Subgroups — #12014008

Order 1280.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10599858	C₂	1	4	yes	maximal
2	#10598858	C₂	1	1	yes	maximal, characteristic
3	#10598857	C₂	1	1	yes	maximal, characteristic
4	C₅×#1311	C₂	1	1	yes	maximal, characteristic
5	#64988	C₂²	1	2	yes	—
6	#64971	—	2	2	no	—
7	#64971	C₂²	1	2	yes	—
8	#64970	—	2	2	no	—
9	C₄×C₅×C₁₆	C₂²	1	2	yes	—
10	C₅×C₈²	C₂²	1	1	yes	characteristic
11	#64723	—	2	8	no	—
12	#12514	—	5	1	no	maximal, Sylow
13	C₂×Dic₂₀	—	2	4	no	—
14	#4015	—	2	2	no	—
15	#4015	—	2	2	no	—
16	C₂×C₅×C₁₆	D₄	1	4	yes	—
17	C₄×C₅×C₈	D₄	1	2	yes	—
18	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
19	#3950	—	2	4	no	—
20	#3950	—	4	4	no	—
21	#3948	—	4	4	no	—
22	#2241	—	5	4	no	—
23	#1713	—	5	1	no	—
24	#1710	—	5	1	no	—
25	#1311	D₅	1	1	yes	characteristic
26	C₂×Dic₁₀	—	4	2	no	—
27	C₂×Dic₁₀	—	4	2	no	—
28	C₂×C₅×C₈	D₈	1	4	yes	—
29	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
30	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
31	C₄⋊Dic₅	—	4	2	no	—
32	C₄⋊Dic₅	—	4	2	no	—
33	C₄⋊Dic₅	—	4	4	no	—
34	Dic₂₀	—	4	8	no	—
35	C₅×C₁₆	—	2	4	no	—
36	#500	—	5	2	no	—
37	#500	—	10	2	no	—
38	#494	—	5	2	no	—
39	#487	—	10	2	no	—
40	#358	—	10	8	no	—
41	C₄×C₁₆	C₂×D₅	1	2	yes	—
42	C₈²	C₂×D₅	1	1	yes	characteristic
43	C₂×C₄×C₅	#127	1	1	yes	characteristic
44	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
45	C₂×Dic₅	—	8	2	no	—
46	C₂×Dic₅	—	8	2	no	—
47	Dic₁₀	—	8	4	no	—
48	Dic₁₀	—	8	4	no	—
49	C₅×C₈	—	2	4	no	—
50	C₅×C₈	#135	1	4	yes	—
51	C₂×Q₁₆	—	10	4	no	—
52	C₄⋊Q₈	—	10	2	no	—
53	C₄⋊Q₈	—	10	2	no	—
54	C₂×C₁₆	D₂₀	1	4	yes	—
55	#107	—	10	4	no	—
56	#107	—	20	4	no	—
57	#103	—	20	4	no	—
58	C₄×C₈	D₂₀	1	2	yes	—
59	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
60	C₂²×C₅	#493	1	1	yes	characteristic
61	C₄×C₅	#495	1	2	yes	—
62	C₄×C₅	#510	1	4	yes	—
63	Dic₅	—	16	2	no	—
64	Dic₅	—	16	2	no	—
65	C₂×Q₈	—	20	2	no	—
66	C₂×Q₈	—	20	2	no	—
67	Q₁₆	—	20	8	no	—
68	C₂×C₈	D₄₀	1	4	yes	—
69	C₂×C₈	C₂×D₂₀	1	2	yes	—
70	C₄⋊C₄	—	20	2	no	—
71	C₄⋊C₄	—	20	2	no	—
72	C₄⋊C₄	—	20	4	no	—
73	C₄²	C₂×D₂₀	1	1	yes	characteristic
74	C₁₆	—	2	4	no	—
75	C₂×C₅	#1709	1	1	yes	characteristic
76	C₂×C₅	#2251	1	2	yes	—
77	Q₈	—	40	4	no	—
78	Q₈	—	40	4	no	—
79	C₂×C₄	—	40	2	no	—
80	C₂×C₄	—	40	2	no	—
81	C₂×C₄	#4020	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄₀	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#4050	1	4	yes	—
85	C₅	#12514	1	1	yes	characteristic, complemented, Sylow
86	C₂²	#64984	1	1	yes	characteristic
87	C₄	—	80	2	no	—
88	C₄	—	80	2	no	—
89	C₄	#64985	1	2	yes	—
90	C₄	#65198	1	4	yes	—
91	C₂	#10598853	1	1	yes	characteristic
92	C₂	#10599877	1	2	yes	—

	subgroup	G/H
center	C₂²	#64984
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1311	C₂
socle, prime radical	C₂²×C₅	#493
hypercenter	#1311	D₅
hypocenter	C₅	#12514
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
85. C₅	#12514	semidirect

Maximal subgroups

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

subgroup	index
#10599858	2
#10598858	2
#10598857	2
C₅×#1311	2
#12514	5