Subgroups — C₂×C₅×#5753 #14510133

Order 1920.

2816 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1010	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5753	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#909	C₂	1	6	yes	maximal, complemented
5	C₂²×C₅×Dic₁₂	C₂	1	3	yes	maximal, complemented
6	C₂×C₅×#904	C₂	1	3	yes	maximal, complemented
7	C₂×C₅×#578	—	3	1	no	maximal, Hall
8	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
9	C₅×#1010	—	2	8	no	—
10	C₅×#1010	C₂²	1	16	yes	complemented
11	C₂×C₅×D₃×Q₈	—	2	2	no	—
12	C₂×C₅×#237	—	2	6	no	—
13	C₂×C₅×#235	C₂²	1	6	yes	—
14	C₂²×C₅×Dic₆	C₂²	1	6	yes	—
15	C₃×C₅×#131	C₂²	1	8	yes	complemented
16	C₂×C₃×C₅×M₁₆	C₂²	1	3	yes	—
17	C₂²×C₃×C₅×C₈	C₂²	1	3	yes	—
18	C₅×#909	C₂²	1	48	yes	complemented
19	C₂×C₅×Dic₁₂	C₂²	1	3	yes	—
20	C₂×C₅×Dic₁₂	C₂²	1	6	yes	—
21	C₂×C₅×Dic₁₂	C₂²	1	24	yes	complemented
22	C₅×#904	C₂²	1	24	yes	complemented
23	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
24	C₂×C₅×#204	C₂²	1	6	yes	—
25	C₂×#5753	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂×C₅×2₋¹⁺⁴	—	3	2	no	—
27	C₅×#578	—	3	16	no	—
28	C₂×C₅×#137	—	3	6	no	—
29	C₂×C₅×#135	—	3	3	no	—
30	C₂²×C₅×Q₁₆	—	3	3	no	—
31	C₂×C₅×#131	D₃	1	1	yes	characteristic, complemented
32	C₃×C₅×#41	D₄	1	4	yes	—
33	C₃×C₅×#41	C₂³	1	4	yes	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
37	C₂×C₅×#67	—	2	6	no	—
38	C₂²×C₅×Dic₃	—	2	6	no	—
39	C₅×D₃×Q₈	—	2	16	no	—
40	C₅×#237	—	2	48	no	—
41	C₅×#235	—	2	12	no	—
42	C₅×#235	C₂³	1	24	yes	—
43	C₂×C₅×D₁₂	C₂³	1	2	yes	—
44	C₂×C₄×C₅×D₃	—	2	6	no	—
45	C₂×C₅×Dic₆	—	2	6	no	—
46	C₂×C₅×Dic₆	—	2	12	no	—
47	C₂×C₅×Dic₆	C₂³	1	6	yes	—
48	C₂×C₅×Dic₆	C₂³	1	24	yes	—
49	C₃×C₅×M₁₆	C₂³	1	12	yes	—
50	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
51	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₂×C₃×C₅×C₈	C₂³	1	12	yes	—
53	C₅×Dic₁₂	C₂³	1	12	yes	—
54	C₅×Dic₁₂	C₂³	1	24	yes	—
55	C₅×D₂₄	C₂³	1	4	yes	—
56	C₅×#204	C₂³	1	24	yes	—
57	C₂×#1010	C₂×C₅	1	2	yes	complemented
58	C₂×C₃×#131	C₂×C₅	1	1	yes	characteristic, complemented
59	#5753	C₂×C₅	1	16	yes	direct product factor
60	C₂×#909	C₂×C₅	1	6	yes	complemented
61	C₂²×Dic₁₂	C₂×C₅	1	3	yes	complemented
62	C₂×#904	C₂×C₅	1	3	yes	complemented
63	C₅×2₋¹⁺⁴	—	3	16	no	—
64	C₅×2₋¹⁺⁴	—	6	8	no	—
65	C₂×C₅×#41	—	3	6	no	—
66	C₂×C₅×#41	—	6	6	no	—
