Subgroups — C₅×#19354 #11942074

Order 1280.

648 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3239	C₂	1	2	yes	maximal
2	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2966	C₂	1	1	yes	maximal, characteristic
4	C₅×#1690	C₂	1	2	yes	maximal, complemented
5	C₅×#1689	C₂	1	2	yes	maximal, complemented
6	C₅×#1670	C₂	1	2	yes	maximal, complemented
7	C₅×#1669	C₂	1	2	yes	maximal, complemented
8	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1556	C₂	1	2	yes	maximal, complemented
10	C₂²×C₅×Q₁₆	—	2	1	no	—
11	C₂²×C₅×Q₁₆	—	2	1	no	—
12	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₅×#529	C₂²	1	1	yes	characteristic
14	C₅×#501	—	2	2	no	—
15	C₅×#494	—	2	1	no	—
16	C₅×#494	—	2	1	no	—
17	C₅×#494	C₂²	1	2	yes	—
18	C₅×#494	C₂²	1	2	yes	—
19	C₅×#489	—	2	2	no	—
20	C₅×#484	—	2	2	no	—
21	C₅×#481	—	2	2	no	—
22	C₅×#475	C₂²	1	4	yes	—
23	C₅×#474	C₂²	1	4	yes	—
24	C₅×#446	C₂²	1	1	yes	characteristic
25	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
26	C₅×#436	C₂²	1	1	yes	characteristic
27	C₅×#435	—	2	1	no	—
28	C₅×C₈×D₄	—	2	1	no	—
29	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
30	C₅×#433	—	2	1	no	—
31	C₅×#432	C₂²	1	2	yes	—
32	C₅×#432	C₂²	1	1	yes	characteristic
33	C₅×#405	—	2	1	no	—
34	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
35	C₅×#335	C₂²	1	2	yes	—
36	C₅×#334	C₂²	1	2	yes	—
37	C₅×#332	C₂²	1	4	yes	—
38	C₅×#332	C₂²	1	4	yes	—
39	#19354	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₅×Q₈	—	2	2	no	—
41	C₂×C₅×Q₁₆	—	2	2	no	—
42	C₂×C₅×Q₁₆	—	2	2	no	—
43	C₂×C₅×Q₁₆	—	2	2	no	—
44	C₂×C₅×Q₁₆	—	2	2	no	—
45	C₂×C₅×Q₁₆	—	4	4	no	—
46	C₂×C₅×Q₁₆	—	4	4	no	—
47	C₂×C₅×M₁₆	—	2	1	no	—
48	C₂²×C₅×C₈	—	2	1	no	—
49	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
50	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
51	C₅×C₄⋊Q₈	—	2	2	no	—
52	C₅×C₄⋊Q₈	—	2	2	no	—
53	C₅×C₄⋊Q₈	—	2	2	no	—
54	C₅×C₄⋊Q₈	C₂³	1	4	yes	—
55	C₅×#126	—	2	1	no	—
56	C₅×#125	D₄	1	1	yes	characteristic
57	C₅×#124	D₄	1	1	yes	characteristic
58	C₅×#123	—	2	1	no	—
59	C₅×C₂²⋊Q₈	D₄	1	1	yes	characteristic
60	C₅×#121	D₄	1	1	yes	characteristic
61	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₄×C₅×D₄	—	2	1	no	—
63	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₅×#117	—	2	1	no	—
65	C₂×C₅×C₄⋊C₄	—	2	2	no	—
66	C₂×C₅×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₅×#107	—	2	4	no	—
69	C₅×#106	—	2	4	no	—
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	—	2	1	no	—
72	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
73	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
74	C₅×C₄⋊C₈	C₂³	1	4	yes	—
75	C₅×#103	—	2	4	no	—
76	C₅×#102	—	2	4	no	—
77	C₅×C₂²⋊C₈	—	2	1	no	—
78	C₅×C₂²⋊C₈	—	2	2	no	—
79	C₅×C₂²⋊C₈	—	2	1	no	—
80	C₅×C₂²⋊C₈	—	2	1	no	—
81	C₅×#97	—	2	1	no	—
82	C₅×#97	—	2	1	no	—
83	C₅×#97	—	2	2	no	—
84	C₄×C₅×C₈	—	2	1	no	—
85	C₄×C₅×C₈	D₄	1	1	yes	characteristic
86	C₄×C₅×C₈	D₄	1	1	yes	characteristic
87	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
