Subgroups — C₇₁×#7542 #31828414

Order 18176.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1425	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₃₂×C₇₁	—	2	1	no	—
5	Q₆₄×C₇₁	—	2	1	no	—
6	Q₆₄×C₇₁	C₄	1	2	yes	—
7	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
8	#368×C₇₁	—	2	1	no	—
9	C₆₄×C₇₁	—	2	1	no	—
10	C₂×Q₁₆×C₇₁	—	4	1	no	—
11	Q₃₂×C₇₁	—	2	2	no	—
12	Q₃₂×C₇₁	—	4	1	no	—
13	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
14	#108×C₇₁	—	4	1	no	—
15	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	D₄	1	1	yes	characteristic
17	C₂×Q₈×C₇₁	—	8	1	no	—
18	Q₁₆×C₇₁	—	4	2	no	—
19	Q₁₆×C₇₁	—	8	1	no	—
20	M₁₆×C₇₁	—	8	1	no	—
21	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
22	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
24	Q₈×C₇₁	—	8	2	no	—
25	Q₈×C₇₁	—	16	1	no	—
26	C₂×C₄×C₇₁	—	16	1	no	—
27	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
28	C₈×C₇₁	—	16	1	no	—
29	C₈×C₇₁	#102	1	1	yes	characteristic
30	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
31	C₂²×C₇₁	SD₆₄	1	1	yes	characteristic
32	C₄×C₇₁	—	16	2	no	—
33	C₄×C₇₁	#357	1	1	yes	characteristic
34	C₄×C₇₁	D₃₂	1	1	yes	characteristic
35	#7542	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₇₁	—	2	1	no	—
37	C₂×C₇₁	#1415	1	1	yes	characteristic
38	C₂×Q₆₄	C₂×C₇₁	1	1	yes	characteristic
39	M₁₂₈	C₂×C₇₁	1	1	yes	characteristic
40	#1425	C₂×C₇₁	1	1	yes	characteristic
41	C₇₁	#7542	1	1	yes	characteristic, direct product factor, Sylow
42	C₂×Q₃₂	—	2	1	no	—
43	Q₆₄	—	2	1	no	—
44	Q₆₄	C₄×C₇₁	1	2	yes	—
45	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
46	#368	—	2	1	no	—
47	C₆₄	—	2	1	no	—
48	C₂×Q₁₆	—	4	1	no	—
49	Q₃₂	—	2	2	no	—
50	Q₃₂	—	4	1	no	—
51	C₂×C₁₆	D₄×C₇₁	1	1	yes	characteristic
52	#108	—	4	1	no	—
53	C₃₂	C₂×C₄×C₇₁	1	1	yes	characteristic
54	C₃₂	D₄×C₇₁	1	1	yes	characteristic
55	C₂×Q₈	—	8	1	no	—
56	Q₁₆	—	4	2	no	—
57	Q₁₆	—	8	1	no	—
58	M₁₆	—	8	1	no	—
59	C₂×C₈	D₈×C₇₁	1	1	yes	characteristic
60	C₁₆	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
61	C₁₆	SD₁₆×C₇₁	1	1	yes	characteristic
62	Q₈	—	8	2	no	—
63	Q₈	—	16	1	no	—
64	C₂×C₄	—	16	1	no	—
65	C₂×C₄	D₁₆×C₇₁	1	1	yes	characteristic
66	C₈	—	16	1	no	—
67	C₈	#102×C₇₁	1	1	yes	characteristic
68	C₈	SD₃₂×C₇₁	1	1	yes	characteristic
69	C₂²	SD₆₄×C₇₁	1	1	yes	characteristic
70	C₄	—	16	2	no	—
71	C₄	#357×C₇₁	1	1	yes	characteristic
72	C₄	D₃₂×C₇₁	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	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	C₂²×C₇₁	SD₆₄
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
35. #7542	C₇₁	direct (N × H)
41. C₇₁	#7542	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₆₄×C₇₁	2
C₇₁×M₁₂₈	2
C₇₁×#1425	2
#7542	71

Subgroups — C71×#7542 #31828414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7542 #31828414