Subgroups — #12014124

Order 1280.

478 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	C₂×#64681	C₂	1	3	yes	maximal
3	C₂×C₄×#672	C₂	1	3	yes	maximal
4	C₂×C₅×#97	C₄	1	3	yes	—
5	C₂×C₄×C₅×C₈	C₂²	1	3	yes	—
6	C₅×#376	C₄	1	1	yes	characteristic
7	C₂²×#672	C₂²	1	1	yes	characteristic
8	C₂²×#672	C₂²	1	3	yes	—
9	#64681	C₄	1	12	yes	complemented
10	C₄×#672	C₄	1	12	yes	complemented
11	#12539	—	5	1	no	maximal, Sylow
12	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
13	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
15	C₅×#97	C₈	1	12	yes	—
16	C₄×C₅×C₈	C₂×C₄	1	6	yes	—
17	C₄×C₅×C₈	C₂×C₄	1	6	yes	—
18	C₅×#95	C₈	1	4	yes	—
19	C₂×#672	C₂×C₄	1	6	yes	—
20	C₂×#672	C₂×C₄	1	6	yes	—
21	C₂×#672	C₂×C₄	1	12	yes	—
22	C₂×#346	—	5	3	no	—
23	C₂×C₄×C₁₆	—	5	3	no	—
24	#1726	D₅	1	1	yes	characteristic
25	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
26	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₅×C₈	C₄²	1	6	yes	—
28	C₂×C₅×C₈	C₂×C₈	1	12	yes	—
29	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
30	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
31	C₄²×C₅	C₄²	1	6	yes	—
32	C₄²×C₅	C₂×C₈	1	6	yes	—
33	#672	C₄²	1	4	yes	—
34	#672	C₄²	1	12	yes	—
35	C₂²×C₁₆	—	5	1	no	—
36	C₂²×C₁₆	—	5	3	no	—
37	C₂×#97	Dic₅	1	3	yes	—
38	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
39	#376	Dic₅	1	1	yes	characteristic
40	#346	—	5	12	no	—
41	C₄×C₁₆	—	5	12	no	—
42	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅	—	2	12	no	—
44	C₂×C₄×C₅	C₄×C₈	1	12	yes	—
45	C₂×C₄×C₅	C₂×C₄²	1	3	yes	—
46	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
47	C₅×C₈	C₄×C₈	1	12	yes	—
48	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
49	C₅×C₈	C₂×C₄²	1	3	yes	—
50	C₂²×C₈	C₂×Dic₅	1	3	yes	—
51	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
52	C₂×C₄²	C₂×Dic₅	1	3	yes	—
53	C₂×C₁₆	—	5	6	no	—
54	C₂×C₁₆	—	5	6	no	—
55	C₂×C₁₆	—	5	12	no	—
56	#97	#168	1	12	yes	—
57	C₄×C₈	C₄×D₅	1	6	yes	—
58	C₄×C₈	C₄×D₅	1	6	yes	—
59	#95	#168	1	4	yes	—
60	C₂²×C₅	C₂×C₄×C₈	1	3	yes	—
61	C₂²×C₅	#504	1	4	yes	—
62	C₄×C₅	—	2	12	no	—
63	C₄×C₅	C₄³	1	1	yes	characteristic
64	C₄×C₅	C₂×C₄×C₈	1	3	yes	—
65	C₂²×C₄	C₂×#168	1	6	yes	—
66	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
67	C₂×C₈	C₂×#168	1	12	yes	—
68	C₂×C₈	C₄×Dic₅	1	6	yes	—
69	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
70	C₂×C₈	C₂×C₄×D₅	1	3	yes	—
71	C₄²	C₂×#168	1	6	yes	—
72	C₄²	C₄×Dic₅	1	6	yes	—
73	C₁₆	—	5	4	no	—
74	C₁₆	—	5	12	no	—
75	C₂×C₅	C₄²×C₈	1	1	yes	characteristic
76	C₂×C₅	#2108	1	6	yes	—
77	C₂³	C₂²×#168	1	1	yes	characteristic
78	C₂×C₄	—	2	12	no	—
79	C₂×C₄	C₄×#168	1	12	yes	—
80	C₂×C₄	C₂²×#168	1	3	yes	—
81	C₂×C₄	C₂×C₄×Dic₅	1	3	yes	—
82	C₈	C₄×#168	1	12	yes	—
83	C₈	C₄²×D₅	1	1	yes	characteristic
84	C₈	C₄²×D₅	1	3	yes	—
85	C₅	#12539	1	1	yes	characteristic, complemented, Sylow
86	C₂²	C₂×C₄×#168	1	3	yes	—
87	C₂²	#65429	1	4	yes	—
88	C₄	—	2	12	no	—
89	C₄	C₂×C₄×#168	1	3	yes	—
90	C₄	C₄²×Dic₅	1	1	yes	characteristic
91	C₂	C₄²×#168	1	1	yes	characteristic
92	C₂	#10600674	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅
commutator	C₂×C₅	C₄²×C₈
Fitting	C₅×#1726	C₂
socle, prime radical	C₂³×C₅	C₂²×C₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1726	D₅
hypocenter	C₅	#12539
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
9. #64681	C₄	semidirect
10. C₄×#672	C₄	semidirect
85. C₅	#12539	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
C₂×#64681	2
C₂×C₄×#672	2
#12539	5