Subgroups — C₅×#19344 #11942064

Order 1280.

612 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3254	C₂	1	2	yes	maximal, complemented
2	C₅×#3169	C₂	1	1	yes	maximal, characteristic
3	C₅×#2960	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1698	C₂	1	4	yes	maximal, complemented
5	C₅×#1676	C₂	1	2	yes	maximal, complemented
6	C₅×#1668	C₂	1	2	yes	maximal, complemented
7	C₅×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1560	C₂	1	2	yes	maximal, complemented
9	C₅×#533	C₂²	1	1	yes	characteristic
10	C₅×#532	C₂²	1	2	yes	—
11	C₅×#500	C₂²	1	1	yes	characteristic
12	C₅×#493	C₂²	1	1	yes	characteristic
13	C₅×#486	C₂²	1	2	yes	—
14	C₅×#477	C₂²	1	4	yes	—
15	C₅×#474	—	2	2	no	—
16	C₅×#468	—	2	4	no	—
17	C₅×#466	—	2	1	no	—
18	C₅×#466	—	2	1	no	—
19	C₅×#461	—	2	2	no	—
20	C₅×#460	C₂²	1	4	yes	—
21	C₅×#446	—	2	1	no	—
22	C₅×#446	C₂²	1	2	yes	—
23	C₅×C₈×Q₈	—	2	1	no	—
24	C₅×C₈×Q₈	C₂²	1	2	yes	—
25	C₄×C₅×D₈	—	2	1	no	—
26	C₄×C₅×D₈	—	2	1	no	—
27	C₅×#424	—	2	1	no	—
28	C₅×#423	C₂²	1	2	yes	—
29	C₅×#423	C₂²	1	1	yes	characteristic
30	C₅×#405	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	2	yes	—
33	C₅×#335	C₂²	1	2	yes	—
34	C₅×#333	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19344	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂×C₅×SD₁₆	—	2	4	no	—
39	C₂×C₅×D₈	—	2	1	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
45	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
46	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₅×#127	C₂³	1	2	yes	—
49	C₅×#125	—	2	1	no	—
50	C₅×#124	—	2	2	no	—
51	C₅×C₂²⋊Q₈	—	2	1	no	—
52	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
53	C₅×#121	—	2	2	no	—
54	C₅×#121	—	2	2	no	—
55	C₄×C₅×Q₈	—	2	1	no	—
56	C₄×C₅×Q₈	C₂³	1	2	yes	—
57	C₄×C₅×D₄	—	2	2	no	—
58	C₄×C₅×D₄	—	2	2	no	—
59	C₅×#117	—	2	1	no	—
60	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
61	C₅×#107	—	2	1	no	—
62	C₅×#107	—	2	1	no	—
63	C₅×#107	—	2	4	no	—
64	C₅×C₄⋊C₈	—	2	2	no	—
65	C₅×C₄⋊C₈	—	2	2	no	—
66	C₅×C₄⋊C₈	—	2	1	no	—
67	C₅×C₄⋊C₈	—	2	2	no	—
68	C₅×C₄⋊C₈	C₂³	1	2	yes	—
69	C₅×C₄⋊C₈	C₂³	1	4	yes	—
70	C₅×#103	—	2	4	no	—
71	C₅×#102	—	2	2	no	—
72	C₅×#102	—	2	2	no	—
73	C₅×#102	—	2	4	no	—
74	C₅×#97	—	2	2	no	—
75	C₄×C₅×C₈	—	2	2	no	—
76	C₄×C₅×C₈	D₄	1	1	yes	characteristic
77	C₄×C₅×C₈	D₄	1	1	yes	characteristic
78	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	#3254	C₂×C₅	1	2	yes	complemented
81	#3169	C₂×C₅	1	1	yes	characteristic
82	#2960	C₂×C₅	1	1	yes	characteristic, complemented
83	#1698	C₂×C₅	1	4	yes	complemented
84	#1676	C₂×C₅	1	2	yes	complemented
85	#1668	C₂×C₅	1	2	yes	complemented
86	#1567	C₂×C₅	1	1	yes	characteristic, complemented
87	#1560	C₂×C₅	1	2	yes	complemented
88	C₅×#41	—	8	4	no	—
89	C₂×C₅×Q₈	—	2	1	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
92	C₂×C₅×D₄	—	2	2	no	—
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	4	2	no	—
96	C₂²×C₄×C₅	—	4	2	no	—
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	#41	1	1	yes	characteristic
100	C₅×SD₁₆	—	8	4	no	—
101	C₅×D₈	—	4	2	no	—
