Subgroups — C₅×#70063 #14297553

Order 1920.

672 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×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#100	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#833	C₂	1	1	yes	maximal, characteristic
4	C₅×#5118	C₂	1	2	yes	maximal, complemented
5	C₅×#4554	C₂	1	2	yes	maximal, complemented
6	C₅×#1800	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂³×C₅×Dic₃	C₄	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₅×#217	C₄	1	2	yes	—
11	C₅×D₄×Dic₃	—	2	2	no	—
12	C₅×D₄×Dic₃	—	2	2	no	—
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₅×#208	C₂²	1	2	yes	—
16	C₃×C₅×#100	—	2	2	no	—
17	C₅×#833	—	2	2	no	—
18	C₅×Dic₃⋊C₈	—	2	2	no	—
19	C₅×Dic₃⋊C₈	—	2	2	no	—
20	#70063	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₄×C₅×D₄	—	3	1	no	—
22	C₅×#423	—	3	2	no	—
23	C₂×C₅×#100	—	3	1	no	—
24	C₂×C₅×#100	D₃	1	1	yes	characteristic
25	C₅×#343	—	3	2	no	—
26	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	2	no	—
29	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
30	C₂×C₃×C₅×D₄	Q₈	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	—	2	2	no	—
33	C₂²×C₅×Dic₃	—	2	2	no	—
34	C₂²×C₅×Dic₃	—	4	4	no	—
35	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
37	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₃×C₅×M₁₆	—	2	2	no	—
40	C₂×C₃×C₅×C₈	—	2	2	no	—
41	C₅×#217	—	2	4	no	—
42	C₅×C₄⋊Dic₃	—	2	2	no	—
43	C₄×C₅×Dic₃	D₄	1	2	yes	—
44	C₄×C₅×Dic₃	D₄	1	2	yes	—
45	C₅×#208	—	2	2	no	—
46	C₅×#208	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×D₄×Dic₃	C₂×C₅	1	1	yes	characteristic
49	C₂×C₃×#100	C₂×C₅	1	1	yes	characteristic
50	C₂×#833	C₂×C₅	1	1	yes	characteristic
51	#5118	C₂×C₅	1	2	yes	complemented
52	#4554	C₂×C₅	1	2	yes	complemented
53	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
54	C₂³×C₄×C₅	—	3	2	no	—
55	C₂×C₅×M₁₆	—	3	2	no	—
56	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
57	C₄×C₅×D₄	—	6	2	no	—
58	C₄×C₅×D₄	—	6	2	no	—
59	C₂×C₅×C₄⋊C₄	—	3	1	no	—
60	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
61	C₂×C₄²×C₅	—	3	1	no	—
62	C₅×C₄⋊C₈	—	6	2	no	—
63	C₅×C₄⋊C₈	—	6	2	no	—
64	C₅×#100	—	2	2	no	—
65	C₅×#100	—	6	2	no	—
66	#1800	—	3	1	no	Sylow
67	C₂³×C₃×C₅	—	2	2	no	—
68	C₂³×C₃×C₅	—	2	2	no	—
69	C₂³×C₃×C₅	—	4	2	no	—
70	C₂³×C₃×C₅	—	4	2	no	—
71	C₂³×C₃×C₅	C₄⋊C₄	1	4	yes	—
72	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	2	4	no	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	#41	1	2	yes	—
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₂×C₅×Dic₃	—	2	2	no	—
81	C₂×C₅×Dic₃	—	4	1	no	—
82	C₂×C₅×Dic₃	—	4	4	no	—
83	C₂×C₅×Dic₃	—	4	4	no	—
84	C₂×C₅×Dic₃	—	4	2	no	—
85	C₂×C₅×Dic₃	C₂²⋊C₄	1	2	yes	—
86	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₅×C₈	—	4	2	no	—
89	C₅×C₃⋊C₈	—	4	2	no	—
90	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
91	C₂³×Dic₃	C₄×C₅	1	2	yes	—
92	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
93	C₂×#217	C₄×C₅	1	2	yes	—
94	D₄×Dic₃	—	2	2	no	—
95	D₄×Dic₃	—	2	2	no	—
96	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
98	C₂×#208	C₂²×C₅	1	2	yes	—
99	C₃×#100	—	2	2	no	—
100	#833	—	2	2	no	—
101	Dic₃⋊C₈	—	2	2	no	—
102	Dic₃⋊C₈	—	2	2	no	—
103	C₂⁴×C₅	C₄×D₃	1	2	yes	—
104	C₂×C₅×D₄	—	2	2	no	—
105	C₂×C₅×D₄	—	2	2	no	—
106	C₂×C₅×D₄	Dic₆	1	2	yes	—
107	C₂×C₅×D₄	#67	1	2	yes	—
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂²×C₄×C₅	—	3	1	no	—
110	C₂²×C₄×C₅	—	3	2	no	—
111	C₂²×C₄×C₅	—	6	2	no	—
112	C₂²×C₄×C₅	—	6	2	no	—
113	C₂²×C₄×C₅	—	12	4	no	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₅×M₁₆	—	2	2	no	—
116	C₅×M₁₆	—	2	2	no	—
117	C₅×M₁₆	—	6	2	no	—
