Subgroups — C₅×#71267 #14291277

Order 1920.

376 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	C₃×C₅×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5087	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5072	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4544	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4457	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1665	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
10	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#127	C₂²	1	1	yes	characteristic
13	C₂×C₅×#213	—	2	1	no	—
14	C₂×C₅×D₃×C₈	—	2	1	no	—
15	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₅×#832	—	2	1	no	—
18	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
19	C₅×#809	—	2	1	no	—
20	C₅×#807	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#71267	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#492	D₃	1	1	yes	characteristic, complemented
26	C₅×#474	—	3	1	no	—
27	C₅×#459	—	3	1	no	—
28	C₅×C₈×D₄	—	3	1	no	—
29	C₅×#334	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	C₅×#331	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₃×C₅×D₄	—	2	1	no	—
36	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	—	2	1	no	—
38	C₃×C₅×SD₁₆	—	4	2	no	—
39	C₃×C₅×SD₁₆	—	4	2	no	—
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#213	—	4	2	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
47	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	—	2	1	no	—
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₅×D₃×C₈	—	4	2	no	—
52	C₃×#492	C₂×C₅	1	1	yes	characteristic, complemented
53	#5087	C₂×C₅	1	1	yes	characteristic, complemented
54	#5072	C₂×C₅	1	1	yes	characteristic
55	C₈×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
56	#4547	C₂×C₅	1	1	yes	characteristic, complemented
57	#4544	C₂×C₅	1	1	yes	characteristic, complemented
58	#4457	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×C₅×SD₁₆	—	2	1	no	—
60	C₂×C₅×SD₁₆	—	2	1	no	—
61	C₂×C₅×D₈	—	6	1	no	—
62	C₂²×C₅×C₈	—	6	1	no	—
63	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
64	C₅×#127	C₂×D₃	1	1	yes	characteristic
65	C₄×C₅×D₄	—	3	1	no	—
66	C₅×#107	—	6	1	no	—
67	C₅×C₄⋊C₈	—	3	1	no	—
68	C₅×C₄⋊C₈	—	3	1	no	—
69	C₅×C₄⋊C₈	—	3	1	no	—
70	C₅×#102	—	6	1	no	—
71	C₅×#102	—	6	1	no	—
72	C₅×C₂²⋊C₈	—	6	1	no	—
73	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	#1665	—	3	1	no	Sylow
75	C₂³×C₃×C₅	—	4	1	no	—
76	C₂²×C₅×D₃	—	2	1	no	—
77	C₃×C₅×Q₈	—	4	1	no	—
78	C₃×C₅×Q₈	—	4	1	no	—
79	C₃×C₅×D₄	—	4	1	no	—
80	C₃×C₅×D₄	—	4	1	no	—
81	C₃×C₅×D₄	—	4	1	no	—
82	C₃×C₅×D₄	—	4	1	no	—
83	C₃×C₅×D₄	—	8	1	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
88	C₂×C₅×Dic₃	—	2	1	no	—
89	C₅×D₁₂	—	2	1	no	—
90	C₅×D₁₂	SD₁₆	1	2	yes	complemented
91	C₄×C₅×D₃	—	4	1	no	—
92	C₄×C₅×D₃	—	4	1	no	—
93	C₃×C₅×C₈	—	2	1	no	—
94	C₃×C₅×C₈	D₈	1	2	yes	complemented