102	C₅×D₈	—	4	2	no	—
103	C₅×M₁₆	—	4	2	no	—
104	C₅×M₁₆	—	8	2	no	—
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	4	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	4	1	no	—
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
121	C₅×C₂²⋊C₄	—	2	2	no	—
122	C₅×C₂²⋊C₄	—	4	2	no	—
123	C₅×C₂²⋊C₄	—	4	2	no	—
124	C₄²×C₅	—	2	2	no	—
125	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₅	#41	1	1	yes	characteristic
128	C₄²×C₅	C₂⁴	1	1	yes	characteristic
129	#533	C₂²×C₅	1	1	yes	characteristic
130	#532	C₂²×C₅	1	2	yes	—
131	#500	C₂²×C₅	1	1	yes	characteristic
132	#493	C₂²×C₅	1	1	yes	characteristic
133	#486	C₂²×C₅	1	2	yes	—
134	#477	C₂²×C₅	1	4	yes	—
135	#474	—	2	2	no	—
136	#468	—	2	4	no	—
137	#466	—	2	1	no	—
138	#466	—	2	1	no	—
139	#461	—	2	2	no	—
140	#460	C₂²×C₅	1	4	yes	—
141	#446	—	2	1	no	—
142	#446	C₂²×C₅	1	2	yes	—
143	C₈×Q₈	—	2	1	no	—
144	C₈×Q₈	C₂²×C₅	1	2	yes	—
145	C₄×D₈	—	2	1	no	—
146	C₄×D₈	—	2	1	no	—
147	#424	—	2	1	no	—
148	#423	C₂²×C₅	1	1	yes	characteristic
149	#423	C₂²×C₅	1	2	yes	—
150	#405	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
152	#335	C₂²×C₅	1	2	yes	—
153	#335	C₂²×C₅	1	2	yes	—
154	#333	C₂²×C₅	1	4	yes	—
155	#331	C₂²×C₅	1	4	yes	—
156	C₂³×C₅	—	4	2	no	—
157	C₂³×C₅	#121	1	1	yes	characteristic
158	C₅×Q₈	—	2	2	no	—
159	C₅×Q₈	—	2	2	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₅×Q₈	—	8	2	no	—
162	C₅×D₄	—	4	2	no	—
163	C₅×D₄	—	4	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₅×D₄	—	4	2	no	—
166	C₅×D₄	—	8	2	no	—
167	C₅×D₄	—	8	4	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	8	4	no	—
177	C₂×C₄×C₅	#121	1	1	yes	characteristic
178	C₂×C₄×C₅	#121	1	1	yes	characteristic
179	C₂×C₄×C₅	#121	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
183	C₅×C₈	—	2	1	no	—
184	C₅×C₈	—	2	1	no	—
185	C₅×C₈	—	2	1	no	—
186	C₅×C₈	—	4	2	no	—
187	C₅×C₈	—	4	4	no	—
188	C₅×C₈	#137	1	2	yes	—
189	C₂×#41	—	2	2	no	—
190	C₂×SD₁₆	—	2	4	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	2	1	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
196	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
197	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
198	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
199	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
200	#127	C₂³×C₅	1	2	yes	—
201	#125	—	2	1	no	—
202	#124	—	2	2	no	—
203	C₂²⋊Q₈	—	2	1	no	—
204	C₂²⋊Q₈	C₅×D₄	1	2	yes	—
205	#121	—	2	2	no	—
206	#121	—	2	2	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	C₂³×C₅	1	2	yes	—
209	C₄×D₄	—	2	2	no	—
210	C₄×D₄	—	2	2	no	—
211	#117	—	2	1	no	—
212	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
213	#107	—	2	1	no	—
214	#107	—	2	1	no	—
215	#107	—	2	4	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	C₂³×C₅	1	2	yes	—
221	C₄⋊C₈	C₂³×C₅	1	4	yes	—
222	#103	—	2	4	no	—
223	#102	—	2	2	no	—
224	#102	—	2	2	no	—
225	#102	—	2	4	no	—
226	#97	—	2	2	no	—
227	C₄×C₈	—	2	2	no	—
228	C₄×C₈	C₅×D₄	1	1	yes	characteristic
229	C₄×C₈	C₅×D₄	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	8	2	no	—