88	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
89	#3239	C₂×C₅	1	2	yes	—
90	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
91	#2966	C₂×C₅	1	1	yes	characteristic
92	#1690	C₂×C₅	1	2	yes	complemented
93	#1689	C₂×C₅	1	2	yes	complemented
94	#1670	C₂×C₅	1	2	yes	complemented
95	#1669	C₂×C₅	1	2	yes	complemented
96	#1568	C₂×C₅	1	1	yes	characteristic, complemented
97	#1556	C₂×C₅	1	2	yes	complemented
98	C₂×C₅×Q₈	—	2	4	no	—
99	C₂×C₅×Q₈	—	4	4	no	—
100	C₂×C₅×Q₈	—	4	4	no	—
101	C₂×C₅×Q₈	—	4	2	no	—
102	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₅×D₄	—	2	1	no	—
104	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	4	2	no	—
107	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	#41	1	1	yes	characteristic
110	C₅×Q₁₆	—	4	4	no	—
111	C₅×Q₁₆	—	4	4	no	—
112	C₅×Q₁₆	—	4	4	no	—
113	C₅×Q₁₆	—	4	4	no	—
114	C₅×M₁₆	—	4	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	4	no	—
120	C₂×C₅×C₈	—	4	2	no	—
121	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	2	1	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	—	2	4	no	—
130	C₅×C₄⋊C₄	—	2	4	no	—
131	C₅×C₄⋊C₄	—	4	2	no	—
132	C₅×C₄⋊C₄	—	4	2	no	—
133	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
134	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
135	C₅×C₂²⋊C₄	—	2	1	no	—
136	C₅×C₂²⋊C₄	—	2	1	no	—
137	C₅×C₂²⋊C₄	—	2	1	no	—
138	C₅×C₂²⋊C₄	—	2	1	no	—
139	C₅×C₂²⋊C₄	—	2	1	no	—
140	C₄²×C₅	—	2	1	no	—
141	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
142	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
143	C₄²×C₅	#41	1	1	yes	characteristic
144	C₄²×C₅	C₂⁴	1	1	yes	characteristic
145	C₂²×Q₁₆	—	2	1	no	—
146	C₂²×Q₁₆	—	2	1	no	—
147	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
148	#529	C₂²×C₅	1	1	yes	characteristic
149	#501	—	2	2	no	—
150	#494	—	2	1	no	—
151	#494	—	2	1	no	—
152	#494	C₂²×C₅	1	2	yes	—
153	#494	C₂²×C₅	1	2	yes	—
154	#489	—	2	2	no	—
155	#484	—	2	2	no	—
156	#481	—	2	2	no	—
157	#475	C₂²×C₅	1	4	yes	—
158	#474	C₂²×C₅	1	4	yes	—
159	#446	C₂²×C₅	1	1	yes	characteristic
160	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
161	#436	C₂²×C₅	1	1	yes	characteristic
162	#435	—	2	1	no	—
163	C₈×D₄	—	2	1	no	—
164	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
165	#433	—	2	1	no	—
166	#432	C₂²×C₅	1	1	yes	characteristic
167	#432	C₂²×C₅	1	2	yes	—
168	#405	—	2	1	no	—
169	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
170	#335	C₂²×C₅	1	2	yes	—
171	#334	C₂²×C₅	1	2	yes	—
172	#332	C₂²×C₅	1	4	yes	—
173	#332	C₂²×C₅	1	4	yes	—
174	C₂³×C₅	—	2	1	no	—
175	C₂³×C₅	#121	1	1	yes	characteristic
176	C₅×Q₈	—	2	2	no	—
177	C₅×Q₈	—	4	4	no	—
178	C₅×Q₈	—	4	4	no	—
179	C₅×Q₈	—	8	4	no	—
180	C₅×D₄	—	2	2	no	—
181	C₅×D₄	—	4	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	1	no	—
184	C₂×C₄×C₅	—	2	1	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	2	1	no	—
187	C₂×C₄×C₅	—	2	1	no	—
188	C₂×C₄×C₅	—	2	1	no	—
189	C₂×C₄×C₅	—	4	1	no	—
190	C₂×C₄×C₅	—	4	1	no	—
191	C₂×C₄×C₅	—	4	4	no	—