67	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₅×Q₈	—	3	6	no	—
69	C₂²×C₅×Q₈	—	6	2	no	—
70	C₅×#137	—	3	48	no	—
71	C₅×#135	—	3	24	no	—
72	C₂×C₅×Q₁₆	—	3	3	no	—
73	C₂×C₅×Q₁₆	—	3	6	no	—
74	C₂×C₅×Q₁₆	—	3	24	no	—
75	C₂×C₅×SD₁₆	—	3	6	no	—
76	C₂×C₅×D₈	—	3	1	no	—
77	C₅×#131	C₂×D₃	1	8	yes	complemented
78	C₂×C₅×M₁₆	C₂×D₃	1	3	yes	—
79	C₂²×C₅×C₈	C₂×D₃	1	3	yes	—
80	C₂×#578	—	3	1	no	Sylow
81	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
82	C₂²×C₅×D₃	—	2	2	no	—
83	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
84	C₃×C₅×D₄	C₂×D₄	1	12	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂⁴	1	6	yes	—
89	C₅×#67	—	2	24	no	—
90	C₂×C₅×Dic₃	—	2	6	no	—
91	C₂×C₅×Dic₃	—	2	24	no	—
92	C₅×D₁₂	—	2	2	no	—
93	C₅×D₁₂	C₂⁴	1	4	yes	—
94	C₄×C₅×D₃	—	2	24	no	—
95	C₅×Dic₆	—	2	6	no	—
96	C₅×Dic₆	—	2	24	no	—
97	C₅×Dic₆	C₂⁴	1	12	yes	—
98	C₃×C₅×C₈	C₂⁴	1	6	yes	—
99	C₃×C₅×C₈	C₂⁴	1	2	yes	—
100	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
101	#1010	—	2	8	no	—
102	#1010	C₂²×C₅	1	16	yes	complemented
103	C₂×D₃×Q₈	—	2	2	no	—
104	C₂×#237	—	2	6	no	—
105	C₂×#235	C₂²×C₅	1	6	yes	—
106	C₂²×Dic₆	C₂²×C₅	1	6	yes	—
107	C₃×#131	C₂²×C₅	1	8	yes	complemented
108	C₂×C₃×M₁₆	C₂²×C₅	1	3	yes	—
109	C₂²×C₃×C₈	C₂²×C₅	1	3	yes	—
110	#909	C₂²×C₅	1	48	yes	complemented
111	C₂×Dic₁₂	C₂²×C₅	1	3	yes	—
112	C₂×Dic₁₂	C₂²×C₅	1	6	yes	—
113	C₂×Dic₁₂	C₂²×C₅	1	24	yes	complemented
114	#904	C₂²×C₅	1	24	yes	complemented
115	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
116	C₂×#204	C₂²×C₅	1	6	yes	—
117	C₅×#41	—	3	24	no	—
118	C₅×#41	—	6	48	no	—
119	C₅×#41	—	6	12	no	—
120	C₅×#41	D₁₂	1	4	yes	—
121	C₅×#41	C₂²×D₃	1	4	yes	—
122	C₂×C₅×Q₈	—	3	6	no	—
123	C₂×C₅×Q₈	—	3	24	no	—
124	C₂×C₅×Q₈	—	6	16	no	—
125	C₂×C₅×Q₈	—	6	6	no	—
126	C₂×C₅×Q₈	—	6	12	no	—
127	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
128	C₂×C₅×D₄	—	3	2	no	—
129	C₂×C₅×D₄	—	6	6	no	—
130	C₂×C₅×D₄	D₁₂	1	3	yes	—
131	C₂²×C₄×C₅	—	6	6	no	—
132	C₂²×C₄×C₅	—	6	6	no	—
133	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
134	C₅×Q₁₆	—	3	12	no	—
135	C₅×Q₁₆	—	3	24	no	—
136	C₅×SD₁₆	—	3	24	no	—
137	C₅×D₈	—	3	4	no	—
138	C₅×M₁₆	C₂²×D₃	1	12	yes	—
139	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
140	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
141	C₂×C₅×C₈	C₂²×D₃	1	12	yes	—
142	C₂×2₋¹⁺⁴	—	3	2	no	—
143	#578	—	3	16	no	—
144	C₂×#137	—	3	6	no	—
145	C₂×#135	—	3	3	no	—
