Subgroups — C₅×#19393 #11942113

Order 1280.

642 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×#3230	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3077	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#476	C₂	1	1	yes	maximal, characteristic
4	C₅×#2972	C₂	1	1	yes	maximal, characteristic
5	C₅×#1683	C₂	1	4	yes	maximal, complemented
6	C₅×#1675	C₂	1	4	yes	maximal, complemented
7	C₅×#1571	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#533	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₅×#500	C₂²	1	2	yes	complemented
14	C₅×#487	C₂²	1	2	yes	complemented
15	C₅×#484	—	2	2	no	—
16	C₅×#478	—	2	2	no	—
17	C₅×#477	—	2	2	no	—
18	C₅×#476	—	2	2	no	—
19	C₅×#476	C₂²	1	4	yes	—
20	C₅×#476	C₂²	1	2	yes	—
21	C₅×#474	C₂²	1	2	yes	—
22	C₅×C₈×D₄	—	2	2	no	—
23	C₅×C₈×D₄	C₂²	1	4	yes	—
24	C₅×#431	C₂²	1	1	yes	characteristic
25	C₅×#431	C₂²	1	1	yes	characteristic
26	C₅×#428	—	2	1	no	—
27	C₂×C₅×#106	—	2	1	no	—
28	C₅×#423	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₅×#418	—	2	1	no	—
31	C₂×C₅×#102	—	2	1	no	—
32	C₂×C₅×C₂²⋊C₈	—	2	2	no	—
33	C₅×#335	C₂²	1	4	yes	complemented
34	C₅×#333	C₂²	1	2	yes	complemented
35	C₅×#333	C₂²	1	4	yes	complemented
36	C₅×#332	C₂²	1	2	yes	complemented
37	#19393	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂²×C₅×Q₈	—	2	1	no	—
39	C₂²×C₅×D₄	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₅	—	2	1	no	—
41	C₂×C₅×M₁₆	—	2	1	no	—
42	C₂²×C₅×C₈	—	2	1	no	—
43	C₂²×C₅×C₈	—	2	4	no	—
44	C₅×C₄⋊Q₈	—	2	1	no	—
45	C₅×C₄⋊Q₈	—	2	1	no	—
46	C₅×C₄⋊Q₈	—	2	1	no	—
47	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	C₅×#125	—	2	1	no	—
51	C₅×C₂²⋊Q₈	—	2	1	no	—
52	C₅×C₂²⋊Q₈	—	2	1	no	—
53	C₄×C₅×Q₈	—	2	1	no	—
54	C₄×C₅×Q₈	—	2	1	no	—
55	C₄×C₅×D₄	—	2	1	no	—
56	C₄×C₅×D₄	—	2	1	no	—
57	C₄×C₅×D₄	D₄	1	2	yes	—
58	C₄×C₅×D₄	C₂³	1	2	yes	—
59	C₅×#117	—	2	1	no	—
60	C₂×C₅×C₄⋊C₄	—	2	1	no	—
61	C₂×C₅×C₄⋊C₄	—	2	1	no	—
62	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
63	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
64	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
65	C₅×#107	—	2	4	no	—
66	C₅×#107	—	2	2	no	—
67	C₅×#106	—	2	4	no	—
68	C₅×#106	—	2	2	no	—
69	C₅×C₄⋊C₈	—	2	1	no	—
70	C₅×C₄⋊C₈	—	2	1	no	—
71	C₅×C₄⋊C₈	D₄	1	2	yes	—
72	C₅×C₄⋊C₈	C₂³	1	2	yes	—
73	C₅×C₄⋊C₈	C₂³	1	2	yes	—
74	C₅×C₄⋊C₈	C₂³	1	2	yes	—
75	C₅×#103	—	2	4	no	—
76	C₅×#102	—	2	4	no	—
77	C₅×#102	—	2	2	no	—
78	C₅×#102	—	2	2	no	—
79	C₅×C₂²⋊C₈	—	2	1	no	—
80	C₅×C₂²⋊C₈	—	2	4	no	—
81	C₅×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₅×C₈	—	2	1	no	—
83	C₄×C₅×C₈	C₂³	1	2	yes	—
84	#3230	C₂×C₅	1	1	yes	characteristic, complemented
85	#3077	C₂×C₅	1	1	yes	characteristic
86	C₂×#476	C₂×C₅	1	1	yes	characteristic
