Subgroups — C₂×C₅×D₅×C₂²⋊C₄ #13106639

Order 1600.

1480 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₅×D₅	C₂	1	2	yes	maximal, complemented
3	C₂×C₅²×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#685	C₂	1	2	yes	maximal, complemented
6	C₅×D₅×C₂²⋊C₄	C₂	1	8	yes	maximal, direct product factor
7	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
8	C₂³×C₅×D₅	—	2	8	no	—
9	C₂³×C₅×D₅	C₄	1	8	yes	complemented
10	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₂³×C₅×D₅	C₂²	1	4	yes	—
12	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅²	C₂²	1	2	yes	complemented
14	C₂²×C₅×Dic₅	C₂²	1	2	yes	complemented
15	C₂×C₄×C₅×D₅	—	2	8	no	—
16	C₂×C₄×C₅×D₅	C₂²	1	8	yes	complemented
17	C₅²×C₂²⋊C₄	C₂²	1	4	yes	complemented
18	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
19	C₅×#685	C₂²	1	8	yes	complemented
20	C₂²×C₅×C₂²⋊C₄	—	5	1	no	maximal
21	C₂×D₅×C₂²⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor
22	C₂³×C₅²	—	2	4	no	—
23	C₂³×C₅²	C₂×C₄	1	4	yes	complemented
24	C₂³×C₅²	C₂³	1	2	yes	—
25	C₂³×C₅²	C₂³	1	1	yes	characteristic
26	C₂²×C₅×D₅	—	2	4	no	—
27	C₂²×C₅×D₅	—	2	32	no	—
28	C₂²×C₅×D₅	—	2	16	no	—
29	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
30	C₂²×C₅×D₅	C₂×C₄	1	16	yes	complemented
31	C₂²×C₅×D₅	C₂×C₄	1	4	yes	complemented
32	C₂²×C₅×D₅	D₄	1	8	yes	—
33	C₂²×C₅×D₅	C₂³	1	2	yes	—
34	C₂²×C₅×D₅	C₂³	1	2	yes	—
35	C₂×C₄×C₅²	—	2	4	no	—
36	C₂×C₄×C₅²	C₂³	1	4	yes	complemented
37	C₂×C₅×Dic₅	—	2	4	no	—
38	C₂×C₅×Dic₅	C₂³	1	4	yes	complemented
39	C₄×C₅×D₅	—	2	16	no	—
40	C₂⁵×C₅	—	5	1	no	—
41	C₂⁴×D₅	C₂×C₅	1	1	yes	characteristic
42	C₂³×C₄×C₅	—	5	2	no	—
43	C₂²×C₄×D₅	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
46	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
47	C₂×C₅×C₂²⋊C₄	—	5	8	no	—
48	C₂×C₅×C₂²⋊C₄	D₅	1	1	yes	characteristic, direct product factor
49	C₂×C₅×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₂²⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×#685	C₂×C₅	1	2	yes	complemented
52	D₅×C₂²⋊C₄	C₂×C₅	1	8	yes	direct product factor
53	C₂²×C₅²	—	2	8	no	—
54	C₂²×C₅²	—	2	4	no	—
55	C₂²×C₅²	C₂²×C₄	1	2	yes	—
56	C₂²×C₅²	C₂²×C₄	1	4	yes	complemented
57	C₂²×C₅²	C₂×D₄	1	4	yes	—
58	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
59	C₂×C₅×D₅	—	2	32	no	—
60	C₂×C₅×D₅	—	2	8	no	—
61	C₂×C₅×D₅	—	2	4	no	—
62	C₂×C₅×D₅	C₂²⋊C₄	1	16	yes	direct product factor
63	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
64	C₂×C₅×D₅	C₂²×C₄	1	4	yes	complemented
65	C₂×C₅×D₅	C₂×D₄	1	8	yes	—
66	C₄×C₅²	—	2	4	no	—
67	C₅×Dic₅	—	2	4	no	—
68	C₂⁴×C₅	—	2	2	no	—
69	C₂⁴×C₅	—	5	1	no	—
70	C₂⁴×C₅	—	5	8	no	—
71	C₂⁴×C₅	—	5	4	no	—
72	C₂⁴×C₅	—	5	1	no	—
73	C₂⁴×C₅	—	10	8	no	—
74	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
