Subgroups — C₃×C₄×C₃₇×C₆₄ #72205366

Order 28416.

172 conjugacy classes of subgroups.

The subgroup lattice has 120 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃₇×C₆₄	C₂	1	2	yes	maximal
3	C₃×C₄×C₁₆×C₃₇	C₄	1	1	yes	characteristic
4	C₂×C₃×C₃₂×C₃₇	C₄	1	1	yes	characteristic
5	C₂×C₃×C₃₂×C₃₇	C₂²	1	1	yes	characteristic
6	C₃×C₃₇×C₆₄	C₄	1	4	yes	direct product factor
7	C₃×C₄×C₈×C₃₇	C₈	1	1	yes	characteristic
8	C₂×C₃×C₁₆×C₃₇	C₈	1	1	yes	characteristic
9	C₂×C₃×C₁₆×C₃₇	C₂×C₄	1	1	yes	characteristic
10	C₃×C₃₂×C₃₇	C₈	1	2	yes	—
11	C₃×C₃₂×C₃₇	C₂×C₄	1	2	yes	—
12	C₄×C₃₇×C₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃₇×C₆₄	C₂×C₃	1	2	yes	—
14	C₄×C₃₂×C₃₇	C₂×C₃	1	1	yes	characteristic
15	C₂×C₃₂×C₃₇	C₃×C₄	1	1	yes	characteristic
16	C₂×C₃₂×C₃₇	C₂²×C₃	1	1	yes	characteristic
17	C₄×C₁₆×C₃₇	C₃×C₄	1	1	yes	characteristic
18	C₃₇×C₆₄	C₃×C₄	1	4	yes	direct product factor
19	C₂×C₃×C₈×C₃₇	C₁₆	1	1	yes	characteristic
20	C₂×C₃×C₈×C₃₇	C₂×C₈	1	1	yes	characteristic
21	C₃×C₄²×C₃₇	C₁₆	1	1	yes	characteristic
22	C₃×C₁₆×C₃₇	C₁₆	1	2	yes	—
23	C₃×C₁₆×C₃₇	C₄²	1	1	yes	characteristic
24	C₃×C₁₆×C₃₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₁₆×C₃₇	C₃×C₈	1	1	yes	characteristic
26	C₂×C₁₆×C₃₇	C₂×C₃×C₄	1	1	yes	characteristic
27	C₄×C₈×C₃₇	C₃×C₈	1	1	yes	characteristic
28	C₃₂×C₃₇	C₃×C₈	1	2	yes	—
29	C₃₂×C₃₇	C₂×C₃×C₄	1	2	yes	—
30	C₂×C₃×C₄×C₃₇	C₃₂	1	2	yes	—
31	C₂×C₃×C₄×C₃₇	C₂×C₁₆	1	1	yes	characteristic
32	C₃×C₈×C₃₇	C₃₂	1	2	yes	—
33	C₃×C₈×C₃₇	C₄×C₈	1	1	yes	characteristic
34	C₃×C₈×C₃₇	C₂×C₁₆	1	1	yes	characteristic
35	C₃×C₄×C₆₄	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₂×C₈×C₃₇	C₃×C₁₆	1	1	yes	characteristic
37	C₂×C₈×C₃₇	C₂×C₃×C₈	1	1	yes	characteristic
38	C₄²×C₃₇	C₃×C₁₆	1	1	yes	characteristic
39	C₁₆×C₃₇	C₃×C₁₆	1	2	yes	—
40	C₁₆×C₃₇	C₃×C₄²	1	1	yes	characteristic
41	C₁₆×C₃₇	C₂×C₃×C₈	1	1	yes	characteristic
42	C₂²×C₃×C₃₇	C₂×C₃₂	1	1	yes	characteristic
43	C₃×C₄×C₃₇	C₆₄	1	4	yes	direct product factor
44	C₃×C₄×C₃₇	C₄×C₁₆	1	1	yes	characteristic
