Subgroups — C₂×C₄²×D₅² #72242709

Order 3200.

6498 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₅²	C₂	1	3	yes	maximal
2	C₂×C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄²×C₅×D₅	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#6036	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×D₅×Dic₅	C₂	1	6	yes	maximal, complemented
6	C₄²×D₅²	C₂	1	16	yes	maximal, direct product factor
7	C₂³×D₅²	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅⋊D₅	C₂²	1	3	yes	—
9	C₂²×C₄×C₅×D₅	C₂²	1	6	yes	—
10	C₂²×#6036	C₂²	1	3	yes	—
11	C₂²×D₅×Dic₅	C₂²	1	6	yes	—
12	C₂×C₄×D₅²	C₄	1	48	yes	direct product factor
13	C₂×C₄×D₅²	C₂²	1	24	yes	—
14	C₂×C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄×#1039	C₂²	1	3	yes	complemented
16	C₄²×C₅⋊D₅	C₂²	1	8	yes	complemented
17	C₂×C₄×C₅×Dic₅	C₂²	1	6	yes	complemented
18	C₄²×C₅×D₅	C₂²	1	16	yes	complemented
19	C₂×Dic₅²	C₂²	1	6	yes	complemented
20	C₄×#6036	C₂²	1	24	yes	complemented
21	C₄×D₅×Dic₅	C₂²	1	48	yes	complemented
22	C₂²×C₄²×D₅	—	5	2	no	maximal
23	C₂³×C₅⋊D₅	C₂³	1	1	yes	characteristic
24	C₂³×C₅×D₅	C₂³	1	2	yes	—
25	C₂²×D₅²	C₂×C₄	1	24	yes	—
26	C₂²×D₅²	C₂³	1	4	yes	—
27	C₂²×C₄×C₅²	C₂³	1	3	yes	—
28	C₂²×#1039	C₂³	1	3	yes	—
29	C₂×C₄×C₅⋊D₅	C₂×C₄	1	24	yes	complemented
30	C₂×C₄×C₅⋊D₅	C₂³	1	12	yes	—
31	C₂²×C₅×Dic₅	C₂³	1	6	yes	—
32	C₂×C₄×C₅×D₅	C₂×C₄	1	48	yes	complemented
33	C₂×C₄×C₅×D₅	C₂³	1	24	yes	—
34	C₂×#6036	C₂×C₄	1	24	yes	complemented
35	C₂×#6036	C₂³	1	12	yes	—
36	C₂×D₅×Dic₅	C₂×C₄	1	48	yes	complemented
37	C₂×D₅×Dic₅	C₂³	1	24	yes	—
38	C₄×D₅²	C₂×C₄	1	192	yes	direct product factor
39	C₄²×C₅²	C₂³	1	4	yes	complemented
40	C₄×#1039	C₂³	1	12	yes	complemented
41	C₄×C₅×Dic₅	C₂³	1	24	yes	complemented
42	Dic₅²	C₂³	1	24	yes	complemented
43	C₂³×C₄×D₅	—	5	6	no	—
44	C₂²×C₄²×C₅	—	5	2	no	—
45	C₂²×C₄×Dic₅	—	5	6	no	—
46	C₂×C₄²×D₅	—	5	6	no	—
47	C₂×C₄²×D₅	—	5	32	no	—
48	C₂×C₄²×D₅	—	5	6	no	—
49	C₂×C₄²×D₅	—	5	2	no	—
50	C₂×C₄²×D₅	—	10	2	no	—
51	C₂×C₄²×D₅	D₅	1	2	yes	direct product factor
52	C₂³×C₅²	C₂⁴	1	1	yes	characteristic
53	C₂²×C₅⋊D₅	C₂²×C₄	1	12	yes	—
54	C₂²×C₅⋊D₅	C₂⁴	1	2	yes	—
55	C₂²×C₅×D₅	C₂²×C₄	1	24	yes	—
56	C₂²×C₅×D₅	C₂⁴	1	4	yes	—
