Subgroups — C₅×#72006 #14313005

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4561	C₂	1	4	yes	maximal, complemented
5	C₅×#2117	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
7	C₃×C₅×#131	—	2	1	no	—
8	C₃×C₅×#131	—	2	1	no	—
9	C₃×C₅×#131	C₄	1	2	yes	—
10	C₃×C₅×#131	C₂²	1	2	yes	—
11	C₂×C₃×C₅×M₁₆	—	2	1	no	—
12	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
13	C₂²×C₃×C₅×C₈	—	2	1	no	—
14	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×#812	—	2	1	no	—
16	C₅×#812	—	2	1	no	—
17	C₅×#812	—	2	1	no	—
18	C₅×#812	C₂²	1	2	yes	complemented
19	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	complemented
21	#72006	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#131	D₃	1	1	yes	characteristic
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂×C₅×M₃₂	—	3	1	no	—
25	C₅×#350	—	3	4	no	—
26	C₃×C₅×#41	—	2	1	no	—
27	C₃×C₅×#41	—	2	1	no	—
28	C₃×C₅×#41	C₈	1	2	yes	—
29	C₃×C₅×#41	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×Q₈	C₈	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	—	2	1	no	—
32	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₅×M₁₆	—	2	1	no	—
36	C₃×C₅×M₁₆	—	2	2	no	—
37	C₃×C₅×M₁₆	—	2	2	no	—
38	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₂×C₃×C₅×C₈	—	2	2	no	—
41	C₂×C₃×C₅×C₈	—	2	2	no	—
42	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×C₃⋊C₁₆	—	2	2	no	—
49	C₅×C₃⋊C₁₆	—	2	2	no	—
50	C₂×C₃×#131	C₂×C₅	1	1	yes	characteristic
51	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
53	#4561	C₂×C₅	1	4	yes	complemented
54	C₂×C₅×#41	Dic₃	1	1	yes	characteristic
55	C₅×#131	—	2	1	no	—
56	C₅×#131	—	2	1	no	—
57	C₅×#131	Dic₃	1	2	yes	—
58	C₅×#131	C₂×D₃	1	2	yes	—
59	C₂×C₅×M₁₆	—	2	1	no	—
60	C₂×C₅×M₁₆	Dic₃	1	1	yes	characteristic
61	C₂²×C₅×C₈	—	2	1	no	—
62	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₅×M₃₂	—	3	2	no	—
64	C₅×M₃₂	—	6	1	no	—
65	C₅×M₃₂	—	6	1	no	—
66	C₅×M₃₂	—	6	1	no	—
67	C₂×C₅×C₁₆	—	3	2	no	—
68	C₂×C₅×C₁₆	—	6	1	no	—
69	#2117	—	3	1	no	Sylow
70	C₂³×C₃×C₅	—	2	1	no	—
71	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	1	no	—
73	C₃×C₅×Q₈	C₂×C₈	1	2	yes	—
74	C₃×C₅×D₄	—	2	2	no	—
75	C₃×C₅×D₄	—	2	2	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	2	yes	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₃×C₅×C₈	—	2	2	no	—
88	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×#41	C₄×C₅	1	1	yes	characteristic
93	C₃×#131	—	2	1	no	—
94	C₃×#131	—	2	1	no	—
95	C₃×#131	C₄×C₅	1	2	yes	—
96	C₃×#131	C₂²×C₅	1	2	yes	—
97	C₂×C₃×M₁₆	—	2	1	no	—
98	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
99	C₂²×C₃×C₈	—	2	1	no	—
100	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
101	#812	—	2	1	no	—
102	#812	—	2	1	no	—
103	#812	—	2	1	no	—
104	#812	C₂²×C₅	1	2	yes	complemented
105	C₂×C₃⋊C₁₆	—	2	1	no	—
106	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	complemented
107	C₅×#41	—	2	1	no	—
108	C₅×#41	—	2	1	no	—
109	C₅×#41	C₃⋊C₈	1	2	yes	—
110	C₅×#41	C₂×Dic₃	1	2	yes	—
111	C₂×C₅×Q₈	C₃⋊C₈	1	1	yes	characteristic
112	C₂×C₅×D₄	—	2	1	no	—
113	C₂×C₅×D₄	C₃⋊C₈	1	1	yes	characteristic
114	C₂²×C₄×C₅	—	2	1	no	—
115	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
116	C₅×M₁₆	—	2	1	no	—
117	C₅×M₁₆	—	2	2	no	—
118	C₅×M₁₆	—	2	2	no	—
119	C₅×M₁₆	C₂×Dic₃	1	2	yes	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
124	C₂×C₅×C₈	#67	1	1	yes	characteristic
125	C₂×C₅×C₈	#67	1	1	yes	characteristic
126	C₂×C₅×C₈	#67	1	1	yes	characteristic
127	C₂×C₅×C₈	#67	1	1	yes	characteristic
128	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
