Subgroups — C₅×#68600 #14295037

Order 1920.

824 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×C₄²×C₅×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₅×#815	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#336	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4524	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1749	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₄	1	2	yes	—
8	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
14	C₄²×C₅×D₃	—	2	4	no	—
15	C₅×(C₂×D₃)⋊C₈	C₄	1	8	yes	complemented
16	C₅×#815	—	2	4	no	—
17	#68600	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₄²×C₅	—	3	1	no	—
19	C₂×C₅×C₂²⋊C₈	—	3	2	no	—
20	C₂×C₅×#97	—	3	2	no	—
21	C₅×#336	—	3	1	no	—
22	C₅×#336	D₃	1	1	yes	characteristic, complemented
23	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
28	C₂×C₄×C₅×D₃	—	2	4	no	—
29	C₂×C₄×C₅×D₃	—	2	4	no	—
30	C₂×C₄×C₅×D₃	—	2	2	no	—
31	C₂×C₄×C₅×D₃	—	2	1	no	—
32	C₂×C₄×C₅×D₃	—	2	1	no	—
33	C₂×C₄×C₅×D₃	—	2	4	no	—
34	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₄×C₅×Dic₃	—	2	4	no	—
40	C₄×C₅×Dic₃	D₄	1	4	yes	—
41	C₂×C₅×C₃⋊C₈	—	2	2	no	—
42	C₂×C₅×C₃⋊C₈	—	2	2	no	—
43	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	complemented
44	C₂×C₄²×D₃	C₂×C₅	1	1	yes	characteristic
45	C₂×(C₂×D₃)⋊C₈	C₂×C₅	1	2	yes	—
46	C₂×#815	C₂×C₅	1	2	yes	complemented
47	C₃×#336	C₂×C₅	1	1	yes	characteristic, complemented
48	#4524	C₂×C₅	1	1	yes	characteristic, complemented
49	C₂³×C₄×C₅	—	3	1	no	—
50	C₂³×C₄×C₅	—	3	2	no	—
51	C₂²×C₅×C₈	—	3	2	no	—
52	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
53	C₂×C₄²×C₅	—	3	1	no	—
54	C₂×C₄²×C₅	—	3	2	no	—
55	C₂×C₄²×C₅	—	6	4	no	—
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	C₅×C₂²⋊C₈	—	3	8	no	—
58	C₅×#97	—	6	4	no	—
59	#1749	—	3	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₅×D₃	—	2	1	no	—
62	C₂²×C₅×D₃	—	2	1	no	—
63	C₂²×C₅×D₃	—	2	2	no	—
64	C₂²×C₅×D₃	C₄²	1	2	yes	—
65	C₂²×C₅×D₃	M₁₆	1	4	yes	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
73	C₂×C₅×Dic₃	—	2	1	no	—
74	C₂×C₅×Dic₃	—	2	1	no	—
75	C₂×C₅×Dic₃	—	2	2	no	—
76	C₂×C₅×Dic₃	—	2	2	no	—
77	C₂×C₅×Dic₃	C₄²	1	2	yes	—
78	C₂×C₅×Dic₃	C₂²⋊C₄	1	4	yes	—
79	C₄×C₅×D₃	—	2	16	no	—
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	C₄×C₅	1	2	yes	—
85	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
87	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₄×Dic₃	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	C₄²×D₃	—	2	4	no	—
92	(C₂×D₃)⋊C₈	C₄×C₅	1	8	yes	complemented
93	#815	—	2	4	no	—
94	C₂⁴×C₅	—	3	1	no	—
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₂²×C₄×C₅	—	3	2	no	—
97	C₂²×C₄×C₅	—	3	4	no	—
98	C₂²×C₄×C₅	—	6	4	no	—
99	C₂²×C₄×C₅	—	6	4	no	—
100	C₂²×C₄×C₅	—	6	4	no	—
101	C₂²×C₄×C₅	—	6	2	no	—
102	C₂²×C₄×C₅	—	6	1	no	—
103	C₂²×C₄×C₅	—	6	1	no	—
104	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂×C₅×C₈	—	2	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	3	4	no	—
109	C₂×C₅×C₈	—	6	2	no	—
110	C₂×C₅×C₈	—	6	2	no	—
111	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	3	4	no	—
114	C₄²×C₅	—	6	4	no	—
115	C₂²×C₄²	—	3	1	no	—
116	C₂×C₂²⋊C₈	—	3	2	no	—
117	C₂×#97	—	3	2	no	—
118	#336	—	3	1	no	—
119	#336	C₅×D₃	1	1	yes	characteristic, complemented
120	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₃×C₅	#117	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
124	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
125	C₂×C₅×D₃	—	2	2	no	—
126	C₂×C₅×D₃	—	2	2	no	—
127	C₂×C₅×D₃	—	2	4	no	—
128	C₂×C₅×D₃	—	2	4	no	—
129	C₂×C₅×D₃	#97	1	4	yes	complemented
