Subgroups — #10604369

Order 640.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#3998	C₂	1	2	yes	maximal
2	#65286	C₂	1	1	yes	maximal, characteristic
3	#64860	C₂	1	4	yes	maximal, complemented
4	C₅×#117	C₄	1	1	yes	characteristic
5	C₂×#681	C₄	1	1	yes	characteristic
6	C₂²×#168	C₂²	1	1	yes	characteristic
7	#3998	C₄	1	4	yes	complemented
8	#3998	C₂²	1	4	yes	—
9	C₂×#674	C₄	1	4	yes	complemented
10	#3935	C₂²	1	2	yes	—
11	C₄×#168	C₄	1	2	yes	—
12	#2110	—	5	1	no	maximal, Sylow
13	C₂²×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₅×C₄⋊C₄	—	2	1	no	—
15	C₅×C₂²⋊C₄	—	2	1	no	—
16	C₄²×C₅	C₂×C₄	1	2	yes	—
17	#681	—	2	2	no	—
18	C₂×#168	C₂×C₄	1	2	yes	—
19	C₂×#168	C₂×C₄	1	1	yes	characteristic
20	C₂×#168	C₂×C₄	1	2	yes	—
21	C₂×#168	C₂×C₄	1	2	yes	—
22	C₂×#168	C₂³	1	1	yes	characteristic
23	#674	C₂×C₄	1	8	yes	—
24	C₂×M₃₂	—	5	2	no	—
25	#405	—	5	1	no	—
26	#347	—	5	4	no	—
27	C₂³×C₅	M₁₆	1	1	yes	characteristic
28	C₂×C₄×C₅	—	2	2	no	—
29	C₂×C₄×C₅	C₄²	1	2	yes	—
30	C₂×C₄×C₅	M₁₆	1	2	yes	—
31	C₂×C₄×C₅	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
33	#168	—	2	2	no	—
34	#168	C₄²	1	1	yes	characteristic
35	#168	C₄²	1	1	yes	characteristic
36	#168	C₂²×C₄	1	2	yes	—
37	C₂×M₁₆	—	5	1	no	—
38	C₂²×C₈	—	5	1	no	—
39	#117	AGL(1,𝔽₅)	1	1	yes	characteristic
40	M₃₂	—	5	4	no	—
41	M₃₂	—	5	4	no	—
42	C₂×C₁₆	—	5	4	no	—
43	#97	—	5	2	no	—
44	C₄×C₈	—	5	2	no	—
45	C₂²×C₅	—	4	1	no	—
46	C₂²×C₅	#97	1	2	yes	—
47	C₂²×C₅	C₂×M₁₆	1	1	yes	characteristic
48	C₄×C₅	—	4	2	no	—
49	C₄×C₅	#97	1	2	yes	—
50	C₄×C₅	C₂×C₄²	1	1	yes	characteristic
51	C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
52	C₂²×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	M₁₆	—	10	2	no	—
54	C₂×C₈	—	5	2	no	—
55	C₂×C₈	—	5	2	no	—
56	C₂×C₈	—	5	1	no	—
57	C₂×C₈	—	5	1	no	—
58	C₂×C₈	—	5	2	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₂²⋊C₄	—	2	1	no	—
61	C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
62	C₁₆	—	5	8	no	—
63	C₂×C₅	—	2	1	no	—
64	C₂×C₅	—	2	2	no	—
65	C₂×C₅	C₂×#97	1	1	yes	characteristic
66	C₂³	#704	1	1	yes	characteristic
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	#700	1	2	yes	—
69	C₂×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
70	C₂×C₄	#704	1	1	yes	characteristic
71	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
72	C₈	—	5	2	no	—
73	C₈	—	5	1	no	—
74	C₈	—	5	1	no	—
75	C₈	—	10	2	no	—
76	C₅	#2110	1	1	yes	characteristic, complemented, Sylow
77	C₂²	—	4	1	no	—
78	C₂²	#4002	1	2	yes	—
79	C₂²	C₂×#704	1	1	yes	characteristic
80	C₄	—	4	2	no	—
81	C₄	#4002	1	2	yes	—
82	C₄	C₂×#700	1	1	yes	characteristic
83	C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	2	2	no	—
86	C₂	C₂×#4002	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#117	C₄
socle	C₂×C₅	C₂×#97
prime radical	C₂³×C₅	M₁₆
cosocle	C₂×#168	C₂³
hypercenter	#117	AGL(1,𝔽₅)
hypocenter	C₅	#2110
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #64860	C₂	semidirect
7. #3998	C₄	semidirect
9. C₂×#674	C₄	semidirect
76. C₅	#2110	semidirect

Maximal subgroups

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

subgroup	index
C₂×#3998	2
#65286	2
#64860	2
#2110	5