192	C₂×C₄×C₅	—	8	4	no	—
193	C₂×C₄×C₅	#121	1	1	yes	characteristic
194	C₂×C₄×C₅	#121	1	1	yes	characteristic
195	C₂×C₄×C₅	#121	1	1	yes	characteristic
196	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
199	C₅×C₈	—	2	1	no	—
200	C₅×C₈	—	2	1	no	—
201	C₅×C₈	—	4	1	no	—
202	C₅×C₈	—	4	1	no	—
203	C₅×C₈	—	4	4	no	—
204	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
205	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
206	C₂²×Q₈	—	2	2	no	—
207	C₂×Q₁₆	—	2	2	no	—
208	C₂×Q₁₆	—	2	2	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	2	no	—
211	C₂×Q₁₆	—	4	4	no	—
212	C₂×Q₁₆	—	4	4	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂²×C₈	—	2	1	no	—
215	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
216	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	C₂³×C₅	1	4	yes	—
221	#126	—	2	1	no	—
222	#125	C₅×D₄	1	1	yes	characteristic
223	#124	C₅×D₄	1	1	yes	characteristic
224	#123	—	2	1	no	—
225	C₂²⋊Q₈	C₅×D₄	1	1	yes	characteristic
226	#121	C₅×D₄	1	1	yes	characteristic
227	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
228	C₄×D₄	—	2	1	no	—
229	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
230	#117	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	—	2	2	no	—
233	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
234	#107	—	2	4	no	—
235	#106	—	2	4	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
239	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
240	C₄⋊C₈	C₂³×C₅	1	4	yes	—
241	#103	—	2	4	no	—
242	#102	—	2	4	no	—
243	C₂²⋊C₈	—	2	1	no	—
244	C₂²⋊C₈	—	2	2	no	—
245	C₂²⋊C₈	—	2	1	no	—
246	C₂²⋊C₈	—	2	1	no	—
247	#97	—	2	1	no	—
248	#97	—	2	1	no	—
249	#97	—	2	2	no	—
250	C₄×C₈	—	2	1	no	—
251	C₄×C₈	C₅×D₄	1	1	yes	characteristic
252	C₄×C₈	C₅×D₄	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
255	C₂²×C₅	—	2	1	no	—
256	C₂²×C₅	—	2	1	no	—
257	C₂²×C₅	—	4	1	no	—
258	C₂²×C₅	—	4	1	no	—
259	C₂²×C₅	—	4	1	no	—
260	C₂²×C₅	#473	1	2	yes	—
261	C₂²×C₅	C₂×#121	1	1	yes	characteristic
262	C₄×C₅	—	2	1	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	2	1	no	—
265	C₄×C₅	—	2	1	no	—
266	C₄×C₅	—	4	1	no	—
267	C₄×C₅	—	4	1	no	—
268	C₄×C₅	—	4	1	no	—
269	C₄×C₅	—	8	4	no	—
270	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
271	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₄×C₅	#578	1	2	yes	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	4	4	no	—
275	C₂×Q₈	—	4	4	no	—
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	4	2	no	—
282	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
283	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂²×C₄	C₅×#41	1	1	yes	characteristic
285	Q₁₆	—	4	4	no	—
286	Q₁₆	—	4	4	no	—
287	Q₁₆	—	4	4	no	—
288	Q₁₆	—	4	4	no	—
289	M₁₆	—	4	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	—	4	2	no	—
296	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
297	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
298	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
