Subgroups — C₅×#79258 #14418116

Order 1920.

916 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	C₅×#5748	C₂	1	1	yes	maximal, characteristic
2	C₅×#5745	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#430	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#825	C₂	1	2	yes	maximal, complemented
5	C₅×Dic₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4895	C₂	1	4	yes	maximal, complemented
7	C₅×#4894	C₂	1	4	yes	maximal, complemented
8	C₅×#2956	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₅×#909	C₄	1	4	yes	complemented
15	C₅×#908	C₄	1	4	yes	complemented
16	C₅×#907	C₂²	1	8	yes	complemented
17	C₅×D₃×M₁₆	—	2	4	no	—
18	C₅×#904	C₂²	1	4	yes	—
19	C₅×#901	—	2	4	no	—
20	C₃×C₅×#108	C₂²	1	4	yes	complemented
21	C₅×#825	C₂²	1	4	yes	complemented
22	C₅×#825	C₂²	1	4	yes	complemented
23	C₅×#815	C₂²	1	2	yes	complemented
24	C₅×C₈×Dic₃	C₂²	1	2	yes	complemented
25	#79258	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₅×#575	—	3	1	no	—
27	C₅×#568	—	3	2	no	—
28	C₅×#444	—	3	4	no	—
29	C₅×#443	—	3	4	no	—
30	C₅×#430	D₃	1	1	yes	characteristic, complemented
31	C₂×C₅×#104	—	3	2	no	—
32	C₄×C₅×M₁₆	—	3	1	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×#67	—	2	2	no	—
35	C₂²×C₅×Dic₃	—	2	1	no	—
36	C₅×#235	—	2	2	no	—
37	C₅×#235	—	2	2	no	—
38	C₅×#235	C₂×C₄	1	4	yes	—
39	C₅×#235	C₂³	1	4	yes	—
40	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₅×D₃	—	2	2	no	—
42	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
43	C₃×C₅×M₁₆	—	2	2	no	—
44	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
45	C₃×C₅×M₁₆	C₂³	1	4	yes	—
46	C₂×C₃×C₅×C₈	—	2	2	no	—
47	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
48	C₄×C₅×Dic₃	—	2	1	no	—
49	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
50	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
51	C₅×#208	D₄	1	4	yes	—
52	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
53	C₅×Dic₁₂	C₂×C₄	1	4	yes	complemented
54	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
55	C₅×#204	C₂×C₄	1	8	yes	complemented
56	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
57	C₅×D₃×C₈	—	2	8	no	—
58	#5748	C₂×C₅	1	1	yes	characteristic
59	#5745	C₂×C₅	1	2	yes	complemented
60	C₃×#430	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×#825	C₂×C₅	1	2	yes	complemented
62	Dic₃×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
63	#4895	C₂×C₅	1	4	yes	complemented
64	#4894	C₂×C₅	1	4	yes	complemented
65	C₂×C₅×#41	—	3	2	no	—
66	C₅×#137	—	3	4	no	—
67	C₅×AGL(1,ℤ/8)	—	3	4	no	—
68	C₅×#135	—	3	4	no	—
69	C₅×#131	—	3	8	no	—
70	C₅×#131	—	6	4	no	—
71	C₂×C₅×M₁₆	—	3	1	no	—
72	C₂×C₅×M₁₆	—	6	4	no	—
73	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
74	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
75	C₂×C₄²×C₅	—	3	1	no	—
76	C₅×#108	C₂×D₃	1	4	yes	complemented
77	C₅×#104	—	3	4	no	—
78	C₅×#104	—	3	4	no	—
79	C₅×#97	—	3	2	no	—
80	C₄×C₅×C₈	—	3	2	no	—
81	#2956	—	3	1	no	Sylow
82	C₂³×C₃×C₅	#41	1	1	yes	characteristic
83	C₂²×C₅×D₃	—	2	2	no	—
84	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₅×#67	—	2	4	no	—
89	C₅×#67	—	2	4	no	—
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₂×C₅×Dic₃	—	2	1	no	—
