Subgroups — C₂⁴×C₃×C₅₁₂ #72201822

Order 24576.

5660 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₂₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₅₁₂	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₅₁₂	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁴×C₃×C₁₂₈	C₄	1	1	yes	characteristic
5	C₂³×C₃×C₂₅₆	C₄	1	15	yes	—
6	C₂³×C₃×C₂₅₆	C₂²	1	15	yes	—
7	C₂²×C₃×C₅₁₂	C₂²	1	140	yes	direct product factor
8	C₂⁴×C₂₅₆	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₅₁₂	C₂×C₃	1	30	yes	direct product factor
10	C₂⁴×C₃×C₆₄	C₈	1	1	yes	characteristic
11	C₂³×C₃×C₁₂₈	C₈	1	15	yes	—
12	C₂³×C₃×C₁₂₈	C₂×C₄	1	15	yes	—
13	C₂²×C₃×C₂₅₆	C₂×C₄	1	105	yes	—
14	C₂²×C₃×C₂₅₆	C₂³	1	35	yes	—
15	C₂×C₃×C₅₁₂	C₂³	1	120	yes	direct product factor
16	C₂⁴×C₁₂₈	C₃×C₄	1	1	yes	characteristic
17	C₂³×C₂₅₆	C₃×C₄	1	15	yes	—
18	C₂³×C₂₅₆	C₂²×C₃	1	15	yes	—
19	C₂²×C₅₁₂	C₂²×C₃	1	140	yes	direct product factor
20	C₂⁴×C₃×C₃₂	C₁₆	1	1	yes	characteristic
21	C₂³×C₃×C₆₄	C₁₆	1	15	yes	—
22	C₂³×C₃×C₆₄	C₂×C₈	1	15	yes	—
23	C₂²×C₃×C₁₂₈	C₂×C₈	1	105	yes	—
24	C₂²×C₃×C₁₂₈	C₂²×C₄	1	35	yes	—
25	C₂×C₃×C₂₅₆	C₂²×C₄	1	105	yes	—
26	C₂×C₃×C₂₅₆	C₂⁴	1	15	yes	—
27	C₃×C₅₁₂	C₂⁴	1	16	yes	direct product factor
28	C₂⁴×C₆₄	C₃×C₈	1	1	yes	characteristic
29	C₂³×C₁₂₈	C₃×C₈	1	15	yes	—
30	C₂³×C₁₂₈	C₂×C₃×C₄	1	15	yes	—
31	C₂²×C₂₅₆	C₂×C₃×C₄	1	105	yes	—
32	C₂²×C₂₅₆	C₂³×C₃	1	35	yes	—
33	C₂×C₅₁₂	C₂³×C₃	1	120	yes	direct product factor
34	C₂⁴×C₃×C₁₆	C₃₂	1	1	yes	characteristic
35	C₂³×C₃×C₃₂	C₃₂	1	15	yes	—
36	C₂³×C₃×C₃₂	C₂×C₁₆	1	15	yes	—
37	C₂²×C₃×C₆₄	C₂×C₁₆	1	105	yes	—
38	C₂²×C₃×C₆₄	C₂²×C₈	1	35	yes	—
39	C₂×C₃×C₁₂₈	C₂²×C₈	1	105	yes	—
40	C₂×C₃×C₁₂₈	C₂³×C₄	1	15	yes	—
41	C₃×C₂₅₆	C₂³×C₄	1	15	yes	—
42	C₃×C₂₅₆	C₂⁵	1	1	yes	characteristic
43	C₂⁴×C₃₂	C₃×C₁₆	1	1	yes	characteristic
44	C₂³×C₆₄	C₃×C₁₆	1	15	yes	—
45	C₂³×C₆₄	C₂×C₃×C₈	1	15	yes	—
46	C₂²×C₁₂₈	C₂×C₃×C₈	1	105	yes	—
47	C₂²×C₁₂₈	C₂²×C₃×C₄	1	35	yes	—
48	C₂×C₂₅₆	C₂²×C₃×C₄	1	105	yes	—
49	C₂×C₂₅₆	C₂⁴×C₃	1	15	yes	—
50	C₅₁₂	C₂⁴×C₃	1	16	yes	direct product factor
51	C₂⁴×C₃×C₈	C₆₄	1	1	yes	characteristic
52	C₂³×C₃×C₁₆	C₆₄	1	15	yes	—
53	C₂³×C₃×C₁₆	C₂×C₃₂	1	15	yes	—
54	C₂²×C₃×C₃₂	C₂×C₃₂	1	105	yes	—
55	C₂²×C₃×C₃₂	C₂²×C₁₆	1	35	yes	—
56	C₂×C₃×C₆₄	C₂²×C₁₆	1	105	yes	—
57	C₂×C₃×C₆₄	C₂³×C₈	1	15	yes	—
