Subgroups — C₂×C₅×#1951 #11943899

Order 1280.

918 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#428	C₂	1	2	yes	maximal
3	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
5	C₅×#1951	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#343	C₂	1	2	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×#108	C₄	1	8	yes	complemented
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₅×#428	—	2	4	no	—
15	C₅×#428	C₂²	1	8	yes	—
16	C₂×C₅×#107	—	2	2	no	—
17	C₂×C₅×#106	—	2	1	no	—
18	C₂×C₅×#106	—	2	1	no	—
19	C₂×C₅×#106	—	2	2	no	—
20	C₂×C₅×#106	—	2	2	no	—
21	C₂×C₅×#106	C₂²	1	4	yes	—
22	C₅×#405	—	2	2	no	—
23	C₅×#405	C₂²	1	4	yes	—
24	C₂×C₅×#97	—	2	1	no	—
25	C₂×C₄×C₅×C₈	—	2	1	no	—
26	C₅×#343	C₂²	1	8	yes	—
27	C₂×#1951	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂³×C₄×C₅	—	2	2	no	—
29	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×M₁₆	—	2	1	no	—
31	C₂×C₅×M₁₆	—	2	2	no	—
32	C₂×C₅×M₁₆	—	2	4	no	—
33	C₂×C₅×M₁₆	—	2	2	no	—
34	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
35	C₂×C₅×M₁₆	Q₈	1	4	yes	—
36	C₂×C₅×M₁₆	C₂³	1	4	yes	—
37	C₂×C₅×M₁₆	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	—	2	1	no	—
39	C₂²×C₅×C₈	—	2	2	no	—
40	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
41	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂²×C₅×C₈	D₄	1	2	yes	—
44	C₂²×C₅×C₈	Q₈	1	4	yes	—
45	C₂²×C₅×C₈	C₂³	1	2	yes	—
46	C₅×#117	—	2	1	no	—
47	C₅×#117	—	2	1	no	—
48	C₅×#117	D₄	1	2	yes	—
49	C₅×#117	C₂³	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	—	2	2	no	—
51	C₂×C₅×C₄⋊C₄	—	2	2	no	—
52	C₂×C₅×C₄⋊C₄	—	2	2	no	—
53	C₂×C₅×C₄⋊C₄	—	2	2	no	—
54	C₂×C₅×C₄⋊C₄	—	2	2	no	—
55	C₂×C₅×C₄⋊C₄	D₄	1	2	yes	—
56	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
57	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
58	C₂×C₄²×C₅	—	2	1	no	—
59	C₅×#108	C₂×C₄	1	16	yes	complemented
60	C₅×#107	—	2	8	no	—
61	C₅×#106	—	2	4	no	—
62	C₅×#106	—	2	4	no	—
63	C₅×#106	—	2	8	no	—
64	C₅×#97	—	2	4	no	—
65	C₄×C₅×C₈	—	2	4	no	—
66	C₂²×#108	C₂×C₅	1	1	yes	characteristic
67	C₂×#428	C₂×C₅	1	2	yes	—
68	C₂²×#106	C₂×C₅	1	1	yes	characteristic
69	C₂×#405	C₂×C₅	1	1	yes	characteristic
70	#1951	C₂×C₅	1	8	yes	direct product factor
71	C₂×#343	C₂×C₅	1	2	yes	—
72	C₂⁴×C₅	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅	—	2	2	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	—	4	2	no	—
76	C₂²×C₄×C₅	—	4	2	no	—
77	C₂²×C₄×C₅	—	8	4	no	—
78	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅	#41	1	2	yes	—
86	C₂²×C₄×C₅	#41	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₅×M₁₆	—	2	8	no	—
89	C₅×M₁₆	—	2	4	no	—
90	C₅×M₁₆	—	2	4	no	—
91	C₅×M₁₆	—	2	2	no	—
92	C₅×M₁₆	C₂²×C₄	1	8	yes	—
93	C₅×M₁₆	C₂×Q₈	1	4	yes	—
94	C₂×C₅×C₈	—	2	8	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	4	no	—
98	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
99	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
100	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
101	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
102	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
103	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
104	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₅×C₄⋊C₄	—	4	2	no	—
110	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
111	C₅×C₂²⋊C₄	—	2	2	no	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
116	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
117	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×#117	C₂²×C₅	1	1	yes	characteristic
119	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
120	C₂×#108	C₄×C₅	1	8	yes	complemented
121	C₂×#108	C₂²×C₅	1	4	yes	—
122	#428	—	2	4	no	—
123	#428	C₂²×C₅	1	8	yes	—
124	C₂×#107	—	2	2	no	—
125	C₂×#106	—	2	1	no	—
126	C₂×#106	—	2	1	no	—
127	C₂×#106	—	2	2	no	—
128	C₂×#106	—	2	2	no	—
129	C₂×#106	C₂²×C₅	1	4	yes	—
130	#405	—	2	2	no	—
131	#405	C₂²×C₅	1	4	yes	—
132	C₂×#97	—	2	1	no	—
133	C₂×C₄×C₈	—	2	1	no	—
134	#343	C₂²×C₅	1	8	yes	—
135	C₂³×C₅	—	2	4	no	—
136	C₂³×C₅	C₄×Q₈	1	2	yes	—
137	C₂³×C₅	#121	1	1	yes	characteristic
138	C₂³×C₅	#121	1	1	yes	characteristic
139	C₂³×C₅	#125	1	2	yes	—
140	C₂³×C₅	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅	—	2	4	no	—
142	C₂×C₄×C₅	—	4	4	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	8	4	no	—
145	C₂×C₄×C₅	—	8	4	no	—
146	C₂×C₄×C₅	—	8	4	no	—
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
149	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
152	C₂×C₄×C₅	#121	1	1	yes	characteristic
153	C₂×C₄×C₅	#121	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
155	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
156	C₂×C₄×C₅	#125	1	2	yes	—
157	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
158	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
159	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
165	C₅×C₈	—	2	8	no	—
166	C₅×C₈	—	2	4	no	—
167	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
168	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	4	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
176	C₂×M₁₆	C₅×Q₈	1	4	yes	—
177	C₂×M₁₆	C₂³×C₅	1	4	yes	—
