Subgroups — C₃²×C₄×C₅×C₈×C₁₃ #72196485

Order 18720.

528 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	C₃²×C₄²×C₅×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×C₅×C₈×C₁₃	C₂	1	2	yes	maximal
3	C₃×C₄×C₅×C₈×C₁₃	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₅×C₁₃	C₄	1	2	yes	—
5	C₂×C₃²×C₄×C₅×C₁₃	C₂²	1	1	yes	characteristic
6	C₃²×C₅×C₈×C₁₃	C₄	1	4	yes	direct product factor
7	C₃²×C₄×C₈×C₁₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×C₈×C₁₃	C₂×C₃	1	8	yes	—
9	C₃×C₄²×C₅×C₁₃	C₂×C₃	1	4	yes	—
10	C₄×C₅×C₈×C₁₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₃²×C₅×C₁₃	C₂×C₄	1	1	yes	characteristic
12	C₃²×C₄×C₅×C₁₃	C₈	1	4	yes	direct product factor
13	C₃²×C₄×C₅×C₁₃	C₂×C₄	1	2	yes	—
14	C₂×C₃²×C₈×C₁₃	C₂×C₅	1	2	yes	—
15	C₃²×C₄²×C₁₃	C₂×C₅	1	1	yes	characteristic
16	C₂×C₃×C₄×C₅×C₁₃	C₃×C₄	1	8	yes	—
17	C₂×C₃×C₄×C₅×C₁₃	C₂²×C₃	1	4	yes	—
18	C₃×C₅×C₈×C₁₃	C₃×C₄	1	16	yes	direct product factor
19	C₃²×C₄×C₅×C₈	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₃×C₄×C₈×C₁₃	C₃×C₅	1	4	yes	direct product factor
21	C₂×C₃²×C₅×C₁₃	C₄²	1	1	yes	characteristic
22	C₂×C₃²×C₅×C₁₃	C₂×C₈	1	2	yes	—
23	C₂×C₅×C₈×C₁₃	C₂×C₃²	1	2	yes	—
24	C₄²×C₅×C₁₃	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₃²×C₄×C₁₃	C₄×C₅	1	2	yes	—
26	C₂×C₃²×C₄×C₁₃	C₂²×C₅	1	1	yes	characteristic
27	C₃²×C₈×C₁₃	C₄×C₅	1	4	yes	direct product factor
28	C₂²×C₃×C₅×C₁₃	C₂×C₃×C₄	1	4	yes	—
29	C₃×C₄×C₅×C₁₃	C₃×C₈	1	16	yes	direct product factor
30	C₃×C₄×C₅×C₁₃	C₂×C₃×C₄	1	8	yes	—
31	C₂×C₃²×C₅×C₈	C₂×C₁₃	1	2	yes	—
32	C₃²×C₄²×C₅	C₂×C₁₃	1	1	yes	characteristic
33	C₂×C₃×C₈×C₁₃	C₂×C₃×C₅	1	8	yes	—
34	C₃×C₄²×C₁₃	C₂×C₃×C₅	1	4	yes	—
35	C₃²×C₅×C₁₃	C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
36	C₂×C₄×C₅×C₁₃	C₃²×C₄	1	2	yes	—
37	C₂×C₄×C₅×C₁₃	C₂²×C₃²	1	1	yes	characteristic
38	C₅×C₈×C₁₃	C₃²×C₄	1	4	yes	direct product factor
39	C₃×C₄×C₅×C₈	C₃×C₁₃	1	4	yes	direct product factor
40	C₂²×C₃²×C₁₃	C₂×C₄×C₅	1	1	yes	characteristic
41	C₃²×C₄×C₁₃	C₅×C₈	1	4	yes	direct product factor
42	C₃²×C₄×C₁₃	C₂×C₄×C₅	1	2	yes	—
43	C₄×C₈×C₁₃	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
44	C₂×C₃×C₅×C₁₃	C₃×C₄²	1	4	yes	—
45	C₂×C₃×C₅×C₁₃	C₂×C₃×C₈	1	8	yes	—
