Subgroups — C₁₆×C₂₇×C₃₂ #72192003

Order 13824.

456 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	C₁₆²×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₈×C₂₇×C₃₂	C₂	1	2	yes	maximal
3	C₉×C₁₆×C₃₂	C₃	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₂₇	C₄	1	2	yes	—
5	C₈×C₁₆×C₂₇	C₂²	1	1	yes	characteristic
6	C₄×C₂₇×C₃₂	C₄	1	4	yes	—
7	C₉×C₁₆²	C₂×C₃	1	1	yes	characteristic
8	C₈×C₉×C₃₂	C₂×C₃	1	2	yes	—
9	C₃×C₁₆×C₃₂	C₉	1	1	yes	characteristic
10	C₂×C₂₇×C₃₂	C₈	1	8	yes	—
11	C₄×C₁₆×C₂₇	C₈	1	4	yes	—
12	C₄×C₁₆×C₂₇	C₂×C₄	1	2	yes	—
13	C₈²×C₂₇	C₂×C₄	1	1	yes	characteristic
14	C₄×C₉×C₃₂	C₃×C₄	1	4	yes	—
15	C₈×C₉×C₁₆	C₃×C₄	1	2	yes	—
16	C₈×C₉×C₁₆	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₁₆×C₂₇	C₁₆	1	8	yes	—
18	C₂×C₁₆×C₂₇	C₂×C₈	1	4	yes	—
19	C₄×C₈×C₂₇	C₄²	1	1	yes	characteristic
20	C₄×C₈×C₂₇	C₂×C₈	1	2	yes	—
21	C₂₇×C₃₂	C₁₆	1	16	yes	direct product factor
22	C₃×C₈×C₃₂	C₂×C₉	1	2	yes	—
23	C₃×C₁₆²	C₂×C₉	1	1	yes	characteristic
24	C₂×C₉×C₃₂	C₃×C₈	1	8	yes	—
25	C₄×C₉×C₁₆	C₃×C₈	1	4	yes	—
26	C₄×C₉×C₁₆	C₂×C₃×C₄	1	2	yes	—
27	C₈²×C₉	C₂×C₃×C₄	1	1	yes	characteristic
28	C₁₆×C₃₂	C₂₇	1	1	yes	characteristic, direct product factor, Sylow
29	C₂×C₈×C₂₇	C₄×C₈	1	2	yes	—
30	C₂×C₈×C₂₇	C₂×C₁₆	1	4	yes	—
31	C₄²×C₂₇	C₄×C₈	1	1	yes	characteristic
32	C₁₆×C₂₇	C₃₂	1	16	yes	direct product factor
33	C₁₆×C₂₇	C₂×C₁₆	1	8	yes	—
34	C₃×C₄×C₃₂	C₄×C₉	1	4	yes	—
35	C₃×C₈×C₁₆	C₄×C₉	1	2	yes	—
36	C₃×C₈×C₁₆	C₂²×C₉	1	1	yes	characteristic
37	C₂×C₉×C₁₆	C₃×C₁₆	1	8	yes	—
38	C₂×C₉×C₁₆	C₂×C₃×C₈	1	4	yes	—
39	C₄×C₈×C₉	C₃×C₄²	1	1	yes	characteristic
40	C₄×C₈×C₉	C₂×C₃×C₈	1	2	yes	—
41	C₉×C₃₂	C₃×C₁₆	1	16	yes	—
42	C₈×C₃₂	C₂×C₂₇	1	2	yes	—
43	C₁₆²	C₂×C₂₇	1	1	yes	characteristic
44	C₂×C₄×C₂₇	C₈²	1	1	yes	characteristic
45	C₂×C₄×C₂₇	C₄×C₁₆	1	2	yes	—
46	C₈×C₂₇	C₄×C₁₆	1	4	yes	—
47	C₈×C₂₇	C₂×C₃₂	1	8	yes	—
48	C₂×C₃×C₃₂	C₈×C₉	1	8	yes	—
49	C₃×C₄×C₁₆	C₈×C₉	1	4	yes	—
50	C₃×C₄×C₁₆	C₂×C₄×C₉	1	2	yes	—
51	C₃×C₈²	C₂×C₄×C₉	1	1	yes	characteristic
52	C₂×C₈×C₉	C₃×C₄×C₈	1	2	yes	—
53	C₂×C₈×C₉	C₂×C₃×C₁₆	1	4	yes	—
54	C₄²×C₉	C₃×C₄×C₈	1	1	yes	characteristic
55	C₉×C₁₆	C₃×C₃₂	1	16	yes	—
56	C₉×C₁₆	C₂×C₃×C₁₆	1	8	yes	—
57	C₄×C₃₂	C₄×C₂₇	1	4	yes	—
58	C₈×C₁₆	C₄×C₂₇	1	2	yes	—
59	C₈×C₁₆	C₂²×C₂₇	1	1	yes	characteristic
60	C₂²×C₂₇	C₈×C₁₆	1	1	yes	characteristic
61	C₄×C₂₇	C₈×C₁₆	1	2	yes	—
62	C₄×C₂₇	C₄×C₃₂	1	4	yes	—
63	C₂×C₃×C₁₆	C₉×C₁₆	1	8	yes	—
64	C₂×C₃×C₁₆	C₂×C₈×C₉	1	4	yes	—
