Subgroups — C₅×#69753 #14295086

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4550	C₂	1	2	yes	maximal, complemented
6	C₅×#1750	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
11	C₂×C₅×Dic₃⋊C₄	C₄	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₅×#872	—	2	2	no	—
15	C₅×#872	—	2	2	no	—
16	C₃×C₄×C₅×C₈	—	2	2	no	—
17	C₅×(C₂×D₃)⋊C₈	C₄	1	8	yes	complemented
18	C₅×#803	—	2	2	no	—
19	#69753	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#117	—	3	1	no	—
21	C₂×C₅×C₂²⋊C₈	—	3	2	no	—
22	C₂×C₅×#97	—	3	1	no	—
23	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
24	C₅×#336	—	3	2	no	—
25	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
29	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅×D₃	—	2	1	no	—
31	C₂×C₄×C₅×D₃	—	2	1	no	—
32	C₂×C₄×C₅×D₃	—	2	2	no	—
33	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
40	C₅×Dic₃⋊C₄	—	2	4	no	—
41	C₄×C₅×Dic₃	—	2	1	no	—
42	C₄×C₅×Dic₃	—	2	1	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
46	C₂×#872	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×(C₂×D₃)⋊C₈	C₂×C₅	1	2	yes	—
49	C₂×#803	C₂×C₅	1	1	yes	characteristic, complemented
50	#4550	C₂×C₅	1	2	yes	complemented
51	C₂³×C₄×C₅	—	3	1	no	—
52	C₂²×C₅×C₈	—	3	2	no	—
53	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
54	C₅×#117	—	6	2	no	—
55	C₅×#117	—	6	2	no	—
56	C₂×C₅×C₄⋊C₄	—	3	2	no	—
57	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
58	C₂×C₄²×C₅	—	3	1	no	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	C₅×C₂²⋊C₈	—	3	8	no	—
61	C₅×#97	—	6	2	no	—
62	C₄×C₅×C₈	—	2	2	no	—
63	#1750	—	3	1	no	Sylow
64	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₅×D₃	—	2	1	no	—
66	C₂²×C₅×D₃	—	2	1	no	—
67	C₂²×C₅×D₃	—	2	4	no	—
68	C₂²×C₅×D₃	C₄²	1	2	yes	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₂×C₅×Dic₃	—	2	1	no	—
76	C₂×C₅×Dic₃	—	2	1	no	—
77	C₂×C₅×Dic₃	—	2	2	no	—
78	C₂×C₅×Dic₃	—	2	4	no	—
79	C₂×C₅×Dic₃	C₄²	1	2	yes	—
80	C₄×C₅×D₃	—	2	4	no	—
81	C₄×C₅×D₃	—	2	4	no	—
82	C₃×C₅×C₈	—	2	4	no	—
83	C₅×C₃⋊C₈	—	2	4	no	—
84	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
86	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
87	C₂×(C₂×D₃)⋊C₄	C₄×C₅	1	2	yes	—
88	C₂×Dic₃⋊C₄	C₄×C₅	1	2	yes	—
89	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
91	#872	—	2	2	no	—
92	#872	—	2	2	no	—
93	C₃×C₄×C₈	—	2	2	no	—
94	(C₂×D₃)⋊C₈	C₄×C₅	1	8	yes	complemented
95	#803	—	2	2	no	—
96	C₂⁴×C₅	—	3	1	no	—
97	C₂²×C₄×C₅	—	3	1	no	—
98	C₂²×C₄×C₅	—	3	4	no	—
99	C₂²×C₄×C₅	—	3	2	no	—
100	C₂²×C₄×C₅	—	6	1	no	—
101	C₂²×C₄×C₅	—	6	2	no	—
102	C₂²×C₄×C₅	—	6	1	no	—
103	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂×C₅×C₈	—	2	2	no	—
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	3	4	no	—
108	C₂×C₅×C₈	—	6	2	no	—
109	C₂×C₅×C₈	—	6	2	no	—
110	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
111	C₅×C₄⋊C₄	—	6	4	no	—
112	C₅×C₂²⋊C₄	—	6	4	no	—
113	C₄²×C₅	—	6	1	no	—
114	C₄²×C₅	—	6	1	no	—
115	C₄²×C₅	D₁₂	1	2	yes	—
116	C₄²×C₅	#67	1	2	yes	—
117	C₂×#117	—	3	1	no	—
118	C₂×C₂²⋊C₈	—	3	2	no	—
119	C₂×#97	—	3	1	no	—
120	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic, complemented
121	#336	—	3	2	no	—
122	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	#117	1	1	yes	characteristic
