Subgroups — C₂⁵×C₅×C₁₂₈ #72198047

Order 20480.

35062 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₅×C₆₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₅×C₁₂₈	C₂	1	62	yes	maximal, direct product factor
3	C₂⁵×C₅×C₃₂	C₄	1	1	yes	characteristic
4	C₂⁴×C₅×C₆₄	C₄	1	31	yes	—
5	C₂⁴×C₅×C₆₄	C₂²	1	31	yes	—
6	C₂³×C₅×C₁₂₈	C₂²	1	620	yes	direct product factor
7	C₂⁵×C₁₂₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂⁵×C₅×C₁₆	C₈	1	1	yes	characteristic
9	C₂⁴×C₅×C₃₂	C₈	1	31	yes	—
10	C₂⁴×C₅×C₃₂	C₂×C₄	1	31	yes	—
11	C₂³×C₅×C₆₄	C₂×C₄	1	465	yes	—
12	C₂³×C₅×C₆₄	C₂³	1	155	yes	—
13	C₂²×C₅×C₁₂₈	C₂³	1	1240	yes	direct product factor
14	C₂⁵×C₆₄	C₂×C₅	1	1	yes	characteristic
15	C₂⁴×C₁₂₈	C₂×C₅	1	62	yes	direct product factor
16	C₂⁵×C₃₂	C₄×C₅	1	1	yes	characteristic
17	C₂⁴×C₆₄	C₄×C₅	1	31	yes	—
18	C₂⁴×C₆₄	C₂²×C₅	1	31	yes	—
19	C₂³×C₁₂₈	C₂²×C₅	1	620	yes	direct product factor
20	C₂⁵×C₅×C₈	C₁₆	1	1	yes	characteristic
21	C₂⁴×C₅×C₁₆	C₁₆	1	31	yes	—
22	C₂⁴×C₅×C₁₆	C₂×C₈	1	31	yes	—
23	C₂³×C₅×C₃₂	C₂×C₈	1	465	yes	—
24	C₂³×C₅×C₃₂	C₂²×C₄	1	155	yes	—
25	C₂²×C₅×C₆₄	C₂²×C₄	1	1085	yes	—
26	C₂²×C₅×C₆₄	C₂⁴	1	155	yes	—
27	C₂×C₅×C₁₂₈	C₂⁴	1	496	yes	direct product factor
28	C₂⁵×C₄×C₅	C₃₂	1	1	yes	characteristic
29	C₂⁴×C₅×C₈	C₃₂	1	31	yes	—
30	C₂⁴×C₅×C₈	C₂×C₁₆	1	31	yes	—
31	C₂³×C₅×C₁₆	C₂×C₁₆	1	465	yes	—
32	C₂³×C₅×C₁₆	C₂²×C₈	1	155	yes	—
33	C₂²×C₅×C₃₂	C₂²×C₈	1	1085	yes	—
34	C₂²×C₅×C₃₂	C₂³×C₄	1	155	yes	—
35	C₂×C₅×C₆₄	C₂³×C₄	1	465	yes	—
36	C₂×C₅×C₆₄	C₂⁵	1	31	yes	—
37	C₅×C₁₂₈	C₂⁵	1	32	yes	direct product factor
38	C₂⁵×C₁₆	C₅×C₈	1	1	yes	characteristic
39	C₂⁴×C₃₂	C₅×C₈	1	31	yes	—
40	C₂⁴×C₃₂	C₂×C₄×C₅	1	31	yes	—
41	C₂³×C₆₄	C₂×C₄×C₅	1	465	yes	—
42	C₂³×C₆₄	C₂³×C₅	1	155	yes	—
43	C₂²×C₁₂₈	C₂³×C₅	1	1240	yes	direct product factor
44	C₂⁶×C₅	C₆₄	1	1	yes	characteristic
45	C₂⁴×C₄×C₅	C₆₄	1	31	yes	—
46	C₂⁴×C₄×C₅	C₂×C₃₂	1	31	yes	—
47	C₂³×C₅×C₈	C₂×C₃₂	1	465	yes	—
48	C₂³×C₅×C₈	C₂²×C₁₆	1	155	yes	—
49	C₂²×C₅×C₁₆	C₂²×C₁₆	1	1085	yes	—
50	C₂²×C₅×C₁₆	C₂³×C₈	1	155	yes	—
51	C₂×C₅×C₃₂	C₂³×C₈	1	465	yes	—
52	C₂×C₅×C₃₂	C₂⁴×C₄	1	31	yes	—
53	C₅×C₆₄	C₂⁴×C₄	1	31	yes	—
54	C₅×C₆₄	C₂⁶	1	1	yes	characteristic
55	C₂⁵×C₈	C₅×C₁₆	1	1	yes	characteristic
