Subgroups — C₅×#11376 #11934096

Order 1280.

654 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#339	C₂	1	2	yes	maximal
4	C₅×#1729	C₂	1	2	yes	maximal
5	C₂×C₅×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#427	—	2	2	no	—
12	C₅×#427	—	2	2	no	—
13	C₂×C₅×#107	C₄	1	1	yes	characteristic, complemented
14	C₂×C₅×#107	C₄	1	1	yes	characteristic, complemented
15	C₂×C₅×#106	C₄	1	2	yes	complemented
16	C₅×#341	—	2	2	no	—
17	C₅×#339	—	2	4	no	—
18	C₅×#336	—	2	2	no	—
19	C₅×#336	—	2	4	no	—
20	#11376	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	—	2	1	no	—
22	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
23	C₂×C₅×M₁₆	—	2	2	no	—
24	C₂×C₅×M₁₆	—	2	2	no	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	Q₈	1	4	yes	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	—	2	1	no	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	—	2	1	no	—
34	C₂²×C₅×C₈	—	2	2	no	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	D₄	1	2	yes	—
39	C₂²×C₅×C₈	D₄	1	2	yes	—
40	C₂²×C₅×C₈	D₄	1	2	yes	—
41	C₅×#117	—	2	2	no	—
42	C₅×#117	—	2	2	no	—
43	C₅×#117	—	2	2	no	—
44	C₅×#117	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₄²×C₅	—	2	2	no	—
49	C₂×C₄²×C₅	—	2	2	no	—
50	C₂×C₄²×C₅	—	2	1	no	—
51	C₂×C₄²×C₅	—	2	2	no	—
52	C₂×C₄²×C₅	D₄	1	2	yes	—
53	C₅×#107	—	2	2	no	—
54	C₅×#107	—	2	2	no	—
55	C₅×#106	—	2	4	no	—
56	C₂×#427	C₂×C₅	1	1	yes	characteristic
57	C₂×#341	C₂×C₅	1	1	yes	characteristic
58	C₂×#339	C₂×C₅	1	2	yes	—
59	#1729	C₂×C₅	1	2	yes	—
60	C₂×#336	C₂×C₅	1	1	yes	characteristic
61	C₂⁴×C₅	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	2	2	no	—
64	C₂²×C₄×C₅	—	2	2	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	—	4	2	no	—
68	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×C₄×C₅	#41	1	2	yes	—
76	C₂²×C₄×C₅	#41	1	1	yes	characteristic
77	C₂²×C₄×C₅	#41	1	2	yes	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	4	2	no	—
81	C₅×M₁₆	—	4	2	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	—	2	2	no	—
84	C₂×C₅×C₈	—	2	2	no	—
85	C₂×C₅×C₈	—	2	2	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	4	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	4	1	no	—
91	C₂×C₅×C₈	—	4	2	no	—
92	C₂×C₅×C₈	—	4	1	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₂×C₅×C₈	—	4	2	no	—
95	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
96	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₂²⋊C₄	—	4	2	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₄²×C₅	—	2	4	no	—
104	C₄²×C₅	—	4	2	no	—
105	C₄²×C₅	—	4	1	no	—
106	C₄²×C₅	—	4	1	no	—
107	C₄²×C₅	—	4	2	no	—
108	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
109	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×#117	C₂²×C₅	1	2	yes	—
112	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
113	#427	—	2	2	no	—
114	#427	—	2	2	no	—
115	C₂×#107	C₄×C₅	1	1	yes	characteristic, complemented
116	C₂×#107	C₄×C₅	1	1	yes	characteristic, complemented
117	C₂×#106	C₄×C₅	1	2	yes	complemented
118	#341	—	2	2	no	—
119	#339	—	2	4	no	—
120	#336	—	2	2	no	—
121	#336	—	2	4	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂³×C₅	#121	1	1	yes	characteristic
126	C₂³×C₅	#124	1	1	yes	characteristic
127	C₂³×C₅	#125	1	2	yes	—
128	C₂³×C₅	#126	1	2	yes	—
129	C₂×C₄×C₅	—	2	4	no	—
130	C₂×C₄×C₅	—	4	8	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	4	no	—
136	C₂×C₄×C₅	—	4	4	no	—
137	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	#117	1	2	yes	—
139	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₅	#121	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
145	C₂×C₄×C₅	#123	1	1	yes	characteristic
146	C₂×C₄×C₅	#123	1	1	yes	characteristic
147	C₂×C₄×C₅	#123	1	2	yes	—
148	C₂×C₄×C₅	#123	1	2	yes	—
149	C₂×C₄×C₅	#123	1	2	yes	—
150	C₂×C₄×C₅	#124	1	1	yes	characteristic
151	C₂×C₄×C₅	#125	1	2	yes	—
152	C₂×C₄×C₅	#126	1	2	yes	—
153	C₂×C₄×C₅	#127	1	1	yes	characteristic
154	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	2	2	no	—
157	C₅×C₈	—	4	2	no	—
158	C₅×C₈	—	4	4	no	—
159	C₂³×C₄	—	2	1	no	—