92	C₂×C₅×Dic₃	—	2	1	no	—
93	C₂×C₅×Dic₃	—	4	2	no	—
94	C₅×D₁₂	—	2	2	no	—
95	C₅×D₁₂	C₂²×C₄	1	4	yes	—
96	C₄×C₅×D₃	—	2	4	no	—
97	C₄×C₅×D₃	—	2	4	no	—
98	C₅×Dic₆	—	2	2	no	—
99	C₅×Dic₆	C₂²×C₄	1	4	yes	—
100	C₃×C₅×C₈	—	2	4	no	—
101	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
102	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
103	C₂×#235	C₂²×C₅	1	2	yes	—
104	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
105	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
106	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
107	C₂×#208	C₂²×C₅	1	1	yes	characteristic
108	#909	C₄×C₅	1	4	yes	complemented
109	#908	C₄×C₅	1	4	yes	complemented
110	#907	C₂²×C₅	1	8	yes	complemented
111	D₃×M₁₆	—	2	4	no	—
112	#904	C₂²×C₅	1	4	yes	—
113	#901	—	2	4	no	—
114	C₃×#108	C₂²×C₅	1	4	yes	complemented
115	#825	C₂²×C₅	1	4	yes	complemented
116	#825	C₂²×C₅	1	4	yes	complemented
117	#815	C₂²×C₅	1	2	yes	complemented
118	C₈×Dic₃	C₂²×C₅	1	2	yes	complemented
119	C₅×#41	—	3	4	no	—
120	C₅×#41	—	3	4	no	—
121	C₅×#41	—	6	2	no	—
122	C₅×#41	—	6	2	no	—
123	C₂×C₅×Q₈	—	3	2	no	—
124	C₂×C₅×D₄	—	3	2	no	—
125	C₂×C₅×D₄	—	6	2	no	—
126	C₂²×C₄×C₅	—	6	2	no	—
127	C₂²×C₄×C₅	—	6	1	no	—
128	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₅×Q₁₆	—	3	4	no	—
130	C₅×SD₁₆	—	3	8	no	—
131	C₅×D₈	—	3	4	no	—
132	C₅×M₁₆	—	2	2	no	—
133	C₅×M₁₆	—	3	4	no	—
134	C₅×M₁₆	—	6	8	no	—
135	C₅×M₁₆	C₄×D₃	1	4	yes	—
136	C₅×M₁₆	C₂²×D₃	1	4	yes	—
137	C₂×C₅×C₈	—	2	2	no	—
138	C₂×C₅×C₈	—	3	2	no	—
139	C₂×C₅×C₈	—	6	8	no	—
140	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
141	C₄²×C₅	—	3	1	no	—
142	C₄²×C₅	—	3	1	no	—
143	C₄²×C₅	—	6	1	no	—
144	#575	—	3	1	no	—
145	#568	—	3	2	no	—
146	#444	—	3	4	no	—
147	#443	—	3	4	no	—
148	#430	C₅×D₃	1	1	yes	characteristic, complemented
149	C₂×#104	—	3	2	no	—
150	C₄×M₁₆	—	3	1	no	—
151	C₂²×C₃×C₅	—	4	1	no	—
152	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
153	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
154	C₂×C₅×D₃	—	2	4	no	—
155	C₂×C₅×D₃	—	8	2	no	—
156	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
159	C₅×Dic₃	—	2	2	no	—
160	C₅×Dic₃	—	2	4	no	—
161	C₅×Dic₃	—	4	1	no	—
162	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
163	C₂×#67	—	2	2	no	—
164	C₂²×Dic₃	—	2	1	no	—
165	#235	—	2	2	no	—
166	#235	—	2	2	no	—
167	#235	C₂×C₄×C₅	1	4	yes	—
168	#235	C₂³×C₅	1	4	yes	—
169	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
170	C₂×C₄×D₃	—	2	2	no	—
171	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
172	C₃×M₁₆	—	2	2	no	—
173	C₃×M₁₆	C₂×C₄×C₅	1	4	yes	—
174	C₃×M₁₆	C₂³×C₅	1	4	yes	—
175	C₂×C₃×C₈	—	2	2	no	—
176	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
177	C₄×Dic₃	—	2	1	no	—
178	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
179	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
180	#208	C₅×D₄	1	4	yes	—
181	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
182	Dic₁₂	C₂×C₄×C₅	1	4	yes	complemented
183	D₂₄	C₂×C₄×C₅	1	4	yes	complemented
184	#204	C₂×C₄×C₅	1	8	yes	complemented
185	ΓL(1,𝔽₂₅)	—	2	8	no	—
186	D₃×C₈	—	2	8	no	—
187	C₂³×C₅	—	6	2	no	—
188	C₂³×C₅	#239	1	1	yes	characteristic
189	C₅×Q₈	—	3	4	no	—
