Subgroups — C₂³×C₅×M₃₂ #11985341

Order 1280.

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×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₅×C₁₆	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
6	C₂³×C₅×C₈	C₄	1	14	yes	—
7	C₂³×C₅×C₈	C₂²	1	14	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×M₃₂	C₂²	1	112	yes	direct product factor
10	C₂²×C₅×C₁₆	C₂²	1	28	yes	complemented
11	C₂³×M₃₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂⁵×C₅	C₈	1	1	yes	characteristic
13	C₂³×C₄×C₅	C₈	1	14	yes	—
14	C₂³×C₄×C₅	C₈	1	1	yes	characteristic
15	C₂³×C₄×C₅	C₂×C₄	1	14	yes	—
16	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×C₈	C₂×C₄	1	14	yes	—
18	C₂²×C₅×C₈	C₂×C₄	1	84	yes	—
19	C₂²×C₅×C₈	C₂×C₄	1	7	yes	—
20	C₂²×C₅×C₈	C₂³	1	7	yes	—
21	C₂²×C₅×C₈	C₂³	1	28	yes	—
22	C₅×M₃₂	C₂³	1	64	yes	direct product factor
23	C₂×C₅×C₁₆	C₂³	1	56	yes	complemented
24	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic
25	C₂²×M₃₂	C₂×C₅	1	28	yes	direct product factor
26	C₂³×C₁₆	C₂×C₅	1	2	yes	complemented
27	C₂⁴×C₅	—	2	8	no	—
28	C₂⁴×C₅	C₂×C₈	1	14	yes	—
29	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅	C₂×C₈	1	84	yes	—
31	C₂²×C₄×C₅	C₂×C₈	1	14	yes	—
32	C₂²×C₄×C₅	C₂×C₈	1	7	yes	—
33	C₂²×C₄×C₅	C₂²×C₄	1	28	yes	—
34	C₂²×C₄×C₅	C₂²×C₄	1	7	yes	—
35	C₂×C₅×C₈	C₂²×C₄	1	56	yes	—
36	C₂×C₅×C₈	C₂²×C₄	1	28	yes	—
37	C₂×C₅×C₈	C₂²×C₄	1	21	yes	—
38	C₂×C₅×C₈	C₂⁴	1	7	yes	—
39	C₂×C₅×C₈	C₂⁴	1	8	yes	—
40	C₅×C₁₆	C₂⁴	1	16	yes	complemented
41	C₂⁴×C₄	C₄×C₅	1	1	yes	characteristic
42	C₂³×C₈	C₄×C₅	1	1	yes	characteristic
43	C₂³×C₈	C₄×C₅	1	14	yes	—
44	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
45	C₂³×C₈	C₂²×C₅	1	14	yes	—
46	C₂×M₃₂	C₂²×C₅	1	112	yes	direct product factor
47	C₂²×C₁₆	C₂²×C₅	1	28	yes	complemented
48	C₂³×C₅	—	2	56	no	—
49	C₂³×C₅	M₃₂	1	8	yes	direct product factor
50	C₂³×C₅	C₂²×C₈	1	28	yes	—
51	C₂³×C₅	C₂²×C₈	1	7	yes	—
52	C₂×C₄×C₅	C₂²×C₈	1	56	yes	—
53	C₂×C₄×C₅	C₂²×C₈	1	21	yes	—
54	C₂×C₄×C₅	C₂²×C₈	1	28	yes	—
55	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
56	C₂×C₄×C₅	C₂³×C₄	1	7	yes	—
57	C₅×C₈	C₂³×C₄	1	7	yes	—
58	C₅×C₈	C₂³×C₄	1	8	yes	—
59	C₅×C₈	C₂⁵	1	1	yes	characteristic
60	C₂⁵	C₅×C₈	1	1	yes	characteristic
61	C₂³×C₄	C₅×C₈	1	14	yes	—
62	C₂³×C₄	C₅×C₈	1	1	yes	characteristic
63	C₂³×C₄	C₂×C₄×C₅	1	14	yes	—
64	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
65	C₂²×C₈	C₂×C₄×C₅	1	84	yes	—
66	C₂²×C₈	C₂×C₄×C₅	1	14	yes	—
67	C₂²×C₈	C₂×C₄×C₅	1	7	yes	—
68	C₂²×C₈	C₂³×C₅	1	7	yes	—
69	C₂²×C₈	C₂³×C₅	1	28	yes	—
70	M₃₂	C₂³×C₅	1	64	yes	direct product factor
71	C₂×C₁₆	C₂³×C₅	1	56	yes	complemented
