Subgroups — C₇₁×#7540 #31828412

Order 18176.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2262	C₂	1	1	yes	maximal, characteristic
2	C₇₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1424	C₂	1	1	yes	maximal, characteristic, complemented
4	#508×C₇₁	—	2	1	no	—
5	Q₆₄×C₇₁	C₄	1	1	yes	characteristic, complemented
6	SD₆₄×C₇₁	—	2	1	no	—
7	D₃₂×C₇₁	C₄	1	1	yes	characteristic, complemented
8	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
9	#366×C₇₁	—	2	1	no	—
10	C₆₄×C₇₁	—	2	1	no	—
11	#135×C₇₁	—	4	1	no	—
12	Q₃₂×C₇₁	—	2	1	no	—
13	SD₃₂×C₇₁	—	4	1	no	—
14	D₁₆×C₇₁	—	2	1	no	—
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	#107×C₇₁	—	4	1	no	—
17	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₃₂×C₇₁	D₄	1	1	yes	characteristic
19	#41×C₇₁	—	8	1	no	—
20	Q₁₆×C₇₁	—	4	1	no	—
21	SD₁₆×C₇₁	—	8	1	no	—
22	D₈×C₇₁	—	4	1	no	—
23	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
24	C₄⋊C₄×C₇₁	—	8	1	no	—
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	—	16	1	no	—
32	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
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	—	32	1	no	—
39	C₄×C₇₁	#357	1	1	yes	characteristic
40	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
41	#7540	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₇₁	—	2	1	no	—
43	C₂×C₇₁	—	32	1	no	—
44	C₂×C₇₁	#1415	1	1	yes	characteristic
45	#2262	C₂×C₇₁	1	1	yes	characteristic
46	M₁₂₈	C₂×C₇₁	1	1	yes	characteristic, complemented
47	#1424	C₂×C₇₁	1	1	yes	characteristic, complemented
48	C₇₁	#7540	1	1	yes	characteristic, direct product factor, Sylow
49	#508	—	2	1	no	—
50	Q₆₄	C₄×C₇₁	1	1	yes	characteristic, complemented
51	SD₆₄	—	2	1	no	—
52	D₃₂	C₄×C₇₁	1	1	yes	characteristic, complemented
53	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
54	#366	—	2	1	no	—
55	C₆₄	—	2	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	D₄×C₇₁	1	1	yes	characteristic
61	#107	—	4	1	no	—
62	C₃₂	C₂×C₄×C₇₁	1	1	yes	characteristic
63	C₃₂	D₄×C₇₁	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	D₈×C₇₁	1	1	yes	characteristic
69	C₄⋊C₄	—	8	1	no	—
70	C₁₆	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
71	C₁₆	SD₁₆×C₇₁	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	—	16	1	no	—
77	C₂×C₄	D₁₆×C₇₁	1	1	yes	characteristic
78	C₈	#102×C₇₁	1	1	yes	characteristic
79	C₈	SD₃₂×C₇₁	1	1	yes	characteristic
80	C₂²	—	16	1	no	—
81	C₂²	D₃₂×C₇₁	1	1	yes	characteristic
82	C₄	—	16	1	no	—
83	C₄	—	32	1	no	—
84	C₄	#357×C₇₁	1	1	yes	characteristic
85	C₄	SD₆₄×C₇₁	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	32	1	no	—
88	C₂	C₇₁×#1415	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₇₁×M₁₂₈	C₂	semidirect
3. C₇₁×#1424	C₂	semidirect
5. Q₆₄×C₇₁	C₄	semidirect
7. D₃₂×C₇₁	C₄	semidirect
41. #7540	C₇₁	direct (N × H)
46. M₁₂₈	C₂×C₇₁	semidirect
47. #1424	C₂×C₇₁	semidirect
48. C₇₁	#7540	direct (N × H)
50. Q₆₄	C₄×C₇₁	semidirect
52. D₃₂	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₇₁×#2262	2
C₇₁×M₁₂₈	2
C₇₁×#1424	2
#7540	71

Subgroups — C71×#7540 #31828412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7540 #31828412