95	C₅×C₃⋊C₈	—	4	1	no	—
96	C₅×C₃⋊C₈	—	4	1	no	—
97	C₅×C₃⋊C₈	—	4	1	no	—
98	C₂×C₃×SD₁₆	—	2	1	no	—
99	C₂×C₃×SD₁₆	—	2	1	no	—
100	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
101	C₃×#127	C₂²×C₅	1	1	yes	characteristic
102	C₂×#213	—	2	1	no	—
103	C₂×D₃×C₈	—	2	1	no	—
104	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
105	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
106	#832	—	2	1	no	—
107	(C₂×D₃)⋊C₈	—	2	1	no	—
108	#809	—	2	1	no	—
109	#807	—	2	1	no	—
110	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
111	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
112	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₅×Q₈	—	2	1	no	—
114	C₂×C₅×D₄	—	2	1	no	—
115	C₂×C₅×D₄	—	2	1	no	—
116	C₂×C₅×D₄	—	2	1	no	—
117	C₂×C₅×D₄	—	3	1	no	—
118	C₂²×C₄×C₅	—	6	1	no	—
119	C₅×SD₁₆	—	4	2	no	—
120	C₅×SD₁₆	—	4	2	no	—
121	C₅×D₈	—	12	2	no	—
122	C₂×C₅×C₈	—	6	1	no	—
123	C₂×C₅×C₈	—	6	1	no	—
124	C₂×C₅×C₈	—	6	1	no	—
125	C₂×C₅×C₈	—	12	2	no	—
126	C₂×C₅×C₈	#67	1	1	yes	characteristic
127	C₂×C₅×C₈	#67	1	1	yes	characteristic
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	—	3	1	no	—
131	C₅×C₂²⋊C₄	—	6	1	no	—
132	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
133	#492	C₅×D₃	1	1	yes	characteristic, complemented
134	#474	—	3	1	no	—
135	#459	—	3	1	no	—
136	C₈×D₄	—	3	1	no	—
137	#334	—	3	1	no	—
138	#332	—	3	1	no	—
139	#331	—	3	1	no	—
140	C₂²×C₃×C₅	—	8	1	no	—
141	C₂²×C₃×C₅	—	8	1	no	—
142	C₂²×C₃×C₅	—	8	1	no	—
143	C₂²×C₃×C₅	#121	1	1	yes	characteristic
144	C₂×C₅×D₃	—	2	2	no	—
145	C₂×C₅×D₃	—	4	1	no	—
146	C₂×C₅×D₃	—	4	1	no	—
147	C₃×C₄×C₅	—	2	1	no	—
148	C₃×C₄×C₅	—	8	1	no	—
149	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
151	C₃×C₄×C₅	#135	1	1	yes	characteristic
152	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₅×Dic₃	—	4	1	no	—
154	C₂×C₃×Q₈	—	2	1	no	—
155	C₂×C₃×D₄	—	2	1	no	—
156	C₂×C₃×D₄	—	2	1	no	—
157	C₂×C₃×D₄	—	2	1	no	—
158	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
159	C₂×C₄×D₃	—	2	1	no	—
160	C₃×SD₁₆	—	4	2	no	—
161	C₃×SD₁₆	—	4	2	no	—
162	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
163	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
164	C₃×C₄⋊C₄	—	2	1	no	—
165	C₃×C₄⋊C₄	—	2	1	no	—
166	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
167	#213	—	4	2	no	—
168	(C₂×D₃)⋊C₄	—	2	1	no	—
169	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
170	C₂×C₃⋊C₈	—	2	1	no	—
171	C₂×C₃⋊C₈	—	2	1	no	—
172	C₂×C₃⋊C₈	—	2	1	no	—
173	D₃×C₈	—	4	2	no	—
174	C₂³×C₅	—	4	1	no	—
175	C₂³×C₅	—	6	1	no	—
176	C₅×Q₈	—	4	1	no	—
177	C₅×Q₈	—	4	1	no	—
178	C₅×D₄	—	3	2	no	—
179	C₅×D₄	—	4	1	no	—
180	C₅×D₄	—	4	1	no	—
181	C₅×D₄	—	4	1	no	—
182	C₅×D₄	—	4	1	no	—
183	C₅×D₄	—	6	1	no	—
184	C₅×D₄	—	8	1	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	6	1	no	—
187	C₂×C₄×C₅	—	12	1	no	—
188	C₂×C₄×C₅	—	12	1	no	—
