Subgroups — C₅×#68918 #14304764

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#825	C₂	1	1	yes	maximal, characteristic
2	C₅×#5129	C₂	1	1	yes	maximal, characteristic
3	C₅×#5119	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5113	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4555	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1937	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₅×#825	—	2	2	no	—
17	C₅×#821	C₄	1	2	yes	complemented
18	C₅×(C₂×D₃)⋊C₈	—	2	2	no	—
19	C₅×#816	C₄	1	2	yes	complemented
20	C₅×Dic₃⋊C₈	—	2	2	no	—
21	C₅×C₈×Dic₃	—	2	2	no	—
22	#68918	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#424	—	3	1	no	—
24	C₂×C₅×#104	—	3	1	no	—
25	C₅×#416	—	3	1	no	—
26	C₅×#408	—	3	1	no	—
27	C₂×C₄×C₅×C₈	—	3	1	no	—
28	C₅×#341	D₃	1	1	yes	characteristic, complemented
29	C₅×#339	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₅×#67	—	2	1	no	—
32	C₂²×C₅×Dic₃	—	2	1	no	—
33	C₅×#235	—	2	1	no	—
34	C₅×#235	—	2	1	no	—
35	C₅×#235	—	2	1	no	—
36	C₅×#235	—	2	1	no	—
37	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	—	2	1	no	—
39	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
40	C₃×C₅×M₁₆	—	2	1	no	—
41	C₃×C₅×M₁₆	—	2	1	no	—
42	C₃×C₅×M₁₆	—	2	1	no	—
43	C₃×C₅×M₁₆	—	2	1	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₂×C₃×C₅×C₈	—	2	1	no	—
46	C₂×C₃×C₅×C₈	—	2	1	no	—
47	C₂×C₃×C₅×C₈	—	2	1	no	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
49	C₅×#217	—	2	1	no	—
50	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
51	C₄×C₅×Dic₃	—	2	1	no	—
52	C₄×C₅×Dic₃	—	2	1	no	—
53	C₄×C₅×Dic₃	D₄	1	2	yes	—
54	C₂×C₅×C₃⋊C₈	—	2	1	no	—
55	C₂×C₅×C₃⋊C₈	—	2	1	no	—
56	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
57	C₂×#825	C₂×C₅	1	1	yes	characteristic
58	#5129	C₂×C₅	1	1	yes	characteristic
59	#5119	C₂×C₅	1	1	yes	characteristic, complemented
60	#5113	C₂×C₅	1	1	yes	characteristic
61	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
62	C₃×#341	C₂×C₅	1	1	yes	characteristic, complemented
63	#4555	C₂×C₅	1	1	yes	characteristic, complemented
64	C₂×C₅×#41	—	3	1	no	—
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₅×C₈	—	3	1	no	—
68	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
69	C₅×#117	—	3	1	no	—
70	C₂×C₄²×C₅	—	3	1	no	—
71	C₅×C₄⋊C₈	—	6	2	no	—
72	C₅×#104	—	6	2	no	—
73	C₅×#103	—	3	2	no	—
74	C₅×#102	—	3	2	no	—
75	C₅×C₂²⋊C₈	—	6	2	no	—
76	C₄×C₅×C₈	—	6	2	no	—
77	#1937	—	3	1	no	Sylow
78	C₂³×C₃×C₅	#41	1	1	yes	characteristic
79	C₂²×C₅×D₃	—	2	1	no	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₅×#67	—	4	1	no	—
87	C₅×#67	—	4	1	no	—
88	C₂×C₅×Dic₃	—	2	4	no	—
89	C₂×C₅×Dic₃	—	2	2	no	—
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₅×D₁₂	—	2	1	no	—
93	C₅×D₁₂	—	2	1	no	—
94	C₄×C₅×D₃	—	4	1	no	—
95	C₄×C₅×D₃	—	4	1	no	—
96	C₅×Dic₆	—	2	1	no	—
97	C₅×Dic₆	—	2	1	no	—
98	C₃×C₅×C₈	—	2	2	no	—
99	C₃×C₅×C₈	—	4	1	no	—
100	C₃×C₅×C₈	—	4	1	no	—