46	C₂×C₃²×C₄×C₅	C₄×C₁₃	1	2	yes	—
47	C₂×C₃²×C₄×C₅	C₂²×C₁₃	1	1	yes	characteristic
48	C₃²×C₅×C₈	C₄×C₁₃	1	4	yes	direct product factor
49	C₂×C₃×C₄×C₁₃	C₃×C₄×C₅	1	8	yes	—
50	C₂×C₃×C₄×C₁₃	C₂²×C₃×C₅	1	4	yes	—
51	C₃×C₈×C₁₃	C₃×C₄×C₅	1	16	yes	direct product factor
52	C₃²×C₄×C₈	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
53	C₂²×C₅×C₁₃	C₂×C₃²×C₄	1	1	yes	characteristic
54	C₄×C₅×C₁₃	C₃²×C₈	1	4	yes	direct product factor
55	C₄×C₅×C₁₃	C₂×C₃²×C₄	1	2	yes	—
56	C₂×C₃×C₅×C₈	C₂×C₃×C₁₃	1	8	yes	—
57	C₃×C₄²×C₅	C₂×C₃×C₁₃	1	4	yes	—
58	C₂×C₃²×C₁₃	C₄²×C₅	1	1	yes	characteristic
59	C₂×C₃²×C₁₃	C₂×C₅×C₈	1	2	yes	—
60	C₂×C₈×C₁₃	C₂×C₃²×C₅	1	2	yes	—
61	C₄²×C₁₃	C₂×C₃²×C₅	1	1	yes	characteristic
62	C₃×C₅×C₁₃	C₃×C₄×C₈	1	4	yes	direct product factor
63	C₂²×C₃²×C₅	C₂×C₄×C₁₃	1	1	yes	characteristic
64	C₃²×C₄×C₅	C₈×C₁₃	1	4	yes	direct product factor
65	C₃²×C₄×C₅	C₂×C₄×C₁₃	1	2	yes	—
66	C₄×C₅×C₈	C₃²×C₁₃	1	1	yes	characteristic, direct product factor, Hall
67	C₂²×C₃×C₁₃	C₂×C₃×C₄×C₅	1	4	yes	—
68	C₃×C₄×C₁₃	C₃×C₅×C₈	1	16	yes	direct product factor
69	C₃×C₄×C₁₃	C₂×C₃×C₄×C₅	1	8	yes	—
70	C₂×C₃²×C₈	C₂×C₅×C₁₃	1	2	yes	—
71	C₃²×C₄²	C₂×C₅×C₁₃	1	1	yes	characteristic
72	C₂×C₅×C₁₃	C₃²×C₄²	1	1	yes	characteristic
73	C₂×C₅×C₁₃	C₂×C₃²×C₈	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₃×C₄×C₁₃	1	8	yes	—
75	C₂×C₃×C₄×C₅	C₂²×C₃×C₁₃	1	4	yes	—
76	C₃×C₅×C₈	C₃×C₄×C₁₃	1	16	yes	direct product factor
77	C₃²×C₁₃	C₄×C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
78	C₂×C₄×C₁₃	C₃²×C₄×C₅	1	2	yes	—
79	C₂×C₄×C₁₃	C₂²×C₃²×C₅	1	1	yes	characteristic
80	C₈×C₁₃	C₃²×C₄×C₅	1	4	yes	direct product factor
81	C₃×C₄×C₈	C₃×C₅×C₁₃	1	4	yes	direct product factor
82	C₂×C₃²×C₅	C₄²×C₁₃	1	1	yes	characteristic
83	C₂×C₃²×C₅	C₂×C₈×C₁₃	1	2	yes	—
84	C₂×C₅×C₈	C₂×C₃²×C₁₃	1	2	yes	—
85	C₄²×C₅	C₂×C₃²×C₁₃	1	1	yes	characteristic
86	C₂×C₃×C₁₃	C₃×C₄²×C₅	1	4	yes	—
87	C₂×C₃×C₁₃	C₂×C₃×C₅×C₈	1	8	yes	—
88	C₂×C₃²×C₄	C₄×C₅×C₁₃	1	2	yes	—
89	C₂×C₃²×C₄	C₂²×C₅×C₁₃	1	1	yes	characteristic
90	C₃²×C₈	C₄×C₅×C₁₃	1	4	yes	direct product factor
91	C₅×C₁₃	C₃²×C₄×C₈	1	1	yes	characteristic, direct product factor, Hall
92	C₂²×C₃×C₅	C₂×C₃×C₄×C₁₃	1	4	yes	—
93	C₃×C₄×C₅	C₃×C₈×C₁₃	1	16	yes	direct product factor
