Subgroups — C₅×#69821 #14305884

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#875	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#821	C₂	1	2	yes	maximal
4	C₂×C₅×#816	C₂	1	2	yes	maximal, complemented
5	C₅×#4935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1959	—	3	1	no	maximal, Hall
7	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
12	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₅×#875	—	2	2	no	—
15	C₅×#875	C₄	1	4	yes	complemented
16	C₃×C₅×#97	—	2	2	no	—
17	C₅×#821	—	2	4	no	—
18	C₅×#816	—	2	4	no	—
19	#69821	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#124	—	3	1	no	—
21	C₂×C₅×#103	—	3	2	no	—
22	C₂×C₅×#102	—	3	2	no	—
23	C₂×C₅×#97	D₃	1	1	yes	characteristic, complemented
24	C₅×#384	—	3	1	no	—
25	C₂³×C₅×D₃	—	2	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×Dic₃	—	2	1	no	—
29	C₂²×C₅×Dic₃	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	2	no	—
31	C₂×C₅×D₁₂	—	2	2	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	2	no	—
36	C₂×C₅×Dic₆	—	2	1	no	—
37	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₃×C₄²×C₅	D₄	1	2	yes	—
43	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
44	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	2	no	—
46	C₅×C₄⋊Dic₃	—	4	4	no	—
47	C₂×#875	C₂×C₅	1	1	yes	characteristic
48	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂×#821	C₂×C₅	1	2	yes	—
50	C₂×#816	C₂×C₅	1	2	yes	complemented
51	#4935	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂²×C₅×Q₈	—	3	1	no	—
53	C₂²×C₅×D₄	—	3	1	no	—
54	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
55	C₅×#124	—	3	4	no	—
56	C₅×#124	—	6	2	no	—
57	C₂×C₅×C₄⋊C₄	—	3	2	no	—
58	C₂×C₅×C₄⋊C₄	—	6	2	no	—
59	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₅×#103	—	6	4	no	—
62	C₅×#102	—	6	4	no	—
63	C₅×#97	—	2	2	no	—
64	#1959	—	3	1	no	Sylow
65	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₅×D₃	—	2	2	no	—
67	C₂²×C₅×D₃	—	4	1	no	—
68	C₂²×C₅×D₃	—	4	1	no	—
69	C₂²×C₅×D₃	—	4	2	no	—
70	C₂²×C₅×D₃	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₂×C₅×Dic₃	—	2	2	no	—
79	C₂×C₅×Dic₃	—	4	1	no	—
80	C₂×C₅×Dic₃	—	4	1	no	—
81	C₂×C₅×Dic₃	—	4	2	no	—
82	C₂×C₅×Dic₃	—	4	4	no	—
83	C₅×D₁₂	—	2	4	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×Dic₆	—	2	4	no	—
86	C₅×Dic₆	—	4	2	no	—
87	C₃×C₅×C₈	—	2	4	no	—
88	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
89	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
93	C₂×C₄⋊Dic₃	—	2	2	no	—
94	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
95	#875	—	2	2	no	—
96	#875	C₄×C₅	1	4	yes	complemented
97	C₃×#97	—	2	2	no	—
98	#821	—	2	4	no	—
99	#816	—	2	4	no	—
100	C₂⁴×C₅	—	6	1	no	—
101	C₂×C₅×Q₈	—	3	2	no	—
102	C₂×C₅×Q₈	—	6	1	no	—
103	C₂×C₅×Q₈	—	6	2	no	—
104	C₂×C₅×Q₈	—	6	2	no	—
105	C₂×C₅×D₄	—	3	2	no	—
106	C₂×C₅×D₄	—	6	2	no	—
107	C₂×C₅×D₄	—	6	2	no	—
108	C₂×C₅×D₄	—	6	1	no	—
109	C₂²×C₄×C₅	—	6	1	no	—
110	C₂²×C₄×C₅	—	6	2	no	—
111	C₂²×C₄×C₅	D₁₂	1	2	yes	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	D₁₂	1	4	yes	—
116	C₅×C₄⋊C₄	—	6	2	no	—
117	C₅×C₄⋊C₄	—	6	2	no	—
118	C₅×C₄⋊C₄	—	12	4	no	—
119	C₅×C₂²⋊C₄	—	6	8	no	—
120	C₄²×C₅	C₄×D₃	1	2	yes	—
121	C₄²×C₅	#67	1	2	yes	—
122	C₂×#124	—	3	1	no	—
123	C₂×#103	—	3	2	no	—
124	C₂×#102	—	3	2	no	—
125	C₂×#97	C₅×D₃	1	1	yes	characteristic, complemented
126	#384	—	3	1	no	—
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	#124	1	1	yes	characteristic
