Subgroups — C₅×#71039 #14292108

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#461	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5088	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5015	C₂	1	1	yes	maximal, characteristic
4	C₅×D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1681	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×#213	—	2	1	no	—
13	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
14	C₅×#874	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#103	—	2	1	no	—
17	C₅×#830	—	2	1	no	—
18	C₅×#809	—	2	1	no	—
19	C₅×#809	—	2	1	no	—
20	C₅×#809	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#71039	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#486	—	3	1	no	—
26	C₅×#461	D₃	1	1	yes	characteristic, complemented
27	C₅×#459	—	3	1	no	—
28	C₅×C₈×D₄	—	3	1	no	—
29	C₅×#335	—	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	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₅×D₁₂	—	2	1	no	—
38	C₃×C₅×SD₁₆	—	4	2	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
43	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
44	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
45	C₅×#213	—	4	2	no	—
46	C₂×C₅×C₃⋊C₈	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	4	2	no	—
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₃×#461	C₂×C₅	1	1	yes	characteristic, complemented
52	#5088	C₂×C₅	1	1	yes	characteristic, complemented
53	#5015	C₂×C₅	1	1	yes	characteristic
54	D₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
55	#4547	C₂×C₅	1	1	yes	characteristic, complemented
56	#4485	C₂×C₅	1	1	yes	characteristic, complemented
57	#4479	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×SD₁₆	—	2	1	no	—
59	C₂×C₅×D₈	—	6	1	no	—
60	C₂²×C₅×C₈	—	6	1	no	—
61	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
62	C₅×#127	—	3	1	no	—
63	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	—	3	1	no	—
65	C₅×C₄⋊C₈	—	3	1	no	—
66	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
67	C₅×#103	—	2	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₅×#102	—	6	1	no	—
70	C₅×#102	—	6	1	no	—
71	C₅×C₂²⋊C₈	—	6	1	no	—
72	C₄×C₅×C₈	—	3	1	no	—
73	#1681	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₂²×C₅×D₃	—	4	1	no	—
76	C₃×C₅×Q₈	—	4	1	no	—
77	C₃×C₅×Q₈	—	4	1	no	—
78	C₃×C₅×D₄	—	2	1	no	—
79	C₃×C₅×D₄	D₈	1	2	yes	—
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	4	1	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
87	C₅×D₁₂	—	4	1	no	—
88	C₅×D₁₂	—	4	1	no	—
89	C₅×D₁₂	—	4	1	no	—
90	C₅×D₁₂	—	4	1	no	—
91	C₅×D₁₂	—	8	1	no	—
92	C₃×C₅×C₈	—	4	1	no	—
93	C₅×C₃⋊C₈	—	2	1	no	—
94	C₅×C₃⋊C₈	—	2	1	no	—
95	C₅×C₃⋊C₈	—	4	1	no	—
96	C₅×C₃⋊C₈	—	4	1	no	—
97	C₂×C₃×SD₁₆	—	2	1	no	—
98	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
100	C₂×#213	—	2	1	no	—
101	C₂²×C₃⋊C₈	—	2	1	no	—
102	#874	C₂²×C₅	1	1	yes	characteristic
103	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×#103	—	2	1	no	—
105	#830	—	2	1	no	—
106	#809	—	2	1	no	—
107	#809	—	2	1	no	—
108	#809	—	2	1	no	—
109	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
110	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
111	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
112	C₂×C₅×Q₈	—	2	1	no	—
113	C₂×C₅×D₄	—	6	1	no	—
114	C₂×C₅×D₄	—	6	1	no	—
115	C₂×C₅×D₄	—	6	1	no	—
116	C₂×C₅×D₄	#67	1	1	yes	characteristic
117	C₂²×C₄×C₅	—	2	1	no	—
118	C₅×SD₁₆	—	4	2	no	—
119	C₅×D₈	—	12	2	no	—
120	C₂×C₅×C₈	—	2	1	no	—
121	C₂×C₅×C₈	—	3	1	no	—
122	C₂×C₅×C₈	—	3	1	no	—
123	C₂×C₅×C₈	—	6	1	no	—
124	C₂×C₅×C₈	—	6	1	no	—