94	C₃×C₄×C₅	C₂×C₃×C₄×C₁₃	1	8	yes	—
95	C₂²×C₁₃	C₂×C₃²×C₄×C₅	1	1	yes	characteristic
96	C₄×C₁₃	C₃²×C₅×C₈	1	4	yes	direct product factor
97	C₄×C₁₃	C₂×C₃²×C₄×C₅	1	2	yes	—
98	C₂×C₃×C₈	C₂×C₃×C₅×C₁₃	1	8	yes	—
99	C₃×C₄²	C₂×C₃×C₅×C₁₃	1	4	yes	—
100	C₃²×C₅	C₄×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Hall
101	C₂×C₄×C₅	C₃²×C₄×C₁₃	1	2	yes	—
102	C₂×C₄×C₅	C₂²×C₃²×C₁₃	1	1	yes	characteristic
103	C₅×C₈	C₃²×C₄×C₁₃	1	4	yes	direct product factor
104	C₃×C₁₃	C₃×C₄×C₅×C₈	1	4	yes	direct product factor
105	C₂²×C₃²	C₂×C₄×C₅×C₁₃	1	1	yes	characteristic
106	C₃²×C₄	C₅×C₈×C₁₃	1	4	yes	direct product factor
107	C₃²×C₄	C₂×C₄×C₅×C₁₃	1	2	yes	—
108	C₄×C₈	C₃²×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×C₃×C₅	C₃×C₄²×C₁₃	1	4	yes	—
110	C₂×C₃×C₅	C₂×C₃×C₈×C₁₃	1	8	yes	—
111	C₂×C₁₃	C₃²×C₄²×C₅	1	1	yes	characteristic
112	C₂×C₁₃	C₂×C₃²×C₅×C₈	1	2	yes	—
113	C₂×C₃×C₄	C₃×C₄×C₅×C₁₃	1	8	yes	—
114	C₂×C₃×C₄	C₂²×C₃×C₅×C₁₃	1	4	yes	—
115	C₃×C₈	C₃×C₄×C₅×C₁₃	1	16	yes	direct product factor
116	C₂²×C₅	C₂×C₃²×C₄×C₁₃	1	1	yes	characteristic
117	C₄×C₅	C₃²×C₈×C₁₃	1	4	yes	direct product factor
118	C₄×C₅	C₂×C₃²×C₄×C₁₃	1	2	yes	—
119	C₂×C₃²	C₄²×C₅×C₁₃	1	1	yes	characteristic
120	C₂×C₃²	C₂×C₅×C₈×C₁₃	1	2	yes	—
121	C₂×C₈	C₂×C₃²×C₅×C₁₃	1	2	yes	—
122	C₄²	C₂×C₃²×C₅×C₁₃	1	1	yes	characteristic
123	C₃×C₅	C₃×C₄×C₈×C₁₃	1	4	yes	direct product factor
124	C₁₃	C₃²×C₄×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
125	C₂²×C₃	C₂×C₃×C₄×C₅×C₁₃	1	4	yes	—
126	C₃×C₄	C₃×C₅×C₈×C₁₃	1	16	yes	direct product factor
127	C₃×C₄	C₂×C₃×C₄×C₅×C₁₃	1	8	yes	—
128	C₂×C₅	C₃²×C₄²×C₁₃	1	1	yes	characteristic
129	C₂×C₅	C₂×C₃²×C₈×C₁₃	1	2	yes	—
130	C₃²	C₄×C₅×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
131	C₂×C₄	C₃²×C₄×C₅×C₁₃	1	2	yes	—
132	C₂×C₄	C₂²×C₃²×C₅×C₁₃	1	1	yes	characteristic
133	C₈	C₃²×C₄×C₅×C₁₃	1	4	yes	direct product factor
134	C₂×C₃	C₃×C₄²×C₅×C₁₃	1	4	yes	—
135	C₂×C₃	C₂×C₃×C₅×C₈×C₁₃	1	8	yes	—
136	C₅	C₃²×C₄×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²	C₂×C₃²×C₄×C₅×C₁₃	1	1	yes	characteristic
138	C₄	C₃²×C₅×C₈×C₁₃	1	4	yes	direct product factor
139	C₄	C₂×C₃²×C₄×C₅×C₁₃	1	2	yes	—
140	C₃	C₃×C₄×C₅×C₈×C₁₃	1	4	yes	direct product factor
141	C₂	C₂×C₃²×C₅×C₈×C₁₃	1	2	yes	—
