Subgroups — C₅×#70690 #14314040

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#904	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4497	C₂	1	4	yes	maximal, complemented
5	C₅×#2143	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	—	2	1	no	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×Dic₁₂	C₄	1	1	yes	characteristic
10	C₅×#904	—	2	1	no	—
11	C₅×#904	—	2	1	no	—
12	C₅×#904	C₄	1	2	yes	—
13	C₅×#904	C₂²	1	2	yes	—
14	C₂×C₅×D₂₄	C₄	1	1	yes	characteristic
15	C₂×C₅×#204	—	2	1	no	—
16	C₃×C₅×#108	—	2	1	no	—
17	C₃×C₅×#108	C₂²	1	2	yes	complemented
18	C₅×#812	—	2	1	no	—
19	C₅×#812	—	2	1	no	—
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	complemented
21	#70690	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#135	—	3	1	no	—
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂×C₅×#108	D₃	1	1	yes	characteristic, complemented
25	C₅×#359	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₅×#67	—	4	1	no	—
28	C₅×#235	—	2	2	no	—
29	C₅×#235	—	2	2	no	—
30	C₅×#235	—	4	1	no	—
31	C₅×#235	—	4	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₄×C₅×D₃	—	4	1	no	—
34	C₂×C₅×Dic₆	—	2	1	no	—
35	C₃×C₅×M₁₆	—	2	2	no	—
36	C₃×C₅×M₁₆	—	4	1	no	—
37	C₂×C₃×C₅×C₈	—	4	1	no	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₅×Dic₁₂	—	2	1	no	—
45	C₅×Dic₁₂	C₂×C₄	1	2	yes	—
46	C₅×D₂₄	—	2	1	no	—
47	C₅×D₂₄	C₂×C₄	1	2	yes	—
48	C₅×#204	—	2	2	no	—
49	C₅×#204	—	2	2	no	—
50	C₅×C₃⋊C₁₆	—	2	2	no	—
51	C₂×#904	C₂×C₅	1	1	yes	characteristic
52	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
54	#4497	C₂×C₅	1	4	yes	complemented
55	C₂×C₅×#41	—	6	1	no	—
56	C₅×#135	—	3	2	no	—
57	C₅×#135	—	3	2	no	—
58	C₅×#135	—	6	1	no	—
59	C₅×#135	—	6	1	no	—
60	C₂×C₅×Q₁₆	—	3	1	no	—
61	C₂×C₅×SD₁₆	—	6	1	no	—
62	C₂×C₅×D₈	—	3	1	no	—
63	C₂×C₅×M₁₆	—	2	1	no	—
64	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₅×M₃₂	—	6	1	no	—
66	C₅×M₃₂	—	6	1	no	—
67	C₂×C₅×C₁₆	—	3	2	no	—
68	C₅×#108	—	2	1	no	—
69	C₅×#108	C₂×D₃	1	2	yes	complemented
70	#2143	—	3	1	no	Sylow
71	C₂³×C₃×C₅	SD₁₆	1	1	yes	characteristic
72	C₂²×C₅×D₃	—	4	1	no	—
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	1	yes	characteristic
76	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	SD₁₆	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×#67	—	4	2	no	—
80	C₅×#67	—	4	2	no	—
81	C₂×C₅×Dic₃	—	4	1	no	—
82	C₅×D₁₂	—	2	2	no	—
83	C₅×D₁₂	—	4	1	no	—
84	C₄×C₅×D₃	—	4	2	no	—
85	C₄×C₅×D₃	—	4	2	no	—
86	C₅×Dic₆	—	2	2	no	—
87	C₅×Dic₆	—	4	1	no	—
88	C₃×C₅×C₈	—	4	2	no	—
89	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
91	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
92	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×#235	—	2	1	no	—
94	C₂×C₃×M₁₆	—	2	1	no	—
95	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
96	C₂×Dic₁₂	C₄×C₅	1	1	yes	characteristic
97	#904	—	2	1	no	—
98	#904	—	2	1	no	—
99	#904	C₄×C₅	1	2	yes	—
100	#904	C₂²×C₅	1	2	yes	—
101	C₂×D₂₄	C₄×C₅	1	1	yes	characteristic
102	C₂×#204	—	2	1	no	—
103	C₃×#108	—	2	1	no	—
104	C₃×#108	C₂²×C₅	1	2	yes	complemented
105	#812	—	2	1	no	—
106	#812	—	2	1	no	—
107	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	complemented
108	C₅×#41	—	6	2	no	—
109	C₅×#41	—	6	2	no	—
110	C₅×#41	—	12	1	no	—
111	C₅×#41	—	12	1	no	—
112	C₂×C₅×Q₈	—	6	1	no	—
113	C₂×C₅×D₄	—	6	1	no	—
114	C₂×C₅×D₄	—	12	1	no	—
115	C₂²×C₄×C₅	—	12	1	no	—
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₅×Q₁₆	—	3	2	no	—
118	C₅×Q₁₆	—	6	1	no	—
119	C₅×SD₁₆	—	6	2	no	—
120	C₅×SD₁₆	—	6	2	no	—
121	C₅×D₈	—	3	2	no	—
122	C₅×D₈	—	6	1	no	—
123	C₅×M₁₆	—	2	2	no	—
124	C₅×M₁₆	—	4	1	no	—
125	C₂×C₅×C₈	—	4	1	no	—
