Subgroups — C₂×C₅×#1994 #11944047

Order 1280.

1422 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	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#423	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
4	C₅×#1994	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	—	2	2	no	—
17	C₂²×C₄²×C₅	C₂²	1	2	yes	—
18	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₅×#423	—	2	2	no	—
20	C₅×#423	C₂²	1	4	yes	—
21	C₂×C₅×C₄⋊C₈	—	2	2	no	—
22	C₄³×C₅	—	2	2	no	—
23	C₄³×C₅	C₂²	1	4	yes	—
24	C₅×#339	C₂²	1	16	yes	—
25	C₂×#1994	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	16	no	—
51	C₂×C₄²×C₅	—	2	2	no	—
52	C₂×C₄²×C₅	—	2	8	no	—
53	C₂×C₄²×C₅	—	2	2	no	—
54	C₂×C₄²×C₅	—	2	4	no	—
55	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₅	D₄	1	2	yes	—
59	C₂×C₄²×C₅	D₄	1	2	yes	—
60	C₂×C₄²×C₅	D₄	1	2	yes	—
61	C₂×C₄²×C₅	Q₈	1	8	yes	complemented
62	C₂×C₄²×C₅	C₂³	1	4	yes	—
63	C₂×C₄²×C₅	C₂³	1	2	yes	—
64	C₅×C₄⋊C₈	—	2	8	no	—
65	C₂×#533	C₂×C₅	1	1	yes	characteristic
66	C₂×#423	C₂×C₅	1	1	yes	characteristic
67	C₂×C₄³	C₂×C₅	1	1	yes	characteristic
68	#1994	C₂×C₅	1	8	yes	direct product factor
69	C₂×#339	C₂×C₅	1	4	yes	—
70	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₅×Q₈	—	2	8	no	—
72	C₂×C₅×Q₈	—	4	8	no	—
73	C₂²×C₄×C₅	—	2	16	no	—
74	C₂²×C₄×C₅	—	2	8	no	—
75	C₂²×C₄×C₅	—	2	4	no	—
76	C₂²×C₄×C₅	—	2	8	no	—
77	C₂²×C₄×C₅	—	2	4	no	—
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	2	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
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×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
88	C₂²×C₄×C₅	#41	1	4	yes	—
89	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₅×M₁₆	—	2	8	no	—
91	C₅×M₁₆	—	4	4	no	—
92	C₂×C₅×C₈	—	2	4	no	—
93	C₂×C₅×C₈	—	4	4	no	—
94	C₅×C₄⋊C₄	—	2	16	no	—
95	C₅×C₄⋊C₄	—	2	8	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
98	C₅×C₂²⋊C₄	—	2	4	no	—
99	C₅×C₂²⋊C₄	—	2	4	no	—
100	C₄²×C₅	—	2	32	no	—
101	C₄²×C₅	—	2	4	no	—
102	C₄²×C₅	—	2	8	no	—
103	C₄²×C₅	—	2	4	no	—
104	C₄²×C₅	—	2	4	no	—
105	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
106	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
107	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
108	C₄²×C₅	C₂²×C₄	1	2	yes	—
109	C₄²×C₅	C₂²×C₄	1	2	yes	—
110	C₄²×C₅	C₂×D₄	1	4	yes	—
111	C₄²×C₅	C₂×Q₈	1	8	yes	complemented
112	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
113	#533	—	2	2	no	—
114	#533	—	2	2	no	—
115	#533	C₄×C₅	1	4	yes	complemented
116	#533	C₂²×C₅	1	4	yes	—
117	C₂×C₄⋊Q₈	C₄×C₅	1	2	yes	complemented
118	C₂×#125	C₄×C₅	1	2	yes	complemented
119	C₂×C₂²⋊Q₈	—	2	2	no	—
120	C₂×C₄×Q₈	—	2	2	no	—
121	C₂×#117	C₂²×C₅	1	2	yes	—
122	C₂²×C₄²	—	2	2	no	—
123	C₂²×C₄²	C₂²×C₅	1	2	yes	—
124	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
125	#423	—	2	2	no	—
126	#423	C₂²×C₅	1	4	yes	—
127	C₂×C₄⋊C₈	—	2	2	no	—
128	C₄³	—	2	2	no	—
129	C₄³	C₂²×C₅	1	4	yes	—
130	#339	C₂²×C₅	1	16	yes	—
131	C₂³×C₅	—	2	4	no	—
132	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂³×C₅	C₂²⋊Q₈	1	4	yes	—
135	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₅×Q₈	—	4	16	no	—
137	C₂×C₄×C₅	—	2	16	no	—
138	C₂×C₄×C₅	—	2	32	no	—
139	C₂×C₄×C₅	—	2	8	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	8	no	—
143	C₂×C₄×C₅	—	2	8	no	—
144	C₂×C₄×C₅	—	4	8	no	—
145	C₂×C₄×C₅	#104	1	8	yes	—
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
147	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
150	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
151	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
152	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
153	C₂×C₄×C₅	#124	1	2	yes	—
154	C₂×C₄×C₅	#124	1	2	yes	—
155	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
156	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
157	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
160	C₂×C₄×C₅	C₂×#41	1	2	yes	—
161	C₅×C₈	—	4	4	no	—
162	C₂²×Q₈	—	2	2	no	—
163	C₂³×C₄	—	2	2	no	—
164	C₂³×C₄	C₅×D₄	1	2	yes	—
165	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	8	no	—
168	C₂×M₁₆	C₂³×C₅	1	4	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	4	no	—
171	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	complemented
172	#125	—	2	4	no	—
173	#125	C₂×C₄×C₅	1	8	yes	complemented
174	C₂²⋊Q₈	—	2	8	no	—
175	C₂²⋊Q₈	—	2	8	no	—
176	C₄×Q₈	—	2	8	no	—
177	C₄×Q₈	—	2	8	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	C₂×C₄×C₅	1	4	yes	—