101	C₅×C₃⋊C₈	—	2	2	no	—
102	C₂×#235	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
105	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
106	#926	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
108	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
109	#825	—	2	2	no	—
110	#821	C₄×C₅	1	2	yes	complemented
111	(C₂×D₃)⋊C₈	—	2	2	no	—
112	#816	C₄×C₅	1	2	yes	complemented
113	Dic₃⋊C₈	—	2	2	no	—
114	C₈×Dic₃	—	2	2	no	—
115	C₅×#41	—	6	1	no	—
116	C₅×#41	—	6	1	no	—
117	C₅×#41	—	6	1	no	—
118	C₅×#41	—	6	1	no	—
119	C₂×C₅×Q₈	—	3	1	no	—
120	C₂×C₅×D₄	—	3	1	no	—
121	C₂×C₅×D₄	—	6	1	no	—
122	C₂²×C₄×C₅	—	6	1	no	—
123	C₂²×C₄×C₅	—	6	1	no	—
124	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₅×M₁₆	—	2	1	no	—
126	C₅×M₁₆	—	2	1	no	—
127	C₅×M₁₆	—	2	1	no	—
128	C₅×M₁₆	—	2	1	no	—
129	C₂×C₅×C₈	—	2	1	no	—
130	C₂×C₅×C₈	—	2	1	no	—
131	C₂×C₅×C₈	—	2	1	no	—
132	C₂×C₅×C₈	—	2	1	no	—
133	C₂×C₅×C₈	—	3	2	no	—
134	C₂×C₅×C₈	—	6	1	no	—
135	C₂×C₅×C₈	—	6	1	no	—
136	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
137	C₅×C₄⋊C₄	—	3	2	no	—
138	C₅×C₂²⋊C₄	—	6	1	no	—
139	C₄²×C₅	—	3	2	no	—
140	C₄²×C₅	—	6	1	no	—
141	C₄²×C₅	—	6	1	no	—
142	#424	—	3	1	no	—
143	C₂×#104	—	3	1	no	—
144	#416	—	3	1	no	—
145	#408	—	3	1	no	—
146	C₂×C₄×C₈	—	3	1	no	—
147	#341	C₅×D₃	1	1	yes	characteristic, complemented
148	#339	—	3	1	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
151	C₂²×C₃×C₅	#123	1	1	yes	characteristic
152	C₂²×C₃×C₅	#126	1	1	yes	characteristic
153	C₂×C₅×D₃	—	4	1	no	—
154	C₂×C₅×D₃	—	4	2	no	—
155	C₃×C₄×C₅	#117	1	1	yes	characteristic
156	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
157	C₃×C₄×C₅	#121	1	1	yes	characteristic
158	C₃×C₄×C₅	#124	1	1	yes	characteristic
159	C₅×Dic₃	—	2	4	no	—
160	C₅×Dic₃	—	4	1	no	—
161	C₅×Dic₃	—	4	2	no	—
162	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
163	C₂×#67	—	2	1	no	—
164	C₂²×Dic₃	—	2	1	no	—
165	#235	—	2	1	no	—
166	#235	—	2	1	no	—
167	#235	—	2	1	no	—
168	#235	—	2	1	no	—
169	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂×C₄×D₃	—	2	1	no	—
171	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
172	C₃×M₁₆	—	2	1	no	—
173	C₃×M₁₆	—	2	1	no	—
174	C₃×M₁₆	—	2	1	no	—
175	C₃×M₁₆	—	2	1	no	—
176	C₂×C₃×C₈	—	2	1	no	—
177	C₂×C₃×C₈	—	2	1	no	—
178	C₂×C₃×C₈	—	2	1	no	—
179	C₂×C₃×C₈	—	2	1	no	—
180	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
181	#217	—	2	1	no	—
182	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
183	C₄×Dic₃	—	2	1	no	—
184	C₄×Dic₃	—	2	1	no	—
185	C₄×Dic₃	C₅×D₄	1	2	yes	—
186	C₂×C₃⋊C₈	—	2	1	no	—
187	C₂×C₃⋊C₈	—	2	1	no	—
188	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
189	C₂³×C₅	—	6	1	no	—
190	C₂³×C₅	#239	1	1	yes	characteristic
191	C₅×Q₈	—	6	1	no	—
192	C₅×Q₈	—	6	1	no	—
193	C₅×D₄	—	6	1	no	—
194	C₅×D₄	—	6	1	no	—
195	C₅×D₄	—	12	1	no	—
196	C₅×D₄	—	12	1	no	—
197	C₂×C₄×C₅	—	6	4	no	—
198	C₂×C₄×C₅	—	6	2	no	—