126	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
127	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
128	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
129	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
130	C₂×C₅×C₈	#67	1	1	yes	characteristic
131	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
132	C₅×C₁₆	—	6	2	no	—
133	C₂×#135	—	3	1	no	—
134	C₂×M₃₂	—	3	1	no	—
135	C₂×#108	C₅×D₃	1	1	yes	characteristic, complemented
136	#359	—	3	4	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	#102	1	1	yes	characteristic
139	C₂²×C₃×C₅	#102	1	1	yes	characteristic
140	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
141	C₂×C₅×D₃	—	4	2	no	—
142	C₂×C₅×D₃	—	8	2	no	—
143	C₃×C₄×C₅	#102	1	1	yes	characteristic
144	C₃×C₄×C₅	#102	1	1	yes	characteristic
145	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
147	C₅×Dic₃	—	4	2	no	—
148	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
149	C₂×#67	—	4	1	no	—
150	#235	—	2	2	no	—
151	#235	—	2	2	no	—
152	#235	—	4	1	no	—
153	#235	—	4	1	no	—
154	C₂×D₁₂	—	2	1	no	—
155	C₂×C₄×D₃	—	4	1	no	—
156	C₂×Dic₆	—	2	1	no	—
157	C₃×M₁₆	—	2	2	no	—
158	C₃×M₁₆	—	4	1	no	—
159	C₂×C₃×C₈	—	4	1	no	—
160	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
162	C₂×C₃×C₈	C₅×D₄	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	C₂³×C₅	1	1	yes	characteristic
166	Dic₁₂	—	2	1	no	—
167	Dic₁₂	C₂×C₄×C₅	1	2	yes	—
168	D₂₄	—	2	1	no	—
169	D₂₄	C₂×C₄×C₅	1	2	yes	—
170	#204	—	2	2	no	—
171	#204	—	2	2	no	—
172	C₃⋊C₁₆	—	2	2	no	—
173	C₂³×C₅	—	12	1	no	—
174	C₂³×C₅	#215	1	1	yes	characteristic
175	C₅×Q₈	—	6	2	no	—
176	C₅×Q₈	—	12	1	no	—
177	C₅×D₄	—	6	2	no	—
178	C₅×D₄	—	12	2	no	—
179	C₅×D₄	—	12	2	no	—
180	C₅×D₄	—	12	1	no	—
181	C₂×C₄×C₅	—	12	1	no	—
182	C₂×C₄×C₅	—	12	2	no	—
183	C₂×C₄×C₅	—	12	2	no	—
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	#213	1	1	yes	characteristic
187	C₂×C₄×C₅	#213	1	1	yes	characteristic
188	C₂×C₄×C₅	#215	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
190	C₅×C₈	—	4	2	no	—
191	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
192	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
193	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
194	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
195	C₂×#41	—	6	1	no	—
196	#135	—	3	2	no	—
197	#135	—	3	2	no	—
198	#135	—	6	1	no	—
199	#135	—	6	1	no	—
200	C₂×Q₁₆	—	3	1	no	—
201	C₂×SD₁₆	—	6	1	no	—
202	C₂×D₈	—	3	1	no	—
203	C₂×M₁₆	—	2	1	no	—
204	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
205	M₃₂	—	6	1	no	—
206	M₃₂	—	6	1	no	—
207	C₂×C₁₆	—	3	2	no	—
208	#108	—	2	1	no	—
209	#108	C₂×C₅×D₃	1	2	yes	complemented
210	C₂×C₃×C₅	—	2	1	no	—
211	C₂×C₃×C₅	—	2	1	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
214	C₅×D₃	—	8	2	no	—
215	C₂³×C₃	C₅×SD₁₆	1	1	yes	characteristic
216	C₂²×D₃	—	4	1	no	—
217	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
220	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
221	C₂×C₃×C₄	C₅×SD₁₆	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	#67	—	4	2	no	—
224	#67	—	4	2	no	—
225	C₂×Dic₃	—	4	1	no	—
226	D₁₂	—	2	2	no	—
227	D₁₂	—	4	1	no	—
228	C₄×D₃	—	4	2	no	—
229	C₄×D₃	—	4	2	no	—
230	Dic₆	—	2	2	no	—
231	Dic₆	—	4	1	no	—
232	C₃×C₈	—	4	2	no	—
233	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
237	C₂²×C₅	—	4	1	no	—
238	C₂²×C₅	—	12	2	no	—
239	C₂²×C₅	—	24	2	no	—
240	C₂²×C₅	#809	1	1	yes	characteristic
241	C₂²×C₅	#809	1	1	yes	characteristic
242	C₂²×C₅	C₂×#215	1	1	yes	characteristic
243	C₄×C₅	—	12	2	no	—
244	C₄×C₅	#809	1	1	yes	characteristic
245	C₄×C₅	#809	1	1	yes	characteristic
246	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
