Subgroups — C₇₁×#7539 #31828411

Order 18176.

88 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	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1425	C₂	1	1	yes	maximal, characteristic, complemented
4	#508×C₇₁	—	2	1	no	—
5	Q₆₄×C₇₁	C₄	1	1	yes	characteristic
6	SD₆₄×C₇₁	—	2	1	no	—
7	D₃₂×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
9	#368×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	#108×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	M₁₆×C₇₁	—	8	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	C₈×C₇₁	—	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	SD₆₄	1	1	yes	characteristic
37	C₄×C₇₁	—	16	1	no	—
38	C₄×C₇₁	#357	1	1	yes	characteristic
39	C₄×C₇₁	D₃₂	1	1	yes	characteristic
40	#7539	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₇₁	—	2	1	no	—
42	C₂×C₇₁	—	32	1	no	—
43	C₂×C₇₁	#1415	1	1	yes	characteristic
44	#2262	C₂×C₇₁	1	1	yes	characteristic
45	C₂×C₆₄	C₂×C₇₁	1	1	yes	characteristic, complemented
46	#1425	C₂×C₇₁	1	1	yes	characteristic, complemented
47	C₇₁	#7539	1	1	yes	characteristic, direct product factor, Sylow
48	#508	—	2	1	no	—
49	Q₆₄	C₄×C₇₁	1	1	yes	characteristic
50	SD₆₄	—	2	1	no	—
51	D₃₂	C₄×C₇₁	1	1	yes	characteristic
52	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
53	#368	—	2	1	no	—
54	C₆₄	—	2	1	no	—
55	#135	—	4	1	no	—
56	Q₃₂	—	2	1	no	—
57	SD₃₂	—	4	1	no	—
58	D₁₆	—	2	1	no	—
59	C₂×C₁₆	D₄×C₇₁	1	1	yes	characteristic
60	#108	—	4	1	no	—
61	C₃₂	C₂×C₄×C₇₁	1	1	yes	characteristic
62	C₃₂	D₄×C₇₁	1	1	yes	characteristic
63	#41	—	8	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	8	1	no	—
66	D₈	—	4	1	no	—
67	M₁₆	—	8	1	no	—
68	C₂×C₈	D₈×C₇₁	1	1	yes	characteristic
69	C₁₆	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
70	C₁₆	SD₁₆×C₇₁	1	1	yes	characteristic
71	Q₈	—	8	1	no	—
72	D₄	—	8	1	no	—
73	D₄	—	16	1	no	—
74	C₂×C₄	—	16	1	no	—
75	C₂×C₄	D₁₆×C₇₁	1	1	yes	characteristic
76	C₈	—	16	1	no	—
77	C₈	#102×C₇₁	1	1	yes	characteristic
78	C₈	SD₃₂×C₇₁	1	1	yes	characteristic
79	C₂²	—	16	1	no	—
80	C₂²	SD₆₄×C₇₁	1	1	yes	characteristic
81	C₄	—	16	1	no	—
82	C₄	#357×C₇₁	1	1	yes	characteristic
83	C₄	D₃₂×C₇₁	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	32	1	no	—
86	C₂	C₇₁×#1415	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₁	D₃₂
commutator	C₃₂	C₂×C₄×C₇₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₇₁
socle	C₂×C₇₁	#1415
prime radical	C₇₁×#2262	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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. C₂×C₆₄×C₇₁	C₂	semidirect
3. C₇₁×#1425	C₂	semidirect
40. #7539	C₇₁	direct (N × H)
45. C₂×C₆₄	C₂×C₇₁	semidirect
46. #1425	C₂×C₇₁	semidirect
47. C₇₁	#7539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₁×#2262	2
C₂×C₆₄×C₇₁	2
C₇₁×#1425	2
#7539	71

Subgroups — C71×#7539 #31828411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7539 #31828411