118	C₅×M₁₆	—	6	2	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₂×C₅×C₈	—	6	2	no	—
121	C₅×C₄⋊C₄	—	6	2	no	—
122	C₅×C₂²⋊C₄	—	6	4	no	—
123	C₄²×C₅	—	3	2	no	—
124	C₄²×C₅	—	3	2	no	—
125	C₂×C₄×D₄	—	3	1	no	—
126	#423	—	3	2	no	—
127	C₂×#100	—	3	1	no	—
128	C₂×#100	C₅×D₃	1	1	yes	characteristic
129	#343	—	3	2	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	4	no	—
133	C₂²×C₃×C₅	—	4	4	no	—
134	C₂²×C₃×C₅	—	4	4	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	—	4	2	no	—
137	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	#117	1	1	yes	characteristic
140	C₃×C₄×C₅	#121	1	2	yes	—
141	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
142	C₅×Dic₃	—	2	1	no	—
143	C₅×Dic₃	—	2	2	no	—
144	C₅×Dic₃	—	4	2	no	—
145	C₅×Dic₃	#100	1	2	yes	complemented
146	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	—	2	2	no	—
149	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
150	C₂×C₃×D₄	C₅×Q₈	1	2	yes	—
151	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂²×Dic₃	—	2	2	no	—
153	C₂²×Dic₃	—	2	2	no	—
154	C₂²×Dic₃	—	4	4	no	—
155	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
157	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
158	C₃×M₁₆	—	2	2	no	—
159	C₃×M₁₆	—	2	2	no	—
160	C₂×C₃×C₈	—	2	2	no	—
161	#217	—	2	4	no	—
162	C₄⋊Dic₃	—	2	2	no	—
163	C₄×Dic₃	C₅×D₄	1	2	yes	—
164	C₄×Dic₃	C₅×D₄	1	2	yes	—
165	#208	—	2	2	no	—
166	#208	—	2	2	no	—
167	C₂×C₃⋊C₈	—	2	2	no	—
168	C₂³×C₅	—	2	2	no	—
169	C₂³×C₅	—	2	2	no	—
170	C₂³×C₅	—	4	2	no	—
171	C₂³×C₅	—	4	2	no	—
172	C₂³×C₅	Dic₃⋊C₄	1	4	yes	—
173	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₅×D₄	—	2	4	no	—
175	C₅×D₄	—	2	4	no	—
176	C₂×C₄×C₅	—	3	1	no	—
177	C₂×C₄×C₅	—	3	2	no	—
178	C₂×C₄×C₅	—	3	1	no	—
179	C₂×C₄×C₅	—	6	2	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	—	12	4	no	—
182	C₂×C₄×C₅	—	12	1	no	—
183	C₂×C₄×C₅	—	12	4	no	—
184	C₂×C₄×C₅	—	12	2	no	—
185	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
186	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
187	C₂×C₄×C₅	#237	1	2	yes	—
188	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
189	C₅×C₈	—	4	2	no	—
190	C₅×C₈	—	12	2	no	—
191	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂³×C₄	—	3	2	no	—
193	C₂×M₁₆	—	3	2	no	—
194	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
195	C₄×D₄	—	6	2	no	—
196	C₄×D₄	—	6	2	no	—
197	C₂×C₄⋊C₄	—	3	1	no	—
198	C₂×C₂²⋊C₄	—	3	2	no	—
199	C₂×C₄²	—	3	1	no	—
200	C₄⋊C₈	—	6	2	no	—
201	C₄⋊C₈	—	6	2	no	—
202	#100	—	2	2	no	—
203	#100	—	6	2	no	—
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	—	2	1	no	—
206	C₂×C₃×C₅	—	4	4	no	—
207	C₂×C₃×C₅	#380	1	1	yes	characteristic
208	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
209	C₂×C₃×C₅	#413	1	1	yes	characteristic
210	C₂³×C₃	—	2	2	no	—
211	C₂³×C₃	—	2	2	no	—
212	C₂³×C₃	—	4	2	no	—
213	C₂³×C₃	—	4	2	no	—
214	C₂³×C₃	C₅×C₄⋊C₄	1	4	yes	—
215	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₃×D₄	—	2	4	no	—
217	C₃×D₄	—	2	4	no	—
218	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₅×#41	1	2	yes	—
222	C₂×Dic₃	—	2	2	no	—
223	C₂×Dic₃	—	2	2	no	—
224	C₂×Dic₃	—	4	1	no	—
225	C₂×Dic₃	—	4	4	no	—
226	C₂×Dic₃	—	4	4	no	—
227	C₂×Dic₃	—	4	2	no	—
228	C₂×Dic₃	C₅×C₂²⋊C₄	1	2	yes	—
229	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₃×C₈	—	4	2	no	—
232	C₃⋊C₈	—	4	2	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	2	1	no	—
235	C₂²×C₅	—	2	4	no	—
236	C₂²×C₅	—	4	4	no	—
237	C₂²×C₅	—	4	4	no	—
238	C₂²×C₅	—	4	2	no	—
239	C₂²×C₅	—	4	2	no	—
240	C₂²×C₅	#877	1	1	yes	characteristic
241	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
243	C₄×C₅	—	3	2	no	—
244	C₄×C₅	—	6	1	no	—
245	C₄×C₅	—	6	2	no	—
246	C₄×C₅	—	12	2	no	—
247	C₄×C₅	#878	1	2	yes	—
