Subgroups — C₂⁴×C₅³×C₁₆ #72208751

Order 32000.

82880 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅³×C₁₆	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₄×C₅³	C₄	1	1	yes	characteristic
4	C₂³×C₅³×C₈	C₄	1	15	yes	—
5	C₂³×C₅³×C₈	C₂²	1	15	yes	—
6	C₂²×C₅³×C₁₆	C₂²	1	140	yes	direct product factor
7	C₂⁴×C₅²×C₁₆	C₅	1	31	yes	maximal, direct product factor
8	C₂⁵×C₅³	C₈	1	1	yes	characteristic
9	C₂³×C₄×C₅³	C₈	1	15	yes	—
10	C₂³×C₄×C₅³	C₂×C₄	1	15	yes	—
11	C₂²×C₅³×C₈	C₂×C₄	1	105	yes	—
12	C₂²×C₅³×C₈	C₂³	1	35	yes	—
13	C₂×C₅³×C₁₆	C₂³	1	120	yes	direct product factor
14	C₂⁴×C₅²×C₈	C₂×C₅	1	31	yes	—
15	C₂³×C₅²×C₁₆	C₂×C₅	1	930	yes	direct product factor
16	C₂⁴×C₅³	C₁₆	1	16	yes	direct product factor
17	C₂⁴×C₅³	C₂×C₈	1	15	yes	—
18	C₂²×C₄×C₅³	C₂×C₈	1	105	yes	—
19	C₂²×C₄×C₅³	C₂²×C₄	1	35	yes	—
20	C₂×C₅³×C₈	C₂²×C₄	1	105	yes	—
21	C₂×C₅³×C₈	C₂⁴	1	15	yes	—
22	C₅³×C₁₆	C₂⁴	1	16	yes	direct product factor
23	C₂⁴×C₄×C₅²	C₄×C₅	1	31	yes	—
24	C₂³×C₅²×C₈	C₄×C₅	1	465	yes	—
25	C₂³×C₅²×C₈	C₂²×C₅	1	465	yes	—
26	C₂²×C₅²×C₁₆	C₂²×C₅	1	4340	yes	direct product factor
27	C₂⁴×C₅×C₁₆	C₅²	1	31	yes	direct product factor
28	C₂³×C₅³	C₂×C₁₆	1	120	yes	direct product factor
29	C₂³×C₅³	C₂²×C₈	1	35	yes	—
30	C₂×C₄×C₅³	C₂²×C₈	1	105	yes	—
31	C₂×C₄×C₅³	C₂³×C₄	1	15	yes	—
32	C₅³×C₈	C₂³×C₄	1	15	yes	—
33	C₅³×C₈	C₂⁵	1	1	yes	characteristic
34	C₂⁵×C₅²	C₅×C₈	1	31	yes	—
35	C₂³×C₄×C₅²	C₅×C₈	1	465	yes	—
36	C₂³×C₄×C₅²	C₂×C₄×C₅	1	465	yes	—
37	C₂²×C₅²×C₈	C₂×C₄×C₅	1	3255	yes	—
38	C₂²×C₅²×C₈	C₂³×C₅	1	1085	yes	—
39	C₂×C₅²×C₁₆	C₂³×C₅	1	3720	yes	direct product factor
40	C₂⁴×C₅×C₈	C₂×C₅²	1	31	yes	—
41	C₂³×C₅×C₁₆	C₂×C₅²	1	930	yes	direct product factor
42	C₂²×C₅³	C₂²×C₁₆	1	140	yes	direct product factor
43	C₂²×C₅³	C₂³×C₈	1	15	yes	—
44	C₄×C₅³	C₂³×C₈	1	15	yes	—
45	C₄×C₅³	C₂⁴×C₄	1	1	yes	characteristic
46	C₂⁴×C₅²	C₅×C₁₆	1	496	yes	direct product factor
47	C₂⁴×C₅²	C₂×C₅×C₈	1	465	yes	—
48	C₂²×C₄×C₅²	C₂×C₅×C₈	1	3255	yes	—
49	C₂²×C₄×C₅²	C₂²×C₄×C₅	1	1085	yes	—
50	C₂×C₅²×C₈	C₂²×C₄×C₅	1	3255	yes	—
51	C₂×C₅²×C₈	C₂⁴×C₅	1	465	yes	—
52	C₅²×C₁₆	C₂⁴×C₅	1	496	yes	direct product factor
53	C₂⁴×C₄×C₅	C₄×C₅²	1	31	yes	—
54	C₂³×C₅×C₈	C₄×C₅²	1	465	yes	—
55	C₂³×C₅×C₈	C₂²×C₅²	1	465	yes	—
56	C₂²×C₅×C₁₆	C₂²×C₅²	1	4340	yes	direct product factor
