Group #10604087 — subgroups

Subgroups — #10604087

Order 640.

196 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 68 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	#65727	C₂	1	2	yes	maximal
2	C₂×#3993	C₂	1	1	yes	maximal, characteristic
3	#65718	C₂	1	1	yes	maximal, characteristic
4	D₅×#108	C₂	1	1	yes	maximal, characteristic
5	#64899	C₂	1	2	yes	maximal
6	#4130	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
8	D₅⋊M₁₆	C₂²	1	1	yes	characteristic
9	D₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
11	#3994	—	2	2	no	—
12	#3993	—	2	2	no	—
13	#3993	—	2	1	no	—
14	#3993	—	2	1	no	—
15	#3992	—	2	1	no	—
16	C₈×AGL(1,𝔽₅)	—	2	1	no	—
17	C₅×#108	C₄	1	1	yes	characteristic, complemented
18	#3954	C₄	1	2	yes	complemented
19	#3951	C₄	1	1	yes	characteristic, complemented
20	#1951	—	5	1	no	maximal, Sylow
21	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
22	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
23	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
24	#704	Q₈	1	2	yes	—
25	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
26	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
27	C₄⋊AGL(1,𝔽₅)	D₄	1	2	yes	—
28	C₄×AGL(1,𝔽₅)	—	2	1	no	—
29	#700	—	2	1	no	—
30	D₅⋊C₈	—	2	1	no	—
31	C₅×M₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	#681	C₂×C₄	1	2	yes	—
34	C₂×#168	C₂×C₄	1	1	yes	characteristic
35	#676	—	2	2	no	—
36	C₈×D₅	—	2	2	no	—
37	C₈×D₅	D₄	1	1	yes	characteristic
38	C₈×D₅	D₄	1	1	yes	characteristic
39	C₈×D₅	Q₈	1	1	yes	characteristic
40	C₈×D₅	Q₈	1	1	yes	characteristic
41	C₂×#108	—	5	1	no	—
42	#428	—	5	2	no	—
43	C₂×#106	—	5	1	no	—
44	#405	—	5	1	no	—
45	#343	—	5	2	no	—
46	C₂²×D₅	#41	1	1	yes	characteristic
47	C₂×AGL(1,𝔽₅)	—	2	2	no	—
48	C₂×AGL(1,𝔽₅)	—	4	2	no	—
49	C₂×AGL(1,𝔽₅)	—	8	2	no	—
50	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₅	C₂×Q₈	1	1	yes	characteristic
52	C₄×D₅	C₂×D₄	1	1	yes	characteristic
53	C₄×D₅	C₂×D₄	1	1	yes	characteristic
54	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
55	C₄×D₅	#41	1	1	yes	characteristic
56	#170	—	2	2	no	—
57	C₅×C₈	—	2	2	no	—
58	C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
60	#168	—	2	2	no	—
61	#168	C₄⋊C₄	1	1	yes	characteristic
62	#168	C₄⋊C₄	1	1	yes	characteristic
63	C₂×M₁₆	—	5	1	no	—
64	C₂×M₁₆	—	5	2	no	—
65	C₂²×C₈	—	5	1	no	—
66	#117	—	5	1	no	—
67	C₂×C₄⋊C₄	—	5	2	no	—
68	#108	—	5	2	no	—
69	#108	—	5	1	no	—
70	#108	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
71	#107	—	10	2	no	—
72	#106	—	10	2	no	—
73	#106	—	10	1	no	—
74	#106	—	10	1	no	—
75	#97	—	10	1	no	—
76	C₄×C₈	—	10	1	no	—
77	C₂²×C₅	C₄×Q₈	1	1	yes	characteristic
78	C₂×D₅	—	2	2	no	—
79	C₂×D₅	#121	1	1	yes	characteristic
80	C₂×D₅	#125	1	1	yes	characteristic
81	AGL(1,𝔽₅)	—	4	2	no	—
82	AGL(1,𝔽₅)	—	8	2	no	—
83	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₅	C₄×D₄	1	1	yes	characteristic
85	Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
86	Dic₅	C₄⋊Q₈	1	1	yes	characteristic
87	C₂²×C₄	—	5	1	no	—
88	C₂²×C₄	—	10	2	no	—
89	M₁₆	—	5	2	no	—
90	M₁₆	—	5	2	no	—
91	M₁₆	—	10	2	no	—
92	M₁₆	—	10	1	no	—
93	M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
94	C₂×C₈	—	5	1	no	—
95	C₂×C₈	—	5	1	no	—
96	C₂×C₈	—	5	1	no	—
97	C₂×C₈	—	5	1	no	—
98	C₂×C₈	—	5	1	no	—
99	C₂×C₈	—	10	2	no	—
100	C₂×C₈	—	10	1	no	—
101	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₄⋊C₄	—	5	2	no	—
103	C₄⋊C₄	—	10	2	no	—
104	C₄⋊C₄	—	10	2	no	—
105	C₂²⋊C₄	—	10	1	no	—
106	C₄²	—	10	1	no	—
107	C₂×C₅	—	2	1	no	—
108	C₂×C₅	#389	1	1	yes	characteristic
109	D₅	—	2	1	no	—
110	D₅	#471	1	2	yes	—
111	C₂³	—	5	1	no	—
112	C₂×C₄	—	5	1	no	—
113	C₂×C₄	—	5	1	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	5	1	no	—
116	C₂×C₄	—	5	1	no	—
117	C₂×C₄	—	10	2	no	—
118	C₂×C₄	—	20	2	no	—
119	C₂×C₄	—	40	2	no	—
120	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	5	1	no	—
123	C₈	—	5	1	no	—
124	C₈	—	10	2	no	—
125	C₈	—	10	2	no	—
126	C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₅	#1951	1	1	yes	characteristic, complemented, Sylow
129	C₂²	—	5	1	no	—
130	C₂²	—	5	1	no	—
131	C₂²	—	10	2	no	—
132	C₂²	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₄	—	5	1	no	—
134	C₄	—	5	1	no	—
135	C₄	—	20	2	no	—
136	C₄	—	40	2	no	—
137	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	5	2	no	—
141	C₂	—	10	1	no	—
142	C₂	#65712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65727

