Subgroups — C₅×#19332 #11942052

Order 1280.

680 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3253	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×#1697	C₂	1	4	yes	maximal, complemented
5	C₅×#1675	C₂	1	2	yes	maximal, complemented
6	C₅×#1667	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	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×#493	C₂²	1	1	yes	characteristic
14	C₅×#486	C₂²	1	2	yes	—
15	C₅×#476	—	2	2	no	—
16	C₅×#476	C₂²	1	4	yes	—
17	C₅×#469	—	2	4	no	—
18	C₅×#466	—	2	1	no	—
19	C₅×#466	—	2	1	no	—
20	C₅×#459	—	2	2	no	—
21	C₅×#459	C₂²	1	4	yes	—
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×#423	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×#335	C₂²	1	2	yes	—
34	C₅×#333	C₂²	1	4	yes	—
35	C₅×#331	C₂²	1	4	yes	—
36	#19332	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×D₈	—	2	4	no	—
40	C₂×C₅×D₈	—	2	1	no	—
41	C₂×C₅×D₈	—	2	1	no	—
42	C₅×#131	—	4	4	no	—
43	C₂×C₅×M₁₆	—	2	1	no	—
44	C₂×C₅×M₁₆	—	2	2	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	1	no	—
54	C₅×#124	—	2	2	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×#106	—	2	4	no	—
67	C₅×C₄⋊C₈	—	2	1	no	—
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	C₂³	1	4	yes	—
71	C₅×#102	—	2	4	no	—
72	C₅×#102	—	2	2	no	—
73	C₅×#102	—	2	2	no	—
74	C₅×#102	—	2	4	no	—
75	C₅×C₂²⋊C₈	—	2	2	no	—
76	C₅×C₂²⋊C₈	—	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	#3253	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	#1697	C₂×C₅	1	4	yes	complemented
85	#1675	C₂×C₅	1	2	yes	complemented
86	#1667	C₂×C₅	1	2	yes	complemented
87	#1567	C₂×C₅	1	1	yes	characteristic, complemented
88	#1560	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×D₈	—	4	2	no	—
108	C₅×D₈	—	4	2	no	—
109	C₅×D₈	—	8	4	no	—
110	C₅×M₁₆	—	4	4	no	—
111	C₅×M₁₆	—	4	2	no	—
112	C₅×M₁₆	—	8	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	#493	C₂²×C₅	1	1	yes	characteristic
139	#486	C₂²×C₅	1	2	yes	—
140	#476	—	2	2	no	—
141	#476	C₂²×C₅	1	4	yes	—
142	#469	—	2	4	no	—
143	#466	—	2	1	no	—
144	#466	—	2	1	no	—
145	#459	—	2	2	no	—
146	#459	C₂²×C₅	1	4	yes	—
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	#423	C₂²×C₅	1	1	yes	characteristic
154	#423	C₂²×C₅	1	2	yes	—
155	#408	—	2	1	no	—
156	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
157	#335	C₂²×C₅	1	2	yes	—
158	#335	C₂²×C₅	1	2	yes	—
159	#333	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	—	8	2	no	—
166	C₅×D₄	—	2	2	no	—
167	C₅×D₄	—	2	2	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	4	no	—
170	C₅×D₄	—	4	2	no	—
171	C₅×D₄	—	4	2	no	—
172	C₅×D₄	—	4	2	no	—
173	C₅×D₄	—	4	2	no	—
174	C₅×D₄	—	4	2	no	—
175	C₅×D₄	—	8	2	no	—
176	C₅×D₄	—	8	4	no	—
177	C₂×C₄×C₅	—	2	1	no	—
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	—	2	1	no	—
180	C₂×C₄×C₅	—	2	1	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	4	2	no	—
186	C₂×C₄×C₅	—	8	4	no	—
187	C₂×C₄×C₅	#121	1	1	yes	characteristic
188	C₂×C₄×C₅	#121	1	1	yes	characteristic
189	C₂×C₄×C₅	#121	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	2	1	no	—
195	C₅×C₈	—	2	1	no	—
196	C₅×C₈	—	4	2	no	—
197	C₅×C₈	—	4	4	no	—
198	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
199	C₂×#41	—	2	1	no	—
200	C₂×#41	—	2	2	no	—
201	C₂×D₈	—	2	4	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	#131	—	4	4	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂×M₁₆	—	2	2	no	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
210	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
211	C₄⋊Q₈	C₅×D₄	1	1	yes	characteristic
212	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
213	#127	C₅×D₄	1	1	yes	characteristic
214	#127	C₂³×C₅	1	2	yes	—
215	#124	—	2	1	no	—
216	#124	—	2	2	no	—
217	#121	—	2	1	no	—
218	#121	—	2	2	no	—
219	#121	—	2	2	no	—
220	#121	C₅×D₄	1	2	yes	—
221	C₄×D₄	—	2	1	no	—
222	C₄×D₄	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	C₂³×C₅	1	2	yes	—
225	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#106	—	2	4	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	C₂³×C₅	1	2	yes	—
