Subgroups — C₂×C₄×C₂₅×C₆₄ #72191111

Order 12800.

486 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₂₅×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₅×C₆₄	C₂	1	2	yes	maximal
3	C₄×C₂₅×C₆₄	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆×C₂₅	C₄	1	1	yes	characteristic
5	C₂²×C₂₅×C₃₂	C₄	1	1	yes	characteristic
6	C₂²×C₂₅×C₃₂	C₂²	1	1	yes	characteristic
7	C₄×C₂₅×C₃₂	C₄	1	2	yes	—
8	C₄×C₂₅×C₃₂	C₂²	1	2	yes	—
9	C₂×C₂₅×C₆₄	C₄	1	8	yes	direct product factor
10	C₂×C₂₅×C₆₄	C₂²	1	4	yes	—
11	C₂×C₄×C₅×C₆₄	C₅	1	1	yes	maximal, characteristic
12	C₂²×C₁₆×C₂₅	C₈	1	1	yes	characteristic
13	C₂²×C₁₆×C₂₅	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₅	C₈	1	1	yes	characteristic
15	C₂×C₂₅×C₃₂	C₈	1	4	yes	—
16	C₂×C₂₅×C₃₂	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂₅×C₃₂	C₂×C₄	1	2	yes	—
18	C₂×C₂₅×C₃₂	C₂×C₄	1	4	yes	—
19	C₂×C₂₅×C₃₂	C₂³	1	1	yes	characteristic
20	C₄×C₁₆×C₂₅	C₈	1	2	yes	—
21	C₄×C₁₆×C₂₅	C₂×C₄	1	2	yes	—
22	C₂₅×C₆₄	C₂×C₄	1	8	yes	direct product factor
23	C₂²×C₅×C₆₄	C₂×C₅	1	2	yes	—
24	C₂×C₄×C₅×C₃₂	C₂×C₅	1	1	yes	characteristic
25	C₄×C₅×C₆₄	C₂×C₅	1	4	yes	—
26	C₂²×C₈×C₂₅	C₁₆	1	1	yes	characteristic
27	C₂²×C₈×C₂₅	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄²×C₂₅	C₁₆	1	1	yes	characteristic
29	C₂×C₁₆×C₂₅	C₁₆	1	4	yes	—
30	C₂×C₁₆×C₂₅	C₄²	1	2	yes	—
31	C₂×C₁₆×C₂₅	C₂×C₈	1	1	yes	characteristic
32	C₂×C₁₆×C₂₅	C₂×C₈	1	2	yes	—
33	C₂×C₁₆×C₂₅	C₂×C₈	1	2	yes	—
34	C₂×C₁₆×C₂₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₈×C₂₅	C₁₆	1	2	yes	—
36	C₄×C₈×C₂₅	C₂×C₈	1	2	yes	—
37	C₂₅×C₃₂	C₄²	1	2	yes	—
38	C₂₅×C₃₂	C₂×C₈	1	4	yes	—
39	C₂₅×C₃₂	C₂²×C₄	1	2	yes	—
40	C₂²×C₅×C₃₂	C₄×C₅	1	1	yes	characteristic
41	C₂²×C₅×C₃₂	C₂²×C₅	1	1	yes	characteristic
42	C₂×C₄×C₅×C₁₆	C₄×C₅	1	1	yes	characteristic
43	C₂×C₅×C₆₄	C₄×C₅	1	8	yes	—
44	C₂×C₅×C₆₄	C₂²×C₅	1	4	yes	—
45	C₄×C₅×C₃₂	C₄×C₅	1	2	yes	—
46	C₄×C₅×C₃₂	C₂²×C₅	1	2	yes	—
47	C₂×C₄×C₆₄	C₂₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂²×C₄×C₂₅	C₃₂	1	2	yes	—
49	C₂²×C₄×C₂₅	C₂×C₁₆	1	1	yes	characteristic
50	C₂×C₈×C₂₅	C₃₂	1	4	yes	—
51	C₂×C₈×C₂₅	C₄×C₈	1	2	yes	—
52	C₂×C₈×C₂₅	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₈×C₂₅	C₂×C₁₆	1	2	yes	—
54	C₂×C₈×C₂₅	C₂×C₁₆	1	2	yes	—
55	C₂×C₈×C₂₅	C₂²×C₈	1	1	yes	characteristic
56	C₄²×C₂₅	C₃₂	1	2	yes	—
57	C₄²×C₂₅	C₂×C₁₆	1	2	yes	—