129	C₂²×C₃×C₅	#127	1	1	yes	characteristic
130	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
131	C₂²×C₃×C₅	#137	1	2	yes	—
132	C₂×C₅×D₃	—	4	2	no	—
133	C₂×C₅×D₃	—	4	4	no	—
134	C₂×C₅×D₃	—	8	2	no	—
135	C₂×C₅×D₃	—	8	1	no	—
136	C₂×C₅×D₃	—	8	1	no	—
137	C₃×C₄×C₅	—	2	4	no	—
138	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₃×C₄×C₅	#121	1	2	yes	—
140	C₅×Dic₃	—	4	2	no	—
141	C₅×Dic₃	—	8	2	no	—
142	C₂³×D₃	—	2	1	no	—
143	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₂²×Dic₃	—	2	1	no	—
146	C₂²×Dic₃	—	2	2	no	—
147	C₂×D₁₂	—	2	2	no	—
148	C₂×D₁₂	—	2	2	no	—
149	C₂×D₁₂	—	2	1	no	—
150	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
151	C₂×Dic₆	—	2	2	no	—
152	C₂×Dic₆	—	2	2	no	—
153	C₂×Dic₆	—	2	1	no	—
154	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₃×C₈	—	2	2	no	—
156	C₂×C₃×C₈	—	2	2	no	—
157	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
158	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
159	C₃×C₄²	C₅×D₄	1	2	yes	—
160	(C₂×D₃)⋊C₄	—	2	8	no	—
161	C₄⋊Dic₃	—	2	2	no	—
162	C₄⋊Dic₃	—	2	2	no	—
163	C₄⋊Dic₃	—	4	4	no	—
164	C₂³×C₅	—	6	2	no	—
165	C₂³×C₅	—	12	1	no	—
166	C₂³×C₅	—	12	1	no	—
167	C₂³×C₅	—	12	2	no	—
168	C₂³×C₅	—	12	2	no	—
169	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
170	C₅×Q₈	—	6	4	no	—
171	C₅×Q₈	—	12	2	no	—
172	C₅×D₄	—	6	4	no	—
173	C₅×D₄	—	12	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	6	2	no	—
177	C₂×C₄×C₅	—	12	1	no	—
178	C₂×C₄×C₅	—	12	1	no	—
179	C₂×C₄×C₅	—	12	2	no	—
180	C₂×C₄×C₅	—	12	4	no	—
181	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
184	C₂×C₄×C₅	#235	1	2	yes	—
185	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
186	C₅×C₈	—	2	4	no	—
187	C₂²×Q₈	—	3	1	no	—
188	C₂²×D₄	—	3	1	no	—
189	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
190	#124	—	3	4	no	—
191	#124	—	6	2	no	—
192	C₂×C₄⋊C₄	—	3	2	no	—
193	C₂×C₄⋊C₄	—	6	2	no	—
194	C₂×C₂²⋊C₄	—	6	2	no	—
195	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
196	#103	—	6	4	no	—
197	#102	—	6	4	no	—
198	#97	—	2	2	no	—
199	C₂×C₃×C₅	#390	1	1	yes	characteristic
200	C₂×C₃×C₅	#417	1	2	yes	—
201	C₂×C₃×C₅	#489	1	2	yes	—
202	C₂×C₃×C₅	#496	1	1	yes	characteristic
203	C₂×C₃×C₅	#497	1	1	yes	characteristic
204	C₅×D₃	—	8	2	no	—
205	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
206	C₂²×D₃	—	2	2	no	—
207	C₂²×D₃	—	4	1	no	—
208	C₂²×D₃	—	4	1	no	—
209	C₂²×D₃	—	4	2	no	—
210	C₂²×D₃	—	4	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
214	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×#41	1	2	yes	—
218	C₂×Dic₃	—	2	2	no	—
219	C₂×Dic₃	—	4	1	no	—
220	C₂×Dic₃	—	4	1	no	—
221	C₂×Dic₃	—	4	2	no	—
222	C₂×Dic₃	—	4	4	no	—
223	D₁₂	—	2	4	no	—
224	D₁₂	—	4	2	no	—
225	Dic₆	—	2	4	no	—
226	Dic₆	—	4	2	no	—
227	C₃×C₈	—	2	4	no	—
228	C₂²×C₅	—	12	4	no	—
229	C₂²×C₅	—	12	2	no	—
230	C₂²×C₅	—	24	1	no	—
231	C₂²×C₅	—	24	2	no	—
232	C₂²×C₅	—	24	1	no	—
233	C₂²×C₅	#874	1	1	yes	characteristic
234	C₂²×C₅	#875	1	1	yes	characteristic
235	C₂²×C₅	#908	1	2	yes	—
236	C₂²×C₅	#909	1	2	yes	—
237	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
238	C₄×C₅	—	2	4	no	—
239	C₄×C₅	—	12	2	no	—
240	C₄×C₅	—	24	2	no	—
241	C₄×C₅	C₄×D₁₂	1	2	yes	—
242	C₄×C₅	#930	1	2	yes	—
243	C₂⁴	—	6	1	no	—
244	C₂×Q₈	—	3	2	no	—
245	C₂×Q₈	—	6	1	no	—
246	C₂×Q₈	—	6	2	no	—
247	C₂×Q₈	—	6	2	no	—
248	C₂×D₄	—	3	2	no	—
249	C₂×D₄	—	6	2	no	—
250	C₂×D₄	—	6	2	no	—