129	C₅×C₁₆	—	6	2	no	—
130	C₅×C₁₆	—	6	2	no	—
131	C₂×#131	C₅×D₃	1	1	yes	characteristic
132	C₂×M₃₂	—	3	1	no	—
133	C₂×M₃₂	—	3	1	no	—
134	#350	—	3	4	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	4	1	no	—
137	C₂²×C₃×C₅	—	4	2	no	—
138	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₃×C₄×C₅	—	2	2	no	—
142	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
146	C₃×#41	—	2	1	no	—
147	C₃×#41	—	2	1	no	—
148	C₃×#41	C₅×C₈	1	2	yes	—
149	C₃×#41	C₂×C₄×C₅	1	2	yes	—
150	C₂×C₃×Q₈	C₅×C₈	1	1	yes	characteristic
151	C₂×C₃×D₄	—	2	1	no	—
152	C₂×C₃×D₄	C₅×C₈	1	1	yes	characteristic
153	C₂²×C₃×C₄	—	2	1	no	—
154	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
155	C₃×M₁₆	—	2	1	no	—
156	C₃×M₁₆	—	2	2	no	—
157	C₃×M₁₆	—	2	2	no	—
158	C₃×M₁₆	C₂×C₄×C₅	1	2	yes	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₂×C₃×C₈	—	2	2	no	—
161	C₂×C₃×C₈	—	2	2	no	—
162	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
163	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
164	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
165	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
166	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
167	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
168	C₃⋊C₁₆	—	2	2	no	—
169	C₃⋊C₁₆	—	2	2	no	—
170	C₂³×C₅	—	2	1	no	—
171	C₂³×C₅	#208	1	1	yes	characteristic
172	C₅×Q₈	—	2	1	no	—
173	C₅×Q₈	C₂×C₃⋊C₈	1	2	yes	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	2	2	no	—
176	C₅×D₄	—	2	1	no	—
177	C₅×D₄	C₂×C₃⋊C₈	1	2	yes	—
178	C₂×C₄×C₅	—	2	1	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	2	2	no	—
181	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
183	C₂×C₄×C₅	#208	1	1	yes	characteristic
184	C₂×C₄×C₅	#217	1	1	yes	characteristic
185	C₂×C₄×C₅	#217	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
187	C₅×C₈	—	2	2	no	—
188	C₅×C₈	#217	1	1	yes	characteristic
189	C₅×C₈	#217	1	1	yes	characteristic
190	C₅×C₈	C₂×#67	1	1	yes	characteristic
191	C₅×C₈	C₂×#67	1	1	yes	characteristic
192	C₂×#41	C₅×Dic₃	1	1	yes	characteristic
193	#131	—	2	1	no	—
194	#131	—	2	1	no	—
195	#131	C₅×Dic₃	1	2	yes	—
196	#131	C₂×C₅×D₃	1	2	yes	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	C₅×Dic₃	1	1	yes	characteristic
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
201	M₃₂	—	3	2	no	—
202	M₃₂	—	6	1	no	—
203	M₃₂	—	6	1	no	—
204	M₃₂	—	6	1	no	—
205	C₂×C₁₆	—	3	2	no	—
206	C₂×C₁₆	—	6	1	no	—
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	—	2	1	no	—
210	C₂×C₃×C₅	—	4	2	no	—
211	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
212	C₂³×C₃	—	2	1	no	—
213	C₂³×C₃	C₅×M₁₆	1	1	yes	characteristic
214	C₃×Q₈	—	2	1	no	—
215	C₃×Q₈	C₂×C₅×C₈	1	2	yes	—
216	C₃×D₄	—	2	2	no	—
217	C₃×D₄	—	2	2	no	—
218	C₃×D₄	—	2	1	no	—
219	C₃×D₄	C₂×C₅×C₈	1	2	yes	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	2	no	—
223	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
227	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
229	C₃×C₈	—	2	2	no	—
230	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
233	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₅	—	2	2	no	—
235	C₂²×C₅	—	4	1	no	—
236	C₂²×C₅	—	4	2	no	—
237	C₂²×C₅	#830	1	1	yes	characteristic
238	C₂²×C₅	#830	1	1	yes	characteristic
239	C₂²×C₅	C₂×#208	1	1	yes	characteristic
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	#830	1	1	yes	characteristic
242	C₄×C₅	#830	1	1	yes	characteristic
243	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
