Subgroups — #10603846

Order 640.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65689	C₂	1	2	yes	maximal
2	C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#64893	C₂	1	4	yes	maximal
4	C₂×C₄⋊AGL(1,𝔽₅)	C₂²	1	4	yes	—
5	C₂×C₈×D₅	C₂²	1	2	yes	—
6	C₄²×D₅	C₂²	1	1	yes	characteristic
7	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	4	no	—
8	C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
9	C₈×Dic₅	C₄	1	2	yes	complemented
10	C₄×#168	C₄	1	1	yes	characteristic, complemented
11	#1835	—	5	1	no	maximal, Sylow
12	C₂²×AGL(1,𝔽₅)	—	2	4	no	—
13	C₂×C₄×D₅	D₄	1	2	yes	—
14	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
15	C₄⋊AGL(1,𝔽₅)	—	2	8	no	—
16	C₂×C₅×C₈	C₂×C₄	1	2	yes	—
17	C₄²×C₅	C₂×C₄	1	1	yes	characteristic
18	C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₅	Q₈	1	2	yes	—
20	C₂×#168	C₂×C₄	1	2	yes	—
21	C₈×D₅	—	2	4	no	—
22	C₂×C₄×C₈	—	5	1	no	—
23	#382	—	5	2	no	—
24	#340	—	5	4	no	—
25	C₂²×D₅	C₂×D₄	1	1	yes	characteristic
26	C₂×AGL(1,𝔽₅)	—	4	4	no	—
27	C₂×AGL(1,𝔽₅)	—	4	8	no	—
28	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₅	C₂×Q₈	1	1	yes	characteristic
32	C₄×D₅	—	2	4	no	—
33	C₄×D₅	#41	1	4	yes	—
34	C₅×C₈	—	2	2	no	—
35	#168	—	2	2	no	—
36	C₂²×C₈	—	5	2	no	—
37	C₂×C₄⋊C₄	—	5	4	no	—
38	C₂×C₄²	—	5	1	no	—
39	C₄×C₈	—	5	2	no	—
40	C₄×C₈	—	5	1	no	—
41	C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
42	#95	—	10	4	no	—
43	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×D₅	#124	1	2	yes	—
45	C₂×D₅	#135	1	4	yes	—
46	AGL(1,𝔽₅)	—	8	4	no	—
47	C₄×C₅	—	2	2	no	—
48	C₄×C₅	#117	1	2	yes	—
49	Dic₅	—	2	2	no	—
50	Dic₅	#125	1	2	yes	—
51	C₂²×C₄	—	5	1	no	—
52	C₂²×C₄	—	5	2	no	—
53	C₂²×C₄	—	10	4	no	—
54	C₂×C₈	—	5	2	no	—
55	C₂×C₈	—	10	4	no	—
56	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
57	C₄⋊C₄	—	10	8	no	—
58	C₄²	—	5	2	no	—
59	C₄²	—	5	1	no	—
60	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂×C₅	#382	1	1	yes	characteristic
62	C₂×C₅	#427	1	2	yes	—
63	D₅	#488	1	4	yes	—
64	C₂³	—	5	1	no	—
65	C₂×C₄	—	5	1	no	—
66	C₂×C₄	—	5	4	no	—
67	C₂×C₄	—	5	2	no	—
68	C₂×C₄	—	10	4	no	—
69	C₂×C₄	—	20	4	no	—
70	C₂×C₄	—	20	8	no	—
71	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
72	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₈	—	10	2	no	—
75	C₅	#1835	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	5	2	no	—
77	C₂²	—	5	4	no	—
78	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	5	2	no	—
81	C₄	—	10	2	no	—
82	C₄	—	40	4	no	—
83	C₄	#4130	1	2	yes	—
84	C₂	—	5	4	no	—
85	C₂	#65689	1	1	yes	characteristic
86	C₂	#65721	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₈	C₄
socle	C₂²×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₅	C₂×D₄
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1835
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
8. C₄×C₅×C₈	C₄	semidirect
9. C₈×Dic₅	C₄	semidirect
10. C₄×#168	C₄	semidirect
41. C₄×C₈	AGL(1,𝔽₅)	semidirect
75. C₅	#1835	semidirect

Maximal subgroups

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

subgroup	index
#65689	2
C₄×C₈×D₅	2
#64893	2
#1835	5