Subgroups — C₂⁵×C₃×C₈×C₃₁ #72201113

Order 23808.

30908 conjugacy classes of subgroups.

The subgroup lattice has 116 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃×C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₃×C₈×C₃₁	C₂	1	62	yes	maximal, direct product factor
3	C₂⁶×C₃×C₃₁	C₄	1	1	yes	characteristic
4	C₂⁴×C₃×C₄×C₃₁	C₄	1	31	yes	—
5	C₂⁴×C₃×C₄×C₃₁	C₂²	1	31	yes	—
6	C₂³×C₃×C₈×C₃₁	C₂²	1	620	yes	direct product factor
7	C₂³×C₃×C₄×C₃₁	C₂×C₄	1	465	yes	—
8	C₂³×C₃×C₄×C₃₁	C₂³	1	155	yes	—
9	C₂⁵×C₃×C₃₁	C₈	1	32	yes	direct product factor
10	C₂⁵×C₃×C₃₁	C₂×C₄	1	31	yes	—
11	C₂²×C₃×C₈×C₃₁	C₂³	1	1240	yes	direct product factor
12	C₂⁵×C₈×C₃₁	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂⁵×C₄×C₃₁	C₂×C₃	1	1	yes	characteristic
14	C₂⁴×C₈×C₃₁	C₂×C₃	1	62	yes	direct product factor
15	C₂⁶×C₃₁	C₃×C₄	1	1	yes	characteristic
16	C₂⁴×C₄×C₃₁	C₃×C₄	1	31	yes	—
17	C₂⁴×C₄×C₃₁	C₂²×C₃	1	31	yes	—
18	C₂³×C₈×C₃₁	C₂²×C₃	1	620	yes	direct product factor
19	C₂⁴×C₃×C₃₁	C₂×C₈	1	496	yes	direct product factor
20	C₂⁴×C₃×C₃₁	C₂²×C₄	1	155	yes	—
21	C₂²×C₃×C₄×C₃₁	C₂²×C₄	1	1085	yes	—
22	C₂²×C₃×C₄×C₃₁	C₂⁴	1	155	yes	—
23	C₂×C₃×C₈×C₃₁	C₂⁴	1	496	yes	direct product factor
24	C₂⁵×C₃₁	C₃×C₈	1	32	yes	direct product factor
25	C₂⁵×C₃₁	C₂×C₃×C₄	1	31	yes	—
26	C₂³×C₄×C₃₁	C₂×C₃×C₄	1	465	yes	—
27	C₂³×C₄×C₃₁	C₂³×C₃	1	155	yes	—
28	C₂²×C₈×C₃₁	C₂³×C₃	1	1240	yes	direct product factor
29	C₂⁵×C₃×C₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂³×C₃×C₃₁	C₂²×C₈	1	1240	yes	direct product factor
31	C₂³×C₃×C₃₁	C₂³×C₄	1	155	yes	—
32	C₂×C₃×C₄×C₃₁	C₂³×C₄	1	465	yes	—
33	C₂×C₃×C₄×C₃₁	C₂⁵	1	31	yes	—
34	C₃×C₈×C₃₁	C₂⁵	1	32	yes	direct product factor
35	C₂⁴×C₃₁	C₂×C₃×C₈	1	496	yes	direct product factor
36	C₂⁴×C₃₁	C₂²×C₃×C₄	1	155	yes	—
37	C₂²×C₄×C₃₁	C₂²×C₃×C₄	1	1085	yes	—
38	C₂²×C₄×C₃₁	C₂⁴×C₃	1	155	yes	—
39	C₂×C₈×C₃₁	C₂⁴×C₃	1	496	yes	direct product factor
40	C₂⁵×C₃×C₄	C₂×C₃₁	1	1	yes	characteristic
41	C₂⁴×C₃×C₈	C₂×C₃₁	1	62	yes	direct product factor
42	C₂²×C₃×C₃₁	C₂³×C₈	1	620	yes	direct product factor
43	C₂²×C₃×C₃₁	C₂⁴×C₄	1	31	yes	—
44	C₃×C₄×C₃₁	C₂⁴×C₄	1	31	yes	—
45	C₃×C₄×C₃₁	C₂⁶	1	1	yes	characteristic
46	C₂⁵×C₈	C₃×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃₁	C₂²×C₃×C₈	1	1240	yes	direct product factor
48	C₂³×C₃₁	C₂³×C₃×C₄	1	155	yes	—
49	C₂×C₄×C₃₁	C₂³×C₃×C₄	1	465	yes	—
50	C₂×C₄×C₃₁	C₂⁵×C₃	1	31	yes	—
51	C₈×C₃₁	C₂⁵×C₃	1	32	yes	direct product factor