58	C₁₆×C₂₅	C₄×C₈	1	2	yes	—
59	C₁₆×C₂₅	C₂×C₁₆	1	4	yes	—
60	C₁₆×C₂₅	C₂×C₄²	1	1	yes	characteristic
61	C₁₆×C₂₅	C₂²×C₈	1	1	yes	characteristic
62	C₂²×C₅×C₁₆	C₅×C₈	1	1	yes	characteristic
63	C₂²×C₅×C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
64	C₂×C₄×C₅×C₈	C₅×C₈	1	1	yes	characteristic
65	C₂×C₅×C₃₂	C₅×C₈	1	4	yes	—
66	C₂×C₅×C₃₂	C₂×C₄×C₅	1	1	yes	characteristic
67	C₂×C₅×C₃₂	C₂×C₄×C₅	1	2	yes	—
68	C₂×C₅×C₃₂	C₂×C₄×C₅	1	4	yes	—
69	C₂×C₅×C₃₂	C₂³×C₅	1	1	yes	characteristic
70	C₄×C₅×C₁₆	C₅×C₈	1	2	yes	—
71	C₄×C₅×C₁₆	C₂×C₄×C₅	1	2	yes	—
72	C₅×C₆₄	C₂×C₄×C₅	1	8	yes	—
73	C₂²×C₆₄	C₂×C₂₅	1	2	yes	—
74	C₂×C₄×C₃₂	C₂×C₂₅	1	1	yes	characteristic
75	C₄×C₆₄	C₂×C₂₅	1	4	yes	direct product factor
76	C₂³×C₂₅	C₂×C₃₂	1	1	yes	characteristic
77	C₂×C₄×C₂₅	C₆₄	1	8	yes	direct product factor
78	C₂×C₄×C₂₅	C₄×C₁₆	1	2	yes	—
79	C₂×C₄×C₂₅	C₂×C₃₂	1	1	yes	characteristic
80	C₂×C₄×C₂₅	C₂×C₃₂	1	2	yes	—
81	C₂×C₄×C₂₅	C₂×C₃₂	1	4	yes	—
82	C₂×C₄×C₂₅	C₂²×C₁₆	1	1	yes	characteristic
83	C₈×C₂₅	C₄×C₁₆	1	2	yes	—
84	C₈×C₂₅	C₂×C₃₂	1	4	yes	—
85	C₈×C₂₅	C₂×C₄×C₈	1	1	yes	characteristic
86	C₈×C₂₅	C₂²×C₁₆	1	1	yes	characteristic
87	C₂²×C₅×C₈	C₅×C₁₆	1	1	yes	characteristic
88	C₂²×C₅×C₈	C₂×C₅×C₈	1	1	yes	characteristic
89	C₂×C₄²×C₅	C₅×C₁₆	1	1	yes	characteristic
90	C₂×C₅×C₁₆	C₅×C₁₆	1	4	yes	—
91	C₂×C₅×C₁₆	C₄²×C₅	1	2	yes	—
92	C₂×C₅×C₁₆	C₂×C₅×C₈	1	1	yes	characteristic
93	C₂×C₅×C₁₆	C₂×C₅×C₈	1	2	yes	—
94	C₂×C₅×C₁₆	C₂×C₅×C₈	1	2	yes	—
95	C₂×C₅×C₁₆	C₂²×C₄×C₅	1	1	yes	characteristic
96	C₄×C₅×C₈	C₅×C₁₆	1	2	yes	—
97	C₄×C₅×C₈	C₂×C₅×C₈	1	2	yes	—
98	C₅×C₃₂	C₄²×C₅	1	2	yes	—
99	C₅×C₃₂	C₂×C₅×C₈	1	4	yes	—
100	C₅×C₃₂	C₂²×C₄×C₅	1	2	yes	—
101	C₂²×C₃₂	C₄×C₂₅	1	1	yes	characteristic
102	C₂²×C₃₂	C₂²×C₂₅	1	1	yes	characteristic
103	C₂×C₄×C₁₆	C₄×C₂₅	1	1	yes	characteristic
104	C₂×C₆₄	C₄×C₂₅	1	8	yes	direct product factor
105	C₂×C₆₄	C₂²×C₂₅	1	4	yes	—
106	C₄×C₃₂	C₄×C₂₅	1	2	yes	—
107	C₄×C₃₂	C₂²×C₂₅	1	2	yes	—
108	C₂²×C₂₅	C₄×C₃₂	1	2	yes	—
109	C₂²×C₂₅	C₂×C₆₄	1	4	yes	—
110	C₂²×C₂₅	C₂²×C₃₂	1	1	yes	characteristic
111	C₄×C₂₅	C₄×C₃₂	1	2	yes	—
112	C₄×C₂₅	C₂×C₆₄	1	8	yes	direct product factor
113	C₄×C₂₅	C₂×C₄×C₁₆	1	1	yes	characteristic
114	C₄×C₂₅	C₂²×C₃₂	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₅×C₃₂	1	2	yes	—
116	C₂²×C₄×C₅	C₂×C₅×C₁₆	1	1	yes	characteristic
117	C₂×C₅×C₈	C₅×C₃₂	1	4	yes	—
