Subgroups — #31885356

Order 18176.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586544	C₂	1	2	yes	maximal
2	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985829	—	2	2	no	—
4	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
5	C₆₄×C₇₁	C₄	1	2	yes	—
6	#14556064	—	4	2	no	—
7	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
8	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
9	C₃₂×C₇₁	Q₈	1	1	yes	characteristic
10	#11765849	—	8	2	no	—
11	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
12	C₁₆×C₇₁	C₄⋊C₄	1	1	yes	characteristic
13	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
14	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
15	C₈×C₇₁	#107	1	1	yes	characteristic
16	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
17	C₇₁⋊C₈	—	16	2	no	—
18	C₂²×C₇₁	Q₆₄	1	1	yes	characteristic
19	C₄×C₇₁	#366	1	1	yes	characteristic
20	C₄×C₇₁	D₃₂	1	1	yes	characteristic
21	#7547	—	71	1	no	maximal, Sylow
22	C₂×C₇₁	—	2	1	no	—
23	C₂×C₇₁	#1423	1	1	yes	characteristic
24	C₂×C₆₄	D₇₁	1	1	yes	characteristic
25	#1425	—	71	2	no	—
26	C₇₁	#7547	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
28	#368	—	142	2	no	—
29	C₆₄	Dic₇₁	1	2	yes	—
30	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
31	#108	—	284	2	no	—
32	C₃₂	Dic₁₄₂	1	1	yes	characteristic
33	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
34	M₁₆	—	568	2	no	—
35	C₂×C₈	D₅₆₈	1	1	yes	characteristic
36	C₁₆	C₄⋊Dic₇₁	1	1	yes	characteristic
37	C₁₆	Dic₂₈₄	1	1	yes	characteristic
38	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
39	C₈	—	1136	2	no	—
40	C₈	#14556062	1	1	yes	characteristic
41	C₈	Dic₅₆₈	1	1	yes	characteristic
42	C₂²	Dic₁₁₃₆	1	1	yes	characteristic
43	C₄	#17985825	1	1	yes	characteristic
44	C₄	D₂₂₇₂	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#22586540	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₂₇₂
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₂×C₆₄×C₇₁	C₂
socle	C₂×C₇₁	#1423
prime radical	C₂²×C₇₁	Q₆₄
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	C₂×C₆₄	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 ◁ C₂×C₆₄

Complements

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

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

Maximal subgroups

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

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