57	C₂⁴×C₁₆	C₅³	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₅³	C₂³×C₁₆	1	30	yes	direct product factor
59	C₂×C₅³	C₂⁴×C₈	1	1	yes	characteristic
60	C₂³×C₅²	C₂×C₅×C₁₆	1	3720	yes	direct product factor
61	C₂³×C₅²	C₂²×C₅×C₈	1	1085	yes	—
62	C₂×C₄×C₅²	C₂²×C₅×C₈	1	3255	yes	—
63	C₂×C₄×C₅²	C₂³×C₄×C₅	1	465	yes	—
64	C₅²×C₈	C₂³×C₄×C₅	1	465	yes	—
65	C₅²×C₈	C₂⁵×C₅	1	31	yes	—
66	C₂⁵×C₅	C₅²×C₈	1	31	yes	—
67	C₂³×C₄×C₅	C₅²×C₈	1	465	yes	—
68	C₂³×C₄×C₅	C₂×C₄×C₅²	1	465	yes	—
69	C₂²×C₅×C₈	C₂×C₄×C₅²	1	3255	yes	—
70	C₂²×C₅×C₈	C₂³×C₅²	1	1085	yes	—
71	C₂×C₅×C₁₆	C₂³×C₅²	1	3720	yes	direct product factor
72	C₂⁴×C₈	C₂×C₅³	1	1	yes	characteristic
73	C₂³×C₁₆	C₂×C₅³	1	30	yes	direct product factor
74	C₅³	C₂⁴×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₅²	C₂²×C₅×C₁₆	1	4340	yes	direct product factor
76	C₂²×C₅²	C₂³×C₅×C₈	1	465	yes	—
77	C₄×C₅²	C₂³×C₅×C₈	1	465	yes	—
78	C₄×C₅²	C₂⁴×C₄×C₅	1	31	yes	—
79	C₂⁴×C₅	C₅²×C₁₆	1	496	yes	direct product factor
80	C₂⁴×C₅	C₂×C₅²×C₈	1	465	yes	—
81	C₂²×C₄×C₅	C₂×C₅²×C₈	1	3255	yes	—
82	C₂²×C₄×C₅	C₂²×C₄×C₅²	1	1085	yes	—
83	C₂×C₅×C₈	C₂²×C₄×C₅²	1	3255	yes	—
84	C₂×C₅×C₈	C₂⁴×C₅²	1	465	yes	—
85	C₅×C₁₆	C₂⁴×C₅²	1	496	yes	direct product factor
86	C₂⁴×C₄	C₄×C₅³	1	1	yes	characteristic
87	C₂³×C₈	C₄×C₅³	1	15	yes	—
88	C₂³×C₈	C₂²×C₅³	1	15	yes	—
89	C₂²×C₁₆	C₂²×C₅³	1	140	yes	direct product factor
90	C₂×C₅²	C₂³×C₅×C₁₆	1	930	yes	direct product factor
91	C₂×C₅²	C₂⁴×C₅×C₈	1	31	yes	—
92	C₂³×C₅	C₂×C₅²×C₁₆	1	3720	yes	direct product factor
93	C₂³×C₅	C₂²×C₅²×C₈	1	1085	yes	—
94	C₂×C₄×C₅	C₂²×C₅²×C₈	1	3255	yes	—
95	C₂×C₄×C₅	C₂³×C₄×C₅²	1	465	yes	—
96	C₅×C₈	C₂³×C₄×C₅²	1	465	yes	—
97	C₅×C₈	C₂⁵×C₅²	1	31	yes	—
98	C₂⁵	C₅³×C₈	1	1	yes	characteristic
99	C₂³×C₄	C₅³×C₈	1	15	yes	—
100	C₂³×C₄	C₂×C₄×C₅³	1	15	yes	—
101	C₂²×C₈	C₂×C₄×C₅³	1	105	yes	—
102	C₂²×C₈	C₂³×C₅³	1	35	yes	—
103	C₂×C₁₆	C₂³×C₅³	1	120	yes	direct product factor
104	C₅²	C₂⁴×C₅×C₁₆	1	31	yes	direct product factor
105	C₂²×C₅	C₂²×C₅²×C₁₆	1	4340	yes	direct product factor
106	C₂²×C₅	C₂³×C₅²×C₈	1	465	yes	—
107	C₄×C₅	C₂³×C₅²×C₈	1	465	yes	—
108	C₄×C₅	C₂⁴×C₄×C₅²	1	31	yes	—
109	C₂⁴	C₅³×C₁₆	1	16	yes	direct product factor
110	C₂⁴	C₂×C₅³×C₈	1	15	yes	—
111	C₂²×C₄	C₂×C₅³×C₈	1	105	yes	—
112	C₂²×C₄	C₂²×C₄×C₅³	1	35	yes	—
113	C₂×C₈	C₂²×C₄×C₅³	1	105	yes	—
114	C₂×C₈	C₂⁴×C₅³	1	15	yes	—
115	C₁₆	C₂⁴×C₅³	1	16	yes	direct product factor
