Subgroups — C₅×#68994 #14297556

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#823	C₂	1	2	yes	maximal
3	C₅×#5118	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#343	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4530	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1800	—	3	1	no	maximal, Hall
7	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅×D₃	C₄	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	2	yes	—
14	C₅×#893	—	2	2	no	—
15	C₅×#893	—	2	2	no	—
16	C₅×#823	—	2	4	no	—
17	C₅×Dic₃⋊C₈	—	2	2	no	—
18	C₅×Dic₃⋊C₈	—	2	2	no	—
19	#68994	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₄×C₅×D₄	—	3	1	no	—
21	C₅×#423	—	3	2	no	—
22	C₂×C₅×#100	—	3	2	no	—
23	C₅×#343	—	3	1	no	—
24	C₅×#343	D₃	1	1	yes	characteristic, complemented
25	C₂³×C₅×D₃	C₂×C₄	1	2	yes	—
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	1	yes	characteristic
29	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₅×D₁₂	—	2	2	no	—
31	C₂×C₅×D₁₂	—	2	2	no	—
32	C₂×C₅×D₁₂	D₄	1	2	yes	—
33	C₂×C₅×D₁₂	Q₈	1	2	yes	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₄×C₅×D₃	—	2	2	no	—
36	C₂×C₄×C₅×D₃	—	4	4	no	—
37	C₃×C₅×M₁₆	—	2	2	no	—
38	C₃×C₅×M₁₆	—	2	2	no	—
39	C₂×C₃×C₅×C₈	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	—	2	2	no	—
41	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
42	C₄×C₅×Dic₃	D₄	1	2	yes	—
43	C₄×C₅×Dic₃	D₄	1	2	yes	—
44	C₅×#208	—	2	2	no	—
45	C₅×#208	—	2	2	no	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×#893	C₂×C₅	1	1	yes	characteristic
48	C₂×#823	C₂×C₅	1	2	yes	—
49	#5118	C₂×C₅	1	2	yes	complemented
50	C₃×#343	C₂×C₅	1	1	yes	characteristic, complemented
51	#4530	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂²×C₅×D₄	—	3	1	no	—
53	C₂³×C₄×C₅	—	3	2	no	—
54	C₂×C₅×M₁₆	—	3	2	no	—
55	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
56	C₄×C₅×D₄	—	6	2	no	—
57	C₄×C₅×D₄	—	6	2	no	—
58	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
59	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
60	C₂×C₄²×C₅	—	3	1	no	—
61	C₅×C₄⋊C₈	—	6	2	no	—
62	C₅×C₄⋊C₈	—	6	2	no	—
63	C₅×#100	—	6	4	no	—
64	#1800	—	3	1	no	Sylow
65	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₅×D₃	—	2	2	no	—
67	C₂²×C₅×D₃	—	2	2	no	—
68	C₂²×C₅×D₃	—	4	2	no	—
69	C₂²×C₅×D₃	—	4	2	no	—
70	C₂²×C₅×D₃	C₄⋊C₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	2	yes	—
77	C₂×C₅×Dic₃	—	2	2	no	—
78	C₂×C₅×Dic₃	—	4	1	no	—
79	C₂×C₅×Dic₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₅×D₁₂	—	2	4	no	—
83	C₅×D₁₂	—	2	4	no	—
84	C₄×C₅×D₃	—	4	4	no	—
85	C₄×C₅×D₃	—	4	4	no	—
86	C₃×C₅×C₈	—	4	2	no	—
87	C₅×C₃⋊C₈	—	4	2	no	—
88	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₄×D₃	C₄×C₅	1	2	yes	—
90	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
91	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
92	C₂×(C₂×D₃)⋊C₄	C₄×C₅	1	2	yes	—
93	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
94	C₂×#208	C₂²×C₅	1	2	yes	—
95	#893	—	2	2	no	—
96	#893	—	2	2	no	—
97	#823	—	2	4	no	—
98	Dic₃⋊C₈	—	2	2	no	—
99	Dic₃⋊C₈	—	2	2	no	—
100	C₂⁴×C₅	—	3	2	no	—
101	C₂×C₅×D₄	—	3	2	no	—
102	C₂×C₅×D₄	—	3	2	no	—
103	C₂×C₅×D₄	—	6	2	no	—
104	C₂×C₅×D₄	—	6	2	no	—
105	C₂²×C₄×C₅	—	3	1	no	—
106	C₂²×C₄×C₅	—	3	1	no	—
107	C₂²×C₄×C₅	—	6	2	no	—
108	C₂²×C₄×C₅	—	6	2	no	—
109	C₂²×C₄×C₅	—	12	4	no	—
110	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
111	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
112	C₅×M₁₆	—	2	2	no	—
113	C₅×M₁₆	—	2	2	no	—
114	C₅×M₁₆	—	6	2	no	—
115	C₅×M₁₆	—	6	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	6	2	no	—
