Subgroups — C₂×C₃²×C₅²×C₃₂ #72192545

Order 14400.

816 conjugacy classes of subgroups.

The subgroup lattice has 135 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₅²×C₃₂	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₅²×C₃₂	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₅²×C₈	C₄	1	1	yes	characteristic
5	C₃²×C₅²×C₁₆	C₄	1	1	yes	characteristic
6	C₃²×C₅²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₃²×C₅×C₃₂	C₅	1	6	yes	maximal, direct product factor
8	C₂×C₃×C₅²×C₁₆	C₂×C₃	1	4	yes	—
9	C₃×C₅²×C₃₂	C₂×C₃	1	8	yes	direct product factor
10	C₂×C₃²×C₄×C₅²	C₈	1	1	yes	characteristic
11	C₃²×C₅²×C₈	C₈	1	1	yes	characteristic
12	C₃²×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₅²×C₃₂	C₃²	1	1	yes	characteristic, direct product factor, Hall
14	C₂×C₃²×C₅×C₁₆	C₂×C₅	1	6	yes	—
15	C₃²×C₅×C₃₂	C₂×C₅	1	12	yes	direct product factor
16	C₂×C₃×C₅²×C₈	C₃×C₄	1	4	yes	—
17	C₃×C₅²×C₁₆	C₃×C₄	1	4	yes	—
18	C₃×C₅²×C₁₆	C₂²×C₃	1	4	yes	—
19	C₂×C₃×C₅×C₃₂	C₃×C₅	1	24	yes	direct product factor
20	C₂²×C₃²×C₅²	C₁₆	1	1	yes	characteristic
21	C₃²×C₄×C₅²	C₁₆	1	1	yes	characteristic
22	C₃²×C₄×C₅²	C₂×C₈	1	1	yes	characteristic
23	C₂×C₅²×C₁₆	C₂×C₃²	1	1	yes	characteristic
24	C₅²×C₃₂	C₂×C₃²	1	2	yes	direct product factor
25	C₂×C₃²×C₅×C₈	C₄×C₅	1	6	yes	—
26	C₃²×C₅×C₁₆	C₄×C₅	1	6	yes	—
27	C₃²×C₅×C₁₆	C₂²×C₅	1	6	yes	—
28	C₂×C₃×C₄×C₅²	C₃×C₈	1	4	yes	—
29	C₃×C₅²×C₈	C₃×C₈	1	4	yes	—
30	C₃×C₅²×C₈	C₂×C₃×C₄	1	4	yes	—
31	C₂×C₃²×C₃₂	C₅²	1	1	yes	characteristic, direct product factor, Hall
32	C₂×C₃×C₅×C₁₆	C₂×C₃×C₅	1	24	yes	—
33	C₃×C₅×C₃₂	C₂×C₃×C₅	1	48	yes	direct product factor
34	C₂×C₃²×C₅²	C₃₂	1	2	yes	direct product factor
35	C₂×C₃²×C₅²	C₂×C₁₆	1	1	yes	characteristic
36	C₂×C₅²×C₈	C₃²×C₄	1	1	yes	characteristic
37	C₅²×C₁₆	C₃²×C₄	1	1	yes	characteristic
38	C₅²×C₁₆	C₂²×C₃²	1	1	yes	characteristic
39	C₂×C₃²×C₄×C₅	C₅×C₈	1	6	yes	—
40	C₃²×C₅×C₈	C₅×C₈	1	6	yes	—
41	C₃²×C₅×C₈	C₂×C₄×C₅	1	6	yes	—
42	C₂×C₅×C₃₂	C₃²×C₅	1	6	yes	direct product factor
43	C₂²×C₃×C₅²	C₃×C₁₆	1	4	yes	—
44	C₃×C₄×C₅²	C₃×C₁₆	1	4	yes	—
45	C₃×C₄×C₅²	C₂×C₃×C₈	1	4	yes	—
46	C₂×C₃²×C₁₆	C₂×C₅²	1	1	yes	characteristic
47	C₃²×C₃₂	C₂×C₅²	1	2	yes	direct product factor
48	C₂×C₃×C₅×C₈	C₃×C₄×C₅	1	24	yes	—
49	C₃×C₅×C₁₆	C₃×C₄×C₅	1	24	yes	—
50	C₃×C₅×C₁₆	C₂²×C₃×C₅	1	24	yes	—
51	C₃²×C₅²	C₂×C₃₂	1	1	yes	characteristic, direct product factor, Hall
52	C₂×C₄×C₅²	C₃²×C₈	1	1	yes	characteristic
53	C₅²×C₈	C₃²×C₈	1	1	yes	characteristic
