Subgroups — C₄²×C₃₂×C₄₁ #72198569

Order 20992.

708 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	C₂×C₄×C₃₂×C₄₁	C₂	1	6	yes	maximal
2	C₄²×C₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₂×C₄₁	C₂²	1	4	yes	—
4	C₂×C₄×C₁₆×C₄₁	C₄	1	3	yes	—
5	C₂×C₄×C₁₆×C₄₁	C₂²	1	3	yes	—
6	C₄²×C₈×C₄₁	C₄	1	1	yes	characteristic
7	C₄×C₃₂×C₄₁	C₄	1	24	yes	direct product factor
8	C₂²×C₁₆×C₄₁	C₂×C₄	1	3	yes	—
9	C₂²×C₁₆×C₄₁	C₂³	1	1	yes	characteristic
10	C₂×C₄×C₈×C₄₁	C₈	1	3	yes	—
11	C₂×C₄×C₈×C₄₁	C₂×C₄	1	3	yes	—
12	C₄³×C₄₁	C₈	1	1	yes	characteristic
13	C₂×C₃₂×C₄₁	C₂×C₄	1	24	yes	—
14	C₄×C₁₆×C₄₁	C₈	1	12	yes	—
15	C₄×C₁₆×C₄₁	C₂×C₄	1	12	yes	—
16	C₂²×C₈×C₄₁	C₂×C₈	1	3	yes	—
17	C₂²×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₄²×C₄₁	C₁₆	1	4	yes	—
19	C₂×C₄²×C₄₁	C₂×C₈	1	3	yes	—
20	C₂×C₁₆×C₄₁	C₄²	1	6	yes	—
21	C₂×C₁₆×C₄₁	C₂×C₈	1	12	yes	—
22	C₂×C₁₆×C₄₁	C₂²×C₄	1	6	yes	—
23	C₄×C₈×C₄₁	C₁₆	1	12	yes	—
24	C₄×C₈×C₄₁	C₄²	1	6	yes	—
25	C₄×C₈×C₄₁	C₂×C₈	1	6	yes	—
26	C₃₂×C₄₁	C₄²	1	16	yes	direct product factor
27	C₂²×C₄×C₄₁	C₂×C₁₆	1	6	yes	—
28	C₂²×C₄×C₄₁	C₂²×C₈	1	1	yes	characteristic
29	C₂×C₈×C₄₁	C₄×C₈	1	6	yes	—
30	C₂×C₈×C₄₁	C₂×C₁₆	1	12	yes	—
31	C₂×C₈×C₄₁	C₂×C₄²	1	3	yes	—
32	C₂×C₈×C₄₁	C₂²×C₈	1	3	yes	—
33	C₄²×C₄₁	C₃₂	1	16	yes	direct product factor
34	C₄²×C₄₁	C₄×C₈	1	6	yes	—
35	C₄²×C₄₁	C₂×C₁₆	1	6	yes	—
36	C₁₆×C₄₁	C₄×C₈	1	12	yes	—
37	C₁₆×C₄₁	C₂×C₄²	1	4	yes	—
38	C₄²×C₃₂	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂³×C₄₁	C₂²×C₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₁	C₄×C₁₆	1	12	yes	—
41	C₂×C₄×C₄₁	C₂×C₃₂	1	24	yes	—
42	C₂×C₄×C₄₁	C₂×C₄×C₈	1	3	yes	—
43	C₂×C₄×C₄₁	C₂²×C₁₆	1	3	yes	—
44	C₈×C₄₁	C₄×C₁₆	1	12	yes	—
45	C₈×C₄₁	C₄³	1	1	yes	characteristic
46	C₈×C₄₁	C₂×C₄×C₈	1	3	yes	—
47	C₂×C₄×C₃₂	C₂×C₄₁	1	6	yes	—
48	C₄²×C₁₆	C₂×C₄₁	1	1	yes	characteristic
49	C₂²×C₄₁	C₂×C₄×C₁₆	1	3	yes	—
50	C₂²×C₄₁	C₂²×C₃₂	1	4	yes	—
51	C₄×C₄₁	C₄×C₃₂	1	24	yes	direct product factor
52	C₄×C₄₁	C₄²×C₈	1	1	yes	characteristic
53	C₄×C₄₁	C₂×C₄×C₁₆	1	3	yes	—
54	C₂²×C₃₂	C₂²×C₄₁	1	4	yes	—
55	C₂×C₄×C₁₆	C₄×C₄₁	1	3	yes	—
56	C₂×C₄×C₁₆	C₂²×C₄₁	1	3	yes	—
57	C₄²×C₈	C₄×C₄₁	1	1	yes	characteristic
58	C₄×C₃₂	C₄×C₄₁	1	24	yes	direct product factor
59	C₂×C₄₁	C₄²×C₁₆	1	1	yes	characteristic
60	C₂×C₄₁	C₂×C₄×C₃₂	1	6	yes	—
