Subgroups — C₂×#11717091 #13106585

Order 1600.

1088 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1039	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×C₂²⋊D₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₂²⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717091	C₂	1	8	yes	maximal, direct product factor
6	C₂×#87780	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#87778	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₅²	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₅²	C₂²	1	1	yes	characteristic
10	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
11	C₂²×#1039	—	2	4	no	—
12	C₂²×#1039	C₂²	1	1	yes	characteristic
13	C₂²×#1039	C₂²	1	1	yes	characteristic
14	C₂²×#1039	C₂²	1	4	yes	complemented
15	C₂×C₅×C₂²⋊D₅	C₄	1	8	yes	complemented
16	C₂×C₅×C₂²⋊D₅	C₂²	1	4	yes	—
17	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
18	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
19	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
20	C₂×D₅×Dic₅	—	2	4	no	—
21	C₂×D₅×Dic₅	C₂²	1	4	yes	complemented
22	C₅×C₂²⋊Dic₅	C₂²	1	4	yes	complemented
23	#87780	C₂²	1	4	yes	complemented
24	#87778	C₂²	1	4	yes	complemented
25	Dic₅²	C₂²	1	4	yes	complemented
26	C₂×D₄×Dic₅	—	5	1	no	maximal
27	C₂×#4027	—	5	1	no	maximal
28	C₂³×C₅²	—	2	2	no	—
29	C₂³×C₅²	—	2	2	no	—
30	C₂³×C₅²	C₂×C₄	1	4	yes	complemented
31	C₂³×C₅²	C₂³	1	2	yes	—
32	C₂³×C₅²	C₂³	1	1	yes	characteristic
33	C₂²×C₅×D₅	—	2	2	no	—
34	C₂²×C₅×D₅	—	2	2	no	—
35	C₂²×C₅×D₅	C₂×C₄	1	4	yes	complemented
36	C₂²×C₅×D₅	C₂³	1	2	yes	—
37	C₂×#1039	—	2	2	no	—
38	C₂×#1039	—	2	8	no	—
39	C₂×#1039	D₄	1	4	yes	complemented
40	C₂×#1039	C₂³	1	2	yes	—
41	C₂×#1039	C₂³	1	2	yes	—
42	C₅×C₂²⋊D₅	C₂×C₄	1	16	yes	complemented
43	C₂×C₅×Dic₅	—	2	2	no	—
44	C₂×C₅×Dic₅	—	2	2	no	—
45	C₂×C₅×Dic₅	C₂×C₄	1	4	yes	complemented
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	2	yes	—
48	C₂×C₅×Dic₅	C₂³	1	2	yes	—
49	D₅×Dic₅	—	2	8	no	—
50	C₂²×C₅×D₄	—	5	1	no	—
51	C₂²×C₂²⋊D₅	D₅	1	1	yes	characteristic
52	C₂³×Dic₅	—	5	1	no	—
53	C₂³×Dic₅	—	5	1	no	—
54	C₂³×Dic₅	—	5	1	no	—
55	C₂³×Dic₅	—	10	2	no	—
56	C₂²×C₄×D₅	—	5	1	no	—
57	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
58	C₂×C₂²⋊Dic₅	—	5	1	no	—
59	C₂×C₂²⋊Dic₅	D₅	1	1	yes	characteristic, complemented