181	#117	C₂³×C₅	1	4	yes	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	16	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	8	no	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
192	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₄²	C₅×D₄	1	2	yes	—
195	C₂×C₄²	C₅×D₄	1	2	yes	—
196	C₂×C₄²	C₅×D₄	1	2	yes	—
197	C₂×C₄²	C₅×Q₈	1	8	yes	complemented
198	C₂×C₄²	C₂³×C₅	1	4	yes	—
199	C₂×C₄²	C₂³×C₅	1	2	yes	—
200	C₄⋊C₈	—	2	8	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	8	no	—
204	C₂²×C₅	#391	1	2	yes	—
205	C₂²×C₅	#391	1	2	yes	—
206	C₂²×C₅	C₂×#104	1	4	yes	—
207	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₅	C₂×C₂²⋊Q₈	1	2	yes	—
209	C₄×C₅	—	2	16	no	—
210	C₄×C₅	—	2	4	no	—
211	C₄×C₅	—	4	8	no	—
212	C₄×C₅	#391	1	2	yes	—
213	C₄×C₅	#391	1	2	yes	—
214	C₄×C₅	C₂×#104	1	8	yes	—
215	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
216	C₄×C₅	C₂×#124	1	1	yes	characteristic
217	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
218	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	—	4	8	no	—
221	C₂²×C₄	—	2	16	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
227	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
228	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
232	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
236	C₂²×C₄	C₅×#41	1	4	yes	—
237	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
238	M₁₆	—	2	8	no	—
239	M₁₆	—	4	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	4	4	no	—
242	C₄⋊C₄	—	2	16	no	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₄²	—	2	32	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
254	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
255	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
256	C₄²	C₂²×C₄×C₅	1	2	yes	—
257	C₄²	C₂²×C₄×C₅	1	2	yes	—
258	C₄²	C₂×C₅×D₄	1	4	yes	—
259	C₄²	C₂×C₅×Q₈	1	8	yes	complemented
260	C₂×C₅	—	2	2	no	—
261	C₂×C₅	—	2	2	no	—
262	C₂×C₅	#1994	1	4	yes	direct product factor
263	C₂×C₅	C₂×#391	1	1	yes	characteristic
264	C₂×C₅	C₂²×#104	1	2	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
267	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
268	C₂³	C₅×C₂²⋊Q₈	1	4	yes	—
269	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
270	Q₈	—	4	16	no	—
271	C₂×C₄	—	2	16	no	—
272	C₂×C₄	—	2	32	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	C₅×#104	1	8	yes	—
280	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
281	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
284	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
285	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
286	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
287	C₂×C₄	C₅×#124	1	2	yes	—
288	C₂×C₄	C₅×#124	1	2	yes	—
289	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
290	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
291	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
294	C₂×C₄	C₂×C₅×#41	1	2	yes	—
295	C₈	—	4	4	no	—
296	C₅	C₂×#1994	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	8	no	—
300	C₂²	C₅×#391	1	2	yes	—
301	C₂²	C₅×#391	1	2	yes	—
302	C₂²	C₂×C₅×#104	1	4	yes	—
303	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
304	C₂²	C₂×C₅×C₂²⋊Q₈	1	2	yes	—
305	C₄	—	2	16	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	8	no	—
308	C₄	C₅×#391	1	2	yes	—
309	C₄	C₅×#391	1	2	yes	—
310	C₄	C₂×C₅×#104	1	8	yes	—
311	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
312	C₄	C₂×C₅×#124	1	1	yes	characteristic
313	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	C₅×#1994	1	4	yes	direct product factor
317	C₂	C₂×C₅×#391	1	1	yes	characteristic
318	C₂	C₂²×C₅×#104	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×#1994	C₂	direct (N × H)
9. C₅×#533	C₄	semidirect
11. C₂×C₅×C₄⋊Q₈	C₄	semidirect
12. C₂×C₅×#125	C₄	semidirect
25. C₂×#1994	C₅	direct (N × H)
35. C₅×C₄⋊Q₈	C₂×C₄	semidirect
37. C₅×#125	C₂×C₄	semidirect
61. C₂×C₄²×C₅	Q₈	semidirect
68. #1994	C₂×C₅	direct (N × H)
111. C₄²×C₅	C₂×Q₈	semidirect
115. #533	C₄×C₅	semidirect
117. C₂×C₄⋊Q₈	C₄×C₅	semidirect
118. C₂×#125	C₄×C₅	semidirect
171. C₄⋊Q₈	C₂×C₄×C₅	semidirect
173. #125	C₂×C₄×C₅	semidirect
197. C₂×C₄²	C₅×Q₈	semidirect
259. C₄²	C₂×C₅×Q₈	semidirect
262. C₂×C₅	#1994	direct (N × H)
296. C₅	C₂×#1994	direct (N × H)
316. C₂	C₅×#1994	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×C₄³×C₅	2
C₅×#1994	2
C₂×C₅×#339	2
C₂×#1994	5

Subgroups — C2×C5×#1994 #11944047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1994 #11944047