Group #10617203 — subgroups

Subgroups — #10617203

Order 640.

432 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 204 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	C₄×#713	C₂	1	1	yes	maximal, characteristic
2	Q₈×#170	C₂	1	2	yes	maximal, complemented
3	#65702	C₂	1	6	yes	maximal, complemented
4	D₅⋊(C₄⋊C₈)	C₂	1	3	yes	maximal, complemented
5	#65678	C₂	1	3	yes	maximal, complemented
6	C₂×#713	C₄	1	1	yes	characteristic
7	C₂×D₅⋊C₈	C₂²	1	6	yes	—
8	C₄×C₅×Q₈	C₄	1	1	yes	characteristic
9	Q₈×Dic₅	C₂²	1	1	yes	characteristic
10	#4039	C₂²	1	3	yes	—
11	#4038	C₄	1	3	yes	—
12	C₄×D₂₀	C₄	1	3	yes	—
13	C₄²×D₅	C₂²	1	3	yes	—
14	#4004	C₂²	1	1	yes	characteristic
15	#4004	C₂²	1	3	yes	—
16	#4003	C₂²	1	6	yes	—
17	C₄⋊#170	C₂²	1	6	yes	—
18	C₄×#170	C₂²	1	6	yes	—
19	#2988	—	5	1	no	maximal, Sylow
20	C₂×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
21	#713	C₈	1	8	yes	complemented
22	C₂×D₂₀	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₅	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₅	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₅	C₂³	1	3	yes	—
26	C₂×#170	C₂³	1	6	yes	—
27	C₂×#170	C₂³	1	2	yes	—
28	D₅⋊C₈	—	4	6	no	—
29	C₅×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₄²×C₅	C₂×C₄	1	3	yes	—
31	#685	C₂×C₄	1	6	yes	—
32	C₄⋊Dic₅	C₂×C₄	1	3	yes	—
33	C₄×Dic₅	C₂×C₄	1	3	yes	—
34	C₄×Dic₅	C₂³	1	3	yes	—
35	C₄×Dic₅	C₂³	1	1	yes	characteristic
36	C₄×#41	—	5	1	no	—
37	C₈×Q₈	—	5	2	no	—
38	C₈×D₄	—	5	6	no	—
39	#431	—	5	3	no	—
40	C₂×C₄⋊C₈	—	5	3	no	—
41	C₂²×D₅	C₂²×C₄	1	3	yes	—
42	C₅×Q₈	C₂×C₈	1	4	yes	complemented
43	C₂×C₄×C₅	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅	C₂²×C₄	1	3	yes	—
46	C₂×Dic₅	C₂²×C₄	1	3	yes	—
47	C₂×Dic₅	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
49	D₂₀	C₂×C₈	1	12	yes	complemented
50	C₄×D₅	—	2	3	no	—
51	C₄×D₅	—	2	3	no	—
52	C₄×D₅	—	4	3	no	—
53	C₄×D₅	C₂×C₈	1	12	yes	complemented
54	#170	—	2	6	no	—
55	#170	—	2	2	no	—
56	C₂×#41	—	5	1	no	—
57	C₂²×C₈	—	5	6	no	—
58	C₄×Q₈	—	5	1	no	—