57	C₂×D₅²	C₄²	1	64	yes	direct product factor
58	C₂×D₅²	C₂²×C₄	1	48	yes	—
59	C₂×C₄×C₅²	C₂²×C₄	1	12	yes	complemented
60	C₂×C₄×C₅²	C₂⁴	1	6	yes	—
61	C₂×#1039	C₂²×C₄	1	12	yes	complemented
62	C₂×#1039	C₂⁴	1	6	yes	—
63	C₄×C₅⋊D₅	C₂²×C₄	1	48	yes	complemented
64	C₂×C₅×Dic₅	C₂²×C₄	1	24	yes	complemented
65	C₂×C₅×Dic₅	C₂⁴	1	12	yes	—
66	C₄×C₅×D₅	C₂²×C₄	1	96	yes	complemented
67	#6036	C₂²×C₄	1	48	yes	complemented
68	D₅×Dic₅	C₂²×C₄	1	96	yes	complemented
69	C₂⁴×D₅	—	5	2	no	—
70	C₂³×C₄×C₅	—	5	6	no	—
71	C₂³×Dic₅	—	5	6	no	—
72	C₂²×C₄×D₅	—	5	6	no	—
73	C₂²×C₄×D₅	—	5	6	no	—
74	C₂²×C₄×D₅	—	5	6	no	—
75	C₂²×C₄×D₅	—	5	96	no	—
76	C₂²×C₄×D₅	—	5	48	no	—
77	C₂²×C₄×D₅	—	10	6	no	—
78	C₂²×C₄×D₅	C₂×D₅	1	6	yes	—
79	C₂×C₄²×C₅	—	2	2	no	—
80	C₂×C₄²×C₅	—	5	6	no	—
81	C₂×C₄²×C₅	—	5	16	no	—
82	C₂×C₄²×C₅	C₂×D₅	1	2	yes	complemented
83	C₂×C₄×Dic₅	—	5	6	no	—
84	C₂×C₄×Dic₅	—	5	12	no	—
85	C₂×C₄×Dic₅	—	5	48	no	—
86	C₂×C₄×Dic₅	—	10	6	no	—
87	C₂×C₄×Dic₅	C₂×D₅	1	6	yes	complemented
88	C₄²×D₅	—	5	48	no	—
89	C₄²×D₅	—	5	16	no	—
90	C₄²×D₅	—	5	48	no	—
91	C₄²×D₅	—	10	16	no	—
92	C₄²×D₅	C₂×D₅	1	16	yes	direct product factor
93	C₂³×C₄²	—	25	1	no	Sylow
94	C₂²×C₅²	C₂³×C₄	1	6	yes	—
95	C₂²×C₅²	C₂⁵	1	1	yes	characteristic
96	C₂×C₅⋊D₅	C₂×C₄²	1	16	yes	complemented
97	C₂×C₅⋊D₅	C₂³×C₄	1	12	yes	—
98	C₂×C₅×D₅	C₂×C₄²	1	32	yes	complemented
99	C₂×C₅×D₅	C₂³×C₄	1	24	yes	—
100	D₅²	C₂×C₄²	1	64	yes	direct product factor
101	C₄×C₅²	C₂³×C₄	1	12	yes	complemented
102	#1039	C₂³×C₄	1	12	yes	complemented
103	C₅×Dic₅	C₂³×C₄	1	24	yes	complemented
104	C₂⁴×C₅	—	5	2	no	—
105	C₂³×D₅	—	5	2	no	—
106	C₂³×D₅	—	5	48	no	—
107	C₂³×D₅	—	5	8	no	—
108	C₂³×D₅	—	10	2	no	—
109	C₂³×D₅	C₂²×D₅	1	2	yes	—
110	C₂²×C₄×C₅	—	2	6	no	—
111	C₂²×C₄×C₅	—	5	6	no	—
112	C₂²×C₄×C₅	—	5	48	no	—
113	C₂²×C₄×C₅	—	5	24	no	—
114	C₂²×C₄×C₅	C₂²×D₅	1	6	yes	—
115	C₂²×Dic₅	—	5	6	no	—
116	C₂²×Dic₅	—	5	48	no	—
117	C₂²×Dic₅	—	5	24	no	—
118	C₂²×Dic₅	—	10	6	no	—
119	C₂²×Dic₅	C₂²×D₅	1	6	yes	—
120	C₂×C₄×D₅	—	5	48	no	—
121	C₂×C₄×D₅	—	5	24	no	—
122	C₂×C₄×D₅	—	5	48	no	—
123	C₂×C₄×D₅	—	5	24	no	—
124	C₂×C₄×D₅	—	5	48	no	—
125	C₂×C₄×D₅	—	5	24	no	—
126	C₂×C₄×D₅	—	5	384	no	—
127	C₂×C₄×D₅	—	10	48	no	—
128	C₂×C₄×D₅	—	10	24	no	—