299	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	1	no	—
302	C₄⋊C₄	—	2	1	no	—
303	C₄⋊C₄	—	2	1	no	—
304	C₄⋊C₄	—	2	4	no	—
305	C₄⋊C₄	—	2	4	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
309	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
310	C₂²⋊C₄	—	2	1	no	—
311	C₂²⋊C₄	—	2	1	no	—
312	C₂²⋊C₄	—	2	1	no	—
313	C₂²⋊C₄	—	2	1	no	—
314	C₂²⋊C₄	—	2	1	no	—
315	C₄²	—	2	1	no	—
316	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
317	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
318	C₄²	C₅×#41	1	1	yes	characteristic
319	C₄²	C₂⁴×C₅	1	1	yes	characteristic
320	C₂×C₅	—	2	2	no	—
321	C₂×C₅	—	8	1	no	—
322	C₂×C₅	#3038	1	1	yes	characteristic
323	C₂×C₅	#3059	1	1	yes	characteristic
324	C₂×C₅	C₂×#473	1	1	yes	characteristic
325	C₂³	—	2	1	no	—
326	C₂³	C₅×#121	1	1	yes	characteristic
327	Q₈	—	2	2	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	4	4	no	—
330	Q₈	—	8	4	no	—
331	D₄	—	2	2	no	—
332	D₄	—	4	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	2	1	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	4	no	—
343	C₂×C₄	—	8	4	no	—
344	C₂×C₄	C₅×#121	1	1	yes	characteristic
345	C₂×C₄	C₅×#121	1	1	yes	characteristic
346	C₂×C₄	C₅×#121	1	1	yes	characteristic
347	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
348	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
349	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	4	1	no	—
353	C₈	—	4	1	no	—
354	C₈	—	4	4	no	—
355	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
356	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
357	C₅	#19354	1	1	yes	characteristic, direct product factor, Sylow
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	4	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	C₅×#473	1	2	yes	—
364	C₂²	C₂×C₅×#121	1	1	yes	characteristic
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	8	4	no	—
373	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
374	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
375	C₄	C₅×#578	1	2	yes	—
376	C₂	—	2	2	no	—
377	C₂	—	8	1	no	—
378	C₂	C₅×#3038	1	1	yes	characteristic
379	C₂	C₅×#3059	1	1	yes	characteristic
380	C₂	C₂×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#121	D₄
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
2. C₂×C₅×#494	C₂	semidirect
4. C₅×#1690	C₂	semidirect
5. C₅×#1689	C₂	semidirect
6. C₅×#1670	C₂	semidirect
7. C₅×#1669	C₂	semidirect
8. C₅×#1568	C₂	semidirect
9. C₅×#1556	C₂	semidirect
39. #19354	C₅	direct (N × H)
90. C₂×#494	C₂×C₅	semidirect
92. #1690	C₂×C₅	semidirect
93. #1689	C₂×C₅	semidirect
94. #1670	C₂×C₅	semidirect
95. #1669	C₂×C₅	semidirect
96. #1568	C₂×C₅	semidirect
97. #1556	C₂×C₅	semidirect
357. C₅	#19354	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×#3239	2
C₂×C₅×#494	2
C₅×#2966	2
C₅×#1690	2
C₅×#1689	2
C₅×#1670	2
C₅×#1669	2
C₅×#1568	2
C₅×#1556	2
#19354	5

Subgroups — C5×#19354 #11942074

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19354 #11942074