Subgroups — C₅×#19283 #11942003

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3109	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1697	C₂	1	4	yes	maximal, complemented
5	C₅×#1695	C₂	1	2	yes	maximal, complemented
6	C₅×#1687	C₂	1	2	yes	maximal, complemented
7	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1551	C₂	1	2	yes	maximal, complemented
9	C₂²×C₅×M₁₆	—	2	1	no	—
10	C₅×#532	C₂²	1	2	yes	—
11	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic, complemented
13	C₅×#493	C₂²	1	1	yes	characteristic, complemented
14	C₅×#486	C₂²	1	2	yes	complemented
15	C₅×#481	—	2	2	no	—
16	C₅×#476	C₂²	1	4	yes	—
17	C₅×#466	—	2	1	no	—
18	C₅×#466	—	2	1	no	—
19	C₅×#459	C₂²	1	4	yes	—
20	C₅×#447	—	2	2	no	—
21	C₅×#442	—	2	4	no	—
22	C₄×C₅×D₈	—	2	1	no	—
23	C₄×C₅×D₈	—	2	1	no	—
24	C₅×#436	—	2	1	no	—
25	C₅×#436	—	2	1	no	—
26	C₅×#436	C₂²	1	2	yes	—
27	C₅×#436	C₂²	1	2	yes	—
28	C₅×#432	C₂²	1	2	yes	—
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₂²⋊C₈	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	2	yes	complemented
33	C₅×#334	C₂²	1	2	yes	complemented
34	C₅×#333	C₂²	1	4	yes	complemented
35	C₅×#331	C₂²	1	4	yes	complemented
36	#19283	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₅	—	2	1	no	—
40	C₂×C₅×D₈	—	2	4	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₂×C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂×C₅×M₁₆	—	4	4	no	—
46	C₂²×C₅×C₈	—	2	1	no	—
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
50	C₅×#127	C₂³	1	2	yes	—
51	C₅×#124	—	2	2	no	—
52	C₅×#121	—	2	2	no	—
53	C₅×#121	—	2	2	no	—
54	C₄×C₅×D₄	—	2	1	no	—
55	C₄×C₅×D₄	—	2	2	no	—
56	C₄×C₅×D₄	—	2	2	no	—
57	C₄×C₅×D₄	—	2	1	no	—
58	C₄×C₅×D₄	D₄	1	2	yes	—
59	C₄×C₅×D₄	C₂³	1	2	yes	—
60	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
62	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
63	C₅×#107	—	2	1	no	—
64	C₅×#107	—	2	1	no	—
65	C₅×#106	—	2	4	no	—
66	C₅×C₄⋊C₈	—	2	1	no	—
67	C₅×C₄⋊C₈	C₂³	1	2	yes	—
68	C₅×C₄⋊C₈	C₂³	1	4	yes	—
69	C₅×#102	—	2	4	no	—
70	C₅×#102	—	2	2	no	—
71	C₅×#102	—	2	2	no	—
72	C₅×#102	—	2	4	no	—
73	C₅×C₂²⋊C₈	—	2	2	no	—
74	C₅×C₂²⋊C₈	—	2	2	no	—
75	C₅×C₂²⋊C₈	—	2	2	no	—
76	C₅×#97	—	2	2	no	—
77	C₄×C₅×C₈	D₄	1	1	yes	characteristic
78	C₄×C₅×C₈	D₄	1	1	yes	characteristic
79	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
81	#3169	C₂×C₅	1	1	yes	characteristic, complemented
82	#3109	C₂×C₅	1	2	yes	complemented
83	C₂×#436	C₂×C₅	1	1	yes	characteristic, complemented
84	#1697	C₂×C₅	1	4	yes	complemented
85	#1695	C₂×C₅	1	2	yes	complemented
86	#1687	C₂×C₅	1	2	yes	complemented
87	#1568	C₂×C₅	1	1	yes	characteristic, complemented
88	#1551	C₂×C₅	1	2	yes	complemented
89	C₂⁴×C₅	—	2	1	no	—
90	C₅×#41	—	8	4	no	—
91	C₂×C₅×Q₈	—	2	2	no	—
92	C₂×C₅×D₄	—	2	1	no	—
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	4	2	no	—
97	C₂×C₅×D₄	—	4	2	no	—
98	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅	—	2	1	no	—
100	C₂²×C₄×C₅	—	2	2	no	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	—	4	2	no	—
103	C₂²×C₄×C₅	—	4	2	no	—
104	C₂²×C₄×C₅	—	4	2	no	—
105	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	#41	1	1	yes	characteristic
108	C₅×D₈	—	4	2	no	—
109	C₅×D₈	—	4	8	no	—
110	C₅×D₈	—	4	2	no	—
111	C₅×M₁₆	—	4	4	no	—
112	C₅×M₁₆	—	4	4	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	4	no	—
116	C₂×C₅×C₈	—	4	1	no	—
117	C₂×C₅×C₈	—	4	2	no	—
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	2	no	—
125	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