58	C₃×C₁₂₈	C₂³×C₈	1	15	yes	—
59	C₃×C₁₂₈	C₂⁴×C₄	1	1	yes	characteristic
60	C₂⁴×C₁₆	C₃×C₃₂	1	1	yes	characteristic
61	C₂³×C₃₂	C₃×C₃₂	1	15	yes	—
62	C₂³×C₃₂	C₂×C₃×C₁₆	1	15	yes	—
63	C₂²×C₆₄	C₂×C₃×C₁₆	1	105	yes	—
64	C₂²×C₆₄	C₂²×C₃×C₈	1	35	yes	—
65	C₂×C₁₂₈	C₂²×C₃×C₈	1	105	yes	—
66	C₂×C₁₂₈	C₂³×C₃×C₄	1	15	yes	—
67	C₂₅₆	C₂³×C₃×C₄	1	15	yes	—
68	C₂₅₆	C₂⁵×C₃	1	1	yes	characteristic
69	C₂⁴×C₃×C₄	C₁₂₈	1	1	yes	characteristic
70	C₂³×C₃×C₈	C₁₂₈	1	15	yes	—
71	C₂³×C₃×C₈	C₂×C₆₄	1	15	yes	—
72	C₂²×C₃×C₁₆	C₂×C₆₄	1	105	yes	—
73	C₂²×C₃×C₁₆	C₂²×C₃₂	1	35	yes	—
74	C₂×C₃×C₃₂	C₂²×C₃₂	1	105	yes	—
75	C₂×C₃×C₃₂	C₂³×C₁₆	1	15	yes	—
76	C₃×C₆₄	C₂³×C₁₆	1	15	yes	—
77	C₃×C₆₄	C₂⁴×C₈	1	1	yes	characteristic
78	C₂⁴×C₈	C₃×C₆₄	1	1	yes	characteristic
79	C₂³×C₁₆	C₃×C₆₄	1	15	yes	—
80	C₂³×C₁₆	C₂×C₃×C₃₂	1	15	yes	—
81	C₂²×C₃₂	C₂×C₃×C₃₂	1	105	yes	—
82	C₂²×C₃₂	C₂²×C₃×C₁₆	1	35	yes	—
83	C₂×C₆₄	C₂²×C₃×C₁₆	1	105	yes	—
84	C₂×C₆₄	C₂³×C₃×C₈	1	15	yes	—
85	C₁₂₈	C₂³×C₃×C₈	1	15	yes	—
86	C₁₂₈	C₂⁴×C₃×C₄	1	1	yes	characteristic
87	C₂⁵×C₃	C₂₅₆	1	1	yes	characteristic
88	C₂³×C₃×C₄	C₂₅₆	1	15	yes	—
89	C₂³×C₃×C₄	C₂×C₁₂₈	1	15	yes	—
90	C₂²×C₃×C₈	C₂×C₁₂₈	1	105	yes	—
91	C₂²×C₃×C₈	C₂²×C₆₄	1	35	yes	—
92	C₂×C₃×C₁₆	C₂²×C₆₄	1	105	yes	—
93	C₂×C₃×C₁₆	C₂³×C₃₂	1	15	yes	—
94	C₃×C₃₂	C₂³×C₃₂	1	15	yes	—
95	C₃×C₃₂	C₂⁴×C₁₆	1	1	yes	characteristic
96	C₂⁴×C₄	C₃×C₁₂₈	1	1	yes	characteristic
97	C₂³×C₈	C₃×C₁₂₈	1	15	yes	—
98	C₂³×C₈	C₂×C₃×C₆₄	1	15	yes	—
99	C₂²×C₁₆	C₂×C₃×C₆₄	1	105	yes	—
100	C₂²×C₁₆	C₂²×C₃×C₃₂	1	35	yes	—
101	C₂×C₃₂	C₂²×C₃×C₃₂	1	105	yes	—
102	C₂×C₃₂	C₂³×C₃×C₁₆	1	15	yes	—
103	C₆₄	C₂³×C₃×C₁₆	1	15	yes	—
104	C₆₄	C₂⁴×C₃×C₈	1	1	yes	characteristic
105	C₂⁴×C₃	C₅₁₂	1	16	yes	direct product factor
106	C₂⁴×C₃	C₂×C₂₅₆	1	15	yes	—
107	C₂²×C₃×C₄	C₂×C₂₅₆	1	105	yes	—
108	C₂²×C₃×C₄	C₂²×C₁₂₈	1	35	yes	—
109	C₂×C₃×C₈	C₂²×C₁₂₈	1	105	yes	—
110	C₂×C₃×C₈	C₂³×C₆₄	1	15	yes	—
111	C₃×C₁₆	C₂³×C₆₄	1	15	yes	—
112	C₃×C₁₆	C₂⁴×C₃₂	1	1	yes	characteristic
113	C₂⁵	C₃×C₂₅₆	1	1	yes	characteristic
114	C₂³×C₄	C₃×C₂₅₆	1	15	yes	—
115	C₂³×C₄	C₂×C₃×C₁₂₈	1	15	yes	—
116	C₂²×C₈	C₂×C₃×C₁₂₈	1	105	yes	—
117	C₂²×C₈	C₂²×C₃×C₆₄	1	35	yes	—
118	C₂×C₁₆	C₂²×C₃×C₆₄	1	105	yes	—
119	C₂×C₁₆	C₂³×C₃×C₃₂	1	15	yes	—
120	C₃₂	C₂³×C₃×C₃₂	1	15	yes	—
121	C₃₂	C₂⁴×C₃×C₁₆	1	1	yes	characteristic