60	D₄×Dic₅	—	5	8	no	—
61	C₂×#685	—	5	1	no	—
62	C₂×C₄⋊Dic₅	—	5	1	no	—
63	C₂×#683	—	5	1	no	—
64	C₂×C₄×Dic₅	—	5	1	no	—
65	C₂×C₄×Dic₅	—	5	1	no	—
66	#4027	—	5	8	no	—
67	C₂²×C₅²	—	2	8	no	—
68	C₂²×C₅²	—	2	2	no	—
69	C₂²×C₅²	—	2	2	no	—
70	C₂²×C₅²	C₂²×C₄	1	2	yes	—
71	C₂²×C₅²	C₂²×C₄	1	4	yes	complemented
72	C₂²×C₅²	C₂×D₄	1	2	yes	—
73	C₂²×C₅²	#41	1	2	yes	—
74	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×D₅	—	2	8	no	—
76	C₂×C₅×D₅	—	2	2	no	—
77	C₂×C₅×D₅	—	2	2	no	—
78	C₂×C₅×D₅	C₂²×C₄	1	4	yes	complemented
79	#1039	—	2	2	no	—
80	#1039	C₂×D₄	1	4	yes	complemented
81	C₅×Dic₅	—	2	2	no	—
82	C₅×Dic₅	—	2	2	no	—
83	C₅×Dic₅	C₂²×C₄	1	4	yes	complemented
84	C₂⁴×C₅	—	2	2	no	—
85	C₂⁴×C₅	—	5	1	no	—
86	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
87	C₂⁴×C₅	C₂×D₅	1	1	yes	characteristic
88	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
89	C₂×C₅×D₄	—	5	8	no	—
90	C₂×C₅×D₄	—	5	4	no	—
91	C₂²×C₄×C₅	—	5	1	no	—
92	C₂²×C₄×C₅	—	5	1	no	—
93	C₂²×C₄×C₅	—	5	1	no	—
94	C₂×C₂²⋊D₅	Dic₅	1	8	yes	complemented
95	C₂×C₂²⋊D₅	C₂×D₅	1	4	yes	—
96	C₂²×Dic₅	—	5	1	no	—
97	C₂²×Dic₅	—	5	1	no	—
98	C₂²×Dic₅	—	5	1	no	—
99	C₂²×Dic₅	—	5	1	no	—
100	C₂²×Dic₅	—	5	4	no	—
101	C₂²×Dic₅	—	5	4	no	—
102	C₂²×Dic₅	—	5	4	no	—
103	C₂²×Dic₅	—	10	2	no	—
104	C₂²×Dic₅	—	10	2	no	—
105	C₂²×Dic₅	—	10	8	no	—
106	C₂²×Dic₅	—	10	4	no	—
107	C₂²×Dic₅	—	10	4	no	—
108	C₂²×Dic₅	—	10	4	no	—
109	C₂²×Dic₅	—	10	16	no	—
110	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
111	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
112	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
113	C₂×C₄×D₅	—	5	4	no	—
114	C₂×C₄×D₅	—	10	4	no	—
115	C₅×C₂²⋊C₄	—	5	4	no	—
116	C₂²⋊Dic₅	—	5	4	no	—
117	C₂²⋊Dic₅	C₂×D₅	1	4	yes	complemented
118	#685	—	5	4	no	—
119	C₄⋊Dic₅	—	5	4	no	—
120	#683	—	5	4	no	—
121	C₄×Dic₅	—	5	4	no	—
122	C₄×Dic₅	—	5	4	no	—
123	C₂×C₄×D₄	—	25	1	no	Sylow
124	C₂×C₅²	—	2	4	no	—
125	C₂×C₅²	C₄×D₄	1	4	yes	complemented
126	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₅²	C₂×#41	1	1	yes	characteristic
129	C₅×D₅	—	2	4	no	—
130	C₂³×C₅	—	2	8	no	—
131	C₂³×C₅	—	2	8	no	—
132	C₂³×C₅	—	2	2	no	—
133	C₂³×C₅	—	2	2	no	—
134	C₂³×C₅	—	2	2	no	—
135	C₂³×C₅	—	2	2	no	—
136	C₂³×C₅	—	2	8	no	—
137	C₂³×C₅	—	2	4	no	—
138	C₂³×C₅	—	2	2	no	—