247	C₄×C₅	C₂×#213	1	1	yes	characteristic
248	#41	—	6	2	no	—
249	#41	—	6	2	no	—
250	#41	—	12	1	no	—
251	#41	—	12	1	no	—
252	C₂×Q₈	—	6	1	no	—
253	C₂×D₄	—	6	1	no	—
254	C₂×D₄	—	12	1	no	—
255	C₂²×C₄	—	12	1	no	—
256	C₂²×C₄	C₅×#67	1	1	yes	characteristic
257	Q₁₆	—	3	2	no	—
258	Q₁₆	—	6	1	no	—
259	SD₁₆	—	6	2	no	—
260	SD₁₆	—	6	2	no	—
261	D₈	—	3	2	no	—
262	D₈	—	6	1	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
267	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
268	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
269	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
270	C₂×C₈	C₅×#67	1	1	yes	characteristic
271	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
272	C₁₆	—	6	2	no	—
273	C₃×C₅	#2143	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	C₅×#102	1	1	yes	characteristic
276	C₂²×C₃	C₅×#102	1	1	yes	characteristic
277	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
278	C₂×D₃	—	4	2	no	—
279	C₂×D₃	—	8	2	no	—
280	C₃×C₄	C₅×#102	1	1	yes	characteristic
281	C₃×C₄	C₅×#102	1	1	yes	characteristic
282	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
284	Dic₃	—	4	2	no	—
285	C₂×C₅	—	2	1	no	—
286	C₂×C₅	—	2	1	no	—
287	C₂×C₅	—	2	1	no	—
288	C₂×C₅	—	24	2	no	—
289	C₂×C₅	C₂×#809	1	1	yes	characteristic
290	C₂³	—	12	1	no	—
291	C₂³	C₅×#215	1	1	yes	characteristic
292	Q₈	—	6	2	no	—
293	Q₈	—	12	1	no	—
294	D₄	—	6	2	no	—
295	D₄	—	12	2	no	—
296	D₄	—	12	2	no	—
297	D₄	—	12	1	no	—
298	C₂×C₄	—	12	1	no	—
299	C₂×C₄	—	12	2	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
303	C₂×C₄	C₅×#213	1	1	yes	characteristic
304	C₂×C₄	C₅×#213	1	1	yes	characteristic
305	C₂×C₄	C₅×#215	1	1	yes	characteristic
306	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
307	C₈	—	4	2	no	—
308	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
309	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
311	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
316	D₃	—	8	2	no	—
317	C₅	#70690	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	4	1	no	—
319	C₂²	—	12	2	no	—
320	C₂²	—	24	2	no	—
321	C₂²	C₅×#809	1	1	yes	characteristic
322	C₂²	C₅×#809	1	1	yes	characteristic
323	C₂²	C₂×C₅×#215	1	1	yes	characteristic
324	C₄	—	12	2	no	—
325	C₄	C₅×#809	1	1	yes	characteristic
326	C₄	C₅×#809	1	1	yes	characteristic
327	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
328	C₄	C₂×C₅×#213	1	1	yes	characteristic
329	C₃	C₅×#2143	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	24	2	no	—
334	C₂	C₂×C₅×#809	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#213
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂×C₃×C₅	C₂×#102
prime radical	C₂×C₅×#904	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×#108	D₃
hypocenter	C₃	C₅×#2143
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×#108

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#108	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. C₅×#4497	C₂	semidirect
17. C₃×C₅×#108	C₂²	semidirect
20. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
21. #70690	C₅	direct (N × H)
24. C₂×C₅×#108	D₃	semidirect
52. C₂×C₃×#108	C₂×C₅	semidirect
53. C₂×#812	C₂×C₅	semidirect
54. #4497	C₂×C₅	semidirect
69. C₅×#108	C₂×D₃	semidirect
104. C₃×#108	C₂²×C₅	semidirect
107. C₂×C₃⋊C₁₆	C₂²×C₅	semidirect
135. C₂×#108	C₅×D₃	semidirect
209. #108	C₂×C₅×D₃	semidirect
273. C₃×C₅	#2143	semidirect
317. C₅	#70690	direct (N × H)
329. C₃	C₅×#2143	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#904	2
C₂×C₃×C₅×#108	2
C₂×C₅×#812	2
C₅×#4497	2
C₅×#2143	3
#70690	5

Subgroups — C5×#70690 #14314040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70690 #14314040