Subgroups — C₅×#71342 #14290627

Order 1920.

468 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5727	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5094	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5083	C₂	1	1	yes	maximal, characteristic
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4459	C₂	1	1	yes	maximal, characteristic
7	C₅×#4456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1651	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₅×#946	—	2	1	no	—
13	C₂×C₅×#215	—	2	1	no	—
14	C₅×#896	—	2	2	no	—
15	C₅×#893	—	2	1	no	—
16	C₅×#888	—	2	1	no	—
17	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
18	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×#106	—	2	2	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#835	—	2	1	no	—
22	C₅×#810	—	2	1	no	—
23	C₅×#808	—	2	1	no	—
24	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
26	#71342	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×D₄×Q₈	—	3	1	no	—
28	C₅×#498	D₃	1	1	yes	characteristic, complemented
29	C₅×#475	—	3	1	no	—
30	C₅×#461	—	3	1	no	—
31	C₅×#332	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×D₄⋊C₈	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₅×Q₈	—	2	1	no	—
36	C₅×D₃×Q₈	—	4	2	no	—
37	C₅×D₃×Q₈	—	4	2	no	—
38	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	—	2	2	no	—
40	C₂×C₄×C₅×D₃	—	2	1	no	—
41	C₂×C₅×Dic₆	—	2	2	no	—
42	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×#215	—	4	2	no	—
50	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
51	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
52	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
53	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
54	C₅×Dic₃⋊C₄	—	2	2	no	—
55	C₅×Dic₃⋊C₄	—	2	1	no	—
56	C₄×C₅×Dic₃	—	2	1	no	—
57	C₂×C₅×C₃⋊C₈	—	2	1	no	—
58	C₂×C₅×C₃⋊C₈	—	2	1	no	—
59	#5727	C₂×C₅	1	1	yes	characteristic
60	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
61	#5094	C₂×C₅	1	1	yes	characteristic, complemented
62	#5083	C₂×C₅	1	1	yes	characteristic
63	#4547	C₂×C₅	1	1	yes	characteristic, complemented
64	#4459	C₂×C₅	1	1	yes	characteristic
65	#4456	C₂×C₅	1	1	yes	characteristic, complemented
66	C₂²×C₅×Q₈	—	6	1	no	—
67	C₂×C₅×SD₁₆	—	6	1	no	—
68	C₅×C₄⋊Q₈	—	6	1	no	—
69	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
70	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
71	C₅×C₂²⋊Q₈	—	6	1	no	—
72	C₅×C₂²⋊Q₈	—	6	2	no	—
73	C₄×C₅×Q₈	—	3	1	no	—
74	C₄×C₅×D₄	—	3	1	no	—
75	C₄×C₅×D₄	—	6	1	no	—
76	C₅×#106	—	2	2	no	—
77	C₅×#106	—	6	1	no	—
78	C₅×C₄⋊C₈	—	3	1	no	—
79	C₅×C₄⋊C₈	—	3	1	no	—
80	C₅×#103	—	6	1	no	—
81	C₅×#103	—	6	1	no	—
82	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
83	#1651	—	3	1	no	Sylow
84	C₂²×C₅×D₃	—	2	1	no	—
85	C₃×C₅×Q₈	—	4	1	no	—
86	C₃×C₅×Q₈	—	4	1	no	—
87	C₃×C₅×Q₈	—	4	1	no	—
88	C₃×C₅×Q₈	—	4	1	no	—
89	C₂×C₃×C₄×C₅	—	2	2	no	—
90	C₂×C₃×C₄×C₅	—	4	1	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₂×C₅×Dic₃	—	2	1	no	—
96	C₂×C₅×Dic₃	—	2	1	no	—
97	C₅×D₁₂	—	2	1	no	—