54	C₅²×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₃₂	C₃×C₅²	1	4	yes	direct product factor
56	C₂²×C₃²×C₅	C₅×C₁₆	1	6	yes	—
57	C₃²×C₄×C₅	C₅×C₁₆	1	6	yes	—
58	C₃²×C₄×C₅	C₂×C₅×C₈	1	6	yes	—
59	C₂×C₅×C₁₆	C₂×C₃²×C₅	1	6	yes	—
60	C₅×C₃₂	C₂×C₃²×C₅	1	12	yes	direct product factor
61	C₂×C₃×C₅²	C₃×C₃₂	1	8	yes	direct product factor
62	C₂×C₃×C₅²	C₂×C₃×C₁₆	1	4	yes	—
63	C₂×C₃²×C₈	C₄×C₅²	1	1	yes	characteristic
64	C₃²×C₁₆	C₄×C₅²	1	1	yes	characteristic
65	C₃²×C₁₆	C₂²×C₅²	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₃×C₅×C₈	1	24	yes	—
67	C₃×C₅×C₈	C₃×C₅×C₈	1	24	yes	—
68	C₃×C₅×C₈	C₂×C₃×C₄×C₅	1	24	yes	—
69	C₂²×C₅²	C₃²×C₁₆	1	1	yes	characteristic
70	C₄×C₅²	C₃²×C₁₆	1	1	yes	characteristic
71	C₄×C₅²	C₂×C₃²×C₈	1	1	yes	characteristic
72	C₂×C₃×C₁₆	C₂×C₃×C₅²	1	4	yes	—
73	C₃×C₃₂	C₂×C₃×C₅²	1	8	yes	direct product factor
74	C₂×C₃²×C₅	C₅×C₃₂	1	12	yes	direct product factor
75	C₂×C₃²×C₅	C₂×C₅×C₁₆	1	6	yes	—
76	C₂×C₅×C₈	C₃²×C₄×C₅	1	6	yes	—
77	C₅×C₁₆	C₃²×C₄×C₅	1	6	yes	—
78	C₅×C₁₆	C₂²×C₃²×C₅	1	6	yes	—
79	C₃×C₅²	C₂×C₃×C₃₂	1	4	yes	direct product factor
80	C₂×C₃²×C₄	C₅²×C₈	1	1	yes	characteristic
81	C₃²×C₈	C₅²×C₈	1	1	yes	characteristic
82	C₃²×C₈	C₂×C₄×C₅²	1	1	yes	characteristic
83	C₂×C₃₂	C₃²×C₅²	1	1	yes	characteristic, direct product factor, Sylow
84	C₂²×C₃×C₅	C₃×C₅×C₁₆	1	24	yes	—
85	C₃×C₄×C₅	C₃×C₅×C₁₆	1	24	yes	—
86	C₃×C₄×C₅	C₂×C₃×C₅×C₈	1	24	yes	—
87	C₂×C₅²	C₃²×C₃₂	1	2	yes	direct product factor
88	C₂×C₅²	C₂×C₃²×C₁₆	1	1	yes	characteristic
89	C₂×C₃×C₈	C₃×C₄×C₅²	1	4	yes	—
90	C₃×C₁₆	C₃×C₄×C₅²	1	4	yes	—
91	C₃×C₁₆	C₂²×C₃×C₅²	1	4	yes	—
92	C₃²×C₅	C₂×C₅×C₃₂	1	6	yes	direct product factor
93	C₂×C₄×C₅	C₃²×C₅×C₈	1	6	yes	—
94	C₅×C₈	C₃²×C₅×C₈	1	6	yes	—
95	C₅×C₈	C₂×C₃²×C₄×C₅	1	6	yes	—
96	C₂²×C₃²	C₅²×C₁₆	1	1	yes	characteristic
97	C₃²×C₄	C₅²×C₁₆	1	1	yes	characteristic
98	C₃²×C₄	C₂×C₅²×C₈	1	1	yes	characteristic
99	C₂×C₁₆	C₂×C₃²×C₅²	1	1	yes	characteristic
100	C₃₂	C₂×C₃²×C₅²	1	2	yes	direct product factor
101	C₂×C₃×C₅	C₃×C₅×C₃₂	1	48	yes	direct product factor
102	C₂×C₃×C₅	C₂×C₃×C₅×C₁₆	1	24	yes	—
103	C₅²	C₂×C₃²×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×C₃×C₄	C₃×C₅²×C₈	1	4	yes	—
105	C₃×C₈	C₃×C₅²×C₈	1	4	yes	—
106	C₃×C₈	C₂×C₃×C₄×C₅²	1	4	yes	—
107	C₂²×C₅	C₃²×C₅×C₁₆	1	6	yes	—
108	C₄×C₅	C₃²×C₅×C₁₆	1	6	yes	—
109	C₄×C₅	C₂×C₃²×C₅×C₈	1	6	yes	—
110	C₂×C₃²	C₅²×C₃₂	1	2	yes	direct product factor
111	C₂×C₃²	C₂×C₅²×C₁₆	1	1	yes	characteristic
112	C₂×C₈	C₃²×C₄×C₅²	1	1	yes	characteristic
113	C₁₆	C₃²×C₄×C₅²	1	1	yes	characteristic
114	C₁₆	C₂²×C₃²×C₅²	1	1	yes	characteristic
