Subgroups — C₅×#70774 #14284768

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×#874	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4924	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4485	C₂	1	4	yes	maximal, complemented
5	C₅×#1545	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	1	no	—
7	C₂²×C₅×D₁₂	—	2	1	no	—
8	C₂²×C₅×D₁₂	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	—	2	1	no	—
11	C₅×#874	—	2	1	no	—
12	C₅×#874	—	2	1	no	—
13	C₅×#874	C₄	1	2	yes	—
14	C₅×#874	C₂²	1	2	yes	—
15	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	complemented
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	—	2	1	no	—
18	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
20	#70774	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×#127	—	3	1	no	—
22	C₅×#431	D₃	1	1	yes	characteristic, complemented
23	C₅×#423	—	3	1	no	—
24	C₅×#331	—	3	4	no	—
25	C₂³×C₅×D₃	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	2	no	—
31	C₂×C₅×D₁₂	—	2	2	no	—
32	C₂×C₅×D₁₂	—	2	2	no	—
33	C₂×C₅×D₁₂	—	2	2	no	—
34	C₂×C₅×D₁₂	—	4	4	no	—
35	C₂×C₅×D₁₂	—	4	2	no	—
36	C₂×C₅×D₁₂	—	4	2	no	—
37	C₂×C₅×D₁₂	—	4	4	no	—
38	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#208	—	4	2	no	—
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	C₂×#874	C₂×C₅	1	1	yes	characteristic
47	C₃×#431	C₂×C₅	1	1	yes	characteristic, complemented
48	#4924	C₂×C₅	1	1	yes	characteristic, complemented
49	#4485	C₂×C₅	1	4	yes	complemented
50	C₂²×C₅×D₄	—	3	2	no	—
51	C₂²×C₅×D₄	—	6	1	no	—
52	C₂²×C₅×D₄	—	6	1	no	—
53	C₂×C₅×M₁₆	—	6	1	no	—
54	C₅×#127	—	3	2	no	—
55	C₅×#127	—	3	2	no	—
56	C₅×#127	—	6	1	no	—
57	C₅×#127	—	6	1	no	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₅×C₄⋊C₈	—	3	2	no	—
60	C₅×C₄⋊C₈	—	6	1	no	—
61	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	complemented
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₄×C₅×C₈	—	2	1	no	—
64	#1545	—	3	1	no	Sylow
65	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₅×D₃	—	2	4	no	—
67	C₂²×C₅×D₃	—	4	4	no	—
68	C₂²×C₅×D₃	—	4	4	no	—
69	C₂²×C₅×D₃	—	4	4	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
76	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₅×D₁₂	—	4	4	no	—
81	C₅×D₁₂	—	4	4	no	—
82	C₅×D₁₂	—	4	4	no	—
83	C₅×D₁₂	—	4	4	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×D₁₂	—	4	2	no	—
86	C₅×D₁₂	—	4	4	no	—
87	C₃×C₅×C₈	—	4	2	no	—
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₂²×D₁₂	—	2	1	no	—
90	C₂²×D₁₂	—	2	1	no	—
91	C₂²×D₁₂	C₄×C₅	1	2	yes	—
92	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
93	C₂×#208	—	2	1	no	—
94	#874	—	2	1	no	—
95	#874	—	2	1	no	—
96	#874	C₄×C₅	1	2	yes	—
97	#874	C₂²×C₅	1	2	yes	—
98	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	complemented
99	C₃×C₂²⋊C₈	—	2	1	no	—
100	C₃×C₄×C₈	—	2	1	no	—
101	C₄⋊(C₃⋊C₈)	—	2	1	no	—
102	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	complemented
103	C₂⁴×C₅	—	6	2	no	—
104	C₂×C₅×D₄	—	3	4	no	—
105	C₂×C₅×D₄	—	6	2	no	—
106	C₂×C₅×D₄	—	6	2	no	—
107	C₂×C₅×D₄	—	6	2	no	—
108	C₂×C₅×D₄	—	6	2	no	—
109	C₂×C₅×D₄	—	6	2	no	—
110	C₂×C₅×D₄	—	12	4	no	—
111	C₂×C₅×D₄	—	12	4	no	—
112	C₂×C₅×D₄	—	12	2	no	—
113	C₂×C₅×D₄	—	12	2	no	—
114	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₅×M₁₆	—	12	2	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	—	6	2	no	—
120	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
121	C₄²×C₅	D₁₂	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
