Subgroups — #65522

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	#4072	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#3967	C₂	1	2	yes	maximal, complemented
5	#3964	C₂	1	2	yes	maximal, complemented
6	C₅×#41	—	2	2	no	—
7	C₅×#41	C₄	1	4	yes	complemented
8	C₂×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×D₄	—	2	1	no	—
10	C₂×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅	—	2	1	no	—
12	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₅	—	2	1	no	—
14	C₄⋊Dic₅	C₂²	1	2	yes	—
15	C₄×Dic₅	—	2	1	no	—
16	C₂×#168	—	2	2	no	—
17	C₂×#168	C₂²	1	2	yes	—
18	#416	—	5	1	no	maximal, Sylow
19	C₂³×C₅	—	2	1	no	—
20	C₂³×C₅	D₄	1	1	yes	characteristic
21	C₅×Q₈	—	2	1	no	—
22	C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₅×D₄	—	2	4	no	—
24	C₅×D₄	—	2	1	no	—
25	C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₅	—	2	1	no	—
27	C₂×C₄×C₅	—	2	4	no	—
28	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₅	D₄	1	2	yes	—
30	C₂×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×Dic₅	—	2	2	no	—
33	#168	—	2	2	no	—
34	C₂×#41	D₅	1	1	yes	characteristic
35	C₂²×C₈	—	5	1	no	—
36	#117	—	5	1	no	—
37	#103	—	5	2	no	—
38	#102	—	5	2	no	—
39	C₂²×C₅	—	2	2	no	—
40	C₂²×C₅	—	2	2	no	—
41	C₂²×C₅	—	4	2	no	—
42	C₂²×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
44	C₄×C₅	—	2	2	no	—
45	C₄×C₅	C₂²⋊C₄	1	2	yes	—
46	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	Dic₅	—	4	2	no	—
49	#41	—	2	2	no	—
50	#41	Dic₅	1	4	yes	complemented
51	C₂×Q₈	C₂×D₅	1	1	yes	characteristic
52	C₂×D₄	—	2	1	no	—
53	C₂×D₄	C₂×D₅	1	1	yes	characteristic
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
56	C₂×C₈	—	5	2	no	—
57	C₂×C₈	—	10	2	no	—
58	C₄⋊C₄	—	5	2	no	—
59	C₂²⋊C₄	—	10	1	no	—
60	C₄²	—	10	1	no	—
61	C₂×C₅	—	2	2	no	—
62	C₂×C₅	—	4	2	no	—
63	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₅	#135	1	2	yes	—
65	C₂³	—	2	1	no	—
66	C₂³	C₂²⋊D₅	1	1	yes	characteristic
67	Q₈	—	2	1	no	—
68	Q₈	C₂×Dic₅	1	2	yes	—
69	D₄	—	2	4	no	—
70	D₄	—	2	1	no	—
71	D₄	C₂×Dic₅	1	2	yes	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	10	2	no	—
75	C₂×C₄	C₂×Dic₅	1	2	yes	—
76	C₂×C₄	C₂²⋊D₅	1	2	yes	—
77	C₂×C₄	C₂²⋊D₅	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
79	C₈	—	10	2	no	—
80	C₅	#416	1	1	yes	characteristic, complemented, Sylow
81	C₂²	—	2	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂²⋊Dic₅	1	2	yes	—
85	C₂²	C₂×C₂²⋊D₅	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	20	2	no	—
88	C₄	C₂²⋊Dic₅	1	2	yes	—
89	C₄	C₂²×Dic₅	1	1	yes	characteristic
90	C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	#4096	1	2	yes	—
94	C₂	C₂×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting, prime radical	C₂×C₅×#41	C₂
socle	C₂²×C₅	C₂×D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×#41	D₅
hypocenter	C₅	#416
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #4072	C₂	semidirect
3. C₂²×#168	C₂	semidirect
4. #3967	C₂	semidirect
5. #3964	C₂	semidirect
7. C₅×#41	C₄	semidirect
50. #41	Dic₅	semidirect
80. C₅	#416	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
#4072	2
C₂²×#168	2
#3967	2
#3964	2
#416	5