126	C₅×C₂²⋊C₄	—	2	2	no	—
127	C₅×C₂²⋊C₄	—	2	2	no	—
128	C₅×C₂²⋊C₄	—	4	2	no	—
129	C₅×C₂²⋊C₄	—	4	2	no	—
130	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₅	#41	1	1	yes	characteristic
133	C₄²×C₅	C₂⁴	1	1	yes	characteristic
134	C₂²×M₁₆	—	2	1	no	—
135	#532	C₂²×C₅	1	2	yes	—
136	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
137	#500	C₂²×C₅	1	1	yes	characteristic, complemented
138	#493	C₂²×C₅	1	1	yes	characteristic, complemented
139	#486	C₂²×C₅	1	2	yes	complemented
140	#481	—	2	2	no	—
141	#476	C₂²×C₅	1	4	yes	—
142	#466	—	2	1	no	—
143	#466	—	2	1	no	—
144	#459	C₂²×C₅	1	4	yes	—
145	#447	—	2	2	no	—
146	#442	—	2	4	no	—
147	C₄×D₈	—	2	1	no	—
148	C₄×D₈	—	2	1	no	—
149	#436	—	2	1	no	—
150	#436	—	2	1	no	—
151	#436	C₂²×C₅	1	2	yes	—
152	#436	C₂²×C₅	1	2	yes	—
153	#432	C₂²×C₅	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
155	C₂×C₂²⋊C₈	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
157	#335	C₂²×C₅	1	2	yes	complemented
158	#334	C₂²×C₅	1	2	yes	complemented
159	#333	C₂²×C₅	1	4	yes	complemented
160	#331	C₂²×C₅	1	4	yes	complemented
161	C₂³×C₅	—	2	2	no	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	—	4	2	no	—
164	C₂³×C₅	—	4	2	no	—
165	C₂³×C₅	—	4	2	no	—
166	C₂³×C₅	#121	1	1	yes	characteristic
167	C₅×Q₈	—	8	2	no	—
168	C₅×D₄	—	2	4	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	4	no	—
173	C₅×D₄	—	8	2	no	—
174	C₅×D₄	—	8	4	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	2	1	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	4	4	no	—
181	C₂×C₄×C₅	—	4	2	no	—
182	C₂×C₄×C₅	—	4	2	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	8	4	no	—
186	C₂×C₄×C₅	#121	1	1	yes	characteristic
187	C₂×C₄×C₅	#121	1	1	yes	characteristic
188	C₂×C₄×C₅	#121	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	2	1	no	—
195	C₅×C₈	—	4	2	no	—
196	C₅×C₈	—	4	4	no	—
197	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
198	C₂×#41	—	2	2	no	—
199	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
200	C₂³×C₄	—	2	1	no	—
201	C₂×D₈	—	2	4	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂×M₁₆	—	4	4	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
209	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
210	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
211	#127	C₂³×C₅	1	2	yes	—
212	#124	—	2	2	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	C₅×D₄	1	2	yes	—
220	C₄×D₄	C₂³×C₅	1	2	yes	—
221	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
222	C₂×C₂²⋊C₄	—	2	1	no	—
223	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	#106	—	2	4	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	C₂³×C₅	1	2	yes	—
229	C₄⋊C₈	C₂³×C₅	1	4	yes	—
230	#102	—	2	4	no	—
231	#102	—	2	2	no	—
232	#102	—	2	2	no	—
233	#102	—	2	4	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	C₂²⋊C₈	—	2	2	no	—
237	#97	—	2	2	no	—
238	C₄×C₈	C₅×D₄	1	1	yes	characteristic
239	C₄×C₈	C₅×D₄	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
242	C₂²×C₅	—	2	1	no	—
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	4	2	no	—
245	C₂²×C₅	—	4	2	no	—
246	C₂²×C₅	—	4	2	no	—
247	C₂²×C₅	—	8	4	no	—
248	C₂²×C₅	—	8	2	no	—
249	C₂²×C₅	—	8	2	no	—
250	C₂²×C₅	—	8	2	no	—
251	C₂²×C₅	#472	1	2	yes	—
252	C₂²×C₅	C₂×#121	1	1	yes	characteristic
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	2	1	no	—
256	C₄×C₅	—	2	1	no	—
257	C₄×C₅	—	4	2	no	—
258	C₄×C₅	—	8	2	no	—
259	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₄×C₅	#575	1	1	yes	characteristic
261	C₄×C₅	#575	1	2	yes	—
262	C₂⁴	—	2	1	no	—