244	C₄×C₅	C₂×#217	1	1	yes	characteristic
245	#41	—	2	1	no	—
246	#41	—	2	1	no	—
247	#41	C₅×C₃⋊C₈	1	2	yes	—
248	#41	C₂×C₅×Dic₃	1	2	yes	—
249	C₂×Q₈	C₅×C₃⋊C₈	1	1	yes	characteristic
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	C₅×C₃⋊C₈	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	2	2	no	—
257	M₁₆	C₂×C₅×Dic₃	1	2	yes	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
262	C₂×C₈	C₅×#67	1	1	yes	characteristic
263	C₂×C₈	C₅×#67	1	1	yes	characteristic
264	C₂×C₈	C₅×#67	1	1	yes	characteristic
265	C₂×C₈	C₅×#67	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₁₆	—	6	2	no	—
268	C₁₆	—	6	2	no	—
269	C₃×C₅	#2117	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	2	2	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	2	no	—
273	C₂²×C₃	C₅×C₂²⋊C₈	1	1	yes	characteristic
274	C₂²×C₃	C₅×C₂²⋊C₈	1	1	yes	characteristic
275	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
276	C₃×C₄	—	2	2	no	—
277	C₃×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
278	C₃×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
279	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	2	1	no	—
284	C₂×C₅	—	4	2	no	—
285	C₂×C₅	C₂×#830	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	C₅×#208	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	C₂×C₅×C₃⋊C₈	1	2	yes	—
290	D₄	—	2	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	2	1	no	—
293	D₄	C₂×C₅×C₃⋊C₈	1	2	yes	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	C₂×C₅×C₃⋊C₈	1	1	yes	characteristic
298	C₂×C₄	C₂×C₅×C₃⋊C₈	1	1	yes	characteristic
299	C₂×C₄	C₅×#208	1	1	yes	characteristic
300	C₂×C₄	C₅×#217	1	1	yes	characteristic
301	C₂×C₄	C₅×#217	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	C₅×#217	1	1	yes	characteristic
305	C₈	C₅×#217	1	1	yes	characteristic
306	C₈	C₂×C₅×#67	1	1	yes	characteristic
307	C₈	C₂×C₅×#67	1	1	yes	characteristic
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	4	2	no	—
312	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
313	C₅	#72006	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	2	2	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	C₅×#830	1	1	yes	characteristic
318	C₂²	C₅×#830	1	1	yes	characteristic
319	C₂²	C₂×C₅×#208	1	1	yes	characteristic
320	C₄	—	2	2	no	—
321	C₄	C₅×#830	1	1	yes	characteristic
322	C₄	C₅×#830	1	1	yes	characteristic
323	C₄	C₂²×C₅×C₃⋊C₈	1	1	yes	characteristic
324	C₄	C₂×C₅×#217	1	1	yes	characteristic
325	C₃	C₅×#2117	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	4	2	no	—
330	C₂	C₂×C₅×#830	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂×#67
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₈
prime radical	C₂×C₃×C₅×#41	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×#131	D₃
hypocenter	C₃	C₅×#2117
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×#131

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#812	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. C₅×#4561	C₂	semidirect
18. C₅×#812	C₂²	semidirect
20. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
21. #72006	C₅	direct (N × H)
51. C₂×#812	C₂×C₅	semidirect
52. C₂×#812	C₂×C₅	semidirect
53. #4561	C₂×C₅	semidirect
104. #812	C₂²×C₅	semidirect
106. C₂×C₃⋊C₁₆	C₂²×C₅	semidirect
269. C₃×C₅	#2117	semidirect
313. C₅	#72006	direct (N × H)
325. C₃	C₅×#2117	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×#131	2
C₂×C₅×#812	2
C₅×#4561	2
C₅×#2117	3
#72006	5

Subgroups — C5×#72006 #14313005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72006 #14313005