98	C₅×D₁₂	SD₁₆	1	2	yes	—
99	C₄×C₅×D₃	—	4	4	no	—
100	C₄×C₅×D₃	—	4	1	no	—
101	C₄×C₅×D₃	—	4	1	no	—
102	C₅×Dic₆	—	2	1	no	—
103	C₅×Dic₆	—	4	4	no	—
104	C₅×Dic₆	SD₁₆	1	2	yes	complemented
105	C₃×C₅×C₈	—	2	2	no	—
106	C₅×C₃⋊C₈	—	4	1	no	—
107	C₅×C₃⋊C₈	—	4	1	no	—
108	C₂×D₃×Q₈	—	2	1	no	—
109	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
110	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
111	#946	—	2	1	no	—
112	C₂×#215	—	2	1	no	—
113	#896	—	2	2	no	—
114	#893	—	2	1	no	—
115	#888	—	2	1	no	—
116	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
117	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
118	C₃×#106	—	2	2	no	—
119	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
120	#835	—	2	1	no	—
121	#810	—	2	1	no	—
122	#808	—	2	1	no	—
123	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
124	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₅×Q₈	—	2	1	no	—
126	C₂×C₅×Q₈	—	2	1	no	—
127	C₂×C₅×Q₈	—	3	1	no	—
128	C₂×C₅×Q₈	—	6	2	no	—
129	C₂×C₅×Q₈	—	12	2	no	—
130	C₂×C₅×Q₈	—	12	2	no	—
131	C₂×C₅×D₄	—	3	1	no	—
132	C₂²×C₄×C₅	—	6	2	no	—
133	C₂²×C₄×C₅	—	6	1	no	—
134	C₅×SD₁₆	—	12	2	no	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	6	1	no	—
137	C₂×C₅×C₈	—	6	1	no	—
138	C₅×C₄⋊C₄	—	2	1	no	—
139	C₅×C₄⋊C₄	—	2	1	no	—
140	C₅×C₄⋊C₄	—	2	1	no	—
141	C₅×C₄⋊C₄	—	3	1	no	—
142	C₅×C₄⋊C₄	—	3	1	no	—
143	C₅×C₄⋊C₄	—	6	2	no	—
144	C₅×C₄⋊C₄	—	6	1	no	—
145	C₅×C₄⋊C₄	#67	1	2	yes	—
146	C₅×C₂²⋊C₄	—	6	2	no	—
147	C₅×C₂²⋊C₄	—	6	1	no	—
148	C₄²×C₅	—	6	1	no	—
149	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
150	D₄×Q₈	—	3	1	no	—
151	#498	C₅×D₃	1	1	yes	characteristic, complemented
152	#475	—	3	1	no	—
153	#461	—	3	1	no	—
154	#332	—	3	1	no	—
155	Q₈⋊C₈	—	3	1	no	—
156	D₄⋊C₈	—	3	1	no	—
157	C₂²×C₃×C₅	#120	1	1	yes	characteristic
158	C₂×C₅×D₃	—	2	2	no	—
159	C₂×C₅×D₃	—	4	1	no	—
160	C₂×C₅×D₃	—	4	1	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	—	4	2	no	—
163	C₃×C₄×C₅	—	8	1	no	—
164	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
165	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
166	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₃×C₄×C₅	#137	1	1	yes	characteristic
168	C₅×Dic₃	—	2	2	no	—
169	C₅×Dic₃	—	4	1	no	—
170	C₅×Dic₃	—	4	1	no	—
171	C₂×C₃×Q₈	—	2	1	no	—
172	C₂×C₃×Q₈	—	2	1	no	—
173	D₃×Q₈	—	4	2	no	—
174	D₃×Q₈	—	4	2	no	—
175	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
176	C₂×C₄×D₃	—	2	2	no	—
177	C₂×C₄×D₃	—	2	1	no	—
178	C₂×Dic₆	—	2	2	no	—
179	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₈	—	2	1	no	—
181	C₃×C₄⋊C₄	—	2	1	no	—
182	C₃×C₄⋊C₄	—	2	1	no	—
183	C₃×C₄⋊C₄	—	2	1	no	—
184	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
185	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
186	#215	—	4	2	no	—
187	(C₂×D₃)⋊C₄	—	2	2	no	—
188	(C₂×D₃)⋊C₄	—	2	1	no	—
189	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
190	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
191	Dic₃⋊C₄	—	2	2	no	—
192	Dic₃⋊C₄	—	2	1	no	—
193	C₄×Dic₃	—	2	1	no	—
