Subgroups — C₅×#21714 #11944434

Order 1280.

810 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#1984	C₂	1	1	yes	maximal, characteristic
5	C₅×#1984	C₂	1	1	yes	maximal, characteristic
6	C₅×#1940	C₂	1	4	yes	maximal
7	C₅×#1842	C₂	1	1	yes	maximal, characteristic
8	C₅×#1771	C₂	1	4	yes	maximal
9	C₄×C₅×#97	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₅×#446	—	2	4	no	—
14	C₅×C₈×Q₈	—	2	4	no	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₅×#97	C₂²	1	2	yes	—
21	C₂×C₅×#97	C₂²	1	2	yes	—
22	C₂×C₅×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
26	C₅×#391	C₄	1	1	yes	characteristic
27	C₅×#391	C₄	1	1	yes	characteristic
28	C₅×#389	C₄	1	4	yes	—
29	C₅×#384	C₄	1	1	yes	characteristic
30	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
31	C₄³×C₅	C₂²	1	1	yes	characteristic
32	C₅×#336	C₂²	1	4	yes	—
33	C₅×#336	C₂²	1	4	yes	—
34	#21714	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Q₈	—	2	4	no	—
43	C₄×C₅×Q₈	—	2	2	no	—
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₄×C₅×Q₈	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
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	C₂³	1	2	yes	—
57	C₂×C₄²×C₅	C₂³	1	2	yes	—
58	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅	C₂³	1	2	yes	—
60	C₅×C₄⋊C₈	—	2	4	no	—
61	C₅×C₄⋊C₈	—	2	4	no	—
62	C₅×C₄⋊C₈	—	2	2	no	—
63	C₅×C₄⋊C₈	—	2	2	no	—
64	C₅×C₄⋊C₈	—	2	8	no	—
65	C₅×#97	—	2	4	no	—
66	C₅×#97	—	2	4	no	—
67	C₅×#97	Q₈	1	8	yes	complemented
68	C₄×C₅×C₈	—	2	4	no	—
69	C₄×C₅×C₈	—	2	2	no	—
70	C₄×C₅×C₈	D₄	1	4	yes	—
71	C₅×#95	C₂×C₄	1	4	yes	—
72	C₅×#95	C₂×C₄	1	4	yes	—
73	C₂×#446	C₂×C₅	1	1	yes	characteristic
74	C₂×C₈×Q₈	C₂×C₅	1	1	yes	characteristic
75	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
76	#1984	C₂×C₅	1	1	yes	characteristic
77	#1984	C₂×C₅	1	1	yes	characteristic
78	#1940	C₂×C₅	1	4	yes	—
79	#1842	C₂×C₅	1	1	yes	characteristic
80	#1771	C₂×C₅	1	4	yes	—
81	C₄×#97	C₂×C₅	1	1	yes	characteristic
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂×C₅×Q₈	—	2	2	no	—
87	C₂×C₅×Q₈	M₁₆	1	4	yes	—
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	4	no	—
103	C₂×C₅×C₈	—	2	4	no	—
104	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
105	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
106	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
107	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	2	8	no	—
113	C₅×C₄⋊C₄	—	4	4	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	2	2	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	—	2	2	no	—
119	C₄²×C₅	—	2	2	no	—
120	C₄²×C₅	—	4	2	no	—
121	C₄²×C₅	C₂×D₄	1	2	yes	—
122	C₄²×C₅	C₂×Q₈	1	4	yes	—
123	C₄²×C₅	#41	1	2	yes	—
124	C₄²×C₅	#41	1	4	yes	—
125	C₂×C₄⋊Q₈	C₄×C₅	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
127	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
128	#446	—	2	4	no	—
129	C₈×Q₈	—	2	4	no	—
130	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
133	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
134	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
135	C₂×#97	C₂²×C₅	1	2	yes	—
136	C₂×#97	C₂²×C₅	1	2	yes	—
137	C₂×#97	C₂²×C₅	1	2	yes	—
138	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	#391	C₄×C₅	1	1	yes	characteristic
142	#391	C₄×C₅	1	1	yes	characteristic
143	#389	C₄×C₅	1	4	yes	—
144	#384	C₄×C₅	1	1	yes	characteristic
145	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
146	C₄³	C₂²×C₅	1	1	yes	characteristic
147	#336	C₂²×C₅	1	4	yes	—
148	#336	C₂²×C₅	1	4	yes	—
149	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
150	C₅×Q₈	—	2	8	no	—
151	C₅×Q₈	—	4	4	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
169	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
170	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
171	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
172	C₂×C₄×C₅	#131	1	2	yes	—
173	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×Q₈	1	2	yes	—
175	C₂×C₄×C₅	C₂×#41	1	2	yes	—
176	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
177	C₅×C₈	—	2	2	no	—
178	C₅×C₈	—	2	4	no	—
179	C₅×C₈	—	4	2	no	—
180	C₅×C₈	—	4	2	no	—
181	C₅×C₈	C₄⋊Q₈	1	4	yes	complemented
