Subgroups — C₂×C₅×#1995 #11944048

Order 1280.

1206 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#376	C₂	1	1	yes	maximal, characteristic
4	C₅×#1995	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#339	C₂	1	4	yes	maximal
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₅×#533	—	2	2	no	—
8	C₅×#533	—	2	2	no	—
9	C₅×#533	C₄	1	4	yes	complemented
10	C₅×#533	C₂²	1	4	yes	—
11	C₂×C₅×C₄⋊Q₈	C₄	1	2	yes	complemented
12	C₂×C₅×#125	C₄	1	2	yes	complemented
13	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
14	C₂×C₄×C₅×Q₈	—	2	2	no	—
15	C₂×C₅×#117	C₂²	1	2	yes	—
16	C₂²×C₄²×C₅	C₂²	1	2	yes	—
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₅×#423	—	2	2	no	—
19	C₅×#423	C₂²	1	4	yes	—
20	C₂×C₅×C₄⋊C₈	—	2	2	no	—
21	C₅×#376	—	2	2	no	—
22	C₅×#376	C₂²	1	4	yes	—
23	C₂×C₅×#95	—	2	2	no	—
24	C₅×#339	C₂²	1	16	yes	—
25	C₂×#1995	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂²×C₅×Q₈	—	2	2	no	—
27	C₂³×C₄×C₅	—	2	2	no	—
28	C₂³×C₄×C₅	D₄	1	2	yes	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×M₁₆	—	2	2	no	—
31	C₂×C₅×M₁₆	—	2	8	no	—
32	C₂×C₅×M₁₆	C₂³	1	4	yes	—
33	C₂²×C₅×C₈	—	2	2	no	—
34	C₅×C₄⋊Q₈	—	2	4	no	—
35	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
36	C₅×#125	—	2	4	no	—
37	C₅×#125	C₂×C₄	1	8	yes	complemented
38	C₅×C₂²⋊Q₈	—	2	8	no	—
39	C₅×C₂²⋊Q₈	—	2	8	no	—
40	C₄×C₅×Q₈	—	2	8	no	—
41	C₄×C₅×Q₈	—	2	8	no	—
42	C₅×#117	—	2	2	no	—
43	C₅×#117	—	2	2	no	—
44	C₅×#117	C₂×C₄	1	4	yes	—
45	C₅×#117	C₂³	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	—	2	4	no	—
47	C₂×C₅×C₄⋊C₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₂×C₄²×C₅	—	2	2	no	—
52	C₂×C₄²×C₅	—	2	4	no	—
53	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₅	D₄	1	2	yes	—
57	C₂×C₄²×C₅	D₄	1	2	yes	—
58	C₂×C₄²×C₅	D₄	1	2	yes	—
59	C₂×C₄²×C₅	Q₈	1	8	yes	complemented
60	C₂×C₄²×C₅	C₂³	1	4	yes	—
61	C₂×C₄²×C₅	C₂³	1	2	yes	—
62	C₅×C₄⋊C₈	—	2	8	no	—
63	C₅×#95	—	2	8	no	—
64	C₂×#533	C₂×C₅	1	1	yes	characteristic
65	C₂×#423	C₂×C₅	1	1	yes	characteristic
66	C₂×#376	C₂×C₅	1	1	yes	characteristic
67	#1995	C₂×C₅	1	8	yes	direct product factor
68	C₂×#339	C₂×C₅	1	4	yes	—
69	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₅×Q₈	—	2	8	no	—
71	C₂×C₅×Q₈	—	4	8	no	—
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	—	2	8	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	—	4	4	no	—
76	C₂²×C₄×C₅	—	4	8	no	—
77	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
87	C₂²×C₄×C₅	#41	1	4	yes	—
88	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₅×M₁₆	—	2	8	no	—
90	C₅×M₁₆	—	4	4	no	—
91	C₂×C₅×C₈	—	2	4	no	—
92	C₂×C₅×C₈	—	4	4	no	—
93	C₅×C₄⋊C₄	—	2	16	no	—
94	C₅×C₄⋊C₄	—	2	8	no	—
95	C₅×C₄⋊C₄	—	2	4	no	—
96	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
97	C₅×C₂²⋊C₄	—	2	4	no	—
98	C₅×C₂²⋊C₄	—	2	4	no	—
99	C₄²×C₅	—	2	4	no	—
100	C₄²×C₅	—	2	8	no	—
101	C₄²×C₅	—	2	4	no	—
102	C₄²×C₅	—	2	4	no	—
103	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
104	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
105	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
106	C₄²×C₅	C₂²×C₄	1	2	yes	—
107	C₄²×C₅	C₂²×C₄	1	2	yes	—
108	C₄²×C₅	C₂×D₄	1	4	yes	—
109	C₄²×C₅	C₂×Q₈	1	8	yes	complemented
110	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
111	#533	—	2	2	no	—
112	#533	—	2	2	no	—
113	#533	C₄×C₅	1	4	yes	complemented