251	C₂×D₄	—	6	1	no	—
252	C₂²×C₄	—	6	1	no	—
253	C₂²×C₄	—	6	2	no	—
254	C₂²×C₄	C₅×D₁₂	1	2	yes	—
255	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₅×D₁₂	1	4	yes	—
259	C₄⋊C₄	—	6	2	no	—
260	C₄⋊C₄	—	6	2	no	—
261	C₄⋊C₄	—	12	4	no	—
262	C₂²⋊C₄	—	6	8	no	—
263	C₄²	C₄×C₅×D₃	1	2	yes	—
264	C₄²	C₅×#67	1	2	yes	—
265	C₃×C₅	#1959	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
267	C₂²×C₃	C₅×#124	1	1	yes	characteristic
268	C₂²×C₃	C₅×#127	1	1	yes	characteristic
269	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
270	C₂²×C₃	C₅×#137	1	2	yes	—
271	C₂×D₃	—	4	2	no	—
272	C₂×D₃	—	4	4	no	—
273	C₂×D₃	—	8	2	no	—
274	C₂×D₃	—	8	1	no	—
275	C₂×D₃	—	8	1	no	—
276	C₃×C₄	—	2	4	no	—
277	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
278	C₃×C₄	C₅×#121	1	2	yes	—
279	Dic₃	—	4	2	no	—
280	Dic₃	—	8	2	no	—
281	C₂×C₅	—	24	2	no	—
282	C₂×C₅	#4712	1	1	yes	characteristic
283	C₂×C₅	#4714	1	2	yes	—
284	C₂×C₅	#4715	1	1	yes	characteristic
285	C₂×C₅	#4939	1	1	yes	characteristic
286	C₂×C₅	#5133	1	2	yes	—
287	C₂³	—	6	2	no	—
288	C₂³	—	12	1	no	—
289	C₂³	—	12	1	no	—
290	C₂³	—	12	2	no	—
291	C₂³	—	12	2	no	—
292	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
293	Q₈	—	6	4	no	—
294	Q₈	—	12	2	no	—
295	D₄	—	6	4	no	—
296	D₄	—	12	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	6	2	no	—
300	C₂×C₄	—	12	1	no	—
301	C₂×C₄	—	12	1	no	—
302	C₂×C₄	—	12	2	no	—
303	C₂×C₄	—	12	4	no	—
304	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
305	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
306	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
307	C₂×C₄	C₅×#235	1	2	yes	—
308	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
309	C₈	—	2	4	no	—
310	C₂×C₃	C₅×#390	1	1	yes	characteristic
311	C₂×C₃	C₅×#417	1	2	yes	—
312	C₂×C₃	C₅×#489	1	2	yes	—
313	C₂×C₃	C₅×#496	1	1	yes	characteristic
314	C₂×C₃	C₅×#497	1	1	yes	characteristic
315	D₃	—	8	2	no	—
316	C₅	#69821	1	1	yes	characteristic, direct product factor, Sylow
317	C₂²	—	12	2	no	—
318	C₂²	—	12	4	no	—
319	C₂²	—	24	1	no	—
320	C₂²	—	24	2	no	—
321	C₂²	—	24	1	no	—
322	C₂²	C₅×#874	1	1	yes	characteristic
323	C₂²	C₅×#875	1	1	yes	characteristic
324	C₂²	C₅×#908	1	2	yes	—
325	C₂²	C₅×#909	1	2	yes	—
326	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
327	C₄	—	2	4	no	—
328	C₄	—	12	2	no	—
329	C₄	—	24	2	no	—
330	C₄	C₄×C₅×D₁₂	1	2	yes	—
331	C₄	C₅×#930	1	2	yes	—
332	C₃	C₅×#1959	1	1	yes	characteristic, complemented, Sylow
333	C₂	—	24	2	no	—
334	C₂	C₅×#4712	1	1	yes	characteristic
335	C₂	C₅×#4714	1	2	yes	—
336	C₂	C₅×#4715	1	1	yes	characteristic
337	C₂	C₅×#4939	1	1	yes	characteristic
338	C₂	C₅×#5133	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₅×#1959
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#97	C₂	semidirect
4. C₂×C₅×#816	C₂	semidirect
5. C₅×#4935	C₂	semidirect
15. C₅×#875	C₄	semidirect
19. #69821	C₅	direct (N × H)
23. C₂×C₅×#97	D₃	semidirect
48. C₂×C₃×#97	C₂×C₅	semidirect
50. C₂×#816	C₂×C₅	semidirect
51. #4935	C₂×C₅	semidirect
96. #875	C₄×C₅	semidirect
125. C₂×#97	C₅×D₃	semidirect
265. C₃×C₅	#1959	semidirect
316. C₅	#69821	direct (N × H)
332. C₃	C₅×#1959	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#875	2
C₂×C₃×C₅×#97	2
C₂×C₅×#821	2
C₂×C₅×#816	2
C₅×#4935	2
C₅×#1959	3
#69821	5

Subgroups — C5×#69821 #14305884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69821 #14305884