Subgroups — C₄×C₅×C₁₆×C₂₇ #72187482

Order 8640.

232 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×C₈×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₁₆×C₂₇	C₂	1	2	yes	maximal
3	C₄×C₅×C₉×C₁₆	C₃	1	1	yes	maximal, characteristic
4	C₄²×C₅×C₂₇	C₄	1	1	yes	characteristic
5	C₂×C₅×C₈×C₂₇	C₄	1	1	yes	characteristic
6	C₂×C₅×C₈×C₂₇	C₂²	1	1	yes	characteristic
7	C₅×C₁₆×C₂₇	C₄	1	4	yes	direct product factor
8	C₄×C₁₆×C₂₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₅×C₉×C₁₆	C₂×C₃	1	2	yes	—
10	C₄×C₅×C₈×C₉	C₂×C₃	1	1	yes	characteristic
11	C₂×C₄×C₅×C₂₇	C₈	1	2	yes	—
12	C₂×C₄×C₅×C₂₇	C₂×C₄	1	1	yes	characteristic
13	C₅×C₈×C₂₇	C₈	1	2	yes	—
14	C₅×C₈×C₂₇	C₂×C₄	1	2	yes	—
15	C₃×C₄×C₅×C₁₆	C₉	1	1	yes	characteristic
16	C₂×C₁₆×C₂₇	C₂×C₅	1	2	yes	—
17	C₄×C₈×C₂₇	C₂×C₅	1	1	yes	characteristic
18	C₂×C₅×C₈×C₉	C₃×C₄	1	1	yes	characteristic
19	C₂×C₅×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
20	C₄²×C₅×C₉	C₃×C₄	1	1	yes	characteristic
21	C₅×C₉×C₁₆	C₃×C₄	1	4	yes	—
22	C₄×C₉×C₁₆	C₃×C₅	1	1	yes	characteristic
23	C₂²×C₅×C₂₇	C₂×C₈	1	1	yes	characteristic
24	C₄×C₅×C₂₇	C₁₆	1	4	yes	direct product factor
25	C₄×C₅×C₂₇	C₄²	1	1	yes	characteristic
26	C₄×C₅×C₂₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₁₆	C₂×C₉	1	2	yes	—
28	C₃×C₄×C₅×C₈	C₂×C₉	1	1	yes	characteristic
29	C₂×C₈×C₂₇	C₄×C₅	1	1	yes	characteristic
30	C₂×C₈×C₂₇	C₂²×C₅	1	1	yes	characteristic
31	C₄²×C₂₇	C₄×C₅	1	1	yes	characteristic
32	C₁₆×C₂₇	C₄×C₅	1	4	yes	direct product factor
33	C₂×C₄×C₅×C₉	C₃×C₈	1	2	yes	—
34	C₂×C₄×C₅×C₉	C₂×C₃×C₄	1	1	yes	characteristic
35	C₅×C₈×C₉	C₃×C₈	1	2	yes	—
36	C₅×C₈×C₉	C₂×C₃×C₄	1	2	yes	—
37	C₄×C₅×C₁₆	C₂₇	1	1	yes	characteristic, direct product factor, Hall
38	C₂×C₉×C₁₆	C₂×C₃×C₅	1	2	yes	—
39	C₄×C₈×C₉	C₂×C₃×C₅	1	1	yes	characteristic
40	C₂×C₅×C₂₇	C₄×C₈	1	1	yes	characteristic
41	C₂×C₅×C₂₇	C₂×C₁₆	1	2	yes	—
42	C₂×C₃×C₅×C₈	C₄×C₉	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂²×C₉	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₄×C₉	1	1	yes	characteristic
45	C₃×C₅×C₁₆	C₄×C₉	1	4	yes	—
46	C₂×C₄×C₂₇	C₅×C₈	1	2	yes	—
47	C₂×C₄×C₂₇	C₂×C₄×C₅	1	1	yes	characteristic
48	C₈×C₂₇	C₅×C₈	1	2	yes	—
49	C₈×C₂₇	C₂×C₄×C₅	1	2	yes	—
50	C₃×C₄×C₁₆	C₅×C₉	1	1	yes	characteristic
51	C₂²×C₅×C₉	C₂×C₃×C₈	1	1	yes	characteristic
52	C₄×C₅×C₉	C₃×C₁₆	1	4	yes	—