194	C₂×C₃⋊C₈	—	2	1	no	—
195	C₂×C₃⋊C₈	—	2	1	no	—
196	C₂³×C₅	—	6	1	no	—
197	C₅×Q₈	—	3	2	no	—
198	C₅×Q₈	—	4	1	no	—
199	C₅×Q₈	—	4	1	no	—
200	C₅×Q₈	—	4	1	no	—
201	C₅×Q₈	—	4	1	no	—
202	C₅×Q₈	—	6	1	no	—
203	C₅×Q₈	—	12	4	no	—
204	C₅×D₄	—	3	2	no	—
205	C₅×D₄	—	6	1	no	—
206	C₂×C₄×C₅	—	2	2	no	—
207	C₂×C₄×C₅	—	4	1	no	—
208	C₂×C₄×C₅	—	6	1	no	—
209	C₂×C₄×C₅	—	6	1	no	—
210	C₂×C₄×C₅	—	6	1	no	—
211	C₂×C₄×C₅	—	12	4	no	—
212	C₂×C₄×C₅	—	12	1	no	—
213	C₂×C₄×C₅	—	12	1	no	—
214	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
215	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
217	C₅×C₈	—	2	2	no	—
218	C₅×C₈	—	12	1	no	—
219	C₅×C₈	—	12	1	no	—
220	C₂²×Q₈	—	6	1	no	—
221	C₂×SD₁₆	—	6	1	no	—
222	C₄⋊Q₈	—	6	1	no	—
223	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
224	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
225	C₂²⋊Q₈	—	6	1	no	—
226	C₂²⋊Q₈	—	6	2	no	—
227	C₄×Q₈	—	3	1	no	—
228	C₄×D₄	—	3	1	no	—
229	C₄×D₄	—	6	1	no	—
230	#106	—	2	2	no	—
231	#106	—	6	1	no	—
232	C₄⋊C₈	—	3	1	no	—
233	C₄⋊C₈	—	3	1	no	—
234	#103	—	6	1	no	—
235	#103	—	6	1	no	—
236	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
237	C₂×C₃×C₅	#448	1	1	yes	characteristic
238	C₂×C₃×C₅	#450	1	1	yes	characteristic
239	C₂×C₃×C₅	#456	1	1	yes	characteristic
240	C₅×D₃	—	4	2	no	—
241	C₂²×D₃	—	2	1	no	—
242	C₃×Q₈	—	4	1	no	—
243	C₃×Q₈	—	4	1	no	—
244	C₃×Q₈	—	4	1	no	—
245	C₃×Q₈	—	4	1	no	—
246	C₂×C₃×C₄	—	2	2	no	—
247	C₂×C₃×C₄	—	4	1	no	—
248	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂×Dic₃	—	2	1	no	—
252	C₂×Dic₃	—	2	1	no	—
253	C₂×Dic₃	—	2	1	no	—
254	D₁₂	—	2	1	no	—
255	D₁₂	C₅×SD₁₆	1	2	yes	—
256	C₄×D₃	—	4	4	no	—
257	C₄×D₃	—	4	1	no	—
258	C₄×D₃	—	4	1	no	—
259	Dic₆	—	2	1	no	—
260	Dic₆	—	4	4	no	—
261	Dic₆	C₅×SD₁₆	1	2	yes	complemented
262	C₃×C₈	—	2	2	no	—
263	C₃⋊C₈	—	4	1	no	—
264	C₃⋊C₈	—	4	1	no	—
265	C₂²×C₅	—	6	2	no	—
266	C₂²×C₅	—	12	1	no	—
267	C₂²×C₅	—	12	1	no	—
268	C₂²×C₅	#937	1	1	yes	characteristic
269	C₄×C₅	—	2	1	no	—
270	C₄×C₅	—	4	2	no	—
271	C₄×C₅	—	6	2	no	—
272	C₄×C₅	—	8	1	no	—
273	C₄×C₅	—	12	1	no	—
274	C₄×C₅	—	12	1	no	—
275	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
276	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
277	C₄×C₅	#914	1	1	yes	characteristic
278	C₄×C₅	#915	1	1	yes	characteristic
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	3	1	no	—
282	C₂×Q₈	—	6	2	no	—
283	C₂×Q₈	—	12	2	no	—
284	C₂×Q₈	—	12	2	no	—
285	C₂×D₄	—	3	1	no	—
286	C₂²×C₄	—	6	2	no	—
287	C₂²×C₄	—	6	1	no	—
288	SD₁₆	—	12	2	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	6	1	no	—
291	C₂×C₈	—	6	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	3	1	no	—
296	C₄⋊C₄	—	3	1	no	—
297	C₄⋊C₄	—	6	2	no	—
298	C₄⋊C₄	—	6	1	no	—
299	C₄⋊C₄	C₅×#67	1	2	yes	—
300	C₂²⋊C₄	—	6	2	no	—
301	C₂²⋊C₄	—	6	1	no	—
302	C₄²	—	6	1	no	—
303	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
304	C₃×C₅	#1651	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	C₅×#120	1	1	yes	characteristic
306	C₂×D₃	—	2	2	no	—
307	C₂×D₃	—	4	1	no	—
