Subgroups — C₃×C₈×C₃₂×C₃₇ #72205367

Order 28416.

268 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×C₈×C₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₃₂×C₃₇	C₂	1	2	yes	maximal
3	C₃×C₈²×C₃₇	C₄	1	1	yes	characteristic
4	C₃×C₄×C₁₆×C₃₇	C₄	1	1	yes	characteristic
5	C₃×C₄×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
6	C₂×C₃×C₃₂×C₃₇	C₄	1	4	yes	—
7	C₃×C₄×C₈×C₃₇	C₈	1	2	yes	—
8	C₃×C₄×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₃×C₁₆×C₃₇	C₈	1	2	yes	—
10	C₂×C₃×C₁₆×C₃₇	C₂×C₄	1	2	yes	—
11	C₃×C₃₂×C₃₇	C₈	1	8	yes	direct product factor
12	C₈×C₃₂×C₃₇	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₄×C₃₂×C₃₇	C₂×C₃	1	2	yes	—
14	C₈×C₁₆×C₃₇	C₂×C₃	1	1	yes	characteristic
15	C₂×C₃₂×C₃₇	C₃×C₄	1	4	yes	—
16	C₄×C₁₆×C₃₇	C₃×C₄	1	1	yes	characteristic
17	C₄×C₁₆×C₃₇	C₂²×C₃	1	1	yes	characteristic
18	C₈²×C₃₇	C₃×C₄	1	1	yes	characteristic
19	C₂×C₃×C₈×C₃₇	C₁₆	1	4	yes	—
20	C₂×C₃×C₈×C₃₇	C₄²	1	1	yes	characteristic
21	C₂×C₃×C₈×C₃₇	C₂×C₈	1	1	yes	characteristic
22	C₃×C₄²×C₃₇	C₂×C₈	1	1	yes	characteristic
23	C₃×C₁₆×C₃₇	C₁₆	1	4	yes	—
24	C₃×C₁₆×C₃₇	C₂×C₈	1	4	yes	—
25	C₂×C₁₆×C₃₇	C₃×C₈	1	2	yes	—
26	C₂×C₁₆×C₃₇	C₂×C₃×C₄	1	2	yes	—
27	C₄×C₈×C₃₇	C₃×C₈	1	2	yes	—
28	C₄×C₈×C₃₇	C₂×C₃×C₄	1	1	yes	characteristic
29	C₃₂×C₃₇	C₃×C₈	1	8	yes	direct product factor
30	C₂×C₃×C₄×C₃₇	C₄×C₈	1	1	yes	characteristic
31	C₂×C₃×C₄×C₃₇	C₂×C₁₆	1	2	yes	—
32	C₃×C₈×C₃₇	C₃₂	1	8	yes	direct product factor
33	C₃×C₈×C₃₇	C₄×C₈	1	2	yes	—
34	C₃×C₈×C₃₇	C₂×C₁₆	1	2	yes	—
35	C₃×C₈×C₃₂	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Hall
36	C₂×C₈×C₃₇	C₃×C₁₆	1	4	yes	—
37	C₂×C₈×C₃₇	C₃×C₄²	1	1	yes	characteristic
38	C₂×C₈×C₃₇	C₂×C₃×C₈	1	1	yes	characteristic
39	C₄²×C₃₇	C₂×C₃×C₈	1	1	yes	characteristic
40	C₁₆×C₃₇	C₃×C₁₆	1	4	yes	—
41	C₁₆×C₃₇	C₂×C₃×C₈	1	4	yes	—
42	C₂²×C₃×C₃₇	C₄×C₁₆	1	1	yes	characteristic
43	C₃×C₄×C₃₇	C₈²	1	1	yes	characteristic
44	C₃×C₄×C₃₇	C₄×C₁₆	1	1	yes	characteristic