236	C₂²×C₅	—	8	2	no	—
237	C₂²×C₅	—	8	2	no	—
238	C₂²×C₅	C₂×#121	1	1	yes	characteristic
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	2	1	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	—	4	2	no	—
244	C₄×C₅	—	8	2	no	—
245	C₄×C₅	#472	1	2	yes	—
246	C₄×C₅	C₂×#137	1	1	yes	characteristic
247	C₄×C₅	#575	1	1	yes	characteristic
248	C₄×C₅	#576	1	1	yes	characteristic
249	C₄×C₅	#578	1	1	yes	characteristic
250	#41	—	8	4	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
261	C₂²×C₄	C₅×#41	1	1	yes	characteristic
262	SD₁₆	—	8	4	no	—
263	D₈	—	4	2	no	—
264	D₈	—	4	2	no	—
265	M₁₆	—	4	2	no	—
266	M₁₆	—	8	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	4	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
288	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
289	C₄²	C₅×#41	1	1	yes	characteristic
290	C₄²	C₂⁴×C₅	1	1	yes	characteristic
291	C₂×C₅	—	4	1	no	—
292	C₂×C₅	—	16	2	no	—
293	C₂×C₅	#3037	1	1	yes	characteristic
294	C₂×C₅	#3061	1	1	yes	characteristic
295	C₂×C₅	C₂×#472	1	1	yes	characteristic
296	C₂³	—	4	2	no	—
297	C₂³	C₅×#121	1	1	yes	characteristic
298	Q₈	—	2	2	no	—
299	Q₈	—	2	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	8	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	D₄	—	4	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	8	2	no	—
307	D₄	—	8	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₅×#121	1	1	yes	characteristic
318	C₂×C₄	C₅×#121	1	1	yes	characteristic
319	C₂×C₄	C₅×#121	1	1	yes	characteristic
320	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
321	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
322	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	4	no	—
328	C₈	C₅×#137	1	2	yes	—
329	C₅	#19344	1	1	yes	characteristic, direct product factor, Sylow
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	C₂×C₅×#121	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	4	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₅×#472	1	2	yes	—
344	C₄	C₂×C₅×#137	1	1	yes	characteristic
345	C₄	C₅×#575	1	1	yes	characteristic
346	C₄	C₅×#576	1	1	yes	characteristic
347	C₄	C₅×#578	1	1	yes	characteristic
348	C₂	—	4	1	no	—
349	C₂	—	16	2	no	—
350	C₂	C₅×#3037	1	1	yes	characteristic
351	C₂	C₅×#3061	1	1	yes	characteristic
352	C₂	C₂×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3254	C₂	semidirect
3. C₅×#2960	C₂	semidirect
4. C₅×#1698	C₂	semidirect
5. C₅×#1676	C₂	semidirect
6. C₅×#1668	C₂	semidirect
7. C₅×#1567	C₂	semidirect
8. C₅×#1560	C₂	semidirect
36. #19344	C₅	direct (N × H)
80. #3254	C₂×C₅	semidirect
82. #2960	C₂×C₅	semidirect
83. #1698	C₂×C₅	semidirect
84. #1676	C₂×C₅	semidirect
85. #1668	C₂×C₅	semidirect
86. #1567	C₂×C₅	semidirect
87. #1560	C₂×C₅	semidirect
329. C₅	#19344	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#3254	2
C₅×#3169	2
C₅×#2960	2
C₅×#1698	2
C₅×#1676	2
C₅×#1668	2
C₅×#1567	2
C₅×#1560	2
#19344	5

Subgroups — C5×#19344 #11942064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19344 #11942064