Subgroups — C₅×#71853 #14302887

Order 1920.

724 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	C₃×C₅×#535	C₂	1	1	yes	maximal, characteristic
2	C₅×#5798	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5236	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#833	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4531	C₂	1	2	yes	maximal, complemented
6	C₅×#1911	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	C₄	1	2	yes	complemented
10	C₃×C₅×#124	C₄	1	2	yes	complemented
11	C₃×C₅×#121	—	2	2	no	—
12	C₃×C₅×#120	—	2	2	no	—
13	C₃×C₅×#117	C₂²	1	1	yes	characteristic
14	C₂×C₅×#217	—	2	2	no	—
15	C₅×D₄×Dic₃	—	2	4	no	—
16	C₅×#926	C₂²	1	2	yes	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₅×#837	—	2	4	no	—
19	C₅×#833	—	2	2	no	—
20	#71853	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#535	D₃	1	1	yes	characteristic
22	C₅×#518	—	3	1	no	—
23	C₅×#421	—	3	2	no	—
24	C₂×C₅×#100	—	3	1	no	—
25	C₅×H(3,ℤ/4)	—	3	2	no	—
26	C₂⁴×C₃×C₅	D₄	1	2	yes	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic, complemented
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	—	2	4	no	—
32	C₂×C₃×C₅×D₄	—	2	2	no	—
33	C₂×C₃×C₅×D₄	—	2	2	no	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic, complemented
36	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
37	C₂²×C₃×C₄×C₅	—	2	1	no	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×C₅×Dic₃	—	2	4	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
42	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	2	yes	complemented
44	C₅×#217	—	2	4	no	—
45	C₅×#217	—	2	2	no	—
46	C₅×C₄⋊Dic₃	—	2	2	no	—
47	C₄×C₅×Dic₃	D₄	1	4	yes	complemented
48	C₅×#208	—	2	2	no	—
49	C₅×#208	—	2	2	no	—
50	C₂×C₅×C₃⋊C₈	—	2	2	no	—
51	C₃×#535	C₂×C₅	1	1	yes	characteristic
52	#5798	C₂×C₅	1	1	yes	characteristic, complemented
53	#5236	C₂×C₅	1	2	yes	complemented
54	C₂×#833	C₂×C₅	1	1	yes	characteristic, complemented
55	#4531	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
58	C₂×C₅×M₁₆	—	3	2	no	—
59	C₅×#127	Dic₃	1	2	yes	complemented
60	C₅×#124	Dic₃	1	2	yes	complemented
61	C₅×#121	—	2	2	no	—
62	C₅×#120	—	2	2	no	—
63	C₄×C₅×D₄	—	6	4	no	—
64	C₅×#117	—	3	2	no	—
65	C₅×#117	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
67	C₅×#104	—	6	4	no	—
68	C₅×#100	—	6	2	no	—
69	#1911	—	3	1	no	Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	—	2	2	no	—
72	C₂³×C₃×C₅	—	2	1	no	—
73	C₂³×C₃×C₅	—	8	2	no	—
74	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×Q₈	—	2	2	no	—
77	C₃×C₅×D₄	—	2	4	no	—
78	C₃×C₅×D₄	—	2	2	no	—
79	C₃×C₅×D₄	—	2	4	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	1	no	—
82	C₃×C₅×D₄	—	8	4	no	—
83	C₂×C₃×C₄×C₅	—	2	2	no	—
84	C₂×C₃×C₄×C₅	—	2	1	no	—
85	C₂×C₃×C₄×C₅	—	4	2	no	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	2	yes	—
90	C₂×C₅×Dic₃	—	2	4	no	—
91	C₂×C₅×Dic₃	—	8	4	no	—
92	C₅×C₃⋊C₈	—	4	2	no	—
93	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
94	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
95	C₃×#127	C₄×C₅	1	2	yes	complemented
96	C₃×#124	C₄×C₅	1	2	yes	complemented
97	C₃×#121	—	2	2	no	—
98	C₃×#120	—	2	2	no	—
99	C₃×#117	C₂²×C₅	1	1	yes	characteristic
100	C₂×#217	—	2	2	no	—
101	D₄×Dic₃	—	2	4	no	—
102	#926	C₂²×C₅	1	2	yes	—
103	C₂×#208	C₂²×C₅	1	2	yes	—
104	#837	—	2	4	no	—
105	#833	—	2	2	no	—
106	C₂⁴×C₅	#67	1	2	yes	—
107	C₅×#41	—	2	2	no	—
108	C₅×#41	—	2	2	no	—
109	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic, complemented
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	—	2	4	no	—
112	C₂×C₅×D₄	—	2	2	no	—
113	C₂×C₅×D₄	—	2	2	no	—