72	C₂²×C₅	—	2	56	no	—
73	C₂²×C₅	C₂×M₃₂	1	28	yes	direct product factor
74	C₂²×C₅	C₂³×C₈	1	8	yes	—
75	C₂²×C₅	C₂³×C₈	1	7	yes	—
76	C₄×C₅	C₂³×C₈	1	7	yes	—
77	C₄×C₅	C₂³×C₈	1	8	yes	—
78	C₄×C₅	C₂⁴×C₄	1	1	yes	characteristic
79	C₂⁴	—	2	8	no	—
80	C₂⁴	C₂×C₅×C₈	1	14	yes	—
81	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×C₅×C₈	1	84	yes	—
83	C₂²×C₄	C₂×C₅×C₈	1	7	yes	—
84	C₂²×C₄	C₂×C₅×C₈	1	14	yes	—
85	C₂²×C₄	C₂²×C₄×C₅	1	28	yes	—
86	C₂²×C₄	C₂²×C₄×C₅	1	7	yes	—
87	C₂×C₈	C₂²×C₄×C₅	1	21	yes	—
88	C₂×C₈	C₂²×C₄×C₅	1	56	yes	—
89	C₂×C₈	C₂²×C₄×C₅	1	28	yes	—
90	C₂×C₈	C₂⁴×C₅	1	7	yes	—
91	C₂×C₈	C₂⁴×C₅	1	8	yes	—
92	C₁₆	C₂⁴×C₅	1	16	yes	complemented
93	C₂×C₅	—	2	8	no	—
94	C₂×C₅	C₂²×M₃₂	1	14	yes	direct product factor
95	C₂×C₅	C₂⁴×C₈	1	1	yes	characteristic
96	C₂³	—	2	56	no	—
97	C₂³	C₅×M₃₂	1	8	yes	direct product factor
98	C₂³	C₂²×C₅×C₈	1	28	yes	—
99	C₂³	C₂²×C₅×C₈	1	7	yes	—
100	C₂×C₄	C₂²×C₅×C₈	1	56	yes	—
101	C₂×C₄	C₂²×C₅×C₈	1	21	yes	—
102	C₂×C₄	C₂²×C₅×C₈	1	28	yes	—
103	C₂×C₄	C₂³×C₄×C₅	1	8	yes	—
104	C₂×C₄	C₂³×C₄×C₅	1	7	yes	—
105	C₈	C₂³×C₄×C₅	1	7	yes	—
106	C₈	C₂³×C₄×C₅	1	8	yes	—
107	C₈	C₂⁵×C₅	1	1	yes	characteristic
108	C₅	C₂³×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
109	C₂²	—	2	56	no	—
110	C₂²	C₂×C₅×M₃₂	1	28	yes	direct product factor
111	C₂²	C₂³×C₅×C₈	1	8	yes	—
112	C₂²	C₂³×C₅×C₈	1	7	yes	—
113	C₄	C₂³×C₅×C₈	1	7	yes	—
114	C₄	C₂³×C₅×C₈	1	8	yes	—
115	C₄	C₂⁴×C₄×C₅	1	1	yes	characteristic
116	C₂	—	2	8	no	—
117	C₂	C₂²×C₅×M₃₂	1	14	yes	direct product factor
118	C₂	C₂⁴×C₅×C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅×C₈	C₂²
commutator	C₂	C₂⁴×C₅×C₈
Frattini, cosocle	C₈	C₂⁵×C₅
socle	C₂⁴×C₅	C₂×C₈
prime radical	C₂⁵×C₅	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×M₃₂	C₂	direct (N × H)
3. C₂³×C₅×C₁₆	C₂	semidirect
9. C₂×C₅×M₃₂	C₂²	direct (N × H)
10. C₂²×C₅×C₁₆	C₂²	semidirect
11. C₂³×M₃₂	C₅	direct (N × H)
22. C₅×M₃₂	C₂³	direct (N × H)
23. C₂×C₅×C₁₆	C₂³	semidirect
25. C₂²×M₃₂	C₂×C₅	direct (N × H)
26. C₂³×C₁₆	C₂×C₅	semidirect
40. C₅×C₁₆	C₂⁴	semidirect
46. C₂×M₃₂	C₂²×C₅	direct (N × H)
47. C₂²×C₁₆	C₂²×C₅	semidirect
49. C₂³×C₅	M₃₂	direct (N × H)
70. M₃₂	C₂³×C₅	direct (N × H)
71. C₂×C₁₆	C₂³×C₅	semidirect
73. C₂²×C₅	C₂×M₃₂	direct (N × H)
92. C₁₆	C₂⁴×C₅	semidirect
94. C₂×C₅	C₂²×M₃₂	direct (N × H)
97. C₂³	C₅×M₃₂	direct (N × H)
108. C₅	C₂³×M₃₂	direct (N × H)
110. C₂²	C₂×C₅×M₃₂	direct (N × H)
117. C₂	C₂²×C₅×M₃₂	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×C₅×C₈	2
C₂²×C₅×M₃₂	2
C₂³×C₅×C₁₆	2
C₂³×M₃₂	5

Subgroups — C23×C5×M32 #11985341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₅×M₃₂ #11985341