124	C₂×#127	—	3	1	no	—
125	#431	C₅×D₃	1	1	yes	characteristic, complemented
126	#423	—	3	1	no	—
127	#331	—	3	4	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	#102	1	2	yes	—
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₅×D₃	—	4	4	no	—
133	C₂×C₅×D₃	—	4	4	no	—
134	C₂×C₅×D₃	—	4	4	no	—
135	C₂×C₅×D₃	—	8	8	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	#100	1	2	yes	—
139	C₃×C₄×C₅	#102	1	2	yes	—
140	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
141	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
143	C₂³×D₃	—	2	2	no	—
144	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
147	C₂×D₁₂	—	2	2	no	—
148	C₂×D₁₂	—	2	2	no	—
149	C₂×D₁₂	—	2	2	no	—
150	C₂×D₁₂	—	2	2	no	—
151	C₂×D₁₂	—	2	2	no	—
152	C₂×D₁₂	—	4	4	no	—
153	C₂×D₁₂	—	4	2	no	—
154	C₂×D₁₂	—	4	2	no	—
155	C₂×D₁₂	—	4	4	no	—
156	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
157	C₂×C₃×C₈	—	2	2	no	—
158	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
159	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
160	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
161	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
162	#208	—	4	2	no	—
163	C₂×C₃⋊C₈	—	2	2	no	—
164	C₂³×C₅	—	6	4	no	—
165	C₂³×C₅	—	12	4	no	—
166	C₂³×C₅	—	12	4	no	—
167	C₂³×C₅	—	12	4	no	—
168	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₅×D₄	—	12	4	no	—
170	C₅×D₄	—	12	4	no	—
171	C₅×D₄	—	12	4	no	—
172	C₅×D₄	—	12	4	no	—
173	C₅×D₄	—	12	2	no	—
174	C₅×D₄	—	12	2	no	—
175	C₅×D₄	—	12	4	no	—
176	C₂×C₄×C₅	—	2	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	#204	1	2	yes	—
179	C₂×C₄×C₅	D₂₄	1	2	yes	—
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
186	C₅×C₈	—	4	2	no	—
187	C₅×C₈	—	12	2	no	—
188	C₂²×D₄	—	3	2	no	—
189	C₂²×D₄	—	6	1	no	—
190	C₂²×D₄	—	6	1	no	—
191	C₂×M₁₆	—	6	1	no	—
192	#127	—	3	2	no	—
193	#127	—	3	2	no	—
194	#127	—	6	1	no	—
195	#127	—	6	1	no	—
196	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
197	C₄⋊C₈	—	3	2	no	—
198	C₄⋊C₈	—	6	1	no	—
199	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	complemented
200	C₂²⋊C₈	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
205	C₂×C₃×C₅	#418	1	1	yes	characteristic
206	C₅×D₃	—	8	4	no	—
207	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×D₃	—	2	4	no	—
209	C₂²×D₃	—	4	4	no	—
210	C₂²×D₃	—	4	4	no	—
211	C₂²×D₃	—	4	4	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
218	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
219	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	D₁₂	—	4	4	no	—
223	D₁₂	—	4	4	no	—
224	D₁₂	—	4	2	no	—
225	D₁₂	—	4	2	no	—
226	D₁₂	—	4	4	no	—
227	D₁₂	—	4	4	no	—
228	D₁₂	—	4	4	no	—
229	C₃×C₈	—	4	2	no	—
230	C₃⋊C₈	—	4	2	no	—
231	C₂²×C₅	—	2	1	no	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	12	4	no	—
234	C₂²×C₅	—	12	4	no	—
235	C₂²×C₅	—	12	4	no	—
236	C₂²×C₅	—	24	8	no	—
237	C₂²×C₅	#821	1	2	yes	—
238	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	#821	1	2	yes	—
242	C₄×C₅	#823	1	2	yes	—
243	C₄×C₅	C₂×#204	1	1	yes	characteristic
244	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
245	C₄×C₅	#949	1	2	yes	—
246	C₂⁴	—	6	2	no	—
247	C₂×D₄	—	3	4	no	—
248	C₂×D₄	—	6	2	no	—
249	C₂×D₄	—	6	2	no	—
250	C₂×D₄	—	6	2	no	—
251	C₂×D₄	—	6	2	no	—
252	C₂×D₄	—	6	2	no	—
253	C₂×D₄	—	12	4	no	—
254	C₂×D₄	—	12	4	no	—
255	C₂×D₄	—	12	2	no	—
256	C₂×D₄	—	12	2	no	—
257	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
