Subgroups — C₂²×C₃×C₂₀₄₈ #72201815

Order 24576.

252 conjugacy classes of subgroups.

The subgroup lattice has 108 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₀₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂₀₄₈	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₂₀₄₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂²×C₃×C₅₁₂	C₄	1	1	yes	characteristic
5	C₂×C₃×C₁₀₂₄	C₄	1	3	yes	—
6	C₂×C₃×C₁₀₂₄	C₂²	1	3	yes	—
7	C₃×C₂₀₄₈	C₂²	1	4	yes	direct product factor
8	C₂²×C₁₀₂₄	C₂×C₃	1	1	yes	characteristic
9	C₂×C₂₀₄₈	C₂×C₃	1	6	yes	direct product factor
10	C₂²×C₃×C₂₅₆	C₈	1	1	yes	characteristic
11	C₂×C₃×C₅₁₂	C₈	1	3	yes	—
12	C₂×C₃×C₅₁₂	C₂×C₄	1	3	yes	—
13	C₃×C₁₀₂₄	C₂×C₄	1	3	yes	—
14	C₃×C₁₀₂₄	C₂³	1	1	yes	characteristic
15	C₂²×C₅₁₂	C₃×C₄	1	1	yes	characteristic
16	C₂×C₁₀₂₄	C₃×C₄	1	3	yes	—
17	C₂×C₁₀₂₄	C₂²×C₃	1	3	yes	—
18	C₂₀₄₈	C₂²×C₃	1	4	yes	direct product factor
19	C₂²×C₃×C₁₂₈	C₁₆	1	1	yes	characteristic
20	C₂×C₃×C₂₅₆	C₁₆	1	3	yes	—
21	C₂×C₃×C₂₅₆	C₂×C₈	1	3	yes	—
22	C₃×C₅₁₂	C₂×C₈	1	3	yes	—
23	C₃×C₅₁₂	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₂₅₆	C₃×C₈	1	1	yes	characteristic
25	C₂×C₅₁₂	C₃×C₈	1	3	yes	—
26	C₂×C₅₁₂	C₂×C₃×C₄	1	3	yes	—
27	C₁₀₂₄	C₂×C₃×C₄	1	3	yes	—
28	C₁₀₂₄	C₂³×C₃	1	1	yes	characteristic
29	C₂²×C₃×C₆₄	C₃₂	1	1	yes	characteristic
30	C₂×C₃×C₁₂₈	C₃₂	1	3	yes	—
31	C₂×C₃×C₁₂₈	C₂×C₁₆	1	3	yes	—
32	C₃×C₂₅₆	C₂×C₁₆	1	3	yes	—
33	C₃×C₂₅₆	C₂²×C₈	1	1	yes	characteristic
34	C₂²×C₁₂₈	C₃×C₁₆	1	1	yes	characteristic
35	C₂×C₂₅₆	C₃×C₁₆	1	3	yes	—
36	C₂×C₂₅₆	C₂×C₃×C₈	1	3	yes	—
37	C₅₁₂	C₂×C₃×C₈	1	3	yes	—
38	C₅₁₂	C₂²×C₃×C₄	1	1	yes	characteristic
39	C₂²×C₃×C₃₂	C₆₄	1	1	yes	characteristic
40	C₂×C₃×C₆₄	C₆₄	1	3	yes	—
41	C₂×C₃×C₆₄	C₂×C₃₂	1	3	yes	—
42	C₃×C₁₂₈	C₂×C₃₂	1	3	yes	—
43	C₃×C₁₂₈	C₂²×C₁₆	1	1	yes	characteristic
44	C₂²×C₆₄	C₃×C₃₂	1	1	yes	characteristic
45	C₂×C₁₂₈	C₃×C₃₂	1	3	yes	—
46	C₂×C₁₂₈	C₂×C₃×C₁₆	1	3	yes	—
47	C₂₅₆	C₂×C₃×C₁₆	1	3	yes	—
48	C₂₅₆	C₂²×C₃×C₈	1	1	yes	characteristic
49	C₂²×C₃×C₁₆	C₁₂₈	1	1	yes	characteristic
50	C₂×C₃×C₃₂	C₁₂₈	1	3	yes	—
51	C₂×C₃×C₃₂	C₂×C₆₄	1	3	yes	—
52	C₃×C₆₄	C₂×C₆₄	1	3	yes	—
53	C₃×C₆₄	C₂²×C₃₂	1	1	yes	characteristic
54	C₂²×C₃₂	C₃×C₆₄	1	1	yes	characteristic
55	C₂×C₆₄	C₃×C₆₄	1	3	yes	—
56	C₂×C₆₄	C₂×C₃×C₃₂	1	3	yes	—
57	C₁₂₈	C₂×C₃×C₃₂	1	3	yes	—
58	C₁₂₈	C₂²×C₃×C₁₆	1	1	yes	characteristic
59	C₂²×C₃×C₈	C₂₅₆	1	1	yes	characteristic
60	C₂×C₃×C₁₆	C₂₅₆	1	3	yes	—
