Group #10603888 — subgroups

Subgroups — #10603888

Order 640.

220 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 92 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×#3996	C₂	1	1	yes	maximal, characteristic
2	C₂×#3995	C₂	1	1	yes	maximal, characteristic
3	C₄×#700	C₂	1	2	yes	maximal
4	D₅×#97	C₂	1	1	yes	maximal, characteristic
5	#64896	C₂	1	2	yes	maximal
6	C₂×#700	C₂²	1	2	yes	—
7	C₂×#700	C₂²	1	2	yes	—
8	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₄²×D₅	C₂²	1	1	yes	characteristic
11	#4002	—	2	2	no	—
12	C₄×#170	—	2	2	no	—
13	#3996	C₄	1	4	yes	complemented
14	#3995	C₄	1	4	yes	complemented
15	C₅×#97	C₄	1	1	yes	characteristic
16	#3947	C₄	1	1	yes	characteristic
17	#3947	C₄	1	1	yes	characteristic
18	#3935	C₄	1	1	yes	characteristic
19	#1778	—	5	1	no	maximal, Sylow
20	C₂×C₄×D₅	Q₈	1	2	yes	—
21	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
22	C₂×#170	—	2	2	no	—
23	C₂×#170	—	2	2	no	—
24	#700	—	2	2	no	—
25	#700	—	2	2	no	—
26	#700	C₂×C₄	1	4	yes	—
27	#700	C₂×C₄	1	4	yes	—
28	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₅	D₄	1	2	yes	—
33	C₂×#168	C₂×C₄	1	1	yes	characteristic
34	C₂×#168	C₂×C₄	1	1	yes	characteristic
35	C₈×D₅	—	2	1	no	—
36	C₈×D₅	—	2	1	no	—
37	C₈×D₅	C₂×C₄	1	2	yes	—
38	C₈×D₅	C₂×C₄	1	2	yes	—
39	C₂×#108	—	5	1	no	—
40	C₂×#108	—	5	1	no	—
41	C₄×M₁₆	—	5	2	no	—
42	C₂×#97	—	5	1	no	—
43	#341	—	5	2	no	—
44	C₂²×D₅	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
48	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
49	C₄×D₅	—	4	2	no	—
50	C₄×D₅	C₂²×C₄	1	1	yes	characteristic
51	C₄×D₅	C₂²×C₄	1	1	yes	characteristic
52	C₄×D₅	#41	1	1	yes	characteristic
53	C₄×D₅	#41	1	1	yes	characteristic
54	#170	—	2	4	no	—
55	#170	—	2	4	no	—
56	C₅×C₈	—	2	1	no	—
57	C₅×C₈	C₄²	1	2	yes	—
58	#168	—	2	1	no	—
59	#168	C₄²	1	2	yes	—
60	C₂×M₁₆	—	5	2	no	—
61	C₂×M₁₆	—	5	2	no	—
62	C₂²×C₈	—	5	1	no	—
63	C₂²×C₈	—	5	1	no	—
64	C₂×C₄²	—	5	1	no	—
65	#108	—	5	4	no	—
66	#108	—	5	4	no	—
67	#97	—	5	1	no	—
68	#97	—	5	1	no	—
69	#97	—	5	1	no	—
70	#97	—	10	2	no	—
71	#97	AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₄×C₈	—	10	2	no	—
73	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×D₅	—	2	2	no	—
75	C₂×D₅	C₄×Q₈	1	1	yes	characteristic
76	C₂×D₅	C₄×Q₈	1	1	yes	characteristic
77	C₄×C₅	—	2	2	no	—
78	C₄×C₅	C₂×C₄²	1	1	yes	characteristic
79	C₄×C₅	#117	1	1	yes	characteristic
80	Dic₅	—	2	2	no	—
81	Dic₅	C₄×D₄	1	1	yes	characteristic
82	Dic₅	C₄×D₄	1	1	yes	characteristic
83	C₂²×C₄	—	5	1	no	—
84	C₂²×C₄	—	5	2	no	—
85	M₁₆	—	5	4	no	—
86	M₁₆	—	5	4	no	—
87	M₁₆	—	10	2	no	—
88	M₁₆	—	10	2	no	—
89	C₂×C₈	—	5	1	no	—
90	C₂×C₈	—	5	2	no	—
91	C₂×C₈	—	5	2	no	—
92	C₂×C₈	—	5	1	no	—
93	C₂×C₈	—	10	1	no	—
94	C₂×C₈	—	10	1	no	—
95	C₂×C₈	—	10	2	no	—
96	C₂×C₈	—	10	2	no	—
97	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
99	C₄²	—	5	2	no	—
100	C₄²	—	5	1	no	—
101	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₂×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₅	#430	1	2	yes	—
104	D₅	—	2	1	no	—
105	D₅	—	2	1	no	—
106	C₂³	—	5	1	no	—
107	C₂×C₄	—	5	1	no	—
108	C₂×C₄	—	5	1	no	—
109	C₂×C₄	—	5	1	no	—
110	C₂×C₄	—	5	1	no	—
111	C₂×C₄	—	5	1	no	—
112	C₂×C₄	—	5	2	no	—
113	C₂×C₄	—	20	2	no	—
114	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
115	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	5	2	no	—
118	C₈	—	10	4	no	—
119	C₈	—	10	4	no	—
120	C₈	—	10	1	no	—
121	C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
122	C₅	#1778	1	1	yes	characteristic, complemented, Sylow
123	C₂²	—	5	1	no	—
124	C₂²	—	5	1	no	—
125	C₂²	—	10	2	no	—
126	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	5	1	no	—
129	C₄	—	5	1	no	—
130	C₄	—	10	2	no	—
131	C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
132	C₄	#4130	1	1	yes	characteristic
133	C₂	—	10	1	no	—
134	C₂	—	10	1	no	—
135	C₂	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₂	#65729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3996