53	C₄×C₅×C₉	C₃×C₄²	1	1	yes	characteristic
54	C₄×C₅×C₉	C₂×C₃×C₈	1	1	yes	characteristic
55	C₂×C₅×C₁₆	C₂×C₂₇	1	2	yes	—
56	C₄×C₅×C₈	C₂×C₂₇	1	1	yes	characteristic
57	C₂×C₈×C₉	C₃×C₄×C₅	1	1	yes	characteristic
58	C₂×C₈×C₉	C₂²×C₃×C₅	1	1	yes	characteristic
59	C₄²×C₉	C₃×C₄×C₅	1	1	yes	characteristic
60	C₉×C₁₆	C₃×C₄×C₅	1	4	yes	—
61	C₅×C₂₇	C₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
62	C₂×C₃×C₄×C₅	C₈×C₉	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×C₄×C₉	1	1	yes	characteristic
64	C₃×C₅×C₈	C₈×C₉	1	2	yes	—
65	C₃×C₅×C₈	C₂×C₄×C₉	1	2	yes	—
66	C₂²×C₂₇	C₂×C₅×C₈	1	1	yes	characteristic
67	C₄×C₂₇	C₅×C₁₆	1	4	yes	direct product factor
68	C₄×C₂₇	C₄²×C₅	1	1	yes	characteristic
69	C₄×C₂₇	C₂×C₅×C₈	1	1	yes	characteristic
70	C₂×C₃×C₁₆	C₂×C₅×C₉	1	2	yes	—
71	C₃×C₄×C₈	C₂×C₅×C₉	1	1	yes	characteristic
72	C₂×C₅×C₉	C₃×C₄×C₈	1	1	yes	characteristic
73	C₂×C₅×C₉	C₂×C₃×C₁₆	1	2	yes	—
74	C₂×C₅×C₈	C₄×C₂₇	1	1	yes	characteristic
75	C₂×C₅×C₈	C₂²×C₂₇	1	1	yes	characteristic
76	C₄²×C₅	C₄×C₂₇	1	1	yes	characteristic
77	C₅×C₁₆	C₄×C₂₇	1	4	yes	direct product factor
78	C₂×C₄×C₉	C₃×C₅×C₈	1	2	yes	—
79	C₂×C₄×C₉	C₂×C₃×C₄×C₅	1	1	yes	characteristic
80	C₈×C₉	C₃×C₅×C₈	1	2	yes	—
81	C₈×C₉	C₂×C₃×C₄×C₅	1	2	yes	—
82	C₄×C₁₆	C₅×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
83	C₂²×C₃×C₅	C₂×C₈×C₉	1	1	yes	characteristic
84	C₃×C₄×C₅	C₉×C₁₆	1	4	yes	—
85	C₃×C₄×C₅	C₄²×C₉	1	1	yes	characteristic
86	C₃×C₄×C₅	C₂×C₈×C₉	1	1	yes	characteristic
87	C₂×C₂₇	C₄×C₅×C₈	1	1	yes	characteristic
88	C₂×C₂₇	C₂×C₅×C₁₆	1	2	yes	—
89	C₂×C₃×C₈	C₄×C₅×C₉	1	1	yes	characteristic
90	C₂×C₃×C₈	C₂²×C₅×C₉	1	1	yes	characteristic
91	C₃×C₄²	C₄×C₅×C₉	1	1	yes	characteristic
92	C₃×C₁₆	C₄×C₅×C₉	1	4	yes	—
93	C₅×C₉	C₃×C₄×C₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅	C₈×C₂₇	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₄×C₂₇	1	1	yes	characteristic
96	C₅×C₈	C₈×C₂₇	1	2	yes	—
97	C₅×C₈	C₂×C₄×C₂₇	1	2	yes	—
98	C₂²×C₉	C₂×C₃×C₅×C₈	1	1	yes	characteristic
99	C₄×C₉	C₃×C₅×C₁₆	1	4	yes	—
100	C₄×C₉	C₃×C₄²×C₅	1	1	yes	characteristic
101	C₄×C₉	C₂×C₃×C₅×C₈	1	1	yes	characteristic
102	C₂×C₁₆	C₂×C₅×C₂₇	1	2	yes	—
103	C₄×C₈	C₂×C₅×C₂₇	1	1	yes	characteristic
104	C₂×C₃×C₅	C₄×C₈×C₉	1	1	yes	characteristic
105	C₂×C₃×C₅	C₂×C₉×C₁₆	1	2	yes	—
106	C₂₇	C₄×C₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×C₃×C₄	C₅×C₈×C₉	1	2	yes	—
