Subgroups — #31885339

Order 18176.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586551	C₂	1	1	yes	maximal, characteristic, complemented
2	#22586542	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#2262	C₂	1	1	yes	maximal, characteristic
4	#17985825	—	2	1	no	—
5	C₇₁⋊C₆₄	—	2	1	no	—
6	#508×C₇₁	—	2	1	no	—
7	Q₆₄×C₇₁	C₄	1	1	yes	characteristic, complemented
8	SD₆₄×C₇₁	—	2	1	no	—
9	D₃₂×C₇₁	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	#14556062	—	4	1	no	—
12	#135×C₇₁	—	4	1	no	—
13	Q₃₂×C₇₁	—	2	1	no	—
14	SD₃₂×C₇₁	—	4	1	no	—
15	D₁₆×C₇₁	—	2	1	no	—
16	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
17	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₁	D₄	1	1	yes	characteristic
19	C₄⋊Dic₇₁	—	8	1	no	—
20	#41×C₇₁	—	8	1	no	—
21	Q₁₆×C₇₁	—	4	1	no	—
22	SD₁₆×C₇₁	—	8	1	no	—
23	D₈×C₇₁	—	4	1	no	—
24	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
25	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
27	Q₈×C₇₁	—	8	1	no	—
28	D₄×C₇₁	—	8	1	no	—
29	D₄×C₇₁	—	16	1	no	—
30	C₂×C₄×C₇₁	—	16	1	no	—
31	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
32	C₂×Dic₇₁	—	16	1	no	—
33	C₈×C₇₁	#102	1	1	yes	characteristic
34	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
35	C₂²×C₇₁	—	16	1	no	—
36	C₂²×C₇₁	D₃₂	1	1	yes	characteristic
37	C₄×C₇₁	—	16	1	no	—
38	C₄×C₇₁	#357	1	1	yes	characteristic
39	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
40	Dic₇₁	—	32	1	no	—
41	#7540	—	71	1	no	maximal, Sylow
42	C₂×C₇₁	—	2	1	no	—
43	C₂×C₇₁	—	32	1	no	—
44	C₂×C₇₁	#1415	1	1	yes	characteristic
45	#2262	D₇₁	1	1	yes	characteristic
46	M₁₂₈	—	71	1	no	—
47	#1424	—	71	1	no	—
48	C₇₁	#7540	1	1	yes	characteristic, complemented, Sylow
49	#508	—	2	1	no	—
50	Q₆₄	Dic₇₁	1	1	yes	characteristic, complemented
51	SD₆₄	—	2	1	no	—
52	D₃₂	Dic₇₁	1	1	yes	characteristic, complemented
53	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
54	#366	—	142	1	no	—
55	C₆₄	—	142	1	no	—
56	#135	—	4	1	no	—
57	Q₃₂	—	2	1	no	—
58	SD₃₂	—	4	1	no	—
59	D₁₆	—	2	1	no	—
60	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
61	#107	—	284	1	no	—
62	C₃₂	C₂×Dic₇₁	1	1	yes	characteristic
63	C₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
64	#41	—	8	1	no	—
65	Q₁₆	—	4	1	no	—
66	SD₁₆	—	8	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	#11765850	1	1	yes	characteristic
69	C₄⋊C₄	—	568	1	no	—
70	C₁₆	C₂²⋊Dic₇₁	1	1	yes	characteristic
71	C₁₆	#11765853	1	1	yes	characteristic
72	Q₈	—	8	1	no	—
73	D₄	—	8	1	no	—
74	D₄	—	16	1	no	—
75	C₂×C₄	—	16	1	no	—
76	C₂×C₄	—	1136	1	no	—
77	C₂×C₄	#14556069	1	1	yes	characteristic
78	C₈	#14556049	1	1	yes	characteristic
79	C₈	#14556072	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	#17985834	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	2272	1	no	—
84	C₄	#17985800	1	1	yes	characteristic
85	C₄	#17985837	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	#22586518	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22586518
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting, prime radical	C₇₁×#2262	C₂
socle	C₂×C₇₁	#1415
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	#2262	D₇₁
hypocenter	C₇₁	#7540
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #2262

Complements

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

N	H ≅ G/N	type
1. #22586551	C₂	semidirect
2. #22586542	C₂	semidirect
7. Q₆₄×C₇₁	C₄	semidirect
9. D₃₂×C₇₁	C₄	semidirect
48. C₇₁	#7540	semidirect
50. Q₆₄	Dic₇₁	semidirect
52. D₃₂	Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22586551	2
#22586542	2
C₇₁×#2262	2
#7540	71