Subgroups — C₂³×C₃×C₅×C₈×C₁₃ #72190825

Order 12480.

1352 conjugacy classes of subgroups.

The subgroup lattice has 152 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₅×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₈×C₁₃	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₅×C₈×C₁₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂²×C₃×C₄×C₅×C₁₃	C₄	1	7	yes	—
5	C₂²×C₃×C₄×C₅×C₁₃	C₂²	1	7	yes	—
6	C₂⁴×C₃×C₅×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₃×C₅×C₈×C₁₃	C₂²	1	28	yes	direct product factor
8	C₂³×C₃×C₈×C₁₃	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂³×C₄×C₅×C₁₃	C₂×C₃	1	1	yes	characteristic
10	C₂²×C₅×C₈×C₁₃	C₂×C₃	1	14	yes	direct product factor
11	C₂³×C₃×C₅×C₁₃	C₈	1	8	yes	direct product factor
12	C₂³×C₃×C₅×C₁₃	C₂×C₄	1	7	yes	—
13	C₂×C₃×C₄×C₅×C₁₃	C₂×C₄	1	21	yes	—
14	C₂×C₃×C₄×C₅×C₁₃	C₂³	1	7	yes	—
15	C₃×C₅×C₈×C₁₃	C₂³	1	8	yes	direct product factor
16	C₂³×C₃×C₄×C₁₃	C₂×C₅	1	1	yes	characteristic
17	C₂²×C₃×C₈×C₁₃	C₂×C₅	1	14	yes	direct product factor
18	C₂⁴×C₅×C₁₃	C₃×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅×C₁₃	C₃×C₄	1	7	yes	—
20	C₂²×C₄×C₅×C₁₃	C₂²×C₃	1	7	yes	—
21	C₂×C₅×C₈×C₁₃	C₂²×C₃	1	28	yes	direct product factor
22	C₂³×C₃×C₅×C₈	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂³×C₈×C₁₃	C₃×C₅	1	1	yes	characteristic, direct product factor, Hall
24	C₂²×C₃×C₅×C₁₃	C₂×C₈	1	28	yes	direct product factor
25	C₂²×C₃×C₅×C₁₃	C₂²×C₄	1	7	yes	—
26	C₃×C₄×C₅×C₁₃	C₂²×C₄	1	7	yes	—
27	C₃×C₄×C₅×C₁₃	C₂⁴	1	1	yes	characteristic
28	C₂⁴×C₃×C₁₃	C₄×C₅	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₁₃	C₄×C₅	1	7	yes	—
30	C₂²×C₃×C₄×C₁₃	C₂²×C₅	1	7	yes	—
31	C₂×C₃×C₈×C₁₃	C₂²×C₅	1	28	yes	direct product factor
32	C₂³×C₅×C₁₃	C₃×C₈	1	8	yes	direct product factor
33	C₂³×C₅×C₁₃	C₂×C₃×C₄	1	7	yes	—
34	C₂×C₄×C₅×C₁₃	C₂×C₃×C₄	1	21	yes	—
35	C₂×C₄×C₅×C₁₃	C₂³×C₃	1	7	yes	—
36	C₅×C₈×C₁₃	C₂³×C₃	1	8	yes	direct product factor
37	C₂³×C₃×C₄×C₅	C₂×C₁₃	1	1	yes	characteristic
38	C₂²×C₃×C₅×C₈	C₂×C₁₃	1	14	yes	direct product factor