129	C₂×C₄×D₅	C₄×D₅	1	48	yes	direct product factor
130	C₂×C₄×D₅	C₂²×D₅	1	24	yes	—
131	C₄²×C₅	—	2	8	no	—
132	C₄²×C₅	—	5	24	no	—
133	C₄²×C₅	C₂²×D₅	1	8	yes	complemented
134	C₄×Dic₅	—	5	24	no	—
135	C₄×Dic₅	—	5	48	no	—
136	C₄×Dic₅	—	10	24	no	—
137	C₄×Dic₅	C₂²×D₅	1	24	yes	complemented
138	C₂⁴×C₄	—	25	3	no	—
139	C₂²×C₄²	—	5	2	no	—
140	C₂²×C₄²	—	25	3	no	—
141	C₂²×C₄²	—	25	16	no	—
142	C₂²×C₄²	—	25	6	no	—
143	C₂²×C₄²	—	25	1	no	—
144	C₂×C₅²	C₂²×C₄²	1	4	yes	complemented
145	C₂×C₅²	C₂⁴×C₄	1	3	yes	—
146	C₅⋊D₅	C₂²×C₄²	1	8	yes	complemented
147	C₅×D₅	C₂²×C₄²	1	16	yes	complemented
148	C₂³×C₅	—	2	2	no	—
149	C₂³×C₅	—	5	24	no	—
150	C₂³×C₅	—	5	4	no	—
151	C₂³×C₅	C₂³×D₅	1	2	yes	—
152	C₂²×D₅	—	5	24	no	—
153	C₂²×D₅	—	5	4	no	—
154	C₂²×D₅	—	5	96	no	—
155	C₂²×D₅	—	5	128	no	—
156	C₂²×D₅	—	10	24	no	—
157	C₂²×D₅	—	10	4	no	—
158	C₂²×D₅	C₂×C₄×D₅	1	24	yes	—
159	C₂²×D₅	C₂³×D₅	1	4	yes	—
160	C₂×C₄×C₅	—	2	24	no	—
161	C₂×C₄×C₅	—	2	12	no	—
162	C₂×C₄×C₅	—	5	24	no	—
163	C₂×C₄×C₅	—	5	12	no	—
164	C₂×C₄×C₅	—	5	96	no	—
165	C₂×C₄×C₅	C₂×C₄×D₅	1	24	yes	complemented
166	C₂×C₄×C₅	C₂³×D₅	1	12	yes	—
167	C₂×Dic₅	—	5	24	no	—
168	C₂×Dic₅	—	5	12	no	—
169	C₂×Dic₅	—	5	96	no	—
170	C₂×Dic₅	—	10	24	no	—
171	C₂×Dic₅	—	10	12	no	—
172	C₂×Dic₅	C₂×C₄×D₅	1	24	yes	complemented
173	C₂×Dic₅	C₂³×D₅	1	12	yes	—
174	C₄×D₅	—	5	96	no	—
175	C₄×D₅	—	5	96	no	—
176	C₄×D₅	—	5	96	no	—
177	C₄×D₅	—	10	96	no	—
178	C₄×D₅	C₂×C₄×D₅	1	96	yes	direct product factor
179	C₂⁵	—	25	1	no	—
180	C₂³×C₄	—	5	6	no	—
181	C₂³×C₄	—	25	48	no	—
182	C₂³×C₄	—	25	3	no	—
183	C₂³×C₄	—	25	24	no	—
184	C₂³×C₄	—	25	6	no	—
185	C₂³×C₄	—	25	3	no	—
186	C₂×C₄²	—	5	6	no	—
187	C₂×C₄²	—	5	16	no	—
188	C₂×C₄²	—	25	24	no	—
189	C₂×C₄²	—	25	48	no	—
190	C₂×C₄²	—	25	6	no	—
191	C₂×C₄²	—	25	3	no	—
192	C₂×C₄²	—	25	8	no	—
193	C₂×C₄²	D₅²	1	1	yes	characteristic, direct product factor
194	C₅²	C₂³×C₄²	1	1	yes	characteristic, complemented, Sylow
195	C₂²×C₅	—	2	12	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	5	32	no	—
198	C₂²×C₅	—	5	24	no	—
199	C₂²×C₅	C₂²×C₄×D₅	1	12	yes	—
200	C₂²×C₅	C₂⁴×D₅	1	2	yes	—
201	C₂×D₅	—	5	32	no	—
202	C₂×D₅	—	5	24	no	—
203	C₂×D₅	—	5	128	no	—
204	C₂×D₅	—	10	32	no	—
