Subgroups — C₅×#68925 #14303598

Order 1920.

372 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5122	C₂	1	1	yes	maximal, characteristic
2	C₅×#5119	C₂	1	1	yes	maximal, characteristic
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic
5	C₅×#4555	C₂	1	1	yes	maximal, characteristic
6	C₅×#4555	C₂	1	1	yes	maximal, characteristic
7	C₅×#4550	C₂	1	1	yes	maximal, characteristic
8	C₅×#1922	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#926	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₅×#818	C₄	1	1	yes	characteristic, complemented
17	C₅×#818	C₄	1	1	yes	characteristic, complemented
18	C₅×#817	C₄	1	1	yes	characteristic, complemented
19	C₅×#817	C₄	1	1	yes	characteristic, complemented
20	C₅×#815	—	2	1	no	—
21	C₅×Dic₃⋊C₈	—	2	1	no	—
22	C₅×Dic₃⋊C₈	—	2	1	no	—
23	C₅×C₈×Dic₃	—	2	1	no	—
24	#68925	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#427	—	3	1	no	—
26	C₅×#424	—	3	1	no	—
27	C₅×#405	—	3	1	no	—
28	C₅×#341	D₃	1	1	yes	characteristic
29	C₅×#339	—	3	1	no	—
30	C₅×#339	—	3	1	no	—
31	C₅×#336	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×Dic₃	—	2	1	no	—
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	—	2	1	no	—
36	C₃×C₅×M₁₆	—	2	1	no	—
37	C₃×C₅×M₁₆	Q₈	1	2	yes	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	1	no	—
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₅×#217	—	2	1	no	—
45	C₅×#217	—	2	1	no	—
46	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
47	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
50	C₄×C₅×Dic₃	—	2	1	no	—
51	C₄×C₅×Dic₃	—	2	1	no	—
52	C₄×C₅×Dic₃	—	2	1	no	—
53	C₄×C₅×Dic₃	—	2	1	no	—
54	C₂×C₅×C₃⋊C₈	—	2	1	no	—
55	C₂×C₅×C₃⋊C₈	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
57	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
58	#5122	C₂×C₅	1	1	yes	characteristic
59	#5119	C₂×C₅	1	1	yes	characteristic
60	#5106	C₂×C₅	1	1	yes	characteristic
61	C₃×#341	C₂×C₅	1	1	yes	characteristic
62	#4555	C₂×C₅	1	1	yes	characteristic
63	#4555	C₂×C₅	1	1	yes	characteristic
64	#4550	C₂×C₅	1	1	yes	characteristic
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₅×C₈	—	3	1	no	—
68	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
69	C₅×#117	—	3	1	no	—
70	C₅×#117	—	3	1	no	—
71	C₂×C₄²×C₅	—	3	1	no	—
72	C₅×#107	—	3	1	no	—
73	C₅×#107	—	3	1	no	—
74	C₅×#106	—	3	1	no	—
75	C₅×#106	—	3	1	no	—
76	C₅×C₄⋊C₈	—	6	1	no	—
77	C₅×C₄⋊C₈	—	6	1	no	—
78	C₅×#97	—	6	1	no	—
79	C₄×C₅×C₈	—	6	1	no	—
80	#1922	—	3	1	no	Sylow
81	C₂³×C₃×C₅	#41	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	C₂×C₅×Dic₃	—	2	2	no	—
89	C₂×C₅×Dic₃	—	2	1	no	—
90	C₂×C₅×Dic₃	—	2	1	no	—
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	4	1	no	—
94	C₂×C₅×Dic₃	—	4	1	no	—
95	C₃×C₅×C₈	—	2	1	no	—
96	C₃×C₅×C₈	—	2	1	no	—
97	C₃×C₅×C₈	—	2	2	no	—
98	C₃×C₅×C₈	—	4	1	no	—
99	C₅×C₃⋊C₈	—	2	1	no	—
100	C₅×C₃⋊C₈	—	2	1	no	—
101	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
