Subgroups — C₅×#19329 #11942049

Order 1280.

684 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3214	C₂	1	2	yes	maximal, complemented
2	C₅×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1701	C₂	1	4	yes	maximal, complemented
5	C₅×#1673	C₂	1	2	yes	maximal, complemented
6	C₅×#1667	C₂	1	2	yes	maximal, complemented
7	C₅×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1557	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#131	—	2	1	no	—
10	C₅×#532	C₂²	1	2	yes	—
11	C₅×#532	C₂²	1	1	yes	characteristic
12	C₅×#500	C₂²	1	1	yes	characteristic
13	C₅×#496	—	2	2	no	—
14	C₅×#493	C₂²	1	1	yes	characteristic
15	C₅×#489	—	2	2	no	—
16	C₅×#486	C₂²	1	2	yes	—
17	C₅×#466	—	2	1	no	—
18	C₅×#466	—	2	1	no	—
19	C₅×#461	C₂²	1	4	yes	—
20	C₅×#459	C₂²	1	4	yes	—
21	C₅×#449	—	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	C₂²	1	2	yes	—
26	C₅×C₈×D₄	—	2	1	no	—
27	C₅×C₈×D₄	C₂²	1	2	yes	—
28	C₅×#432	C₂²	1	2	yes	—
29	C₅×#423	C₂²	1	1	yes	characteristic
30	C₅×#408	—	2	1	no	—
31	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
32	C₅×#335	C₂²	1	2	yes	—
33	C₅×#334	C₂²	1	2	yes	—
34	C₅×#332	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19329	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	1	no	—
38	C₂×C₅×#41	—	2	2	no	—
39	C₂×C₅×SD₁₆	—	2	4	no	—
40	C₂×C₅×D₈	—	2	4	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₂×C₅×D₈	—	2	1	no	—
43	C₅×#131	—	4	4	no	—
44	C₂×C₅×M₁₆	—	2	1	no	—
45	C₂×C₅×M₁₆	—	2	2	no	—
46	C₂²×C₅×C₈	—	2	2	no	—
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	D₄	1	1	yes	characteristic
50	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
51	C₅×#127	D₄	1	1	yes	characteristic
52	C₅×#127	C₂³	1	2	yes	—
53	C₅×#124	—	2	2	no	—
54	C₅×#124	—	2	1	no	—
55	C₅×#121	—	2	1	no	—
56	C₅×#121	—	2	2	no	—
57	C₅×#121	—	2	2	no	—
58	C₅×#121	D₄	1	2	yes	—
59	C₄×C₅×D₄	—	2	1	no	—
60	C₄×C₅×D₄	—	2	2	no	—
61	C₄×C₅×D₄	—	2	2	no	—
62	C₄×C₅×D₄	C₂³	1	2	yes	—
63	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
64	C₅×#107	—	2	1	no	—
65	C₅×#107	—	2	1	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×#103	—	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	#3214	C₂×C₅	1	2	yes	complemented
82	#3169	C₂×C₅	1	1	yes	characteristic, complemented
83	#2931	C₂×C₅	1	1	yes	characteristic, complemented
84	#1701	C₂×C₅	1	4	yes	complemented
85	#1673	C₂×C₅	1	2	yes	complemented
86	#1667	C₂×C₅	1	2	yes	complemented
87	#1568	C₂×C₅	1	1	yes	characteristic, complemented
88	#1557	C₂×C₅	1	2	yes	complemented
89	C₅×#41	—	4	2	no	—
90	C₅×#41	—	4	2	no	—
91	C₅×#41	—	8	4	no	—
92	C₂×C₅×Q₈	—	2	1	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×D₄	—	2	2	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	2	1	no	—
97	C₂×C₅×D₄	—	2	2	no	—
98	C₂×C₅×D₄	—	2	2	no	—
99	C₂×C₅×D₄	—	2	2	no	—
100	C₂×C₅×D₄	—	4	2	no	—
101	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₅	—	2	2	no	—
103	C₂²×C₄×C₅	—	4	2	no	—
104	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₅	#41	1	1	yes	characteristic
107	C₅×SD₁₆	—	8	4	no	—
108	C₅×D₈	—	4	2	no	—
109	C₅×D₈	—	4	2	no	—
110	C₅×D₈	—	8	4	no	—
111	C₅×M₁₆	—	4	4	no	—