114	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
115	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic, complemented
116	C₂×C₅×D₄	#67	1	2	yes	—
117	C₂²×C₄×C₅	—	2	1	no	—
118	C₂²×C₄×C₅	—	6	4	no	—
119	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₅×M₁₆	—	6	2	no	—
121	C₅×M₁₆	—	6	2	no	—
122	C₂×C₅×C₈	—	6	2	no	—
123	C₅×C₄⋊C₄	—	2	1	no	—
124	C₅×C₄⋊C₄	—	6	2	no	—
125	C₅×C₂²⋊C₄	—	2	4	no	—
126	C₅×C₂²⋊C₄	—	2	1	no	—
127	C₅×C₂²⋊C₄	—	6	4	no	—
128	C₅×C₂²⋊C₄	—	6	2	no	—
129	C₄²×C₅	—	3	4	no	—
130	C₄²×C₅	C₂×Dic₃	1	2	yes	complemented
131	#535	C₅×D₃	1	1	yes	characteristic
132	#518	—	3	1	no	—
133	#421	—	3	2	no	—
134	C₂×#100	—	3	1	no	—
135	H(3,ℤ/4)	—	3	2	no	—
136	C₂²×C₃×C₅	—	2	4	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	—	4	4	no	—
139	C₂²×C₃×C₅	—	4	4	no	—
140	C₂²×C₃×C₅	—	4	2	no	—
141	C₂²×C₃×C₅	—	4	1	no	—
142	C₂²×C₃×C₅	—	8	2	no	—
143	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₃×C₅	#120	1	1	yes	characteristic
145	C₂²×C₃×C₅	#123	1	1	yes	characteristic
146	C₃×C₄×C₅	—	4	2	no	—
147	C₃×C₄×C₅	—	4	1	no	—
148	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
149	C₃×C₄×C₅	#121	1	2	yes	—
150	C₅×Dic₃	—	4	4	no	—
151	C₂⁴×C₃	C₅×D₄	1	2	yes	—
152	C₃×#41	—	2	2	no	—
153	C₃×#41	—	2	2	no	—
154	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic, complemented
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	2	4	no	—
157	C₂×C₃×D₄	—	2	2	no	—
158	C₂×C₃×D₄	—	2	2	no	—
159	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
160	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic, complemented
161	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
162	C₂²×C₃×C₄	—	2	1	no	—
163	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂²×Dic₃	—	2	4	no	—
165	C₃×C₄⋊C₄	—	2	1	no	—
166	C₃×C₂²⋊C₄	—	2	4	no	—
167	C₃×C₂²⋊C₄	—	2	1	no	—
168	C₃×C₄²	C₂×C₄×C₅	1	2	yes	complemented
169	#217	—	2	4	no	—
170	#217	—	2	2	no	—
171	C₄⋊Dic₃	—	2	2	no	—
172	C₄×Dic₃	C₅×D₄	1	4	yes	complemented
173	#208	—	2	2	no	—
174	#208	—	2	2	no	—
175	C₂×C₃⋊C₈	—	2	2	no	—
176	C₂³×C₅	—	2	2	no	—
177	C₂³×C₅	—	2	2	no	—
178	C₂³×C₅	—	2	1	no	—
179	C₂³×C₅	—	8	2	no	—
180	C₂³×C₅	#217	1	4	yes	—
181	C₂³×C₅	C₂×#67	1	1	yes	characteristic
182	C₅×Q₈	—	2	2	no	—
183	C₅×D₄	—	2	4	no	—
184	C₅×D₄	—	2	2	no	—
185	C₅×D₄	—	2	4	no	—
186	C₅×D₄	—	4	1	no	—
187	C₅×D₄	—	4	1	no	—
188	C₅×D₄	—	8	4	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	1	no	—
191	C₂×C₄×C₅	—	4	2	no	—
192	C₂×C₄×C₅	—	6	4	no	—
193	C₂×C₄×C₅	—	24	4	no	—
194	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
195	C₂×C₄×C₅	#237	1	2	yes	—
196	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
198	C₅×C₈	—	12	2	no	—
199	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
201	C₂×M₁₆	—	3	2	no	—
202	#127	C₅×Dic₃	1	2	yes	complemented
203	#124	C₅×Dic₃	1	2	yes	complemented
204	#121	—	2	2	no	—
205	#120	—	2	2	no	—
206	C₄×D₄	—	6	4	no	—
207	#117	—	3	2	no	—
208	#117	C₂×C₅×D₃	1	1	yes	characteristic
209	C₂×C₂²⋊C₄	—	6	2	no	—
210	#104	—	6	4	no	—
211	#100	—	6	2	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	2	1	no	—
214	C₂×C₃×C₅	—	2	1	no	—
215	C₂×C₃×C₅	—	4	4	no	—
216	C₂×C₃×C₅	—	8	1	no	—
217	C₂×C₃×C₅	#386	1	1	yes	characteristic
218	C₂³×C₃	—	2	2	no	—
219	C₂³×C₃	—	2	2	no	—
220	C₂³×C₃	—	2	1	no	—
221	C₂³×C₃	—	8	2	no	—
222	C₂³×C₃	C₅×C₂²⋊C₄	1	4	yes	—
223	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
224	C₃×Q₈	—	2	2	no	—
225	C₃×D₄	—	2	4	no	—
226	C₃×D₄	—	2	2	no	—
227	C₃×D₄	—	2	4	no	—
228	C₃×D₄	—	4	1	no	—
229	C₃×D₄	—	4	1	no	—
230	C₃×D₄	—	8	4	no	—
231	C₂×C₃×C₄	—	2	2	no	—
232	C₂×C₃×C₄	—	2	1	no	—
233	C₂×C₃×C₄	—	4	2	no	—
