Subgroups — C₂³×M₃₂×C₃₇ #22683874

Order 9472.

2334 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	C₂⁴×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₃₇	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₆×C₃₇	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₄×C₃₇	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₃₇	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₃₇	C₄	1	14	yes	—
7	C₂³×C₈×C₃₇	C₂²	1	14	yes	—
8	C₂³×C₈×C₃₇	C₂²	1	1	yes	characteristic
9	C₂×M₃₂×C₃₇	C₂²	1	112	yes	direct product factor
10	C₂²×C₁₆×C₃₇	C₂²	1	28	yes	complemented
11	C₂⁵×C₃₇	C₈	1	1	yes	characteristic
12	C₂³×C₄×C₃₇	C₈	1	14	yes	—
13	C₂³×C₄×C₃₇	C₈	1	1	yes	characteristic
14	C₂³×C₄×C₃₇	C₂×C₄	1	14	yes	—
15	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₇	C₂×C₄	1	14	yes	—
17	C₂²×C₈×C₃₇	C₂×C₄	1	84	yes	—
18	C₂²×C₈×C₃₇	C₂×C₄	1	7	yes	—
19	C₂²×C₈×C₃₇	C₂³	1	28	yes	—
20	C₂²×C₈×C₃₇	C₂³	1	7	yes	—
21	M₃₂×C₃₇	C₂³	1	64	yes	direct product factor
22	C₂×C₁₆×C₃₇	C₂³	1	56	yes	complemented
23	C₂⁴×C₃₇	—	2	8	no	—
24	C₂⁴×C₃₇	C₂×C₈	1	14	yes	—
25	C₂⁴×C₃₇	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₃₇	C₂×C₈	1	84	yes	—
27	C₂²×C₄×C₃₇	C₂×C₈	1	14	yes	—
28	C₂²×C₄×C₃₇	C₂×C₈	1	7	yes	—
29	C₂²×C₄×C₃₇	C₂²×C₄	1	28	yes	—
30	C₂²×C₄×C₃₇	C₂²×C₄	1	7	yes	—
31	C₂×C₈×C₃₇	C₂²×C₄	1	56	yes	—
32	C₂×C₈×C₃₇	C₂²×C₄	1	28	yes	—
33	C₂×C₈×C₃₇	C₂²×C₄	1	21	yes	—
34	C₂×C₈×C₃₇	C₂⁴	1	7	yes	—
35	C₂×C₈×C₃₇	C₂⁴	1	8	yes	—
36	C₁₆×C₃₇	C₂⁴	1	16	yes	complemented
37	C₂³×C₃₇	—	2	56	no	—
38	C₂³×C₃₇	M₃₂	1	8	yes	direct product factor
39	C₂³×C₃₇	C₂²×C₈	1	28	yes	—
40	C₂³×C₃₇	C₂²×C₈	1	7	yes	—
41	C₂×C₄×C₃₇	C₂²×C₈	1	56	yes	—
42	C₂×C₄×C₃₇	C₂²×C₈	1	21	yes	—
43	C₂×C₄×C₃₇	C₂²×C₈	1	28	yes	—
44	C₂×C₄×C₃₇	C₂³×C₄	1	8	yes	—
45	C₂×C₄×C₃₇	C₂³×C₄	1	7	yes	—
46	C₈×C₃₇	C₂³×C₄	1	7	yes	—
47	C₈×C₃₇	C₂³×C₄	1	8	yes	—
48	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
49	C₂³×M₃₂	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₃₇	—	2	56	no	—
51	C₂²×C₃₇	C₂×M₃₂	1	28	yes	direct product factor
52	C₂²×C₃₇	C₂³×C₈	1	8	yes	—
53	C₂²×C₃₇	C₂³×C₈	1	7	yes	—
54	C₄×C₃₇	C₂³×C₈	1	7	yes	—
55	C₄×C₃₇	C₂³×C₈	1	8	yes	—
56	C₄×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
57	C₂⁴×C₈	C₂×C₃₇	1	1	yes	characteristic
58	C₂²×M₃₂	C₂×C₃₇	1	28	yes	direct product factor
59	C₂³×C₁₆	C₂×C₃₇	1	2	yes	complemented
60	C₂×C₃₇	—	2	8	no	—
61	C₂×C₃₇	C₂²×M₃₂	1	14	yes	direct product factor
62	C₂×C₃₇	C₂⁴×C₈	1	1	yes	characteristic
63	C₂⁴×C₄	C₄×C₃₇	1	1	yes	characteristic
64	C₂³×C₈	C₄×C₃₇	1	1	yes	characteristic
65	C₂³×C₈	C₄×C₃₇	1	14	yes	—
66	C₂³×C₈	C₂²×C₃₇	1	14	yes	—
67	C₂³×C₈	C₂²×C₃₇	1	1	yes	characteristic
68	C₂×M₃₂	C₂²×C₃₇	1	112	yes	direct product factor
69	C₂²×C₁₆	C₂²×C₃₇	1	28	yes	complemented
70	C₃₇	C₂³×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
