Subgroups — C₅×#71723 #14301441

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic
2	C₅×#5798	C₂	1	1	yes	maximal, characteristic
3	C₅×#5225	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#833	C₂	1	1	yes	maximal, characteristic
5	C₅×#4523	C₂	1	2	yes	maximal, complemented
6	C₅×#1885	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₃×C₅×#124	—	2	2	no	—
10	C₃×C₅×#124	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#217	—	2	2	no	—
14	C₅×D₄×Dic₃	—	2	4	no	—
15	C₅×#926	C₂²	1	2	yes	—
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₅×#835	—	2	4	no	—
18	C₅×#833	—	2	2	no	—
19	#71723	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#124	D₃	1	1	yes	characteristic
21	C₅×#518	—	3	1	no	—
22	C₅×#419	—	3	2	no	—
23	C₂×C₅×#100	—	3	1	no	—
24	C₅×#339	—	3	2	no	—
25	C₂⁴×C₃×C₅	D₄	1	2	yes	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	complemented
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₅×Dic₃	—	2	4	no	—
38	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
39	C₃×C₄²×C₅	—	2	1	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	2	yes	complemented
41	C₅×#217	—	2	4	no	—
42	C₅×#217	—	2	2	no	—
43	C₅×C₄⋊Dic₃	D₄	1	4	yes	—
44	C₄×C₅×Dic₃	—	2	2	no	—
45	C₅×#208	—	2	2	no	—
46	C₅×#208	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic
49	#5798	C₂×C₅	1	1	yes	characteristic
50	#5225	C₂×C₅	1	2	yes	complemented
51	C₂×#833	C₂×C₅	1	1	yes	characteristic
52	#4523	C₂×C₅	1	2	yes	complemented
53	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
55	C₂×C₅×M₁₆	—	3	2	no	—
56	C₅×#124	—	2	2	no	—
57	C₅×#124	Dic₃	1	4	yes	complemented
58	C₄×C₅×D₄	—	6	4	no	—
59	C₅×#117	—	3	2	no	—
60	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
61	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	C₅×#103	—	6	4	no	—
64	C₅×#100	—	6	2	no	—
65	#1885	—	3	1	no	Sylow
66	C₂³×C₃×C₅	—	2	2	no	—
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	4	4	no	—
69	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	4	no	—
72	C₃×C₅×Q₈	—	4	2	no	—
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₅×Dic₃	—	2	4	no	—
85	C₂×C₅×Dic₃	—	8	4	no	—
86	C₅×C₃⋊C₈	—	4	2	no	—
87	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
88	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
89	C₃×#124	—	2	2	no	—
90	C₃×#124	C₄×C₅	1	4	yes	complemented
91	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₂×#217	—	2	2	no	—
94	D₄×Dic₃	—	2	4	no	—
95	#926	C₂²×C₅	1	2	yes	—
96	C₂×#208	C₂²×C₅	1	2	yes	—
97	#835	—	2	4	no	—
98	#833	—	2	2	no	—
99	C₂⁴×C₅	#67	1	2	yes	—
100	C₂×C₅×Q₈	—	2	1	no	—
101	C₂×C₅×Q₈	—	2	2	no	—
102	C₂×C₅×Q₈	—	2	2	no	—
103	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	complemented
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	C₂×Dic₃	1	2	yes	—
107	C₂×C₅×D₄	#67	1	2	yes	—
108	C₂²×C₄×C₅	—	2	1	no	—
109	C₂²×C₄×C₅	—	2	1	no	—
110	C₂²×C₄×C₅	—	6	4	no	—
111	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₅×M₁₆	—	6	2	no	—
113	C₅×M₁₆	—	6	2	no	—
114	C₂×C₅×C₈	—	6	2	no	—
115	C₅×C₄⋊C₄	—	3	4	no	—
116	C₅×C₂²⋊C₄	—	2	8	no	—
117	C₅×C₂²⋊C₄	—	6	4	no	—
118	C₅×C₂²⋊C₄	—	6	2	no	—
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	—	6	2	no	—
121	C₄²×C₅	C₂×Dic₃	1	2	yes	complemented
122	C₂×#124	C₅×D₃	1	1	yes	characteristic
123	#518	—	3	1	no	—
124	#419	—	3	2	no	—
125	C₂×#100	—	3	1	no	—
126	#339	—	3	2	no	—
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	—	2	4	no	—
129	C₂²×C₃×C₅	—	4	4	no	—
130	C₂²×C₃×C₅	—	4	4	no	—
131	C₂²×C₃×C₅	—	8	2	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#120	1	1	yes	characteristic
134	C₂²×C₃×C₅	#123	1	1	yes	characteristic
135	C₃×C₄×C₅	—	4	2	no	—
136	C₃×C₄×C₅	—	4	2	no	—
137	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
138	C₃×C₄×C₅	#121	1	2	yes	—
139	C₅×Dic₃	—	4	4	no	—
140	C₂⁴×C₃	C₅×D₄	1	2	yes	—
141	C₂×C₃×Q₈	—	2	1	no	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	C₂×C₃×Q₈	—	2	2	no	—
144	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	complemented
145	C₂×C₃×D₄	—	2	2	no	—
146	C₂×C₃×D₄	—	2	2	no	—
147	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
148	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂²×C₃×C₄	—	2	1	no	—
151	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂²×Dic₃	—	2	4	no	—
153	C₃×C₂²⋊C₄	—	2	8	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	C₂×C₄×C₅	1	2	yes	complemented
156	#217	—	2	4	no	—
157	#217	—	2	2	no	—
158	C₄⋊Dic₃	C₅×D₄	1	4	yes	—
159	C₄×Dic₃	—	2	2	no	—
