Subgroups — C₅×#69190 #14313930

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×D₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#798	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4880	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4562	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4491	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2140	—	3	1	no	maximal, Hall
9	C₂×C₅×D₃×D₄	—	2	1	no	—
10	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#213	—	2	1	no	—
12	C₅×D₃×D₈	—	2	2	no	—
13	C₅×D₃×D₈	C₄	1	4	yes	complemented
14	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
15	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
16	C₅×#892	—	2	1	no	—
17	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
18	C₃×C₅×#107	C₂²	1	1	yes	characteristic, complemented
19	C₅×#818	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
21	C₅×#808	—	2	1	no	—
22	C₅×#798	—	2	2	no	—
23	#69190	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₅×D₈	—	3	1	no	—
25	C₂×C₅×M₃₂	—	3	1	no	—
26	C₅×#427	—	3	1	no	—
27	C₅×#357	—	3	1	no	—
28	C₅×#357	—	3	1	no	—
29	C₅×#357	—	3	1	no	—
30	C₅×#357	D₃	1	1	yes	characteristic, complemented
31	C₂³×C₅×D₃	—	4	1	no	—
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₅×#67	—	4	1	no	—
34	C₅×D₃×D₄	—	2	4	no	—
35	C₅×D₃×D₄	—	4	2	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₃×C₅×D₈	—	2	1	no	—
39	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₅×#213	—	2	4	no	—
43	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₄×C₅×Dic₃	—	4	1	no	—
46	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₅×D₂₄	—	2	1	no	—
48	C₅×D₂₄	C₂×C₄	1	2	yes	—
49	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
50	C₅×D₃×C₈	D₄	1	2	yes	—
51	C₃×C₅×C₁₆	—	2	1	no	—
52	C₅×C₃⋊C₁₆	—	2	1	no	—
53	C₂×D₃×D₈	C₂×C₅	1	1	yes	characteristic
54	C₂×#798	C₂×C₅	1	1	yes	characteristic, complemented
55	#4880	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#357	C₂×C₅	1	1	yes	characteristic, complemented
57	#4562	C₂×C₅	1	1	yes	characteristic, complemented
58	#4509	C₂×C₅	1	1	yes	characteristic, complemented
59	#4491	C₂×C₅	1	1	yes	characteristic, complemented
60	C₂²×C₅×D₄	—	6	1	no	—
61	C₂×C₅×D₈	—	3	4	no	—
62	C₂×C₅×D₈	—	3	1	no	—
63	C₂×C₅×D₈	—	6	2	no	—
64	C₂×C₅×D₈	—	6	1	no	—
65	C₂×C₅×D₈	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	—	3	1	no	—
67	C₅×#117	—	6	1	no	—
68	C₅×M₃₂	—	6	2	no	—
69	C₂×C₅×C₁₆	—	3	1	no	—
70	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic, complemented
71	C₅×#107	—	3	1	no	—
72	C₅×#107	C₂×D₃	1	1	yes	characteristic, complemented
73	C₅×#106	—	6	1	no	—
74	#2140	—	3	1	no	Sylow
75	C₂³×C₃×C₅	—	4	1	no	—
76	C₂²×C₅×D₃	—	4	1	no	—
77	C₂²×C₅×D₃	—	4	4	no	—
78	C₂²×C₅×D₃	—	8	2	no	—
79	C₂²×C₅×D₃	—	8	2	no	—
80	C₂²×C₅×D₃	D₈	1	1	yes	characteristic
81	C₃×C₅×D₄	—	2	2	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₅×#67	—	4	4	no	—
86	C₂×C₅×Dic₃	—	4	1	no	—
87	C₂×C₅×Dic₃	D₈	1	1	yes	characteristic
88	C₅×D₁₂	—	2	2	no	—
89	C₅×D₁₂	—	4	1	no	—