102	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
103	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
104	#926	C₂²×C₅	1	1	yes	characteristic
105	#926	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
107	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
108	#818	C₄×C₅	1	1	yes	characteristic, complemented
109	#818	C₄×C₅	1	1	yes	characteristic, complemented
110	#817	C₄×C₅	1	1	yes	characteristic, complemented
111	#817	C₄×C₅	1	1	yes	characteristic, complemented
112	#815	—	2	1	no	—
113	Dic₃⋊C₈	—	2	1	no	—
114	Dic₃⋊C₈	—	2	1	no	—
115	C₈×Dic₃	—	2	1	no	—
116	C₂²×C₄×C₅	—	6	1	no	—
117	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₅×M₁₆	—	2	1	no	—
119	C₅×M₁₆	—	2	1	no	—
120	C₅×M₁₆	—	2	1	no	—
121	C₅×M₁₆	Dic₆	1	2	yes	—
122	C₂×C₅×C₈	—	2	1	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	2	1	no	—
126	C₂×C₅×C₈	—	3	1	no	—
127	C₂×C₅×C₈	—	3	1	no	—
128	C₂×C₅×C₈	—	6	1	no	—
129	C₂×C₅×C₈	—	6	1	no	—
130	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
131	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
132	C₅×C₄⋊C₄	—	3	1	no	—
133	C₅×C₄⋊C₄	—	3	1	no	—
134	C₅×C₄⋊C₄	—	3	1	no	—
135	C₅×C₄⋊C₄	—	3	1	no	—
136	C₅×C₂²⋊C₄	—	6	1	no	—
137	C₅×C₂²⋊C₄	—	6	1	no	—
138	C₄²×C₅	—	6	1	no	—
139	C₄²×C₅	—	6	1	no	—
140	C₄²×C₅	—	6	1	no	—
141	C₄²×C₅	—	6	1	no	—
142	#427	—	3	1	no	—
143	#424	—	3	1	no	—
144	#405	—	3	1	no	—
145	#341	C₅×D₃	1	1	yes	characteristic
146	#339	—	3	1	no	—
147	#339	—	3	1	no	—
148	#336	—	3	1	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	2	1	no	—
151	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
152	C₂²×C₃×C₅	#125	1	1	yes	characteristic
153	C₂²×C₃×C₅	#126	1	1	yes	characteristic
154	C₃×C₄×C₅	#117	1	1	yes	characteristic
155	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
157	C₃×C₄×C₅	#123	1	1	yes	characteristic
158	C₅×Dic₃	—	4	2	no	—
159	C₅×Dic₃	—	4	1	no	—
160	C₅×Dic₃	—	4	1	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₅×Dic₃	—	4	1	no	—
163	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂²×Dic₃	—	2	1	no	—
165	C₃×M₁₆	—	2	1	no	—
166	C₃×M₁₆	—	2	1	no	—
167	C₃×M₁₆	—	2	1	no	—
168	C₃×M₁₆	C₅×Q₈	1	2	yes	—
169	C₂×C₃×C₈	—	2	1	no	—
170	C₂×C₃×C₈	—	2	1	no	—
171	C₂×C₃×C₈	—	2	1	no	—
172	C₂×C₃×C₈	—	2	1	no	—
173	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
174	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
175	#217	—	2	1	no	—
176	#217	—	2	1	no	—
177	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
178	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
179	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
180	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
181	C₄×Dic₃	—	2	1	no	—
182	C₄×Dic₃	—	2	1	no	—
183	C₄×Dic₃	—	2	1	no	—
184	C₄×Dic₃	—	2	1	no	—
185	C₂×C₃⋊C₈	—	2	1	no	—
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
188	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