118	C₅×C₄⋊C₄	—	2	2	no	—
119	C₅×C₂²⋊C₄	—	6	4	no	—
120	C₄²×C₅	—	3	2	no	—
121	C₄²×C₅	—	3	2	no	—
122	C₂×C₄×D₄	—	3	1	no	—
123	#423	—	3	2	no	—
124	C₂×#100	—	3	2	no	—
125	#343	—	3	1	no	—
126	#343	C₅×D₃	1	1	yes	characteristic, complemented
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₃×C₅	#117	1	1	yes	characteristic
132	C₂×C₅×D₃	—	2	4	no	—
133	C₂×C₅×D₃	—	4	4	no	—
134	C₂×C₅×D₃	—	4	2	no	—
135	C₂×C₅×D₃	—	4	2	no	—
136	C₂×C₅×D₃	—	4	4	no	—
137	C₃×C₄×C₅	—	4	2	no	—
138	C₃×C₄×C₅	#121	1	2	yes	—
139	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
140	C₅×Dic₃	—	2	1	no	—
141	C₅×Dic₃	—	2	2	no	—
142	C₅×Dic₃	#100	1	2	yes	complemented
143	C₂³×D₃	C₂×C₄×C₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
145	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂×D₁₂	—	2	2	no	—
149	C₂×D₁₂	—	2	2	no	—
150	C₂×D₁₂	C₅×D₄	1	2	yes	—
151	C₂×D₁₂	C₅×Q₈	1	2	yes	—
152	C₂×C₄×D₃	—	2	2	no	—
153	C₂×C₄×D₃	—	2	2	no	—
154	C₂×C₄×D₃	—	4	4	no	—
155	C₃×M₁₆	—	2	2	no	—
156	C₃×M₁₆	—	2	2	no	—
157	C₂×C₃×C₈	—	2	2	no	—
158	C₃×C₄⋊C₄	—	2	2	no	—
159	(C₂×D₃)⋊C₄	—	2	4	no	—
160	C₄×Dic₃	C₅×D₄	1	2	yes	—
161	C₄×Dic₃	C₅×D₄	1	2	yes	—
162	#208	—	2	2	no	—
163	#208	—	2	2	no	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂³×C₅	—	3	4	no	—
166	C₂³×C₅	—	6	2	no	—
167	C₂³×C₅	—	6	2	no	—
168	C₂³×C₅	—	12	2	no	—
169	C₂³×C₅	—	12	2	no	—
170	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
171	C₅×D₄	—	6	4	no	—
172	C₅×D₄	—	6	4	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	3	1	no	—
175	C₂×C₄×C₅	—	3	2	no	—
176	C₂×C₄×C₅	—	3	1	no	—
177	C₂×C₄×C₅	—	4	2	no	—
178	C₂×C₄×C₅	—	6	2	no	—
179	C₂×C₄×C₅	—	12	4	no	—
180	C₂×C₄×C₅	—	12	4	no	—
181	C₂×C₄×C₅	—	12	1	no	—
182	C₂×C₄×C₅	#235	1	2	yes	—
183	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
185	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
186	C₅×C₈	—	4	2	no	—
187	C₅×C₈	—	12	2	no	—
188	C₂²×D₄	—	3	1	no	—
189	C₂³×C₄	—	3	2	no	—
190	C₂×M₁₆	—	3	2	no	—
191	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
192	C₄×D₄	—	6	2	no	—
193	C₄×D₄	—	6	2	no	—
194	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
195	C₂×C₂²⋊C₄	—	3	2	no	—
196	C₂×C₄²	—	3	1	no	—
197	C₄⋊C₈	—	6	2	no	—
198	C₄⋊C₈	—	6	2	no	—
199	#100	—	6	4	no	—
200	C₂×C₃×C₅	—	2	1	no	—
201	C₂×C₃×C₅	—	2	1	no	—
202	C₂×C₃×C₅	#380	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
204	C₂×C₃×C₅	#413	1	1	yes	characteristic
205	C₅×D₃	—	4	4	no	—
206	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
207	C₂²×D₃	—	2	2	no	—
208	C₂²×D₃	—	2	2	no	—
209	C₂²×D₃	—	4	2	no	—
210	C₂²×D₃	—	4	2	no	—
211	C₂²×D₃	C₅×C₄⋊C₄	1	4	yes	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×#41	1	2	yes	—
218	C₂×Dic₃	—	2	2	no	—
219	C₂×Dic₃	—	4	1	no	—
220	C₂×Dic₃	C₅×C₂²⋊C₄	1	2	yes	—
221	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	D₁₂	—	2	4	no	—
224	D₁₂	—	2	4	no	—
225	C₄×D₃	—	4	4	no	—
226	C₄×D₃	—	4	4	no	—
227	C₃×C₈	—	4	2	no	—
228	C₃⋊C₈	—	4	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	6	4	no	—
232	C₂²×C₅	—	12	4	no	—
233	C₂²×C₅	—	12	4	no	—
234	C₂²×C₅	—	12	2	no	—
235	C₂²×C₅	—	12	2	no	—
236	C₂²×C₅	#872	1	1	yes	characteristic
237	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
239	C₄×C₅	—	3	2	no	—
240	C₄×C₅	—	4	2	no	—
241	C₄×C₅	—	6	1	no	—
242	C₄×C₅	—	6	2	no	—
243	C₄×C₅	#884	1	2	yes	—
244	C₄×C₅	#896	1	2	yes	—
245	C₂⁴	—	3	2	no	—
246	C₂×D₄	—	3	2	no	—
247	C₂×D₄	—	3	2	no	—
248	C₂×D₄	—	6	2	no	—
249	C₂×D₄	—	6	2	no	—
250	C₂²×C₄	—	3	1	no	—
251	C₂²×C₄	—	3	1	no	—
252	C₂²×C₄	—	6	2	no	—
253	C₂²×C₄	—	6	2	no	—
254	C₂²×C₄	—	12	4	no	—
