Subgroups — C₅×#70546 #14300352

Order 1920.

520 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×#5745	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5110	C₂	1	1	yes	maximal, characteristic
4	C₅×#5102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5006	C₂	1	1	yes	maximal, characteristic
6	C₅×#4556	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1868	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	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	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	1	yes	characteristic
14	C₂×C₅×#208	C₂²	1	1	yes	characteristic
15	C₂×C₅×#208	C₂²	1	1	yes	characteristic
16	C₅×#907	—	2	2	no	—
17	C₅×D₃×M₁₆	—	2	2	no	—
18	C₅×#901	—	2	2	no	—
19	C₃×C₅×C₄⋊C₈	—	2	2	no	—
20	C₅×(C₂×D₃)⋊C₈	—	2	2	no	—
21	C₅×Dic₃⋊C₈	—	2	2	no	—
22	C₅×#806	C₄	1	4	yes	—
23	#70546	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₅×#568	—	3	1	no	—
25	C₅×#424	—	3	1	no	—
26	C₅×#424	D₃	1	1	yes	characteristic, complemented
27	C₅×#410	—	3	1	no	—
28	C₅×#408	—	3	1	no	—
29	C₅×#344	—	3	1	no	—
30	C₅×#344	—	3	1	no	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₅×#67	C₂×C₄	1	2	yes	—
33	C₅×#235	—	2	1	no	—
34	C₅×#235	—	2	1	no	—
35	C₅×#235	—	2	1	no	—
36	C₅×#235	—	2	1	no	—
37	C₂×C₅×D₁₂	Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
39	C₂×C₅×Dic₆	Q₈	1	1	yes	characteristic
40	C₃×C₅×M₁₆	—	2	1	no	—
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
46	C₃×C₅×C₄⋊C₄	—	2	1	no	—
47	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
48	C₃×C₄²×C₅	—	2	1	no	—
49	C₅×#217	C₂×C₄	1	2	yes	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₄×C₅×Dic₃	—	2	1	no	—
52	C₅×#208	—	2	1	no	—
53	C₅×#208	—	2	1	no	—
54	C₅×#208	—	2	1	no	—
55	C₅×#208	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	—	2	1	no	—
57	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
58	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
59	C₅×D₃×C₈	—	2	4	no	—
60	#5745	C₂×C₅	1	1	yes	characteristic
61	C₃×#424	C₂×C₅	1	1	yes	characteristic, complemented
62	#5110	C₂×C₅	1	1	yes	characteristic
63	#5102	C₂×C₅	1	1	yes	characteristic, complemented
64	#5006	C₂×C₅	1	1	yes	characteristic
65	#4556	C₂×C₅	1	1	yes	characteristic, complemented
66	#4533	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×C₅×#41	—	3	1	no	—
68	C₅×#131	—	6	2	no	—
69	C₅×#131	—	6	2	no	—
70	C₂×C₅×M₁₆	—	3	1	no	—
71	C₂×C₅×M₁₆	—	3	1	no	—
72	C₂×C₅×M₁₆	—	6	2	no	—
73	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
74	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
75	C₅×#117	—	3	1	no	—
76	C₅×#117	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	—	2	2	no	—
78	C₅×C₄⋊C₈	—	6	2	no	—
79	C₅×#99	—	3	4	no	—
80	C₅×C₂²⋊C₈	—	6	2	no	—
81	#1868	—	3	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₅×D₃	C₄⋊C₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
91	C₅×#67	—	2	2	no	—
92	C₅×#67	—	2	2	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
95	C₅×D₁₂	—	2	1	no	—
96	C₅×D₁₂	—	2	1	no	—