122	C₂³×C₃	C₂×C₅₁₂	1	120	yes	direct product factor
123	C₂³×C₃	C₂²×C₂₅₆	1	35	yes	—
124	C₂×C₃×C₄	C₂²×C₂₅₆	1	105	yes	—
125	C₂×C₃×C₄	C₂³×C₁₂₈	1	15	yes	—
126	C₃×C₈	C₂³×C₁₂₈	1	15	yes	—
127	C₃×C₈	C₂⁴×C₆₄	1	1	yes	characteristic
128	C₂⁴	C₃×C₅₁₂	1	16	yes	direct product factor
129	C₂⁴	C₂×C₃×C₂₅₆	1	15	yes	—
130	C₂²×C₄	C₂×C₃×C₂₅₆	1	105	yes	—
131	C₂²×C₄	C₂²×C₃×C₁₂₈	1	35	yes	—
132	C₂×C₈	C₂²×C₃×C₁₂₈	1	105	yes	—
133	C₂×C₈	C₂³×C₃×C₆₄	1	15	yes	—
134	C₁₆	C₂³×C₃×C₆₄	1	15	yes	—
135	C₁₆	C₂⁴×C₃×C₃₂	1	1	yes	characteristic
136	C₂²×C₃	C₂²×C₅₁₂	1	140	yes	direct product factor
137	C₂²×C₃	C₂³×C₂₅₆	1	15	yes	—
138	C₃×C₄	C₂³×C₂₅₆	1	15	yes	—
139	C₃×C₄	C₂⁴×C₁₂₈	1	1	yes	characteristic
140	C₂³	C₂×C₃×C₅₁₂	1	120	yes	direct product factor
141	C₂³	C₂²×C₃×C₂₅₆	1	35	yes	—
142	C₂×C₄	C₂²×C₃×C₂₅₆	1	105	yes	—
143	C₂×C₄	C₂³×C₃×C₁₂₈	1	15	yes	—
144	C₈	C₂³×C₃×C₁₂₈	1	15	yes	—
145	C₈	C₂⁴×C₃×C₆₄	1	1	yes	characteristic
146	C₂×C₃	C₂³×C₅₁₂	1	30	yes	direct product factor
147	C₂×C₃	C₂⁴×C₂₅₆	1	1	yes	characteristic
148	C₂²	C₂²×C₃×C₅₁₂	1	140	yes	direct product factor
149	C₂²	C₂³×C₃×C₂₅₆	1	15	yes	—
150	C₄	C₂³×C₃×C₂₅₆	1	15	yes	—
151	C₄	C₂⁴×C₃×C₁₂₈	1	1	yes	characteristic
152	C₃	C₂⁴×C₅₁₂	1	1	yes	characteristic, direct product factor, Sylow
153	C₂	C₂³×C₃×C₅₁₂	1	30	yes	direct product factor
154	C₂	C₂⁴×C₃×C₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂₅₆	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₂₅₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×C₅₁₂	C₂	direct (N × H)
3. C₂⁴×C₅₁₂	C₃	direct (N × H)
7. C₂²×C₃×C₅₁₂	C₂²	direct (N × H)
9. C₂³×C₅₁₂	C₂×C₃	direct (N × H)
15. C₂×C₃×C₅₁₂	C₂³	direct (N × H)
19. C₂²×C₅₁₂	C₂²×C₃	direct (N × H)
27. C₃×C₅₁₂	C₂⁴	direct (N × H)
33. C₂×C₅₁₂	C₂³×C₃	direct (N × H)
50. C₅₁₂	C₂⁴×C₃	direct (N × H)
105. C₂⁴×C₃	C₅₁₂	direct (N × H)
122. C₂³×C₃	C₂×C₅₁₂	direct (N × H)
128. C₂⁴	C₃×C₅₁₂	direct (N × H)
136. C₂²×C₃	C₂²×C₅₁₂	direct (N × H)
140. C₂³	C₂×C₃×C₅₁₂	direct (N × H)
146. C₂×C₃	C₂³×C₅₁₂	direct (N × H)
148. C₂²	C₂²×C₃×C₅₁₂	direct (N × H)
152. C₃	C₂⁴×C₅₁₂	direct (N × H)
153. C₂	C₂³×C₃×C₅₁₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₂₅₆	2
C₂³×C₃×C₅₁₂	2
C₂⁴×C₅₁₂	3

Subgroups — C24×C3×C512 #72201822

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×C₅₁₂ #72201822