45	C₃×C₄×C₃₇	C₂×C₃₂	1	1	yes	characteristic
46	C₂×C₃×C₆₄	C₂×C₃₇	1	2	yes	—
47	C₃×C₄×C₃₂	C₂×C₃₇	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₃×C₃₂	1	2	yes	—
49	C₂×C₄×C₃₇	C₂×C₃×C₁₆	1	1	yes	characteristic
50	C₈×C₃₇	C₃×C₃₂	1	2	yes	—
51	C₈×C₃₇	C₃×C₄×C₈	1	1	yes	characteristic
52	C₈×C₃₇	C₂×C₃×C₁₆	1	1	yes	characteristic
53	C₄×C₆₄	C₃×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×C₃×C₃₇	C₄×C₃₂	1	1	yes	characteristic
55	C₂×C₃×C₃₇	C₂×C₆₄	1	2	yes	—
56	C₂×C₃×C₃₂	C₄×C₃₇	1	1	yes	characteristic
57	C₂×C₃×C₃₂	C₂²×C₃₇	1	1	yes	characteristic
58	C₃×C₄×C₁₆	C₄×C₃₇	1	1	yes	characteristic
59	C₃×C₆₄	C₄×C₃₇	1	4	yes	direct product factor
60	C₂²×C₃₇	C₂×C₃×C₃₂	1	1	yes	characteristic
61	C₄×C₃₇	C₃×C₆₄	1	4	yes	direct product factor
62	C₄×C₃₇	C₃×C₄×C₁₆	1	1	yes	characteristic
63	C₄×C₃₇	C₂×C₃×C₃₂	1	1	yes	characteristic
64	C₂×C₆₄	C₂×C₃×C₃₇	1	2	yes	—
65	C₄×C₃₂	C₂×C₃×C₃₇	1	1	yes	characteristic
66	C₃×C₃₇	C₄×C₆₄	1	1	yes	characteristic, direct product factor, Hall
67	C₂×C₃×C₁₆	C₈×C₃₇	1	1	yes	characteristic
68	C₂×C₃×C₁₆	C₂×C₄×C₃₇	1	1	yes	characteristic
69	C₃×C₄×C₈	C₈×C₃₇	1	1	yes	characteristic
70	C₃×C₃₂	C₈×C₃₇	1	2	yes	—
71	C₃×C₃₂	C₂×C₄×C₃₇	1	2	yes	—
72	C₂×C₃₇	C₃×C₄×C₃₂	1	1	yes	characteristic
73	C₂×C₃₇	C₂×C₃×C₆₄	1	2	yes	—
74	C₂×C₃₂	C₃×C₄×C₃₇	1	1	yes	characteristic
75	C₂×C₃₂	C₂²×C₃×C₃₇	1	1	yes	characteristic
76	C₄×C₁₆	C₃×C₄×C₃₇	1	1	yes	characteristic
77	C₆₄	C₃×C₄×C₃₇	1	4	yes	direct product factor
78	C₂×C₃×C₈	C₁₆×C₃₇	1	1	yes	characteristic
79	C₂×C₃×C₈	C₂×C₈×C₃₇	1	1	yes	characteristic
80	C₃×C₄²	C₁₆×C₃₇	1	1	yes	characteristic
81	C₃×C₁₆	C₁₆×C₃₇	1	2	yes	—
82	C₃×C₁₆	C₄²×C₃₇	1	1	yes	characteristic
83	C₃×C₁₆	C₂×C₈×C₃₇	1	1	yes	characteristic
84	C₃₇	C₃×C₄×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×C₁₆	C₃×C₈×C₃₇	1	1	yes	characteristic
86	C₂×C₁₆	C₂×C₃×C₄×C₃₇	1	1	yes	characteristic
87	C₄×C₈	C₃×C₈×C₃₇	1	1	yes	characteristic
88	C₃₂	C₃×C₈×C₃₇	1	2	yes	—
89	C₃₂	C₂×C₃×C₄×C₃₇	1	2	yes	—
90	C₂×C₃×C₄	C₃₂×C₃₇	1	2	yes	—
91	C₂×C₃×C₄	C₂×C₁₆×C₃₇	1	1	yes	characteristic
