Subgroups — C₇₁×#2264 #22584826

Order 9088.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₆₄×C₇₁	C₂	1	2	yes	maximal, complemented
4	SD₆₄×C₇₁	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₇₁	—	2	1	no	—
7	C₂×Q₁₆×C₇₁	—	2	1	no	—
8	Q₃₂×C₇₁	—	2	1	no	—
9	Q₃₂×C₇₁	C₂²	1	2	yes	—
10	Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	SD₃₂×C₇₁	—	2	1	no	—
12	D₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₃₂×C₇₁	C₂²	1	2	yes	—
15	#41×C₇₁	—	4	1	no	—
16	C₂×Q₈×C₇₁	—	4	1	no	—
17	Q₁₆×C₇₁	—	2	1	no	—
18	Q₁₆×C₇₁	—	2	2	no	—
19	Q₁₆×C₇₁	—	4	1	no	—
20	SD₁₆×C₇₁	—	4	1	no	—
21	D₈×C₇₁	—	2	1	no	—
22	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₁₆×C₇₁	D₄	1	1	yes	characteristic
24	C₁₆×C₇₁	C₂³	1	1	yes	characteristic
25	Q₈×C₇₁	—	4	1	no	—
26	Q₈×C₇₁	—	4	2	no	—
27	Q₈×C₇₁	—	8	1	no	—
28	D₄×C₇₁	—	4	1	no	—
29	D₄×C₇₁	—	8	1	no	—
30	C₂×C₄×C₇₁	—	8	1	no	—
31	C₂×C₄×C₇₁	—	8	1	no	—
32	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
33	C₈×C₇₁	D₈	1	1	yes	characteristic
34	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₇₁	—	8	1	no	—
36	C₂²×C₇₁	D₁₆	1	1	yes	characteristic
37	C₄×C₇₁	—	8	1	no	—
38	C₄×C₇₁	—	8	2	no	—
39	C₄×C₇₁	D₁₆	1	1	yes	characteristic
40	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
41	C₂×C₇₁	—	2	1	no	—
42	C₂×C₇₁	—	16	1	no	—
43	C₂×C₇₁	C₂×D₁₆	1	1	yes	characteristic
44	#2264	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₇₁	#2264	1	1	yes	characteristic, direct product factor, Sylow
46	#508	C₂×C₇₁	1	1	yes	characteristic
47	C₂×Q₃₂	C₂×C₇₁	1	1	yes	characteristic, complemented
48	Q₆₄	C₂×C₇₁	1	2	yes	complemented
49	SD₆₄	C₂×C₇₁	1	2	yes	complemented
50	M₆₄	C₂×C₇₁	1	1	yes	characteristic, complemented
51	#135	—	2	1	no	—
52	C₂×Q₁₆	—	2	1	no	—
53	Q₃₂	—	2	1	no	—
54	Q₃₂	C₂²×C₇₁	1	2	yes	—
55	Q₃₂	C₂²×C₇₁	1	1	yes	characteristic
56	SD₃₂	—	2	1	no	—
57	D₁₆	C₂²×C₇₁	1	1	yes	characteristic
58	C₂×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
59	C₃₂	C₂²×C₇₁	1	2	yes	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	4	1	no	—
62	Q₁₆	—	2	1	no	—
63	Q₁₆	—	2	2	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	1	no	—
67	C₂×C₈	D₄×C₇₁	1	1	yes	characteristic
68	C₁₆	D₄×C₇₁	1	1	yes	characteristic
69	C₁₆	C₂³×C₇₁	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	8	1	no	—
73	D₄	—	4	1	no	—
74	D₄	—	8	1	no	—
75	C₂×C₄	—	8	1	no	—
76	C₂×C₄	—	8	1	no	—
77	C₂×C₄	D₈×C₇₁	1	1	yes	characteristic
78	C₈	D₈×C₇₁	1	1	yes	characteristic
79	C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
80	C₂²	—	8	1	no	—
81	C₂²	D₁₆×C₇₁	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	2	no	—
84	C₄	D₁₆×C₇₁	1	1	yes	characteristic
85	C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	16	1	no	—
88	C₂	C₂×D₁₆×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇₁	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₇₁
prime radical	#508×C₇₁	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇₁ ◁ C₂×C₄×C₇₁ ◁ C₂×C₈×C₇₁ ◁ C₂×C₁₆×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₃₂×C₇₁	C₂	semidirect
3. Q₆₄×C₇₁	C₂	semidirect
4. SD₆₄×C₇₁	C₂	semidirect
5. M₆₄×C₇₁	C₂	semidirect
44. #2264	C₇₁	direct (N × H)
45. C₇₁	#2264	direct (N × H)
47. C₂×Q₃₂	C₂×C₇₁	semidirect
48. Q₆₄	C₂×C₇₁	semidirect
49. SD₆₄	C₂×C₇₁	semidirect
50. M₆₄	C₂×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
#508×C₇₁	2
C₂×Q₃₂×C₇₁	2
Q₆₄×C₇₁	2
SD₆₄×C₇₁	2
M₆₄×C₇₁	2
#2264	71

Subgroups — C71×#2264 #22584826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#2264 #22584826