Group #10604673 — subgroups

Subgroups — #10604673

Order 640.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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	D₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4009	C₂	1	4	yes	maximal
3	#65689	C₂	1	1	yes	maximal, characteristic
4	#65676	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₈×D₅	C₂²	1	2	yes	—
6	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
7	C₂×D₅⋊C₈	C₂²	1	2	yes	—
8	C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
9	#4090	C₄	1	2	yes	complemented
10	D₅×C₄⋊C₄	—	2	2	no	—
11	#4034	—	2	2	no	—
12	C₄²×D₅	C₂²	1	1	yes	characteristic
13	#4015	C₄	1	1	yes	characteristic, complemented
14	#4009	—	2	8	no	—
15	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	2	no	—
16	#4002	—	2	2	no	—
17	#1960	—	5	1	no	maximal, Sylow
18	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
19	C₂×C₅×Q₈	C₂×C₄	1	2	yes	—
20	Q₈×D₅	—	2	4	no	—
21	Q₈×D₅	—	2	4	no	—
22	C₂×C₄×D₅	—	2	2	no	—
23	C₂×C₄×D₅	D₄	1	2	yes	—
24	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
25	C₂×Dic₁₀	—	2	2	no	—
26	C₂×Dic₁₀	C₂×C₄	1	2	yes	—
27	C₂×#170	—	2	2	no	—
28	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
29	D₅⋊C₈	—	2	2	no	—
30	D₅⋊C₈	D₄	1	4	yes	—
31	C₅×C₄⋊C₄	—	2	2	no	—
32	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₅	—	2	2	no	—
34	#683	—	2	4	no	—
35	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₅	D₄	1	2	yes	—
37	C₂×C₄⋊Q₈	—	5	1	no	—
38	C₂×#103	—	5	4	no	—
39	C₂×#97	—	5	1	no	—
40	#382	—	5	1	no	—
41	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
42	C₂×AGL(1,𝔽₅)	—	4	2	no	—
43	C₂×AGL(1,𝔽₅)	—	4	4	no	—
44	C₅×Q₈	—	2	4	no	—
45	C₂×C₄×C₅	—	2	2	no	—
46	C₂×C₄×C₅	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅	—	2	2	no	—
49	C₂×Dic₅	C₂²⋊C₄	1	2	yes	—
50	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
51	C₄×D₅	—	2	4	no	—
52	C₄×D₅	—	4	2	no	—
53	C₄×D₅	—	4	2	no	—
54	C₄×D₅	C₂×D₄	1	2	yes	—
55	C₄×D₅	#41	1	2	yes	—
56	Dic₁₀	—	2	4	no	—
57	Dic₁₀	—	4	4	no	—
58	#170	—	2	4	no	—
59	C₂²×Q₈	—	5	2	no	—
60	C₂²×C₈	—	5	2	no	—
61	C₄⋊Q₈	—	5	2	no	—
62	C₄⋊Q₈	—	5	1	no	—
63	C₄⋊Q₈	—	10	2	no	—
64	C₄⋊Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
65	C₂×C₄⋊C₄	—	5	2	no	—
66	C₂×C₄⋊C₄	—	10	2	no	—
67	C₂×C₄²	—	5	1	no	—
68	#103	—	10	8	no	—
69	#97	—	10	2	no	—
70	#95	—	10	2	no	—
71	C₂²×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂×D₅	#124	1	1	yes	characteristic
73	C₂×D₅	#127	1	1	yes	characteristic
74	C₂×D₅	#137	1	4	yes	—
75	AGL(1,𝔽₅)	—	8	2	no	—
76	C₄×C₅	—	2	2	no	—
77	C₄×C₅	—	4	2	no	—
78	C₄×C₅	C₄×D₄	1	2	yes	—
79	Dic₅	—	2	2	no	—
80	Dic₅	—	4	2	no	—
81	Dic₅	#121	1	2	yes	—
82	C₂×Q₈	—	5	2	no	—
83	C₂×Q₈	—	10	2	no	—
84	C₂×Q₈	—	10	4	no	—
85	C₂×Q₈	—	10	4	no	—
86	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
87	C₂²×C₄	—	5	1	no	—
88	C₂²×C₄	—	5	2	no	—
89	C₂²×C₄	—	10	2	no	—
90	C₂²×C₄	—	10	2	no	—
91	C₂×C₈	—	5	4	no	—
92	C₂×C₈	—	10	2	no	—
93	C₂×C₈	—	10	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	10	4	no	—
96	C₄⋊C₄	—	10	4	no	—
97	C₄⋊C₄	—	10	2	no	—
98	C₄²	—	5	2	no	—
99	C₄²	—	5	1	no	—
100	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₂×C₅	#390	1	1	yes	characteristic
102	C₂×C₅	#419	1	2	yes	—
103	D₅	#491	1	2	yes	—
104	D₅	#497	1	2	yes	—
105	C₂³	—	5	1	no	—
106	Q₈	—	2	4	no	—
107	Q₈	—	10	4	no	—
108	Q₈	—	20	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	5	1	no	—
111	C₂×C₄	—	5	2	no	—
112	C₂×C₄	—	5	2	no	—
113	C₂×C₄	—	5	2	no	—
114	C₂×C₄	—	10	4	no	—
115	C₂×C₄	—	10	2	no	—
116	C₂×C₄	—	20	2	no	—
117	C₂×C₄	—	20	2	no	—
118	C₂×C₄	—	20	2	no	—
119	C₂×C₄	—	20	4	no	—
120	C₂×C₄	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
121	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
122	C₈	—	10	4	no	—
123	C₅	#1960	1	1	yes	characteristic, complemented, Sylow
124	C₂²	—	5	1	no	—
125	C₂²	—	5	1	no	—
126	C₂²	—	5	4	no	—
127	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	5	2	no	—
131	C₄	—	10	2	no	—
132	C₄	—	20	2	no	—
133	C₄	—	40	2	no	—
134	C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
135	C₂	—	5	2	no	—
136	C₂	—	5	2	no	—
137	C₂	#65780	1	1	yes	characteristic
138	C₂	#65782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