146	C₂²×Q₁₆	—	3	3	no	—
147	C₂×#131	C₅×D₃	1	1	yes	characteristic, complemented
148	C₂²×C₃×C₅	—	2	6	no	—
149	C₂²×C₃×C₅	C₂²×D₄	1	6	yes	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₅×D₃	—	2	4	no	—
152	C₂×C₅×D₃	—	4	4	no	—
153	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₃×C₄×C₅	C₂²×D₄	1	6	yes	—
155	C₃×C₄×C₅	C₂⁵	1	1	yes	characteristic
156	C₅×Dic₃	—	2	12	no	—
157	C₃×#41	C₅×D₄	1	4	yes	—
158	C₃×#41	C₂³×C₅	1	4	yes	—
159	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×D₄	C₅×D₄	1	3	yes	—
161	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
162	C₂×#67	—	2	6	no	—
163	C₂²×Dic₃	—	2	6	no	—
164	D₃×Q₈	—	2	16	no	—
165	#237	—	2	48	no	—
166	#235	—	2	12	no	—
167	#235	C₂³×C₅	1	24	yes	—
168	C₂×D₁₂	C₂³×C₅	1	2	yes	—
169	C₂×C₄×D₃	—	2	6	no	—
170	C₂×Dic₆	—	2	6	no	—
171	C₂×Dic₆	—	2	12	no	—
172	C₂×Dic₆	C₂³×C₅	1	6	yes	—
173	C₂×Dic₆	C₂³×C₅	1	24	yes	—
174	C₃×M₁₆	C₂³×C₅	1	12	yes	—
175	C₂×C₃×C₈	C₂³×C₅	1	3	yes	—
176	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
177	C₂×C₃×C₈	C₂³×C₅	1	12	yes	—
178	Dic₁₂	C₂³×C₅	1	12	yes	—
179	Dic₁₂	C₂³×C₅	1	24	yes	—
180	D₂₄	C₂³×C₅	1	4	yes	—
181	#204	C₂³×C₅	1	24	yes	—
182	C₂³×C₅	—	6	2	no	—
183	C₂³×C₅	C₂×D₁₂	1	3	yes	—
184	C₅×Q₈	—	3	12	no	—
185	C₅×Q₈	—	6	24	no	—
186	C₅×Q₈	—	6	6	no	—
187	C₅×Q₈	C₂×D₁₂	1	4	yes	—
188	C₅×D₄	—	3	4	no	—
189	C₅×D₄	—	6	24	no	—
190	C₅×D₄	—	6	2	no	—
191	C₅×D₄	C₂×D₁₂	1	12	yes	—
192	C₂×C₄×C₅	—	6	24	no	—
193	C₂×C₄×C₅	—	6	6	no	—
194	C₂×C₄×C₅	—	6	24	no	—
195	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
196	C₂×C₄×C₅	C₂×D₁₂	1	3	yes	—
197	C₂×C₄×C₅	C₂³×D₃	1	6	yes	—
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₅×C₈	C₂³×D₃	1	2	yes	—
200	C₅×C₈	C₂³×D₃	1	6	yes	—
201	2₋¹⁺⁴	—	3	16	no	—
202	2₋¹⁺⁴	—	6	8	no	—
203	C₂×#41	—	3	6	no	—
204	C₂×#41	—	6	6	no	—
205	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
206	C₂²×Q₈	—	3	6	no	—
207	C₂²×Q₈	—	6	2	no	—
208	#137	—	3	48	no	—
209	#135	—	3	24	no	—
210	C₂×Q₁₆	—	3	3	no	—
211	C₂×Q₁₆	—	3	6	no	—
212	C₂×Q₁₆	—	3	24	no	—
213	C₂×SD₁₆	—	3	6	no	—
214	C₂×D₈	—	3	1	no	—
215	#131	C₂×C₅×D₃	1	8	yes	complemented
216	C₂×M₁₆	C₂×C₅×D₃	1	3	yes	—
217	C₂²×C₈	C₂×C₅×D₃	1	3	yes	—
218	C₂×C₃×C₅	—	2	6	no	—
219	C₂×C₃×C₅	#578	1	2	yes	complemented
220	C₂×C₃×C₅	C₂³×D₄	1	1	yes	characteristic
221	C₅×D₃	—	4	4	no	—
222	C₂³×C₃	C₂×C₅×D₄	1	3	yes	—
223	C₂²×D₃	—	2	2	no	—
