Subgroups — C₂×C₃³×#127 #13179985

Order 1728.

6972 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃³×D₄	C₂	1	6	yes	maximal, complemented
2	C₃³×#127	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃²×#127	C₃	1	13	yes	maximal, direct product factor
5	C₂⁴×C₃³	C₂²	1	4	yes	—
6	C₂×C₃³×D₄	—	2	24	no	—
7	C₂×C₃³×D₄	C₂²	1	24	yes	complemented
8	C₂²×C₃³×C₄	C₂²	1	3	yes	—
9	C₃³×C₄²	C₂²	1	4	yes	complemented
10	C₂²×C₃²×D₄	C₂×C₃	1	78	yes	complemented
11	C₃²×#127	C₂×C₃	1	104	yes	direct product factor
12	C₂×C₃²×C₄²	C₂×C₃	1	13	yes	complemented
13	C₂³×C₃³	—	2	24	no	—
14	C₂³×C₃³	C₂³	1	8	yes	—
15	C₂³×C₃³	C₂³	1	1	yes	characteristic
16	C₃³×D₄	—	2	48	no	—
17	C₂×C₃³×C₄	D₄	1	12	yes	complemented
18	C₂×C₃³×C₄	C₂³	1	6	yes	—
19	C₂×C₃×#127	C₃²	1	13	yes	direct product factor
20	C₂⁴×C₃²	C₂²×C₃	1	52	yes	—
21	C₂×C₃²×D₄	—	2	312	no	—
22	C₂×C₃²×D₄	C₂²×C₃	1	312	yes	complemented
23	C₂²×C₃²×C₄	C₂²×C₃	1	39	yes	—
24	C₃²×C₄²	C₂²×C₃	1	52	yes	complemented
25	C₂²×C₃³	—	2	24	no	—
26	C₂²×C₃³	—	4	16	no	—
27	C₂²×C₃³	C₂×D₄	1	6	yes	—
28	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
29	C₃³×C₄	C₂×D₄	1	12	yes	complemented
30	C₂²×C₃×D₄	C₂×C₃²	1	78	yes	complemented
31	C₃×#127	C₂×C₃²	1	104	yes	direct product factor
32	C₂×C₃×C₄²	C₂×C₃²	1	13	yes	complemented
33	C₂³×C₃²	—	2	312	no	—
34	C₂³×C₃²	C₂³×C₃	1	104	yes	—
35	C₂³×C₃²	C₂³×C₃	1	13	yes	—
36	C₃²×D₄	—	2	624	no	—
37	C₂×C₃²×C₄	C₃×D₄	1	156	yes	complemented
38	C₂×C₃²×C₄	C₂³×C₃	1	78	yes	—
39	C₂×#127	C₃³	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃³	—	4	8	no	—
41	C₂×C₃³	#127	1	4	yes	direct product factor
42	C₂×C₃³	C₂²×D₄	1	3	yes	—
43	C₂⁴×C₃	C₂²×C₃²	1	52	yes	—
44	C₂×C₃×D₄	—	2	312	no	—
45	C₂×C₃×D₄	C₂²×C₃²	1	312	yes	complemented
46	C₂²×C₃×C₄	C₂²×C₃²	1	39	yes	—
47	C₃×C₄²	C₂²×C₃²	1	52	yes	complemented
48	C₂²×C₃²	—	2	312	no	—
49	C₂²×C₃²	—	4	208	no	—
50	C₂²×C₃²	C₂×C₃×D₄	1	78	yes	—
51	C₂²×C₃²	C₂⁴×C₃	1	13	yes	—
52	C₃²×C₄	C₂×C₃×D₄	1	156	yes	complemented
53	C₂²×D₄	C₂×C₃³	1	6	yes	complemented
54	#127	C₂×C₃³	1	8	yes	direct product factor
55	C₂×C₄²	C₂×C₃³	1	1	yes	characteristic, complemented
56	C₃³	C₂×#127	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃	—	2	312	no	—
58	C₂³×C₃	C₂³×C₃²	1	104	yes	—
59	C₂³×C₃	C₂³×C₃²	1	13	yes	—
60	C₃×D₄	—	2	624	no	—
61	C₂×C₃×C₄	C₃²×D₄	1	156	yes	complemented
62	C₂×C₃×C₄	C₂³×C₃²	1	78	yes	—
63	C₂×C₃²	—	4	104	no	—
64	C₂×C₃²	C₃×#127	1	52	yes	direct product factor