139	C₂³×C₅	—	5	4	no	—
140	C₂³×C₅	—	5	2	no	—
141	C₂³×C₅	—	10	2	no	—
142	C₂³×C₅	—	10	2	no	—
143	C₂³×C₅	C₄×D₅	1	4	yes	complemented
144	C₂³×C₅	C₂×Dic₅	1	4	yes	complemented
145	C₂³×C₅	C₂²×D₅	1	2	yes	—
146	C₂³×C₅	C₂²×D₅	1	2	yes	—
147	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
148	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
149	C₂²×D₅	—	2	2	no	—
150	C₂²×D₅	—	2	2	no	—
151	C₂²×D₅	C₂×Dic₅	1	4	yes	complemented
152	C₂²×D₅	C₂²×D₅	1	2	yes	—
153	C₅×D₄	—	5	16	no	—
154	C₂×C₄×C₅	—	5	4	no	—
155	C₂×C₄×C₅	—	5	2	no	—
156	C₂×C₄×C₅	—	5	2	no	—
157	C₂×C₄×C₅	—	5	2	no	—
158	C₂×C₄×C₅	—	10	2	no	—
159	C₂×C₄×C₅	—	10	2	no	—
160	C₂²⋊D₅	C₂×Dic₅	1	16	yes	complemented
161	C₂×Dic₅	—	2	2	no	—
162	C₂×Dic₅	—	2	2	no	—
163	C₂×Dic₅	—	5	4	no	—
164	C₂×Dic₅	—	5	4	no	—
165	C₂×Dic₅	—	5	2	no	—
166	C₂×Dic₅	—	5	2	no	—
167	C₂×Dic₅	—	5	2	no	—
168	C₂×Dic₅	—	5	2	no	—
169	C₂×Dic₅	—	10	8	no	—
170	C₂×Dic₅	—	10	4	no	—
171	C₂×Dic₅	—	10	4	no	—
172	C₂×Dic₅	—	10	8	no	—
173	C₂×Dic₅	—	10	8	no	—
174	C₂×Dic₅	—	10	8	no	—
175	C₂×Dic₅	—	10	32	no	—
176	C₂×Dic₅	—	10	8	no	—
177	C₂×Dic₅	—	10	2	no	—
178	C₂×Dic₅	—	10	2	no	—
179	C₂×Dic₅	C₂×Dic₅	1	4	yes	complemented
180	C₂×Dic₅	C₂²×D₅	1	2	yes	—
181	C₂×Dic₅	C₂²×D₅	1	2	yes	—
182	C₂×Dic₅	C₂²×D₅	1	2	yes	—
183	C₄×D₅	—	10	8	no	—
184	C₂²×D₄	—	5	1	no	—
185	C₂³×C₄	—	25	1	no	—
186	C₂³×C₄	—	25	1	no	—
187	C₄×D₄	—	25	8	no	—
188	C₂×C₄⋊C₄	—	25	1	no	—
189	C₂×C₂²⋊C₄	—	5	1	no	—
190	C₂×C₂²⋊C₄	—	25	1	no	—
191	C₂×C₄²	—	25	1	no	—
192	C₅²	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
193	C₂²×C₅	—	2	8	no	—
194	C₂²×C₅	—	2	8	no	—
195	C₂²×C₅	—	2	32	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	8	no	—
201	C₂²×C₅	—	2	8	no	—
202	C₂²×C₅	—	2	4	no	—
203	C₂²×C₅	—	2	8	no	—
204	C₂²×C₅	—	2	4	no	—
205	C₂²×C₅	—	2	4	no	—
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	5	4	no	—
208	C₂²×C₅	—	10	2	no	—
209	C₂²×C₅	—	10	2	no	—
210	C₂²×C₅	—	10	8	no	—
211	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
212	C₂²×C₅	C₂×C₄×D₅	1	4	yes	complemented
213	C₂²×C₅	D₄×D₅	1	2	yes	—
214	C₂²×C₅	D₄×D₅	1	2	yes	—
215	C₂²×C₅	#711	1	2	yes	—
216	C₂²×C₅	#711	1	2	yes	—
217	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
218	C₂²×C₅	C₂²×Dic₅	1	4	yes	complemented
219	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