39	C₂³×C₄×C₁₃	C₂×C₃×C₅	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	C₂×C₃×C₅	1	14	yes	direct product factor
41	C₂×C₃×C₅×C₁₃	C₂²×C₈	1	14	yes	direct product factor
42	C₂×C₃×C₅×C₁₃	C₂³×C₄	1	1	yes	characteristic
43	C₂³×C₅×C₈	C₃×C₁₃	1	1	yes	characteristic, direct product factor, Hall
44	C₂³×C₃×C₁₃	C₅×C₈	1	8	yes	direct product factor
45	C₂³×C₃×C₁₃	C₂×C₄×C₅	1	7	yes	—
46	C₂×C₃×C₄×C₁₃	C₂×C₄×C₅	1	21	yes	—
47	C₂×C₃×C₄×C₁₃	C₂³×C₅	1	7	yes	—
48	C₃×C₈×C₁₃	C₂³×C₅	1	8	yes	direct product factor
49	C₂²×C₅×C₁₃	C₂×C₃×C₈	1	28	yes	direct product factor
50	C₂²×C₅×C₁₃	C₂²×C₃×C₄	1	7	yes	—
51	C₄×C₅×C₁₃	C₂²×C₃×C₄	1	7	yes	—
52	C₄×C₅×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
53	C₂⁴×C₃×C₅	C₄×C₁₃	1	1	yes	characteristic
54	C₂²×C₃×C₄×C₅	C₄×C₁₃	1	7	yes	—
55	C₂²×C₃×C₄×C₅	C₂²×C₁₃	1	7	yes	—
56	C₂×C₃×C₅×C₈	C₂²×C₁₃	1	28	yes	direct product factor
57	C₂⁴×C₁₃	C₃×C₄×C₅	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₃×C₄×C₅	1	7	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₃×C₅	1	7	yes	—
60	C₂×C₈×C₁₃	C₂²×C₃×C₅	1	28	yes	direct product factor
61	C₃×C₅×C₁₃	C₂³×C₈	1	1	yes	characteristic, direct product factor, Hall
62	C₂³×C₃×C₈	C₅×C₁₃	1	1	yes	characteristic, direct product factor, Hall
63	C₂³×C₄×C₅	C₂×C₃×C₁₃	1	1	yes	characteristic
64	C₂²×C₅×C₈	C₂×C₃×C₁₃	1	14	yes	direct product factor
65	C₂²×C₃×C₁₃	C₂×C₅×C₈	1	28	yes	direct product factor
66	C₂²×C₃×C₁₃	C₂²×C₄×C₅	1	7	yes	—
67	C₃×C₄×C₁₃	C₂²×C₄×C₅	1	7	yes	—
68	C₃×C₄×C₁₃	C₂⁴×C₅	1	1	yes	characteristic
69	C₂×C₅×C₁₃	C₂²×C₃×C₈	1	14	yes	direct product factor
70	C₂×C₅×C₁₃	C₂³×C₃×C₄	1	1	yes	characteristic
71	C₂³×C₃×C₅	C₈×C₁₃	1	8	yes	direct product factor
72	C₂³×C₃×C₅	C₂×C₄×C₁₃	1	7	yes	—
73	C₂×C₃×C₄×C₅	C₂×C₄×C₁₃	1	21	yes	—
74	C₂×C₃×C₄×C₅	C₂³×C₁₃	1	7	yes	—
75	C₃×C₅×C₈	C₂³×C₁₃	1	8	yes	direct product factor
76	C₂³×C₁₃	C₃×C₅×C₈	1	8	yes	direct product factor
77	C₂³×C₁₃	C₂×C₃×C₄×C₅	1	7	yes	—
78	C₂×C₄×C₁₃	C₂×C₃×C₄×C₅	1	21	yes	—
79	C₂×C₄×C₁₃	C₂³×C₃×C₅	1	7	yes	—
80	C₈×C₁₃	C₂³×C₃×C₅	1	8	yes	direct product factor
81	C₂³×C₃×C₄	C₂×C₅×C₁₃	1	1	yes	characteristic
