Group #10617330 — subgroups

Subgroups — #10617330

Order 640.

348 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 162 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	#66022	C₂	1	1	yes	maximal, characteristic
2	#65709	C₂	1	4	yes	maximal, complemented
3	#65708	C₂	1	2	yes	maximal
4	#65707	C₂	1	4	yes	maximal, complemented
5	#65701	C₂	1	2	yes	maximal
6	#65681	C₂	1	1	yes	maximal, characteristic
7	#65676	C₂	1	1	yes	maximal, characteristic
8	C₂×#700	C₂²	1	2	yes	—
9	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₅⋊C₈	C₂²	1	1	yes	characteristic
11	C₅×#125	C₄	1	1	yes	characteristic
12	#4042	C₄	1	4	yes	—
13	#4038	C₂²	1	2	yes	—
14	#4034	C₄	1	2	yes	—
15	#4033	C₂²	1	4	yes	—
16	C₄²×D₅	C₂²	1	1	yes	characteristic
17	#4016	C₄	1	1	yes	characteristic
18	#4004	C₂²	1	4	yes	—
19	#4002	C₂²	1	2	yes	—
20	#4002	C₂²	1	2	yes	—
21	#4002	C₂²	1	4	yes	—
22	C₄⋊#170	C₂²	1	4	yes	—
23	C₄⋊#170	C₂²	1	4	yes	—
24	C₄×#170	C₂²	1	4	yes	—
25	#2961	—	5	1	no	maximal, Sylow
26	C₂×C₄×D₅	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₀	C₂×C₄	1	4	yes	—
29	C₂×#170	C₂³	1	1	yes	characteristic
30	C₂×#170	C₂³	1	1	yes	characteristic
31	C₂×#170	C₂³	1	4	yes	—
32	C₂×#170	C₂³	1	2	yes	—
33	#700	—	2	4	no	—
34	D₅⋊C₈	—	2	2	no	—
35	D₅⋊C₈	Q₈	1	4	yes	complemented
36	C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	#685	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₅	C₂×C₄	1	2	yes	—
41	#683	C₂×C₄	1	4	yes	—
42	#683	C₂×C₄	1	4	yes	—
43	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₄×Dic₅	C₂³	1	4	yes	—
45	C₄×Dic₅	C₂³	1	2	yes	—
46	#533	—	5	1	no	—
47	#446	—	5	4	no	—
48	#446	—	5	4	no	—
49	#424	—	5	2	no	—
50	#423	—	5	1	no	—
51	#405	—	5	2	no	—
52	C₂×#97	—	5	1	no	—
53	C₂²×D₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₅	C₂²×C₄	1	2	yes	—
55	C₂×C₄×C₅	C₂²×C₄	1	4	yes	—
56	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂×Dic₅	C₂²×C₄	1	2	yes	—
58	C₂×Dic₅	C₂²×C₄	1	4	yes	—
59	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
60	C₄×D₅	—	4	2	no	—
61	C₄×D₅	C₂×Q₈	1	2	yes	—
62	C₄×D₅	#41	1	2	yes	—
63	Dic₁₀	—	4	4	no	—
64	#170	—	2	1	no	—
65	#170	—	2	1	no	—
66	#170	—	2	4	no	—
67	#170	C₂×Q₈	1	4	yes	—
68	C₂×M₁₆	—	5	2	no	—
69	C₂²×C₈	—	5	1	no	—
70	C₂²×C₈	—	5	1	no	—
71	C₄⋊Q₈	—	5	2	no	—
72	#125	—	5	1	no	—
73	#125	AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂²⋊Q₈	—	5	4	no	—
75	C₄×Q₈	—	5	4	no	—
76	#117	—	5	2	no	—
77	C₂×C₄²	—	5	1	no	—
78	C₄⋊C₈	—	5	4	no	—
79	C₄⋊C₈	—	5	4	no	—
80	C₄⋊C₈	—	5	4	no	—
81	#97	—	5	2	no	—
82	#97	—	5	2	no	—
83	#97	—	5	4	no	—
84	C₄×C₈	—	5	4	no	—
85	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
86	C₂×D₅	—	2	2	no	—
87	C₂×D₅	C₄×Q₈	1	2	yes	—
88	C₄×C₅	—	2	4	no	—
89	C₄×C₅	—	2	2	no	—
90	C₄×C₅	C₄×Q₈	1	2	yes	—
91	Dic₅	—	2	2	no	—
92	Dic₅	—	2	4	no	—
93	Dic₅	C₂²×Q₈	1	1	yes	characteristic
94	Dic₅	C₂×#41	1	1	yes	characteristic
95	C₂×Q₈	—	5	4	no	—
96	C₂²×C₄	—	5	1	no	—
97	C₂²×C₄	—	5	2	no	—
98	M₁₆	—	10	4	no	—
99	C₂×C₈	—	5	2	no	—
100	C₂×C₈	—	5	4	no	—
101	C₂×C₈	—	5	4	no	—
102	C₂×C₈	—	5	1	no	—
103	C₂×C₈	—	5	1	no	—
104	C₂×C₈	—	10	2	no	—
105	C₄⋊C₄	—	5	2	no	—
106	C₄⋊C₄	—	5	4	no	—
107	C₄⋊C₄	—	5	4	no	—
108	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
109	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
110	C₂²⋊C₄	—	5	4	no	—
111	C₄²	—	5	4	no	—
112	C₄²	—	5	2	no	—
113	C₄²	—	5	1	no	—
114	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
115	C₂×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
116	C₂×C₅	#568	1	2	yes	—
117	D₅	—	2	2	no	—
118	C₂³	—	5	1	no	—
119	Q₈	—	20	4	no	—
120	C₂×C₄	—	5	1	no	—
121	C₂×C₄	—	5	2	no	—
122	C₂×C₄	—	5	2	no	—
123	C₂×C₄	—	5	2	no	—
124	C₂×C₄	—	5	4	no	—
125	C₂×C₄	—	20	2	no	—
126	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
127	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
128	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₈	—	5	4	no	—
130	C₈	—	10	1	no	—
131	C₈	—	10	1	no	—
132	C₈	—	10	4	no	—
133	C₅	#2961	1	1	yes	characteristic, complemented, Sylow
134	C₂²	—	5	2	no	—
135	C₂²	—	10	2	no	—
136	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	5	1	no	—
140	C₄	—	5	1	no	—
141	C₄	—	10	2	no	—
142	C₄	—	10	4	no	—
143	C₄	Q₈×AGL(1,𝔽₅)	1	2	yes	—
144	C₂	—	10	2	no	—
145	C₂	C₂×Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂	#66264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#66022

