Subgroups — #17986398

Order 4544.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14556120	C₂	1	1	yes	maximal, characteristic
2	#14556061	C₂	1	2	yes	maximal
3	#14556059	C₂	1	2	yes	maximal
4	C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#125×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₇₁	C₂²	1	2	yes	—
9	Dic₇₁⋊C₄	—	2	2	no	—
10	C₄⋊C₄×C₇₁	—	2	2	no	—
11	C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
12	C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇₁	—	2	2	no	—
14	C₂×C₄×C₇₁	D₄	1	2	yes	—
15	C₂×C₄×C₇₁	C₂³	1	1	yes	characteristic
16	C₂×Dic₇₁	—	2	2	no	—
17	C₇₁⋊C₈	Q₈	1	2	yes	—
18	C₇₁⋊C₈	Q₈	1	2	yes	—
19	C₂²×C₇₁	C₂×D₄	1	1	yes	characteristic
20	C₄×C₇₁	—	2	2	no	—
21	C₄×C₇₁	—	4	2	no	—
22	C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
24	Dic₇₁	—	4	2	no	—
25	C₂×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
26	C₂×C₇₁	#135	1	2	yes	—
27	C₇₁	#499	1	1	yes	characteristic, complemented, Sylow
28	#499	—	71	1	no	maximal, Sylow
29	#125	—	71	1	no	—
30	#125	D₇₁	1	1	yes	characteristic
31	#107	—	71	2	no	—
32	#106	—	71	2	no	—
33	C₄×C₈	—	71	1	no	—
34	C₂×C₈	—	71	1	no	—
35	C₂×C₈	—	71	1	no	—
36	C₄⋊C₄	—	2	2	no	—
37	C₄⋊C₄	—	71	2	no	—
38	C₄⋊C₄	—	142	2	no	—
39	C₄⋊C₄	C₂×D₇₁	1	2	yes	—
40	C₄²	C₂×D₇₁	1	1	yes	characteristic
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	—	142	2	no	—
43	C₂×C₄	C₂²⋊D₇₁	1	2	yes	—
44	C₂×C₄	C₂²×D₇₁	1	1	yes	characteristic
45	C₈	—	71	2	no	—
46	C₈	—	71	2	no	—
47	C₂²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₄	—	4	2	no	—
50	C₄	—	284	2	no	—
51	C₄	Q₈×D₇₁	1	1	yes	characteristic
52	C₄	Q₈×D₇₁	1	1	yes	characteristic
53	C₂	#14556126	1	1	yes	characteristic
54	C₂	#14556148	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₇₁
commutator, cosocle	C₂×C₄×C₇₁	C₂³
Frattini	C₂×C₄	C₂²×D₇₁
Fitting	#125×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₄
hypercenter	#125	D₇₁
hypocenter	C₇₁	#499
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
27. C₇₁	#499	semidirect

Maximal subgroups

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

subgroup	index
#14556120	2
#14556061	2
#14556059	2
C₄×C₇₁⋊C₈	2
#125×C₇₁	2
#499	71