220	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
221	C₂×D₅	—	2	2	no	—
222	C₂×D₅	—	2	2	no	—
223	C₂×D₅	—	2	8	no	—
224	C₂×D₅	C₂²×Dic₅	1	4	yes	complemented
225	C₄×C₅	—	5	4	no	—
226	C₄×C₅	—	10	2	no	—
227	C₄×C₅	—	10	2	no	—
228	Dic₅	—	2	2	no	—
229	Dic₅	—	2	2	no	—
230	Dic₅	—	5	4	no	—
231	Dic₅	—	5	4	no	—
232	Dic₅	—	10	8	no	—
233	Dic₅	—	10	8	no	—
234	Dic₅	—	10	2	no	—
235	Dic₅	—	10	2	no	—
236	Dic₅	C₂²×Dic₅	1	4	yes	complemented
237	C₂⁴	—	5	1	no	—
238	C₂⁴	D₅²	1	1	yes	characteristic
239	C₂×D₄	—	5	8	no	—
240	C₂×D₄	—	5	4	no	—
241	C₂²×C₄	—	5	1	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	25	4	no	—
245	C₂²×C₄	—	25	1	no	—
246	C₂²×C₄	—	25	4	no	—
247	C₂²×C₄	—	25	1	no	—
248	C₂²×C₄	—	50	4	no	—
249	C₂²×C₄	—	50	4	no	—
250	C₄⋊C₄	—	25	4	no	—
251	C₂²⋊C₄	—	5	4	no	—
252	C₂²⋊C₄	—	25	4	no	—
253	C₄²	—	25	4	no	—
254	C₂×C₅	—	2	16	no	—
255	C₂×C₅	—	2	4	no	—
256	C₂×C₅	—	2	4	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	2	2	no	—
260	C₂×C₅	—	2	8	no	—
261	C₂×C₅	—	10	4	no	—
262	C₂×C₅	#4027	1	4	yes	complemented
263	C₂×C₅	D₄×Dic₅	1	4	yes	complemented
264	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
265	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
266	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
267	C₂×C₅	C₂×#711	1	1	yes	characteristic
268	C₂×C₅	C₂×#711	1	1	yes	characteristic
269	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
270	D₅	—	2	4	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	5	4	no	—
274	C₂³	—	5	2	no	—
275	C₂³	—	10	2	no	—
276	C₂³	—	10	2	no	—
277	C₂³	D₅×Dic₅	1	4	yes	complemented
278	C₂³	C₂×D₅²	1	2	yes	—
279	C₂³	C₂×D₅²	1	1	yes	characteristic
280	D₄	—	5	16	no	—
281	C₂×C₄	—	5	4	no	—
282	C₂×C₄	—	5	2	no	—
283	C₂×C₄	—	5	2	no	—
284	C₂×C₄	—	5	2	no	—
285	C₂×C₄	—	10	2	no	—
286	C₂×C₄	—	10	2	no	—
287	C₂×C₄	—	25	4	no	—
288	C₂×C₄	—	25	2	no	—
289	C₂×C₄	—	25	2	no	—
290	C₂×C₄	—	50	8	no	—
291	C₂×C₄	—	50	8	no	—
292	C₂×C₄	—	50	2	no	—
293	C₅	—	2	2	no	—
294	C₅	C₂×#4027	1	1	yes	characteristic, complemented
295	C₅	C₂×D₄×Dic₅	1	1	yes	characteristic, complemented
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	8	no	—
299	C₂²	—	5	4	no	—
300	C₂²	—	10	2	no	—
301	C₂²	—	10	2	no	—
302	C₂²	—	10	8	no	—
303	C₂²	C₂×D₅×Dic₅	1	2	yes	—
