Subgroups — #12015722

Order 1280.

278 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1747	C₂	1	1	yes	maximal, characteristic
2	#10600678	C₂	1	3	yes	maximal
3	#10600675	C₂	1	3	yes	maximal
4	C₅×#405	C₂²	1	3	yes	—
5	C₂×#3944	C₂²	1	3	yes	—
6	C₂²×#672	C₂²	1	1	yes	characteristic
7	C₅×#338	C₄	1	1	yes	characteristic
8	C₅×H(3,ℤ/4)	C₄	1	3	yes	—
9	#64683	—	2	6	no	—
10	C₄⋊#672	—	2	6	no	—
11	#12902	—	5	1	no	maximal, Sylow
12	C₂×C₅×M₁₆	C₂×C₄	1	3	yes	—
13	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
14	C₅×#117	C₂×C₄	1	3	yes	—
15	C₅×#97	Q₈	1	6	yes	—
16	C₄×C₅×C₈	D₄	1	6	yes	—
17	#3944	—	2	6	no	—
18	C₂×#672	—	2	3	no	—
19	C₂×#672	—	2	3	no	—
20	#2153	—	5	3	no	—
21	#2151	—	5	3	no	—
22	#1747	D₅	1	1	yes	characteristic
23	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
24	C₅×M₁₆	—	2	6	no	—
25	C₂×C₅×C₈	C₄⋊C₄	1	6	yes	—
26	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
27	C₂×C₅×C₈	C₂×Q₈	1	3	yes	—
28	C₅×C₄⋊C₄	—	2	3	no	—
29	C₅×C₂²⋊C₄	—	2	3	no	—
30	C₄²×C₅	C₄⋊C₄	1	6	yes	—
31	#672	—	4	1	no	—
32	#672	—	4	3	no	—
33	C₂×M₃₂	—	5	3	no	—
34	C₂²×C₁₆	—	5	1	no	—
35	#405	C₂×D₅	1	3	yes	—
36	#364	—	10	6	no	—
37	C₄⋊C₁₆	—	10	6	no	—
38	#338	Dic₅	1	1	yes	characteristic
39	H(3,ℤ/4)	Dic₅	1	3	yes	—
40	C₂³×C₅	#131	1	1	yes	characteristic
41	C₂×C₄×C₅	—	2	6	no	—
42	C₂×C₄×C₅	C₂×C₄⋊C₄	1	3	yes	—
43	C₂×C₄×C₅	#131	1	3	yes	—
44	C₅×C₈	—	2	6	no	—
45	C₅×C₈	#127	1	1	yes	characteristic
46	C₅×C₈	C₄⋊Q₈	1	3	yes	—
47	C₂×M₁₆	C₂×Dic₅	1	3	yes	—
48	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
49	#117	C₂×Dic₅	1	3	yes	—
50	M₃₂	—	10	6	no	—
51	C₂×C₁₆	—	10	3	no	—
52	C₂×C₁₆	—	10	3	no	—
53	#97	Dic₁₀	1	6	yes	—
54	C₄×C₈	D₂₀	1	6	yes	—
55	C₂²×C₅	—	4	1	no	—
56	C₂²×C₅	#424	1	3	yes	—
57	C₄×C₅	—	4	6	no	—
58	C₄×C₅	#384	1	1	yes	characteristic
59	C₄×C₅	#424	1	3	yes	—
60	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
61	M₁₆	—	2	6	no	—
62	C₂×C₈	C₄⋊Dic₅	1	6	yes	—
63	C₂×C₈	C₂×Dic₁₀	1	3	yes	—
64	C₂×C₈	C₂×D₂₀	1	3	yes	—
65	C₄⋊C₄	—	2	3	no	—
66	C₂²⋊C₄	—	2	3	no	—
67	C₄²	C₄⋊Dic₅	1	6	yes	—
68	C₁₆	—	20	1	no	—
69	C₁₆	—	20	3	no	—
70	C₂×C₅	—	2	3	no	—
71	C₂×C₅	#1843	1	1	yes	characteristic
72	C₂³	#4094	1	1	yes	characteristic
73	C₂×C₄	—	2	6	no	—
74	C₂×C₄	C₂×C₄⋊Dic₅	1	3	yes	—
75	C₂×C₄	#4094	1	3	yes	—
76	C₈	—	2	6	no	—
77	C₈	#4015	1	3	yes	—
78	C₈	#4020	1	1	yes	characteristic
79	C₅	#12902	1	1	yes	characteristic, complemented, Sylow
80	C₂²	—	4	1	no	—
81	C₂²	#65288	1	3	yes	—
82	C₄	—	4	6	no	—
83	C₄	#65224	1	1	yes	characteristic
84	C₄	#65288	1	3	yes	—
85	C₂	—	2	3	no	—
86	C₂	#10600232	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#4020
commutator	C₂²×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1747	C₂
socle	C₂×C₅	#1843
prime radical	C₂³×C₅	#131
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1747	D₅
hypocenter	C₅	#12902
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
79. C₅	#12902	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1747	2
#10600678	2
#10600675	2
#12902	5