234	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
236	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂×C₃×C₄	C₅×#41	1	2	yes	—
238	C₂×Dic₃	—	2	4	no	—
239	C₂×Dic₃	—	8	4	no	—
240	C₃⋊C₈	—	4	2	no	—
241	C₂²×C₅	—	2	4	no	—
242	C₂²×C₅	—	4	1	no	—
243	C₂²×C₅	—	4	4	no	—
244	C₂²×C₅	—	4	4	no	—
245	C₂²×C₅	—	4	2	no	—
246	C₂²×C₅	—	4	1	no	—
247	C₂²×C₅	—	8	2	no	—
248	C₂²×C₅	#935	1	1	yes	characteristic
249	C₂²×C₅	#937	1	1	yes	characteristic
250	C₂²×C₅	C₂×#217	1	1	yes	characteristic
251	C₄×C₅	—	4	2	no	—
252	C₄×C₅	—	4	1	no	—
253	C₄×C₅	—	12	4	no	—
254	C₄×C₅	D₄×Dic₃	1	2	yes	—
255	C₄×C₅	#939	1	2	yes	—
256	C₂⁴	C₅×#67	1	2	yes	—
257	#41	—	2	2	no	—
258	#41	—	2	2	no	—
259	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic, complemented
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
265	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic, complemented
266	C₂×D₄	C₅×#67	1	2	yes	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	6	4	no	—
269	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
270	M₁₆	—	6	2	no	—
271	M₁₆	—	6	2	no	—
272	C₂×C₈	—	6	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	6	2	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	6	4	no	—
278	C₂²⋊C₄	—	6	2	no	—
279	C₄²	—	3	4	no	—
280	C₄²	C₂×C₅×Dic₃	1	2	yes	complemented
281	C₃×C₅	#1911	1	1	yes	characteristic, complemented, Hall
282	C₂²×C₃	—	2	4	no	—
283	C₂²×C₃	—	4	1	no	—
284	C₂²×C₃	—	4	4	no	—
285	C₂²×C₃	—	4	4	no	—
286	C₂²×C₃	—	4	2	no	—
287	C₂²×C₃	—	4	1	no	—
288	C₂²×C₃	—	8	2	no	—
289	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
290	C₂²×C₃	C₅×#120	1	1	yes	characteristic
291	C₂²×C₃	C₅×#123	1	1	yes	characteristic
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	—	4	1	no	—
294	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
295	C₃×C₄	C₅×#121	1	2	yes	—
296	Dic₃	—	4	4	no	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	—	4	4	no	—
301	C₂×C₅	—	8	1	no	—
302	C₂×C₅	#5220	1	1	yes	characteristic
303	C₂³	—	2	2	no	—
304	C₂³	—	2	2	no	—
305	C₂³	—	2	1	no	—
306	C₂³	—	8	2	no	—
307	C₂³	C₅×#217	1	4	yes	—
308	C₂³	C₂×C₅×#67	1	1	yes	characteristic
309	Q₈	—	2	2	no	—
310	D₄	—	2	4	no	—
311	D₄	—	2	2	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	1	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	6	4	no	—
320	C₂×C₄	—	24	4	no	—
321	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
322	C₂×C₄	C₅×#237	1	2	yes	—
323	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
324	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
325	C₈	—	12	2	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	4	4	no	—
330	C₂×C₃	—	8	1	no	—
331	C₂×C₃	C₅×#386	1	1	yes	characteristic
332	C₅	#71853	1	1	yes	characteristic, direct product factor, Sylow
333	C₂²	—	2	4	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	4	no	—
336	C₂²	—	4	4	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₅×#935	1	1	yes	characteristic
341	C₂²	C₅×#937	1	1	yes	characteristic
342	C₂²	C₂×C₅×#217	1	1	yes	characteristic
343	C₄	—	4	2	no	—
344	C₄	—	4	1	no	—
345	C₄	—	12	4	no	—
346	C₄	C₅×D₄×Dic₃	1	2	yes	—
347	C₄	C₅×#939	1	2	yes	—
348	C₃	C₅×#1911	1	1	yes	characteristic, complemented, Sylow
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	4	no	—
353	C₂	—	8	1	no	—
354	C₂	C₅×#5220	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#5220
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₃×C₅×#535	C₂
socle	C₂×C₃×C₅	#386
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#535	D₃
hypocenter	C₃	C₅×#1911
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#535

