Subgroups — C₃×C₆₄×C₁₂₈ #72201830

Order 24576.

988 conjugacy classes of subgroups.

The subgroup lattice has 112 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₆₄²	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃₂×C₁₂₈	C₂	1	2	yes	maximal
3	C₆₄×C₁₂₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃×C₃₂×C₆₄	C₄	1	2	yes	—
5	C₃×C₃₂×C₆₄	C₂²	1	1	yes	characteristic
6	C₃×C₁₆×C₁₂₈	C₄	1	4	yes	—
7	C₆₄²	C₂×C₃	1	1	yes	characteristic
8	C₃₂×C₁₂₈	C₂×C₃	1	2	yes	—
9	C₃×C₃₂²	C₂×C₄	1	1	yes	characteristic
10	C₃×C₁₆×C₆₄	C₈	1	4	yes	—
11	C₃×C₁₆×C₆₄	C₂×C₄	1	2	yes	—
12	C₃×C₈×C₁₂₈	C₈	1	8	yes	—
13	C₃₂×C₆₄	C₃×C₄	1	2	yes	—
14	C₃₂×C₆₄	C₂²×C₃	1	1	yes	characteristic
15	C₁₆×C₁₂₈	C₃×C₄	1	4	yes	—
16	C₃×C₁₆×C₃₂	C₄²	1	1	yes	characteristic
17	C₃×C₁₆×C₃₂	C₂×C₈	1	2	yes	—
18	C₃×C₈×C₆₄	C₁₆	1	8	yes	—
19	C₃×C₈×C₆₄	C₂×C₈	1	4	yes	—
20	C₃×C₄×C₁₂₈	C₁₆	1	16	yes	—
21	C₃₂²	C₂×C₃×C₄	1	1	yes	characteristic
22	C₁₆×C₆₄	C₃×C₈	1	4	yes	—
23	C₁₆×C₆₄	C₂×C₃×C₄	1	2	yes	—
24	C₈×C₁₂₈	C₃×C₈	1	8	yes	—
25	C₂×C₃×C₁₂₈	C₃₂	1	32	yes	—
26	C₃×C₄×C₆₄	C₃₂	1	16	yes	—
27	C₃×C₄×C₆₄	C₂×C₁₆	1	8	yes	—
28	C₃×C₈×C₃₂	C₄×C₈	1	2	yes	—
29	C₃×C₈×C₃₂	C₂×C₁₆	1	4	yes	—
30	C₃×C₁₆²	C₄×C₈	1	1	yes	characteristic
31	C₄×C₁₂₈	C₃×C₁₆	1	16	yes	—
32	C₈×C₆₄	C₃×C₁₆	1	8	yes	—
33	C₈×C₆₄	C₂×C₃×C₈	1	4	yes	—
34	C₁₆×C₃₂	C₃×C₄²	1	1	yes	characteristic
35	C₁₆×C₃₂	C₂×C₃×C₈	1	2	yes	—
36	C₂×C₃×C₆₄	C₆₄	1	32	yes	—
37	C₂×C₃×C₆₄	C₂×C₃₂	1	16	yes	—
38	C₃×C₄×C₃₂	C₄×C₁₆	1	4	yes	—
39	C₃×C₄×C₃₂	C₂×C₃₂	1	8	yes	—
40	C₃×C₈×C₁₆	C₈²	1	1	yes	characteristic
41	C₃×C₈×C₁₆	C₄×C₁₆	1	2	yes	—
42	C₃×C₁₂₈	C₆₄	1	64	yes	direct product factor
43	C₂×C₁₂₈	C₃×C₃₂	1	32	yes	—
44	C₄×C₆₄	C₃×C₃₂	1	16	yes	—
45	C₄×C₆₄	C₂×C₃×C₁₆	1	8	yes	—
46	C₈×C₃₂	C₃×C₄×C₈	1	2	yes	—
47	C₈×C₃₂	C₂×C₃×C₁₆	1	4	yes	—
48	C₁₆²	C₃×C₄×C₈	1	1	yes	characteristic
49	C₂×C₃×C₃₂	C₄×C₃₂	1	8	yes	—
50	C₂×C₃×C₃₂	C₂×C₆₄	1	16	yes	—
51	C₃×C₄×C₁₆	C₈×C₁₆	1	2	yes	—
52	C₃×C₄×C₁₆	C₄×C₃₂	1	4	yes	—
53	C₃×C₈²	C₈×C₁₆	1	1	yes	characteristic
54	C₃×C₆₄	C₁₂₈	1	64	yes	direct product factor
55	C₃×C₆₄	C₂×C₆₄	1	32	yes	—
56	C₂×C₆₄	C₃×C₆₄	1	32	yes	—
57	C₂×C₆₄	C₂×C₃×C₃₂	1	16	yes	—
58	C₄×C₃₂	C₃×C₄×C₁₆	1	4	yes	—
59	C₄×C₃₂	C₂×C₃×C₃₂	1	8	yes	—
60	C₈×C₁₆	C₃×C₈²	1	1	yes	characteristic