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	—	2	4	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	1	yes	characteristic
139	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
140	C₂×C₄×D₃	—	2	4	no	—
141	C₂×C₄×D₃	—	2	4	no	—
142	C₂×C₄×D₃	—	2	2	no	—
143	C₂×C₄×D₃	—	2	1	no	—
144	C₂×C₄×D₃	—	2	1	no	—
145	C₂×C₄×D₃	—	2	4	no	—
146	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
147	C₂×C₃×C₈	—	2	2	no	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	complemented
150	C₃×C₄²	—	2	2	no	—
151	C₄×Dic₃	—	2	4	no	—
152	C₄×Dic₃	C₅×D₄	1	4	yes	—
153	C₂×C₃⋊C₈	—	2	2	no	—
154	C₂×C₃⋊C₈	—	2	2	no	—
155	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
156	C₂³×C₅	—	3	2	no	—
157	C₂³×C₅	—	3	4	no	—
158	C₂³×C₅	—	6	1	no	—
159	C₂³×C₅	—	6	1	no	—
160	C₂³×C₅	—	6	2	no	—
161	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	3	2	no	—
165	C₂×C₄×C₅	—	3	4	no	—
166	C₂×C₄×C₅	—	6	16	no	—
167	C₂×C₄×C₅	—	6	1	no	—
168	C₂×C₄×C₅	—	6	1	no	—
169	C₂×C₄×C₅	—	6	2	no	—
170	C₂×C₄×C₅	—	6	2	no	—
171	C₂×C₄×C₅	—	6	4	no	—
172	C₂×C₄×C₅	—	6	4	no	—
173	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	4	yes	—
174	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
175	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	#237	1	1	yes	characteristic
177	C₂×C₄×C₅	#239	1	1	yes	characteristic
178	C₅×C₈	—	2	4	no	—
179	C₅×C₈	—	6	4	no	—
180	C₂³×C₄	—	3	1	no	—
181	C₂³×C₄	—	3	2	no	—
182	C₂²×C₈	—	3	2	no	—
183	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
184	C₂×C₄²	—	3	1	no	—
185	C₂×C₄²	—	3	2	no	—
186	C₂×C₄²	—	6	4	no	—
187	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
188	C₂²⋊C₈	—	3	8	no	—
189	#97	—	6	4	no	—
190	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
192	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
193	C₂×C₃×C₅	#407	1	2	yes	—
194	C₂×C₃×C₅	#432	1	2	yes	—
195	C₅×D₃	—	2	4	no	—
196	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
197	C₂²×D₃	—	2	1	no	—
198	C₂²×D₃	—	2	1	no	—
199	C₂²×D₃	—	2	2	no	—
200	C₂²×D₃	C₄²×C₅	1	2	yes	—
201	C₂²×D₃	C₅×M₁₆	1	4	yes	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
207	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
208	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
209	C₂×Dic₃	—	2	1	no	—
210	C₂×Dic₃	—	2	1	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	—	2	2	no	—
213	C₂×Dic₃	C₄²×C₅	1	2	yes	—
214	C₂×Dic₃	C₅×C₂²⋊C₄	1	4	yes	—
215	C₄×D₃	—	2	16	no	—
216	C₄×D₃	—	2	4	no	—
217	C₄×D₃	—	2	4	no	—
218	C₃×C₈	—	2	4	no	—
219	C₃⋊C₈	—	2	4	no	—
220	C₂²×C₅	—	3	4	no	—
221	C₂²×C₅	—	6	2	no	—
222	C₂²×C₅	—	6	2	no	—
223	C₂²×C₅	—	6	4	no	—
224	C₂²×C₅	—	6	4	no	—
225	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
226	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
227	C₂²×C₅	#892	1	1	yes	characteristic
228	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
229	C₂²×C₅	D₃×M₁₆	1	2	yes	—
230	C₄×C₅	—	2	4	no	—
231	C₄×C₅	—	6	2	no	—
232	C₄×C₅	—	6	4	no	—
233	C₄×C₅	#67⋊C₄	1	2	yes	—
234	C₄×C₅	#893	1	2	yes	—
235	C₂⁴	—	3	1	no	—
236	C₂²×C₄	—	3	1	no	—
237	C₂²×C₄	—	3	2	no	—
238	C₂²×C₄	—	3	4	no	—
239	C₂²×C₄	—	6	4	no	—
240	C₂²×C₄	—	6	4	no	—
241	C₂²×C₄	—	6	4	no	—
242	C₂²×C₄	—	6	2	no	—
243	C₂²×C₄	—	6	1	no	—
244	C₂²×C₄	—	6	1	no	—
245	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
246	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	3	4	no	—
250	C₂×C₈	—	6	2	no	—
251	C₂×C₈	—	6	2	no	—
252	C₂×C₈	C₄×C₅×D₃	1	4	yes	complemented
253	C₄²	—	2	2	no	—
254	C₄²	—	3	4	no	—
255	C₄²	—	6	4	no	—
256	C₃×C₅	#1749	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
258	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
259	C₂²×C₃	C₅×#117	1	1	yes	characteristic