82	C₂²×C₃×C₈	C₂×C₅×C₁₃	1	14	yes	direct product factor
83	C₂⁴×C₅	C₃×C₄×C₁₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₃×C₄×C₁₃	1	7	yes	—
85	C₂²×C₄×C₅	C₂²×C₃×C₁₃	1	7	yes	—
86	C₂×C₅×C₈	C₂²×C₃×C₁₃	1	28	yes	direct product factor
87	C₂×C₃×C₁₃	C₂²×C₅×C₈	1	14	yes	direct product factor
88	C₂×C₃×C₁₃	C₂³×C₄×C₅	1	1	yes	characteristic
89	C₅×C₁₃	C₂³×C₃×C₈	1	1	yes	characteristic, direct product factor, Hall
90	C₂³×C₈	C₃×C₅×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
91	C₂²×C₃×C₅	C₂×C₈×C₁₃	1	28	yes	direct product factor
92	C₂²×C₃×C₅	C₂²×C₄×C₁₃	1	7	yes	—
93	C₃×C₄×C₅	C₂²×C₄×C₁₃	1	7	yes	—
94	C₃×C₄×C₅	C₂⁴×C₁₃	1	1	yes	characteristic
95	C₂²×C₁₃	C₂×C₃×C₅×C₈	1	28	yes	direct product factor
96	C₂²×C₁₃	C₂²×C₃×C₄×C₅	1	7	yes	—
97	C₄×C₁₃	C₂²×C₃×C₄×C₅	1	7	yes	—
98	C₄×C₁₃	C₂⁴×C₃×C₅	1	1	yes	characteristic
99	C₂⁴×C₃	C₄×C₅×C₁₃	1	1	yes	characteristic
100	C₂²×C₃×C₄	C₄×C₅×C₁₃	1	7	yes	—
101	C₂²×C₃×C₄	C₂²×C₅×C₁₃	1	7	yes	—
102	C₂×C₃×C₈	C₂²×C₅×C₁₃	1	28	yes	direct product factor
103	C₂³×C₅	C₃×C₈×C₁₃	1	8	yes	direct product factor
104	C₂³×C₅	C₂×C₃×C₄×C₁₃	1	7	yes	—
105	C₂×C₄×C₅	C₂×C₃×C₄×C₁₃	1	21	yes	—
106	C₂×C₄×C₅	C₂³×C₃×C₁₃	1	7	yes	—
107	C₅×C₈	C₂³×C₃×C₁₃	1	8	yes	direct product factor
108	C₃×C₁₃	C₂³×C₅×C₈	1	1	yes	characteristic, direct product factor, Hall
109	C₂³×C₄	C₂×C₃×C₅×C₁₃	1	1	yes	characteristic
110	C₂²×C₈	C₂×C₃×C₅×C₁₃	1	14	yes	direct product factor
111	C₂×C₃×C₅	C₂²×C₈×C₁₃	1	14	yes	direct product factor
112	C₂×C₃×C₅	C₂³×C₄×C₁₃	1	1	yes	characteristic
113	C₂×C₁₃	C₂²×C₃×C₅×C₈	1	14	yes	direct product factor
114	C₂×C₁₃	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
115	C₂³×C₃	C₅×C₈×C₁₃	1	8	yes	direct product factor
116	C₂³×C₃	C₂×C₄×C₅×C₁₃	1	7	yes	—
117	C₂×C₃×C₄	C₂×C₄×C₅×C₁₃	1	21	yes	—
118	C₂×C₃×C₄	C₂³×C₅×C₁₃	1	7	yes	—
119	C₃×C₈	C₂³×C₅×C₁₃	1	8	yes	direct product factor
120	C₂²×C₅	C₂×C₃×C₈×C₁₃	1	28	yes	direct product factor
121	C₂²×C₅	C₂²×C₃×C₄×C₁₃	1	7	yes	—
122	C₄×C₅	C₂²×C₃×C₄×C₁₃	1	7	yes	—
123	C₄×C₅	C₂⁴×C₃×C₁₃	1	1	yes	characteristic
124	C₂⁴	C₃×C₄×C₅×C₁₃	1	1	yes	characteristic
