Subgroups — C₅×#74153 #14324371

Order 1920.

1480 conjugacy classes of subgroups.

The subgroup lattice has 316 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	2	yes	maximal, complemented
2	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4988	C₂	1	2	yes	maximal, complemented
5	C₅×#4932	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4665	C₂	1	8	yes	maximal, complemented
7	C₅×#2328	—	3	1	no	maximal, Hall
8	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
9	C₂²×C₅×D₁₂	C₂²	1	4	yes	—
10	C₂²×C₄×C₅×D₃	C₂²	1	4	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
14	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
15	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₅×#893	—	2	8	no	—
18	C₅×#874	C₄	1	8	yes	complemented
19	C₃×C₅×#95	C₂²	1	4	yes	—
20	C₅×C₃⋊#95	C₂²	1	4	yes	—
21	#74153	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#127	—	3	1	no	—
23	C₂×C₄×C₅×D₄	—	3	2	no	—
24	C₅×#390	—	3	2	no	—
25	C₅×#386	—	3	8	no	—
26	C₅×#382	D₃	1	1	yes	characteristic, complemented
27	C₅×#376	—	3	1	no	—
28	C₂³×C₅×D₃	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
33	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
34	C₂×C₅×D₁₂	—	2	4	no	—
35	C₂×C₅×D₁₂	—	2	4	no	—
36	C₂×C₅×D₁₂	—	4	8	no	—
37	C₂×C₅×D₁₂	C₂×C₄	1	8	yes	complemented
38	C₂×C₅×D₁₂	C₂×C₄	1	16	yes	complemented
39	C₂×C₄×C₅×D₃	—	2	8	no	—
40	C₂×C₄×C₅×D₃	—	4	8	no	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
43	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
44	C₅×(C₂×D₃)⋊C₄	—	4	8	no	—
45	C₄×C₅×Dic₃	—	2	2	no	—
46	C₄×C₅×Dic₃	D₄	1	4	yes	complemented
47	C₂×#893	C₂×C₅	1	2	yes	complemented
48	C₂×#874	C₂×C₅	1	1	yes	characteristic
49	C₃×#382	C₂×C₅	1	1	yes	characteristic, complemented
50	#4988	C₂×C₅	1	2	yes	complemented
51	#4932	C₂×C₅	1	1	yes	characteristic, complemented
52	#4665	C₂×C₅	1	8	yes	complemented
53	C₂²×C₅×D₄	—	3	2	no	—
54	C₂²×C₅×D₄	—	3	4	no	—
55	C₂³×C₄×C₅	—	3	4	no	—
56	C₅×#127	—	3	8	no	—
57	C₄×C₅×D₄	—	6	8	no	—
58	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
59	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
60	C₂×C₅×C₂²⋊C₄	—	3	8	no	—
61	C₂×C₄²×C₅	—	3	1	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	C₅×#95	—	3	4	no	—
65	C₅×#95	C₂×D₃	1	4	yes	—
66	#2328	—	3	1	no	Sylow
67	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
68	C₂²×C₅×D₃	—	2	4	no	—
69	C₂²×C₅×D₃	—	2	4	no	—
70	C₂²×C₅×D₃	—	2	8	no	—
71	C₂²×C₅×D₃	—	2	8	no	—
72	C₂²×C₅×D₃	C₂²×C₄	1	8	yes	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	8	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₅×Dic₃	—	2	4	no	—
81	C₂×C₅×Dic₃	—	2	2	no	—
82	C₂×C₅×Dic₃	—	2	4	no	—
83	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
84	C₅×D₁₂	—	2	16	no	—
85	C₅×D₁₂	—	4	16	no	—
86	C₄×C₅×D₃	—	4	16	no	—
87	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
88	C₂²×D₁₂	C₂²×C₅	1	4	yes	—
89	C₂²×C₄×D₃	C₂²×C₅	1	4	yes	—
90	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
93	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	8	yes	—
94	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
96	#893	—	2	8	no	—
97	#874	C₄×C₅	1	8	yes	complemented
98	C₃×#95	C₂²×C₅	1	4	yes	—
99	C₃⋊#95	C₂²×C₅	1	4	yes	—
100	C₂⁴×C₅	—	3	4	no	—
101	C₂×C₅×D₄	—	3	8	no	—
102	C₂×C₅×D₄	—	3	16	no	—
103	C₂×C₅×D₄	—	6	4	no	—
104	C₂×C₅×D₄	—	6	4	no	—
105	C₂×C₅×D₄	—	12	8	no	—
106	C₂²×C₄×C₅	—	3	2	no	—
107	C₂²×C₄×C₅	—	3	2	no	—
108	C₂²×C₄×C₅	—	6	8	no	—
109	C₂²×C₄×C₅	—	12	8	no	—
110	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
111	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	4	no	—
114	C₅×C₂²⋊C₄	—	6	8	no	—
115	C₅×C₂²⋊C₄	—	12	8	no	—
116	C₄²×C₅	—	3	4	no	—
117	C₄²×C₅	—	6	2	no	—
118	C₄²×C₅	C₄×D₃	1	4	yes	complemented
119	C₂×#127	—	3	1	no	—
120	C₂×C₄×D₄	—	3	2	no	—
121	#390	—	3	2	no	—
122	#386	—	3	8	no	—
123	#382	C₅×D₃	1	1	yes	characteristic, complemented
124	#376	—	3	1	no	—
125	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
126	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
128	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
129	C₂×C₅×D₃	—	2	8	no	—
130	C₂×C₅×D₃	—	2	16	no	—
131	C₂×C₅×D₃	—	4	4	no	—
132	C₂×C₅×D₃	—	4	4	no	—
133	C₂×C₅×D₃	—	4	8	no	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	4	4	no	—
136	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
137	C₅×Dic₃	—	2	4	no	—
138	C₅×Dic₃	—	4	2	no	—
139	C₂³×D₃	C₂³×C₅	1	4	yes	—
140	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
141	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
143	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
144	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
145	C₂×D₁₂	—	2	4	no	—
146	C₂×D₁₂	—	2	4	no	—
147	C₂×D₁₂	—	4	8	no	—
148	C₂×D₁₂	C₂×C₄×C₅	1	8	yes	complemented
149	C₂×D₁₂	C₂×C₄×C₅	1	16	yes	complemented
150	C₂×C₄×D₃	—	2	8	no	—
151	C₂×C₄×D₃	—	4	8	no	—
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
154	(C₂×D₃)⋊C₄	—	2	8	no	—
155	(C₂×D₃)⋊C₄	—	4	8	no	—
156	C₄×Dic₃	—	2	2	no	—
157	C₄×Dic₃	C₅×D₄	1	4	yes	complemented
158	C₂³×C₅	—	3	8	no	—