87	#2972	C₂×C₅	1	1	yes	characteristic
88	#1683	C₂×C₅	1	4	yes	complemented
89	#1675	C₂×C₅	1	4	yes	complemented
90	#1571	C₂×C₅	1	1	yes	characteristic, complemented
91	#1548	C₂×C₅	1	1	yes	characteristic, complemented
92	#1542	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂⁴×C₅	—	2	1	no	—
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	2	2	no	—
97	C₂×C₅×Q₈	—	4	1	no	—
98	C₂×C₅×Q₈	—	8	2	no	—
99	C₂×C₅×D₄	—	2	1	no	—
100	C₂×C₅×D₄	—	2	2	no	—
101	C₂×C₅×D₄	Q₁₆	1	4	yes	—
102	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅	—	2	1	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	—	2	2	no	—
106	C₂²×C₄×C₅	—	4	1	no	—
107	C₂²×C₄×C₅	—	8	2	no	—
108	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₅	#41	1	1	yes	characteristic
111	C₅×M₁₆	—	4	2	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	4	2	no	—
116	C₂×C₅×C₈	—	8	4	no	—
117	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	—	2	1	no	—
125	C₅×C₄⋊C₄	—	4	1	no	—
126	C₅×C₄⋊C₄	—	4	1	no	—
127	C₅×C₄⋊C₄	—	4	1	no	—
128	C₅×C₄⋊C₄	—	4	1	no	—
129	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
130	C₅×C₂²⋊C₄	—	2	2	no	—
131	C₅×C₂²⋊C₄	—	2	2	no	—
132	C₅×C₂²⋊C₄	—	4	1	no	—
133	C₄²×C₅	—	4	1	no	—
134	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₅	#41	1	1	yes	characteristic
137	C₄²×C₅	C₂⁴	1	1	yes	characteristic
138	#533	C₂²×C₅	1	1	yes	characteristic
139	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
140	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
141	#500	C₂²×C₅	1	2	yes	complemented
142	#487	C₂²×C₅	1	2	yes	complemented
143	#484	—	2	2	no	—
144	#478	—	2	2	no	—
145	#477	—	2	2	no	—
146	#476	—	2	2	no	—
147	#476	C₂²×C₅	1	2	yes	—
148	#476	C₂²×C₅	1	4	yes	—
149	#474	C₂²×C₅	1	2	yes	—
150	C₈×D₄	—	2	2	no	—
151	C₈×D₄	C₂²×C₅	1	4	yes	—
152	#431	C₂²×C₅	1	1	yes	characteristic
153	#431	C₂²×C₅	1	1	yes	characteristic
154	#428	—	2	1	no	—
155	C₂×#106	—	2	1	no	—
156	#423	C₂²×C₅	1	1	yes	characteristic
157	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
158	#418	—	2	1	no	—
159	C₂×#102	—	2	1	no	—
160	C₂×C₂²⋊C₈	—	2	2	no	—
161	#335	C₂²×C₅	1	4	yes	complemented
162	#333	C₂²×C₅	1	2	yes	complemented
163	#333	C₂²×C₅	1	4	yes	complemented
164	#332	C₂²×C₅	1	2	yes	complemented
165	C₂³×C₅	—	2	2	no	—
166	C₂³×C₅	—	2	4	no	—
167	C₂³×C₅	—	8	1	no	—
168	C₂³×C₅	—	8	1	no	—
169	C₂³×C₅	#121	1	1	yes	characteristic
170	C₅×Q₈	—	4	2	no	—
171	C₅×Q₈	—	8	1	no	—
172	C₅×Q₈	—	8	2	no	—
173	C₅×Q₈	—	8	2	no	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	2	4	no	—
176	C₅×D₄	C₂×Q₁₆	1	4	yes	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	4	1	no	—
180	C₂×C₄×C₅	—	4	4	no	—
181	C₂×C₄×C₅	—	4	1	no	—
182	C₂×C₄×C₅	—	4	1	no	—