160	#208	—	2	2	no	—
161	#208	—	2	2	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂³×C₅	—	2	2	no	—
164	C₂³×C₅	—	2	2	no	—
165	C₂³×C₅	—	4	4	no	—
166	C₂³×C₅	#217	1	4	yes	—
167	C₂³×C₅	C₂×#67	1	1	yes	characteristic
168	C₅×Q₈	—	2	4	no	—
169	C₅×Q₈	—	4	2	no	—
170	C₅×D₄	—	2	4	no	—
171	C₅×D₄	—	2	4	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	—	6	4	no	—
178	C₂×C₄×C₅	—	24	4	no	—
179	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
180	C₂×C₄×C₅	#237	1	2	yes	—
181	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
183	C₅×C₈	—	12	2	no	—
184	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
185	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
186	C₂×M₁₆	—	3	2	no	—
187	#124	—	2	2	no	—
188	#124	C₅×Dic₃	1	4	yes	complemented
189	C₄×D₄	—	6	4	no	—
190	#117	—	3	2	no	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	6	2	no	—
193	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
194	#103	—	6	4	no	—
195	#100	—	6	2	no	—
196	C₂×C₃×C₅	—	2	1	no	—
197	C₂×C₃×C₅	—	2	1	no	—
198	C₂×C₃×C₅	—	4	4	no	—
199	C₂×C₃×C₅	#386	1	1	yes	characteristic
200	C₂×C₃×C₅	#455	1	2	yes	—
201	C₂³×C₃	—	2	2	no	—
202	C₂³×C₃	—	2	2	no	—
203	C₂³×C₃	—	4	4	no	—
204	C₂³×C₃	C₅×C₂²⋊C₄	1	4	yes	—
205	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
206	C₃×Q₈	—	2	4	no	—
207	C₃×Q₈	—	4	2	no	—
208	C₃×D₄	—	2	4	no	—
209	C₃×D₄	—	2	4	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₅×#41	1	2	yes	—
219	C₂×Dic₃	—	2	4	no	—
220	C₂×Dic₃	—	8	4	no	—
221	C₃⋊C₈	—	4	2	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	2	4	no	—
224	C₂²×C₅	—	4	4	no	—
225	C₂²×C₅	—	4	4	no	—
226	C₂²×C₅	—	8	2	no	—
227	C₂²×C₅	#935	1	1	yes	characteristic
228	C₂²×C₅	#937	1	1	yes	characteristic
229	C₂²×C₅	C₂×#217	1	1	yes	characteristic
230	C₄×C₅	—	4	2	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	—	12	4	no	—
233	C₄×C₅	D₄×Dic₃	1	2	yes	—
234	C₄×C₅	#939	1	2	yes	—
235	C₂⁴	C₅×#67	1	2	yes	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	C₂×C₅×Dic₃	1	2	yes	complemented
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	C₂×C₅×Dic₃	1	2	yes	—
243	C₂×D₄	C₅×#67	1	2	yes	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	6	4	no	—
247	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
248	M₁₆	—	6	2	no	—
249	M₁₆	—	6	2	no	—
250	C₂×C₈	—	6	2	no	—
251	C₄⋊C₄	—	3	4	no	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	6	4	no	—
254	C₂²⋊C₄	—	6	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	6	2	no	—
257	C₄²	C₂×C₅×Dic₃	1	2	yes	complemented
258	C₃×C₅	#1885	1	1	yes	characteristic, complemented, Hall
259	C₂²×C₃	—	2	2	no	—
260	C₂²×C₃	—	2	4	no	—
261	C₂²×C₃	—	4	4	no	—
262	C₂²×C₃	—	4	4	no	—
263	C₂²×C₃	—	8	2	no	—
264	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₂²×C₃	C₅×#120	1	1	yes	characteristic
266	C₂²×C₃	C₅×#123	1	1	yes	characteristic
267	C₃×C₄	—	4	2	no	—
268	C₃×C₄	—	4	2	no	—
269	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
270	C₃×C₄	C₅×#121	1	2	yes	—
271	Dic₃	—	4	4	no	—
272	C₂×C₅	—	2	1	no	—
273	C₂×C₅	—	2	1	no	—
274	C₂×C₅	—	4	4	no	—
275	C₂×C₅	#5061	1	2	yes	—
276	C₂×C₅	#5220	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₅×#217	1	4	yes	—
281	C₂³	C₂×C₅×#67	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	6	4	no	—
292	C₂×C₄	—	24	4	no	—
293	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
294	C₂×C₄	C₅×#237	1	2	yes	—
295	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
296	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
297	C₈	—	12	2	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	2	1	no	—
300	C₂×C₃	—	4	4	no	—
301	C₂×C₃	C₅×#386	1	1	yes	characteristic
302	C₂×C₃	C₅×#455	1	2	yes	—
303	C₅	#71723	1	1	yes	characteristic, direct product factor, Sylow
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₅×#935	1	1	yes	characteristic
310	C₂²	C₅×#937	1	1	yes	characteristic
311	C₂²	C₂×C₅×#217	1	1	yes	characteristic
312	C₄	—	4	2	no	—
313	C₄	—	4	2	no	—
314	C₄	—	12	4	no	—
315	C₄	C₅×D₄×Dic₃	1	2	yes	—
316	C₄	C₅×#939	1	2	yes	—
317	C₃	C₅×#1885	1	1	yes	characteristic, complemented, Sylow
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	C₅×#5061	1	2	yes	—
322	C₂	C₅×#5220	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#937
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#124	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#124	D₃
hypocenter	C₃	C₅×#1885
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#124