182	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
183	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₅	1	2	yes	—
185	C₂²×C₈	C₂³×C₅	1	2	yes	—
186	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₅	1	2	yes	—
188	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	4	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
200	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
202	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
203	C₂×C₄²	C₂³×C₅	1	2	yes	—
204	C₂×C₄²	C₂³×C₅	1	2	yes	—
205	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₅	1	2	yes	—
207	C₄⋊C₈	—	2	4	no	—
208	C₄⋊C₈	—	2	4	no	—
209	C₄⋊C₈	—	2	2	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	8	no	—
212	#97	—	2	4	no	—
213	#97	—	2	4	no	—
214	#97	C₅×Q₈	1	8	yes	complemented
215	C₄×C₈	—	2	4	no	—
216	C₄×C₈	—	2	2	no	—
217	C₄×C₈	C₅×D₄	1	4	yes	—
218	#95	C₂×C₄×C₅	1	4	yes	—
219	#95	C₂×C₄×C₅	1	4	yes	—
220	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₅	C₂×C₄×Q₈	1	2	yes	—
222	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
223	C₂²×C₅	C₂×#131	1	1	yes	characteristic
224	C₂²×C₅	#568	1	2	yes	—
225	C₄×C₅	—	2	4	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	2	4	no	—
229	C₄×C₅	—	4	2	no	—
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	#435	1	4	yes	—
233	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
234	C₄×C₅	#532	1	1	yes	characteristic
235	C₄×C₅	#533	1	2	yes	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	C₅×M₁₆	1	4	yes	—
242	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
245	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
249	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
259	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
260	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
261	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×C₅×D₄	1	2	yes	—
276	C₄²	C₂×C₅×Q₈	1	4	yes	—
277	C₄²	C₅×#41	1	2	yes	—
278	C₄²	C₅×#41	1	4	yes	—
279	C₂×C₅	C₄×C₄⋊Q₈	1	1	yes	characteristic
280	C₂×C₅	C₂×#435	1	1	yes	characteristic
281	C₂×C₅	#2932	1	1	yes	characteristic
282	C₂×C₅	#2962	1	2	yes	—
283	C₂×C₅	Q₈×M₁₆	1	2	yes	—
284	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
285	Q₈	—	2	8	no	—
286	Q₈	—	4	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
302	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
303	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
304	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
305	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
306	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
307	C₂×C₄	C₅×#131	1	2	yes	—
308	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
309	C₂×C₄	C₂²×C₅×Q₈	1	2	yes	—
310	C₂×C₄	C₂×C₅×#41	1	2	yes	—
311	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
312	C₈	—	2	2	no	—
313	C₈	—	2	4	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	2	no	—
316	C₈	C₅×C₄⋊Q₈	1	4	yes	complemented
317	C₅	#21714	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
319	C₂²	C₂×C₄×C₅×Q₈	1	2	yes	—
320	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
321	C₂²	C₂×C₅×#131	1	1	yes	characteristic
322	C₂²	C₅×#568	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	4	no	—
326	C₄	—	2	4	no	—
327	C₄	—	4	2	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	C₅×#435	1	4	yes	—
331	C₄	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
332	C₄	C₅×#532	1	1	yes	characteristic
333	C₄	C₅×#533	1	2	yes	—
334	C₂	C₄×C₅×C₄⋊Q₈	1	1	yes	characteristic
335	C₂	C₂×C₅×#435	1	1	yes	characteristic
336	C₂	C₅×#2932	1	1	yes	characteristic
337	C₂	C₅×#2962	1	2	yes	—
338	C₂	C₅×Q₈×M₁₆	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
34. #21714	C₅	direct (N × H)
67. C₅×#97	Q₈	semidirect
181. C₅×C₈	C₄⋊Q₈	semidirect
214. #97	C₅×Q₈	semidirect
316. C₈	C₅×C₄⋊Q₈	semidirect
317. C₅	#21714	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×#446	2
C₂×C₅×C₈×Q₈	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1984	2
C₅×#1940	2
C₅×#1842	2
C₅×#1771	2
C₄×C₅×#97	2
#21714	5

Subgroups — C5×#21714 #11944434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21714 #11944434