114	#533	C₂²×C₅	1	4	yes	—
115	C₂×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
116	C₂×#125	C₄×C₅	1	2	yes	complemented
117	C₂×C₂²⋊Q₈	—	2	2	no	—
118	C₂×C₄×Q₈	—	2	2	no	—
119	C₂×#117	C₂²×C₅	1	2	yes	—
120	C₂²×C₄²	C₂²×C₅	1	2	yes	—
121	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
122	#423	—	2	2	no	—
123	#423	C₂²×C₅	1	4	yes	—
124	C₂×C₄⋊C₈	—	2	2	no	—
125	#376	—	2	2	no	—
126	#376	C₂²×C₅	1	4	yes	—
127	C₂×#95	—	2	2	no	—
128	#339	C₂²×C₅	1	16	yes	—
129	C₂³×C₅	—	2	4	no	—
130	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂³×C₅	C₂²⋊Q₈	1	4	yes	—
133	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₅×Q₈	—	4	16	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	8	no	—
138	C₂×C₄×C₅	—	2	8	no	—
139	C₂×C₄×C₅	—	4	4	no	—
140	C₂×C₄×C₅	—	4	4	no	—
141	C₂×C₄×C₅	—	4	8	no	—
142	C₂×C₄×C₅	—	4	16	no	—
143	C₂×C₄×C₅	—	4	8	no	—
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
149	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
150	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
151	C₂×C₄×C₅	#124	1	2	yes	—
152	C₂×C₄×C₅	#124	1	2	yes	—
153	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
154	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
155	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
158	C₂×C₄×C₅	C₂×#41	1	2	yes	—
159	C₅×C₈	—	4	4	no	—
160	C₂²×Q₈	—	2	2	no	—
161	C₂³×C₄	—	2	2	no	—
162	C₂³×C₄	C₅×D₄	1	2	yes	—
163	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	8	no	—
166	C₂×M₁₆	C₂³×C₅	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₄⋊Q₈	—	2	4	no	—
169	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	complemented
170	#125	—	2	4	no	—
171	#125	C₂×C₄×C₅	1	8	yes	complemented
172	C₂²⋊Q₈	—	2	8	no	—
173	C₂²⋊Q₈	—	2	8	no	—
174	C₄×Q₈	—	2	8	no	—
175	C₄×Q₈	—	2	8	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	C₂×C₄×C₅	1	4	yes	—
179	#117	C₂³×C₅	1	4	yes	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	2	no	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	—	2	4	no	—
187	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
188	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
190	C₂×C₄²	C₅×D₄	1	2	yes	—
191	C₂×C₄²	C₅×D₄	1	2	yes	—
192	C₂×C₄²	C₅×D₄	1	2	yes	—
193	C₂×C₄²	C₅×Q₈	1	8	yes	complemented
194	C₂×C₄²	C₂³×C₅	1	4	yes	—
195	C₂×C₄²	C₂³×C₅	1	2	yes	—
196	C₄⋊C₈	—	2	8	no	—
197	#95	—	2	8	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	8	no	—
201	C₂²×C₅	#391	1	2	yes	—
202	C₂²×C₅	#391	1	2	yes	—
203	C₂²×C₅	#421	1	4	yes	—
204	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
205	C₂²×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
206	C₄×C₅	—	2	4	no	—
207	C₄×C₅	—	2	4	no	—
208	C₄×C₅	—	4	8	no	—
209	C₄×C₅	—	4	8	no	—
210	C₄×C₅	#391	1	2	yes	—
211	C₄×C₅	#391	1	2	yes	—
212	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
213	C₄×C₅	C₂×#124	1	1	yes	characteristic
214	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
215	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×Q₈	—	2	8	no	—
217	C₂×Q₈	—	4	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	8	no	—
223	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
224	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
225	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
229	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