116	C₂×C₅	C₂³×C₅²×C₁₆	1	930	yes	direct product factor
117	C₂×C₅	C₂⁴×C₅²×C₈	1	31	yes	—
118	C₂³	C₂×C₅³×C₁₆	1	120	yes	direct product factor
119	C₂³	C₂²×C₅³×C₈	1	35	yes	—
120	C₂×C₄	C₂²×C₅³×C₈	1	105	yes	—
121	C₂×C₄	C₂³×C₄×C₅³	1	15	yes	—
122	C₈	C₂³×C₄×C₅³	1	15	yes	—
123	C₈	C₂⁵×C₅³	1	1	yes	characteristic
124	C₅	C₂⁴×C₅²×C₁₆	1	31	yes	direct product factor
125	C₂²	C₂²×C₅³×C₁₆	1	140	yes	direct product factor
126	C₂²	C₂³×C₅³×C₈	1	15	yes	—
127	C₄	C₂³×C₅³×C₈	1	15	yes	—
128	C₄	C₂⁴×C₄×C₅³	1	1	yes	characteristic
129	C₂	C₂³×C₅³×C₁₆	1	30	yes	direct product factor
130	C₂	C₂⁴×C₅³×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂⁵×C₅³
socle, prime radical	C₂⁵×C₅³	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₅³×C₁₆	C₂	direct (N × H)
6. C₂²×C₅³×C₁₆	C₂²	direct (N × H)
7. C₂⁴×C₅²×C₁₆	C₅	direct (N × H)
13. C₂×C₅³×C₁₆	C₂³	direct (N × H)
15. C₂³×C₅²×C₁₆	C₂×C₅	direct (N × H)
16. C₂⁴×C₅³	C₁₆	direct (N × H)
22. C₅³×C₁₆	C₂⁴	direct (N × H)
26. C₂²×C₅²×C₁₆	C₂²×C₅	direct (N × H)
27. C₂⁴×C₅×C₁₆	C₅²	direct (N × H)
28. C₂³×C₅³	C₂×C₁₆	direct (N × H)
39. C₂×C₅²×C₁₆	C₂³×C₅	direct (N × H)
41. C₂³×C₅×C₁₆	C₂×C₅²	direct (N × H)
42. C₂²×C₅³	C₂²×C₁₆	direct (N × H)
46. C₂⁴×C₅²	C₅×C₁₆	direct (N × H)
52. C₅²×C₁₆	C₂⁴×C₅	direct (N × H)
56. C₂²×C₅×C₁₆	C₂²×C₅²	direct (N × H)
57. C₂⁴×C₁₆	C₅³	direct (N × H)
58. C₂×C₅³	C₂³×C₁₆	direct (N × H)
60. C₂³×C₅²	C₂×C₅×C₁₆	direct (N × H)
71. C₂×C₅×C₁₆	C₂³×C₅²	direct (N × H)
73. C₂³×C₁₆	C₂×C₅³	direct (N × H)
74. C₅³	C₂⁴×C₁₆	direct (N × H)
75. C₂²×C₅²	C₂²×C₅×C₁₆	direct (N × H)
79. C₂⁴×C₅	C₅²×C₁₆	direct (N × H)
85. C₅×C₁₆	C₂⁴×C₅²	direct (N × H)
89. C₂²×C₁₆	C₂²×C₅³	direct (N × H)
90. C₂×C₅²	C₂³×C₅×C₁₆	direct (N × H)
92. C₂³×C₅	C₂×C₅²×C₁₆	direct (N × H)
103. C₂×C₁₆	C₂³×C₅³	direct (N × H)
104. C₅²	C₂⁴×C₅×C₁₆	direct (N × H)
105. C₂²×C₅	C₂²×C₅²×C₁₆	direct (N × H)
109. C₂⁴	C₅³×C₁₆	direct (N × H)
115. C₁₆	C₂⁴×C₅³	direct (N × H)
116. C₂×C₅	C₂³×C₅²×C₁₆	direct (N × H)
118. C₂³	C₂×C₅³×C₁₆	direct (N × H)
124. C₅	C₂⁴×C₅²×C₁₆	direct (N × H)
125. C₂²	C₂²×C₅³×C₁₆	direct (N × H)
129. C₂	C₂³×C₅³×C₁₆	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×C₅³×C₈	2
C₂³×C₅³×C₁₆	2
C₂⁴×C₅²×C₁₆	5

Subgroups — C24×C53×C16 #72208751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₅³×C₁₆ #72208751