159	C₂³×C₅	—	6	4	no	—
160	C₂³×C₅	—	6	4	no	—
161	C₂³×C₅	—	6	8	no	—
162	C₂³×C₅	—	6	8	no	—
163	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
164	C₅×D₄	—	6	16	no	—
165	C₅×D₄	—	12	16	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	4	no	—
168	C₂×C₄×C₅	—	2	8	no	—
169	C₂×C₄×C₅	—	3	4	no	—
170	C₂×C₄×C₅	—	6	4	no	—
171	C₂×C₄×C₅	—	6	2	no	—
172	C₂×C₄×C₅	—	6	4	no	—
173	C₂×C₄×C₅	—	12	16	no	—
174	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
175	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
176	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
177	C₂×C₄×C₅	#239	1	2	yes	—
178	C₂²×D₄	—	3	2	no	—
179	C₂²×D₄	—	3	4	no	—
180	C₂³×C₄	—	3	4	no	—
181	#127	—	3	8	no	—
182	C₄×D₄	—	6	8	no	—
183	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
184	C₂×C₂²⋊C₄	—	3	4	no	—
185	C₂×C₂²⋊C₄	—	3	8	no	—
186	C₂×C₄²	—	3	1	no	—
187	C₂×C₄²	—	3	1	no	—
188	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
189	#95	—	3	4	no	—
190	#95	C₂×C₅×D₃	1	4	yes	—
191	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂×C₃×C₅	#518	1	2	yes	—
193	C₂×C₃×C₅	#535	1	2	yes	—
194	C₂×C₃×C₅	#540	1	2	yes	—
195	C₅×D₃	—	4	8	no	—
196	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
197	C₂²×D₃	—	2	4	no	—
198	C₂²×D₃	—	2	4	no	—
199	C₂²×D₃	—	2	8	no	—
200	C₂²×D₃	—	2	8	no	—
201	C₂²×D₃	C₂²×C₄×C₅	1	8	yes	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	2	8	no	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
206	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₅×#41	1	2	yes	—
209	C₂×Dic₃	—	2	4	no	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	4	no	—
212	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
213	D₁₂	—	2	16	no	—
214	D₁₂	—	4	16	no	—
215	C₄×D₃	—	4	16	no	—
216	C₂²×C₅	—	6	8	no	—
217	C₂²×C₅	—	6	16	no	—
218	C₂²×C₅	—	12	4	no	—
219	C₂²×C₅	—	12	4	no	—
220	C₂²×C₅	—	12	8	no	—
221	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
222	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
223	C₂²×C₅	C₂×#239	1	1	yes	characteristic
224	C₂²×C₅	#1009	1	4	yes	—
225	C₄×C₅	—	2	4	no	—
226	C₄×C₅	—	4	4	no	—
227	C₄×C₅	—	6	4	no	—
228	C₄×C₅	—	12	2	no	—
229	C₄×C₅	#893	1	4	yes	—
230	C₂⁴	—	3	4	no	—
231	C₂×D₄	—	3	8	no	—
232	C₂×D₄	—	3	16	no	—
233	C₂×D₄	—	6	4	no	—
234	C₂×D₄	—	6	4	no	—
235	C₂×D₄	—	12	8	no	—
236	C₂²×C₄	—	3	2	no	—
237	C₂²×C₄	—	3	2	no	—
238	C₂²×C₄	—	6	8	no	—
239	C₂²×C₄	—	12	8	no	—
240	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
241	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
242	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
243	C₄⋊C₄	—	2	4	no	—
244	C₂²⋊C₄	—	6	8	no	—
245	C₂²⋊C₄	—	12	8	no	—
246	C₄²	—	3	4	no	—
247	C₄²	—	6	2	no	—
248	C₄²	C₄×C₅×D₃	1	4	yes	complemented
249	C₃×C₅	#2328	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
251	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
252	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
253	C₂²×C₃	C₅×H(5,𝔽₂)	1	4	yes	—
254	C₂×D₃	—	2	8	no	—
255	C₂×D₃	—	2	16	no	—
256	C₂×D₃	—	4	4	no	—
257	C₂×D₃	—	4	4	no	—
258	C₂×D₃	—	4	8	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	4	4	no	—
261	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
262	Dic₃	—	2	4	no	—
263	Dic₃	—	4	2	no	—
264	C₂×C₅	—	12	8	no	—
265	C₂×C₅	C₂×#893	1	1	yes	characteristic
266	C₂×C₅	#5521	1	2	yes	—
267	C₂×C₅	#5674	1	2	yes	—
268	C₂×C₅	#5697	1	2	yes	—
269	C₂³	—	3	8	no	—
270	C₂³	—	6	4	no	—
271	C₂³	—	6	4	no	—
272	C₂³	—	6	8	no	—
273	C₂³	—	6	8	no	—
274	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
275	D₄	—	6	16	no	—
276	D₄	—	12	16	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	3	4	no	—
281	C₂×C₄	—	6	4	no	—
282	C₂×C₄	—	6	2	no	—
283	C₂×C₄	—	6	4	no	—
284	C₂×C₄	—	12	16	no	—
285	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
286	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
287	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
288	C₂×C₄	C₅×#239	1	2	yes	—
289	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
290	C₂×C₃	C₅×#518	1	2	yes	—
291	C₂×C₃	C₅×#535	1	2	yes	—
292	C₂×C₃	C₅×#540	1	2	yes	—
293	D₃	—	4	8	no	—
294	C₅	#74153	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	6	8	no	—
296	C₂²	—	6	16	no	—
297	C₂²	—	12	4	no	—
298	C₂²	—	12	4	no	—
299	C₂²	—	12	8	no	—
300	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
301	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
302	C₂²	C₂×C₅×#239	1	1	yes	characteristic
303	C₂²	C₅×#1009	1	4	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	4	4	no	—
306	C₄	—	6	4	no	—
307	C₄	—	12	2	no	—
308	C₄	C₅×#893	1	4	yes	—
309	C₃	C₅×#2328	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	12	8	no	—
311	C₂	C₂×C₅×#893	1	1	yes	characteristic
312	C₂	C₅×#5521	1	2	yes	—
313	C₂	C₅×#5674	1	2	yes	—
314	C₂	C₅×#5697	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#382	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#874	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#382	D₃
hypocenter	C₃	C₅×#2328
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#382