233	C₂²×C₄	C₅×#41	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
235	M₁₆	—	2	8	no	—
236	M₁₆	—	4	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	4	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
243	C₂²⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	8	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
250	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
251	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
252	C₄²	C₂²×C₄×C₅	1	2	yes	—
253	C₄²	C₂²×C₄×C₅	1	2	yes	—
254	C₄²	C₂×C₅×D₄	1	4	yes	—
255	C₄²	C₂×C₅×Q₈	1	8	yes	complemented
256	C₂×C₅	—	2	2	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	#1995	1	4	yes	direct product factor
259	C₂×C₅	C₂×#391	1	1	yes	characteristic
260	C₂×C₅	C₂×#421	1	2	yes	—
261	C₂³	—	2	4	no	—
262	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
263	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂³	C₅×C₂²⋊Q₈	1	4	yes	—
265	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
266	Q₈	—	4	16	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	8	no	—
274	C₂×C₄	—	4	16	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
277	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
278	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
281	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
282	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
283	C₂×C₄	C₅×#124	1	2	yes	—
284	C₂×C₄	C₅×#124	1	2	yes	—
285	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
286	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
287	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
288	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
290	C₂×C₄	C₂×C₅×#41	1	2	yes	—
291	C₈	—	4	4	no	—
292	C₅	C₂×#1995	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	8	no	—
296	C₂²	C₅×#391	1	2	yes	—
297	C₂²	C₅×#391	1	2	yes	—
298	C₂²	C₅×#421	1	4	yes	—
299	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
300	C₂²	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
301	C₄	—	2	4	no	—
302	C₄	—	2	4	no	—
303	C₄	—	4	8	no	—
304	C₄	—	4	8	no	—
305	C₄	C₅×#391	1	2	yes	—
306	C₄	C₅×#391	1	2	yes	—
307	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
308	C₄	C₂×C₅×#124	1	1	yes	characteristic
309	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
310	C₂	—	2	2	no	—
311	C₂	—	2	2	no	—
312	C₂	C₅×#1995	1	4	yes	direct product factor
313	C₂	C₂×C₅×#391	1	1	yes	characteristic
314	C₂	C₂×C₅×#421	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1995	C₂	direct (N × H)
9. C₅×#533	C₄	semidirect
11. C₂×C₅×C₄⋊Q₈	C₄	semidirect
12. C₂×C₅×#125	C₄	semidirect
25. C₂×#1995	C₅	direct (N × H)
35. C₅×C₄⋊Q₈	C₂×C₄	semidirect
37. C₅×#125	C₂×C₄	semidirect
59. C₂×C₄²×C₅	Q₈	semidirect
67. #1995	C₂×C₅	direct (N × H)
109. C₄²×C₅	C₂×Q₈	semidirect
113. #533	C₄×C₅	semidirect
115. C₂×C₄⋊Q₈	C₄×C₅	semidirect
116. C₂×#125	C₄×C₅	semidirect
169. C₄⋊Q₈	C₂×C₄×C₅	semidirect
171. #125	C₂×C₄×C₅	semidirect
193. C₂×C₄²	C₅×Q₈	semidirect
255. C₄²	C₂×C₅×Q₈	semidirect
258. C₂×C₅	#1995	direct (N × H)
292. C₅	C₂×#1995	direct (N × H)
312. C₂	C₅×#1995	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×C₅×#533	2
C₂×C₅×#423	2
C₂×C₅×#376	2
C₅×#1995	2
C₂×C₅×#339	2
C₂×#1995	5

Subgroups — C2×C5×#1995 #11944048

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1995 #11944048