Subgroups — #11986974

Order 1280.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1422	C₂	1	1	yes	maximal, characteristic
2	#10598001	C₂	1	1	yes	maximal, characteristic
3	C₁₆×#168	C₂	1	1	yes	maximal, characteristic
4	C₅×M₆₄	—	2	1	no	—
5	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
6	#64719	—	2	1	no	—
7	C₄×#672	C₄	1	1	yes	characteristic
8	C₈×#168	C₄	1	1	yes	characteristic
9	#7459	—	5	1	no	maximal, Sylow
10	C₂×C₅×C₁₆	D₄	1	1	yes	characteristic
11	C₂×C₅×C₁₆	Q₈	1	1	yes	characteristic
12	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×#672	C₈	1	1	yes	characteristic
14	C₂×#672	C₈	1	1	yes	characteristic
15	C₄×#168	—	2	1	no	—
16	C₅×C₃₂	—	4	1	no	—
17	#3926	—	4	1	no	—
18	#1422	—	5	1	no	—
19	#1422	D₅	1	1	yes	characteristic
20	C₈×C₁₆	—	5	1	no	—
21	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
23	C₄²×C₅	M₁₆	1	1	yes	characteristic
24	C₂×#168	—	2	2	no	—
25	C₅×C₁₆	—	2	1	no	—
26	C₅×C₁₆	D₈	1	1	yes	characteristic
27	C₅×C₁₆	Q₁₆	1	1	yes	characteristic
28	#672	—	2	1	no	—
29	#672	C₁₆	1	2	yes	—
30	M₆₄	—	2	1	no	—
31	M₆₄	—	10	1	no	—
32	C₄×C₁₆	—	5	1	no	—
33	C₄×C₁₆	C₂×D₅	1	1	yes	characteristic
34	C₈²	—	5	1	no	—
35	C₂×C₄×C₅	—	2	1	no	—
36	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
37	C₅×C₈	#107	1	1	yes	characteristic
38	C₅×C₈	#108	1	1	yes	characteristic
39	C₅×C₈	C₂×C₁₆	1	1	yes	characteristic
40	C₅×C₈	M₃₂	1	1	yes	characteristic
41	#168	—	2	4	no	—
42	C₂×C₁₆	—	5	1	no	—
43	C₂×C₁₆	—	5	1	no	—
44	C₂×C₁₆	Dic₁₀	1	1	yes	characteristic
45	C₂×C₁₆	C₂²⋊D₅	1	1	yes	characteristic
46	C₄×C₈	—	10	1	no	—
47	C₄×C₈	C₄×D₅	1	1	yes	characteristic
48	C₃₂	—	4	1	no	—
49	C₃₂	—	20	1	no	—
50	C₂²×C₅	#364	1	1	yes	characteristic
51	C₄×C₅	—	2	2	no	—
52	C₄×C₅	#335	1	1	yes	characteristic
53	C₄×C₅	C₄⋊C₁₆	1	1	yes	characteristic
54	C₂×C₈	—	10	2	no	—
55	C₂×C₈	C₈×D₅	1	1	yes	characteristic
56	C₂×C₈	#683	1	1	yes	characteristic
57	C₄²	#676	1	1	yes	characteristic
58	C₁₆	—	2	1	no	—
59	C₁₆	—	5	2	no	—
60	C₁₆	—	10	1	no	—
61	C₁₆	#686	1	1	yes	characteristic
62	C₁₆	#689	1	1	yes	characteristic
63	C₂×C₅	—	2	1	no	—
64	C₂×C₅	#1370	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	#3946	1	1	yes	characteristic
67	C₈	—	10	4	no	—
68	C₈	D₅×C₁₆	1	1	yes	characteristic
69	C₈	#3930	1	1	yes	characteristic
70	C₈	#3939	1	1	yes	characteristic
71	C₈	#3954	1	1	yes	characteristic
72	C₅	#7459	1	1	yes	characteristic, complemented, Sylow
73	C₂²	#64731	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	#64699	1	1	yes	characteristic
76	C₄	#64721	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	#10598046	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#686
commutator	C₅×C₈	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₅
Fitting	C₅×#1422	C₂
socle	C₂×C₅	#1370
prime radical	C₂²×C₅	#364
cosocle	C₄×C₅×C₁₆	C₂²
hypercenter	#1422	D₅
hypocenter	C₅	#7459
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
72. C₅	#7459	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1422	2
#10598001	2
C₁₆×#168	2
#7459	5