45	C₃×C₄×C₃₇	C₂×C₃₂	1	4	yes	—
46	C₃×C₄×C₃₂	C₂×C₃₇	1	2	yes	—
47	C₃×C₈×C₁₆	C₂×C₃₇	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₃×C₄×C₈	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂×C₃×C₁₆	1	2	yes	—
50	C₈×C₃₇	C₃×C₃₂	1	8	yes	direct product factor
51	C₈×C₃₇	C₃×C₄×C₈	1	2	yes	—
52	C₈×C₃₇	C₂×C₃×C₁₆	1	2	yes	—
53	C₈×C₃₂	C₃×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×C₃×C₃₇	C₈×C₁₆	1	1	yes	characteristic
55	C₂×C₃×C₃₇	C₄×C₃₂	1	2	yes	—
56	C₂×C₃×C₃₂	C₄×C₃₇	1	4	yes	—
57	C₃×C₄×C₁₆	C₄×C₃₇	1	1	yes	characteristic
58	C₃×C₄×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
59	C₃×C₈²	C₄×C₃₇	1	1	yes	characteristic
60	C₂²×C₃₇	C₃×C₄×C₁₆	1	1	yes	characteristic
61	C₄×C₃₇	C₃×C₈²	1	1	yes	characteristic
62	C₄×C₃₇	C₃×C₄×C₁₆	1	1	yes	characteristic
63	C₄×C₃₇	C₂×C₃×C₃₂	1	4	yes	—
64	C₄×C₃₂	C₂×C₃×C₃₇	1	2	yes	—
65	C₈×C₁₆	C₂×C₃×C₃₇	1	1	yes	characteristic
66	C₃×C₃₇	C₈×C₃₂	1	1	yes	characteristic, direct product factor, Hall
67	C₂×C₃×C₁₆	C₈×C₃₇	1	2	yes	—
68	C₂×C₃×C₁₆	C₂×C₄×C₃₇	1	2	yes	—
69	C₃×C₄×C₈	C₈×C₃₇	1	2	yes	—
70	C₃×C₄×C₈	C₂×C₄×C₃₇	1	1	yes	characteristic
71	C₃×C₃₂	C₈×C₃₇	1	8	yes	direct product factor
72	C₂×C₃₇	C₃×C₈×C₁₆	1	1	yes	characteristic
73	C₂×C₃₇	C₃×C₄×C₃₂	1	2	yes	—
74	C₂×C₃₂	C₃×C₄×C₃₇	1	4	yes	—
75	C₄×C₁₆	C₃×C₄×C₃₇	1	1	yes	characteristic
76	C₄×C₁₆	C₂²×C₃×C₃₇	1	1	yes	characteristic
77	C₈²	C₃×C₄×C₃₇	1	1	yes	characteristic
78	C₂×C₃×C₈	C₁₆×C₃₇	1	4	yes	—
79	C₂×C₃×C₈	C₄²×C₃₇	1	1	yes	characteristic
80	C₂×C₃×C₈	C₂×C₈×C₃₇	1	1	yes	characteristic
81	C₃×C₄²	C₂×C₈×C₃₇	1	1	yes	characteristic
82	C₃×C₁₆	C₁₆×C₃₇	1	4	yes	—
83	C₃×C₁₆	C₂×C₈×C₃₇	1	4	yes	—
84	C₃₇	C₃×C₈×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
85	C₂×C₁₆	C₃×C₈×C₃₇	1	2	yes	—
86	C₂×C₁₆	C₂×C₃×C₄×C₃₇	1	2	yes	—
87	C₄×C₈	C₃×C₈×C₃₇	1	2	yes	—
88	C₄×C₈	C₂×C₃×C₄×C₃₇	1	1	yes	characteristic
89	C₃₂	C₃×C₈×C₃₇	1	8	yes	direct product factor
90	C₂×C₃×C₄	C₄×C₈×C₃₇	1	1	yes	characteristic