224	C₃×Q₈	C₂×C₅×D₄	1	4	yes	—
225	C₃×D₄	C₂×C₅×D₄	1	12	yes	—
226	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
227	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
228	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂⁴×C₅	1	6	yes	—
230	#67	—	2	24	no	—
231	C₂×Dic₃	—	2	6	no	—
232	C₂×Dic₃	—	2	24	no	—
233	D₁₂	—	2	2	no	—
234	D₁₂	C₂⁴×C₅	1	4	yes	—
235	C₄×D₃	—	2	24	no	—
236	Dic₆	—	2	6	no	—
237	Dic₆	—	2	24	no	—
238	Dic₆	C₂⁴×C₅	1	12	yes	—
239	C₃×C₈	C₂⁴×C₅	1	6	yes	—
240	C₃×C₈	C₂⁴×C₅	1	2	yes	—
241	C₂²×C₅	—	2	6	no	—
242	C₂²×C₅	—	6	4	no	—
243	C₂²×C₅	—	12	4	no	—
244	C₂²×C₅	C₂²×D₁₂	1	6	yes	—
245	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
246	C₄×C₅	—	6	12	no	—
247	C₄×C₅	C₂²×D₁₂	1	6	yes	—
248	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
249	C₄×C₅	C₂⁴×D₃	1	1	yes	characteristic
250	#41	—	3	24	no	—
251	#41	—	6	48	no	—
252	#41	—	6	12	no	—
253	#41	C₅×D₁₂	1	4	yes	—
254	#41	C₂²×C₅×D₃	1	4	yes	—
255	C₂×Q₈	—	3	6	no	—
256	C₂×Q₈	—	3	24	no	—
257	C₂×Q₈	—	6	16	no	—
258	C₂×Q₈	—	6	6	no	—
259	C₂×Q₈	—	6	12	no	—
260	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
261	C₂×D₄	—	3	2	no	—
262	C₂×D₄	—	6	6	no	—
263	C₂×D₄	C₅×D₁₂	1	3	yes	—
264	C₂²×C₄	—	6	6	no	—
265	C₂²×C₄	—	6	6	no	—
266	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
267	Q₁₆	—	3	12	no	—
268	Q₁₆	—	3	24	no	—
269	SD₁₆	—	3	24	no	—
270	D₈	—	3	4	no	—
271	M₁₆	C₂²×C₅×D₃	1	12	yes	—
272	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×C₅×D₃	1	3	yes	—
274	C₂×C₈	C₂²×C₅×D₃	1	12	yes	—
275	C₃×C₅	C₂×#578	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	2	6	no	—
277	C₂²×C₃	C₂²×C₅×D₄	1	6	yes	—
278	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
279	C₂×D₃	—	2	4	no	—
280	C₂×D₃	—	4	4	no	—
281	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
282	C₃×C₄	C₂²×C₅×D₄	1	6	yes	—
283	C₃×C₄	C₂⁵×C₅	1	1	yes	characteristic
284	Dic₃	—	2	12	no	—
285	C₂×C₅	—	2	6	no	—
286	C₂×C₅	—	12	4	no	—
287	C₂×C₅	#5753	1	2	yes	direct product factor
288	C₂×C₅	C₂³×D₁₂	1	1	yes	characteristic
289	C₂³	—	6	2	no	—
290	C₂³	C₂×C₅×D₁₂	1	3	yes	—
291	Q₈	—	3	12	no	—
292	Q₈	—	6	24	no	—
293	Q₈	—	6	6	no	—
294	Q₈	C₂×C₅×D₁₂	1	4	yes	—
295	D₄	—	3	4	no	—
296	D₄	—	6	24	no	—
297	D₄	—	6	2	no	—
298	D₄	C₂×C₅×D₁₂	1	12	yes	—
299	C₂×C₄	—	6	6	no	—
300	C₂×C₄	—	6	24	no	—
301	C₂×C₄	—	6	24	no	—
302	C₂×C₄	C₂×C₅×D₁₂	1	3	yes	—
303	C₂×C₄	C₂×C₅×D₁₂	1	6	yes	—
