Subgroups — C₇₁×#7514 #31828386

Order 18176.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2258	C₂	1	1	yes	maximal, characteristic
2	C₇₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#504×C₇₁	C₄	1	1	yes	characteristic
5	M₆₄×C₇₁	—	2	1	no	—
6	M₆₄×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₇₁	—	2	1	no	—
8	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
9	C₆₄×C₇₁	—	2	1	no	—
10	C₆₄×C₇₁	—	2	1	no	—
11	#131×C₇₁	C₈	1	1	yes	characteristic
12	M₃₂×C₇₁	—	2	1	no	—
13	M₃₂×C₇₁	C₈	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	—	2	1	no	—
15	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	—	2	1	no	—
17	C₃₂×C₇₁	D₄	1	1	yes	characteristic
18	C₃₂×C₇₁	D₄	1	1	yes	characteristic
19	#41×C₇₁	C₁₆	1	1	yes	characteristic
20	M₁₆×C₇₁	—	2	1	no	—
21	M₁₆×C₇₁	C₁₆	1	1	yes	characteristic
22	C₂×C₈×C₇₁	—	2	1	no	—
23	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₇₁	—	2	1	no	—
25	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₁₆×C₇₁	M₁₆	1	1	yes	characteristic
27	Q₈×C₇₁	C₃₂	1	1	yes	characteristic
28	D₄×C₇₁	—	2	1	no	—
29	D₄×C₇₁	C₃₂	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	1	no	—
31	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
32	C₈×C₇₁	—	2	1	no	—
33	C₈×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
34	C₈×C₇₁	M₃₂	1	1	yes	characteristic
35	C₂²×C₇₁	—	2	1	no	—
36	C₂²×C₇₁	M₆₄	1	1	yes	characteristic
37	C₄×C₇₁	—	2	1	no	—
38	C₄×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
39	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
40	#7514	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×C₇₁	—	2	1	no	—
42	C₂×C₇₁	—	4	1	no	—
43	C₂×C₇₁	C₂²⋊C₃₂	1	1	yes	characteristic
44	#2258	C₂×C₇₁	1	1	yes	characteristic
45	M₁₂₈	C₂×C₇₁	1	1	yes	characteristic, complemented
46	C₂×C₆₄	C₂×C₇₁	1	1	yes	characteristic, complemented
47	C₇₁	#7514	1	1	yes	characteristic, direct product factor, Sylow
48	#504	C₄×C₇₁	1	1	yes	characteristic
49	M₆₄	—	2	1	no	—
50	M₆₄	C₄×C₇₁	1	1	yes	characteristic
51	C₂×C₃₂	—	2	1	no	—
52	C₂×C₃₂	C₂²×C₇₁	1	1	yes	characteristic
53	C₆₄	—	2	1	no	—
54	C₆₄	—	2	1	no	—
55	#131	C₈×C₇₁	1	1	yes	characteristic
56	M₃₂	—	2	1	no	—
57	M₃₂	C₈×C₇₁	1	1	yes	characteristic
58	C₂×C₁₆	—	2	1	no	—
59	C₂×C₁₆	C₂×C₄×C₇₁	1	1	yes	characteristic
60	C₃₂	—	2	1	no	—
61	C₃₂	D₄×C₇₁	1	1	yes	characteristic
62	C₃₂	D₄×C₇₁	1	1	yes	characteristic
63	#41	C₁₆×C₇₁	1	1	yes	characteristic
64	M₁₆	—	2	1	no	—
65	M₁₆	C₁₆×C₇₁	1	1	yes	characteristic
66	C₂×C₈	—	2	1	no	—
67	C₂×C₈	C₂×C₈×C₇₁	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
70	C₁₆	M₁₆×C₇₁	1	1	yes	characteristic
71	Q₈	C₃₂×C₇₁	1	1	yes	characteristic
72	D₄	—	2	1	no	—
73	D₄	C₃₂×C₇₁	1	1	yes	characteristic
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	C₂×C₁₆×C₇₁	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	C₂²⋊C₈×C₇₁	1	1	yes	characteristic
78	C₈	M₃₂×C₇₁	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	M₆₄×C₇₁	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	C₂²⋊C₁₆×C₇₁	1	1	yes	characteristic
83	C₄	C₂×C₃₂×C₇₁	1	1	yes	characteristic
84	C₂	—	2	1	no	—
85	C₂	—	4	1	no	—
86	C₂	C₇₁×C₂²⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂×C₇₁	D₄
commutator	C₄	C₂×C₃₂×C₇₁
Frattini, cosocle	C₂×C₃₂	C₂²×C₇₁
socle	C₂×C₇₁	C₂²⋊C₃₂
prime radical	#41×C₇₁	C₁₆
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×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. C₂×C₆₄×C₇₁	C₂	semidirect
40. #7514	C₇₁	direct (N × H)
45. M₁₂₈	C₂×C₇₁	semidirect
46. C₂×C₆₄	C₂×C₇₁	semidirect
47. C₇₁	#7514	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×#2258	2
C₇₁×M₁₂₈	2
C₂×C₆₄×C₇₁	2
#7514	71

Subgroups — C71×#7514 #31828386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7514 #31828386