125	C₂²×C₄	C₃×C₄×C₅×C₁₃	1	7	yes	—
126	C₂²×C₄	C₂²×C₃×C₅×C₁₃	1	7	yes	—
127	C₂×C₈	C₂²×C₃×C₅×C₁₃	1	28	yes	direct product factor
128	C₃×C₅	C₂³×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Hall
129	C₁₃	C₂³×C₃×C₅×C₈	1	1	yes	characteristic, direct product factor, Sylow
130	C₂²×C₃	C₂×C₅×C₈×C₁₃	1	28	yes	direct product factor
131	C₂²×C₃	C₂²×C₄×C₅×C₁₃	1	7	yes	—
132	C₃×C₄	C₂²×C₄×C₅×C₁₃	1	7	yes	—
133	C₃×C₄	C₂⁴×C₅×C₁₃	1	1	yes	characteristic
134	C₂×C₅	C₂²×C₃×C₈×C₁₃	1	14	yes	direct product factor
135	C₂×C₅	C₂³×C₃×C₄×C₁₃	1	1	yes	characteristic
136	C₂³	C₃×C₅×C₈×C₁₃	1	8	yes	direct product factor
137	C₂³	C₂×C₃×C₄×C₅×C₁₃	1	7	yes	—
138	C₂×C₄	C₂×C₃×C₄×C₅×C₁₃	1	21	yes	—
139	C₂×C₄	C₂³×C₃×C₅×C₁₃	1	7	yes	—
140	C₈	C₂³×C₃×C₅×C₁₃	1	8	yes	direct product factor
141	C₂×C₃	C₂²×C₅×C₈×C₁₃	1	14	yes	direct product factor
142	C₂×C₃	C₂³×C₄×C₅×C₁₃	1	1	yes	characteristic
143	C₅	C₂³×C₃×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
144	C₂²	C₂×C₃×C₅×C₈×C₁₃	1	28	yes	direct product factor
145	C₂²	C₂²×C₃×C₄×C₅×C₁₃	1	7	yes	—
146	C₄	C₂⁴×C₃×C₅×C₁₃	1	1	yes	characteristic
147	C₄	C₂²×C₃×C₄×C₅×C₁₃	1	7	yes	—
148	C₃	C₂³×C₅×C₈×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
149	C₂	C₂²×C₃×C₅×C₈×C₁₃	1	14	yes	direct product factor
150	C₂	C₂³×C₃×C₄×C₅×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂⁴×C₃×C₅×C₁₃
socle, prime radical	C₂⁴×C₃×C₅×C₁₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₈×C₁₃	C₂	direct (N × H)
3. C₂³×C₅×C₈×C₁₃	C₃	direct (N × H)
7. C₂×C₃×C₅×C₈×C₁₃	C₂²	direct (N × H)
8. C₂³×C₃×C₈×C₁₃	C₅	direct (N × H)
10. C₂²×C₅×C₈×C₁₃	C₂×C₃	direct (N × H)
11. C₂³×C₃×C₅×C₁₃	C₈	direct (N × H)
15. C₃×C₅×C₈×C₁₃	C₂³	direct (N × H)
17. C₂²×C₃×C₈×C₁₃	C₂×C₅	direct (N × H)
21. C₂×C₅×C₈×C₁₃	C₂²×C₃	direct (N × H)
22. C₂³×C₃×C₅×C₈	C₁₃	direct (N × H)
23. C₂³×C₈×C₁₃	C₃×C₅	direct (N × H)
24. C₂²×C₃×C₅×C₁₃	C₂×C₈	direct (N × H)
31. C₂×C₃×C₈×C₁₃	C₂²×C₅	direct (N × H)
32. C₂³×C₅×C₁₃	C₃×C₈	direct (N × H)
36. C₅×C₈×C₁₃	C₂³×C₃	direct (N × H)
38. C₂²×C₃×C₅×C₈	C₂×C₁₃	direct (N × H)