75	C₂⁴×C₅	C₂²×C₅	1	1	yes	characteristic
76	C₂³×D₅	—	2	8	no	—
77	C₂³×D₅	C₄×C₅	1	8	yes	complemented
78	C₂³×D₅	C₂²×C₅	1	4	yes	—
79	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
80	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	—	5	2	no	—
83	C₂²×C₄×C₅	—	5	8	no	—
84	C₂²×C₄×C₅	—	10	8	no	—
85	C₂²×C₄×C₅	C₂×D₅	1	2	yes	complemented
86	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	complemented
87	C₂²×Dic₅	C₂²×C₅	1	2	yes	complemented
88	C₂×C₄×D₅	—	2	8	no	—
89	C₂×C₄×D₅	C₂²×C₅	1	8	yes	complemented
90	C₅×C₂²⋊C₄	—	2	8	no	—
91	C₅×C₂²⋊C₄	—	5	4	no	—
92	C₅×C₂²⋊C₄	—	5	8	no	—
93	C₅×C₂²⋊C₄	C₂×D₅	1	4	yes	direct product factor
94	C₅×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
95	C₂²⋊Dic₅	C₂²×C₅	1	4	yes	complemented
96	#685	C₂²×C₅	1	8	yes	complemented
97	C₂²×C₂²⋊C₄	—	5	1	no	Sylow
98	C₂×C₅²	—	2	4	no	—
99	C₂×C₅²	C₂×C₂²⋊C₄	1	4	yes	complemented
100	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₅²	C₂²×D₄	1	2	yes	—
102	C₅×D₅	—	2	4	no	—
103	C₅×D₅	C₂×C₂²⋊C₄	1	8	yes	direct product factor
104	C₂³×C₅	—	2	4	no	—
105	C₂³×C₅	—	2	4	no	—
106	C₂³×C₅	—	2	8	no	—
107	C₂³×C₅	—	2	2	no	—
108	C₂³×C₅	—	2	4	no	—
109	C₂³×C₅	—	4	8	no	—
110	C₂³×C₅	—	5	4	no	—
111	C₂³×C₅	—	5	8	no	—
112	C₂³×C₅	—	5	2	no	—
113	C₂³×C₅	—	5	4	no	—
114	C₂³×C₅	—	5	16	no	—
115	C₂³×C₅	—	5	2	no	—
116	C₂³×C₅	—	10	16	no	—
117	C₂³×C₅	—	10	32	no	—
118	C₂³×C₅	—	10	4	no	—
119	C₂³×C₅	C₄×D₅	1	4	yes	complemented
120	C₂³×C₅	C₂×C₄×C₅	1	4	yes	complemented
121	C₂³×C₅	C₂²×D₅	1	2	yes	—
122	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
123	C₂³×C₅	C₂³×C₅	1	2	yes	—
124	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
125	C₂²×D₅	—	2	4	no	—
126	C₂²×D₅	—	2	32	no	—
127	C₂²×D₅	—	2	16	no	—
128	C₂²×D₅	C₂×C₄×C₅	1	4	yes	—
129	C₂²×D₅	C₂×C₄×C₅	1	16	yes	complemented
130	C₂²×D₅	C₂×C₄×C₅	1	4	yes	complemented
131	C₂²×D₅	C₅×D₄	1	8	yes	—
132	C₂²×D₅	C₂³×C₅	1	2	yes	—
133	C₂²×D₅	C₂³×C₅	1	2	yes	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	8	no	—
137	C₂×C₄×C₅	—	4	8	no	—
138	C₂×C₄×C₅	—	5	4	no	—
139	C₂×C₄×C₅	—	10	16	no	—
140	C₂×C₄×C₅	—	10	4	no	—
141	C₂×C₄×C₅	C₂²×D₅	1	4	yes	complemented
142	C₂×C₄×C₅	C₂³×C₅	1	4	yes	complemented
143	C₂×Dic₅	—	2	4	no	—
144	C₂×Dic₅	C₂³×C₅	1	4	yes	complemented
145	C₄×D₅	—	2	16	no	—
146	C₂⁵	—	5	1	no	—
147	C₂³×C₄	—	5	2	no	—
148	C₂×C₂²⋊C₄	—	5	1	no	—
149	C₂×C₂²⋊C₄	—	5	2	no	—
150	C₂×C₂²⋊C₄	—	5	8	no	—
151	C₂×C₂²⋊C₄	C₅×D₅	1	1	yes	characteristic, direct product factor
152	C₅²	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
153	C₂²×C₅	—	2	8	no	—
154	C₂²×C₅	—	2	4	no	—
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	2	8	no	—
157	C₂²×C₅	—	2	4	no	—
158	C₂²×C₅	—	2	8	no	—
159	C₂²×C₅	—	2	8	no	—
160	C₂²×C₅	—	2	2	no	—
161	C₂²×C₅	—	4	8	no	—
162	C₂²×C₅	—	4	16	no	—