205	C₂×D₅	—	10	24	no	—
206	C₂×D₅	C₄²×D₅	1	32	yes	direct product factor
207	C₂×D₅	C₂²×C₄×D₅	1	24	yes	—
208	C₄×C₅	—	2	24	no	—
209	C₄×C₅	—	5	24	no	—
210	C₄×C₅	C₂²×C₄×D₅	1	24	yes	complemented
211	Dic₅	—	5	24	no	—
212	Dic₅	—	10	24	no	—
213	Dic₅	C₂²×C₄×D₅	1	24	yes	complemented
214	C₂⁴	—	5	2	no	—
215	C₂⁴	—	25	4	no	—
216	C₂⁴	—	25	24	no	—
217	C₂⁴	—	25	1	no	—
218	C₂²×C₄	—	5	6	no	—
219	C₂²×C₄	—	5	48	no	—
220	C₂²×C₄	—	5	24	no	—
221	C₂²×C₄	—	25	12	no	—
222	C₂²×C₄	—	25	24	no	—
223	C₂²×C₄	—	25	192	no	—
224	C₂²×C₄	—	25	24	no	—
225	C₂²×C₄	—	25	48	no	—
226	C₂²×C₄	—	25	3	no	—
227	C₂²×C₄	—	25	24	no	—
228	C₂²×C₄	—	25	12	no	—
229	C₂²×C₄	C₂×D₅²	1	3	yes	—
230	C₄²	—	5	24	no	—
231	C₄²	—	25	24	no	—
232	C₄²	—	25	12	no	—
233	C₄²	C₂×D₅²	1	4	yes	direct product factor
234	C₂×C₅	—	2	6	no	—
235	C₂×C₅	—	2	8	no	—
236	C₂×C₅	—	5	16	no	—
237	C₂×C₅	C₂×C₄²×D₅	1	8	yes	complemented
238	C₂×C₅	C₂³×C₄×D₅	1	6	yes	—
239	D₅	—	5	16	no	—
240	D₅	—	10	16	no	—
241	D₅	C₂×C₄²×D₅	1	16	yes	direct product factor
242	C₂³	—	5	24	no	—
243	C₂³	—	5	4	no	—
244	C₂³	—	25	48	no	—
245	C₂³	—	25	64	no	—
246	C₂³	—	25	12	no	—
247	C₂³	—	25	2	no	—
248	C₂³	C₂²×D₅²	1	1	yes	characteristic
249	C₂×C₄	—	5	24	no	—
250	C₂×C₄	—	5	12	no	—
251	C₂×C₄	—	5	96	no	—
252	C₂×C₄	—	25	48	no	—
253	C₂×C₄	—	25	96	no	—
254	C₂×C₄	—	25	12	no	—
255	C₂×C₄	—	25	6	no	—
256	C₂×C₄	—	25	48	no	—
257	C₂×C₄	C₄×D₅²	1	12	yes	direct product factor
258	C₂×C₄	C₂²×D₅²	1	6	yes	—
259	C₅	—	2	2	no	—
260	C₅	C₂²×C₄²×D₅	1	2	yes	complemented
261	C₂²	—	5	24	no	—
262	C₂²	—	5	32	no	—
263	C₂²	—	25	64	no	—
264	C₂²	—	25	12	no	—
265	C₂²	—	25	16	no	—
266	C₂²	C₂×C₄×D₅²	1	6	yes	—
267	C₂²	C₂³×D₅²	1	1	yes	characteristic
268	C₄	—	5	24	no	—
269	C₄	—	25	12	no	—
270	C₄	C₂×C₄×D₅²	1	12	yes	direct product factor
271	C₂	—	5	16	no	—
272	C₂	—	25	8	no	—
273	C₂	C₄²×D₅²	1	4	yes	direct product factor
274	C₂	C₂²×C₄×D₅²	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₅²
commutator, hypocenter	C₅²	C₂³×C₄²
Frattini	C₂²	C₂³×D₅²
Fitting	C₂×C₄²×C₅²	C₂²
socle	C₂³×C₅²	C₂⁴
prime radical	C₂³×D₅²	C₂²
cosocle	C₂²×C₅²	C₂⁵
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄²×C₅⋊D₅	C₂	semidirect