C₂

yes

maximal, characteristic

C₂×#3995

C₂

yes

maximal, characteristic

C₄×#700

C₂

yes

maximal

D₅×#97

C₂

yes

maximal, characteristic

#64896

C₂

yes

maximal

C₂×#700

C₂²

yes

—

C₂×#700

C₂²

yes

—

C₂×C₈×D₅

C₂²

yes

characteristic

C₂×C₈×D₅

C₂²

yes

characteristic

C₄²×D₅

C₂²

yes

—

—

C₄

yes

complemented

#3995

C₄

yes

complemented

C₅×#97

C₄

yes

characteristic

#3947

C₄

yes

characteristic

#3947

C₄

yes

characteristic

#3935

C₄

yes

characteristic

#1778

—

maximal, Sylow

C₂×C₄×D₅

Q₈

yes

—

C₂×C₄×D₅

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

#700

C₂×C₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₂×C₅×C₈

C₂×C₄

yes

characteristic

C₄²×C₅

C₂×C₄

yes

characteristic

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

—

C₂×#168

C₂×C₄

yes

characteristic

C₂×#168

C₂×C₄

yes

characteristic

C₈×D₅

—

C₈×D₅

—

C₈×D₅

C₂×C₄

yes

—

C₈×D₅

C₂×C₄

yes

—

C₂×#108

—

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

C₂²×D₅

C₂×Q₈

yes

characteristic

C₂×C₄×C₅

C₄⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₄⋊C₄

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

C₂²×C₄

yes

characteristic

C₄×D₅

C₂²×C₄

yes

C₄×D₅

yes

C₄×D₅

yes

—

—

C₅×C₈

—

C₅×C₈

C₄²

yes

—

#168

—

#168

C₄²

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

—

—

—

—

—

—

yes

C₄×C₈

—

C₂²×C₅

C₂×C₄⋊C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

C₄×Q₈

yes

characteristic

C₂×D₅

C₄×Q₈

yes

characteristic

C₄×C₅

—

C₄×C₅

C₂×C₄²

yes

characteristic

C₄×C₅

#117

yes

characteristic

Dic₅

—

Dic₅

C₄×D₄

yes

characteristic

Dic₅

C₄×D₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

100

C₄²

—

101

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

102

C₂×C₅

C₄×C₄⋊C₄

yes

characteristic

103

C₂×C₅

#430

yes

—

104

D₅

—

105

D₅

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₄⋊AGL(1,𝔽₅)

yes

—

115

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

C₄×AGL(1,𝔽₅)

yes

—

122

C₅

#1778

yes

characteristic, complemented, Sylow

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×C₄×AGL(1,𝔽₅)

yes

characteristic

132

C₄

#4130

yes

characteristic

133

C₂

—

134

C₂

—

135

C₂

C₄×C₄⋊AGL(1,𝔽₅)

yes

characteristic

136

C₂

#65729

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#97	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₂×Q₈
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#97	AGL(1,𝔽₅)
hypocenter	C₅	#1778
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,𝔽₅)

commutator

C₄×C₅

C₂×C₄²

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×#97

C₄

socle

C₂²×C₅

C₂×C₄⋊C₄

prime radical

C₂²×D₅

C₂×Q₈

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
13. #3996	C₄	semidirect
14. #3995	C₄	semidirect
122. C₅	#1778	semidirect

H ≅ G/N

type

13. #3996

C₄

semidirect

14. #3995

C₄

semidirect

122. C₅

#1778

semidirect

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

subgroup	index
C₂×#3996	2
C₂×#3995	2
C₄×#700	2
D₅×#97	2
#64896	2
#1778	5

subgroup

index

D₅×#97

Subgroups — #10603888

Conjugacy classes of subgroups

Complements

Maximal subgroups