163	C₂²×C₅	—	5	4	no	—
164	C₂²×C₅	—	5	8	no	—
165	C₂²×C₅	—	5	16	no	—
166	C₂²×C₅	—	5	4	no	—
167	C₂²×C₅	—	10	32	no	—
168	C₂²×C₅	—	10	4	no	—
169	C₂²×C₅	—	10	8	no	—
170	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
171	C₂²×C₅	C₂×C₄×D₅	1	4	yes	complemented
172	C₂²×C₅	D₄×D₅	1	4	yes	—
173	C₂²×C₅	C₂²×C₄×C₅	1	2	yes	—
174	C₂²×C₅	C₂²×C₄×C₅	1	4	yes	complemented
175	C₂²×C₅	C₂×C₅×D₄	1	4	yes	—
176	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
177	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
178	C₂×D₅	—	2	8	no	—
179	C₂×D₅	—	2	4	no	—
180	C₂×D₅	—	2	32	no	—
181	C₂×D₅	C₅×C₂²⋊C₄	1	16	yes	direct product factor
182	C₂×D₅	C₂²×C₄×C₅	1	4	yes	—
183	C₂×D₅	C₂²×C₄×C₅	1	4	yes	complemented
184	C₂×D₅	C₂×C₅×D₄	1	8	yes	—
185	C₄×C₅	—	2	4	no	—
186	C₄×C₅	—	2	4	no	—
187	C₄×C₅	—	4	8	no	—
188	C₄×C₅	—	10	4	no	—
189	Dic₅	—	2	4	no	—
190	C₂⁴	—	5	1	no	—
191	C₂⁴	—	5	8	no	—
192	C₂⁴	—	5	4	no	—
193	C₂⁴	—	5	1	no	—
194	C₂⁴	—	10	8	no	—
195	C₂⁴	C₂×C₅×D₅	1	1	yes	characteristic
196	C₂²×C₄	—	5	2	no	—
197	C₂²×C₄	—	5	8	no	—
198	C₂²×C₄	—	10	8	no	—
199	C₂²×C₄	C₂×C₅×D₅	1	2	yes	complemented
200	C₂²⋊C₄	—	5	4	no	—
201	C₂²⋊C₄	—	5	8	no	—
202	C₂²⋊C₄	C₂×C₅×D₅	1	4	yes	direct product factor
203	C₂×C₅	—	2	4	no	—
204	C₂×C₅	—	2	4	no	—
205	C₂×C₅	—	2	2	no	—
206	C₂×C₅	—	2	4	no	—
207	C₂×C₅	—	2	8	no	—
208	C₂×C₅	—	4	8	no	—
209	C₂×C₅	—	5	8	no	—
210	C₂×C₅	—	10	4	no	—
211	C₂×C₅	D₅×C₂²⋊C₄	1	4	yes	direct product factor
212	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	4	yes	complemented
213	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
214	C₂×C₅	C₂×D₄×D₅	1	2	yes	—
215	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
216	C₂×C₅	C₂²×C₅×D₄	1	2	yes	—
217	D₅	—	2	4	no	—
218	D₅	C₂×C₅×C₂²⋊C₄	1	8	yes	direct product factor
219	C₂³	—	2	4	no	—
220	C₂³	—	5	4	no	—
221	C₂³	—	5	8	no	—
222	C₂³	—	5	2	no	—
223	C₂³	—	5	4	no	—
224	C₂³	—	5	16	no	—
225	C₂³	—	5	2	no	—
226	C₂³	—	10	16	no	—
227	C₂³	—	10	32	no	—
228	C₂³	—	10	4	no	—
229	C₂³	C₄×C₅×D₅	1	4	yes	complemented
230	C₂³	C₂²×C₅×D₅	1	2	yes	—
231	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	5	4	no	—
234	C₂×C₄	—	10	16	no	—
235	C₂×C₄	—	10	4	no	—
236	C₂×C₄	C₂²×C₅×D₅	1	4	yes	complemented
237	C₅	—	2	2	no	—
238	C₅	C₂×D₅×C₂²⋊C₄	1	1	yes	characteristic, direct product factor
239	C₅	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic, complemented
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	5	4	no	—
243	C₂²	—	5	8	no	—
244	C₂²	—	5	16	no	—
245	C₂²	—	5	4	no	—
246	C₂²	—	10	32	no	—
247	C₂²	—	10	4	no	—
248	C₂²	—	10	8	no	—
249	C₂²	C₂×C₄×C₅×D₅	1	2	yes	—
250	C₂²	C₂×C₄×C₅×D₅	1	4	yes	complemented
251	C₂²	C₅×D₄×D₅	1	4	yes	—
252	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	10	4	no	—
255	C₂	—	2	4	no	—
256	C₂	—	5	8	no	—