160	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	C₅×Q₈	1	4	yes	—
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂²×C₈	C₅×D₄	1	2	yes	—
177	C₂²×C₈	C₅×D₄	1	2	yes	—
178	C₂²×C₈	C₅×D₄	1	2	yes	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	1	no	—
190	C₂×C₄²	C₅×D₄	1	2	yes	—
191	#107	—	2	2	no	—
192	#107	—	2	2	no	—
193	#106	—	2	4	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	—	2	4	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	#387	1	2	yes	—
201	C₂²×C₅	#390	1	1	yes	characteristic
202	C₂²×C₅	#397	1	1	yes	characteristic
203	C₂²×C₅	#398	1	1	yes	characteristic
204	C₂²×C₅	#400	1	2	yes	—
205	C₂²×C₅	#443	1	2	yes	—
206	C₂²×C₅	#444	1	2	yes	—
207	C₄×C₅	—	4	4	no	—
208	C₄×C₅	—	8	2	no	—
209	C₄×C₅	—	8	2	no	—
210	C₄×C₅	#381	1	1	yes	characteristic
211	C₄×C₅	#387	1	2	yes	—
212	C₄×C₅	#391	1	1	yes	characteristic
213	C₄×C₅	#397	1	1	yes	characteristic
214	C₄×C₅	#398	1	1	yes	characteristic
215	C₄×C₅	#399	1	2	yes	—
216	C₂⁴	C₅×#41	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
229	C₂²×C₄	C₅×#41	1	2	yes	—
230	C₂²×C₄	C₅×#41	1	2	yes	—
231	C₂²×C₄	C₅×#41	1	1	yes	characteristic
232	C₂²×C₄	C₅×#41	1	2	yes	—
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	4	no	—
235	M₁₆	—	4	2	no	—
236	M₁₆	—	4	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
251	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	4	2	no	—
260	C₄²	—	4	1	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	2	no	—
263	C₂×C₅	—	2	1	no	—
264	C₂×C₅	—	2	1	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	#1445	1	1	yes	characteristic
267	C₂×C₅	#1922	1	4	yes	—
268	C₂×C₅	#1974	1	1	yes	characteristic
269	C₂×C₅	#1975	1	1	yes	characteristic
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
273	C₂³	C₅×#121	1	1	yes	characteristic
274	C₂³	C₅×#124	1	1	yes	characteristic
275	C₂³	C₅×#125	1	2	yes	—
276	C₂³	C₅×#126	1	2	yes	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
286	C₂×C₄	C₅×#117	1	2	yes	—
287	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
288	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
289	C₂×C₄	C₅×#121	1	1	yes	characteristic
290	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
291	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
292	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
293	C₂×C₄	C₅×#123	1	1	yes	characteristic
294	C₂×C₄	C₅×#123	1	1	yes	characteristic
295	C₂×C₄	C₅×#123	1	2	yes	—
296	C₂×C₄	C₅×#123	1	2	yes	—
297	C₂×C₄	C₅×#123	1	2	yes	—
298	C₂×C₄	C₅×#124	1	1	yes	characteristic
299	C₂×C₄	C₅×#125	1	2	yes	—
300	C₂×C₄	C₅×#126	1	2	yes	—
301	C₂×C₄	C₅×#127	1	1	yes	characteristic
302	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	2	2	no	—
305	C₈	—	4	2	no	—
306	C₈	—	4	4	no	—
307	C₅	#11376	1	1	yes	characteristic, direct product factor, Sylow
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	C₅×#387	1	2	yes	—
315	C₂²	C₅×#390	1	1	yes	characteristic
316	C₂²	C₅×#397	1	1	yes	characteristic
317	C₂²	C₅×#398	1	1	yes	characteristic
318	C₂²	C₅×#400	1	2	yes	—
319	C₂²	C₅×#443	1	2	yes	—
320	C₂²	C₅×#444	1	2	yes	—
321	C₄	—	4	4	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₅×#381	1	1	yes	characteristic
325	C₄	C₅×#387	1	2	yes	—
326	C₄	C₅×#391	1	1	yes	characteristic
327	C₄	C₅×#397	1	1	yes	characteristic
328	C₄	C₅×#398	1	1	yes	characteristic
329	C₄	C₅×#399	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	C₅×#1445	1	1	yes	characteristic
334	C₂	C₅×#1922	1	4	yes	—
335	C₂	C₅×#1974	1	1	yes	characteristic
336	C₂	C₅×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₄×D₄
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×C₅×#107	C₄	semidirect
14. C₂×C₅×#107	C₄	semidirect
15. C₂×C₅×#106	C₄	semidirect
20. #11376	C₅	direct (N × H)
115. C₂×#107	C₄×C₅	semidirect
116. C₂×#107	C₄×C₅	semidirect
117. C₂×#106	C₄×C₅	semidirect
307. C₅	#11376	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×#427	2
C₂×C₅×#341	2
C₂×C₅×#339	2
C₅×#1729	2
C₂×C₅×#336	2
#11376	5

Subgroups — C5×#11376 #11934096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11376 #11934096