Subgroups — #12015714

Order 1280.

370 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×C₄⋊#672	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	C₄⋊#672	—	2	12	no	—
8	#12899	—	5	1	no	maximal, Sylow
9	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
10	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₅	C₂×C₄	1	3	yes	—
12	C₅×#97	—	2	6	no	—
13	C₄×C₅×C₈	D₄	1	6	yes	—
14	C₄×C₅×C₈	Q₈	1	6	yes	—
15	C₅×#95	C₈	1	4	yes	—
16	C₂×#672	—	2	12	no	—
17	C₂×C₄⋊C₁₆	—	5	6	no	—
18	#1726	D₅	1	1	yes	characteristic
19	C₂²×C₄×C₅	C₂×C₈	1	6	yes	—
20	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
21	C₂×C₅×C₈	—	2	6	no	—
22	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
23	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
24	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
25	C₄²×C₅	C₄⋊C₄	1	6	yes	—
26	C₄²×C₅	M₁₆	1	6	yes	—
27	#672	—	4	4	no	—
28	C₂²×C₁₆	—	5	4	no	—
29	C₂×#97	Dic₅	1	3	yes	—
30	C₂×C₄×C₈	C₂×D₅	1	3	yes	—
31	#376	Dic₅	1	1	yes	characteristic
32	C₄⋊C₁₆	—	10	12	no	—
33	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
34	C₂×C₄×C₅	—	2	12	no	—
35	C₂×C₄×C₅	C₄⋊C₈	1	12	yes	—
36	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
37	C₂×C₄×C₅	C₂×M₁₆	1	3	yes	—
38	C₅×C₈	—	2	6	no	—
39	C₅×C₈	#127	1	1	yes	characteristic
40	C₅×C₈	C₄⋊Q₈	1	3	yes	—
41	C₂²×C₈	C₂×Dic₅	1	3	yes	—
42	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
43	C₂×C₄²	C₂×Dic₅	1	3	yes	—
44	C₂×C₁₆	—	10	12	no	—
45	#97	—	2	6	no	—
46	C₄×C₈	Dic₁₀	1	6	yes	—
47	C₄×C₈	D₂₀	1	6	yes	—
48	#95	#168	1	4	yes	—
49	C₂²×C₅	C₂×C₄⋊C₈	1	3	yes	—
50	C₂²×C₅	#504	1	4	yes	—
51	C₄×C₅	—	2	12	no	—
52	C₄×C₅	#384	1	1	yes	characteristic
53	C₄×C₅	#423	1	3	yes	—
54	C₂²×C₄	C₂×#168	1	6	yes	—
55	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
56	C₂×C₈	—	2	6	no	—
57	C₂×C₈	C₄⋊Dic₅	1	6	yes	—
58	C₂×C₈	C₂×Dic₁₀	1	3	yes	—
59	C₂×C₈	C₂×D₂₀	1	3	yes	—
60	C₄²	#681	1	6	yes	—
61	C₄²	C₄⋊Dic₅	1	6	yes	—
62	C₁₆	—	20	4	no	—
63	C₂×C₅	#1842	1	1	yes	characteristic
64	C₂×C₅	#2151	1	6	yes	—
65	C₂³	C₂²×#168	1	1	yes	characteristic
66	C₂×C₄	—	2	12	no	—
67	C₂×C₄	C₄⋊#168	1	12	yes	—
68	C₂×C₄	C₂×#681	1	3	yes	—
69	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
70	C₈	—	2	6	no	—
71	C₈	#4015	1	3	yes	—
72	C₈	#4020	1	1	yes	characteristic
73	C₅	#12899	1	1	yes	characteristic, complemented, Sylow
74	C₂²	C₂×C₄⋊#168	1	3	yes	—
75	C₂²	#65429	1	4	yes	—
76	C₄	—	2	12	no	—
77	C₄	#65213	1	3	yes	—
78	C₄	#65224	1	1	yes	characteristic
79	C₂	#10600187	1	1	yes	characteristic
80	C₂	#10600675	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	#12899
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
73. C₅	#12899	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×C₄⋊#672	2
#12899	5