263	#41	—	8	4	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	4	2	no	—
276	C₂²×C₄	—	4	2	no	—
277	C₂²×C₄	—	4	2	no	—
278	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂²×C₄	C₅×#41	1	1	yes	characteristic
281	D₈	—	4	2	no	—
282	D₈	—	4	8	no	—
283	D₈	—	4	2	no	—
284	M₁₆	—	4	4	no	—
285	M₁₆	—	4	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
292	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
293	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
294	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
299	C₂²⋊C₄	—	2	2	no	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
304	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
305	C₄²	C₅×#41	1	1	yes	characteristic
306	C₄²	C₂⁴×C₅	1	1	yes	characteristic
307	C₂×C₅	—	2	2	no	—
308	C₂×C₅	—	8	2	no	—
309	C₂×C₅	—	16	2	no	—
310	C₂×C₅	#3036	1	1	yes	characteristic
311	C₂×C₅	#3054	1	1	yes	characteristic
312	C₂×C₅	C₂×#472	1	1	yes	characteristic
313	C₂³	—	2	2	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	C₅×#121	1	1	yes	characteristic
319	Q₈	—	8	2	no	—
320	D₄	—	2	4	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	4	no	—
325	D₄	—	8	2	no	—
326	D₄	—	8	4	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	4	no	—
338	C₂×C₄	C₅×#121	1	1	yes	characteristic
339	C₂×C₄	C₅×#121	1	1	yes	characteristic
340	C₂×C₄	C₅×#121	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
342	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
343	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	4	2	no	—
348	C₈	—	4	4	no	—
349	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
350	C₅	#19283	1	1	yes	characteristic, direct product factor, Sylow
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	4	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	8	4	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	C₅×#472	1	2	yes	—
361	C₂²	C₂×C₅×#121	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	4	2	no	—
367	C₄	—	8	2	no	—
368	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
369	C₄	C₅×#575	1	1	yes	characteristic
370	C₄	C₅×#575	1	2	yes	—
371	C₂	—	2	2	no	—
372	C₂	—	8	2	no	—
373	C₂	—	16	2	no	—
374	C₂	C₅×#3036	1	1	yes	characteristic
375	C₂	C₅×#3054	1	1	yes	characteristic
376	C₂	C₂×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
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×#3169	C₂	semidirect
2. C₅×#3109	C₂	semidirect
3. C₂×C₅×#436	C₂	semidirect
4. C₅×#1697	C₂	semidirect
5. C₅×#1695	C₂	semidirect
6. C₅×#1687	C₂	semidirect
7. C₅×#1568	C₂	semidirect
8. C₅×#1551	C₂	semidirect
12. C₅×#500	C₂²	semidirect
13. C₅×#493	C₂²	semidirect
14. C₅×#486	C₂²	semidirect
32. C₅×#335	C₂²	semidirect
33. C₅×#334	C₂²	semidirect
34. C₅×#333	C₂²	semidirect
35. C₅×#331	C₂²	semidirect
36. #19283	C₅	direct (N × H)
81. #3169	C₂×C₅	semidirect
82. #3109	C₂×C₅	semidirect
83. C₂×#436	C₂×C₅	semidirect
84. #1697	C₂×C₅	semidirect
85. #1695	C₂×C₅	semidirect
86. #1687	C₂×C₅	semidirect
87. #1568	C₂×C₅	semidirect
88. #1551	C₂×C₅	semidirect
137. #500	C₂²×C₅	semidirect
138. #493	C₂²×C₅	semidirect
139. #486	C₂²×C₅	semidirect
157. #335	C₂²×C₅	semidirect
158. #334	C₂²×C₅	semidirect
159. #333	C₂²×C₅	semidirect
160. #331	C₂²×C₅	semidirect
350. C₅	#19283	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×#3169	2
C₅×#3109	2
C₂×C₅×#436	2
C₅×#1697	2
C₅×#1695	2
C₅×#1687	2
C₅×#1568	2
C₅×#1551	2
#19283	5

Subgroups — C5×#19283 #11942003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19283 #11942003