

119

C₂×C₅

#568

yes

characteristic

120

C₂×C₅

#568

yes

characteristic

121

D₅

—

122

C₂³

—

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

133

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

134

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

135

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

136

C₈

—

137

C₈

—

138

C₅

#2987

yes

characteristic, complemented, Sylow

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×AGL(1,𝔽₅)

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

152

C₂

#66256

yes

characteristic

153

C₂

#66260

yes

characteristic

154

C₂

#66262

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×C₄⋊Q₈	C₄
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2987
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂³×C₄

Fitting

C₅×C₄⋊Q₈

C₄

prime radical

C₂²×D₅

C₂²×C₄

cosocle

C₂×Dic₅

C₂⁴

C₄⋊Q₈

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #65788	C₂	semidirect
3. #65707	C₂	semidirect
138. C₅	#2987	semidirect

H ≅ G/N

type

2. #65788

C₂

semidirect

3. #65707

C₂

semidirect

138. C₅

#2987

semidirect

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

subgroup	index
#66062	2
#65788	2
#65707	2
#65706	2
#65684	2
#2987	5

subgroup

index

Subgroups — #10617384

Conjugacy classes of subgroups

Complements

Maximal subgroups