59	C₄×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
60	C₄×D₄	—	5	3	no	—
61	C₄×D₄	—	5	3	no	—
62	#117	—	5	3	no	—
63	C₂×C₄²	—	5	3	no	—
64	C₄⋊C₈	—	5	6	no	—
65	C₄⋊C₈	—	5	3	no	—
66	C₄⋊C₈	—	5	1	no	—
67	C₂²⋊C₈	—	5	6	no	—
68	C₄×C₈	—	5	6	no	—
69	C₂²×C₅	C₂³×C₄	1	1	yes	characteristic
70	C₂×D₅	—	2	3	no	—
71	C₂×D₅	—	2	3	no	—
72	C₂×D₅	C₂²×C₈	1	6	yes	—
73	C₄×C₅	—	2	3	no	—
74	C₄×C₅	—	2	1	no	—
75	C₄×C₅	C₂²×C₈	1	6	yes	—
76	Dic₅	—	2	3	no	—
77	Dic₅	C₂²×C₈	1	2	yes	—
78	Dic₅	H(5,𝔽₂)	1	1	yes	characteristic
79	Dic₅	2₋¹⁺⁴	1	1	yes	characteristic
80	#41	—	5	8	no	—
81	C₂×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×D₄	—	5	3	no	—
83	C₂²×C₄	—	5	3	no	—
84	C₂²×C₄	—	5	3	no	—
85	C₂²×C₄	—	5	3	no	—
86	C₂×C₈	—	5	2	no	—
87	C₂×C₈	—	5	6	no	—
88	C₂×C₈	—	20	6	no	—
89	C₄⋊C₄	—	5	3	no	—
90	C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	3	yes	—
91	C₂²⋊C₄	—	5	6	no	—
92	C₄²	—	5	3	no	—
93	C₄²	—	5	1	no	—
94	C₄²	—	5	3	no	—
95	C₄²	C₂×AGL(1,𝔽₅)	1	3	yes	—
96	C₂×C₅	#520	1	1	yes	characteristic
97	C₂×C₅	C₂³×C₈	1	1	yes	characteristic
98	C₂×C₅	#568	1	1	yes	characteristic
99	D₅	—	2	6	no	—
100	C₂³	—	5	3	no	—
101	Q₈	D₅⋊C₈	1	4	yes	complemented
102	D₄	—	5	12	no	—
103	C₂×C₄	—	5	1	no	—
104	C₂×C₄	—	5	1	no	—
105	C₂×C₄	—	5	12	no	—
106	C₂×C₄	—	5	3	no	—
107	C₂×C₄	—	10	3	no	—
108	C₂×C₄	—	10	3	no	—
109	C₂×C₄	—	20	3	no	—
110	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
111	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
112	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
113	C₈	—	10	6	no	—
114	C₈	—	10	2	no	—
115	C₅	#2988	1	1	yes	characteristic, complemented, Sylow
116	C₂²	—	5	6	no	—
117	C₂²	—	10	3	no	—
118	C₂²	—	10	3	no	—
119	C₂²	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	2	3	no	—
122	C₄	—	5	2	no	—
123	C₄	—	5	1	no	—
124	C₄	—	5	1	no	—
125	C₄	—	10	3	no	—
126	C₄	C₂×D₅⋊C₈	1	6	yes	—
127	C₂	—	10	6	no	—
128	C₂	C₂²×D₅⋊C₈	1	1	yes	characteristic
129	C₂	#66260	1	1	yes	characteristic
130	C₂	#66266	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#713