56	C₂⁴×C₁₆	C₅×C₁₆	1	31	yes	—
57	C₂⁴×C₁₆	C₂×C₅×C₈	1	31	yes	—
58	C₂³×C₃₂	C₂×C₅×C₈	1	465	yes	—
59	C₂³×C₃₂	C₂²×C₄×C₅	1	155	yes	—
60	C₂²×C₆₄	C₂²×C₄×C₅	1	1085	yes	—
61	C₂²×C₆₄	C₂⁴×C₅	1	155	yes	—
62	C₂×C₁₂₈	C₂⁴×C₅	1	496	yes	direct product factor
63	C₂⁵×C₅	C₁₂₈	1	32	yes	direct product factor
64	C₂⁵×C₅	C₂×C₆₄	1	31	yes	—
65	C₂³×C₄×C₅	C₂×C₆₄	1	465	yes	—
66	C₂³×C₄×C₅	C₂²×C₃₂	1	155	yes	—
67	C₂²×C₅×C₈	C₂²×C₃₂	1	1085	yes	—
68	C₂²×C₅×C₈	C₂³×C₁₆	1	155	yes	—
69	C₂×C₅×C₁₆	C₂³×C₁₆	1	465	yes	—
70	C₂×C₅×C₁₆	C₂⁴×C₈	1	31	yes	—
71	C₅×C₃₂	C₂⁴×C₈	1	31	yes	—
72	C₅×C₃₂	C₂⁵×C₄	1	1	yes	characteristic
73	C₂⁵×C₄	C₅×C₃₂	1	1	yes	characteristic
74	C₂⁴×C₈	C₅×C₃₂	1	31	yes	—
75	C₂⁴×C₈	C₂×C₅×C₁₆	1	31	yes	—
76	C₂³×C₁₆	C₂×C₅×C₁₆	1	465	yes	—
77	C₂³×C₁₆	C₂²×C₅×C₈	1	155	yes	—
78	C₂²×C₃₂	C₂²×C₅×C₈	1	1085	yes	—
79	C₂²×C₃₂	C₂³×C₄×C₅	1	155	yes	—
80	C₂×C₆₄	C₂³×C₄×C₅	1	465	yes	—
81	C₂×C₆₄	C₂⁵×C₅	1	31	yes	—
82	C₁₂₈	C₂⁵×C₅	1	32	yes	direct product factor
83	C₂⁴×C₅	C₂×C₁₂₈	1	496	yes	direct product factor
84	C₂⁴×C₅	C₂²×C₆₄	1	155	yes	—
85	C₂²×C₄×C₅	C₂²×C₆₄	1	1085	yes	—
86	C₂²×C₄×C₅	C₂³×C₃₂	1	155	yes	—
87	C₂×C₅×C₈	C₂³×C₃₂	1	465	yes	—
88	C₂×C₅×C₈	C₂⁴×C₁₆	1	31	yes	—
89	C₅×C₁₆	C₂⁴×C₁₆	1	31	yes	—
90	C₅×C₁₆	C₂⁵×C₈	1	1	yes	characteristic
91	C₂⁶	C₅×C₆₄	1	1	yes	characteristic
92	C₂⁴×C₄	C₅×C₆₄	1	31	yes	—
93	C₂⁴×C₄	C₂×C₅×C₃₂	1	31	yes	—
94	C₂³×C₈	C₂×C₅×C₃₂	1	465	yes	—
95	C₂³×C₈	C₂²×C₅×C₁₆	1	155	yes	—
96	C₂²×C₁₆	C₂²×C₅×C₁₆	1	1085	yes	—
97	C₂²×C₁₆	C₂³×C₅×C₈	1	155	yes	—
98	C₂×C₃₂	C₂³×C₅×C₈	1	465	yes	—
99	C₂×C₃₂	C₂⁴×C₄×C₅	1	31	yes	—
100	C₆₄	C₂⁴×C₄×C₅	1	31	yes	—
101	C₆₄	C₂⁶×C₅	1	1	yes	characteristic
102	C₂³×C₅	C₂²×C₁₂₈	1	1240	yes	direct product factor
103	C₂³×C₅	C₂³×C₆₄	1	155	yes	—
104	C₂×C₄×C₅	C₂³×C₆₄	1	465	yes	—
105	C₂×C₄×C₅	C₂⁴×C₃₂	1	31	yes	—
106	C₅×C₈	C₂⁴×C₃₂	1	31	yes	—
107	C₅×C₈	C₂⁵×C₁₆	1	1	yes	characteristic
108	C₂⁵	C₅×C₁₂₈	1	32	yes	direct product factor
109	C₂⁵	C₂×C₅×C₆₄	1	31	yes	—
110	C₂³×C₄	C₂×C₅×C₆₄	1	465	yes	—
111	C₂³×C₄	C₂²×C₅×C₃₂	1	155	yes	—
112	C₂²×C₈	C₂²×C₅×C₃₂	1	1085	yes	—
113	C₂²×C₈	C₂³×C₅×C₁₆	1	155	yes	—
114	C₂×C₁₆	C₂³×C₅×C₁₆	1	465	yes	—
115	C₂×C₁₆	C₂⁴×C₅×C₈	1	31	yes	—
116	C₃₂	C₂⁴×C₅×C₈	1	31	yes	—
117	C₃₂	C₂⁵×C₄×C₅	1	1	yes	characteristic