91	C₂×C₃×C₄	C₂×C₁₆×C₃₇	1	2	yes	—
92	C₃×C₈	C₃₂×C₃₇	1	8	yes	direct product factor
93	C₃×C₈	C₄×C₈×C₃₇	1	2	yes	—
94	C₃×C₈	C₂×C₁₆×C₃₇	1	2	yes	—
95	C₂×C₈	C₃×C₁₆×C₃₇	1	4	yes	—
96	C₂×C₈	C₃×C₄²×C₃₇	1	1	yes	characteristic
97	C₂×C₈	C₂×C₃×C₈×C₃₇	1	1	yes	characteristic
98	C₄²	C₂×C₃×C₈×C₃₇	1	1	yes	characteristic
99	C₁₆	C₃×C₁₆×C₃₇	1	4	yes	—
100	C₁₆	C₂×C₃×C₈×C₃₇	1	4	yes	—
101	C₂²×C₃	C₄×C₁₆×C₃₇	1	1	yes	characteristic
102	C₃×C₄	C₈²×C₃₇	1	1	yes	characteristic
103	C₃×C₄	C₄×C₁₆×C₃₇	1	1	yes	characteristic
104	C₃×C₄	C₂×C₃₂×C₃₇	1	4	yes	—
105	C₂×C₄	C₂×C₃×C₁₆×C₃₇	1	2	yes	—
106	C₂×C₄	C₃×C₄×C₈×C₃₇	1	1	yes	characteristic
107	C₈	C₃×C₃₂×C₃₇	1	8	yes	direct product factor
108	C₈	C₂×C₃×C₁₆×C₃₇	1	2	yes	—
109	C₈	C₃×C₄×C₈×C₃₇	1	2	yes	—
110	C₂×C₃	C₈×C₁₆×C₃₇	1	1	yes	characteristic
111	C₂×C₃	C₄×C₃₂×C₃₇	1	2	yes	—
112	C₂²	C₃×C₄×C₁₆×C₃₇	1	1	yes	characteristic
113	C₄	C₂×C₃×C₃₂×C₃₇	1	4	yes	—
114	C₄	C₃×C₄×C₁₆×C₃₇	1	1	yes	characteristic
115	C₄	C₃×C₈²×C₃₇	1	1	yes	characteristic
116	C₃	C₈×C₃₂×C₃₇	1	1	yes	characteristic, direct product factor, Sylow
117	C₂	C₃×C₄×C₃₂×C₃₇	1	2	yes	—
118	C₂	C₃×C₈×C₁₆×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₁₆	C₂²×C₃×C₃₇
socle, prime radical	C₂²×C₃×C₃₇	C₄×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₃×C₃₂×C₃₇	C₈	direct (N × H)
12. C₈×C₃₂×C₃₇	C₃	direct (N × H)
29. C₃₂×C₃₇	C₃×C₈	direct (N × H)
32. C₃×C₈×C₃₇	C₃₂	direct (N × H)
35. C₃×C₈×C₃₂	C₃₇	direct (N × H)
50. C₈×C₃₇	C₃×C₃₂	direct (N × H)
53. C₈×C₃₂	C₃×C₃₇	direct (N × H)
66. C₃×C₃₇	C₈×C₃₂	direct (N × H)
71. C₃×C₃₂	C₈×C₃₇	direct (N × H)
84. C₃₇	C₃×C₈×C₃₂	direct (N × H)
89. C₃₂	C₃×C₈×C₃₇	direct (N × H)
92. C₃×C₈	C₃₂×C₃₇	direct (N × H)
107. C₈	C₃×C₃₂×C₃₇	direct (N × H)
116. C₃	C₈×C₃₂×C₃₇	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×C₈×C₁₆×C₃₇	2
C₃×C₄×C₃₂×C₃₇	2
C₈×C₃₂×C₃₇	3
C₃×C₈×C₃₂	37

Subgroups — C3×C8×C32×C37 #72205367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×C₃₂×C₃₇ #72205367