Subgroups — C₂²×C₄²×D₂₇₁ #46849518

Order 34688.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₂₇₁	C₂	1	3	yes	maximal
2	C₂²×C₄×Dic₂₇₁	C₂	1	3	yes	maximal, complemented
3	C₂×C₄²×D₂₇₁	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₄²×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₂₇₁	C₂²	1	1	yes	characteristic
6	C₂³×Dic₂₇₁	C₂²	1	3	yes	—
7	C₂²×C₄×D₂₇₁	C₄	1	48	yes	direct product factor
8	C₂²×C₄×D₂₇₁	C₂²	1	36	yes	—
9	C₂×C₄×Dic₂₇₁	C₂²	1	36	yes	complemented
10	C₄²×D₂₇₁	C₂²	1	64	yes	direct product factor
11	C₂³×C₄×C₂₇₁	C₂²	1	3	yes	—
12	C₂×C₄²×C₂₇₁	C₂²	1	12	yes	complemented
13	C₂³×D₂₇₁	C₂×C₄	1	24	yes	—
14	C₂³×D₂₇₁	C₂³	1	6	yes	—
15	C₂²×Dic₂₇₁	C₂×C₄	1	24	yes	complemented
16	C₂²×Dic₂₇₁	C₂³	1	18	yes	—
17	C₂×C₄×D₂₇₁	C₂×C₄	1	288	yes	direct product factor
18	C₂×C₄×D₂₇₁	C₂³	1	48	yes	—
19	C₄×Dic₂₇₁	C₂³	1	48	yes	complemented
20	C₂⁴×C₂₇₁	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₂₇₁	C₂×C₄	1	24	yes	complemented
22	C₂²×C₄×C₂₇₁	C₂³	1	18	yes	—
23	C₄²×C₂₇₁	C₂³	1	16	yes	complemented
24	C₂²×D₂₇₁	C₄²	1	64	yes	direct product factor
25	C₂²×D₂₇₁	C₂²×C₄	1	72	yes	—
26	C₂²×D₂₇₁	C₂⁴	1	4	yes	—
27	C₂×Dic₂₇₁	C₂²×C₄	1	72	yes	complemented
28	C₂×Dic₂₇₁	C₂⁴	1	12	yes	—
29	C₄×D₂₇₁	C₂²×C₄	1	192	yes	direct product factor
30	C₂³×C₂₇₁	C₂²×C₄	1	12	yes	—
31	C₂³×C₂₇₁	C₂⁴	1	3	yes	—
32	C₂×C₄×C₂₇₁	C₂²×C₄	1	72	yes	complemented
33	C₂×C₄×C₂₇₁	C₂⁴	1	12	yes	—
34	C₂²×C₂₇₁	C₂×C₄²	1	16	yes	complemented
35	C₂²×C₂₇₁	C₂³×C₄	1	18	yes	—
36	C₂²×C₂₇₁	C₂⁵	1	1	yes	characteristic
37	C₂×D₂₇₁	C₂×C₄²	1	96	yes	direct product factor
38	C₂×D₂₇₁	C₂³×C₄	1	24	yes	—
39	C₄×C₂₇₁	C₂³×C₄	1	24	yes	complemented
40	Dic₂₇₁	C₂³×C₄	1	24	yes	complemented
41	C₂×C₂₇₁	C₂²×C₄²	1	12	yes	complemented
42	C₂×C₂₇₁	C₂⁴×C₄	1	3	yes	—
43	D₂₇₁	C₂²×C₄²	1	16	yes	direct product factor
44	C₂₇₁	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
45	C₂³×C₄²	—	271	1	no	maximal, Sylow
46	C₂⁴×C₄	—	271	3	no	—
47	C₂²×C₄²	—	271	3	no	—
48	C₂²×C₄²	—	271	24	no	—
49	C₂²×C₄²	D₂₇₁	1	1	yes	characteristic, direct product factor
50	C₂⁵	—	271	1	no	—
51	C₂³×C₄	—	271	3	no	—
52	C₂³×C₄	—	271	48	no	—
53	C₂³×C₄	—	271	36	no	—
54	C₂³×C₄	C₂×D₂₇₁	1	3	yes	—
55	C₂×C₄²	—	271	36	no	—
56	C₂×C₄²	—	271	64	no	—
57	C₂×C₄²	C₂×D₂₇₁	1	12	yes	direct product factor
58	C₂⁴	—	271	24	no	—
59	C₂⁴	—	271	6	no	—
60	C₂⁴	C₂²×D₂₇₁	1	1	yes	characteristic
61	C₂²×C₄	—	271	24	no	—