52	C₂⁶×C₃	C₄×C₃₁	1	1	yes	characteristic
53	C₂⁴×C₃×C₄	C₄×C₃₁	1	31	yes	—
54	C₂⁴×C₃×C₄	C₂²×C₃₁	1	31	yes	—
55	C₂³×C₃×C₈	C₂²×C₃₁	1	620	yes	direct product factor
56	C₂×C₃×C₃₁	C₂⁴×C₈	1	62	yes	direct product factor
57	C₂×C₃×C₃₁	C₂⁵×C₄	1	1	yes	characteristic
58	C₂⁵×C₄	C₂×C₃×C₃₁	1	1	yes	characteristic
59	C₂⁴×C₈	C₂×C₃×C₃₁	1	62	yes	direct product factor
60	C₂²×C₃₁	C₂³×C₃×C₈	1	620	yes	direct product factor
61	C₂²×C₃₁	C₂⁴×C₃×C₄	1	31	yes	—
62	C₄×C₃₁	C₂⁴×C₃×C₄	1	31	yes	—
63	C₄×C₃₁	C₂⁶×C₃	1	1	yes	characteristic
64	C₂⁵×C₃	C₈×C₃₁	1	32	yes	direct product factor
65	C₂⁵×C₃	C₂×C₄×C₃₁	1	31	yes	—
66	C₂³×C₃×C₄	C₂×C₄×C₃₁	1	465	yes	—
67	C₂³×C₃×C₄	C₂³×C₃₁	1	155	yes	—
68	C₂²×C₃×C₈	C₂³×C₃₁	1	1240	yes	direct product factor
69	C₃×C₃₁	C₂⁵×C₈	1	1	yes	characteristic, direct product factor, Hall
70	C₂⁶	C₃×C₄×C₃₁	1	1	yes	characteristic
71	C₂⁴×C₄	C₃×C₄×C₃₁	1	31	yes	—
72	C₂⁴×C₄	C₂²×C₃×C₃₁	1	31	yes	—
73	C₂³×C₈	C₂²×C₃×C₃₁	1	620	yes	direct product factor
74	C₂×C₃₁	C₂⁴×C₃×C₈	1	62	yes	direct product factor
75	C₂×C₃₁	C₂⁵×C₃×C₄	1	1	yes	characteristic
76	C₂⁴×C₃	C₂×C₈×C₃₁	1	496	yes	direct product factor
77	C₂⁴×C₃	C₂²×C₄×C₃₁	1	155	yes	—
78	C₂²×C₃×C₄	C₂²×C₄×C₃₁	1	1085	yes	—
79	C₂²×C₃×C₄	C₂⁴×C₃₁	1	155	yes	—
80	C₂×C₃×C₈	C₂⁴×C₃₁	1	496	yes	direct product factor
81	C₂⁵	C₃×C₈×C₃₁	1	32	yes	direct product factor
82	C₂⁵	C₂×C₃×C₄×C₃₁	1	31	yes	—
83	C₂³×C₄	C₂×C₃×C₄×C₃₁	1	465	yes	—
84	C₂³×C₄	C₂³×C₃×C₃₁	1	155	yes	—
85	C₂²×C₈	C₂³×C₃×C₃₁	1	1240	yes	direct product factor
86	C₃₁	C₂⁵×C₃×C₈	1	1	yes	characteristic, direct product factor, Sylow
87	C₂³×C₃	C₂²×C₈×C₃₁	1	1240	yes	direct product factor
88	C₂³×C₃	C₂³×C₄×C₃₁	1	155	yes	—
89	C₂×C₃×C₄	C₂³×C₄×C₃₁	1	465	yes	—
90	C₂×C₃×C₄	C₂⁵×C₃₁	1	31	yes	—
91	C₃×C₈	C₂⁵×C₃₁	1	32	yes	direct product factor
92	C₂⁴	C₂×C₃×C₈×C₃₁	1	496	yes	direct product factor
93	C₂⁴	C₂²×C₃×C₄×C₃₁	1	155	yes	—
94	C₂²×C₄	C₂²×C₃×C₄×C₃₁	1	1085	yes	—
95	C₂²×C₄	C₂⁴×C₃×C₃₁	1	155	yes	—
96	C₂×C₈	C₂⁴×C₃×C₃₁	1	496	yes	direct product factor
97	C₂²×C₃	C₂³×C₈×C₃₁	1	620	yes	direct product factor
98	C₂²×C₃	C₂⁴×C₄×C₃₁	1	31	yes	—
99	C₃×C₄	C₂⁴×C₄×C₃₁	1	31	yes	—
100	C₃×C₄	C₂⁶×C₃₁	1	1	yes	characteristic
101	C₂³	C₂²×C₃×C₈×C₃₁	1	1240	yes	direct product factor
102	C₂³	C₂³×C₃×C₄×C₃₁	1	155	yes	—
103	C₂×C₄	C₂⁵×C₃×C₃₁	1	31	yes	—
104	C₂×C₄	C₂³×C₃×C₄×C₃₁	1	465	yes	—
105	C₈	C₂⁵×C₃×C₃₁	1	32	yes	direct product factor
106	C₂×C₃	C₂⁴×C₈×C₃₁	1	62	yes	direct product factor