40. C₂²×C₈×C₁₃	C₂×C₃×C₅	direct (N × H)
41. C₂×C₃×C₅×C₁₃	C₂²×C₈	direct (N × H)
43. C₂³×C₅×C₈	C₃×C₁₃	direct (N × H)
44. C₂³×C₃×C₁₃	C₅×C₈	direct (N × H)
48. C₃×C₈×C₁₃	C₂³×C₅	direct (N × H)
49. C₂²×C₅×C₁₃	C₂×C₃×C₈	direct (N × H)
56. C₂×C₃×C₅×C₈	C₂²×C₁₃	direct (N × H)
60. C₂×C₈×C₁₃	C₂²×C₃×C₅	direct (N × H)
61. C₃×C₅×C₁₃	C₂³×C₈	direct (N × H)
62. C₂³×C₃×C₈	C₅×C₁₃	direct (N × H)
64. C₂²×C₅×C₈	C₂×C₃×C₁₃	direct (N × H)
65. C₂²×C₃×C₁₃	C₂×C₅×C₈	direct (N × H)
69. C₂×C₅×C₁₃	C₂²×C₃×C₈	direct (N × H)
71. C₂³×C₃×C₅	C₈×C₁₃	direct (N × H)
75. C₃×C₅×C₈	C₂³×C₁₃	direct (N × H)
76. C₂³×C₁₃	C₃×C₅×C₈	direct (N × H)
80. C₈×C₁₃	C₂³×C₃×C₅	direct (N × H)
82. C₂²×C₃×C₈	C₂×C₅×C₁₃	direct (N × H)
86. C₂×C₅×C₈	C₂²×C₃×C₁₃	direct (N × H)
87. C₂×C₃×C₁₃	C₂²×C₅×C₈	direct (N × H)
89. C₅×C₁₃	C₂³×C₃×C₈	direct (N × H)
90. C₂³×C₈	C₃×C₅×C₁₃	direct (N × H)
91. C₂²×C₃×C₅	C₂×C₈×C₁₃	direct (N × H)
95. C₂²×C₁₃	C₂×C₃×C₅×C₈	direct (N × H)
102. C₂×C₃×C₈	C₂²×C₅×C₁₃	direct (N × H)
103. C₂³×C₅	C₃×C₈×C₁₃	direct (N × H)
107. C₅×C₈	C₂³×C₃×C₁₃	direct (N × H)
108. C₃×C₁₃	C₂³×C₅×C₈	direct (N × H)
110. C₂²×C₈	C₂×C₃×C₅×C₁₃	direct (N × H)
111. C₂×C₃×C₅	C₂²×C₈×C₁₃	direct (N × H)
113. C₂×C₁₃	C₂²×C₃×C₅×C₈	direct (N × H)
115. C₂³×C₃	C₅×C₈×C₁₃	direct (N × H)
119. C₃×C₈	C₂³×C₅×C₁₃	direct (N × H)
120. C₂²×C₅	C₂×C₃×C₈×C₁₃	direct (N × H)
127. C₂×C₈	C₂²×C₃×C₅×C₁₃	direct (N × H)
128. C₃×C₅	C₂³×C₈×C₁₃	direct (N × H)
129. C₁₃	C₂³×C₃×C₅×C₈	direct (N × H)
130. C₂²×C₃	C₂×C₅×C₈×C₁₃	direct (N × H)
134. C₂×C₅	C₂²×C₃×C₈×C₁₃	direct (N × H)
136. C₂³	C₃×C₅×C₈×C₁₃	direct (N × H)
140. C₈	C₂³×C₃×C₅×C₁₃	direct (N × H)
141. C₂×C₃	C₂²×C₅×C₈×C₁₃	direct (N × H)
143. C₅	C₂³×C₃×C₈×C₁₃	direct (N × H)
144. C₂²	C₂×C₃×C₅×C₈×C₁₃	direct (N × H)
148. C₃	C₂³×C₅×C₈×C₁₃	direct (N × H)
149. C₂	C₂²×C₃×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
C₂³×C₃×C₄×C₅×C₁₃	2
C₂²×C₃×C₅×C₈×C₁₃	2
C₂³×C₅×C₈×C₁₃	3
C₂³×C₃×C₈×C₁₃	5
C₂³×C₃×C₅×C₈	13

Subgroups — C23×C3×C5×C8×C13 #72190825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₅×C₈×C₁₃ #72190825