Complements

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

N	H ≅ G/N	type
3. C₅×#5225	C₂	semidirect
5. C₅×#4523	C₂	semidirect
10. C₃×C₅×#124	C₄	semidirect
19. #71723	C₅	direct (N × H)
29. C₂×C₃×C₅×Q₈	C₂×C₄	semidirect
40. C₃×C₄²×C₅	C₂×C₄	semidirect
50. #5225	C₂×C₅	semidirect
52. #4523	C₂×C₅	semidirect
57. C₅×#124	Dic₃	semidirect
90. C₃×#124	C₄×C₅	semidirect
103. C₂×C₅×Q₈	C₂×Dic₃	semidirect
121. C₄²×C₅	C₂×Dic₃	semidirect
144. C₂×C₃×Q₈	C₂×C₄×C₅	semidirect
155. C₃×C₄²	C₂×C₄×C₅	semidirect
188. #124	C₅×Dic₃	semidirect
239. C₂×Q₈	C₂×C₅×Dic₃	semidirect
257. C₄²	C₂×C₅×Dic₃	semidirect
258. C₃×C₅	#1885	semidirect
303. C₅	#71723	direct (N × H)
317. C₃	C₅×#1885	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#124	2
C₅×#5798	2
C₅×#5225	2
C₂×C₅×#833	2
C₅×#4523	2
C₅×#1885	3
#71723	5

Subgroups — C5×#71723 #14301441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71723 #14301441