3. C₂×C₄²×C₅×D₅	C₂	semidirect
4. C₂×C₄×#6036	C₂	semidirect
5. C₂×C₄×D₅×Dic₅	C₂	semidirect
6. C₄²×D₅²	C₂	direct (N × H)
12. C₂×C₄×D₅²	C₄	direct (N × H)
14. C₂×C₄²×C₅²	C₂²	semidirect
15. C₂×C₄×#1039	C₂²	semidirect
16. C₄²×C₅⋊D₅	C₂²	semidirect
17. C₂×C₄×C₅×Dic₅	C₂²	semidirect
18. C₄²×C₅×D₅	C₂²	semidirect
19. C₂×Dic₅²	C₂²	semidirect
20. C₄×#6036	C₂²	semidirect
21. C₄×D₅×Dic₅	C₂²	semidirect
29. C₂×C₄×C₅⋊D₅	C₂×C₄	semidirect
32. C₂×C₄×C₅×D₅	C₂×C₄	semidirect
34. C₂×#6036	C₂×C₄	semidirect
36. C₂×D₅×Dic₅	C₂×C₄	semidirect
38. C₄×D₅²	C₂×C₄	direct (N × H)
39. C₄²×C₅²	C₂³	semidirect
40. C₄×#1039	C₂³	semidirect
41. C₄×C₅×Dic₅	C₂³	semidirect
42. Dic₅²	C₂³	semidirect
51. C₂×C₄²×D₅	D₅	direct (N × H)
57. C₂×D₅²	C₄²	direct (N × H)
59. C₂×C₄×C₅²	C₂²×C₄	semidirect
61. C₂×#1039	C₂²×C₄	semidirect
63. C₄×C₅⋊D₅	C₂²×C₄	semidirect
64. C₂×C₅×Dic₅	C₂²×C₄	semidirect
66. C₄×C₅×D₅	C₂²×C₄	semidirect
67. #6036	C₂²×C₄	semidirect
68. D₅×Dic₅	C₂²×C₄	semidirect
82. C₂×C₄²×C₅	C₂×D₅	semidirect
87. C₂×C₄×Dic₅	C₂×D₅	semidirect
92. C₄²×D₅	C₂×D₅	direct (N × H)
96. C₂×C₅⋊D₅	C₂×C₄²	semidirect
98. C₂×C₅×D₅	C₂×C₄²	semidirect
100. D₅²	C₂×C₄²	direct (N × H)
101. C₄×C₅²	C₂³×C₄	semidirect
102. #1039	C₂³×C₄	semidirect
103. C₅×Dic₅	C₂³×C₄	semidirect
129. C₂×C₄×D₅	C₄×D₅	direct (N × H)
133. C₄²×C₅	C₂²×D₅	semidirect
137. C₄×Dic₅	C₂²×D₅	semidirect
144. C₂×C₅²	C₂²×C₄²	semidirect
146. C₅⋊D₅	C₂²×C₄²	semidirect
147. C₅×D₅	C₂²×C₄²	semidirect
165. C₂×C₄×C₅	C₂×C₄×D₅	semidirect
172. C₂×Dic₅	C₂×C₄×D₅	semidirect
178. C₄×D₅	C₂×C₄×D₅	direct (N × H)
193. C₂×C₄²	D₅²	direct (N × H)
194. C₅²	C₂³×C₄²	semidirect
206. C₂×D₅	C₄²×D₅	direct (N × H)
210. C₄×C₅	C₂²×C₄×D₅	semidirect
213. Dic₅	C₂²×C₄×D₅	semidirect
233. C₄²	C₂×D₅²	direct (N × H)
237. C₂×C₅	C₂×C₄²×D₅	semidirect
241. D₅	C₂×C₄²×D₅	direct (N × H)
257. C₂×C₄	C₄×D₅²	direct (N × H)
260. C₅	C₂²×C₄²×D₅	semidirect
270. C₄	C₂×C₄×D₅²	direct (N × H)
273. C₂	C₄²×D₅²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₅²	2
C₂×C₄²×C₅⋊D₅	2
C₂×C₄²×C₅×D₅	2
C₂×C₄×#6036	2
C₂×C₄×D₅×Dic₅	2
C₄²×D₅²	2
C₂²×C₄²×D₅	5

Subgroups — C2×C42×D52 #72242709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×D₅² #72242709