108	C₂×C₃×C₄	C₂×C₄×C₅×C₉	1	1	yes	characteristic
109	C₃×C₈	C₅×C₈×C₉	1	2	yes	—
110	C₃×C₈	C₂×C₄×C₅×C₉	1	2	yes	—
111	C₂²×C₅	C₂×C₈×C₂₇	1	1	yes	characteristic
112	C₄×C₅	C₁₆×C₂₇	1	4	yes	direct product factor
113	C₄×C₅	C₄²×C₂₇	1	1	yes	characteristic
114	C₄×C₅	C₂×C₈×C₂₇	1	1	yes	characteristic
115	C₂×C₉	C₃×C₄×C₅×C₈	1	1	yes	characteristic
116	C₂×C₉	C₂×C₃×C₅×C₁₆	1	2	yes	—
117	C₂×C₈	C₄×C₅×C₂₇	1	1	yes	characteristic
118	C₂×C₈	C₂²×C₅×C₂₇	1	1	yes	characteristic
119	C₄²	C₄×C₅×C₂₇	1	1	yes	characteristic
120	C₁₆	C₄×C₅×C₂₇	1	4	yes	direct product factor
121	C₃×C₅	C₄×C₉×C₁₆	1	1	yes	characteristic
122	C₂²×C₃	C₂×C₅×C₈×C₉	1	1	yes	characteristic
123	C₃×C₄	C₅×C₉×C₁₆	1	4	yes	—
124	C₃×C₄	C₄²×C₅×C₉	1	1	yes	characteristic
125	C₃×C₄	C₂×C₅×C₈×C₉	1	1	yes	characteristic
126	C₂×C₅	C₄×C₈×C₂₇	1	1	yes	characteristic
127	C₂×C₅	C₂×C₁₆×C₂₇	1	2	yes	—
128	C₉	C₃×C₄×C₅×C₁₆	1	1	yes	characteristic
129	C₂×C₄	C₅×C₈×C₂₇	1	2	yes	—
130	C₂×C₄	C₂×C₄×C₅×C₂₇	1	1	yes	characteristic
131	C₈	C₅×C₈×C₂₇	1	2	yes	—
132	C₈	C₂×C₄×C₅×C₂₇	1	2	yes	—
133	C₂×C₃	C₄×C₅×C₈×C₉	1	1	yes	characteristic
134	C₂×C₃	C₂×C₅×C₉×C₁₆	1	2	yes	—
135	C₅	C₄×C₁₆×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
136	C₂²	C₂×C₅×C₈×C₂₇	1	1	yes	characteristic
137	C₄	C₅×C₁₆×C₂₇	1	4	yes	direct product factor
138	C₄	C₂×C₅×C₈×C₂₇	1	1	yes	characteristic
139	C₄	C₄²×C₅×C₂₇	1	1	yes	characteristic
140	C₃	C₄×C₅×C₉×C₁₆	1	1	yes	characteristic
141	C₂	C₂×C₅×C₁₆×C₂₇	1	2	yes	—
142	C₂	C₄×C₅×C₈×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈×C₉	C₂²×C₃×C₅
socle, prime radical	C₂²×C₃×C₅	C₂×C₈×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₅×C₁₆×C₂₇	C₄	direct (N × H)
8. C₄×C₁₆×C₂₇	C₅	direct (N × H)
24. C₄×C₅×C₂₇	C₁₆	direct (N × H)
32. C₁₆×C₂₇	C₄×C₅	direct (N × H)
37. C₄×C₅×C₁₆	C₂₇	direct (N × H)
61. C₅×C₂₇	C₄×C₁₆	direct (N × H)
67. C₄×C₂₇	C₅×C₁₆	direct (N × H)
77. C₅×C₁₆	C₄×C₂₇	direct (N × H)
82. C₄×C₁₆	C₅×C₂₇	direct (N × H)
106. C₂₇	C₄×C₅×C₁₆	direct (N × H)
112. C₄×C₅	C₁₆×C₂₇	direct (N × H)
120. C₁₆	C₄×C₅×C₂₇	direct (N × H)
135. C₅	C₄×C₁₆×C₂₇	direct (N × H)
137. C₄	C₅×C₁₆×C₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×C₈×C₂₇	2
C₂×C₅×C₁₆×C₂₇	2
C₄×C₅×C₉×C₁₆	3
C₄×C₁₆×C₂₇	5

Subgroups — C4×C5×C16×C27 #72187482

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₁₆×C₂₇ #72187482