260	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
261	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
262	C₂×D₃	—	2	2	no	—
263	C₂×D₃	—	2	2	no	—
264	C₂×D₃	—	2	4	no	—
265	C₂×D₃	—	2	4	no	—
266	C₂×D₃	C₅×#97	1	4	yes	complemented
267	C₃×C₄	—	2	4	no	—
268	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
269	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
270	Dic₃	—	2	2	no	—
271	Dic₃	—	2	4	no	—
272	C₂×C₅	—	6	4	no	—
273	C₂×C₅	#4666	1	1	yes	characteristic
274	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
275	C₂×C₅	D₃×#97	1	1	yes	characteristic
276	C₂×C₅	#4726	1	2	yes	—
277	C₂×C₅	#4835	1	2	yes	—
278	C₂³	—	3	2	no	—
279	C₂³	—	3	4	no	—
280	C₂³	—	6	1	no	—
281	C₂³	—	6	1	no	—
282	C₂³	—	6	2	no	—
283	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	3	2	no	—
287	C₂×C₄	—	3	4	no	—
288	C₂×C₄	—	6	16	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	6	2	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	6	4	no	—
295	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	4	yes	—
296	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
297	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
298	C₂×C₄	C₅×#237	1	1	yes	characteristic
299	C₂×C₄	C₅×#239	1	1	yes	characteristic
300	C₈	—	2	4	no	—
301	C₈	—	6	4	no	—
302	C₂×C₃	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
303	C₂×C₃	C₂×C₅×#97	1	1	yes	characteristic
304	C₂×C₃	C₄×C₅×M₁₆	1	1	yes	characteristic
305	C₂×C₃	C₅×#407	1	2	yes	—
306	C₂×C₃	C₅×#432	1	2	yes	—
307	D₃	—	2	4	no	—
308	C₅	#68600	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	3	4	no	—
310	C₂²	—	6	2	no	—
311	C₂²	—	6	2	no	—
312	C₂²	—	6	4	no	—
313	C₂²	—	6	4	no	—
314	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
315	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
316	C₂²	C₅×#892	1	1	yes	characteristic
317	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
318	C₂²	C₅×D₃×M₁₆	1	2	yes	—
319	C₄	—	2	4	no	—
320	C₄	—	6	2	no	—
321	C₄	—	6	4	no	—
322	C₄	C₅×#67⋊C₄	1	2	yes	—
323	C₄	C₅×#893	1	2	yes	—
324	C₃	C₅×#1749	1	1	yes	characteristic, complemented, Sylow
325	C₂	—	6	4	no	—
326	C₂	C₅×#4666	1	1	yes	characteristic
327	C₂	C₄×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
328	C₂	C₅×D₃×#97	1	1	yes	characteristic
329	C₂	C₅×#4726	1	2	yes	—
330	C₂	C₅×#4835	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	C₃×C₅×#336	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₅×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#336	D₃
hypocenter	C₃	C₅×#1749
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₅×#336

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×#815	C₂	semidirect
4. C₃×C₅×#336	C₂	semidirect
5. C₅×#4524	C₂	semidirect
15. C₅×(C₂×D₃)⋊C₈	C₄	semidirect
17. #68600	C₅	direct (N × H)
22. C₅×#336	D₃	semidirect
37. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
43. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
46. C₂×#815	C₂×C₅	semidirect
47. C₃×#336	C₂×C₅	semidirect
48. #4524	C₂×C₅	semidirect
92. (C₂×D₃)⋊C₈	C₄×C₅	semidirect
111. C₂×C₅×C₈	C₄×D₃	semidirect
119. #336	C₅×D₃	semidirect
129. C₂×C₅×D₃	#97	semidirect
149. C₂×C₃×C₈	C₂×C₄×C₅	semidirect
155. C₂×C₃⋊C₈	C₂×C₄×C₅	semidirect
252. C₂×C₈	C₄×C₅×D₃	semidirect
256. C₃×C₅	#1749	semidirect
266. C₂×D₃	C₅×#97	semidirect
308. C₅	#68600	direct (N × H)
324. C₃	C₅×#1749	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄²×C₅×D₃	2
C₂×C₅×(C₂×D₃)⋊C₈	2
C₂×C₅×#815	2
C₃×C₅×#336	2
C₅×#4524	2
C₅×#1749	3
#68600	5

Subgroups — C5×#68600 #14295037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68600 #14295037