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#893	C₂	semidirect
3. C₃×C₅×#382	C₂	semidirect
4. C₅×#4988	C₂	semidirect
5. C₅×#4932	C₂	semidirect
6. C₅×#4665	C₂	semidirect
18. C₅×#874	C₄	semidirect
21. #74153	C₅	direct (N × H)
26. C₅×#382	D₃	semidirect
37. C₂×C₅×D₁₂	C₂×C₄	semidirect
38. C₂×C₅×D₁₂	C₂×C₄	semidirect
42. C₃×C₄²×C₅	C₂×C₄	semidirect
46. C₄×C₅×Dic₃	D₄	semidirect
47. C₂×#893	C₂×C₅	semidirect
49. C₃×#382	C₂×C₅	semidirect
50. #4988	C₂×C₅	semidirect
51. #4932	C₂×C₅	semidirect
52. #4665	C₂×C₅	semidirect
97. #874	C₄×C₅	semidirect
118. C₄²×C₅	C₄×D₃	semidirect
123. #382	C₅×D₃	semidirect
148. C₂×D₁₂	C₂×C₄×C₅	semidirect
149. C₂×D₁₂	C₂×C₄×C₅	semidirect
153. C₃×C₄²	C₂×C₄×C₅	semidirect
157. C₄×Dic₃	C₅×D₄	semidirect
248. C₄²	C₄×C₅×D₃	semidirect
249. C₃×C₅	#2328	semidirect
294. C₅	#74153	direct (N × H)
309. C₃	C₅×#2328	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×#874	2
C₃×C₅×#382	2
C₅×#4988	2
C₅×#4932	2
C₅×#4665	2
C₅×#2328	3
#74153	5

Subgroups — C5×#74153 #14324371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#74153 #14324371