61	C₂×C₃×C₁₆	C₂×C₁₂₈	1	3	yes	—
62	C₃×C₃₂	C₂×C₁₂₈	1	3	yes	—
63	C₃×C₃₂	C₂²×C₆₄	1	1	yes	characteristic
64	C₂²×C₁₆	C₃×C₁₂₈	1	1	yes	characteristic
65	C₂×C₃₂	C₃×C₁₂₈	1	3	yes	—
66	C₂×C₃₂	C₂×C₃×C₆₄	1	3	yes	—
67	C₆₄	C₂×C₃×C₆₄	1	3	yes	—
68	C₆₄	C₂²×C₃×C₃₂	1	1	yes	characteristic
69	C₂²×C₃×C₄	C₅₁₂	1	1	yes	characteristic
70	C₂×C₃×C₈	C₅₁₂	1	3	yes	—
71	C₂×C₃×C₈	C₂×C₂₅₆	1	3	yes	—
72	C₃×C₁₆	C₂×C₂₅₆	1	3	yes	—
73	C₃×C₁₆	C₂²×C₁₂₈	1	1	yes	characteristic
74	C₂²×C₈	C₃×C₂₅₆	1	1	yes	characteristic
75	C₂×C₁₆	C₃×C₂₅₆	1	3	yes	—
76	C₂×C₁₆	C₂×C₃×C₁₂₈	1	3	yes	—
77	C₃₂	C₂×C₃×C₁₂₈	1	3	yes	—
78	C₃₂	C₂²×C₃×C₆₄	1	1	yes	characteristic
79	C₂³×C₃	C₁₀₂₄	1	1	yes	characteristic
80	C₂×C₃×C₄	C₁₀₂₄	1	3	yes	—
81	C₂×C₃×C₄	C₂×C₅₁₂	1	3	yes	—
82	C₃×C₈	C₂×C₅₁₂	1	3	yes	—
83	C₃×C₈	C₂²×C₂₅₆	1	1	yes	characteristic
84	C₂²×C₄	C₃×C₅₁₂	1	1	yes	characteristic
85	C₂×C₈	C₃×C₅₁₂	1	3	yes	—
86	C₂×C₈	C₂×C₃×C₂₅₆	1	3	yes	—
87	C₁₆	C₂×C₃×C₂₅₆	1	3	yes	—
88	C₁₆	C₂²×C₃×C₁₂₈	1	1	yes	characteristic
89	C₂²×C₃	C₂₀₄₈	1	4	yes	direct product factor
90	C₂²×C₃	C₂×C₁₀₂₄	1	3	yes	—
91	C₃×C₄	C₂×C₁₀₂₄	1	3	yes	—
92	C₃×C₄	C₂²×C₅₁₂	1	1	yes	characteristic
93	C₂³	C₃×C₁₀₂₄	1	1	yes	characteristic
94	C₂×C₄	C₃×C₁₀₂₄	1	3	yes	—
95	C₂×C₄	C₂×C₃×C₅₁₂	1	3	yes	—
96	C₈	C₂×C₃×C₅₁₂	1	3	yes	—
97	C₈	C₂²×C₃×C₂₅₆	1	1	yes	characteristic
98	C₂×C₃	C₂×C₂₀₄₈	1	6	yes	direct product factor
99	C₂×C₃	C₂²×C₁₀₂₄	1	1	yes	characteristic
100	C₂²	C₃×C₂₀₄₈	1	4	yes	direct product factor
101	C₂²	C₂×C₃×C₁₀₂₄	1	3	yes	—
102	C₄	C₂×C₃×C₁₀₂₄	1	3	yes	—
103	C₄	C₂²×C₃×C₅₁₂	1	1	yes	characteristic
104	C₃	C₂²×C₂₀₄₈	1	1	yes	characteristic, direct product factor, Sylow
105	C₂	C₂×C₃×C₂₀₄₈	1	6	yes	direct product factor
106	C₂	C₂²×C₃×C₁₀₂₄	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₀₂₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₁₀₂₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₂₀₄₈	C₂	direct (N × H)
3. C₂²×C₂₀₄₈	C₃	direct (N × H)
7. C₃×C₂₀₄₈	C₂²	direct (N × H)
9. C₂×C₂₀₄₈	C₂×C₃	direct (N × H)
18. C₂₀₄₈	C₂²×C₃	direct (N × H)
89. C₂²×C₃	C₂₀₄₈	direct (N × H)
98. C₂×C₃	C₂×C₂₀₄₈	direct (N × H)
100. C₂²	C₃×C₂₀₄₈	direct (N × H)
104. C₃	C₂²×C₂₀₄₈	direct (N × H)
105. C₂	C₂×C₃×C₂₀₄₈	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×C₃×C₁₀₂₄	2
C₂×C₃×C₂₀₄₈	2
C₂²×C₂₀₄₈	3

Subgroups — C22×C3×C2048 #72201815

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₂₀₄₈ #72201815