190	C₅×Q₈	—	6	2	no	—
191	C₅×D₄	—	3	4	no	—
192	C₅×D₄	—	6	4	no	—
193	C₅×D₄	—	6	4	no	—
194	C₅×D₄	—	6	2	no	—
195	C₂×C₄×C₅	—	6	2	no	—
196	C₂×C₄×C₅	—	6	1	no	—
197	C₂×C₄×C₅	—	6	4	no	—
198	C₂×C₄×C₅	—	6	4	no	—
199	C₂×C₄×C₅	—	6	1	no	—
200	C₂×C₄×C₅	—	12	2	no	—
201	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
202	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
203	C₂×C₄×C₅	#239	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₅×C₈	—	2	4	no	—
206	C₅×C₈	—	3	4	no	—
207	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
208	C₂×#41	—	3	2	no	—
209	#137	—	3	4	no	—
210	AGL(1,ℤ/8)	—	3	4	no	—
211	#135	—	3	4	no	—
212	#131	—	3	8	no	—
213	#131	—	6	4	no	—
214	C₂×M₁₆	—	3	1	no	—
215	C₂×M₁₆	—	6	4	no	—
216	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
217	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
218	C₂×C₄²	—	3	1	no	—
219	#108	C₂×C₅×D₃	1	4	yes	complemented
220	#104	—	3	4	no	—
221	#104	—	3	4	no	—
222	#97	—	3	2	no	—
223	C₄×C₈	—	3	2	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	2	no	—
226	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₅×D₃	—	4	4	no	—
228	C₂³×C₃	C₅×#41	1	1	yes	characteristic
229	C₂²×D₃	—	2	2	no	—
230	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
231	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
232	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
234	#67	—	2	4	no	—
235	#67	—	2	4	no	—
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	—	2	1	no	—
238	C₂×Dic₃	—	2	1	no	—
239	C₂×Dic₃	—	4	2	no	—
240	D₁₂	—	2	2	no	—
241	D₁₂	C₂²×C₄×C₅	1	4	yes	—
242	C₄×D₃	—	2	4	no	—
243	C₄×D₃	—	2	4	no	—
244	Dic₆	—	2	2	no	—
245	Dic₆	C₂²×C₄×C₅	1	4	yes	—
246	C₃×C₈	—	2	4	no	—
247	C₃×C₈	C₂²×C₄×C₅	1	4	yes	—
248	C₃⋊C₈	C₂×C₅×D₄	1	4	yes	—
249	C₂²×C₅	—	4	1	no	—
250	C₂²×C₅	—	6	4	no	—
251	C₂²×C₅	—	24	2	no	—
252	C₂²×C₅	#893	1	2	yes	—
253	C₂²×C₅	C₂×#239	1	1	yes	characteristic
254	C₄×C₅	—	6	2	no	—
255	C₄×C₅	—	6	4	no	—
256	C₄×C₅	—	12	1	no	—
257	C₄×C₅	#893	1	2	yes	—
258	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
259	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
260	#41	—	3	4	no	—
261	#41	—	3	4	no	—
262	#41	—	6	2	no	—
263	#41	—	6	2	no	—
264	C₂×Q₈	—	3	2	no	—
265	C₂×D₄	—	3	2	no	—
266	C₂×D₄	—	6	2	no	—
267	C₂²×C₄	—	6	2	no	—
268	C₂²×C₄	—	6	1	no	—
269	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
270	Q₁₆	—	3	4	no	—
271	SD₁₆	—	3	8	no	—
272	D₈	—	3	4	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	3	4	no	—
275	M₁₆	—	6	8	no	—
276	M₁₆	C₄×C₅×D₃	1	4	yes	—
277	M₁₆	C₂²×C₅×D₃	1	4	yes	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	3	2	no	—
280	C₂×C₈	—	6	8	no	—
281	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
282	C₄²	—	3	1	no	—
283	C₄²	—	3	1	no	—
284	C₄²	—	6	1	no	—
285	C₃×C₅	#2956	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
288	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
289	C₂×D₃	—	2	4	no	—
290	C₂×D₃	—	8	2	no	—
291	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
292	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
293	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
294	Dic₃	—	2	2	no	—
295	Dic₃	—	2	4	no	—