199	C₂×C₄×C₅	—	6	2	no	—
200	C₂×C₄×C₅	—	6	1	no	—
201	C₂×C₄×C₅	—	12	1	no	—
202	C₂×C₄×C₅	—	12	1	no	—
203	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
204	C₂×C₄×C₅	#235	1	1	yes	characteristic
205	C₂×C₄×C₅	#235	1	1	yes	characteristic
206	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
207	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
208	C₂×C₄×C₅	#237	1	1	yes	characteristic
209	C₅×C₈	—	2	2	no	—
210	C₅×C₈	—	4	1	no	—
211	C₅×C₈	—	4	1	no	—
212	C₅×C₈	—	6	2	no	—
213	C₂×#41	—	3	1	no	—
214	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
215	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
216	C₂²×C₈	—	3	1	no	—
217	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
218	#117	—	3	1	no	—
219	C₂×C₄²	—	3	1	no	—
220	C₄⋊C₈	—	6	2	no	—
221	#104	—	6	2	no	—
222	#103	—	3	2	no	—
223	#102	—	3	2	no	—
224	C₂²⋊C₈	—	6	2	no	—
225	C₄×C₈	—	6	2	no	—
226	C₂×C₃×C₅	—	2	1	no	—
227	C₂×C₃×C₅	—	2	1	no	—
228	C₂×C₃×C₅	#388	1	1	yes	characteristic
229	C₂×C₃×C₅	#443	1	2	yes	—
230	C₅×D₃	—	8	1	no	—
231	C₂³×C₃	C₅×#41	1	1	yes	characteristic
232	C₂²×D₃	—	2	1	no	—
233	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
236	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
237	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
238	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
239	#67	—	4	1	no	—
240	#67	—	4	1	no	—
241	C₂×Dic₃	—	2	4	no	—
242	C₂×Dic₃	—	2	2	no	—
243	C₂×Dic₃	—	2	2	no	—
244	C₂×Dic₃	—	2	1	no	—
245	D₁₂	—	2	1	no	—
246	D₁₂	—	2	1	no	—
247	C₄×D₃	—	4	1	no	—
248	C₄×D₃	—	4	1	no	—
249	Dic₆	—	2	1	no	—
250	Dic₆	—	2	1	no	—
251	C₃×C₈	—	2	2	no	—
252	C₃×C₈	—	4	1	no	—
253	C₃×C₈	—	4	1	no	—
254	C₃⋊C₈	—	2	2	no	—
255	C₂²×C₅	—	2	2	no	—
256	C₂²×C₅	—	12	2	no	—
257	C₂²×C₅	—	12	1	no	—
258	C₂²×C₅	#893	1	1	yes	characteristic
259	C₂²×C₅	#894	1	1	yes	characteristic
260	C₂²×C₅	#898	1	1	yes	characteristic
261	C₄×C₅	—	6	4	no	—
262	C₄×C₅	—	12	2	no	—
263	C₄×C₅	—	12	1	no	—
264	C₄×C₅	#872	1	1	yes	characteristic
265	C₄×C₅	#67⋊C₄	1	1	yes	characteristic
266	C₄×C₅	#884	1	1	yes	characteristic
267	C₄×C₅	#885	1	1	yes	characteristic
268	#41	—	6	1	no	—
269	#41	—	6	1	no	—
270	#41	—	6	1	no	—
271	#41	—	6	1	no	—
272	C₂×Q₈	—	3	1	no	—
273	C₂×D₄	—	3	1	no	—
274	C₂×D₄	—	6	1	no	—
275	C₂²×C₄	—	6	1	no	—
276	C₂²×C₄	—	6	1	no	—
277	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
278	M₁₆	—	2	1	no	—
279	M₁₆	—	2	1	no	—
280	M₁₆	—	2	1	no	—
281	M₁₆	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	3	2	no	—
287	C₂×C₈	—	6	1	no	—
288	C₂×C₈	—	6	1	no	—
289	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
290	C₄⋊C₄	—	3	2	no	—
291	C₂²⋊C₄	—	6	1	no	—
292	C₄²	—	3	2	no	—
293	C₄²	—	6	1	no	—
294	C₄²	—	6	1	no	—
295	C₃×C₅	#1937	1	1	yes	characteristic, complemented, Hall
296	C₂²×C₃	—	2	2	no	—
297	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
298	C₂²×C₃	C₅×#123	1	1	yes	characteristic