118	C₂×C₅×C₈	C₄×C₅×C₈	1	2	yes	—
119	C₂×C₅×C₈	C₂×C₅×C₁₆	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×C₅×C₁₆	1	2	yes	—
121	C₂×C₅×C₈	C₂×C₅×C₁₆	1	2	yes	—
122	C₂×C₅×C₈	C₂²×C₅×C₈	1	1	yes	characteristic
123	C₄²×C₅	C₅×C₃₂	1	2	yes	—
124	C₄²×C₅	C₂×C₅×C₁₆	1	2	yes	—
125	C₅×C₁₆	C₄×C₅×C₈	1	2	yes	—
126	C₅×C₁₆	C₂×C₅×C₁₆	1	4	yes	—
127	C₅×C₁₆	C₂×C₄²×C₅	1	1	yes	characteristic
128	C₅×C₁₆	C₂²×C₅×C₈	1	1	yes	characteristic
129	C₂²×C₁₆	C₈×C₂₅	1	1	yes	characteristic
130	C₂²×C₁₆	C₂×C₄×C₂₅	1	1	yes	characteristic
131	C₂×C₄×C₈	C₈×C₂₅	1	1	yes	characteristic
132	C₂×C₃₂	C₈×C₂₅	1	4	yes	—
133	C₂×C₃₂	C₂×C₄×C₂₅	1	1	yes	characteristic
134	C₂×C₃₂	C₂×C₄×C₂₅	1	2	yes	—
135	C₂×C₃₂	C₂×C₄×C₂₅	1	4	yes	—
136	C₂×C₃₂	C₂³×C₂₅	1	1	yes	characteristic
137	C₄×C₁₆	C₈×C₂₅	1	2	yes	—
138	C₄×C₁₆	C₂×C₄×C₂₅	1	2	yes	—
139	C₆₄	C₂×C₄×C₂₅	1	8	yes	direct product factor
140	C₂×C₂₅	C₄×C₆₄	1	4	yes	direct product factor
141	C₂×C₂₅	C₂×C₄×C₃₂	1	1	yes	characteristic
142	C₂×C₂₅	C₂²×C₆₄	1	2	yes	—
143	C₂³×C₅	C₂×C₅×C₃₂	1	1	yes	characteristic
144	C₂×C₄×C₅	C₅×C₆₄	1	8	yes	—
145	C₂×C₄×C₅	C₄×C₅×C₁₆	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₅×C₃₂	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₅×C₃₂	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₅×C₃₂	1	4	yes	—
149	C₂×C₄×C₅	C₂²×C₅×C₁₆	1	1	yes	characteristic
150	C₅×C₈	C₄×C₅×C₁₆	1	2	yes	—
151	C₅×C₈	C₂×C₅×C₃₂	1	4	yes	—
152	C₅×C₈	C₂×C₄×C₅×C₈	1	1	yes	characteristic
153	C₅×C₈	C₂²×C₅×C₁₆	1	1	yes	characteristic
154	C₂²×C₈	C₁₆×C₂₅	1	1	yes	characteristic
155	C₂²×C₈	C₂×C₈×C₂₅	1	1	yes	characteristic
156	C₂×C₄²	C₁₆×C₂₅	1	1	yes	characteristic
157	C₂×C₁₆	C₁₆×C₂₅	1	4	yes	—
158	C₂×C₁₆	C₄²×C₂₅	1	2	yes	—
159	C₂×C₁₆	C₂×C₈×C₂₅	1	1	yes	characteristic
160	C₂×C₁₆	C₂×C₈×C₂₅	1	2	yes	—
161	C₂×C₁₆	C₂×C₈×C₂₅	1	2	yes	—
162	C₂×C₁₆	C₂²×C₄×C₂₅	1	1	yes	characteristic
163	C₄×C₈	C₁₆×C₂₅	1	2	yes	—
164	C₄×C₈	C₂×C₈×C₂₅	1	2	yes	—
165	C₃₂	C₄²×C₂₅	1	2	yes	—
166	C₃₂	C₂×C₈×C₂₅	1	4	yes	—
167	C₃₂	C₂²×C₄×C₂₅	1	2	yes	—
168	C₂₅	C₂×C₄×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
169	C₂²×C₅	C₄×C₅×C₃₂	1	2	yes	—
170	C₂²×C₅	C₂×C₅×C₆₄	1	4	yes	—
171	C₂²×C₅	C₂²×C₅×C₃₂	1	1	yes	characteristic
172	C₄×C₅	C₄×C₅×C₃₂	1	2	yes	—
173	C₄×C₅	C₂×C₅×C₆₄	1	8	yes	—
174	C₄×C₅	C₂×C₄×C₅×C₁₆	1	1	yes	characteristic
175	C₄×C₅	C₂²×C₅×C₃₂	1	1	yes	characteristic