189	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	#237	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
192	C₅×C₈	—	2	1	no	—
193	C₅×C₈	—	12	1	no	—
194	C₅×C₈	—	12	1	no	—
195	C₅×C₈	—	12	1	no	—
196	C₅×C₈	#213	1	2	yes	complemented
197	C₂×SD₁₆	—	2	1	no	—
198	C₂×SD₁₆	—	2	1	no	—
199	C₂×D₈	—	6	1	no	—
200	C₂²×C₈	—	6	1	no	—
201	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
202	#127	C₂×C₅×D₃	1	1	yes	characteristic
203	C₄×D₄	—	3	1	no	—
204	#107	—	6	1	no	—
205	C₄⋊C₈	—	3	1	no	—
206	C₄⋊C₈	—	3	1	no	—
207	C₄⋊C₈	—	3	1	no	—
208	#102	—	6	1	no	—
209	#102	—	6	1	no	—
210	C₂²⋊C₈	—	6	1	no	—
211	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
212	C₂×C₃×C₅	—	16	1	no	—
213	C₂×C₃×C₅	#459	1	1	yes	characteristic
214	C₂×C₃×C₅	#465	1	1	yes	characteristic
215	C₂×C₃×C₅	#471	1	1	yes	characteristic
216	C₅×D₃	—	4	2	no	—
217	C₂³×C₃	—	4	1	no	—
218	C₂²×D₃	—	2	1	no	—
219	C₃×Q₈	—	4	1	no	—
220	C₃×Q₈	—	4	1	no	—
221	C₃×D₄	—	4	1	no	—
222	C₃×D₄	—	4	1	no	—
223	C₃×D₄	—	4	1	no	—
224	C₃×D₄	—	4	1	no	—
225	C₃×D₄	—	8	1	no	—
226	C₂×C₃×C₄	—	4	1	no	—
227	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
229	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
230	C₂×Dic₃	—	2	1	no	—
231	D₁₂	—	2	1	no	—
232	D₁₂	C₅×SD₁₆	1	2	yes	complemented
233	C₄×D₃	—	4	1	no	—
234	C₄×D₃	—	4	1	no	—
235	C₃×C₈	—	2	1	no	—
236	C₃×C₈	C₅×D₈	1	2	yes	complemented
237	C₃⋊C₈	—	4	1	no	—
238	C₃⋊C₈	—	4	1	no	—
239	C₃⋊C₈	—	4	1	no	—
240	C₂²×C₅	—	6	2	no	—
241	C₂²×C₅	—	8	1	no	—
242	C₂²×C₅	—	8	1	no	—
243	C₂²×C₅	—	8	1	no	—
244	C₂²×C₅	—	12	1	no	—
245	C₂²×C₅	—	12	1	no	—
246	C₂²×C₅	#938	1	1	yes	characteristic
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	8	1	no	—
249	C₄×C₅	—	12	1	no	—
250	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
251	C₄×C₅	#916	1	1	yes	characteristic
252	C₄×C₅	C₂×#213	1	1	yes	characteristic
253	C₄×C₅	#932	1	1	yes	characteristic
254	C₂×Q₈	—	2	1	no	—
255	C₂×D₄	—	2	1	no	—
256	C₂×D₄	—	2	1	no	—
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	3	1	no	—
259	C₂²×C₄	—	6	1	no	—
260	SD₁₆	—	4	2	no	—
261	SD₁₆	—	4	2	no	—
262	D₈	—	12	2	no	—
263	C₂×C₈	—	6	1	no	—
264	C₂×C₈	—	6	1	no	—
265	C₂×C₈	—	6	1	no	—
266	C₂×C₈	—	12	2	no	—
267	C₂×C₈	C₅×#67	1	1	yes	characteristic
268	C₂×C₈	C₅×#67	1	1	yes	characteristic
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	3	1	no	—
272	C₂²⋊C₄	—	6	1	no	—
273	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
274	C₃×C₅	#1665	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	—	8	1	no	—
276	C₂²×C₃	—	8	1	no	—
277	C₂²×C₃	—	8	1	no	—
278	C₂²×C₃	C₅×#121	1	1	yes	characteristic
279	C₂×D₃	—	2	2	no	—
280	C₂×D₃	—	4	1	no	—
281	C₂×D₃	—	4	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	8	1	no	—
284	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
285	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
286	C₃×C₄	C₅×#135	1	1	yes	characteristic