C₂

yes

maximal

C₂×#3993

C₂

yes

maximal, characteristic

#65718

C₂

yes

maximal, characteristic

D₅×#108

C₂

yes

maximal, characteristic

#64899

C₂

yes

maximal

#4130

C₂²

yes

characteristic

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

—

D₅⋊M₁₆

C₂²

yes

characteristic

D₅×M₁₆

C₂²

yes

—

C₂×C₈×D₅

C₂²

yes

—

—

—

—

—

—

C₄

yes

characteristic, complemented

#3954

C₄

yes

complemented

#3951

C₄

yes

characteristic, complemented

#1951

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂²⋊AGL(1,𝔽₅)

—

#704

Q₈

yes

—

C₄⋊AGL(1,𝔽₅)

—

C₄⋊AGL(1,𝔽₅)

—

C₄⋊AGL(1,𝔽₅)

D₄

yes

—

C₄×AGL(1,𝔽₅)

—

#700

—

D₅⋊C₈

—

C₅×M₁₆

C₂×C₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

#681

C₂×C₄

yes

—

C₂×#168

C₂×C₄

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

D₄

yes

characteristic

C₈×D₅

D₄

yes

characteristic

C₈×D₅

Q₈

yes

characteristic

C₈×D₅

Q₈

yes

—

—

—

—

—

C₂²×D₅

yes

—

—

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂×Q₈

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×Q₈

yes

C₄×D₅

yes

—

C₅×C₈

—

C₅×C₈

C₄⋊C₄

yes

characteristic

C₅×C₈

C₄⋊C₄

yes

characteristic

#168

—

#168

C₄⋊C₄

yes

characteristic

#168

C₄⋊C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

—

—

yes

characteristic, complemented

—

—

—

—

—

C₄×C₈

—

C₂²×C₅

C₄×Q₈

yes

characteristic

C₂×D₅

—

C₂×D₅

yes

C₂×D₅

yes

—

—

C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅

C₄×D₄

yes

characteristic

Dic₅

C₂²⋊Q₈

yes

characteristic

Dic₅

C₄⋊Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₂×C₅

—

108

C₂×C₅

#389

yes

characteristic

109

D₅

—

110

D₅

#471

yes

—

111

C₂³

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

127

C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

128

C₅

#1951

yes

characteristic, complemented, Sylow

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

Q₈×AGL(1,𝔽₅)

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

138

C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

142

C₂

#65712

yes

characteristic

	subgroup	G/H
center	C₂	#65712
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#108	C₄
socle	C₂×C₅	#389
prime radical	C₂²×D₅	#41
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#108	AGL(1,𝔽₅)
hypocenter	C₅	#1951
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#65712

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

C₄

C₂×C₅

C₂²×D₅

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. C₅×#108	C₄	semidirect
18. #3954	C₄	semidirect
19. #3951	C₄	semidirect
70. #108	AGL(1,𝔽₅)	semidirect
128. C₅	#1951	semidirect

H ≅ G/N

type

17. C₅×#108

C₄

semidirect

18. #3954

C₄

semidirect

19. #3951

C₄

semidirect

70. #108

AGL(1,𝔽₅)

semidirect

128. C₅

#1951

semidirect

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

subgroup	index
#65727	2
C₂×#3993	2
#65718	2
D₅×#108	2
#64899	2
#1951	5

subgroup

index

Subgroups — #10604087

Conjugacy classes of subgroups

Complements

Maximal subgroups