Group #10616236 — subgroups

Subgroups — C₅×#2641 #10616236

Order 640.

584 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 200 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×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#394	C₂	1	8	yes	maximal, complemented
4	C₅×#390	C₂	1	4	yes	maximal, complemented
5	C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₅×D₄	C₂²	1	2	yes	—
7	C₂²×C₅×D₄	C₂²	1	4	yes	—
8	C₅×#127	—	2	4	no	—
9	C₅×#125	—	2	4	no	—
10	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₅×C₂²⋊C₄	C₂²	1	16	yes	—
13	C₂×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₅×#95	C₂²	1	4	yes	—
16	#2641	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂⁴×C₅	C₂³	1	4	yes	—
18	C₂×C₅×D₄	—	2	8	no	—
19	C₂×C₅×D₄	—	2	4	no	—
20	C₂×C₅×D₄	—	2	8	no	—
21	C₂×C₅×D₄	—	4	8	no	—
22	C₂²×C₄×C₅	C₂³	1	4	yes	—
23	C₂²×C₄×C₅	C₂³	1	4	yes	—
24	C₂²×C₄×C₅	C₂³	1	2	yes	—
25	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₅×C₄⋊C₄	—	2	8	no	—
27	C₅×C₄⋊C₄	—	2	4	no	—
28	C₅×C₂²⋊C₄	—	4	16	no	—
29	C₄²×C₅	—	2	4	no	—
30	C₄²×C₅	D₄	1	4	yes	—
31	C₂×#127	C₂×C₅	1	1	yes	characteristic
32	C₂×#125	C₂×C₅	1	1	yes	characteristic, complemented
33	#394	C₂×C₅	1	8	yes	complemented
34	#390	C₂×C₅	1	4	yes	complemented
35	#376	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂³×C₅	—	2	4	no	—
37	C₂³×C₅	—	2	8	no	—
38	C₂³×C₅	—	2	16	no	—
39	C₂³×C₅	C₂⁴	1	1	yes	characteristic
40	C₅×D₄	—	4	16	no	—
41	C₅×D₄	—	4	8	no	—
42	C₂×C₄×C₅	—	2	4	no	—
43	C₂×C₄×C₅	—	2	4	no	—
44	C₂×C₄×C₅	—	2	4	no	—
45	C₂×C₄×C₅	—	2	8	no	—
46	C₂×C₄×C₅	—	2	8	no	—
47	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
48	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₅	#41	1	4	yes	—
50	C₂²×D₄	C₂²×C₅	1	2	yes	—
51	C₂²×D₄	C₂²×C₅	1	4	yes	—
52	#127	—	2	4	no	—
53	#125	—	2	4	no	—
54	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
55	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
56	C₂×C₂²⋊C₄	C₂²×C₅	1	16	yes	—
57	C₂×C₄²	C₂²×C₅	1	2	yes	—
58	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
59	#95	C₂²×C₅	1	4	yes	—
60	C₂²×C₅	—	4	4	no	—
61	C₂²×C₅	—	4	8	no	—
62	C₂²×C₅	—	4	16	no	—
63	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
64	C₂²×C₅	C₂×#41	1	2	yes	—
65	C₂²×C₅	H(5,𝔽₂)	1	4	yes	—
66	C₄×C₅	—	2	4	no	—
67	C₄×C₅	—	4	4	no	—
68	C₄×C₅	—	4	4	no	—
69	C₄×C₅	#124	1	4	yes	—
70	C₂⁴	C₂³×C₅	1	4	yes	—
71	C₂×D₄	—	2	8	no	—
72	C₂×D₄	—	2	4	no	—
73	C₂×D₄	—	2	8	no	—
74	C₂×D₄	—	4	8	no	—
75	C₂²×C₄	C₂³×C₅	1	4	yes	—
76	C₂²×C₄	C₂³×C₅	1	4	yes	—
77	C₂²×C₄	C₂³×C₅	1	2	yes	—
78	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₂²⋊C₄	—	4	16	no	—
82	C₄²	—	2	4	no	—
83	C₄²	C₅×D₄	1	4	yes	—
84	C₂×C₅	—	8	4	no	—
85	C₂×C₅	C₂×#124	1	1	yes	characteristic
86	C₂×C₅	#535	1	2	yes	—
87	C₂×C₅	#540	1	4	yes	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	8	no	—
90	C₂³	—	2	16	no	—
91	C₂³	C₂⁴×C₅	1	1	yes	characteristic
92	D₄	—	4	16	no	—
93	D₄	—	4	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	8	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	C₂×C₅×D₄	1	4	yes	—
100	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
101	C₂×C₄	C₅×#41	1	4	yes	—
102	C₅	#2641	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²	—	4	4	no	—
104	C₂²	—	4	8	no	—
105	C₂²	—	4	16	no	—
106	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
107	C₂²	C₂×C₅×#41	1	2	yes	—
108	C₂²	C₅×H(5,𝔽₂)	1	4	yes	—
109	C₄	—	2	4	no	—
110	C₄	—	4	4	no	—
111	C₄	—	4	4	no	—
112	C₄	C₅×#124	1	4	yes	—
113	C₂	—	8	4	no	—
114	C₂	C₂×C₅×#124	1	1	yes	characteristic
115	C₂	C₅×#535	1	2	yes	—
116	C₂	C₅×#540	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#127