178	C₂×M₁₆	C₂³×C₅	1	2	yes	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
182	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
183	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
184	C₂²×C₈	C₅×D₄	1	2	yes	—
185	C₂²×C₈	C₅×Q₈	1	4	yes	—
186	C₂²×C₈	C₂³×C₅	1	2	yes	—
187	#117	—	2	1	no	—
188	#117	—	2	1	no	—
189	#117	C₅×D₄	1	2	yes	—
190	#117	C₂³×C₅	1	2	yes	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₅×D₄	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	#108	C₂×C₄×C₅	1	16	yes	complemented
201	#107	—	2	8	no	—
202	#106	—	2	4	no	—
203	#106	—	2	4	no	—
204	#106	—	2	8	no	—
205	#97	—	2	4	no	—
206	C₄×C₈	—	2	4	no	—
207	C₂²×C₅	—	2	4	no	—
208	C₂²×C₅	—	2	4	no	—
209	C₂²×C₅	—	2	2	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	#389	1	2	yes	—
212	C₂²×C₅	#389	1	2	yes	—
213	C₂²×C₅	#471	1	4	yes	—
214	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
215	C₂²×C₅	C₂×#121	1	1	yes	characteristic
216	C₂²×C₅	C₂×#125	1	1	yes	characteristic
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	—	8	4	no	—
219	C₄×C₅	#389	1	2	yes	—
220	C₄×C₅	#389	1	2	yes	—
221	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
222	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
223	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
224	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
225	C₂⁴	C₅×#41	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	8	4	no	—
231	C₂²×C₄	C₂²×C₄×C₅	1	2	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×D₄	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
237	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
238	C₂²×C₄	C₅×#41	1	2	yes	—
239	C₂²×C₄	C₅×#41	1	1	yes	characteristic
240	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
241	M₁₆	—	2	8	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	2	4	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	C₂²×C₄×C₅	1	8	yes	—
246	M₁₆	C₂×C₅×Q₈	1	4	yes	—
247	C₂×C₈	—	2	8	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
252	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
253	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
254	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
255	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
256	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
257	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₂×C₅	—	2	1	no	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	2	no	—
271	C₂×C₅	#1951	1	4	yes	direct product factor
272	C₂×C₅	C₂×#389	1	1	yes	characteristic
273	C₂×C₅	C₂×#471	1	2	yes	—
274	C₂³	—	2	4	no	—
275	C₂³	C₄×C₅×Q₈	1	2	yes	—
276	C₂³	C₅×#121	1	1	yes	characteristic
277	C₂³	C₅×#121	1	1	yes	characteristic
278	C₂³	C₅×#125	1	2	yes	—
279	C₂³	C₂×C₅×#41	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	8	4	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
287	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
288	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
289	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
290	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
291	C₂×C₄	C₅×#121	1	1	yes	characteristic
292	C₂×C₄	C₅×#121	1	1	yes	characteristic
293	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
294	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
295	C₂×C₄	C₅×#125	1	2	yes	—
296	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
297	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
298	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
303	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
304	C₈	—	2	8	no	—
305	C₈	—	2	4	no	—
306	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
307	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
308	C₅	C₂×#1951	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	C₅×#389	1	2	yes	—
314	C₂²	C₅×#389	1	2	yes	—
315	C₂²	C₅×#471	1	4	yes	—
316	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
317	C₂²	C₂×C₅×#121	1	1	yes	characteristic
318	C₂²	C₂×C₅×#125	1	1	yes	characteristic
319	C₄	—	4	4	no	—
320	C₄	—	8	4	no	—
321	C₄	C₅×#389	1	2	yes	—
322	C₄	C₅×#389	1	2	yes	—
323	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
324	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
325	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
326	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
327	C₂	—	2	2	no	—
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	C₅×#1951	1	4	yes	direct product factor
331	C₂	C₂×C₅×#389	1	1	yes	characteristic
332	C₂	C₂×C₅×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	#121
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1951	C₂	direct (N × H)
12. C₂×C₅×#108	C₄	semidirect
27. C₂×#1951	C₅	direct (N × H)
59. C₅×#108	C₂×C₄	semidirect
70. #1951	C₂×C₅	direct (N × H)
120. C₂×#108	C₄×C₅	semidirect
200. #108	C₂×C₄×C₅	semidirect
271. C₂×C₅	#1951	direct (N × H)
308. C₅	C₂×#1951	direct (N × H)
330. C₂	C₅×#1951	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×C₅×#108	2
C₂×C₅×#428	2
C₂²×C₅×#106	2
C₂×C₅×#405	2
C₅×#1951	2
C₂×C₅×#343	2
C₂×#1951	5

Subgroups — C2×C5×#1951 #11943899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1951 #11943899