308	C₂×D₃	—	4	1	no	—
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	—	4	2	no	—
311	C₃×C₄	—	8	1	no	—
312	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
313	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
314	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
315	C₃×C₄	C₅×#137	1	1	yes	characteristic
316	Dic₃	—	2	2	no	—
317	Dic₃	—	4	1	no	—
318	Dic₃	—	4	1	no	—
319	C₂×C₅	—	12	2	no	—
320	C₂×C₅	#5095	1	1	yes	characteristic
321	C₂×C₅	#5169	1	1	yes	characteristic
322	C₂×C₅	#5170	1	1	yes	characteristic
323	C₂³	—	6	1	no	—
324	Q₈	—	3	2	no	—
325	Q₈	—	4	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	4	1	no	—
329	Q₈	—	6	1	no	—
330	Q₈	—	12	4	no	—
331	D₄	—	3	2	no	—
332	D₄	—	6	1	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	6	1	no	—
336	C₂×C₄	—	6	1	no	—
337	C₂×C₄	—	6	1	no	—
338	C₂×C₄	—	12	4	no	—
339	C₂×C₄	—	12	1	no	—
340	C₂×C₄	—	12	1	no	—
341	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
342	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
343	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
344	C₈	—	2	2	no	—
345	C₈	—	12	1	no	—
346	C₈	—	12	1	no	—
347	C₂×C₃	C₅×#448	1	1	yes	characteristic
348	C₂×C₃	C₅×#450	1	1	yes	characteristic
349	C₂×C₃	C₅×#456	1	1	yes	characteristic
350	D₃	—	4	2	no	—
351	C₅	#71342	1	1	yes	characteristic, direct product factor, Sylow
352	C₂²	—	6	2	no	—
353	C₂²	—	12	1	no	—
354	C₂²	—	12	1	no	—
355	C₂²	C₅×#937	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	4	2	no	—
358	C₄	—	6	2	no	—
359	C₄	—	8	1	no	—
360	C₄	—	12	1	no	—
361	C₄	—	12	1	no	—
362	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
363	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
364	C₄	C₅×#914	1	1	yes	characteristic
365	C₄	C₅×#915	1	1	yes	characteristic
366	C₃	C₅×#1651	1	1	yes	characteristic, complemented, Sylow
367	C₂	—	12	2	no	—
368	C₂	C₅×#5095	1	1	yes	characteristic
369	C₂	C₅×#5169	1	1	yes	characteristic
370	C₂	C₅×#5170	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#937
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#498	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂×C₅×D₁₂	D₄
hypercenter	C₅×#498	D₃
hypocenter	C₃	C₅×#1651
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#498

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×C₅×#498	C₂	semidirect
3. C₅×#5094	C₂	semidirect
5. C₅×#4547	C₂	semidirect
7. C₅×#4456	C₂	semidirect
26. #71342	C₅	direct (N × H)
28. C₅×#498	D₃	semidirect
60. C₃×#498	C₂×C₅	semidirect
61. #5094	C₂×C₅	semidirect
63. #4547	C₂×C₅	semidirect
65. #4456	C₂×C₅	semidirect
104. C₅×Dic₆	SD₁₆	semidirect
151. #498	C₅×D₃	semidirect
261. Dic₆	C₅×SD₁₆	semidirect
304. C₃×C₅	#1651	semidirect
351. C₅	#71342	direct (N × H)
366. C₃	C₅×#1651	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#5727	2
C₃×C₅×#498	2
C₅×#5094	2
C₅×#5083	2
C₅×#4547	2
C₅×#4459	2
C₅×#4456	2
C₅×#1651	3
#71342	5

Subgroups — C5×#71342 #14290627

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71342 #14290627