Subgroups — #31918475

Order 18176.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590617	C₂	1	2	yes	maximal
2	#22586542	C₂	1	1	yes	maximal, characteristic
3	#22586539	C₂	1	2	yes	maximal
4	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1424	C₂	1	1	yes	maximal, characteristic
6	#17986400	—	2	2	no	—
7	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
8	#17985826	—	2	2	no	—
9	#17985825	C₂²	1	2	yes	—
10	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
12	#366×C₇₁	C₂²	1	2	yes	—
13	#14556119	—	4	2	no	—
14	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
15	#14556062	—	2	2	no	—
16	#14556059	—	4	2	no	—
17	C₈×Dic₇₁	D₄	1	1	yes	characteristic
18	C₇₁⋊C₃₂	Q₈	1	2	yes	—
19	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
20	#107×C₇₁	—	2	2	no	—
21	C₃₂×C₇₁	Q₈	1	2	yes	—
22	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
23	C₄⋊Dic₇₁	—	4	2	no	—
24	C₄⋊Dic₇₁	—	8	2	no	—
25	Dic₇₁⋊C₄	—	8	2	no	—
26	C₄×Dic₇₁	D₈	1	1	yes	characteristic
27	C₇₁⋊C₁₆	Q₁₆	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
31	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	8	2	no	—
33	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₇₁	—	8	2	no	—
35	C₂×Dic₇₁	D₁₆	1	2	yes	—
36	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
37	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
38	C₇₁⋊C₈	Q₃₂	1	2	yes	—
39	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
40	C₄×C₇₁	—	16	2	no	—
41	C₄×C₇₁	#500	1	1	yes	characteristic
42	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
43	Dic₇₁	—	2	2	no	—
44	Dic₇₁	—	16	2	no	—
45	#13732	—	71	1	no	maximal, Sylow
46	C₂×C₇₁	#2252	1	1	yes	characteristic
47	C₂×C₇₁	#2262	1	2	yes	—
48	#2253	—	71	2	no	—
49	#1424	—	71	1	no	—
50	#1424	D₇₁	1	1	yes	characteristic
51	#1423	—	71	2	no	—
52	C₄×C₃₂	—	71	1	no	—
53	C₇₁	#13732	1	1	yes	characteristic, complemented, Sylow
54	#499	—	142	2	no	—
55	C₂×C₃₂	—	71	1	no	—
56	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
57	#367	—	142	2	no	—
58	#366	—	71	2	no	—
59	#366	C₂×D₇₁	1	2	yes	—
60	C₄×C₁₆	—	71	1	no	—
61	#125	—	284	2	no	—
62	C₂×C₁₆	—	71	1	no	—
63	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
64	#107	—	2	2	no	—
65	#107	—	142	2	no	—
66	#106	—	284	2	no	—
67	C₄×C₈	—	71	1	no	—
68	C₃₂	—	71	2	no	—
69	C₃₂	Dic₁₄₂	1	2	yes	—
70	C₂×C₈	—	71	1	no	—
71	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	284	2	no	—
74	C₄⋊C₄	—	568	2	no	—
75	C₄⋊C₄	—	568	2	no	—
76	C₄²	—	71	1	no	—
77	C₁₆	—	71	2	no	—
78	C₁₆	Q₈×D₇₁	1	1	yes	characteristic
79	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	568	2	no	—
82	C₂×C₄	—	71	2	no	—
83	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
84	C₈	—	71	2	no	—
85	C₈	#14556127	1	1	yes	characteristic
86	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
87	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
88	C₄	—	16	2	no	—
89	C₄	—	1136	2	no	—
90	C₄	—	142	2	no	—
91	C₄	#17986403	1	1	yes	characteristic
92	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
93	C₂	#22590614	1	1	yes	characteristic
94	C₂	#22590648	1	2	yes	—

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1424	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#1424	D₇₁
hypocenter	C₇₁	#13732
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
53. C₇₁	#13732	semidirect

Maximal subgroups

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

subgroup	index
#22590617	2
#22586542	2
#22586539	2
C₃₂×Dic₇₁	2
C₇₁×#1424	2
#13732	71