C₂

yes

maximal, characteristic

C₂×C₅×#125

C₂

yes

maximal, characteristic, complemented

C₅×#394

C₂

yes

maximal, complemented

C₅×#390

C₂

yes

maximal, complemented

C₅×#376

C₂

yes

maximal, characteristic, complemented

C₂²×C₅×D₄

C₂²

yes

—

C₂²×C₅×D₄

C₂²

yes

—

C₅×#127

—

C₅×#125

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₄⋊C₄

C₂²

yes

—

C₂×C₅×C₂²⋊C₄

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

—

C₂×C₄²×C₅

C₂²

yes

characteristic

C₅×#95

C₂²

yes

—

#2641

C₅

yes

maximal, characteristic, direct product factor, Sylow

C₂⁴×C₅

C₂³

yes

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₂²⋊C₄

—

C₄²×C₅

—

C₄²×C₅

D₄

yes

—

C₂×#127

C₂×C₅

yes

characteristic

C₂×#125

C₂×C₅

yes

characteristic, complemented

#394

C₂×C₅

yes

complemented

#390

C₂×C₅

yes

complemented

#376

C₂×C₅

yes

characteristic, complemented

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

—

C₂³×C₅

C₂⁴

yes

characteristic

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

C₂×D₄

yes

—

C₂×C₄×C₅

#41

yes

—

C₂²×D₄

C₂²×C₅

yes

—

C₂²×D₄

C₂²×C₅

yes

—

#127

—

#125

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₄⋊C₄

C₂²×C₅

yes

—

C₂×C₂²⋊C₄

C₂²×C₅

yes

—

C₂×C₄²

C₂²×C₅

yes

—

C₂×C₄²

C₂²×C₅

yes

characteristic

#95

C₂²×C₅

yes

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

C₂²×D₄

yes

characteristic

C₂²×C₅

C₂×#41

yes

—

C₂²×C₅

H(5,𝔽₂)

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#124

yes

—

C₂⁴

C₂³×C₅

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

—

C₂²×C₄

C₂³×C₅

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₅×D₄

yes

—

C₂×C₅

—

C₂×C₅

C₂×#124

yes

characteristic

C₂×C₅

#535

yes

—

C₂×C₅

#540

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂⁴×C₅

yes

characteristic

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₅×D₄

yes

—

100

C₂×C₄

C₂×C₅×D₄

yes

—

101

C₂×C₄

C₅×#41

yes

—

102

C₅

#2641

yes

characteristic, direct product factor, Sylow

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

C₂²×C₅×D₄

yes

characteristic

107

C₂²

C₂×C₅×#41

yes

—

108

C₂²

C₅×H(5,𝔽₂)

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₅×#124

yes

—

113

C₂

—

114

C₂

C₂×C₅×#124

yes

characteristic

115

C₂

C₅×#535

yes

—

116

C₂

C₅×#540

yes

—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

subgroup

G/H

center, socle

C₂³×C₅

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₅

prime radical

C₂×C₅×#127

C₂

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₅ ◁ G

N	H ≅ G/N	type
2. C₂×C₅×#125	C₂	semidirect
3. C₅×#394	C₂	semidirect
4. C₅×#390	C₂	semidirect
5. C₅×#376	C₂	semidirect
16. #2641	C₅	direct (N × H)
32. C₂×#125	C₂×C₅	semidirect
33. #394	C₂×C₅	semidirect
34. #390	C₂×C₅	semidirect
35. #376	C₂×C₅	semidirect
102. C₅	#2641	direct (N × H)

H ≅ G/N

type

2. C₂×C₅×#125

C₂

semidirect

3. C₅×#394

C₂

semidirect

4. C₅×#390

C₂

semidirect

5. C₅×#376

C₂

semidirect

16. #2641

C₅

direct (N × H)

32. C₂×#125

C₂×C₅

semidirect

33. #394

C₂×C₅

semidirect

34. #390

C₂×C₅

semidirect

35. #376

C₂×C₅

semidirect

102. C₅

#2641

direct (N × H)

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

subgroup	index
C₂×C₅×#127	2
C₂×C₅×#125	2
C₅×#394	2
C₅×#390	2
C₅×#376	2
#2641	5

subgroup

index

C₂×C₅×#127

C₂×C₅×#125

Subgroups — C₅×#2641 #10616236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C5×#2641 #10616236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2641 #10616236