Subgroups — #31885355

Order 18176.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic
2	#22586544	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1425	C₂	1	1	yes	maximal, characteristic
4	#17985829	—	2	1	no	—
5	C₇₁⋊C₆₄	C₄	1	2	yes	—
6	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
7	#368×C₇₁	—	2	1	no	—
8	#14556064	—	4	1	no	—
9	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
10	#108×C₇₁	—	4	1	no	—
11	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
12	C₃₂×C₇₁	Q₈	1	1	yes	characteristic
13	#11765849	—	8	1	no	—
14	M₁₆×C₇₁	—	8	1	no	—
15	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
16	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
17	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
18	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
19	C₈×C₇₁	—	16	1	no	—
20	C₈×C₇₁	#107	1	1	yes	characteristic
21	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
22	C₇₁⋊C₈	—	16	1	no	—
23	C₂²×C₇₁	Q₆₄	1	1	yes	characteristic
24	C₄×C₇₁	#366	1	1	yes	characteristic
25	C₄×C₇₁	D₃₂	1	1	yes	characteristic
26	#7547	—	71	1	no	maximal, Sylow
27	C₂×C₇₁	—	2	1	no	—
28	C₂×C₇₁	#1423	1	1	yes	characteristic
29	C₂×C₆₄	—	71	1	no	—
30	#1425	—	71	1	no	—
31	#1425	D₇₁	1	1	yes	characteristic
32	C₇₁	#7547	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
34	#368	—	2	1	no	—
35	#368	—	142	1	no	—
36	C₆₄	—	71	2	no	—
37	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
38	#108	—	4	1	no	—
39	#108	—	284	1	no	—
40	C₃₂	Dic₁₄₂	1	1	yes	characteristic
41	C₃₂	C₄×D₇₁	1	1	yes	characteristic
42	M₁₆	—	8	1	no	—
43	M₁₆	—	568	1	no	—
44	C₂×C₈	#11765850	1	1	yes	characteristic
45	C₁₆	Dic₇₁⋊C₄	1	1	yes	characteristic
46	C₁₆	#11765855	1	1	yes	characteristic
47	C₂×C₄	#14556069	1	1	yes	characteristic
48	C₈	—	16	1	no	—
49	C₈	—	1136	1	no	—
50	C₈	#14556061	1	1	yes	characteristic
51	C₈	#14556074	1	1	yes	characteristic
52	C₂²	#17985839	1	1	yes	characteristic
53	C₄	#17985824	1	1	yes	characteristic
54	C₄	#17985834	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#22586539	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17985834
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₇₁×#1425	C₂
socle	C₂×C₇₁	#1423
prime radical	C₂²×C₇₁	Q₆₄
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	#1425	D₇₁
hypocenter	C₇₁	#7547
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1425

Complements

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

N	H ≅ G/N	type
32. C₇₁	#7547	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇₁⋊C₆₄	2
#22586544	2
C₇₁×#1425	2
#7547	71