248	C₄×C₅	#939	1	2	yes	—
249	C₂⁴	C₄×C₅×D₃	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	C₅×Dic₆	1	2	yes	—
253	C₂×D₄	C₅×#67	1	2	yes	—
254	C₂²×C₄	—	3	1	no	—
255	C₂²×C₄	—	3	1	no	—
256	C₂²×C₄	—	3	2	no	—
257	C₂²×C₄	—	6	2	no	—
258	C₂²×C₄	—	6	2	no	—
259	C₂²×C₄	—	12	4	no	—
260	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	6	2	no	—
264	M₁₆	—	6	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	6	2	no	—
267	C₄⋊C₄	—	6	2	no	—
268	C₂²⋊C₄	—	6	4	no	—
269	C₄²	—	3	2	no	—
270	C₄²	—	3	2	no	—
271	C₃×C₅	#1800	1	1	yes	characteristic, complemented, Hall
272	C₂²×C₃	—	2	1	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	4	no	—
275	C₂²×C₃	—	4	4	no	—
276	C₂²×C₃	—	4	4	no	—
277	C₂²×C₃	—	4	2	no	—
278	C₂²×C₃	—	4	2	no	—
279	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
281	C₂²×C₃	C₅×#117	1	1	yes	characteristic
282	C₃×C₄	C₅×#121	1	2	yes	—
283	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
284	Dic₃	—	2	1	no	—
285	Dic₃	—	2	2	no	—
286	Dic₃	—	4	2	no	—
287	Dic₃	C₅×#100	1	2	yes	complemented
288	C₂×C₅	—	2	1	no	—
289	C₂×C₅	—	2	1	no	—
290	C₂×C₅	—	4	4	no	—
291	C₂×C₅	D₃×#100	1	1	yes	characteristic
292	C₂×C₅	#4745	1	1	yes	characteristic
293	C₂×C₅	#4942	1	1	yes	characteristic
294	C₂³	—	2	2	no	—
295	C₂³	—	2	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	C₅×Dic₃⋊C₄	1	4	yes	—
299	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
300	D₄	—	2	4	no	—
301	D₄	—	2	4	no	—
302	C₂×C₄	—	3	1	no	—
303	C₂×C₄	—	3	2	no	—
304	C₂×C₄	—	3	1	no	—
305	C₂×C₄	—	6	2	no	—
306	C₂×C₄	—	6	2	no	—
307	C₂×C₄	—	12	4	no	—
308	C₂×C₄	—	12	1	no	—
309	C₂×C₄	—	12	4	no	—
310	C₂×C₄	—	12	2	no	—
311	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
312	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
313	C₂×C₄	C₅×#237	1	2	yes	—
314	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
315	C₈	—	4	2	no	—
316	C₈	—	12	2	no	—
317	C₂×C₃	—	2	1	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	4	4	no	—
320	C₂×C₃	C₅×#380	1	1	yes	characteristic
321	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
322	C₂×C₃	C₅×#413	1	1	yes	characteristic
323	C₅	#70063	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	4	4	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	2	no	—
331	C₂²	C₅×#877	1	1	yes	characteristic
332	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
333	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
334	C₄	—	3	2	no	—
335	C₄	—	6	1	no	—
336	C₄	—	6	2	no	—
337	C₄	—	12	2	no	—
338	C₄	C₅×#878	1	2	yes	—
339	C₄	C₅×#939	1	2	yes	—
340	C₃	C₅×#1800	1	1	yes	characteristic, complemented, Sylow
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	4	no	—
344	C₂	C₅×D₃×#100	1	1	yes	characteristic
345	C₂	C₅×#4745	1	1	yes	characteristic
346	C₂	C₅×#4942	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#100	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₃×C₅×D₄	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#100	D₃
hypocenter	C₃	C₅×#1800
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#100

Complements

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

N	H ≅ G/N	type
4. C₅×#5118	C₂	semidirect
5. C₅×#4554	C₂	semidirect
20. #70063	C₅	direct (N × H)
51. #5118	C₂×C₅	semidirect
52. #4554	C₂×C₅	semidirect
145. C₅×Dic₃	#100	semidirect
271. C₃×C₅	#1800	semidirect
287. Dic₃	C₅×#100	semidirect
323. C₅	#70063	direct (N × H)
340. C₃	C₅×#1800	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×D₄×Dic₃	2
C₂×C₃×C₅×#100	2
C₂×C₅×#833	2
C₅×#5118	2
C₅×#4554	2
C₅×#1800	3
#70063	5

Subgroups — C5×#70063 #14297553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70063 #14297553