Subgroups — C₇₁×#7454 #31828326

Order 18176.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2253	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1422	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1373	C₂	1	1	yes	maximal, characteristic
4	#499×C₇₁	—	2	1	no	—
5	M₆₄×C₇₁	—	2	1	no	—
6	#367×C₇₁	C₄	1	2	yes	—
7	#366×C₇₁	—	2	1	no	—
8	#364×C₇₁	—	2	1	no	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	#125×C₇₁	—	4	1	no	—
11	M₃₂×C₇₁	—	4	1	no	—
12	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
14	#107×C₇₁	—	2	2	no	—
15	#106×C₇₁	—	4	1	no	—
16	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₃₂×C₇₁	—	4	1	no	—
18	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
20	C₄⋊C₄×C₇₁	—	4	2	no	—
21	C₄⋊C₄×C₇₁	—	8	2	no	—
22	C₄²×C₇₁	SD₁₆	1	1	yes	characteristic
23	C₁₆×C₇₁	—	2	1	no	—
24	C₁₆×C₇₁	—	8	1	no	—
25	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
26	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	2	1	no	—
28	C₂×C₄×C₇₁	—	8	2	no	—
29	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
30	C₈×C₇₁	#101	1	1	yes	characteristic
31	C₈×C₇₁	#103	1	1	yes	characteristic
32	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
33	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
34	C₂²×C₇₁	#361	1	1	yes	characteristic
35	C₄×C₇₁	—	4	1	no	—
36	C₄×C₇₁	—	16	2	no	—
37	C₄×C₇₁	#332	1	1	yes	characteristic
38	C₄×C₇₁	#358	1	1	yes	characteristic
39	#7454	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₇₁	—	2	1	no	—
41	C₂×C₇₁	#1365	1	1	yes	characteristic
42	#2253	C₂×C₇₁	1	1	yes	characteristic
43	#1422	C₂×C₇₁	1	1	yes	characteristic
44	#1373	C₂×C₇₁	1	1	yes	characteristic
45	C₇₁	#7454	1	1	yes	characteristic, direct product factor, Sylow
46	#499	—	2	1	no	—
47	M₆₄	—	2	1	no	—
48	#367	C₄×C₇₁	1	2	yes	—
49	#366	—	2	1	no	—
50	#364	—	2	1	no	—
51	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
52	#125	—	4	1	no	—
53	M₃₂	—	4	1	no	—
54	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
55	C₂×C₁₆	D₄×C₇₁	1	1	yes	characteristic
56	#107	—	2	2	no	—
57	#106	—	4	1	no	—
58	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
59	C₃₂	—	4	1	no	—
60	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
61	C₂×C₈	D₈×C₇₁	1	1	yes	characteristic
62	C₄⋊C₄	—	4	2	no	—
63	C₄⋊C₄	—	8	2	no	—
64	C₄²	SD₁₆×C₇₁	1	1	yes	characteristic
65	C₁₆	—	2	1	no	—
66	C₁₆	—	8	1	no	—
67	C₁₆	SD₁₆×C₇₁	1	1	yes	characteristic
68	C₁₆	Q₁₆×C₇₁	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	8	2	no	—
71	C₂×C₄	#102×C₇₁	1	1	yes	characteristic
72	C₈	#101×C₇₁	1	1	yes	characteristic
73	C₈	#103×C₇₁	1	1	yes	characteristic
74	C₈	SD₃₂×C₇₁	1	1	yes	characteristic
75	C₈	Q₃₂×C₇₁	1	1	yes	characteristic
76	C₂²	#361×C₇₁	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	16	2	no	—
79	C₄	#332×C₇₁	1	1	yes	characteristic
80	C₄	#358×C₇₁	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	C₇₁×#1365	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇₁	#1365
commutator	C₂×C₁₆	C₂×C₄×C₇₁
Frattini, cosocle	C₄×C₁₆	C₂²×C₇₁
prime radical	C₂²×C₇₁	#361
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₄×C₈×C₇₁ ◁ C₄×C₁₆×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
39. #7454	C₇₁	direct (N × H)
45. C₇₁	#7454	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×#2253	2
C₇₁×#1422	2
C₇₁×#1373	2
#7454	71

Subgroups — C71×#7454 #31828326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7454 #31828326