255	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
256	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
257	M₁₆	—	2	2	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	6	2	no	—
260	M₁₆	—	6	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	6	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	6	4	no	—
265	C₄²	—	3	2	no	—
266	C₄²	—	3	2	no	—
267	C₃×C₅	#1800	1	1	yes	characteristic, complemented, Hall
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
272	C₂²×C₃	C₅×#117	1	1	yes	characteristic
273	C₂×D₃	—	2	4	no	—
274	C₂×D₃	—	4	4	no	—
275	C₂×D₃	—	4	2	no	—
276	C₂×D₃	—	4	2	no	—
277	C₂×D₃	—	4	4	no	—
278	C₃×C₄	—	4	2	no	—
279	C₃×C₄	C₅×#121	1	2	yes	—
280	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
281	Dic₃	—	2	1	no	—
282	Dic₃	—	2	2	no	—
283	Dic₃	C₅×#100	1	2	yes	complemented
284	C₂×C₅	—	2	1	no	—
285	C₂×C₅	—	2	1	no	—
286	C₂×C₅	—	12	4	no	—
287	C₂×C₅	#4667	1	1	yes	characteristic
288	C₂×C₅	D₃×#100	1	1	yes	characteristic
289	C₂×C₅	#4751	1	1	yes	characteristic
290	C₂³	—	3	4	no	—
291	C₂³	—	6	2	no	—
292	C₂³	—	6	2	no	—
293	C₂³	—	12	2	no	—
294	C₂³	—	12	2	no	—
295	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
296	D₄	—	6	4	no	—
297	D₄	—	6	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	3	1	no	—
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	3	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	12	4	no	—
305	C₂×C₄	—	12	4	no	—
306	C₂×C₄	—	12	1	no	—
307	C₂×C₄	C₅×#235	1	2	yes	—
308	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
309	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
310	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
311	C₈	—	4	2	no	—
312	C₈	—	12	2	no	—
313	C₂×C₃	—	2	1	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	C₅×#380	1	1	yes	characteristic
316	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
317	C₂×C₃	C₅×#413	1	1	yes	characteristic
318	D₃	—	4	4	no	—
319	C₅	#68994	1	1	yes	characteristic, direct product factor, Sylow
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	6	4	no	—
323	C₂²	—	12	4	no	—
324	C₂²	—	12	4	no	—
325	C₂²	—	12	2	no	—
326	C₂²	—	12	2	no	—
327	C₂²	C₅×#872	1	1	yes	characteristic
328	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
329	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
330	C₄	—	3	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	6	1	no	—
333	C₄	—	6	2	no	—
334	C₄	C₅×#884	1	2	yes	—
335	C₄	C₅×#896	1	2	yes	—
336	C₃	C₅×#1800	1	1	yes	characteristic, complemented, Sylow
337	C₂	—	2	1	no	—
338	C₂	—	2	1	no	—
339	C₂	—	12	4	no	—
340	C₂	C₅×#4667	1	1	yes	characteristic
341	C₂	C₅×D₃×#100	1	1	yes	characteristic
342	C₂	C₅×#4751	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#343	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#343	D₃
hypocenter	C₃	C₅×#1800
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#343

Complements

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

N	H ≅ G/N	type
3. C₅×#5118	C₂	semidirect
4. C₃×C₅×#343	C₂	semidirect
5. C₅×#4530	C₂	semidirect
19. #68994	C₅	direct (N × H)
24. C₅×#343	D₃	semidirect
49. #5118	C₂×C₅	semidirect
50. C₃×#343	C₂×C₅	semidirect
51. #4530	C₂×C₅	semidirect
126. #343	C₅×D₃	semidirect
142. C₅×Dic₃	#100	semidirect
267. C₃×C₅	#1800	semidirect
283. Dic₃	C₅×#100	semidirect
319. C₅	#68994	direct (N × H)
336. C₃	C₅×#1800	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#893	2
C₂×C₅×#823	2
C₅×#5118	2
C₃×C₅×#343	2
C₅×#4530	2
C₅×#1800	3
#68994	5

Subgroups — C5×#68994 #14297556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68994 #14297556