Subgroups — C₇₁×#7544 #31828416

Order 18176.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×M₁₂₈	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₇₁	C₄	1	1	yes	characteristic
4	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
5	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
6	C₆₄×C₇₁	—	2	2	no	—
7	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
8	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
9	C₄×C₈×C₇₁	C₈	1	1	yes	characteristic
10	C₃₂×C₇₁	—	2	1	no	—
11	C₃₂×C₇₁	D₄	1	1	yes	characteristic
12	C₃₂×C₇₁	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₇₁	C₁₆	1	1	yes	characteristic
14	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₇₁	C₁₆	1	1	yes	characteristic
16	C₁₆×C₇₁	—	2	1	no	—
17	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
18	C₁₆×C₇₁	M₁₆	1	1	yes	characteristic
19	C₂×C₄×C₇₁	—	2	1	no	—
20	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
21	C₈×C₇₁	C₃₂	1	2	yes	—
22	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
23	C₈×C₇₁	M₃₂	1	1	yes	characteristic
24	C₂²×C₇₁	M₆₄	1	1	yes	characteristic
25	C₄×C₇₁	—	2	2	no	—
26	C₄×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
27	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
28	#7544	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂×C₇₁	—	2	1	no	—
30	C₂×C₇₁	C₄⋊C₃₂	1	1	yes	characteristic
31	M₁₂₈	C₂×C₇₁	1	2	yes	—
32	C₄×C₃₂	C₂×C₇₁	1	1	yes	characteristic
33	C₇₁	#7544	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×C₃₂	C₄×C₇₁	1	1	yes	characteristic
35	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
36	C₄×C₁₆	C₄×C₇₁	1	1	yes	characteristic
37	C₆₄	—	2	2	no	—
38	C₂×C₁₆	C₈×C₇₁	1	1	yes	characteristic
39	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
40	C₄×C₈	C₈×C₇₁	1	1	yes	characteristic
41	C₃₂	—	2	1	no	—
42	C₃₂	D₄×C₇₁	1	1	yes	characteristic
43	C₃₂	Q₈×C₇₁	1	1	yes	characteristic
44	C₂×C₈	C₁₆×C₇₁	1	1	yes	characteristic
45	C₂×C₈	C₂×C₈×C₇₁	1	1	yes	characteristic
46	C₄²	C₁₆×C₇₁	1	1	yes	characteristic
47	C₁₆	—	2	1	no	—
48	C₁₆	C₄⋊C₄×C₇₁	1	1	yes	characteristic
49	C₁₆	M₁₆×C₇₁	1	1	yes	characteristic
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	C₂×C₁₆×C₇₁	1	1	yes	characteristic
52	C₈	C₃₂×C₇₁	1	2	yes	—
53	C₈	C₄⋊C₈×C₇₁	1	1	yes	characteristic
54	C₈	M₃₂×C₇₁	1	1	yes	characteristic
55	C₂²	M₆₄×C₇₁	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	C₄⋊C₁₆×C₇₁	1	1	yes	characteristic
58	C₄	C₂×C₃₂×C₇₁	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	C₇₁×C₄⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂×C₇₁	D₄
commutator	C₄	C₂×C₃₂×C₇₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₇₁
socle	C₂×C₇₁	C₄⋊C₃₂
prime radical	C₂²×C₇₁	M₆₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₇₁ ◁ C₂×C₃₂×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
28. #7544	C₇₁	direct (N × H)
33. C₇₁	#7544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₁×M₁₂₈	2
C₄×C₃₂×C₇₁	2
#7544	71

Subgroups — C71×#7544 #31828416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7544 #31828416