71	C₂⁵	C₈×C₃₇	1	1	yes	characteristic
72	C₂³×C₄	C₈×C₃₇	1	14	yes	—
73	C₂³×C₄	C₈×C₃₇	1	1	yes	characteristic
74	C₂³×C₄	C₂×C₄×C₃₇	1	14	yes	—
75	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
76	C₂²×C₈	C₂×C₄×C₃₇	1	84	yes	—
77	C₂²×C₈	C₂×C₄×C₃₇	1	14	yes	—
78	C₂²×C₈	C₂×C₄×C₃₇	1	7	yes	—
79	C₂²×C₈	C₂³×C₃₇	1	7	yes	—
80	C₂²×C₈	C₂³×C₃₇	1	28	yes	—
81	M₃₂	C₂³×C₃₇	1	64	yes	direct product factor
82	C₂×C₁₆	C₂³×C₃₇	1	56	yes	complemented
83	C₂⁴	—	2	8	no	—
84	C₂⁴	C₂×C₈×C₃₇	1	14	yes	—
85	C₂⁴	C₂×C₈×C₃₇	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₈×C₃₇	1	84	yes	—
87	C₂²×C₄	C₂×C₈×C₃₇	1	14	yes	—
88	C₂²×C₄	C₂×C₈×C₃₇	1	7	yes	—
89	C₂²×C₄	C₂²×C₄×C₃₇	1	28	yes	—
90	C₂²×C₄	C₂²×C₄×C₃₇	1	7	yes	—
91	C₂×C₈	C₂²×C₄×C₃₇	1	21	yes	—
92	C₂×C₈	C₂²×C₄×C₃₇	1	56	yes	—
93	C₂×C₈	C₂²×C₄×C₃₇	1	28	yes	—
94	C₂×C₈	C₂⁴×C₃₇	1	7	yes	—
95	C₂×C₈	C₂⁴×C₃₇	1	8	yes	—
96	C₁₆	C₂⁴×C₃₇	1	16	yes	complemented
97	C₂³	—	2	56	no	—
98	C₂³	M₃₂×C₃₇	1	8	yes	direct product factor
99	C₂³	C₂²×C₈×C₃₇	1	28	yes	—
100	C₂³	C₂²×C₈×C₃₇	1	7	yes	—
101	C₂×C₄	C₂²×C₈×C₃₇	1	56	yes	—
102	C₂×C₄	C₂²×C₈×C₃₇	1	21	yes	—
103	C₂×C₄	C₂²×C₈×C₃₇	1	28	yes	—
104	C₂×C₄	C₂³×C₄×C₃₇	1	8	yes	—
105	C₂×C₄	C₂³×C₄×C₃₇	1	7	yes	—
106	C₈	C₂³×C₄×C₃₇	1	7	yes	—
107	C₈	C₂³×C₄×C₃₇	1	8	yes	—
108	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
109	C₂²	—	2	56	no	—
110	C₂²	C₂×M₃₂×C₃₇	1	28	yes	direct product factor
111	C₂²	C₂³×C₈×C₃₇	1	8	yes	—
112	C₂²	C₂³×C₈×C₃₇	1	7	yes	—
113	C₄	C₂³×C₈×C₃₇	1	7	yes	—
114	C₄	C₂³×C₈×C₃₇	1	8	yes	—
115	C₄	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
116	C₂	—	2	8	no	—
117	C₂	C₂²×M₃₂×C₃₇	1	14	yes	direct product factor
118	C₂	C₂⁴×C₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈×C₃₇	C₂²
commutator	C₂	C₂⁴×C₈×C₃₇
Frattini, cosocle	C₈	C₂⁵×C₃₇
socle	C₂⁴×C₃₇	C₂×C₈
prime radical	C₂⁵×C₃₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₃₇	C₂	direct (N × H)
3. C₂³×C₁₆×C₃₇	C₂	semidirect
9. C₂×M₃₂×C₃₇	C₂²	direct (N × H)
10. C₂²×C₁₆×C₃₇	C₂²	semidirect
21. M₃₂×C₃₇	C₂³	direct (N × H)
22. C₂×C₁₆×C₃₇	C₂³	semidirect
36. C₁₆×C₃₇	C₂⁴	semidirect
38. C₂³×C₃₇	M₃₂	direct (N × H)
49. C₂³×M₃₂	C₃₇	direct (N × H)
51. C₂²×C₃₇	C₂×M₃₂	direct (N × H)
58. C₂²×M₃₂	C₂×C₃₇	direct (N × H)
59. C₂³×C₁₆	C₂×C₃₇	semidirect
61. C₂×C₃₇	C₂²×M₃₂	direct (N × H)
68. C₂×M₃₂	C₂²×C₃₇	direct (N × H)
69. C₂²×C₁₆	C₂²×C₃₇	semidirect
70. C₃₇	C₂³×M₃₂	direct (N × H)
81. M₃₂	C₂³×C₃₇	direct (N × H)
82. C₂×C₁₆	C₂³×C₃₇	semidirect
96. C₁₆	C₂⁴×C₃₇	semidirect
98. C₂³	M₃₂×C₃₇	direct (N × H)
110. C₂²	C₂×M₃₂×C₃₇	direct (N × H)
117. C₂	C₂²×M₃₂×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
C₂⁴×C₈×C₃₇	2
C₂²×M₃₂×C₃₇	2
C₂³×C₁₆×C₃₇	2
C₂³×M₃₂	37

Subgroups — C23×M32×C37 #22683874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×M₃₂×C₃₇ #22683874