299	C₂²×C₃	C₅×#126	1	1	yes	characteristic
300	C₂×D₃	—	4	1	no	—
301	C₂×D₃	—	4	2	no	—
302	C₃×C₄	C₅×#117	1	1	yes	characteristic
303	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
304	C₃×C₄	C₅×#121	1	1	yes	characteristic
305	C₃×C₄	C₅×#124	1	1	yes	characteristic
306	Dic₃	—	2	4	no	—
307	Dic₃	—	4	1	no	—
308	Dic₃	—	4	2	no	—
309	C₂×C₅	—	2	1	no	—
310	C₂×C₅	—	2	1	no	—
311	C₂×C₅	—	24	1	no	—
312	C₂×C₅	#4669	1	1	yes	characteristic
313	C₂×C₅	#4895	1	2	yes	—
314	C₂³	—	6	1	no	—
315	C₂³	C₅×#239	1	1	yes	characteristic
316	Q₈	—	6	1	no	—
317	Q₈	—	6	1	no	—
318	D₄	—	6	1	no	—
319	D₄	—	6	1	no	—
320	D₄	—	12	1	no	—
321	D₄	—	12	1	no	—
322	C₂×C₄	—	6	4	no	—
323	C₂×C₄	—	6	2	no	—
324	C₂×C₄	—	6	2	no	—
325	C₂×C₄	—	6	1	no	—
326	C₂×C₄	—	12	1	no	—
327	C₂×C₄	—	12	1	no	—
328	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
329	C₂×C₄	C₅×#235	1	1	yes	characteristic
330	C₂×C₄	C₅×#235	1	1	yes	characteristic
331	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
332	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
333	C₂×C₄	C₅×#237	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	1	no	—
337	C₈	—	6	2	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	C₅×#388	1	1	yes	characteristic
341	C₂×C₃	C₅×#443	1	2	yes	—
342	D₃	—	8	1	no	—
343	C₅	#68918	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	12	1	no	—
346	C₂²	—	12	2	no	—
347	C₂²	C₅×#893	1	1	yes	characteristic
348	C₂²	C₅×#894	1	1	yes	characteristic
349	C₂²	C₅×#898	1	1	yes	characteristic
350	C₄	—	6	4	no	—
351	C₄	—	12	1	no	—
352	C₄	—	12	2	no	—
353	C₄	C₅×#872	1	1	yes	characteristic
354	C₄	C₅×#67⋊C₄	1	1	yes	characteristic
355	C₄	C₅×#884	1	1	yes	characteristic
356	C₄	C₅×#885	1	1	yes	characteristic
357	C₃	C₅×#1937	1	1	yes	characteristic, complemented, Sylow
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	24	1	no	—
361	C₂	C₅×#4669	1	1	yes	characteristic
362	C₂	C₅×#4895	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#341	D₃
hypocenter	C₃	C₅×#1937
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#341

Complements

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

N	H ≅ G/N	type
3. C₅×#5119	C₂	semidirect
5. C₂×C₅×C₈×Dic₃	C₂	semidirect
6. C₃×C₅×#341	C₂	semidirect
7. C₅×#4555	C₂	semidirect
17. C₅×#821	C₄	semidirect
19. C₅×#816	C₄	semidirect
22. #68918	C₅	direct (N × H)
28. C₅×#341	D₃	semidirect
59. #5119	C₂×C₅	semidirect
61. C₂×C₈×Dic₃	C₂×C₅	semidirect
62. C₃×#341	C₂×C₅	semidirect
63. #4555	C₂×C₅	semidirect
110. #821	C₄×C₅	semidirect
112. #816	C₄×C₅	semidirect
147. #341	C₅×D₃	semidirect
295. C₃×C₅	#1937	semidirect
343. C₅	#68918	direct (N × H)
357. C₃	C₅×#1937	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#825	2
C₅×#5129	2
C₅×#5119	2
C₅×#5113	2
C₂×C₅×C₈×Dic₃	2
C₃×C₅×#341	2
C₅×#4555	2
C₅×#1937	3
#68918	5

Subgroups — C5×#68918 #14304764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68918 #14304764