61	C₂²×C₁₆	C₂×C₄×C₄₁	1	3	yes	—
62	C₂²×C₁₆	C₂³×C₄₁	1	1	yes	characteristic
63	C₂×C₄×C₈	C₈×C₄₁	1	3	yes	—
64	C₂×C₄×C₈	C₂×C₄×C₄₁	1	3	yes	—
65	C₄³	C₈×C₄₁	1	1	yes	characteristic
66	C₂×C₃₂	C₂×C₄×C₄₁	1	24	yes	—
67	C₄×C₁₆	C₈×C₄₁	1	12	yes	—
68	C₄×C₁₆	C₂×C₄×C₄₁	1	12	yes	—
69	C₄₁	C₄²×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×C₈	C₂×C₈×C₄₁	1	3	yes	—
71	C₂²×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
72	C₂×C₄²	C₁₆×C₄₁	1	4	yes	—
73	C₂×C₄²	C₂×C₈×C₄₁	1	3	yes	—
74	C₂×C₁₆	C₄²×C₄₁	1	6	yes	—
75	C₂×C₁₆	C₂×C₈×C₄₁	1	12	yes	—
76	C₂×C₁₆	C₂²×C₄×C₄₁	1	6	yes	—
77	C₄×C₈	C₁₆×C₄₁	1	12	yes	—
78	C₄×C₈	C₄²×C₄₁	1	6	yes	—
79	C₄×C₈	C₂×C₈×C₄₁	1	6	yes	—
80	C₃₂	C₄²×C₄₁	1	16	yes	direct product factor
81	C₂²×C₄	C₂×C₁₆×C₄₁	1	6	yes	—
82	C₂²×C₄	C₂²×C₈×C₄₁	1	1	yes	characteristic
83	C₂×C₈	C₄×C₈×C₄₁	1	6	yes	—
84	C₂×C₈	C₂×C₁₆×C₄₁	1	12	yes	—
85	C₂×C₈	C₂×C₄²×C₄₁	1	3	yes	—
86	C₂×C₈	C₂²×C₈×C₄₁	1	3	yes	—
87	C₄²	C₃₂×C₄₁	1	16	yes	direct product factor
88	C₄²	C₄×C₈×C₄₁	1	6	yes	—
89	C₄²	C₂×C₁₆×C₄₁	1	6	yes	—
90	C₁₆	C₄×C₈×C₄₁	1	12	yes	—
91	C₁₆	C₂×C₄²×C₄₁	1	4	yes	—
92	C₂³	C₂²×C₁₆×C₄₁	1	1	yes	characteristic
93	C₂×C₄	C₄×C₁₆×C₄₁	1	12	yes	—
94	C₂×C₄	C₂×C₃₂×C₄₁	1	24	yes	—
95	C₂×C₄	C₂×C₄×C₈×C₄₁	1	3	yes	—
96	C₂×C₄	C₂²×C₁₆×C₄₁	1	3	yes	—
97	C₈	C₄×C₁₆×C₄₁	1	12	yes	—
98	C₈	C₄³×C₄₁	1	1	yes	characteristic
99	C₈	C₂×C₄×C₈×C₄₁	1	3	yes	—
100	C₂²	C₂×C₄×C₁₆×C₄₁	1	3	yes	—
101	C₂²	C₂²×C₃₂×C₄₁	1	4	yes	—
102	C₄	C₄×C₃₂×C₄₁	1	24	yes	direct product factor
103	C₄	C₄²×C₈×C₄₁	1	1	yes	characteristic
104	C₄	C₂×C₄×C₁₆×C₄₁	1	3	yes	—
105	C₂	C₄²×C₁₆×C₄₁	1	1	yes	characteristic
106	C₂	C₂×C₄×C₃₂×C₄₁	1	6	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₁₆	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂²×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₄×C₃₂×C₄₁	C₄	direct (N × H)
26. C₃₂×C₄₁	C₄²	direct (N × H)
33. C₄²×C₄₁	C₃₂	direct (N × H)
38. C₄²×C₃₂	C₄₁	direct (N × H)
51. C₄×C₄₁	C₄×C₃₂	direct (N × H)
58. C₄×C₃₂	C₄×C₄₁	direct (N × H)
69. C₄₁	C₄²×C₃₂	direct (N × H)
80. C₃₂	C₄²×C₄₁	direct (N × H)
87. C₄²	C₃₂×C₄₁	direct (N × H)
102. C₄	C₄×C₃₂×C₄₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₃₂×C₄₁	2
C₄²×C₁₆×C₄₁	2
C₄²×C₃₂	41

Subgroups — C42×C32×C41 #72198569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₃₂×C₄₁ #72198569