125	C₂×C₅×C₈	—	12	2	no	—
126	C₅×C₄⋊C₄	—	2	1	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	#67	1	1	yes	characteristic
129	C₅×C₂²⋊C₄	—	2	1	no	—
130	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
131	#486	—	3	1	no	—
132	#461	C₅×D₃	1	1	yes	characteristic, complemented
133	#459	—	3	1	no	—
134	C₈×D₄	—	3	1	no	—
135	#335	—	3	1	no	—
136	#332	—	3	1	no	—
137	#331	—	3	1	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	1	no	—
141	C₂²×C₃×C₅	#121	1	1	yes	characteristic
142	C₂×C₅×D₃	—	8	1	no	—
143	C₂×C₅×D₃	—	8	1	no	—
144	C₂×C₅×D₃	—	8	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	—	4	1	no	—
147	C₃×C₄×C₅	—	8	1	no	—
148	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
149	C₃×C₄×C₅	#135	1	1	yes	characteristic
150	C₃×C₄×C₅	#135	1	1	yes	characteristic
151	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×C₃×Q₈	—	2	1	no	—
153	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	2	1	no	—
155	C₂×D₁₂	—	2	1	no	—
156	C₂×D₁₂	—	2	1	no	—
157	C₂×D₁₂	—	2	1	no	—
158	C₃×SD₁₆	—	4	2	no	—
159	C₂×C₃×C₈	—	2	1	no	—
160	C₃×C₄⋊C₄	—	2	1	no	—
161	C₃×C₄⋊C₄	—	2	1	no	—
162	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
163	C₃×C₂²⋊C₄	—	2	1	no	—
164	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
165	#213	—	4	2	no	—
166	C₂×C₃⋊C₈	—	2	1	no	—
167	C₂×C₃⋊C₈	—	2	1	no	—
168	C₂×C₃⋊C₈	—	4	2	no	—
169	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
170	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
171	C₂³×C₅	—	2	1	no	—
172	C₂³×C₅	—	12	1	no	—
173	C₅×Q₈	—	4	1	no	—
174	C₅×Q₈	—	4	1	no	—
175	C₅×D₄	—	2	1	no	—
176	C₅×D₄	—	12	1	no	—
177	C₅×D₄	—	12	1	no	—
178	C₅×D₄	—	12	1	no	—
179	C₅×D₄	—	12	1	no	—
180	C₅×D₄	—	24	1	no	—
181	C₅×D₄	#213	1	2	yes	—
182	C₂×C₄×C₅	—	2	1	no	—
183	C₂×C₄×C₅	—	4	1	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
187	C₂×C₄×C₅	#237	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
189	C₅×C₈	—	4	1	no	—
190	C₅×C₈	—	6	1	no	—
191	C₅×C₈	—	6	1	no	—
192	C₅×C₈	—	12	1	no	—
193	C₅×C₈	—	12	1	no	—
194	C₂×SD₁₆	—	2	1	no	—
195	C₂×D₈	—	6	1	no	—
196	C₂²×C₈	—	6	1	no	—
197	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
198	#127	—	3	1	no	—
199	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
200	C₄⋊C₈	—	3	1	no	—
201	C₄⋊C₈	—	3	1	no	—
202	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
203	#103	—	2	1	no	—
204	#102	—	6	1	no	—
205	#102	—	6	1	no	—
206	#102	—	6	1	no	—
207	C₂²⋊C₈	—	6	1	no	—
208	C₄×C₈	—	3	1	no	—
209	C₂×C₃×C₅	—	4	2	no	—
210	C₂×C₃×C₅	#463	1	1	yes	characteristic
211	C₂×C₃×C₅	#466	1	1	yes	characteristic
212	C₂×C₃×C₅	#471	1	1	yes	characteristic
213	C₅×D₃	—	16	1	no	—
214	C₂³×C₃	—	2	1	no	—
215	C₂²×D₃	—	4	1	no	—
216	C₃×Q₈	—	4	1	no	—
217	C₃×Q₈	—	4	1	no	—
218	C₃×D₄	—	2	1	no	—
219	C₃×D₄	C₅×D₈	1	2	yes	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	—	4	1	no	—
223	C₂×C₃×C₄	—	4	1	no	—
224	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
227	D₁₂	—	4	1	no	—
228	D₁₂	—	4	1	no	—
229	D₁₂	—	4	1	no	—
230	D₁₂	—	4	1	no	—
231	D₁₂	—	8	1	no	—
232	C₃×C₈	—	4	1	no	—
233	C₃⋊C₈	—	2	1	no	—
234	C₃⋊C₈	—	2	1	no	—
235	C₃⋊C₈	—	4	1	no	—
236	C₃⋊C₈	—	4	1	no	—
237	C₂²×C₅	—	2	2	no	—
238	C₂²×C₅	—	4	1	no	—
239	C₂²×C₅	—	4	1	no	—
240	C₂²×C₅	—	24	1	no	—
241	C₂²×C₅	—	24	1	no	—
242	C₂²×C₅	—	24	1	no	—
243	C₂²×C₅	#939	1	1	yes	characteristic
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	4	1	no	—
246	C₄×C₅	—	8	1	no	—
247	C₄×C₅	#914	1	1	yes	characteristic
248	C₄×C₅	#916	1	1	yes	characteristic
249	C₄×C₅	C₂×#213	1	1	yes	characteristic
250	C₄×C₅	#950	1	1	yes	characteristic
251	C₂×Q₈	—	2	1	no	—