189	C₂³×C₅	#239	1	1	yes	characteristic
190	C₂×C₄×C₅	—	6	2	no	—
191	C₂×C₄×C₅	—	6	1	no	—
192	C₂×C₄×C₅	—	6	1	no	—
193	C₂×C₄×C₅	—	6	1	no	—
194	C₂×C₄×C₅	—	6	1	no	—
195	C₂×C₄×C₅	—	12	1	no	—
196	C₂×C₄×C₅	—	12	1	no	—
197	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	#235	1	1	yes	characteristic
200	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
201	C₂×C₄×C₅	#237	1	1	yes	characteristic
202	C₂×C₄×C₅	#237	1	1	yes	characteristic
203	C₅×C₈	—	2	1	no	—
204	C₅×C₈	—	2	1	no	—
205	C₅×C₈	—	2	2	no	—
206	C₅×C₈	—	4	1	no	—
207	C₅×C₈	—	6	1	no	—
208	C₅×C₈	—	6	1	no	—
209	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
210	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
211	C₂²×C₈	—	3	1	no	—
212	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
213	#117	—	3	1	no	—
214	#117	—	3	1	no	—
215	C₂×C₄²	—	3	1	no	—
216	#107	—	3	1	no	—
217	#107	—	3	1	no	—
218	#106	—	3	1	no	—
219	#106	—	3	1	no	—
220	C₄⋊C₈	—	6	1	no	—
221	C₄⋊C₈	—	6	1	no	—
222	#97	—	6	1	no	—
223	C₄×C₈	—	6	1	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	1	no	—
226	C₂×C₃×C₅	#387	1	1	yes	characteristic
227	C₂×C₃×C₅	#443	1	1	yes	characteristic
228	C₂×C₃×C₅	#444	1	1	yes	characteristic
229	C₂³×C₃	C₅×#41	1	1	yes	characteristic
230	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
233	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
234	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
235	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	—	2	1	no	—
238	C₂×Dic₃	—	2	1	no	—
239	C₂×Dic₃	—	2	1	no	—
240	C₂×Dic₃	—	2	1	no	—
241	C₂×Dic₃	—	4	1	no	—
242	C₂×Dic₃	—	4	1	no	—
243	C₃×C₈	—	2	1	no	—
244	C₃×C₈	—	2	1	no	—
245	C₃×C₈	—	2	2	no	—
246	C₃×C₈	—	4	1	no	—
247	C₃⋊C₈	—	2	1	no	—
248	C₃⋊C₈	—	2	1	no	—
249	C₂²×C₅	—	2	1	no	—
250	C₂²×C₅	—	2	1	no	—
251	C₂²×C₅	#890	1	1	yes	characteristic
252	C₂²×C₅	#893	1	1	yes	characteristic
253	C₂²×C₅	#898	1	1	yes	characteristic
254	C₄×C₅	—	12	2	no	—
255	C₄×C₅	—	12	1	no	—
256	C₄×C₅	—	12	1	no	—
257	C₄×C₅	—	12	1	no	—
258	C₄×C₅	—	12	1	no	—
259	C₄×C₅	C₄×Dic₆	1	1	yes	characteristic
260	C₄×C₅	#877	1	1	yes	characteristic
261	C₄×C₅	#878	1	1	yes	characteristic
262	C₄×C₅	#883	1	1	yes	characteristic
263	C₂²×C₄	—	6	1	no	—
264	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
265	M₁₆	—	2	1	no	—
266	M₁₆	—	2	1	no	—
267	M₁₆	—	2	1	no	—
268	M₁₆	C₅×Dic₆	1	2	yes	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	3	1	no	—
274	C₂×C₈	—	3	1	no	—
275	C₂×C₈	—	6	1	no	—
276	C₂×C₈	—	6	1	no	—
277	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
278	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
279	C₄⋊C₄	—	3	1	no	—
280	C₄⋊C₄	—	3	1	no	—
281	C₄⋊C₄	—	3	1	no	—
282	C₄⋊C₄	—	3	1	no	—
283	C₂²⋊C₄	—	6	1	no	—
284	C₂²⋊C₄	—	6	1	no	—
285	C₄²	—	6	1	no	—
286	C₄²	—	6	1	no	—
287	C₄²	—	6	1	no	—
288	C₄²	—	6	1	no	—
289	C₃×C₅	#1922	1	1	yes	characteristic, complemented, Hall