112	C₅×M₁₆	—	4	2	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	4	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	C₂×#131	—	2	1	no	—
135	#532	C₂²×C₅	1	2	yes	—
136	#532	C₂²×C₅	1	1	yes	characteristic
137	#500	C₂²×C₅	1	1	yes	characteristic
138	#496	—	2	2	no	—
139	#493	C₂²×C₅	1	1	yes	characteristic
140	#489	—	2	2	no	—
141	#486	C₂²×C₅	1	2	yes	—
142	#466	—	2	1	no	—
143	#466	—	2	1	no	—
144	#461	C₂²×C₅	1	4	yes	—
145	#459	C₂²×C₅	1	4	yes	—
146	#449	—	2	4	no	—
147	C₄×D₈	—	2	1	no	—
148	C₄×D₈	—	2	1	no	—
149	#436	—	2	1	no	—
150	#436	C₂²×C₅	1	2	yes	—
151	C₈×D₄	—	2	1	no	—
152	C₈×D₄	C₂²×C₅	1	2	yes	—
153	#432	C₂²×C₅	1	2	yes	—
154	#423	C₂²×C₅	1	1	yes	characteristic
155	#408	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
157	#335	C₂²×C₅	1	2	yes	—
158	#334	C₂²×C₅	1	2	yes	—
159	#332	C₂²×C₅	1	4	yes	—
160	#331	C₂²×C₅	1	4	yes	—
161	C₂³×C₅	—	2	2	no	—
162	C₂³×C₅	—	4	2	no	—
163	C₂³×C₅	#121	1	1	yes	characteristic
164	C₅×Q₈	—	4	2	no	—
165	C₅×Q₈	—	4	4	no	—
166	C₅×D₄	—	2	2	no	—
167	C₅×D₄	—	2	2	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	4	4	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	4	no	—
174	C₅×D₄	—	8	2	no	—
175	C₅×D₄	—	8	4	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	1	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	—	8	4	no	—
185	C₂×C₄×C₅	#121	1	1	yes	characteristic
186	C₂×C₄×C₅	#121	1	1	yes	characteristic
187	C₂×C₄×C₅	#121	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
191	C₅×C₈	—	2	1	no	—
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	4	2	no	—
195	C₅×C₈	—	4	4	no	—
196	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
197	C₂×#41	—	2	1	no	—
198	C₂×#41	—	2	2	no	—
199	C₂×SD₁₆	—	2	4	no	—
200	C₂×D₈	—	2	4	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	#131	—	4	4	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
208	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
209	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
210	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
211	#127	C₅×D₄	1	1	yes	characteristic
212	#127	C₂³×C₅	1	2	yes	—
213	#124	—	2	2	no	—
214	#124	—	2	1	no	—
215	#121	—	2	1	no	—
216	#121	—	2	2	no	—
217	#121	—	2	2	no	—
218	#121	C₅×D₄	1	2	yes	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₅	1	2	yes	—
223	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
224	#107	—	2	1	no	—
225	#107	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	C₂³×C₅	1	2	yes	—
228	C₄⋊C₈	C₂³×C₅	1	4	yes	—
229	#103	—	2	4	no	—
230	#102	—	2	2	no	—
231	#102	—	2	2	no	—
232	#102	—	2	4	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	2	2	no	—
235	C₂²⋊C₈	—	2	2	no	—
236	#97	—	2	2	no	—
237	C₄×C₈	C₅×D₄	1	1	yes	characteristic
238	C₄×C₈	C₅×D₄	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
241	C₂²×C₅	—	2	1	no	—
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	2	no	—
248	C₂²×C₅	—	8	2	no	—
249	C₂²×C₅	—	8	2	no	—
250	C₂²×C₅	C₂×#121	1	1	yes	characteristic
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	1	no	—
253	C₄×C₅	—	2	1	no	—
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	4	2	no	—
256	C₄×C₅	—	8	2	no	—