257	C₂	—	10	4	no	—
258	C₂	C₅×D₅×C₂²⋊C₄	1	4	yes	direct product factor
259	C₂	C₂²×C₄×C₅×D₅	1	1	yes	characteristic
260	C₂	C₂×C₅×D₄×D₅	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂×C₅²×C₂²⋊C₄	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂⁴×C₅×D₅	C₂
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×C₂²⋊C₄	D₅
hypocenter	C₅	C₂²×C₅×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₅×D₅	C₂	semidirect
3. C₂×C₅²×C₂²⋊C₄	C₂	semidirect
4. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
5. C₂×C₅×#685	C₂	semidirect
6. C₅×D₅×C₂²⋊C₄	C₂	direct (N × H)
9. C₂³×C₅×D₅	C₄	semidirect
13. C₂²×C₄×C₅²	C₂²	semidirect
14. C₂²×C₅×Dic₅	C₂²	semidirect
16. C₂×C₄×C₅×D₅	C₂²	semidirect
17. C₅²×C₂²⋊C₄	C₂²	semidirect
18. C₅×C₂²⋊Dic₅	C₂²	semidirect
19. C₅×#685	C₂²	semidirect
21. C₂×D₅×C₂²⋊C₄	C₅	direct (N × H)
23. C₂³×C₅²	C₂×C₄	semidirect
30. C₂²×C₅×D₅	C₂×C₄	semidirect
31. C₂²×C₅×D₅	C₂×C₄	semidirect
36. C₂×C₄×C₅²	C₂³	semidirect
38. C₂×C₅×Dic₅	C₂³	semidirect
43. C₂²×C₄×D₅	C₂×C₅	semidirect
48. C₂×C₅×C₂²⋊C₄	D₅	direct (N × H)
49. C₂×C₅×C₂²⋊C₄	C₂×C₅	semidirect
50. C₂×C₂²⋊Dic₅	C₂×C₅	semidirect
51. C₂×#685	C₂×C₅	semidirect
52. D₅×C₂²⋊C₄	C₂×C₅	direct (N × H)
56. C₂²×C₅²	C₂²×C₄	semidirect
62. C₂×C₅×D₅	C₂²⋊C₄	direct (N × H)
64. C₂×C₅×D₅	C₂²×C₄	semidirect
77. C₂³×D₅	C₄×C₅	semidirect
85. C₂²×C₄×C₅	C₂×D₅	semidirect
86. C₂²×C₄×C₅	C₂²×C₅	semidirect
87. C₂²×Dic₅	C₂²×C₅	semidirect
89. C₂×C₄×D₅	C₂²×C₅	semidirect
93. C₅×C₂²⋊C₄	C₂×D₅	direct (N × H)
94. C₅×C₂²⋊C₄	C₂²×C₅	semidirect
95. C₂²⋊Dic₅	C₂²×C₅	semidirect
96. #685	C₂²×C₅	semidirect
99. C₂×C₅²	C₂×C₂²⋊C₄	semidirect
103. C₅×D₅	C₂×C₂²⋊C₄	direct (N × H)
119. C₂³×C₅	C₄×D₅	semidirect
120. C₂³×C₅	C₂×C₄×C₅	semidirect
129. C₂²×D₅	C₂×C₄×C₅	semidirect
130. C₂²×D₅	C₂×C₄×C₅	semidirect
141. C₂×C₄×C₅	C₂²×D₅	semidirect
142. C₂×C₄×C₅	C₂³×C₅	semidirect
144. C₂×Dic₅	C₂³×C₅	semidirect
151. C₂×C₂²⋊C₄	C₅×D₅	direct (N × H)
152. C₅²	C₂²×C₂²⋊C₄	semidirect
171. C₂²×C₅	C₂×C₄×D₅	semidirect
174. C₂²×C₅	C₂²×C₄×C₅	semidirect
181. C₂×D₅	C₅×C₂²⋊C₄	direct (N × H)
183. C₂×D₅	C₂²×C₄×C₅	semidirect
199. C₂²×C₄	C₂×C₅×D₅	semidirect
202. C₂²⋊C₄	C₂×C₅×D₅	direct (N × H)
211. C₂×C₅	D₅×C₂²⋊C₄	direct (N × H)
212. C₂×C₅	C₂×C₅×C₂²⋊C₄	semidirect
218. D₅	C₂×C₅×C₂²⋊C₄	direct (N × H)
229. C₂³	C₄×C₅×D₅	semidirect
236. C₂×C₄	C₂²×C₅×D₅	semidirect
238. C₅	C₂×D₅×C₂²⋊C₄	direct (N × H)
239. C₅	C₂²×C₅×C₂²⋊C₄	semidirect
250. C₂²	C₂×C₄×C₅×D₅	semidirect
258. C₂	C₅×D₅×C₂²⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×D₅	2
C₂²×C₄×C₅×D₅	2
C₂×C₅²×C₂²⋊C₄	2
C₂×C₅×C₂²⋊Dic₅	2
C₂×C₅×#685	2
C₅×D₅×C₂²⋊C₄	2
C₂²×C₅×C₂²⋊C₄	5
C₂×D₅×C₂²⋊C₄	5

Subgroups — C2×C5×D5×C22⋊C4 #13106639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×D₅×C₂²⋊C₄ #13106639