183	C₂×C₄×C₅	—	4	2	no	—
184	C₂×C₄×C₅	—	8	2	no	—
185	C₂×C₄×C₅	—	8	1	no	—
186	C₂×C₄×C₅	—	8	2	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	C₂×Q₁₆	1	2	yes	—
191	C₂×C₄×C₅	#135	1	2	yes	—
192	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
195	C₅×C₈	—	2	4	no	—
196	C₅×C₈	—	4	2	no	—
197	C₅×C₈	—	4	2	no	—
198	C₅×C₈	—	4	2	no	—
199	C₂²×Q₈	—	2	1	no	—
200	C₂²×D₄	C₅×D₄	1	1	yes	characteristic
201	C₂³×C₄	—	2	1	no	—
202	C₂×M₁₆	—	2	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	2	4	no	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
209	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
210	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
211	#125	—	2	1	no	—
212	C₂²⋊Q₈	—	2	1	no	—
213	C₂²⋊Q₈	—	2	1	no	—
214	C₄×Q₈	—	2	1	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×D₄	—	2	1	no	—
217	C₄×D₄	—	2	1	no	—
218	C₄×D₄	C₅×D₄	1	2	yes	—
219	C₄×D₄	C₂³×C₅	1	2	yes	—
220	#117	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
224	C₂×C₂²⋊C₄	—	2	1	no	—
225	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
226	#107	—	2	4	no	—
227	#107	—	2	2	no	—
228	#106	—	2	4	no	—
229	#106	—	2	2	no	—
230	C₄⋊C₈	—	2	1	no	—
231	C₄⋊C₈	—	2	1	no	—
232	C₄⋊C₈	C₅×D₄	1	2	yes	—
233	C₄⋊C₈	C₂³×C₅	1	2	yes	—
234	C₄⋊C₈	C₂³×C₅	1	2	yes	—
235	C₄⋊C₈	C₂³×C₅	1	2	yes	—
236	#103	—	2	4	no	—
237	#102	—	2	4	no	—
238	#102	—	2	2	no	—
239	#102	—	2	2	no	—
240	C₂²⋊C₈	—	2	1	no	—
241	C₂²⋊C₈	—	2	4	no	—
242	C₂²⋊C₈	C₅×D₄	1	2	yes	—
243	C₄×C₈	—	2	1	no	—
244	C₄×C₈	C₂³×C₅	1	2	yes	—
245	C₂²×C₅	—	2	1	no	—
246	C₂²×C₅	—	2	1	no	—
247	C₂²×C₅	—	2	4	no	—
248	C₂²×C₅	—	4	2	no	—
249	C₂²×C₅	—	4	2	no	—
250	C₂²×C₅	—	8	4	no	—
251	C₂²×C₅	#467	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	—	4	2	no	—
257	C₄×C₅	—	8	1	no	—
258	C₄×C₅	—	8	2	no	—
259	C₄×C₅	—	8	1	no	—
260	C₄×C₅	#467	1	2	yes	—
261	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
262	C₄×C₅	C₂×#135	1	1	yes	characteristic
263	C₄×C₅	#577	1	2	yes	—
264	C₂⁴	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	4	1	no	—
269	C₂×Q₈	—	8	2	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	C₅×Q₁₆	1	4	yes	—
273	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
274	C₂²×C₄	—	2	1	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	4	1	no	—
278	C₂²×C₄	—	8	2	no	—
279	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
281	C₂²×C₄	C₅×#41	1	1	yes	characteristic
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	8	4	no	—
288	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
306	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
307	C₄²	C₅×#41	1	1	yes	characteristic
308	C₄²	C₂⁴×C₅	1	1	yes	characteristic
309	C₂×C₅	—	2	2	no	—
310	C₂×C₅	—	4	4	no	—