C₂

yes

maximal, characteristic

#65709

C₂

yes

maximal, complemented

#65708

C₂

yes

maximal

#65707

C₂

yes

maximal, complemented

#65701

C₂

yes

maximal

#65681

C₂

yes

maximal, characteristic

#65676

C₂

yes

maximal, characteristic

C₂×#700

C₂²

yes

—

C₂×D₅⋊C₈

C₂²

yes

characteristic

C₂×D₅⋊C₈

C₂²

yes

characteristic

C₅×#125

C₄

yes

characteristic

#4042

C₄

yes

—

#4038

C₂²

yes

—

#4034

C₄

yes

—

#4033

C₂²

yes

—

C₄²×D₅

C₂²

yes

characteristic

#4016

C₄

yes

characteristic

#4004

C₂²

yes

—

#4002

C₂²

yes

—

#4002

C₂²

yes

—

#4002

C₂²

yes

—

C₄⋊#170

C₂²

yes

—

C₄⋊#170

C₂²

yes

—

C₄×#170

C₂²

yes

—

#2961

—

maximal, Sylow

C₂×C₄×D₅

C₂×C₄

yes

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

C₂³

yes

characteristic

C₂×#170

C₂³

yes

characteristic

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

#700

—

D₅⋊C₈

—

D₅⋊C₈

Q₈

yes

complemented

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

—

#683

C₂×C₄

yes

—

#683

C₂×C₄

yes

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂³

yes

—

C₄×Dic₅

C₂³

yes

—

—

—

—

—

—

—

C₂×#97

—

C₂²×D₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂⁴

yes

characteristic

C₄×D₅

—

C₄×D₅

C₂×Q₈

yes

—

C₄×D₅

#41

yes

—

Dic₁₀

—

—

—

—

C₂×Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄×Q₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×Q₈

yes

—

Dic₅

—

Dic₅

—

Dic₅

C₂²×Q₈

yes

characteristic

Dic₅

C₂×#41

yes

characteristic

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

109

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

115

C₂×C₅

C₂×C₄×Q₈

yes

characteristic

116

C₂×C₅

#568

yes

—

117

D₅

—

118

C₂³

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

127

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

128

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₅

#2961

yes

characteristic, complemented, Sylow

134

C₂²

—

135

C₂²

—

136

C₂²

C₂³×AGL(1,𝔽₅)

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

Q₈×AGL(1,𝔽₅)

yes

—

144

C₂

—

145

C₂

C₂×Q₈×AGL(1,𝔽₅)

yes

characteristic

146

C₂

#66264

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₄
Fitting	C₅×#125	C₄
prime radical	C₂²×D₅	C₂²×C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	#125	AGL(1,𝔽₅)
hypocenter	C₅	#2961
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₅)

commutator, socle

C₂²×C₅

C₂³×C₄

Fitting

C₅×#125

C₄

prime radical

C₂²×D₅

C₂²×C₄

cosocle

C₂×Dic₅

C₂⁴

C₅

G ▷ C₂²×C₅ ▷ C₅

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
2. #65709	C₂	semidirect
4. #65707	C₂	semidirect
35. D₅⋊C₈	Q₈	semidirect
133. C₅	#2961	semidirect

H ≅ G/N

type

2. #65709

C₂

semidirect

4. #65707

C₂

semidirect

35. D₅⋊C₈

Q₈

semidirect

133. C₅

#2961

semidirect

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

subgroup	index
#66022	2
#65709	2
#65708	2
#65707	2
#65701	2
#65681	2
#65676	2
#2961	5

subgroup

index

Subgroups — #10617330

Conjugacy classes of subgroups

Complements

Maximal subgroups