Subgroups — #31885120

Order 18176.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586538	C₂	1	2	yes	maximal
2	C₇₁×#1311	C₂	1	1	yes	maximal, characteristic
3	#17985833	—	2	2	no	—
4	C₄×C₁₆×C₇₁	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
6	C₈²×C₇₁	C₄	1	1	yes	characteristic
7	C₇₁⋊C₃₂	—	4	2	no	—
8	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
9	C₂×C₁₆×C₇₁	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
12	C₄×C₈×C₇₁	—	2	1	no	—
13	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₇₁	—	2	2	no	—
15	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
16	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₇₁	M₁₆	1	1	yes	characteristic
18	C₁₆×C₇₁	—	2	1	no	—
19	C₁₆×C₇₁	—	2	1	no	—
20	C₁₆×C₇₁	C₁₆	1	2	yes	—
21	C₁₆×C₇₁	SD₁₆	1	2	yes	—
22	C₂×C₄×C₇₁	—	2	1	no	—
23	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
24	C₈×C₇₁	—	4	2	no	—
25	C₈×C₇₁	#106	1	1	yes	characteristic
26	C₈×C₇₁	#108	1	1	yes	characteristic
27	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
28	C₈×C₇₁	M₃₂	1	1	yes	characteristic
29	C₂²×C₇₁	#364	1	1	yes	characteristic
30	C₄×C₇₁	—	2	2	no	—
31	C₄×C₇₁	#334	1	1	yes	characteristic
32	C₄×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
33	#7456	—	71	1	no	maximal, Sylow
34	C₂×C₇₁	—	2	1	no	—
35	C₂×C₇₁	#1367	1	1	yes	characteristic
36	#1422	—	71	2	no	—
37	#1311	D₇₁	1	1	yes	characteristic
38	C₇₁	#7456	1	1	yes	characteristic, complemented, Sylow
39	M₆₄	—	142	2	no	—
40	C₄×C₁₆	Dic₇₁	1	1	yes	characteristic
41	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
42	C₈²	Dic₇₁	1	1	yes	characteristic
43	C₂×C₁₆	C₇₁⋊C₈	1	1	yes	characteristic
44	C₂×C₁₆	C₇₁⋊C₈	1	1	yes	characteristic
45	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
46	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
47	C₄×C₈	—	2	1	no	—
48	C₄×C₈	C₂×Dic₇₁	1	1	yes	characteristic
49	C₃₂	—	284	2	no	—
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
52	C₂×C₈	C₂×C₇₁⋊C₈	1	1	yes	characteristic
53	C₄²	#11765849	1	1	yes	characteristic
54	C₁₆	—	2	1	no	—
55	C₁₆	—	2	1	no	—
56	C₁₆	C₇₁⋊C₁₆	1	2	yes	—
57	C₁₆	#11765854	1	2	yes	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
60	C₈	—	4	2	no	—
61	C₈	#14556060	1	1	yes	characteristic
62	C₈	#14556064	1	1	yes	characteristic
63	C₈	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
64	C₈	#14556068	1	1	yes	characteristic
65	C₂²	#17985820	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	#17985750	1	1	yes	characteristic
68	C₄	C₄⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#22586404	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556060
commutator	C₈×C₇₁	C₂×C₁₆
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1311	C₂
socle	C₂×C₇₁	#1367
prime radical	C₂²×C₇₁	#364
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1311	D₇₁
hypocenter	C₇₁	#7456
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ #1311

Complements

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

N	H ≅ G/N	type
38. C₇₁	#7456	semidirect

Maximal subgroups

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

subgroup	index
#22586538	2
C₇₁×#1311	2
#7456	71