118	C₂²×C₅	C₂³×C₁₂₈	1	620	yes	direct product factor
119	C₂²×C₅	C₂⁴×C₆₄	1	31	yes	—
120	C₄×C₅	C₂⁴×C₆₄	1	31	yes	—
121	C₄×C₅	C₂⁵×C₃₂	1	1	yes	characteristic
122	C₂⁴	C₂×C₅×C₁₂₈	1	496	yes	direct product factor
123	C₂⁴	C₂²×C₅×C₆₄	1	155	yes	—
124	C₂²×C₄	C₂²×C₅×C₆₄	1	1085	yes	—
125	C₂²×C₄	C₂³×C₅×C₃₂	1	155	yes	—
126	C₂×C₈	C₂³×C₅×C₃₂	1	465	yes	—
127	C₂×C₈	C₂⁴×C₅×C₁₆	1	31	yes	—
128	C₁₆	C₂⁴×C₅×C₁₆	1	31	yes	—
129	C₁₆	C₂⁵×C₅×C₈	1	1	yes	characteristic
130	C₂×C₅	C₂⁴×C₁₂₈	1	62	yes	direct product factor
131	C₂×C₅	C₂⁵×C₆₄	1	1	yes	characteristic
132	C₂³	C₂²×C₅×C₁₂₈	1	1240	yes	direct product factor
133	C₂³	C₂³×C₅×C₆₄	1	155	yes	—
134	C₂×C₄	C₂³×C₅×C₆₄	1	465	yes	—
135	C₂×C₄	C₂⁴×C₅×C₃₂	1	31	yes	—
136	C₈	C₂⁴×C₅×C₃₂	1	31	yes	—
137	C₈	C₂⁵×C₅×C₁₆	1	1	yes	characteristic
138	C₅	C₂⁵×C₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
139	C₂²	C₂³×C₅×C₁₂₈	1	620	yes	direct product factor
140	C₂²	C₂⁴×C₅×C₆₄	1	31	yes	—
141	C₄	C₂⁴×C₅×C₆₄	1	31	yes	—
142	C₄	C₂⁵×C₅×C₃₂	1	1	yes	characteristic
143	C₂	C₂⁴×C₅×C₁₂₈	1	62	yes	direct product factor
144	C₂	C₂⁵×C₅×C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₆₄	C₂⁶×C₅
socle, prime radical	C₂⁶×C₅	C₆₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₅×C₁₂₈	C₂	direct (N × H)
6. C₂³×C₅×C₁₂₈	C₂²	direct (N × H)
7. C₂⁵×C₁₂₈	C₅	direct (N × H)
13. C₂²×C₅×C₁₂₈	C₂³	direct (N × H)
15. C₂⁴×C₁₂₈	C₂×C₅	direct (N × H)
19. C₂³×C₁₂₈	C₂²×C₅	direct (N × H)
27. C₂×C₅×C₁₂₈	C₂⁴	direct (N × H)
37. C₅×C₁₂₈	C₂⁵	direct (N × H)
43. C₂²×C₁₂₈	C₂³×C₅	direct (N × H)
62. C₂×C₁₂₈	C₂⁴×C₅	direct (N × H)
63. C₂⁵×C₅	C₁₂₈	direct (N × H)
82. C₁₂₈	C₂⁵×C₅	direct (N × H)
83. C₂⁴×C₅	C₂×C₁₂₈	direct (N × H)
102. C₂³×C₅	C₂²×C₁₂₈	direct (N × H)
108. C₂⁵	C₅×C₁₂₈	direct (N × H)
118. C₂²×C₅	C₂³×C₁₂₈	direct (N × H)
122. C₂⁴	C₂×C₅×C₁₂₈	direct (N × H)
130. C₂×C₅	C₂⁴×C₁₂₈	direct (N × H)
132. C₂³	C₂²×C₅×C₁₂₈	direct (N × H)
138. C₅	C₂⁵×C₁₂₈	direct (N × H)
139. C₂²	C₂³×C₅×C₁₂₈	direct (N × H)
143. C₂	C₂⁴×C₅×C₁₂₈	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×C₅×C₆₄	2
C₂⁴×C₅×C₁₂₈	2
C₂⁵×C₁₂₈	5

Subgroups — C25×C5×C128 #72198047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₅×C₁₂₈ #72198047