Subgroups — #12015702

Order 1280.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1726	C₂	1	1	yes	maximal, characteristic
2	#10598148	C₂	1	6	yes	maximal
3	C₂×C₅×#97	C₄	1	3	yes	—
4	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
5	C₅×#376	C₄	1	1	yes	characteristic
6	C₂²×#672	C₂²	1	4	yes	—
7	#12895	—	5	1	no	maximal, Sylow
8	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
9	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
11	C₅×#97	—	2	6	no	—
12	C₄×C₅×C₈	—	2	6	no	—
13	C₅×#95	C₈	1	4	yes	—
14	C₂×#672	—	2	12	no	—
15	#1726	D₅	1	1	yes	characteristic
16	#1374	—	5	6	no	—
17	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
18	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈	—	2	3	no	—
20	C₂×C₅×C₈	—	2	3	no	—
21	C₂×C₅×C₈	M₁₆	1	6	yes	—
22	C₂×C₅×C₈	#41	1	6	yes	—
23	C₄²×C₅	—	2	3	no	—
24	C₄²×C₅	—	2	3	no	—
25	#672	—	4	4	no	—
26	C₂²×C₁₆	—	5	4	no	—
27	C₂×#97	Dic₅	1	3	yes	—
28	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
29	#376	Dic₅	1	1	yes	characteristic
30	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅	—	2	6	no	—
32	C₂×C₄×C₅	—	2	12	no	—
33	C₂×C₄×C₅	#117	1	3	yes	—
34	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
35	C₅×C₈	—	2	6	no	—
36	C₅×C₈	#126	1	4	yes	—
37	C₂²×C₈	C₂×Dic₅	1	3	yes	—
38	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
39	C₂×C₄²	C₂×Dic₅	1	3	yes	—
40	C₂×C₁₆	—	10	12	no	—
41	#97	—	2	6	no	—
42	C₄×C₈	—	2	6	no	—
43	#95	#168	1	4	yes	—
44	C₂²×C₅	#431	1	3	yes	—
45	C₂²×C₅	#504	1	4	yes	—
46	C₄×C₅	—	4	6	no	—
47	C₄×C₅	#383	1	1	yes	characteristic
48	C₄×C₅	#432	1	3	yes	—
49	C₂²×C₄	C₂×#168	1	6	yes	—
50	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
51	C₂×C₈	—	2	3	no	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	#681	1	6	yes	—
54	C₂×C₈	#709	1	6	yes	—
55	C₄²	—	2	3	no	—
56	C₄²	—	2	3	no	—
57	C₁₆	—	20	4	no	—
58	C₂×C₅	#1839	1	1	yes	characteristic
59	C₂×C₅	#2164	1	6	yes	—
60	C₂³	C₂²×#168	1	1	yes	characteristic
61	C₂×C₄	—	2	12	no	—
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	C₂×#681	1	3	yes	—
64	C₂×C₄	#4072	1	3	yes	—
65	C₈	—	2	6	no	—
66	C₈	#4022	1	4	yes	—
67	C₅	#12895	1	1	yes	characteristic, complemented, Sylow
68	C₂²	#65214	1	3	yes	—
69	C₂²	#65429	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#65215	1	3	yes	—
72	C₄	#65226	1	1	yes	characteristic
73	C₂	#10600186	1	1	yes	characteristic
74	C₂	#10600676	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator, socle, prime radical	C₂³×C₅	C₂²×C₈
Fitting	C₅×#1726	C₂
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1726	D₅
hypocenter	C₅	#12895
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
67. C₅	#12895	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1726	2
#10598148	2
#12895	5