107	C₂×C₃	C₂⁵×C₄×C₃₁	1	1	yes	characteristic
108	C₂²	C₂³×C₃×C₈×C₃₁	1	620	yes	direct product factor
109	C₂²	C₂⁴×C₃×C₄×C₃₁	1	31	yes	—
110	C₄	C₂⁴×C₃×C₄×C₃₁	1	31	yes	—
111	C₄	C₂⁶×C₃×C₃₁	1	1	yes	characteristic
112	C₃	C₂⁵×C₈×C₃₁	1	1	yes	characteristic, direct product factor, Sylow
113	C₂	C₂⁴×C₃×C₈×C₃₁	1	62	yes	direct product factor
114	C₂	C₂⁵×C₃×C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂⁶×C₃×C₃₁
socle, prime radical	C₂⁶×C₃×C₃₁	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₃×C₈×C₃₁	C₂	direct (N × H)
6. C₂³×C₃×C₈×C₃₁	C₂²	direct (N × H)
9. C₂⁵×C₃×C₃₁	C₈	direct (N × H)
11. C₂²×C₃×C₈×C₃₁	C₂³	direct (N × H)
12. C₂⁵×C₈×C₃₁	C₃	direct (N × H)
14. C₂⁴×C₈×C₃₁	C₂×C₃	direct (N × H)
18. C₂³×C₈×C₃₁	C₂²×C₃	direct (N × H)
19. C₂⁴×C₃×C₃₁	C₂×C₈	direct (N × H)
23. C₂×C₃×C₈×C₃₁	C₂⁴	direct (N × H)
24. C₂⁵×C₃₁	C₃×C₈	direct (N × H)
28. C₂²×C₈×C₃₁	C₂³×C₃	direct (N × H)
29. C₂⁵×C₃×C₈	C₃₁	direct (N × H)
30. C₂³×C₃×C₃₁	C₂²×C₈	direct (N × H)
34. C₃×C₈×C₃₁	C₂⁵	direct (N × H)
35. C₂⁴×C₃₁	C₂×C₃×C₈	direct (N × H)
39. C₂×C₈×C₃₁	C₂⁴×C₃	direct (N × H)
41. C₂⁴×C₃×C₈	C₂×C₃₁	direct (N × H)
42. C₂²×C₃×C₃₁	C₂³×C₈	direct (N × H)
46. C₂⁵×C₈	C₃×C₃₁	direct (N × H)
47. C₂³×C₃₁	C₂²×C₃×C₈	direct (N × H)
51. C₈×C₃₁	C₂⁵×C₃	direct (N × H)
55. C₂³×C₃×C₈	C₂²×C₃₁	direct (N × H)
56. C₂×C₃×C₃₁	C₂⁴×C₈	direct (N × H)
59. C₂⁴×C₈	C₂×C₃×C₃₁	direct (N × H)
60. C₂²×C₃₁	C₂³×C₃×C₈	direct (N × H)
64. C₂⁵×C₃	C₈×C₃₁	direct (N × H)
68. C₂²×C₃×C₈	C₂³×C₃₁	direct (N × H)
69. C₃×C₃₁	C₂⁵×C₈	direct (N × H)
73. C₂³×C₈	C₂²×C₃×C₃₁	direct (N × H)
74. C₂×C₃₁	C₂⁴×C₃×C₈	direct (N × H)
76. C₂⁴×C₃	C₂×C₈×C₃₁	direct (N × H)
80. C₂×C₃×C₈	C₂⁴×C₃₁	direct (N × H)
81. C₂⁵	C₃×C₈×C₃₁	direct (N × H)
85. C₂²×C₈	C₂³×C₃×C₃₁	direct (N × H)
86. C₃₁	C₂⁵×C₃×C₈	direct (N × H)
87. C₂³×C₃	C₂²×C₈×C₃₁	direct (N × H)
91. C₃×C₈	C₂⁵×C₃₁	direct (N × H)
92. C₂⁴	C₂×C₃×C₈×C₃₁	direct (N × H)
96. C₂×C₈	C₂⁴×C₃×C₃₁	direct (N × H)
97. C₂²×C₃	C₂³×C₈×C₃₁	direct (N × H)
101. C₂³	C₂²×C₃×C₈×C₃₁	direct (N × H)
105. C₈	C₂⁵×C₃×C₃₁	direct (N × H)
106. C₂×C₃	C₂⁴×C₈×C₃₁	direct (N × H)
108. C₂²	C₂³×C₃×C₈×C₃₁	direct (N × H)
112. C₃	C₂⁵×C₈×C₃₁	direct (N × H)
113. C₂	C₂⁴×C₃×C₈×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃×C₄×C₃₁	2
C₂⁴×C₃×C₈×C₃₁	2
C₂⁵×C₈×C₃₁	3
C₂⁵×C₃×C₈	31

Subgroups — C25×C3×C8×C31 #72201113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₃×C₈×C₃₁ #72201113