115	C₃×C₅	C₂×C₃×C₅×C₃₂	1	24	yes	direct product factor
116	C₂²×C₃	C₃×C₅²×C₁₆	1	4	yes	—
117	C₃×C₄	C₃×C₅²×C₁₆	1	4	yes	—
118	C₃×C₄	C₂×C₃×C₅²×C₈	1	4	yes	—
119	C₂×C₅	C₃²×C₅×C₃₂	1	12	yes	direct product factor
120	C₂×C₅	C₂×C₃²×C₅×C₁₆	1	6	yes	—
121	C₃²	C₂×C₅²×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
122	C₂×C₄	C₃²×C₅²×C₈	1	1	yes	characteristic
123	C₈	C₃²×C₅²×C₈	1	1	yes	characteristic
124	C₈	C₂×C₃²×C₄×C₅²	1	1	yes	characteristic
125	C₂×C₃	C₃×C₅²×C₃₂	1	8	yes	direct product factor
126	C₂×C₃	C₂×C₃×C₅²×C₁₆	1	4	yes	—
127	C₅	C₂×C₃²×C₅×C₃₂	1	6	yes	direct product factor
128	C₂²	C₃²×C₅²×C₁₆	1	1	yes	characteristic
129	C₄	C₃²×C₅²×C₁₆	1	1	yes	characteristic
130	C₄	C₂×C₃²×C₅²×C₈	1	1	yes	characteristic
131	C₃	C₂×C₃×C₅²×C₃₂	1	4	yes	direct product factor
132	C₂	C₃²×C₅²×C₃₂	1	2	yes	direct product factor
133	C₂	C₂×C₃²×C₅²×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂²×C₃²×C₅²
socle, prime radical	C₂²×C₃²×C₅²	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₅²×C₃₂	C₂	direct (N × H)
3. C₂×C₃×C₅²×C₃₂	C₃	direct (N × H)
7. C₂×C₃²×C₅×C₃₂	C₅	direct (N × H)
9. C₃×C₅²×C₃₂	C₂×C₃	direct (N × H)
13. C₂×C₅²×C₃₂	C₃²	direct (N × H)
15. C₃²×C₅×C₃₂	C₂×C₅	direct (N × H)
19. C₂×C₃×C₅×C₃₂	C₃×C₅	direct (N × H)
24. C₅²×C₃₂	C₂×C₃²	direct (N × H)
31. C₂×C₃²×C₃₂	C₅²	direct (N × H)
33. C₃×C₅×C₃₂	C₂×C₃×C₅	direct (N × H)
34. C₂×C₃²×C₅²	C₃₂	direct (N × H)
42. C₂×C₅×C₃₂	C₃²×C₅	direct (N × H)
47. C₃²×C₃₂	C₂×C₅²	direct (N × H)
51. C₃²×C₅²	C₂×C₃₂	direct (N × H)
55. C₂×C₃×C₃₂	C₃×C₅²	direct (N × H)
60. C₅×C₃₂	C₂×C₃²×C₅	direct (N × H)
61. C₂×C₃×C₅²	C₃×C₃₂	direct (N × H)
73. C₃×C₃₂	C₂×C₃×C₅²	direct (N × H)
74. C₂×C₃²×C₅	C₅×C₃₂	direct (N × H)
79. C₃×C₅²	C₂×C₃×C₃₂	direct (N × H)
83. C₂×C₃₂	C₃²×C₅²	direct (N × H)
87. C₂×C₅²	C₃²×C₃₂	direct (N × H)
92. C₃²×C₅	C₂×C₅×C₃₂	direct (N × H)
100. C₃₂	C₂×C₃²×C₅²	direct (N × H)
101. C₂×C₃×C₅	C₃×C₅×C₃₂	direct (N × H)
103. C₅²	C₂×C₃²×C₃₂	direct (N × H)
110. C₂×C₃²	C₅²×C₃₂	direct (N × H)
115. C₃×C₅	C₂×C₃×C₅×C₃₂	direct (N × H)
119. C₂×C₅	C₃²×C₅×C₃₂	direct (N × H)
121. C₃²	C₂×C₅²×C₃₂	direct (N × H)
125. C₂×C₃	C₃×C₅²×C₃₂	direct (N × H)
127. C₅	C₂×C₃²×C₅×C₃₂	direct (N × H)
131. C₃	C₂×C₃×C₅²×C₃₂	direct (N × H)
132. C₂	C₃²×C₅²×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
C₂×C₃²×C₅²×C₁₆	2
C₃²×C₅²×C₃₂	2
C₂×C₃×C₅²×C₃₂	3
C₂×C₃²×C₅×C₃₂	5

Subgroups — C2×C32×C52×C32 #72192545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅²×C₃₂ #72192545