90	C₄×C₅×D₃	C₂²⋊C₄	1	2	yes	—
91	C₄×C₅×D₃	SD₁₆	1	2	yes	—
92	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
94	C₅×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
95	C₂×D₃×D₄	—	2	1	no	—
96	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
97	C₂×#213	—	2	1	no	—
98	D₃×D₈	—	2	2	no	—
99	D₃×D₈	C₄×C₅	1	4	yes	complemented
100	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
101	C₂×D₃×C₈	C₂²×C₅	1	1	yes	characteristic
102	#892	—	2	1	no	—
103	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
104	C₃×#107	C₂²×C₅	1	1	yes	characteristic, complemented
105	#818	C₂²×C₅	1	1	yes	characteristic, complemented
106	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
107	#808	—	2	1	no	—
108	#798	—	2	2	no	—
109	C₂⁴×C₅	—	12	1	no	—
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	—	6	4	no	—
112	C₂×C₅×D₄	—	6	1	no	—
113	C₂×C₅×D₄	—	12	2	no	—
114	C₂×C₅×D₄	—	12	1	no	—
115	C₂²×C₄×C₅	—	3	1	no	—
116	C₅×D₈	—	2	1	no	—
117	C₅×D₈	—	3	2	no	—
118	C₅×D₈	—	6	4	no	—
119	C₅×D₈	—	6	1	no	—
120	C₅×D₈	C₄×D₃	1	2	yes	—
121	C₂×C₅×C₈	—	3	1	no	—
122	C₂×C₅×C₈	—	3	2	no	—
123	C₂×C₅×C₈	—	3	2	no	—
124	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	2	1	no	—
126	C₅×C₄⋊C₄	—	6	1	no	—
127	C₅×C₂²⋊C₄	—	12	1	no	—
128	C₄²×C₅	—	12	1	no	—
129	C₅×C₁₆	—	2	1	no	—
130	C₅×C₁₆	—	6	1	no	—
131	C₂²×D₈	—	3	1	no	—
132	C₂×M₃₂	—	3	1	no	—
133	#427	—	3	1	no	—
134	#357	—	3	1	no	—
135	#357	—	3	1	no	—
136	#357	—	3	1	no	—
137	#357	C₅×D₃	1	1	yes	characteristic, complemented
138	C₂²×C₃×C₅	—	4	2	no	—
139	C₂²×C₃×C₅	—	8	1	no	—
140	C₂²×C₃×C₅	—	8	1	no	—
141	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
142	C₂×C₅×D₃	—	2	1	no	—
143	C₂×C₅×D₃	—	2	1	no	—
144	C₂×C₅×D₃	—	4	2	no	—
145	C₂×C₅×D₃	—	8	8	no	—
146	C₂×C₅×D₃	—	8	1	no	—
147	C₂×C₅×D₃	—	8	1	no	—
148	C₂×C₅×D₃	#102	1	2	yes	—
149	C₃×C₄×C₅	—	8	1	no	—
150	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
152	C₅×Dic₃	—	8	1	no	—
153	C₅×Dic₃	#102	1	2	yes	—
154	C₂³×D₃	—	4	1	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×#67	—	4	1	no	—
157	D₃×D₄	—	2	4	no	—
158	D₃×D₄	—	4	2	no	—
159	C₂×D₁₂	—	2	1	no	—
160	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
161	C₃×D₈	—	2	1	no	—
162	C₃×D₈	C₂×C₄×C₅	1	2	yes	—
163	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	2	1	no	—
165	#213	—	2	4	no	—
166	(C₂×D₃)⋊C₄	—	4	1	no	—
167	C₄⋊Dic₃	—	2	1	no	—
168	C₄×Dic₃	—	4	1	no	—
169	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
170	D₂₄	—	2	1	no	—
171	D₂₄	C₂×C₄×C₅	1	2	yes	—
172	D₃×C₈	C₂×C₄×C₅	1	2	yes	—
173	D₃×C₈	C₅×D₄	1	2	yes	—
174	C₃×C₁₆	—	2	1	no	—
175	C₃⋊C₁₆	—	2	1	no	—
176	C₂³×C₅	—	3	1	no	—
177	C₂³×C₅	—	4	1	no	—
178	C₂³×C₅	—	12	4	no	—
179	C₂³×C₅	—	12	1	no	—
180	C₂³×C₅	—	24	2	no	—
181	C₂³×C₅	—	24	2	no	—
182	C₅×D₄	—	2	2	no	—
183	C₅×D₄	—	4	1	no	—
184	C₅×D₄	—	6	2	no	—
185	C₅×D₄	—	12	4	no	—
186	C₅×D₄	—	12	1	no	—
187	C₂×C₄×C₅	—	3	1	no	—
188	C₂×C₄×C₅	—	3	2	no	—
189	C₂×C₄×C₅	—	3	2	no	—
190	C₂×C₄×C₅	—	4	1	no	—
191	C₂×C₄×C₅	—	12	1	no	—
192	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