92	C₃×C₈	C₃₂×C₃₇	1	2	yes	—
93	C₃×C₈	C₄×C₈×C₃₇	1	1	yes	characteristic
94	C₃×C₈	C₂×C₁₆×C₃₇	1	1	yes	characteristic
95	C₂×C₈	C₃×C₁₆×C₃₇	1	1	yes	characteristic
96	C₂×C₈	C₂×C₃×C₈×C₃₇	1	1	yes	characteristic
97	C₄²	C₃×C₁₆×C₃₇	1	1	yes	characteristic
98	C₁₆	C₃×C₁₆×C₃₇	1	2	yes	—
99	C₁₆	C₃×C₄²×C₃₇	1	1	yes	characteristic
100	C₁₆	C₂×C₃×C₈×C₃₇	1	1	yes	characteristic
101	C₂²×C₃	C₂×C₃₂×C₃₇	1	1	yes	characteristic
102	C₃×C₄	C₃₇×C₆₄	1	4	yes	direct product factor
103	C₃×C₄	C₄×C₁₆×C₃₇	1	1	yes	characteristic
104	C₃×C₄	C₂×C₃₂×C₃₇	1	1	yes	characteristic
105	C₂×C₄	C₃×C₃₂×C₃₇	1	2	yes	—
106	C₂×C₄	C₂×C₃×C₁₆×C₃₇	1	1	yes	characteristic
107	C₈	C₃×C₃₂×C₃₇	1	2	yes	—
108	C₈	C₂×C₃×C₁₆×C₃₇	1	1	yes	characteristic
109	C₈	C₃×C₄×C₈×C₃₇	1	1	yes	characteristic
110	C₂×C₃	C₄×C₃₂×C₃₇	1	1	yes	characteristic
111	C₂×C₃	C₂×C₃₇×C₆₄	1	2	yes	—
112	C₂²	C₂×C₃×C₃₂×C₃₇	1	1	yes	characteristic
113	C₄	C₃×C₃₇×C₆₄	1	4	yes	direct product factor
114	C₄	C₂×C₃×C₃₂×C₃₇	1	1	yes	characteristic
115	C₄	C₃×C₄×C₁₆×C₃₇	1	1	yes	characteristic
116	C₃	C₄×C₃₇×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
117	C₂	C₂×C₃×C₃₇×C₆₄	1	2	yes	—
118	C₂	C₃×C₄×C₃₂×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₃₂	C₂²×C₃×C₃₇
socle, prime radical	C₂²×C₃×C₃₇	C₂×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃×C₃₇×C₆₄	C₄	direct (N × H)
12. C₄×C₃₇×C₆₄	C₃	direct (N × H)
18. C₃₇×C₆₄	C₃×C₄	direct (N × H)
35. C₃×C₄×C₆₄	C₃₇	direct (N × H)
43. C₃×C₄×C₃₇	C₆₄	direct (N × H)
53. C₄×C₆₄	C₃×C₃₇	direct (N × H)
59. C₃×C₆₄	C₄×C₃₇	direct (N × H)
61. C₄×C₃₇	C₃×C₆₄	direct (N × H)
66. C₃×C₃₇	C₄×C₆₄	direct (N × H)
77. C₆₄	C₃×C₄×C₃₇	direct (N × H)
84. C₃₇	C₃×C₄×C₆₄	direct (N × H)
102. C₃×C₄	C₃₇×C₆₄	direct (N × H)
113. C₄	C₃×C₃₇×C₆₄	direct (N × H)
116. C₃	C₄×C₃₇×C₆₄	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×C₄×C₃₂×C₃₇	2
C₂×C₃×C₃₇×C₆₄	2
C₄×C₃₇×C₆₄	3
C₃×C₄×C₆₄	37

Subgroups — C3×C4×C37×C64 #72205366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₃₇×C₆₄ #72205366