176	C₂²×C₄	C₂₅×C₃₂	1	2	yes	—
177	C₂²×C₄	C₂×C₁₆×C₂₅	1	1	yes	characteristic
178	C₂×C₈	C₂₅×C₃₂	1	4	yes	—
179	C₂×C₈	C₄×C₈×C₂₅	1	2	yes	—
180	C₂×C₈	C₂×C₁₆×C₂₅	1	1	yes	characteristic
181	C₂×C₈	C₂×C₁₆×C₂₅	1	2	yes	—
182	C₂×C₈	C₂×C₁₆×C₂₅	1	2	yes	—
183	C₂×C₈	C₂²×C₈×C₂₅	1	1	yes	characteristic
184	C₄²	C₂₅×C₃₂	1	2	yes	—
185	C₄²	C₂×C₁₆×C₂₅	1	2	yes	—
186	C₁₆	C₄×C₈×C₂₅	1	2	yes	—
187	C₁₆	C₂×C₁₆×C₂₅	1	4	yes	—
188	C₁₆	C₂×C₄²×C₂₅	1	1	yes	characteristic
189	C₁₆	C₂²×C₈×C₂₅	1	1	yes	characteristic
190	C₂×C₅	C₄×C₅×C₆₄	1	4	yes	—
191	C₂×C₅	C₂×C₄×C₅×C₃₂	1	1	yes	characteristic
192	C₂×C₅	C₂²×C₅×C₆₄	1	2	yes	—
193	C₂³	C₂×C₂₅×C₃₂	1	1	yes	characteristic
194	C₂×C₄	C₂₅×C₆₄	1	8	yes	direct product factor
195	C₂×C₄	C₄×C₁₆×C₂₅	1	2	yes	—
196	C₂×C₄	C₂×C₂₅×C₃₂	1	1	yes	characteristic
197	C₂×C₄	C₂×C₂₅×C₃₂	1	2	yes	—
198	C₂×C₄	C₂×C₂₅×C₃₂	1	4	yes	—
199	C₂×C₄	C₂²×C₁₆×C₂₅	1	1	yes	characteristic
200	C₈	C₄×C₁₆×C₂₅	1	2	yes	—
201	C₈	C₂×C₂₅×C₃₂	1	4	yes	—
202	C₈	C₂×C₄×C₈×C₂₅	1	1	yes	characteristic
203	C₈	C₂²×C₁₆×C₂₅	1	1	yes	characteristic
204	C₅	C₂×C₄×C₅×C₆₄	1	1	yes	characteristic
205	C₂²	C₂×C₂₅×C₆₄	1	4	yes	—
206	C₂²	C₄×C₂₅×C₃₂	1	2	yes	—
207	C₂²	C₂²×C₂₅×C₃₂	1	1	yes	characteristic
208	C₄	C₂×C₂₅×C₆₄	1	8	yes	direct product factor
209	C₄	C₄×C₂₅×C₃₂	1	2	yes	—
210	C₄	C₂²×C₂₅×C₃₂	1	1	yes	characteristic
211	C₄	C₂×C₄×C₁₆×C₂₅	1	1	yes	characteristic
212	C₂	C₄×C₂₅×C₆₄	1	4	yes	direct product factor
213	C₂	C₂²×C₂₅×C₆₄	1	2	yes	—
214	C₂	C₂×C₄×C₂₅×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₅×C₃₂	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₅×C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₂₅×C₆₄	C₂	direct (N × H)
9. C₂×C₂₅×C₆₄	C₄	direct (N × H)
22. C₂₅×C₆₄	C₂×C₄	direct (N × H)
47. C₂×C₄×C₆₄	C₂₅	direct (N × H)
75. C₄×C₆₄	C₂×C₂₅	direct (N × H)
77. C₂×C₄×C₂₅	C₆₄	direct (N × H)
104. C₂×C₆₄	C₄×C₂₅	direct (N × H)
112. C₄×C₂₅	C₂×C₆₄	direct (N × H)
139. C₆₄	C₂×C₄×C₂₅	direct (N × H)
140. C₂×C₂₅	C₄×C₆₄	direct (N × H)
168. C₂₅	C₂×C₄×C₆₄	direct (N × H)
194. C₂×C₄	C₂₅×C₆₄	direct (N × H)
208. C₄	C₂×C₂₅×C₆₄	direct (N × H)
212. C₂	C₄×C₂₅×C₆₄	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×C₄×C₂₅×C₃₂	2
C₂²×C₂₅×C₆₄	2
C₄×C₂₅×C₆₄	2
C₂×C₄×C₅×C₆₄	5

Subgroups — C2×C4×C25×C64 #72191111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₂₅×C₆₄ #72191111