97	C₄×C₅×D₃	—	2	2	no	—
98	C₄×C₅×D₃	—	2	2	no	—
99	C₅×Dic₆	—	2	1	no	—
100	C₅×Dic₆	—	2	1	no	—
101	C₃×C₅×C₈	—	2	2	no	—
102	C₃×C₅×C₈	—	4	1	no	—
103	C₅×C₃⋊C₈	—	2	2	no	—
104	C₅×C₃⋊C₈	—	4	1	no	—
105	C₂×#235	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
108	C₃×#117	C₂²×C₅	1	1	yes	characteristic
109	#926	C₂²×C₅	1	1	yes	characteristic
110	C₂×#208	C₂²×C₅	1	1	yes	characteristic
111	C₂×#208	C₂²×C₅	1	1	yes	characteristic
112	#907	—	2	2	no	—
113	D₃×M₁₆	—	2	2	no	—
114	#901	—	2	2	no	—
115	C₃×C₄⋊C₈	—	2	2	no	—
116	(C₂×D₃)⋊C₈	—	2	2	no	—
117	Dic₃⋊C₈	—	2	2	no	—
118	#806	C₄×C₅	1	4	yes	—
119	C₅×#41	—	6	1	no	—
120	C₅×#41	—	6	1	no	—
121	C₅×#41	—	6	1	no	—
122	C₅×#41	—	6	1	no	—
123	C₂×C₅×Q₈	—	3	1	no	—
124	C₂×C₅×D₄	—	3	2	no	—
125	C₂×C₅×D₄	—	3	1	no	—
126	C₂²×C₄×C₅	—	3	2	no	—
127	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₅×M₁₆	—	2	1	no	—
129	C₅×M₁₆	—	2	1	no	—
130	C₅×M₁₆	—	6	4	no	—
131	C₅×M₁₆	—	6	1	no	—
132	C₅×M₁₆	—	6	1	no	—
133	C₅×M₁₆	—	6	1	no	—
134	C₅×M₁₆	—	6	1	no	—
135	C₂×C₅×C₈	—	2	1	no	—
136	C₂×C₅×C₈	—	2	1	no	—
137	C₂×C₅×C₈	—	2	1	no	—
138	C₂×C₅×C₈	—	3	2	no	—
139	C₂×C₅×C₈	—	6	4	no	—
140	C₂×C₅×C₈	—	6	1	no	—
141	C₂×C₅×C₈	#67	1	2	yes	—
142	C₅×C₄⋊C₄	—	2	1	no	—
143	C₅×C₄⋊C₄	—	6	1	no	—
144	C₅×C₂²⋊C₄	—	3	2	no	—
145	C₅×C₂²⋊C₄	C₄×D₃	1	2	yes	—
146	C₄²×C₅	—	2	1	no	—
147	C₄²×C₅	—	6	1	no	—
148	#568	—	3	1	no	—
149	#424	—	3	1	no	—
150	#424	C₅×D₃	1	1	yes	characteristic, complemented
151	#410	—	3	1	no	—
152	#408	—	3	1	no	—
153	#344	—	3	1	no	—
154	#344	—	3	1	no	—
155	C₂²×C₃×C₅	—	4	1	no	—
156	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂×C₅×D₃	—	2	2	no	—
160	C₂×C₅×D₃	—	4	2	no	—
161	C₃×C₄×C₅	—	4	2	no	—
162	C₃×C₄×C₅	#120	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
164	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
165	C₃×C₄×C₅	#123	1	1	yes	characteristic
166	C₅×Dic₃	—	2	2	no	—
167	C₅×Dic₃	—	4	2	no	—
168	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
169	C₂×#67	C₂×C₄×C₅	1	2	yes	—
170	#235	—	2	1	no	—
171	#235	—	2	1	no	—
172	#235	—	2	1	no	—
173	#235	—	2	1	no	—
174	C₂×D₁₂	C₅×Q₈	1	1	yes	characteristic
175	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
176	C₂×Dic₆	C₅×Q₈	1	1	yes	characteristic
177	C₃×M₁₆	—	2	1	no	—
178	C₃×M₁₆	—	2	1	no	—
179	C₂×C₃×C₈	—	2	1	no	—
180	C₂×C₃×C₈	—	2	1	no	—
181	C₂×C₃×C₈	—	2	1	no	—
182	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
183	C₃×C₄⋊C₄	—	2	1	no	—
184	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
185	C₃×C₄²	—	2	1	no	—
186	#217	C₂×C₄×C₅	1	2	yes	—
187	C₄⋊Dic₃	—	2	1	no	—
188	C₄×Dic₃	—	2	1	no	—
189	#208	—	2	1	no	—
190	#208	—	2	1	no	—
191	#208	—	2	1	no	—
192	#208	—	2	1	no	—
193	C₂×C₃⋊C₈	—	2	1	no	—
194	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
195	ΓL(1,𝔽₂₅)	—	2	4	no	—
196	D₃×C₈	—	2	4	no	—
197	C₂³×C₅	—	3	2	no	—
198	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₅×Q₈	—	6	1	no	—
200	C₅×Q₈	—	6	1	no	—
201	C₅×D₄	—	6	2	no	—
202	C₅×D₄	—	6	2	no	—
203	C₅×D₄	—	6	1	no	—
204	C₅×D₄	—	6	1	no	—