125	C₂²×C₃×C₅	#131	1	4	yes	—
126	C₂×C₅×D₃	—	2	2	no	—
127	C₂×C₅×D₃	—	2	4	no	—
128	C₂×C₅×D₃	—	4	2	no	—
129	C₂×C₅×D₃	—	4	2	no	—
130	C₃×C₄×C₅	—	2	4	no	—
131	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₅×Dic₃	—	2	2	no	—
134	C₅×Dic₃	—	4	2	no	—
135	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
137	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
138	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
139	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
140	C₂×C₄×D₃	—	2	1	no	—
141	C₂×C₄×D₃	—	2	1	no	—
142	C₂×C₄×D₃	—	2	2	no	—
143	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
147	C₃×C₄²	C₅×D₄	1	2	yes	—
148	C₃×C₄²	C₅×D₄	1	2	yes	—
149	(C₂×D₃)⋊C₄	—	2	4	no	—
150	Dic₃⋊C₄	—	2	4	no	—
151	C₄×Dic₃	—	2	1	no	—
152	C₄×Dic₃	—	2	1	no	—
153	C₂×C₃⋊C₈	—	2	2	no	—
154	C₂×C₃⋊C₈	—	2	2	no	—
155	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
156	C₂³×C₅	—	3	2	no	—
157	C₂³×C₅	—	6	1	no	—
158	C₂³×C₅	—	6	4	no	—
159	C₂³×C₅	—	6	1	no	—
160	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	3	2	no	—
163	C₂×C₄×C₅	—	6	1	no	—
164	C₂×C₄×C₅	—	6	1	no	—
165	C₂×C₄×C₅	—	6	4	no	—
166	C₂×C₄×C₅	—	6	4	no	—
167	C₂×C₄×C₅	—	6	4	no	—
168	C₂×C₄×C₅	—	6	2	no	—
169	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
170	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
171	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₄×C₅	#235	1	2	yes	—
173	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
174	C₅×C₈	—	2	4	no	—
175	C₅×C₈	—	6	4	no	—
176	C₂³×C₄	—	3	1	no	—
177	C₂²×C₈	—	3	2	no	—
178	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
179	#117	—	6	2	no	—
180	#117	—	6	2	no	—
181	C₂×C₄⋊C₄	—	3	2	no	—
182	C₂×C₂²⋊C₄	—	3	2	no	—
183	C₂×C₄²	—	3	1	no	—
184	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
185	C₂²⋊C₈	—	3	8	no	—
186	#97	—	6	2	no	—
187	C₄×C₈	—	2	2	no	—
188	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₅	#405	1	2	yes	—
190	C₂×C₃×C₅	#408	1	2	yes	—
191	C₂×C₃×C₅	#433	1	2	yes	—
192	C₅×D₃	—	4	2	no	—
193	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
194	C₂²×D₃	—	2	1	no	—
195	C₂²×D₃	—	2	1	no	—
196	C₂²×D₃	—	2	4	no	—
197	C₂²×D₃	C₄²×C₅	1	2	yes	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
200	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
201	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₅×#41	1	2	yes	—
204	C₂×Dic₃	—	2	1	no	—
205	C₂×Dic₃	—	2	1	no	—
206	C₂×Dic₃	—	2	2	no	—
207	C₂×Dic₃	—	2	4	no	—
208	C₂×Dic₃	C₄²×C₅	1	2	yes	—
209	C₄×D₃	—	2	4	no	—
210	C₄×D₃	—	2	4	no	—
211	C₃×C₈	—	2	4	no	—
212	C₃⋊C₈	—	2	4	no	—
213	C₂²×C₅	—	6	2	no	—
214	C₂²×C₅	—	6	4	no	—
215	C₂²×C₅	—	12	2	no	—
216	C₂²×C₅	—	12	2	no	—
217	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
218	C₂²×C₅	#872	1	1	yes	characteristic
219	C₂²×C₅	#901	1	4	yes	—
220	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
221	C₄×C₅	—	2	4	no	—
222	C₄×C₅	—	6	2	no	—
223	C₄×C₅	—	12	2	no	—
224	C₄×C₅	C₄×D₁₂	1	2	yes	—
225	C₄×C₅	C₄×#67	1	2	yes	—
226	C₂⁴	—	3	1	no	—
227	C₂²×C₄	—	3	1	no	—
228	C₂²×C₄	—	3	4	no	—
229	C₂²×C₄	—	3	2	no	—
230	C₂²×C₄	—	6	1	no	—
231	C₂²×C₄	—	6	2	no	—
232	C₂²×C₄	—	6	1	no	—
233	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