D₅×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#4009

C₂

yes

maximal

#65689

C₂

yes

maximal, characteristic

#65676

C₂

yes

maximal, characteristic

C₂×Q₈×D₅

C₂²

yes

—

C₂×C₄⋊AGL(1,𝔽₅)

C₂²

yes

—

C₂×D₅⋊C₈

C₂²

yes

—

C₅×C₄⋊Q₈

C₄

yes

characteristic, complemented

#4090

C₄

yes

complemented

D₅×C₄⋊C₄

—

#4034

—

C₄²×D₅

C₂²

yes

characteristic

#4015

C₄

yes

characteristic, complemented

#4009

—

(C₂×AGL(1,𝔽₅))⋊C₄

—

#4002

—

#1960

—

maximal, Sylow

C₂²×AGL(1,𝔽₅)

—

C₂×C₅×Q₈

C₂×C₄

yes

—

Q₈×D₅

—

Q₈×D₅

—

C₂×C₄×D₅

—

C₂×C₄×D₅

D₄

yes

—

C₂×C₄×D₅

C₂³

yes

characteristic

C₂×Dic₁₀

—

C₂×Dic₁₀

C₂×C₄

yes

—

C₂×#170

—

C₄⋊AGL(1,𝔽₅)

—

D₅⋊C₈

—

D₅⋊C₈

D₄

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×C₄

yes

characteristic

C₄⋊Dic₅

—

#683

—

C₄×Dic₅

C₂×C₄

yes

characteristic

C₄×Dic₅

D₄

yes

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×#97

—

#382

—

C₂²×D₅

C₂×D₄

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×AGL(1,𝔽₅)

—

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

—

C₂×Dic₅

C₂²⋊C₄

yes

—

C₂×Dic₅

C₂×D₄

yes

characteristic

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×D₄

yes

—

C₄×D₅

#41

yes

—

Dic₁₀

—

Dic₁₀

—

#170

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

AGL(1,𝔽₅)

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#103

—

#97

—

#95

—

C₂²×C₅

C₂×C₂²⋊C₄

yes

C₂×D₅

yes

C₂×D₅

yes

C₂×D₅

yes

—

AGL(1,𝔽₅)

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₄×D₄

yes

—

Dic₅

—

Dic₅

—

Dic₅

#121

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

100

C₄²

C₂×AGL(1,𝔽₅)

yes

characteristic

101

C₂×C₅

#390

yes

characteristic

102

C₂×C₅

#419

yes

—

103

D₅

#491

yes

—

104

D₅

#497

yes

—

105

C₂³

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

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

—

121

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

122

C₈

—

123

C₅

#1960

yes

characteristic, complemented, Sylow

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

D₄×AGL(1,𝔽₅)

yes

—

135

C₂

—

136

C₂

—

137

C₂

#65780

yes

characteristic

138

C₂

#65782

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×C₄⋊Q₈	C₄
socle	C₂²×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄⋊Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#1960
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊AGL(1,𝔽₅)

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

Fitting

C₅×C₄⋊Q₈

C₄

socle

C₂²×C₅

C₂×C₂²⋊C₄

prime radical

C₂²×D₅

C₂×D₄

cosocle

C₂×C₄×D₅

C₂³

C₄⋊Q₈

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
8. C₅×C₄⋊Q₈	C₄	semidirect
9. #4090	C₄	semidirect
13. #4015	C₄	semidirect
64. C₄⋊Q₈	AGL(1,𝔽₅)	semidirect
123. C₅	#1960	semidirect

H ≅ G/N

type

8. C₅×C₄⋊Q₈

C₄

semidirect

9. #4090

C₄

semidirect

13. #4015

C₄

semidirect

64. C₄⋊Q₈

AGL(1,𝔽₅)

semidirect

123. C₅

#1960

semidirect

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

subgroup	index
D₅×C₄⋊Q₈	2
C₂×#4009	2
#65689	2
#65676	2
#1960	5

subgroup

index

D₅×C₄⋊Q₈

Subgroups — #10604673

Conjugacy classes of subgroups

Complements

Maximal subgroups