Subgroups — #10603857

Order 640.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#65682	C₂	1	2	yes	maximal
2	C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	#64896	C₂	1	4	yes	maximal
4	C₂×#700	C₂²	1	4	yes	—
5	C₂×C₈×D₅	C₂²	1	2	yes	—
6	C₄²×D₅	C₂²	1	1	yes	characteristic
7	#4004	—	2	4	no	—
8	C₄×C₅×C₈	C₄	1	1	yes	characteristic
9	C₈×Dic₅	C₄	1	2	yes	—
10	C₄×#168	C₄	1	1	yes	characteristic
11	#1837	—	5	1	no	maximal, Sylow
12	C₂×C₄×D₅	Q₈	1	2	yes	—
13	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
14	C₂×#170	—	2	4	no	—
15	#700	—	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	D₄	1	2	yes	—
20	C₂×#168	C₂×C₄	1	2	yes	—
21	C₈×D₅	—	2	4	no	—
22	#423	—	5	2	no	—
23	C₂×C₄×C₈	—	5	1	no	—
24	#341	—	5	4	no	—
25	C₂²×D₅	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₅	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₅	C₄⋊C₄	1	2	yes	—
29	C₂×Dic₅	C₂×D₄	1	1	yes	characteristic
30	C₄×D₅	—	2	4	no	—
31	C₄×D₅	#41	1	4	yes	—
32	#170	—	4	4	no	—
33	C₅×C₈	—	2	2	no	—
34	#168	—	2	2	no	—
35	C₂×M₁₆	—	5	4	no	—
36	C₂²×C₈	—	5	2	no	—
37	C₂×C₄²	—	5	1	no	—
38	C₄⋊C₈	—	10	4	no	—
39	C₄×C₈	—	5	2	no	—
40	C₄×C₈	—	5	1	no	—
41	C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₂²×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×D₅	—	2	2	no	—
44	C₂×D₅	#125	1	2	yes	—
45	C₄×C₅	—	2	2	no	—
46	C₄×C₅	#117	1	2	yes	—
47	Dic₅	#108	1	4	yes	—
48	Dic₅	#124	1	2	yes	—
49	C₂²×C₄	—	5	1	no	—
50	C₂²×C₄	—	5	2	no	—
51	M₁₆	—	10	8	no	—
52	C₂×C₈	—	5	2	no	—
53	C₂×C₈	—	10	4	no	—
54	C₂×C₈	—	10	4	no	—
55	C₂×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
56	C₄²	—	5	2	no	—
57	C₄²	—	5	1	no	—
58	C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
59	C₂×C₅	#382	1	1	yes	characteristic
60	C₂×C₅	C₂×#108	1	2	yes	—
61	D₅	—	2	2	no	—
62	C₂³	—	5	1	no	—
63	C₂×C₄	—	5	1	no	—
64	C₂×C₄	—	5	2	no	—
65	C₂×C₄	—	5	4	no	—
66	C₂×C₄	—	10	4	no	—
67	C₂×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
68	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₈	—	2	2	no	—
70	C₈	—	10	2	no	—
71	C₈	—	20	4	no	—
72	C₅	#1837	1	1	yes	characteristic, complemented, Sylow
73	C₂²	—	5	2	no	—
74	C₂²	—	10	2	no	—
75	C₂²	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	5	2	no	—
78	C₄	—	5	4	no	—
79	C₄	#4130	1	2	yes	—
80	C₂	—	10	2	no	—
81	C₂	#65689	1	1	yes	characteristic
82	C₂	D₅⋊#108	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×Q₈
cosocle	C₂×C₄×D₅	C₂³
hypercenter	C₄×C₈	AGL(1,𝔽₅)
hypocenter	C₅	#1837
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
72. C₅	#1837	semidirect

Maximal subgroups

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

subgroup	index
#65682	2
C₄×C₈×D₅	2
#64896	2
#1837	5