205	C₂×C₄×C₅	—	2	2	no	—
206	C₂×C₄×C₅	—	3	2	no	—
207	C₂×C₄×C₅	—	6	2	no	—
208	C₂×C₄×C₅	—	6	2	no	—
209	C₂×C₄×C₅	—	6	2	no	—
210	C₂×C₄×C₅	#235	1	1	yes	characteristic
211	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
212	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
213	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
214	C₂×C₄×C₅	#239	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
216	C₅×C₈	—	2	2	no	—
217	C₅×C₈	—	4	1	no	—
218	C₅×C₈	—	6	2	no	—
219	C₅×C₈	—	12	1	no	—
220	C₂×#41	—	3	1	no	—
221	#131	—	6	2	no	—
222	#131	—	6	2	no	—
223	C₂×M₁₆	—	3	1	no	—
224	C₂×M₁₆	—	3	1	no	—
225	C₂×M₁₆	—	6	2	no	—
226	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
227	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
228	#117	—	3	1	no	—
229	#117	C₂×C₅×D₃	1	1	yes	characteristic
230	C₄⋊C₈	—	2	2	no	—
231	C₄⋊C₈	—	6	2	no	—
232	#99	—	3	4	no	—
233	C₂²⋊C₈	—	6	2	no	—
234	C₂×C₃×C₅	—	2	1	no	—
235	C₂×C₃×C₅	—	2	1	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	#385	1	1	yes	characteristic
238	C₅×D₃	—	4	2	no	—
239	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₂²×D₃	C₅×C₄⋊C₄	1	2	yes	—
241	C₂×C₃×C₄	—	2	2	no	—
242	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
243	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
244	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
246	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
247	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
248	#67	—	2	2	no	—
249	#67	—	2	2	no	—
250	C₂×Dic₃	—	2	2	no	—
251	C₂×Dic₃	C₅×C₄⋊C₄	1	2	yes	—
252	D₁₂	—	2	1	no	—
253	D₁₂	—	2	1	no	—
254	C₄×D₃	—	2	2	no	—
255	C₄×D₃	—	2	2	no	—
256	Dic₆	—	2	1	no	—
257	Dic₆	—	2	1	no	—
258	C₃×C₈	—	2	2	no	—
259	C₃×C₈	—	4	1	no	—
260	C₃⋊C₈	—	2	2	no	—
261	C₃⋊C₈	—	4	1	no	—
262	C₂²×C₅	—	4	1	no	—
263	C₂²×C₅	—	6	2	no	—
264	C₂²×C₅	—	12	2	no	—
265	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
266	C₂²×C₅	#893	1	1	yes	characteristic
267	C₂²×C₅	C₄×#67	1	1	yes	characteristic
268	C₄×C₅	—	4	2	no	—
269	C₄×C₅	—	6	2	no	—
270	C₄×C₅	—	12	2	no	—
271	C₄×C₅	#894	1	1	yes	characteristic
272	C₄×C₅	#896	1	1	yes	characteristic
273	C₄×C₅	#937	1	1	yes	characteristic
274	C₄×C₅	#946	1	1	yes	characteristic
275	#41	—	6	1	no	—
276	#41	—	6	1	no	—
277	#41	—	6	1	no	—
278	#41	—	6	1	no	—
279	C₂×Q₈	—	3	1	no	—
280	C₂×D₄	—	3	2	no	—
281	C₂×D₄	—	3	1	no	—
282	C₂²×C₄	—	3	2	no	—
283	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
284	M₁₆	—	2	1	no	—
285	M₁₆	—	2	1	no	—
286	M₁₆	—	6	4	no	—
287	M₁₆	—	6	1	no	—
288	M₁₆	—	6	1	no	—
289	M₁₆	—	6	1	no	—
290	M₁₆	—	6	1	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	3	2	no	—
295	C₂×C₈	—	6	4	no	—
296	C₂×C₈	—	6	1	no	—
297	C₂×C₈	C₅×#67	1	2	yes	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	6	1	no	—
300	C₂²⋊C₄	—	3	2	no	—
301	C₂²⋊C₄	C₄×C₅×D₃	1	2	yes	—
302	C₄²	—	2	1	no	—
303	C₄²	—	6	1	no	—
304	C₃×C₅	#1868	1	1	yes	characteristic, complemented, Hall
305	C₂²×C₃	—	4	1	no	—
306	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
