Group #10605576 — subgroups

Subgroups — #10605576

Order 640.

548 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₅×#379	C₂	1	1	yes	maximal, characteristic
2	#65536	C₂	1	2	yes	maximal
3	#65238	C₂	1	4	yes	maximal, complemented
4	#65237	C₂	1	8	yes	maximal, complemented
5	C₂⁵×C₅	C₂²	1	1	yes	characteristic
6	C₂×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
7	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
8	C₂×C₂²⋊Dic₅	C₂²	1	4	yes	—
9	C₂×C₂²⋊Dic₅	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₅	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₅	C₂²	1	4	yes	complemented
12	C₂×#683	C₂²	1	8	yes	complemented
13	#3963	C₂²	1	4	yes	complemented
14	#2842	—	5	1	no	maximal, Sylow
15	C₂⁴×C₅	—	4	4	no	—
16	C₂⁴×C₅	—	4	2	no	—
17	C₂⁴×C₅	Q₈	1	4	yes	—
18	C₂⁴×C₅	C₂³	1	2	yes	—
19	C₂⁴×C₅	C₂³	1	1	yes	characteristic
20	C₂²×C₄×C₅	C₂³	1	4	yes	—
21	C₂²×Dic₅	C₂³	1	8	yes	—
22	C₅×C₂²⋊C₄	—	4	4	no	—
23	C₅×C₂²⋊C₄	—	4	2	no	—
24	C₂²⋊Dic₅	—	4	4	no	—
25	C₂²⋊Dic₅	—	4	4	no	—
26	C₂²⋊Dic₅	—	4	4	no	—
27	C₄⋊Dic₅	—	4	4	no	—
28	#683	—	4	8	no	—
29	#399	—	5	4	no	—
30	#396	—	5	8	no	—
31	#379	—	5	2	no	—
32	#379	D₅	1	1	yes	characteristic
33	C₂³×C₅	—	2	4	no	—
34	C₂³×C₅	—	2	4	no	—
35	C₂³×C₅	—	2	4	no	—
36	C₂³×C₅	—	2	4	no	—
37	C₂³×C₅	—	2	2	no	—
38	C₂³×C₅	—	4	8	no	—
39	C₂³×C₅	—	4	4	no	—
40	C₂³×C₅	—	4	16	no	—
41	C₂³×C₅	C₂×Q₈	1	4	yes	—
42	C₂³×C₅	C₂×Q₈	1	2	yes	—
43	C₂³×C₅	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₅	—	2	8	no	—
45	C₂×C₄×C₅	—	2	4	no	—
46	C₂×Dic₅	—	2	8	no	—
47	C₂×Dic₅	—	2	8	no	—
48	C₂×Dic₅	—	2	8	no	—
49	C₂⁵	C₂×D₅	1	1	yes	characteristic
50	C₂×C₄⋊C₄	—	5	8	no	—
51	C₂×C₄⋊C₄	—	5	4	no	—
52	C₂×C₂²⋊C₄	—	5	4	no	—
53	C₂×C₂²⋊C₄	—	5	4	no	—
54	C₂×C₂²⋊C₄	—	5	4	no	—
55	C₂×C₂²⋊C₄	C₂×D₅	1	4	yes	—
56	C₂×C₂²⋊C₄	C₂×D₅	1	2	yes	—
57	#95	—	5	4	no	—
58	C₂²×C₅	—	2	4	no	—
59	C₂²×C₅	—	2	4	no	—
60	C₂²×C₅	—	2	4	no	—
61	C₂²×C₅	—	2	4	no	—
62	C₂²×C₅	—	2	2	no	—
63	C₂²×C₅	—	4	16	no	—
64	C₂²×C₅	—	4	8	no	—
65	C₂²×C₅	—	4	4	no	—
66	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₅	H(5,𝔽₂)	1	4	yes	—
68	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
69	C₄×C₅	—	4	4	no	—
70	Dic₅	—	4	8	no	—
71	C₂⁴	—	4	4	no	—
72	C₂⁴	—	4	2	no	—
73	C₂⁴	Dic₁₀	1	4	yes	—
74	C₂⁴	C₂²×D₅	1	2	yes	—
75	C₂⁴	C₂²×D₅	1	1	yes	characteristic
76	C₂²×C₄	—	5	8	no	—
77	C₂²×C₄	C₂²×D₅	1	4	yes	—
78	C₄⋊C₄	—	20	8	no	—
79	C₄⋊C₄	—	20	4	no	—
80	C₂²⋊C₄	—	4	4	no	—
81	C₂²⋊C₄	—	4	2	no	—
82	C₂²⋊C₄	—	20	4	no	—
83	C₂²⋊C₄	—	20	4	no	—
84	C₂²⋊C₄	—	20	4	no	—
85	C₂×C₅	—	4	4	no	—
86	C₂×C₅	—	4	2	no	—
87	C₂×C₅	#543	1	2	yes	—
88	C₂×C₅	#543	1	1	yes	characteristic
89	C₂×C₅	#560	1	4	yes	—
90	C₂³	—	2	4	no	—
91	C₂³	—	2	2	no	—
92	C₂³	—	2	4	no	—
93	C₂³	—	2	4	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	4	8	no	—
96	C₂³	—	4	4	no	—
97	C₂³	—	4	16	no	—
98	C₂³	C₂×Dic₁₀	1	4	yes	—
99	C₂³	C₂×Dic₁₀	1	2	yes	—
100	C₂³	C₂³×D₅	1	1	yes	characteristic
101	C₂×C₄	—	2	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	10	8	no	—
104	C₂×C₄	—	10	8	no	—
105	C₂×C₄	—	10	8	no	—
106	C₅	#2842	1	1	yes	characteristic, complemented, Sylow
107	C₂²	—	2	4	no	—
108	C₂²	—	2	2	no	—
109	C₂²	—	2	4	no	—
110	C₂²	—	2	4	no	—
111	C₂²	—	2	4	no	—
112	C₂²	—	4	16	no	—
113	C₂²	—	4	8	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂²×Dic₁₀	1	1	yes	characteristic
116	C₂²	#4144	1	4	yes	—
117	C₂²	#4144	1	2	yes	—
118	C₄	—	4	4	no	—
119	C₄	—	20	8	no	—
120	C₂	—	4	4	no	—
121	C₂	—	4	2	no	—
122	C₂	#65817	1	2	yes	—
123	C₂	#65817	1	1	yes	characteristic
124	C₂	#65929	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#379