290	C₂²×C₃	—	2	1	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
293	C₂²×C₃	C₅×#125	1	1	yes	characteristic
294	C₂²×C₃	C₅×#126	1	1	yes	characteristic
295	C₃×C₄	C₅×#117	1	1	yes	characteristic
296	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
297	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
298	C₃×C₄	C₅×#123	1	1	yes	characteristic
299	Dic₃	—	4	2	no	—
300	Dic₃	—	4	1	no	—
301	Dic₃	—	4	1	no	—
302	Dic₃	—	4	1	no	—
303	Dic₃	—	4	1	no	—
304	C₂×C₅	—	2	1	no	—
305	C₂×C₅	—	2	1	no	—
306	C₂×C₅	#4654	1	1	yes	characteristic
307	C₂×C₅	#4894	1	1	yes	characteristic
308	C₂×C₅	#4895	1	1	yes	characteristic
309	C₂³	C₅×#239	1	1	yes	characteristic
310	C₂×C₄	—	6	2	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	—	6	1	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	6	1	no	—
315	C₂×C₄	—	12	1	no	—
316	C₂×C₄	—	12	1	no	—
317	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
319	C₂×C₄	C₅×#235	1	1	yes	characteristic
320	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
321	C₂×C₄	C₅×#237	1	1	yes	characteristic
322	C₂×C₄	C₅×#237	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	—	4	1	no	—
327	C₈	—	6	1	no	—
328	C₈	—	6	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	C₅×#387	1	1	yes	characteristic
332	C₂×C₃	C₅×#443	1	1	yes	characteristic
333	C₂×C₃	C₅×#444	1	1	yes	characteristic
334	C₅	#68925	1	1	yes	characteristic, direct product factor, Sylow
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	C₅×#890	1	1	yes	characteristic
338	C₂²	C₅×#893	1	1	yes	characteristic
339	C₂²	C₅×#898	1	1	yes	characteristic
340	C₄	—	12	2	no	—
341	C₄	—	12	1	no	—
342	C₄	—	12	1	no	—
343	C₄	—	12	1	no	—
344	C₄	—	12	1	no	—
345	C₄	C₄×C₅×Dic₆	1	1	yes	characteristic
346	C₄	C₅×#877	1	1	yes	characteristic
347	C₄	C₅×#878	1	1	yes	characteristic
348	C₄	C₅×#883	1	1	yes	characteristic
349	C₃	C₅×#1922	1	1	yes	characteristic, complemented, Sylow
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	C₅×#4654	1	1	yes	characteristic
353	C₂	C₅×#4894	1	1	yes	characteristic
354	C₂	C₅×#4895	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#341	D₃
hypocenter	C₃	C₅×#1922
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#341

Complements

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

N	H ≅ G/N	type
16. C₅×#818	C₄	semidirect
17. C₅×#818	C₄	semidirect
18. C₅×#817	C₄	semidirect
19. C₅×#817	C₄	semidirect
24. #68925	C₅	direct (N × H)
108. #818	C₄×C₅	semidirect
109. #818	C₄×C₅	semidirect
110. #817	C₄×C₅	semidirect
111. #817	C₄×C₅	semidirect
289. C₃×C₅	#1922	semidirect
334. C₅	#68925	direct (N × H)
349. C₃	C₅×#1922	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5122	2
C₅×#5119	2
C₅×#5106	2
C₃×C₅×#341	2
C₅×#4555	2
C₅×#4550	2
C₅×#1922	3
#68925	5

Subgroups — C5×#68925 #14303598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68925 #14303598