Subgroups — #65523

Order 320.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#681	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#168	C₂	1	4	yes	maximal, complemented
5	C₅×#41	—	2	2	no	—
6	C₅×#41	—	2	2	no	—
7	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₅×D₄	C₄	1	2	yes	—
9	C₂×C₅×D₄	C₄	1	1	yes	characteristic
10	C₂²×C₄×C₅	C₂²	1	2	yes	—
11	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
12	#681	—	2	2	no	—
13	C₂×#168	—	2	2	no	—
14	C₂×#168	C₂²	1	2	yes	—
15	C₂×#168	C₂²	1	2	yes	—
16	#408	—	5	1	no	maximal, Sylow
17	C₂³×C₅	C₂×C₄	1	2	yes	—
18	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
19	C₅×Q₈	—	2	2	no	—
20	C₅×D₄	—	2	4	no	—
21	C₅×D₄	—	2	2	no	—
22	C₂×C₄×C₅	—	2	2	no	—
23	C₂×C₄×C₅	—	2	2	no	—
24	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅	D₄	1	2	yes	—
27	C₂×C₄×C₅	D₄	1	2	yes	—
28	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
29	#168	—	2	2	no	—
30	#168	—	2	2	no	—
31	C₂×#41	D₅	1	1	yes	characteristic
32	C₂×M₁₆	—	5	1	no	—
33	C₂²×C₈	—	5	1	no	—
34	C₂²⋊C₈	—	5	4	no	—
35	C₂²×C₅	—	2	2	no	—
36	C₂²×C₅	—	2	2	no	—
37	C₂²×C₅	—	2	4	no	—
38	C₂²×C₅	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₅	—	2	2	no	—
41	C₄×C₅	C₂²⋊C₄	1	2	yes	—
42	C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	#41	—	2	2	no	—
45	#41	—	2	2	no	—
46	C₂×Q₈	Dic₅	1	1	yes	characteristic
47	C₂×D₄	Dic₅	1	2	yes	—
48	C₂×D₄	Dic₅	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₅	1	2	yes	—
50	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
51	M₁₆	—	10	2	no	—
52	C₂×C₈	—	5	2	no	—
53	C₂×C₈	—	5	2	no	—
54	C₂×C₈	—	10	2	no	—
55	C₂×C₅	—	2	2	no	—
56	C₂×C₅	—	4	2	no	—
57	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₅	#131	1	2	yes	—
59	C₂³	C₂×Dic₅	1	2	yes	—
60	C₂³	C₂×Dic₅	1	1	yes	characteristic
61	Q₈	—	2	2	no	—
62	D₄	—	2	4	no	—
63	D₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	C₂×Dic₅	1	2	yes	—
67	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₅	1	2	yes	—
69	C₂×C₄	C₂²⋊D₅	1	2	yes	—
70	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
71	C₈	—	10	2	no	—
72	C₈	—	10	2	no	—
73	C₅	#408	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	2	2	no	—
75	C₂²	—	2	2	no	—
76	C₂²	—	2	4	no	—
77	C₂²	C₂²⋊Dic₅	1	2	yes	—
78	C₂²	C₂²×Dic₅	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	C₂²⋊Dic₅	1	2	yes	—
81	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
82	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	4	2	no	—
85	C₂	#4094	1	2	yes	—
86	C₂	C₂×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₅
commutator, socle	C₂²×C₅	C₂²×C₄
Fitting, prime radical	C₂×C₅×#41	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×#41	D₅
hypocenter	C₅	#408
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. C₂×#681	C₂	semidirect
3. C₂²×#168	C₂	semidirect
4. C₂²⋊#168	C₂	semidirect
73. C₅	#408	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#41	2
C₂×#681	2
C₂²×#168	2
C₂²⋊#168	2
#408	5