65	C₃×C₄×C₈	C₄²×C₉	1	1	yes	characteristic
66	C₃×C₄×C₈	C₂×C₈×C₉	1	2	yes	—
67	C₃×C₃₂	C₉×C₁₆	1	16	yes	—
68	C₂×C₄×C₉	C₃×C₈²	1	1	yes	characteristic
69	C₂×C₄×C₉	C₃×C₄×C₁₆	1	2	yes	—
70	C₈×C₉	C₃×C₄×C₁₆	1	4	yes	—
71	C₈×C₉	C₂×C₃×C₃₂	1	8	yes	—
72	C₂×C₃₂	C₈×C₂₇	1	8	yes	—
73	C₄×C₁₆	C₈×C₂₇	1	4	yes	—
74	C₄×C₁₆	C₂×C₄×C₂₇	1	2	yes	—
75	C₈²	C₂×C₄×C₂₇	1	1	yes	characteristic
76	C₂×C₂₇	C₁₆²	1	1	yes	characteristic
77	C₂×C₂₇	C₈×C₃₂	1	2	yes	—
78	C₂×C₃×C₈	C₄×C₈×C₉	1	2	yes	—
79	C₂×C₃×C₈	C₂×C₉×C₁₆	1	4	yes	—
80	C₃×C₄²	C₄×C₈×C₉	1	1	yes	characteristic
81	C₃×C₁₆	C₉×C₃₂	1	16	yes	—
82	C₃×C₁₆	C₂×C₉×C₁₆	1	8	yes	—
83	C₂²×C₉	C₃×C₈×C₁₆	1	1	yes	characteristic
84	C₄×C₉	C₃×C₈×C₁₆	1	2	yes	—
85	C₄×C₉	C₃×C₄×C₃₂	1	4	yes	—
86	C₂×C₁₆	C₁₆×C₂₇	1	8	yes	—
87	C₂×C₁₆	C₂×C₈×C₂₇	1	4	yes	—
88	C₄×C₈	C₄²×C₂₇	1	1	yes	characteristic
89	C₄×C₈	C₂×C₈×C₂₇	1	2	yes	—
90	C₃₂	C₁₆×C₂₇	1	16	yes	direct product factor
91	C₂₇	C₁₆×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×C₃×C₄	C₈²×C₉	1	1	yes	characteristic
93	C₂×C₃×C₄	C₄×C₉×C₁₆	1	2	yes	—
94	C₃×C₈	C₄×C₉×C₁₆	1	4	yes	—
95	C₃×C₈	C₂×C₉×C₃₂	1	8	yes	—
96	C₂×C₉	C₃×C₁₆²	1	1	yes	characteristic
97	C₂×C₉	C₃×C₈×C₃₂	1	2	yes	—
98	C₂×C₈	C₄×C₈×C₂₇	1	2	yes	—
99	C₂×C₈	C₂×C₁₆×C₂₇	1	4	yes	—
100	C₄²	C₄×C₈×C₂₇	1	1	yes	characteristic
101	C₁₆	C₂₇×C₃₂	1	16	yes	direct product factor
102	C₁₆	C₂×C₁₆×C₂₇	1	8	yes	—
103	C₂²×C₃	C₈×C₉×C₁₆	1	1	yes	characteristic
104	C₃×C₄	C₈×C₉×C₁₆	1	2	yes	—
105	C₃×C₄	C₄×C₉×C₃₂	1	4	yes	—
106	C₉	C₃×C₁₆×C₃₂	1	1	yes	characteristic
107	C₂×C₄	C₈²×C₂₇	1	1	yes	characteristic
108	C₂×C₄	C₄×C₁₆×C₂₇	1	2	yes	—
109	C₈	C₄×C₁₆×C₂₇	1	4	yes	—
110	C₈	C₂×C₂₇×C₃₂	1	8	yes	—
111	C₂×C₃	C₈×C₉×C₃₂	1	2	yes	—
112	C₂×C₃	C₉×C₁₆²	1	1	yes	characteristic
113	C₂²	C₈×C₁₆×C₂₇	1	1	yes	characteristic
114	C₄	C₄×C₂₇×C₃₂	1	4	yes	—
115	C₄	C₈×C₁₆×C₂₇	1	2	yes	—
116	C₃	C₉×C₁₆×C₃₂	1	1	yes	characteristic
117	C₂	C₈×C₂₇×C₃₂	1	2	yes	—
118	C₂	C₁₆²×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈×C₉×C₁₆	C₂²×C₃
socle, prime radical	C₂²×C₃	C₈×C₉×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
21. C₂₇×C₃₂	C₁₆	direct (N × H)
28. C₁₆×C₃₂	C₂₇	direct (N × H)
32. C₁₆×C₂₇	C₃₂	direct (N × H)
90. C₃₂	C₁₆×C₂₇	direct (N × H)
91. C₂₇	C₁₆×C₃₂	direct (N × H)
101. C₁₆	C₂₇×C₃₂	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₆²×C₂₇	2
C₈×C₂₇×C₃₂	2
C₉×C₁₆×C₃₂	3

Subgroups — C16×C27×C32 #72192003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×C₂₇×C₃₂ #72192003