65	C₂×C₃²	C₂²×C₃×D₄	1	39	yes	—
66	C₂⁴	C₂²×C₃³	1	4	yes	—
67	C₂×D₄	—	2	24	no	—
68	C₂×D₄	C₂²×C₃³	1	24	yes	complemented
69	C₂²×C₄	C₂²×C₃³	1	3	yes	—
70	C₄²	C₂²×C₃³	1	4	yes	complemented
71	C₂²×C₃	—	2	312	no	—
72	C₂²×C₃	—	4	208	no	—
73	C₂²×C₃	C₂×C₃²×D₄	1	78	yes	—
74	C₂²×C₃	C₂⁴×C₃²	1	13	yes	—
75	C₃×C₄	C₂×C₃²×D₄	1	156	yes	complemented
76	C₃²	C₂×C₃×#127	1	13	yes	direct product factor
77	C₂³	—	2	24	no	—
78	C₂³	C₂³×C₃³	1	8	yes	—
79	C₂³	C₂³×C₃³	1	1	yes	characteristic
80	D₄	—	2	48	no	—
81	C₂×C₄	C₃³×D₄	1	12	yes	complemented
82	C₂×C₄	C₂³×C₃³	1	6	yes	—
83	C₂×C₃	—	4	104	no	—
84	C₂×C₃	C₃²×#127	1	52	yes	direct product factor
85	C₂×C₃	C₂²×C₃²×D₄	1	39	yes	—
86	C₂²	—	2	24	no	—
87	C₂²	—	4	16	no	—
88	C₂²	C₂×C₃³×D₄	1	6	yes	—
89	C₂²	C₂⁴×C₃³	1	1	yes	characteristic
90	C₄	C₂×C₃³×D₄	1	12	yes	complemented
91	C₃	C₂×C₃²×#127	1	13	yes	direct product factor
92	C₂	—	4	8	no	—
93	C₂	C₃³×#127	1	4	yes	direct product factor
94	C₂	C₂²×C₃³×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃³×D₄	C₂	semidirect
2. C₃³×#127	C₂	direct (N × H)
3. C₂×C₃³×C₄²	C₂	semidirect
4. C₂×C₃²×#127	C₃	direct (N × H)
7. C₂×C₃³×D₄	C₂²	semidirect
9. C₃³×C₄²	C₂²	semidirect
10. C₂²×C₃²×D₄	C₂×C₃	semidirect
11. C₃²×#127	C₂×C₃	direct (N × H)
12. C₂×C₃²×C₄²	C₂×C₃	semidirect
17. C₂×C₃³×C₄	D₄	semidirect
19. C₂×C₃×#127	C₃²	direct (N × H)
22. C₂×C₃²×D₄	C₂²×C₃	semidirect
24. C₃²×C₄²	C₂²×C₃	semidirect
29. C₃³×C₄	C₂×D₄	semidirect
30. C₂²×C₃×D₄	C₂×C₃²	semidirect
31. C₃×#127	C₂×C₃²	direct (N × H)
32. C₂×C₃×C₄²	C₂×C₃²	semidirect
37. C₂×C₃²×C₄	C₃×D₄	semidirect
39. C₂×#127	C₃³	direct (N × H)
41. C₂×C₃³	#127	direct (N × H)
45. C₂×C₃×D₄	C₂²×C₃²	semidirect
47. C₃×C₄²	C₂²×C₃²	semidirect
52. C₃²×C₄	C₂×C₃×D₄	semidirect
53. C₂²×D₄	C₂×C₃³	semidirect
54. #127	C₂×C₃³	direct (N × H)
55. C₂×C₄²	C₂×C₃³	semidirect
56. C₃³	C₂×#127	direct (N × H)
61. C₂×C₃×C₄	C₃²×D₄	semidirect
64. C₂×C₃²	C₃×#127	direct (N × H)
68. C₂×D₄	C₂²×C₃³	semidirect
70. C₄²	C₂²×C₃³	semidirect
75. C₃×C₄	C₂×C₃²×D₄	semidirect
76. C₃²	C₂×C₃×#127	direct (N × H)
81. C₂×C₄	C₃³×D₄	semidirect
84. C₂×C₃	C₃²×#127	direct (N × H)
90. C₄	C₂×C₃³×D₄	semidirect
91. C₃	C₂×C₃²×#127	direct (N × H)
93. C₂	C₃³×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃³×D₄	2
C₃³×#127	2
C₂×C₃³×C₄²	2
C₂×C₃²×#127	3

Subgroups — C2×C33×#127 #13179985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#127 #13179985