61	C₈×C₁₆	C₃×C₄×C₁₆	1	2	yes	—
62	C₁₂₈	C₃×C₆₄	1	64	yes	direct product factor
63	C₂×C₃×C₁₆	C₈×C₃₂	1	4	yes	—
64	C₂×C₃×C₁₆	C₄×C₆₄	1	8	yes	—
65	C₃×C₄×C₈	C₁₆²	1	1	yes	characteristic
66	C₃×C₄×C₈	C₈×C₃₂	1	2	yes	—
67	C₃×C₃₂	C₄×C₆₄	1	16	yes	—
68	C₃×C₃₂	C₂×C₁₂₈	1	32	yes	—
69	C₂×C₃₂	C₃×C₄×C₃₂	1	8	yes	—
70	C₂×C₃₂	C₂×C₃×C₆₄	1	16	yes	—
71	C₄×C₁₆	C₃×C₈×C₁₆	1	2	yes	—
72	C₄×C₁₆	C₃×C₄×C₃₂	1	4	yes	—
73	C₈²	C₃×C₈×C₁₆	1	1	yes	characteristic
74	C₆₄	C₃×C₁₂₈	1	64	yes	direct product factor
75	C₆₄	C₂×C₃×C₆₄	1	32	yes	—
76	C₂×C₃×C₈	C₁₆×C₃₂	1	2	yes	—
77	C₂×C₃×C₈	C₈×C₆₄	1	4	yes	—
78	C₃×C₄²	C₁₆×C₃₂	1	1	yes	characteristic
79	C₃×C₁₆	C₈×C₆₄	1	8	yes	—
80	C₃×C₁₆	C₄×C₁₂₈	1	16	yes	—
81	C₂×C₁₆	C₃×C₈×C₃₂	1	4	yes	—
82	C₂×C₁₆	C₃×C₄×C₆₄	1	8	yes	—
83	C₄×C₈	C₃×C₁₆²	1	1	yes	characteristic
84	C₄×C₈	C₃×C₈×C₃₂	1	2	yes	—
85	C₃₂	C₃×C₄×C₆₄	1	16	yes	—
86	C₃₂	C₂×C₃×C₁₂₈	1	32	yes	—
87	C₂×C₃×C₄	C₁₆×C₆₄	1	2	yes	—
88	C₂×C₃×C₄	C₃₂²	1	1	yes	characteristic
89	C₃×C₈	C₈×C₁₂₈	1	8	yes	—
90	C₃×C₈	C₁₆×C₆₄	1	4	yes	—
91	C₂×C₈	C₃×C₈×C₆₄	1	4	yes	—
92	C₂×C₈	C₃×C₁₆×C₃₂	1	2	yes	—
93	C₄²	C₃×C₁₆×C₃₂	1	1	yes	characteristic
94	C₁₆	C₃×C₄×C₁₂₈	1	16	yes	—
95	C₁₆	C₃×C₈×C₆₄	1	8	yes	—
96	C₂²×C₃	C₃₂×C₆₄	1	1	yes	characteristic
97	C₃×C₄	C₁₆×C₁₂₈	1	4	yes	—
98	C₃×C₄	C₃₂×C₆₄	1	2	yes	—
99	C₂×C₄	C₃×C₁₆×C₆₄	1	2	yes	—
100	C₂×C₄	C₃×C₃₂²	1	1	yes	characteristic
101	C₈	C₃×C₈×C₁₂₈	1	8	yes	—
102	C₈	C₃×C₁₆×C₆₄	1	4	yes	—
103	C₂×C₃	C₃₂×C₁₂₈	1	2	yes	—
104	C₂×C₃	C₆₄²	1	1	yes	characteristic
105	C₂²	C₃×C₃₂×C₆₄	1	1	yes	characteristic
106	C₄	C₃×C₁₆×C₁₂₈	1	4	yes	—
107	C₄	C₃×C₃₂×C₆₄	1	2	yes	—
108	C₃	C₆₄×C₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
109	C₂	C₃×C₃₂×C₁₂₈	1	2	yes	—
110	C₂	C₃×C₆₄²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂×C₆₄	C₂²×C₃
socle, prime radical	C₂²×C₃	C₃₂×C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₄×C₁₂₈	C₃	direct (N × H)
42. C₃×C₁₂₈	C₆₄	direct (N × H)
54. C₃×C₆₄	C₁₂₈	direct (N × H)
62. C₁₂₈	C₃×C₆₄	direct (N × H)
74. C₆₄	C₃×C₁₂₈	direct (N × H)
108. C₃	C₆₄×C₁₂₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₆₄²	2
C₃×C₃₂×C₁₂₈	2
C₆₄×C₁₂₈	3

Subgroups — C3×C64×C128 #72201830

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₆₄×C₁₂₈ #72201830