232	C₄⋊C₈	C₂³×C₅	1	4	yes	—
233	#102	—	2	4	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	#102	—	2	4	no	—
237	C₂²⋊C₈	—	2	2	no	—
238	C₂²⋊C₈	—	2	2	no	—
239	C₄×C₈	C₅×D₄	1	1	yes	characteristic
240	C₄×C₈	C₅×D₄	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
243	C₂²×C₅	—	2	1	no	—
244	C₂²×C₅	—	2	1	no	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	4	2	no	—
247	C₂²×C₅	—	4	2	no	—
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	8	2	no	—
250	C₂²×C₅	—	8	2	no	—
251	C₂²×C₅	—	8	2	no	—
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	—	4	2	no	—
257	C₄×C₅	—	8	2	no	—
258	C₄×C₅	#472	1	2	yes	—
259	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₄×C₅	#575	1	1	yes	characteristic
261	C₄×C₅	#576	1	1	yes	characteristic
262	C₄×C₅	#578	1	1	yes	characteristic
263	#41	—	4	2	no	—
264	#41	—	4	2	no	—
265	#41	—	8	4	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	1	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	4	2	no	—
275	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
276	C₂²×C₄	—	2	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	2	no	—
283	D₈	—	8	4	no	—
284	M₁₆	—	4	4	no	—
285	M₁₆	—	4	2	no	—
286	M₁₆	—	8	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	4	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	4	no	—
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	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
296	C₄⋊C₄	—	2	1	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
305	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
306	C₄²	C₅×#41	1	1	yes	characteristic
307	C₄²	C₂⁴×C₅	1	1	yes	characteristic
308	C₂×C₅	—	4	1	no	—
309	C₂×C₅	—	8	2	no	—
310	C₂×C₅	—	16	2	no	—
311	C₂×C₅	#3036	1	1	yes	characteristic
312	C₂×C₅	#3061	1	1	yes	characteristic
313	C₂×C₅	C₂×#472	1	1	yes	characteristic
314	C₂³	—	2	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	C₅×#121	1	1	yes	characteristic
317	Q₈	—	4	2	no	—
318	Q₈	—	8	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	4	no	—
323	D₄	—	4	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	8	4	no	—
340	C₂×C₄	C₅×#121	1	1	yes	characteristic
341	C₂×C₄	C₅×#121	1	1	yes	characteristic
342	C₂×C₄	C₅×#121	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
344	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
345	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	2	1	no	—
349	C₈	—	4	2	no	—
350	C₈	—	4	4	no	—
351	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
352	C₅	#19332	1	1	yes	characteristic, direct product factor, Sylow
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	2	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	8	2	no	—
361	C₂²	—	8	2	no	—
362	C₂²	C₂×C₅×#121	1	1	yes	characteristic
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	C₅×#472	1	2	yes	—
369	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
370	C₄	C₅×#575	1	1	yes	characteristic
371	C₄	C₅×#576	1	1	yes	characteristic
372	C₄	C₅×#578	1	1	yes	characteristic
373	C₂	—	4	1	no	—
374	C₂	—	8	2	no	—
375	C₂	—	16	2	no	—
376	C₂	C₅×#3036	1	1	yes	characteristic
377	C₂	C₅×#3061	1	1	yes	characteristic
378	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×#3253	C₂	semidirect
2. C₅×#3169	C₂	semidirect
3. C₅×#2931	C₂	semidirect
4. C₅×#1697	C₂	semidirect
5. C₅×#1675	C₂	semidirect
6. C₅×#1667	C₂	semidirect
7. C₅×#1567	C₂	semidirect
8. C₅×#1560	C₂	semidirect
36. #19332	C₅	direct (N × H)
81. #3253	C₂×C₅	semidirect
82. #3169	C₂×C₅	semidirect
83. #2931	C₂×C₅	semidirect
84. #1697	C₂×C₅	semidirect
85. #1675	C₂×C₅	semidirect
86. #1667	C₂×C₅	semidirect
87. #1567	C₂×C₅	semidirect
88. #1560	C₂×C₅	semidirect
352. C₅	#19332	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×#3253	2
C₅×#3169	2
C₅×#2931	2
C₅×#1697	2
C₅×#1675	2
C₅×#1667	2
C₅×#1567	2
C₅×#1560	2
#19332	5

Subgroups — C5×#19332 #11942052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19332 #11942052