C₂

yes

maximal, characteristic

Q₈×#170

C₂

yes

maximal, complemented

#65702

C₂

yes

maximal, complemented

D₅⋊(C₄⋊C₈)

C₂

yes

maximal, complemented

#65678

C₂

yes

maximal, complemented

C₂×#713

C₄

yes

characteristic

C₂×D₅⋊C₈

C₂²

yes

—

C₄×C₅×Q₈

C₄

yes

characteristic

Q₈×Dic₅

C₂²

yes

characteristic

#4039

C₂²

yes

—

#4038

C₄

yes

—

C₄×D₂₀

C₄

yes

—

C₄²×D₅

C₂²

yes

—

#4004

C₂²

yes

characteristic

#4004

C₂²

yes

—

#4003

C₂²

yes

—

C₄⋊#170

C₂²

yes

—

C₄×#170

C₂²

yes

—

#2988

—

maximal, Sylow

C₂×C₅×Q₈

C₂×C₄

yes

characteristic

#713

C₈

yes

complemented

C₂×D₂₀

C₂×C₄

yes

—

C₂×C₄×D₅

C₂×C₄

yes

—

C₂×C₄×D₅

C₂×C₄

yes

—

C₂×C₄×D₅

C₂³

yes

—

C₂×#170

C₂³

yes

—

C₂×#170

C₂³

yes

—

D₅⋊C₈

—

C₅×C₄⋊C₄

C₂×C₄

yes

—

C₄²×C₅

C₂×C₄

yes

—

#685

C₂×C₄

yes

—

C₄⋊Dic₅

C₂×C₄

yes

—

C₄×Dic₅

C₂×C₄

yes

—

C₄×Dic₅

C₂³

yes

—

C₄×Dic₅

C₂³

yes

characteristic

C₄×#41

—

C₈×Q₈

—

C₈×D₄

—

#431

—

C₂×C₄⋊C₈

—

C₂²×D₅

C₂²×C₄

yes

—

C₅×Q₈

C₂×C₈

yes

complemented

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×C₄×C₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂²×C₄

yes

—

C₂×Dic₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

C₂⁴

yes

characteristic

D₂₀

C₂×C₈

yes

complemented

C₄×D₅

—

C₄×D₅

—

C₄×D₅

—

C₄×D₅

C₂×C₈

yes

complemented

#170

—

#170

—

C₂×#41

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

AGL(1,𝔽₅)

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₅

C₂³×C₄

yes

characteristic

C₂×D₅

—

C₂×D₅

—

C₂×D₅

C₂²×C₈

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

C₂²×C₈

yes

—

Dic₅

—

Dic₅

C₂²×C₈

yes

—

Dic₅

H(5,𝔽₂)

yes

characteristic

Dic₅

2₋¹⁺⁴

yes

characteristic

#41

—

C₂×Q₈

C₂×AGL(1,𝔽₅)

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×AGL(1,𝔽₅)

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₅

#520

yes

characteristic

C₂×C₅

C₂³×C₈

yes

characteristic

C₂×C₅

#568

yes

characteristic

D₅

—

100

C₂³

—

101

Q₈

D₅⋊C₈

yes

complemented

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

111

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

112

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

—

113

C₈

—

114

C₈

—

115

C₅

#2988

yes

characteristic, complemented, Sylow

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×AGL(1,𝔽₅)

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×D₅⋊C₈

yes

—

127

C₂

—

128

C₂

C₂²×D₅⋊C₈

yes

characteristic

129

C₂

#66260

yes

characteristic

130

C₂

#66266

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₈
Fitting	C₄×C₅×Q₈	C₄
socle	C₂²×C₅	C₂³×C₄
prime radical	C₂×#713	C₄
cosocle	C₂×Dic₅	C₂⁴
hypercenter	C₄×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#2988
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₅)

commutator

C₂×C₅

C₂³×C₈

Fitting

C₄×C₅×Q₈

C₄

socle

C₂²×C₅

C₂³×C₄

prime radical

C₂×#713

C₄

cosocle

C₂×Dic₅

C₂⁴

C₄×Q₈

C₅

G ▷ C₂×C₅ ▷ C₅

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
2. Q₈×#170	C₂	semidirect
3. #65702	C₂	semidirect
4. D₅⋊(C₄⋊C₈)	C₂	semidirect
5. #65678	C₂	semidirect
21. #713	C₈	semidirect
42. C₅×Q₈	C₂×C₈	semidirect
49. D₂₀	C₂×C₈	semidirect
53. C₄×D₅	C₂×C₈	semidirect
101. Q₈	D₅⋊C₈	semidirect
115. C₅	#2988	semidirect

H ≅ G/N

type

2. Q₈×#170

C₂

semidirect

3. #65702

C₂

semidirect

4. D₅⋊(C₄⋊C₈)

C₂

semidirect

5. #65678

C₂

semidirect

21. #713

C₈

semidirect

42. C₅×Q₈

C₂×C₈

semidirect

49. D₂₀

C₂×C₈

semidirect

53. C₄×D₅

C₂×C₈

semidirect

101. Q₈

D₅⋊C₈

semidirect

115. C₅

#2988

semidirect

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

subgroup	index
C₄×#713	2
Q₈×#170	2
#65702	2
D₅⋊(C₄⋊C₈)	2
#65678	2
#2988	5

subgroup

index

D₅⋊(C₄⋊C₈)

Subgroups — #10617203

Conjugacy classes of subgroups

Complements

Maximal subgroups