C₂

yes

maximal, characteristic

#65536

C₂

yes

maximal

#65238

C₂

yes

maximal, complemented

#65237

C₂

yes

maximal, complemented

C₂⁵×C₅

C₂²

yes

characteristic

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₅

C₂²

yes

—

C₂×C₂²⋊Dic₅

C₂²

yes

—

C₂×C₂²⋊Dic₅

C₂²

yes

—

C₂×C₄⋊Dic₅

C₂²

yes

complemented

C₂×#683

C₂²

yes

complemented

#3963

C₂²

yes

complemented

#2842

—

maximal, Sylow

C₂⁴×C₅

—

C₂⁴×C₅

—

C₂⁴×C₅

Q₈

yes

—

C₂⁴×C₅

C₂³

yes

—

C₂⁴×C₅

C₂³

yes

characteristic

C₂²×C₄×C₅

C₂³

yes

—

C₂²×Dic₅

C₂³

yes

—

C₅×C₂²⋊C₄

—

C₅×C₂²⋊C₄

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

D₅

yes

characteristic

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂×Q₈

yes

—

C₂³×C₅

C₂×Q₈

yes

—

C₂³×C₅

C₂⁴

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂⁵

C₂×D₅

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₅

yes

—

C₂×C₂²⋊C₄

C₂×D₅

yes

—

#95

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×Q₈

yes

characteristic

C₂²×C₅

H(5,𝔽₂)

yes

—

C₂²×C₅

H(5,𝔽₂)

yes

—

C₄×C₅

—

Dic₅

—

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₁₀

yes

—

C₂⁴

C₂²×D₅

yes

—

C₂⁴

C₂²×D₅

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

#543

yes

—

C₂×C₅

#543

yes

characteristic

C₂×C₅

#560

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×Dic₁₀

yes

—

C₂³

C₂×Dic₁₀

yes

—

100

C₂³

C₂³×D₅

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₅

#2842

yes

characteristic, complemented, Sylow

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂²×Dic₁₀

yes

characteristic

116

C₂²

#4144

yes

—

117

C₂²

#4144

yes

—

118

C₄

—

119

C₄

—

120

C₂

—

121

C₂

—

122

C₂

#65817

yes

—

123

C₂

#65817

yes

characteristic

124

C₂

#65929

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅
commutator, socle, cosocle	C₂³×C₅	C₂⁴
Fitting	C₅×#379	C₂
prime radical	C₂⁵×C₅	C₂²
hypercenter	#379	D₅
hypocenter	C₅	#2842
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₅

commutator, socle, cosocle

C₂³×C₅

C₂⁴

Fitting

C₅×#379

C₂

prime radical

C₂⁵×C₅

C₂²

D₅

C₅

G ▷ C₂³×C₅ ▷ C₅

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
3. #65238	C₂	semidirect
4. #65237	C₂	semidirect
11. C₂×C₄⋊Dic₅	C₂²	semidirect
12. C₂×#683	C₂²	semidirect
13. #3963	C₂²	semidirect
106. C₅	#2842	semidirect

H ≅ G/N

type

3. #65238

C₂

semidirect

4. #65237

C₂

semidirect

11. C₂×C₄⋊Dic₅

C₂²

semidirect

12. C₂×#683

C₂²

semidirect

13. #3963

C₂²

semidirect

106. C₅

#2842

semidirect

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

subgroup	index
C₅×#379	2
#65536	2
#65238	2
#65237	2
#2842	5

subgroup

index

Subgroups — #10605576

Conjugacy classes of subgroups

Complements

Maximal subgroups