311	C₂×C₅	#3042	1	1	yes	characteristic
312	C₂×C₅	C₂×#467	1	1	yes	characteristic
313	C₂×C₅	#3068	1	1	yes	characteristic
314	C₂³	—	2	2	no	—
315	C₂³	—	2	4	no	—
316	C₂³	—	8	1	no	—
317	C₂³	—	8	1	no	—
318	C₂³	C₅×#121	1	1	yes	characteristic
319	Q₈	—	4	2	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	8	1	no	—
322	Q₈	—	8	2	no	—
323	D₄	—	2	4	no	—
324	D₄	—	2	2	no	—
325	D₄	C₂×C₅×Q₁₆	1	4	yes	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	4	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	2	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	2	no	—
336	C₂×C₄	C₅×#121	1	1	yes	characteristic
337	C₂×C₄	C₅×#121	1	1	yes	characteristic
338	C₂×C₄	C₅×#121	1	1	yes	characteristic
339	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
340	C₂×C₄	C₅×#135	1	2	yes	—
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	4	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	4	2	no	—
348	C₅	#19393	1	1	yes	characteristic, direct product factor, Sylow
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	4	no	—
352	C₂²	—	4	2	no	—
353	C₂²	—	4	2	no	—
354	C₂²	—	8	4	no	—
355	C₂²	C₅×#467	1	2	yes	—
356	C₂²	C₂×C₅×#121	1	1	yes	characteristic
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	1	no	—
360	C₄	—	4	2	no	—
361	C₄	—	8	1	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	C₅×#467	1	2	yes	—
365	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
366	C₄	C₂×C₅×#135	1	1	yes	characteristic
367	C₄	C₅×#577	1	2	yes	—
368	C₂	—	2	2	no	—
369	C₂	—	4	4	no	—
370	C₂	C₅×#3042	1	1	yes	characteristic
371	C₂	C₂×C₅×#467	1	1	yes	characteristic
372	C₂	C₅×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	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
1. C₅×#3230	C₂	semidirect
5. C₅×#1683	C₂	semidirect
6. C₅×#1675	C₂	semidirect
7. C₅×#1571	C₂	semidirect
8. C₅×#1548	C₂	semidirect
9. C₅×#1542	C₂	semidirect
13. C₅×#500	C₂²	semidirect
14. C₅×#487	C₂²	semidirect
33. C₅×#335	C₂²	semidirect
34. C₅×#333	C₂²	semidirect
35. C₅×#333	C₂²	semidirect
36. C₅×#332	C₂²	semidirect
37. #19393	C₅	direct (N × H)
84. #3230	C₂×C₅	semidirect
88. #1683	C₂×C₅	semidirect
89. #1675	C₂×C₅	semidirect
90. #1571	C₂×C₅	semidirect
91. #1548	C₂×C₅	semidirect
92. #1542	C₂×C₅	semidirect
141. #500	C₂²×C₅	semidirect
142. #487	C₂²×C₅	semidirect
161. #335	C₂²×C₅	semidirect
162. #333	C₂²×C₅	semidirect
163. #333	C₂²×C₅	semidirect
164. #332	C₂²×C₅	semidirect
348. C₅	#19393	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×#3230	2
C₅×#3077	2
C₂×C₅×#476	2
C₅×#2972	2
C₅×#1683	2
C₅×#1675	2
C₅×#1571	2
C₅×#1548	2
C₅×#1542	2
#19393	5

Subgroups — C5×#19393 #11942113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19393 #11942113