252	C₂×D₄	—	6	1	no	—
253	C₂×D₄	—	6	1	no	—
254	C₂×D₄	—	6	1	no	—
255	C₂×D₄	C₅×#67	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	SD₁₆	—	4	2	no	—
258	D₈	—	12	2	no	—
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	3	1	no	—
261	C₂×C₈	—	3	1	no	—
262	C₂×C₈	—	6	1	no	—
263	C₂×C₈	—	6	1	no	—
264	C₂×C₈	—	12	2	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	C₅×#67	1	1	yes	characteristic
268	C₂²⋊C₄	—	2	1	no	—
269	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
270	C₃×C₅	#1681	1	1	yes	characteristic, complemented, Hall
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	4	1	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	C₅×#121	1	1	yes	characteristic
275	C₂×D₃	—	8	1	no	—
276	C₂×D₃	—	8	1	no	—
277	C₂×D₃	—	8	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	4	1	no	—
280	C₃×C₄	—	8	1	no	—
281	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
282	C₃×C₄	C₅×#135	1	1	yes	characteristic
283	C₃×C₄	C₅×#135	1	1	yes	characteristic
284	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
285	C₂×C₅	—	4	2	no	—
286	C₂×C₅	—	48	1	no	—
287	C₂×C₅	#5038	1	1	yes	characteristic
288	C₂×C₅	#5165	1	1	yes	characteristic
289	C₂×C₅	#5203	1	1	yes	characteristic
290	C₂³	—	2	1	no	—
291	C₂³	—	12	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	4	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	12	1	no	—
296	D₄	—	12	1	no	—
297	D₄	—	12	1	no	—
298	D₄	—	12	1	no	—
299	D₄	—	24	1	no	—
300	D₄	C₅×#213	1	2	yes	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
306	C₂×C₄	C₅×#237	1	1	yes	characteristic
307	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
308	C₈	—	4	1	no	—
309	C₈	—	6	1	no	—
310	C₈	—	6	1	no	—
311	C₈	—	12	1	no	—
312	C₈	—	12	1	no	—
313	C₂×C₃	—	4	2	no	—
314	C₂×C₃	C₅×#463	1	1	yes	characteristic
315	C₂×C₃	C₅×#466	1	1	yes	characteristic
316	C₂×C₃	C₅×#471	1	1	yes	characteristic
317	D₃	—	16	1	no	—
318	C₅	#71039	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	24	1	no	—
323	C₂²	—	24	1	no	—
324	C₂²	—	24	1	no	—
325	C₂²	C₅×#939	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	8	1	no	—
329	C₄	C₅×#914	1	1	yes	characteristic
330	C₄	C₅×#916	1	1	yes	characteristic
331	C₄	C₂×C₅×#213	1	1	yes	characteristic
332	C₄	C₅×#950	1	1	yes	characteristic
333	C₃	C₅×#1681	1	1	yes	characteristic, complemented, Sylow
334	C₂	—	4	2	no	—
335	C₂	—	48	1	no	—
336	C₂	C₅×#5038	1	1	yes	characteristic
337	C₂	C₅×#5165	1	1	yes	characteristic
338	C₂	C₅×#5203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#939
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#461	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#5015	C₂
hypercenter	C₅×#461	D₃
hypocenter	C₃	C₅×#1681
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#461

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#461	C₂	semidirect
2. C₅×#5088	C₂	semidirect
4. C₅×D₄×C₃⋊C₈	C₂	semidirect
5. C₅×#4547	C₂	semidirect
6. C₅×#4485	C₂	semidirect
7. C₅×#4479	C₂	semidirect
24. #71039	C₅	direct (N × H)
26. C₅×#461	D₃	semidirect
51. C₃×#461	C₂×C₅	semidirect
52. #5088	C₂×C₅	semidirect
54. D₄×C₃⋊C₈	C₂×C₅	semidirect
55. #4547	C₂×C₅	semidirect
56. #4485	C₂×C₅	semidirect
57. #4479	C₂×C₅	semidirect
132. #461	C₅×D₃	semidirect
270. C₃×C₅	#1681	semidirect
318. C₅	#71039	direct (N × H)
333. C₃	C₅×#1681	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#461	2
C₅×#5088	2
C₅×#5015	2
C₅×D₄×C₃⋊C₈	2
C₅×#4547	2
C₅×#4485	2
C₅×#4479	2
C₅×#1681	3
#71039	5

Subgroups — C5×#71039 #14292108

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71039 #14292108