296	Dic₃	—	4	1	no	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	—	12	4	no	—
300	C₂×C₅	C₂×#893	1	1	yes	characteristic
301	C₂³	—	6	2	no	—
302	C₂³	C₅×#239	1	1	yes	characteristic
303	Q₈	—	3	4	no	—
304	Q₈	—	6	2	no	—
305	D₄	—	3	4	no	—
306	D₄	—	6	4	no	—
307	D₄	—	6	4	no	—
308	D₄	—	6	2	no	—
309	C₂×C₄	—	6	2	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	4	no	—
312	C₂×C₄	—	6	4	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	12	2	no	—
315	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
316	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
317	C₂×C₄	C₅×#239	1	1	yes	characteristic
318	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	3	4	no	—
321	C₈	C₂×C₄×C₅×D₃	1	4	yes	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	2	no	—
324	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
325	D₃	—	4	4	no	—
326	C₅	#79258	1	1	yes	characteristic, direct product factor, Sylow
327	C₂²	—	4	1	no	—
328	C₂²	—	6	4	no	—
329	C₂²	—	24	2	no	—
330	C₂²	C₅×#893	1	2	yes	—
331	C₂²	C₂×C₅×#239	1	1	yes	characteristic
332	C₄	—	6	2	no	—
333	C₄	—	6	4	no	—
334	C₄	—	12	1	no	—
335	C₄	C₅×#893	1	2	yes	—
336	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
337	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
338	C₃	C₅×#2956	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	—	12	4	no	—
342	C₂	C₂×C₅×#893	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#430	C₂
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₅×#5748	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#430	D₃
hypocenter	C₃	C₅×#2956
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#430

Complements

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

N	H ≅ G/N	type
2. C₅×#5745	C₂	semidirect
3. C₃×C₅×#430	C₂	semidirect
4. C₂×C₅×#825	C₂	semidirect
5. C₅×Dic₃×M₁₆	C₂	semidirect
6. C₅×#4895	C₂	semidirect
7. C₅×#4894	C₂	semidirect
14. C₅×#909	C₄	semidirect
15. C₅×#908	C₄	semidirect
16. C₅×#907	C₂²	semidirect
20. C₃×C₅×#108	C₂²	semidirect
21. C₅×#825	C₂²	semidirect
22. C₅×#825	C₂²	semidirect
23. C₅×#815	C₂²	semidirect
24. C₅×C₈×Dic₃	C₂²	semidirect
25. #79258	C₅	direct (N × H)
30. C₅×#430	D₃	semidirect
53. C₅×Dic₁₂	C₂×C₄	semidirect
54. C₅×D₂₄	C₂×C₄	semidirect
55. C₅×#204	C₂×C₄	semidirect
59. #5745	C₂×C₅	semidirect
60. C₃×#430	C₂×C₅	semidirect
61. C₂×#825	C₂×C₅	semidirect
62. Dic₃×M₁₆	C₂×C₅	semidirect
63. #4895	C₂×C₅	semidirect
64. #4894	C₂×C₅	semidirect
76. C₅×#108	C₂×D₃	semidirect
108. #909	C₄×C₅	semidirect
109. #908	C₄×C₅	semidirect
110. #907	C₂²×C₅	semidirect
114. C₃×#108	C₂²×C₅	semidirect
115. #825	C₂²×C₅	semidirect
116. #825	C₂²×C₅	semidirect
117. #815	C₂²×C₅	semidirect
118. C₈×Dic₃	C₂²×C₅	semidirect
148. #430	C₅×D₃	semidirect
182. Dic₁₂	C₂×C₄×C₅	semidirect
183. D₂₄	C₂×C₄×C₅	semidirect
184. #204	C₂×C₄×C₅	semidirect
219. #108	C₂×C₅×D₃	semidirect
285. C₃×C₅	#2956	semidirect
326. C₅	#79258	direct (N × H)
338. C₃	C₅×#2956	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5748	2
C₅×#5745	2
C₃×C₅×#430	2
C₂×C₅×#825	2
C₅×Dic₃×M₁₆	2
C₅×#4895	2
C₅×#4894	2
C₅×#2956	3
#79258	5

Subgroups — C5×#79258 #14418116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#79258 #14418116