304	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
305	C₂×C₄	C₂³×C₅×D₃	1	6	yes	—
306	C₈	C₂³×C₅×D₃	1	2	yes	—
307	C₈	C₂³×C₅×D₃	1	6	yes	—
308	C₂×C₃	—	2	6	no	—
309	C₂×C₃	C₅×#578	1	2	yes	complemented
310	C₂×C₃	C₂³×C₅×D₄	1	1	yes	characteristic
311	D₃	—	4	4	no	—
312	C₅	C₂×#5753	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	2	6	no	—
314	C₂²	—	6	4	no	—
315	C₂²	—	12	4	no	—
316	C₂²	C₂²×C₅×D₁₂	1	6	yes	—
317	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
318	C₄	—	6	12	no	—
319	C₄	C₂²×C₅×D₁₂	1	6	yes	—
320	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
321	C₄	C₂⁴×C₅×D₃	1	1	yes	characteristic
322	C₃	C₂×C₅×#578	1	1	yes	characteristic, complemented, Sylow
323	C₂	—	2	6	no	—
324	C₂	—	12	4	no	—
325	C₂	C₅×#5753	1	2	yes	direct product factor
326	C₂	C₂³×C₅×D₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂²×D₁₂
commutator, cosocle	C₃×C₄	C₂⁵×C₅
Frattini	C₄	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₂×C₅×#131	D₃
hypocenter	C₃	C₂×C₅×#578
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#1010	C₂	semidirect
2. C₂×C₃×C₅×#131	C₂	semidirect
3. C₅×#5753	C₂	direct (N × H)
4. C₂×C₅×#909	C₂	semidirect
5. C₂²×C₅×Dic₁₂	C₂	semidirect
6. C₂×C₅×#904	C₂	semidirect
10. C₅×#1010	C₂²	semidirect
15. C₃×C₅×#131	C₂²	semidirect
18. C₅×#909	C₂²	semidirect
21. C₂×C₅×Dic₁₂	C₂²	semidirect
22. C₅×#904	C₂²	semidirect
25. C₂×#5753	C₅	direct (N × H)
31. C₂×C₅×#131	D₃	semidirect
57. C₂×#1010	C₂×C₅	semidirect
58. C₂×C₃×#131	C₂×C₅	semidirect
59. #5753	C₂×C₅	direct (N × H)
60. C₂×#909	C₂×C₅	semidirect
61. C₂²×Dic₁₂	C₂×C₅	semidirect
62. C₂×#904	C₂×C₅	semidirect
77. C₅×#131	C₂×D₃	semidirect
102. #1010	C₂²×C₅	semidirect
107. C₃×#131	C₂²×C₅	semidirect
110. #909	C₂²×C₅	semidirect
113. C₂×Dic₁₂	C₂²×C₅	semidirect
114. #904	C₂²×C₅	semidirect
147. C₂×#131	C₅×D₃	semidirect
215. #131	C₂×C₅×D₃	semidirect
219. C₂×C₃×C₅	#578	semidirect
275. C₃×C₅	C₂×#578	semidirect
287. C₂×C₅	#5753	direct (N × H)
309. C₂×C₃	C₅×#578	semidirect
312. C₅	C₂×#5753	direct (N × H)
322. C₃	C₂×C₅×#578	semidirect
325. C₂	C₅×#5753	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#1010	2
C₂×C₃×C₅×#131	2
C₅×#5753	2
C₂×C₅×#909	2
C₂²×C₅×Dic₁₂	2
C₂×C₅×#904	2
C₂×C₅×#578	3
C₂×#5753	5

Subgroups — C2×C5×#5753 #14510133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5753 #14510133