304	C₂²	C₂×D₅×Dic₅	1	4	yes	complemented
305	C₂²	#87880	1	2	yes	—
306	C₂²	#87886	1	2	yes	—
307	C₂²	C₂²×D₅²	1	1	yes	characteristic
308	C₄	—	5	4	no	—
309	C₄	—	10	2	no	—
310	C₄	—	10	2	no	—
311	C₄	—	25	4	no	—
312	C₄	—	50	2	no	—
313	C₂	—	2	4	no	—
314	C₂	—	10	4	no	—
315	C₂	#11717091	1	4	yes	direct product factor
316	C₂	C₂²×D₅×Dic₅	1	1	yes	characteristic
317	C₂	C₂×#87880	1	1	yes	characteristic
318	C₂	C₂×#87886	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅²
Fitting	C₂⁴×C₅²	C₂²
socle	C₂³×C₅²	C₂³
prime radical	C₂²×C₅×C₂²⋊D₅	C₂
cosocle	C₂²×C₅²	C₂⁴
hypercenter	C₂⁴	D₅²
hypocenter	C₅²	C₂×C₄×D₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×#1039	C₂	semidirect
3. C₂²×D₅×Dic₅	C₂	semidirect
4. C₂×C₅×C₂²⋊Dic₅	C₂	semidirect
5. #11717091	C₂	direct (N × H)
6. C₂×#87780	C₂	semidirect
7. C₂×#87778	C₂	semidirect
8. C₂×Dic₅²	C₂	semidirect
14. C₂²×#1039	C₂²	semidirect
15. C₂×C₅×C₂²⋊D₅	C₄	semidirect
21. C₂×D₅×Dic₅	C₂²	semidirect
22. C₅×C₂²⋊Dic₅	C₂²	semidirect
23. #87780	C₂²	semidirect
24. #87778	C₂²	semidirect
25. Dic₅²	C₂²	semidirect
30. C₂³×C₅²	C₂×C₄	semidirect
35. C₂²×C₅×D₅	C₂×C₄	semidirect
39. C₂×#1039	D₄	semidirect
42. C₅×C₂²⋊D₅	C₂×C₄	semidirect
45. C₂×C₅×Dic₅	C₂×C₄	semidirect
59. C₂×C₂²⋊Dic₅	D₅	semidirect
71. C₂²×C₅²	C₂²×C₄	semidirect
78. C₂×C₅×D₅	C₂²×C₄	semidirect
80. #1039	C₂×D₄	semidirect
83. C₅×Dic₅	C₂²×C₄	semidirect
94. C₂×C₂²⋊D₅	Dic₅	semidirect
117. C₂²⋊Dic₅	C₂×D₅	semidirect
125. C₂×C₅²	C₄×D₄	semidirect
143. C₂³×C₅	C₄×D₅	semidirect
144. C₂³×C₅	C₂×Dic₅	semidirect
151. C₂²×D₅	C₂×Dic₅	semidirect
160. C₂²⋊D₅	C₂×Dic₅	semidirect
179. C₂×Dic₅	C₂×Dic₅	semidirect
192. C₅²	C₂×C₄×D₄	semidirect
212. C₂²×C₅	C₂×C₄×D₅	semidirect
218. C₂²×C₅	C₂²×Dic₅	semidirect
224. C₂×D₅	C₂²×Dic₅	semidirect
236. Dic₅	C₂²×Dic₅	semidirect
262. C₂×C₅	#4027	semidirect
263. C₂×C₅	D₄×Dic₅	semidirect
277. C₂³	D₅×Dic₅	semidirect
294. C₅	C₂×#4027	semidirect
295. C₅	C₂×D₄×Dic₅	semidirect
304. C₂²	C₂×D₅×Dic₅	semidirect
315. C₂	#11717091	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1039	2
C₂²×C₅×C₂²⋊D₅	2
C₂²×D₅×Dic₅	2
C₂×C₅×C₂²⋊Dic₅	2
#11717091	2
C₂×#87780	2
C₂×#87778	2
C₂×Dic₅²	2
C₂×D₄×Dic₅	5
C₂×#4027	5

Subgroups — C2×#11717091 #13106585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11717091 #13106585