193	C₅×C₈	—	3	2	no	—
194	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
195	C₅×C₈	D₃×D₄	1	1	yes	characteristic
196	C₂²×D₄	—	6	1	no	—
197	C₂×D₈	—	3	4	no	—
198	C₂×D₈	—	3	1	no	—
199	C₂×D₈	—	6	2	no	—
200	C₂×D₈	—	6	1	no	—
201	C₂×D₈	C₂×C₅×D₃	1	1	yes	characteristic
202	C₂²×C₈	—	3	1	no	—
203	#117	—	6	1	no	—
204	M₃₂	—	6	2	no	—
205	C₂×C₁₆	—	3	1	no	—
206	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic, complemented
207	#107	—	3	1	no	—
208	#107	C₂×C₅×D₃	1	1	yes	characteristic, complemented
209	#106	—	6	1	no	—
210	C₂×C₃×C₅	—	8	2	no	—
211	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
212	C₂×C₃×C₅	#509	1	1	yes	characteristic
213	C₂×C₃×C₅	#509	1	1	yes	characteristic
214	C₅×D₃	—	2	2	no	—
215	C₅×D₃	—	8	2	no	—
216	C₂³×C₃	—	4	1	no	—
217	C₂²×D₃	—	4	1	no	—
218	C₂²×D₃	—	4	4	no	—
219	C₂²×D₃	—	8	2	no	—
220	C₂²×D₃	—	8	2	no	—
221	C₂²×D₃	C₅×D₈	1	1	yes	characteristic
222	C₃×D₄	—	2	2	no	—
223	C₃×D₄	—	4	1	no	—
224	C₂×C₃×C₄	—	4	1	no	—
225	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	#67	—	4	4	no	—
227	C₂×Dic₃	—	4	1	no	—
228	C₂×Dic₃	C₅×D₈	1	1	yes	characteristic
229	D₁₂	—	2	2	no	—
230	D₁₂	—	4	1	no	—
231	C₄×D₃	C₅×C₂²⋊C₄	1	2	yes	—
232	C₄×D₃	C₅×SD₁₆	1	2	yes	—
233	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
235	C₃⋊C₈	C₅×C₂²⋊C₄	1	2	yes	—
236	C₂²×C₅	—	3	2	no	—
237	C₂²×C₅	—	4	2	no	—
238	C₂²×C₅	—	6	1	no	—
239	C₂²×C₅	—	6	1	no	—
240	C₂²×C₅	—	8	1	no	—
241	C₂²×C₅	—	8	1	no	—
242	C₂²×C₅	—	12	2	no	—
243	C₂²×C₅	—	24	8	no	—
244	C₂²×C₅	—	24	1	no	—
245	C₂²×C₅	—	24	1	no	—
246	C₂²×C₅	D₃×D₈	1	1	yes	characteristic
247	C₄×C₅	—	3	2	no	—
248	C₄×C₅	—	8	1	no	—
249	C₄×C₅	—	24	1	no	—
250	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
251	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
252	C₂⁴	—	12	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	6	4	no	—
255	C₂×D₄	—	6	1	no	—
256	C₂×D₄	—	12	2	no	—
257	C₂×D₄	—	12	1	no	—
258	C₂²×C₄	—	3	1	no	—
259	D₈	—	2	1	no	—
260	D₈	—	3	2	no	—
261	D₈	—	6	4	no	—
262	D₈	—	6	1	no	—
263	D₈	C₄×C₅×D₃	1	2	yes	—
264	C₂×C₈	—	3	1	no	—
265	C₂×C₈	—	3	2	no	—
266	C₂×C₈	—	3	2	no	—
267	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	6	1	no	—
270	C₂²⋊C₄	—	12	1	no	—
271	C₄²	—	12	1	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	—	6	1	no	—
274	C₃×C₅	#2140	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	—	4	2	no	—
276	C₂²×C₃	—	8	1	no	—
277	C₂²×C₃	—	8	1	no	—
278	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
279	C₂×D₃	—	2	1	no	—
280	C₂×D₃	—	2	1	no	—
281	C₂×D₃	—	4	2	no	—
282	C₂×D₃	—	8	8	no	—
283	C₂×D₃	—	8	1	no	—
284	C₂×D₃	—	8	1	no	—
285	C₂×D₃	C₅×#102	1	2	yes	—
286	C₃×C₄	—	8	1	no	—
287	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
289	Dic₃	—	8	1	no	—
290	Dic₃	C₅×#102	1	2	yes	—
291	C₂×C₅	—	6	2	no	—
292	C₂×C₅	—	8	2	no	—
293	C₂×C₅	—	24	2	no	—
294	C₂×C₅	D₃×#102	1	1	yes	characteristic
295	C₂×C₅	#4911	1	1	yes	characteristic
296	C₂×C₅	#4914	1	1	yes	characteristic
297	C₂³	—	3	1	no	—
298	C₂³	—	4	1	no	—
299	C₂³	—	12	4	no	—
