Subgroups — #31885135

Order 18176.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586414	C₂	1	2	yes	maximal
2	C₇₁×#1397	C₂	1	1	yes	maximal, characteristic
3	#17985752	—	2	2	no	—
4	C₂×C₃₂×C₇₁	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₇₁	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
7	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
8	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
11	C₃₂×C₇₁	—	2	1	no	—
12	C₃₂×C₇₁	C₈	1	2	yes	complemented
13	C₂×C₇₁⋊C₈	—	8	2	no	—
14	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
16	C₄²×C₇₁	D₈	1	1	yes	characteristic
17	C₁₆×C₇₁	—	2	1	no	—
18	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₇₁	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
21	C₂×C₄×C₇₁	SD₃₂	1	2	yes	—
22	C₈×C₇₁	—	2	1	no	—
23	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₇₁	#108	1	1	yes	characteristic
25	C₇₁⋊C₈	—	16	2	no	—
26	C₂²×C₇₁	#367	1	1	yes	characteristic
27	C₄×C₇₁	—	2	2	no	—
28	C₄×C₇₁	#335	1	1	yes	characteristic
29	C₄×C₇₁	#368	1	1	yes	characteristic
30	#7463	—	71	1	no	maximal, Sylow
31	C₂×C₇₁	#1372	1	1	yes	characteristic
32	C₂×C₇₁	#1426	1	2	yes	—
33	#1397	D₇₁	1	1	yes	characteristic
34	#1372	—	71	2	no	—
35	C₇₁	#7463	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₃₂	Dic₇₁	1	1	yes	characteristic
37	C₂×C₃₂	Dic₇₁	1	1	yes	characteristic
38	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
39	#335	—	142	2	no	—
40	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
41	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
42	C₄⋊C₈	—	284	2	no	—
43	C₄×C₈	D₂₈₄	1	1	yes	characteristic
44	C₃₂	—	2	1	no	—
45	C₃₂	C₇₁⋊C₈	1	2	yes	complemented
46	C₂×C₈	—	568	2	no	—
47	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
48	C₂×C₈	Dic₂₈₄	1	1	yes	characteristic
49	C₄²	D₅₆₈	1	1	yes	characteristic
50	C₁₆	—	2	1	no	—
51	C₁₆	C₂×C₇₁⋊C₈	1	1	yes	characteristic
52	C₁₆	#11765849	1	1	yes	characteristic
53	C₂×C₄	#14556062	1	1	yes	characteristic
54	C₂×C₄	#14556073	1	2	yes	—
55	C₈	—	2	1	no	—
56	C₈	—	1136	2	no	—
57	C₈	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
58	C₈	#14556064	1	1	yes	characteristic
59	C₂²	#17985827	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	#17985752	1	1	yes	characteristic
62	C₄	#17985829	1	1	yes	characteristic
63	C₂	#22586414	1	1	yes	characteristic
64	C₂	#22586546	1	2	yes	—

	subgroup	G/H
center	C₂²	#17985827
commutator	C₁₆×C₇₁	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1397	C₂
socle, prime radical	C₂²×C₇₁	#367
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1397	D₇₁
hypocenter	C₇₁	#7463
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₇₁	C₈	semidirect
35. C₇₁	#7463	semidirect
45. C₃₂	C₇₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22586414	2
C₇₁×#1397	2
#7463	71