142	C₂	C₃²×C₄²×C₅×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₄	C₂²×C₃²×C₅×C₁₃
socle, prime radical	C₂²×C₃²×C₅×C₁₃	C₂×C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₅×C₈×C₁₃	C₃	direct (N × H)
6. C₃²×C₅×C₈×C₁₃	C₄	direct (N × H)
7. C₃²×C₄×C₈×C₁₃	C₅	direct (N × H)
10. C₄×C₅×C₈×C₁₃	C₃²	direct (N × H)
12. C₃²×C₄×C₅×C₁₃	C₈	direct (N × H)
18. C₃×C₅×C₈×C₁₃	C₃×C₄	direct (N × H)
19. C₃²×C₄×C₅×C₈	C₁₃	direct (N × H)
20. C₃×C₄×C₈×C₁₃	C₃×C₅	direct (N × H)
27. C₃²×C₈×C₁₃	C₄×C₅	direct (N × H)
29. C₃×C₄×C₅×C₁₃	C₃×C₈	direct (N × H)
35. C₃²×C₅×C₁₃	C₄×C₈	direct (N × H)
38. C₅×C₈×C₁₃	C₃²×C₄	direct (N × H)
39. C₃×C₄×C₅×C₈	C₃×C₁₃	direct (N × H)
41. C₃²×C₄×C₁₃	C₅×C₈	direct (N × H)
43. C₄×C₈×C₁₃	C₃²×C₅	direct (N × H)
48. C₃²×C₅×C₈	C₄×C₁₃	direct (N × H)
51. C₃×C₈×C₁₃	C₃×C₄×C₅	direct (N × H)
52. C₃²×C₄×C₈	C₅×C₁₃	direct (N × H)
54. C₄×C₅×C₁₃	C₃²×C₈	direct (N × H)
62. C₃×C₅×C₁₃	C₃×C₄×C₈	direct (N × H)
64. C₃²×C₄×C₅	C₈×C₁₃	direct (N × H)
66. C₄×C₅×C₈	C₃²×C₁₃	direct (N × H)
68. C₃×C₄×C₁₃	C₃×C₅×C₈	direct (N × H)
76. C₃×C₅×C₈	C₃×C₄×C₁₃	direct (N × H)
77. C₃²×C₁₃	C₄×C₅×C₈	direct (N × H)
80. C₈×C₁₃	C₃²×C₄×C₅	direct (N × H)
81. C₃×C₄×C₈	C₃×C₅×C₁₃	direct (N × H)
90. C₃²×C₈	C₄×C₅×C₁₃	direct (N × H)
91. C₅×C₁₃	C₃²×C₄×C₈	direct (N × H)
93. C₃×C₄×C₅	C₃×C₈×C₁₃	direct (N × H)
96. C₄×C₁₃	C₃²×C₅×C₈	direct (N × H)
100. C₃²×C₅	C₄×C₈×C₁₃	direct (N × H)
103. C₅×C₈	C₃²×C₄×C₁₃	direct (N × H)
104. C₃×C₁₃	C₃×C₄×C₅×C₈	direct (N × H)
106. C₃²×C₄	C₅×C₈×C₁₃	direct (N × H)
108. C₄×C₈	C₃²×C₅×C₁₃	direct (N × H)
115. C₃×C₈	C₃×C₄×C₅×C₁₃	direct (N × H)
117. C₄×C₅	C₃²×C₈×C₁₃	direct (N × H)
123. C₃×C₅	C₃×C₄×C₈×C₁₃	direct (N × H)
124. C₁₃	C₃²×C₄×C₅×C₈	direct (N × H)
126. C₃×C₄	C₃×C₅×C₈×C₁₃	direct (N × H)
130. C₃²	C₄×C₅×C₈×C₁₃	direct (N × H)
133. C₈	C₃²×C₄×C₅×C₁₃	direct (N × H)
136. C₅	C₃²×C₄×C₈×C₁₃	direct (N × H)
138. C₄	C₃²×C₅×C₈×C₁₃	direct (N × H)
140. C₃	C₃×C₄×C₅×C₈×C₁₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₄²×C₅×C₁₃	2
C₂×C₃²×C₅×C₈×C₁₃	2
C₃×C₄×C₅×C₈×C₁₃	3
C₃²×C₄×C₈×C₁₃	5
C₃²×C₄×C₅×C₈	13

Subgroups — C32×C4×C5×C8×C13 #72196485

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×C₅×C₈×C₁₃ #72196485