300	C₂³	—	12	1	no	—
301	C₂³	—	24	2	no	—
302	C₂³	—	24	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	1	no	—
305	D₄	—	6	2	no	—
306	D₄	—	12	4	no	—
307	D₄	—	12	1	no	—
308	C₂×C₄	—	3	1	no	—
309	C₂×C₄	—	3	2	no	—
310	C₂×C₄	—	3	2	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	12	1	no	—
313	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
314	C₈	—	3	2	no	—
315	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
316	C₈	C₅×D₃×D₄	1	1	yes	characteristic
317	C₂×C₃	—	8	2	no	—
318	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
319	C₂×C₃	C₅×#509	1	1	yes	characteristic
320	C₂×C₃	C₅×#509	1	1	yes	characteristic
321	D₃	—	2	2	no	—
322	D₃	—	8	2	no	—
323	C₅	#69190	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	3	2	no	—
325	C₂²	—	4	2	no	—
326	C₂²	—	6	1	no	—
327	C₂²	—	6	1	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	—	12	2	no	—
331	C₂²	—	24	8	no	—
332	C₂²	—	24	1	no	—
333	C₂²	—	24	1	no	—
334	C₂²	C₅×D₃×D₈	1	1	yes	characteristic
335	C₄	—	3	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	24	1	no	—
338	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
339	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
340	C₃	C₅×#2140	1	1	yes	characteristic, complemented, Sylow
341	C₂	—	6	2	no	—
342	C₂	—	8	2	no	—
343	C₂	—	24	2	no	—
344	C₂	C₅×D₃×#102	1	1	yes	characteristic
345	C₂	C₅×#4911	1	1	yes	characteristic
346	C₂	C₅×#4914	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×D₈
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#357	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×D₃×D₈	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#357	D₃
hypocenter	C₃	C₅×#2140
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₅×#357

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#798	C₂	semidirect
3. C₅×#4880	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₅×#4562	C₂	semidirect
6. C₅×#4509	C₂	semidirect
7. C₅×#4491	C₂	semidirect
13. C₅×D₃×D₈	C₄	semidirect
17. C₂×C₃×C₅×C₁₆	C₂²	semidirect
18. C₃×C₅×#107	C₂²	semidirect
19. C₅×#818	C₂²	semidirect
20. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
23. #69190	C₅	direct (N × H)
30. C₅×#357	D₃	semidirect
54. C₂×#798	C₂×C₅	semidirect
55. #4880	C₂×C₅	semidirect
56. C₃×#357	C₂×C₅	semidirect
57. #4562	C₂×C₅	semidirect
58. #4509	C₂×C₅	semidirect
59. #4491	C₂×C₅	semidirect
70. C₂×C₅×C₁₆	C₂×D₃	semidirect
72. C₅×#107	C₂×D₃	semidirect
99. D₃×D₈	C₄×C₅	semidirect
103. C₂×C₃×C₁₆	C₂²×C₅	semidirect
104. C₃×#107	C₂²×C₅	semidirect
105. #818	C₂²×C₅	semidirect
106. C₂×C₃⋊C₁₆	C₂²×C₅	semidirect
137. #357	C₅×D₃	semidirect
206. C₂×C₁₆	C₂×C₅×D₃	semidirect
208. #107	C₂×C₅×D₃	semidirect
274. C₃×C₅	#2140	semidirect
323. C₅	#69190	direct (N × H)
340. C₃	C₅×#2140	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₃×D₈	2
C₂×C₅×#798	2
C₅×#4880	2
C₃×C₅×#357	2
C₅×#4562	2
C₅×#4509	2
C₅×#4491	2
C₅×#2140	3
#69190	5

Subgroups — C5×#69190 #14313930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69190 #14313930