Subgroups — C₄×#700 #65677

Order 320.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#700	C₂	1	2	yes	maximal
2	C₄²×D₅	C₂	1	1	yes	maximal, characteristic
3	#4002	C₂	1	2	yes	maximal, complemented
4	C₄×#170	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₅	C₄	1	1	yes	characteristic
6	C₂×C₄×D₅	C₄	1	1	yes	characteristic
7	C₂×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂×#170	C₂²	1	4	yes	—
9	#700	C₄	1	8	yes	direct product factor
10	C₄²×C₅	C₄	1	1	yes	characteristic
11	C₄×Dic₅	C₄	1	1	yes	characteristic
12	C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₄×Dic₅	C₂²	1	1	yes	characteristic
14	C₄×M₁₆	—	5	1	no	maximal, Sylow
15	C₂²×D₅	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₅	C₂³	1	1	yes	characteristic
22	C₄×D₅	—	2	2	no	—
23	C₄×D₅	—	2	2	no	—
24	C₄×D₅	C₂×C₄	1	4	yes	—
25	#170	C₂×C₄	1	8	yes	complemented
26	C₂×M₁₆	—	5	2	no	—
27	C₂×C₄²	—	5	1	no	—
28	#97	—	5	2	no	—
29	C₄×C₈	—	5	2	no	—
30	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₅	—	2	1	no	—
32	C₂×D₅	—	2	1	no	—
33	C₂×D₅	C₄²	1	2	yes	—
34	C₄×C₅	—	2	1	no	—
35	C₄×C₅	C₄²	1	2	yes	—
36	C₄×C₅	M₁₆	1	2	yes	complemented
37	Dic₅	—	2	1	no	—
38	Dic₅	M₁₆	1	2	yes	complemented
39	Dic₅	C₂²×C₄	1	2	yes	—
40	C₂²×C₄	—	5	1	no	—
41	C₂²×C₄	—	5	1	no	—
42	C₂²×C₄	—	5	1	no	—
43	M₁₆	—	5	8	no	—
44	C₂×C₈	—	5	4	no	—
45	C₄²	—	5	1	no	—
46	C₄²	—	5	1	no	—
47	C₄²	—	5	1	no	—
48	C₄²	AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
50	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
51	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
52	D₅	—	2	2	no	—
53	C₂³	—	5	1	no	—
54	C₂×C₄	—	5	1	no	—
55	C₂×C₄	—	5	1	no	—
56	C₂×C₄	—	5	4	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	—	10	2	no	—
59	C₂×C₄	—	10	2	no	—
60	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
61	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
62	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
63	C₈	—	5	8	no	—
64	C₅	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
65	C₂²	—	5	2	no	—
66	C₂²	—	10	1	no	—
67	C₂²	—	10	1	no	—
68	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	5	2	no	—
71	C₄	—	5	2	no	—
72	C₄	—	10	1	no	—
73	C₄	#700	1	2	yes	direct product factor
74	C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
75	C₂	—	10	2	no	—
76	C₂	C₂×#700	1	1	yes	characteristic
77	C₂	D₅⋊M₁₆	1	1	yes	characteristic
78	C₂	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₄²×C₅	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₄²	AGL(1,𝔽₅)
hypocenter	C₅	C₄×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
3. #4002	C₂	semidirect
4. C₄×#170	C₂	semidirect
9. #700	C₄	direct (N × H)
25. #170	C₂×C₄	semidirect
36. C₄×C₅	M₁₆	semidirect
38. Dic₅	M₁₆	semidirect
64. C₅	C₄×M₁₆	semidirect
73. C₄	#700	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#700	2
C₄²×D₅	2
#4002	2
C₄×#170	2
C₄×M₁₆	5

Subgroups — C4×#700 #65677

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#700 #65677