287	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
288	Dic₃	—	4	1	no	—
289	C₂×C₅	—	12	2	no	—
290	C₂×C₅	—	16	1	no	—
291	C₂×C₅	#5072	1	1	yes	characteristic
292	C₂×C₅	#5171	1	1	yes	characteristic
293	C₂×C₅	#5177	1	1	yes	characteristic
294	C₂³	—	4	1	no	—
295	C₂³	—	6	1	no	—
296	Q₈	—	4	1	no	—
297	Q₈	—	4	1	no	—
298	D₄	—	3	2	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	4	1	no	—
302	D₄	—	4	1	no	—
303	D₄	—	6	1	no	—
304	D₄	—	8	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	6	1	no	—
307	C₂×C₄	—	12	1	no	—
308	C₂×C₄	—	12	1	no	—
309	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
310	C₂×C₄	C₅×#237	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	12	1	no	—
314	C₈	—	12	1	no	—
315	C₈	—	12	1	no	—
316	C₈	C₅×#213	1	2	yes	complemented
317	C₂×C₃	—	16	1	no	—
318	C₂×C₃	C₅×#459	1	1	yes	characteristic
319	C₂×C₃	C₅×#465	1	1	yes	characteristic
320	C₂×C₃	C₅×#471	1	1	yes	characteristic
321	D₃	—	4	2	no	—
322	C₅	#71267	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	6	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	12	1	no	—
328	C₂²	—	12	1	no	—
329	C₂²	C₅×#938	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	12	1	no	—
333	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
334	C₄	C₅×#916	1	1	yes	characteristic
335	C₄	C₂×C₅×#213	1	1	yes	characteristic
336	C₄	C₅×#932	1	1	yes	characteristic
337	C₃	C₅×#1665	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	12	2	no	—
339	C₂	—	16	1	no	—
340	C₂	C₅×#5072	1	1	yes	characteristic
341	C₂	C₅×#5171	1	1	yes	characteristic
342	C₂	C₅×#5177	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#938
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#492	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#5072	C₂
hypercenter	C₅×#492	D₃
hypocenter	C₃	C₅×#1665
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#492

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#492	C₂	semidirect
2. C₅×#5087	C₂	semidirect
4. C₅×C₈×D₁₂	C₂	semidirect
5. C₅×#4547	C₂	semidirect
6. C₅×#4544	C₂	semidirect
7. C₅×#4457	C₂	semidirect
24. #71267	C₅	direct (N × H)
25. C₅×#492	D₃	semidirect
52. C₃×#492	C₂×C₅	semidirect
53. #5087	C₂×C₅	semidirect
55. C₈×D₁₂	C₂×C₅	semidirect
56. #4547	C₂×C₅	semidirect
57. #4544	C₂×C₅	semidirect
58. #4457	C₂×C₅	semidirect
90. C₅×D₁₂	SD₁₆	semidirect
94. C₃×C₅×C₈	D₈	semidirect
133. #492	C₅×D₃	semidirect
196. C₅×C₈	#213	semidirect
232. D₁₂	C₅×SD₁₆	semidirect
236. C₃×C₈	C₅×D₈	semidirect
274. C₃×C₅	#1665	semidirect
316. C₈	C₅×#213	semidirect
322. C₅	#71267	direct (N × H)
337. C₃	C₅×#1665	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#492	2
C₅×#5087	2
C₅×#5072	2
C₅×C₈×D₁₂	2
C₅×#4547	2
C₅×#4544	2
C₅×#4457	2
C₅×#1665	3
#71267	5

Subgroups — C5×#71267 #14291277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71267 #14291277