Complements

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

N	H ≅ G/N	type
2. C₅×#5798	C₂	semidirect
3. C₅×#5236	C₂	semidirect
4. C₂×C₅×#833	C₂	semidirect
5. C₅×#4531	C₂	semidirect
9. C₃×C₅×#127	C₄	semidirect
10. C₃×C₅×#124	C₄	semidirect
20. #71853	C₅	direct (N × H)
29. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
35. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
43. C₃×C₄²×C₅	C₂×C₄	semidirect
47. C₄×C₅×Dic₃	D₄	semidirect
52. #5798	C₂×C₅	semidirect
53. #5236	C₂×C₅	semidirect
54. C₂×#833	C₂×C₅	semidirect
55. #4531	C₂×C₅	semidirect
59. C₅×#127	Dic₃	semidirect
60. C₅×#124	Dic₃	semidirect
95. C₃×#127	C₄×C₅	semidirect
96. C₃×#124	C₄×C₅	semidirect
109. C₂×C₅×Q₈	C₂×Dic₃	semidirect
115. C₂×C₅×D₄	C₂×Dic₃	semidirect
130. C₄²×C₅	C₂×Dic₃	semidirect
154. C₂×C₃×Q₈	C₂×C₄×C₅	semidirect
160. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
168. C₃×C₄²	C₂×C₄×C₅	semidirect
172. C₄×Dic₃	C₅×D₄	semidirect
202. #127	C₅×Dic₃	semidirect
203. #124	C₅×Dic₃	semidirect
259. C₂×Q₈	C₂×C₅×Dic₃	semidirect
265. C₂×D₄	C₂×C₅×Dic₃	semidirect
280. C₄²	C₂×C₅×Dic₃	semidirect
281. C₃×C₅	#1911	semidirect
332. C₅	#71853	direct (N × H)
348. C₃	C₅×#1911	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#535	2
C₅×#5798	2
C₅×#5236	2
C₂×C₅×#833	2
C₅×#4531	2
C₅×#1911	3
#71853	5

Subgroups — C5×#71853 #14302887

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71853 #14302887