307	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
308	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
309	C₂×D₃	—	2	2	no	—
310	C₂×D₃	—	4	2	no	—
311	C₃×C₄	—	4	2	no	—
312	C₃×C₄	C₅×#120	1	1	yes	characteristic
313	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
314	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
315	C₃×C₄	C₅×#123	1	1	yes	characteristic
316	Dic₃	—	2	2	no	—
317	Dic₃	—	4	2	no	—
318	C₂×C₅	—	2	1	no	—
319	C₂×C₅	—	2	1	no	—
320	C₂×C₅	—	2	1	no	—
321	C₂×C₅	—	12	2	no	—
322	C₂×C₅	#4989	1	1	yes	characteristic
323	C₂³	—	3	2	no	—
324	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
325	Q₈	—	6	1	no	—
326	Q₈	—	6	1	no	—
327	D₄	—	6	2	no	—
328	D₄	—	6	2	no	—
329	D₄	—	6	1	no	—
330	D₄	—	6	1	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	3	2	no	—
333	C₂×C₄	—	6	2	no	—
334	C₂×C₄	—	6	2	no	—
335	C₂×C₄	—	6	2	no	—
336	C₂×C₄	C₅×#235	1	1	yes	characteristic
337	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
338	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
339	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
340	C₂×C₄	C₅×#239	1	1	yes	characteristic
341	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
342	C₈	—	2	2	no	—
343	C₈	—	4	1	no	—
344	C₈	—	6	2	no	—
345	C₈	—	12	1	no	—
346	C₂×C₃	—	2	1	no	—
347	C₂×C₃	—	2	1	no	—
348	C₂×C₃	—	2	1	no	—
349	C₂×C₃	C₅×#385	1	1	yes	characteristic
350	D₃	—	4	2	no	—
351	C₅	#70546	1	1	yes	characteristic, direct product factor, Sylow
352	C₂²	—	4	1	no	—
353	C₂²	—	6	2	no	—
354	C₂²	—	12	2	no	—
355	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
356	C₂²	C₅×#893	1	1	yes	characteristic
357	C₂²	C₄×C₅×#67	1	1	yes	characteristic
358	C₄	—	4	2	no	—
359	C₄	—	6	2	no	—
360	C₄	—	12	2	no	—
361	C₄	C₅×#894	1	1	yes	characteristic
362	C₄	C₅×#896	1	1	yes	characteristic
363	C₄	C₅×#937	1	1	yes	characteristic
364	C₄	C₅×#946	1	1	yes	characteristic
365	C₃	C₅×#1868	1	1	yes	characteristic, complemented, Sylow
366	C₂	—	2	1	no	—
367	C₂	—	2	1	no	—
368	C₂	—	2	1	no	—
369	C₂	—	12	2	no	—
370	C₂	C₅×#4989	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#937
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#424	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#424	D₃
hypocenter	C₃	C₅×#1868
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#424

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#424	C₂	semidirect
4. C₅×#5102	C₂	semidirect
6. C₅×#4556	C₂	semidirect
7. C₅×#4533	C₂	semidirect
23. #70546	C₅	direct (N × H)
26. C₅×#424	D₃	semidirect
61. C₃×#424	C₂×C₅	semidirect
63. #5102	C₂×C₅	semidirect
65. #4556	C₂×C₅	semidirect
66. #4533	C₂×C₅	semidirect
150. #424	C₅×D₃	semidirect
304. C₃×C₅	#1868	semidirect
351. C₅	#70546	direct (N × H)
365. C₃	C₅×#1868	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5745	2
C₃×C₅×#424	2
C₅×#5110	2
C₅×#5102	2
C₅×#5006	2
C₅×#4556	2
C₅×#4533	2
C₅×#1868	3
#70546	5

Subgroups — C5×#70546 #14300352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70546 #14300352