234	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	3	4	no	—
238	C₂×C₈	—	6	2	no	—
239	C₂×C₈	—	6	2	no	—
240	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
241	C₄⋊C₄	—	6	4	no	—
242	C₂²⋊C₄	—	6	4	no	—
243	C₄²	—	6	1	no	—
244	C₄²	—	6	1	no	—
245	C₄²	C₅×D₁₂	1	2	yes	—
246	C₄²	C₅×#67	1	2	yes	—
247	C₃×C₅	#1750	1	1	yes	characteristic, complemented, Hall
248	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
249	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂²×C₃	C₅×#117	1	1	yes	characteristic
251	C₂²×C₃	C₅×#131	1	4	yes	—
252	C₂×D₃	—	2	2	no	—
253	C₂×D₃	—	2	4	no	—
254	C₂×D₃	—	4	2	no	—
255	C₂×D₃	—	4	2	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
258	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
259	Dic₃	—	2	2	no	—
260	Dic₃	—	4	2	no	—
261	C₂×C₅	—	12	2	no	—
262	C₂×C₅	#4689	1	2	yes	—
263	C₂×C₅	#4690	1	2	yes	—
264	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
265	C₂×C₅	#5110	1	2	yes	—
266	C₂³	—	3	2	no	—
267	C₂³	—	6	1	no	—
268	C₂³	—	6	4	no	—
269	C₂³	—	6	1	no	—
270	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	3	2	no	—
273	C₂×C₄	—	6	1	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	4	no	—
276	C₂×C₄	—	6	4	no	—
277	C₂×C₄	—	6	4	no	—
278	C₂×C₄	—	6	2	no	—
279	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
280	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
281	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
282	C₂×C₄	C₅×#235	1	2	yes	—
283	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	6	4	no	—
286	C₂×C₃	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×C₃	C₅×#405	1	2	yes	—
288	C₂×C₃	C₅×#408	1	2	yes	—
289	C₂×C₃	C₅×#433	1	2	yes	—
290	D₃	—	4	2	no	—
291	C₅	#69753	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	6	2	no	—
293	C₂²	—	6	4	no	—
294	C₂²	—	12	2	no	—
295	C₂²	—	12	2	no	—
296	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
297	C₂²	C₅×#872	1	1	yes	characteristic
298	C₂²	C₅×#901	1	4	yes	—
299	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	6	2	no	—
302	C₄	—	12	2	no	—
303	C₄	C₄×C₅×D₁₂	1	2	yes	—
304	C₄	C₄×C₅×#67	1	2	yes	—
305	C₃	C₅×#1750	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	12	2	no	—
307	C₂	C₅×#4689	1	2	yes	—
308	C₂	C₅×#4690	1	2	yes	—
309	C₂	C₄×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₂	C₅×#5110	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₅×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₅×#1750
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. C₂×C₅×#803	C₂	semidirect
5. C₅×#4550	C₂	semidirect
17. C₅×(C₂×D₃)⋊C₈	C₄	semidirect
19. #69753	C₅	direct (N × H)
23. C₂×C₄×C₅×C₈	D₃	semidirect
47. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
49. C₂×#803	C₂×C₅	semidirect
50. #4550	C₂×C₅	semidirect
94. (C₂×D₃)⋊C₈	C₄×C₅	semidirect
120. C₂×C₄×C₈	C₅×D₃	semidirect
247. C₃×C₅	#1750	semidirect
291. C₅	#69753	direct (N × H)
305. C₃	C₅×#1750	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#872	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×(C₂×D₃)⋊C₈	2
C₂×C₅×#803	2
C₅×#4550	2
C₅×#1750	3
#69753	5

Subgroups — C5×#69753 #14295086

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69753 #14295086