257	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₄×C₅	#575	1	1	yes	characteristic
259	C₄×C₅	#576	1	2	yes	—
260	#41	—	4	2	no	—
261	#41	—	4	2	no	—
262	#41	—	8	4	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	4	2	no	—
275	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
276	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂²×C₄	C₅×#41	1	1	yes	characteristic
278	SD₁₆	—	8	4	no	—
279	D₈	—	4	2	no	—
280	D₈	—	4	2	no	—
281	D₈	—	8	4	no	—
282	M₁₆	—	4	4	no	—
283	M₁₆	—	4	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	4	4	no	—
289	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
290	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
291	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
292	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
302	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
303	C₄²	C₅×#41	1	1	yes	characteristic
304	C₄²	C₂⁴×C₅	1	1	yes	characteristic
305	C₂×C₅	—	4	1	no	—
306	C₂×C₅	—	8	2	no	—
307	C₂×C₅	—	16	2	no	—
308	C₂×C₅	#3036	1	1	yes	characteristic
309	C₂×C₅	#3058	1	1	yes	characteristic
310	C₂×C₅	#3067	1	1	yes	characteristic
311	C₂³	—	2	2	no	—
312	C₂³	—	4	2	no	—
313	C₂³	C₅×#121	1	1	yes	characteristic
314	Q₈	—	4	2	no	—
315	Q₈	—	4	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	4	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	4	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	4	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	4	no	—
335	C₂×C₄	C₅×#121	1	1	yes	characteristic
336	C₂×C₄	C₅×#121	1	1	yes	characteristic
337	C₂×C₄	C₅×#121	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
339	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
340	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	2	no	—
345	C₈	—	4	4	no	—
346	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
347	C₅	#19329	1	1	yes	characteristic, direct product factor, Sylow
348	C₂²	—	2	1	no	—
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	C₂×C₅×#121	1	1	yes	characteristic
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	8	2	no	—
364	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
365	C₄	C₅×#575	1	1	yes	characteristic
366	C₄	C₅×#576	1	2	yes	—
367	C₂	—	4	1	no	—
368	C₂	—	8	2	no	—
369	C₂	—	16	2	no	—
370	C₂	C₅×#3036	1	1	yes	characteristic
371	C₂	C₅×#3058	1	1	yes	characteristic
372	C₂	C₅×#3067	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×#3214	C₂	semidirect
2. C₅×#3169	C₂	semidirect
3. C₅×#2931	C₂	semidirect
4. C₅×#1701	C₂	semidirect
5. C₅×#1673	C₂	semidirect
6. C₅×#1667	C₂	semidirect
7. C₅×#1568	C₂	semidirect
8. C₅×#1557	C₂	semidirect
36. #19329	C₅	direct (N × H)
81. #3214	C₂×C₅	semidirect
82. #3169	C₂×C₅	semidirect
83. #2931	C₂×C₅	semidirect
84. #1701	C₂×C₅	semidirect
85. #1673	C₂×C₅	semidirect
86. #1667	C₂×C₅	semidirect
87. #1568	C₂×C₅	semidirect
88. #1557	C₂×C₅	semidirect
347. C₅	#19329	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×#3214	2
C₅×#3169	2
C₅×#2931	2
C₅×#1701	2
C₅×#1673	2
C₅×#1667	2
C₅×#1568	2
C₅×#1557	2
#19329	5

Subgroups — C5×#19329 #11942049

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19329 #11942049