258	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
259	C₂²×C₄	C₅×#67	1	1	yes	characteristic
260	M₁₆	—	12	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	6	2	no	—
263	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
264	C₄²	C₅×D₁₂	1	1	yes	characteristic
265	C₄²	C₅×#67	1	1	yes	characteristic
266	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₃×C₅	#1545	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	C₅×#102	1	2	yes	—
271	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×D₃	—	4	4	no	—
273	C₂×D₃	—	4	4	no	—
274	C₂×D₃	—	4	4	no	—
275	C₂×D₃	—	8	8	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	C₅×#100	1	2	yes	—
279	C₃×C₄	C₅×#102	1	2	yes	—
280	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
281	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
282	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
283	C₂×C₅	—	2	2	no	—
284	C₂×C₅	—	24	4	no	—
285	C₂×C₅	#5001	1	1	yes	characteristic
286	C₂×C₅	C₂×#821	1	1	yes	characteristic
287	C₂×C₅	C₂×#823	1	1	yes	characteristic
288	C₂³	—	6	4	no	—
289	C₂³	—	12	4	no	—
290	C₂³	—	12	4	no	—
291	C₂³	—	12	4	no	—
292	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
293	D₄	—	12	4	no	—
294	D₄	—	12	4	no	—
295	D₄	—	12	2	no	—
296	D₄	—	12	2	no	—
297	D₄	—	12	4	no	—
298	D₄	—	12	4	no	—
299	D₄	—	12	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₅×#204	1	2	yes	—
303	C₂×C₄	C₅×D₂₄	1	2	yes	—
304	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
307	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
309	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
310	C₈	—	4	2	no	—
311	C₈	—	12	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
314	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
315	C₂×C₃	C₅×#418	1	1	yes	characteristic
316	D₃	—	8	4	no	—
317	C₅	#70774	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	12	4	no	—
321	C₂²	—	12	4	no	—
322	C₂²	—	12	4	no	—
323	C₂²	—	24	8	no	—
324	C₂²	C₅×#821	1	2	yes	—
325	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	C₅×#821	1	2	yes	—
329	C₄	C₅×#823	1	2	yes	—
330	C₄	C₂×C₅×#204	1	1	yes	characteristic
331	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
332	C₄	C₅×#949	1	2	yes	—
333	C₃	C₅×#1545	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	2	2	no	—
335	C₂	—	24	4	no	—
336	C₂	C₅×#5001	1	1	yes	characteristic
337	C₂	C₂×C₅×#821	1	1	yes	characteristic
338	C₂	C₂×C₅×#823	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#874	C₂
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#431	D₃
hypocenter	C₃	C₅×#1545
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#431

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#431	C₂	semidirect
3. C₅×#4924	C₂	semidirect
4. C₅×#4485	C₂	semidirect
15. C₃×C₅×C₄⋊C₈	C₂²	semidirect
19. C₅×C₄⋊(C₃⋊C₈)	C₂²	semidirect
20. #70774	C₅	direct (N × H)
22. C₅×#431	D₃	semidirect
47. C₃×#431	C₂×C₅	semidirect
48. #4924	C₂×C₅	semidirect
49. #4485	C₂×C₅	semidirect
61. C₅×C₄⋊C₈	C₂×D₃	semidirect
98. C₃×C₄⋊C₈	C₂²×C₅	semidirect
102. C₄⋊(C₃⋊C₈)	C₂²×C₅	semidirect
125. #431	C₅×D₃	semidirect
199. C₄⋊C₈	C₂×C₅×D₃	semidirect
267. C₃×C₅	#1545	semidirect
317. C₅	#70774	direct (N × H)
333. C₃	C₅×#1545	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#874	2
C₃×C₅×#431	2
C₅×#4924	2
C₅×#4485	2
C₅×#1545	3
#70774	5

Subgroups — C5×#70774 #14284768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70774 #14284768