62	C₂²×C₄	—	271	18	no	—
63	C₂²×C₄	—	271	288	no	—
64	C₂²×C₄	—	271	48	no	—
65	C₂²×C₄	C₄×D₂₇₁	1	24	yes	direct product factor
66	C₂²×C₄	C₂²×D₂₇₁	1	18	yes	—
67	C₄²	—	271	48	no	—
68	C₄²	C₂²×D₂₇₁	1	16	yes	direct product factor
69	C₂³	—	271	72	no	—
70	C₂³	—	271	4	no	—
71	C₂³	—	271	64	no	—
72	C₂³	C₂×C₄×D₂₇₁	1	12	yes	—
73	C₂³	C₂³×D₂₇₁	1	3	yes	—
74	C₂×C₄	—	271	72	no	—
75	C₂×C₄	—	271	12	no	—
76	C₂×C₄	—	271	192	no	—
77	C₂×C₄	C₂×C₄×D₂₇₁	1	72	yes	direct product factor
78	C₂×C₄	C₂³×D₂₇₁	1	12	yes	—
79	C₂²	—	271	96	no	—
80	C₂²	—	271	24	no	—
81	C₂²	C₄²×D₂₇₁	1	16	yes	direct product factor
82	C₂²	C₂²×C₄×D₂₇₁	1	18	yes	—
83	C₂²	C₂⁴×D₂₇₁	1	1	yes	characteristic
84	C₄	—	271	24	no	—
85	C₄	C₂²×C₄×D₂₇₁	1	24	yes	direct product factor
86	C₂	—	271	16	no	—
87	C₂	C₂×C₄²×D₂₇₁	1	12	yes	direct product factor
88	C₂	C₂³×C₄×D₂₇₁	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂²×C₄²	D₂₇₁
commutator, hypocenter	C₂₇₁	C₂³×C₄²
Frattini	C₂²	C₂⁴×D₂₇₁
Fitting	C₂²×C₄²×C₂₇₁	C₂
socle	C₂⁴×C₂₇₁	C₂³
prime radical	C₂⁴×D₂₇₁	C₂²
cosocle	C₂²×C₂₇₁	C₂⁵
LCS	G ▷ C₂₇₁
UCS	C₁ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Dic₂₇₁	C₂	semidirect
3. C₂×C₄²×D₂₇₁	C₂	direct (N × H)
4. C₂²×C₄²×C₂₇₁	C₂	semidirect
7. C₂²×C₄×D₂₇₁	C₄	direct (N × H)
9. C₂×C₄×Dic₂₇₁	C₂²	semidirect
10. C₄²×D₂₇₁	C₂²	direct (N × H)
12. C₂×C₄²×C₂₇₁	C₂²	semidirect
15. C₂²×Dic₂₇₁	C₂×C₄	semidirect
17. C₂×C₄×D₂₇₁	C₂×C₄	direct (N × H)
19. C₄×Dic₂₇₁	C₂³	semidirect
21. C₂²×C₄×C₂₇₁	C₂×C₄	semidirect
23. C₄²×C₂₇₁	C₂³	semidirect
24. C₂²×D₂₇₁	C₄²	direct (N × H)
27. C₂×Dic₂₇₁	C₂²×C₄	semidirect
29. C₄×D₂₇₁	C₂²×C₄	direct (N × H)
32. C₂×C₄×C₂₇₁	C₂²×C₄	semidirect
34. C₂²×C₂₇₁	C₂×C₄²	semidirect
37. C₂×D₂₇₁	C₂×C₄²	direct (N × H)
39. C₄×C₂₇₁	C₂³×C₄	semidirect
40. Dic₂₇₁	C₂³×C₄	semidirect
41. C₂×C₂₇₁	C₂²×C₄²	semidirect
43. D₂₇₁	C₂²×C₄²	direct (N × H)
44. C₂₇₁	C₂³×C₄²	semidirect
49. C₂²×C₄²	D₂₇₁	direct (N × H)
57. C₂×C₄²	C₂×D₂₇₁	direct (N × H)
65. C₂²×C₄	C₄×D₂₇₁	direct (N × H)
68. C₄²	C₂²×D₂₇₁	direct (N × H)
77. C₂×C₄	C₂×C₄×D₂₇₁	direct (N × H)
81. C₂²	C₄²×D₂₇₁	direct (N × H)
85. C₄	C₂²×C₄×D₂₇₁	direct (N × H)
87. C₂	C₂×C₄²×D₂₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₂₇₁	2
C₂²×C₄×Dic₂₇₁	2
C₂×C₄²×D₂₇₁	2
C₂²×C₄²×C₂₇₁	2
C₂³×C₄²	271

Subgroups — C22×C42×D271 #46849518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×D₂₇₁ #46849518