Subgroups — C₅×#83012 #14438664

Order 1920.

784 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5477	C₂	1	1	yes	maximal, characteristic
3	C₅×#5069	C₂	1	2	yes	maximal, complemented
4	C₅×#5068	C₂	1	4	yes	maximal, complemented
5	C₅×#5067	C₂	1	2	yes	maximal, complemented
6	C₅×#4966	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3142	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₃×C₅×#125	—	2	2	no	—
11	C₃×C₅×#125	C₂²	1	2	yes	—
12	C₃×C₅×#125	C₂²	1	2	yes	—
13	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₅×#930	—	2	2	no	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₅×#875	C₂²	1	2	yes	—
19	C₅×#874	C₂²	1	1	yes	characteristic
20	C₄×C₅×D₁₂	—	2	2	no	—
21	C₅×#869	C₂²	1	1	yes	characteristic
22	C₅×#810	C₂²	1	8	yes	—
23	C₅×#809	C₂²	1	8	yes	—
24	C₅×#803	C₂²	1	2	yes	—
25	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
26	#83012	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₅×#532	—	3	1	no	—
28	C₂×C₅×#125	D₃	1	1	yes	characteristic, complemented
29	C₅×#491	—	3	2	no	—
30	C₅×#490	—	3	2	no	—
31	C₅×#488	—	3	4	no	—
32	C₅×#417	—	3	4	no	—
33	C₄×C₅×M₁₆	—	3	1	no	—
34	C₂²×C₃×C₄×C₅	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₅×#67	—	2	2	no	—
38	C₅×#235	—	2	8	no	—
39	C₂×C₅×D₁₂	—	2	2	no	—
40	C₂×C₅×D₁₂	C₂³	1	2	yes	—
41	C₂×C₄×C₅×D₃	—	2	2	no	—
42	C₂×C₅×Dic₆	C₂³	1	2	yes	—
43	C₃×C₅×C₄⋊C₄	—	2	4	no	—
44	C₃×C₅×C₄⋊C₄	—	2	4	no	—
45	C₃×C₅×C₄⋊C₄	—	2	2	no	—
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×C₄²×C₅	D₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
51	C₅×C₄⋊Dic₃	—	2	2	no	—
52	C₅×#208	—	2	4	no	—
53	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
54	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic, complemented
55	#5477	C₂×C₅	1	1	yes	characteristic
56	#5069	C₂×C₅	1	2	yes	complemented
57	#5068	C₂×C₅	1	4	yes	complemented
58	#5067	C₂×C₅	1	2	yes	complemented
59	#4966	C₂×C₅	1	4	yes	complemented
60	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
61	C₂×C₅×#41	—	3	2	no	—
62	C₂×C₅×M₁₆	—	3	2	no	—
63	C₅×C₄⋊Q₈	—	3	1	no	—
64	C₅×#127	—	3	1	no	—
65	C₅×#125	—	2	2	no	—
66	C₅×#125	C₂×D₃	1	2	yes	—
67	C₅×#125	C₂×D₃	1	2	yes	—
68	C₅×#124	—	3	2	no	—
69	C₅×#121	—	6	2	no	—
70	C₄×C₅×D₄	—	6	2	no	—
71	C₂×C₅×C₄⋊C₄	—	2	2	no	—
72	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
73	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	C₅×#103	—	3	8	no	—
75	C₅×#102	—	3	8	no	—
76	C₅×#97	—	3	2	no	—
77	C₄×C₅×C₈	—	3	2	no	—
78	#3142	—	3	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₅×D₃	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	4	2	no	—
83	C₂×C₃×C₄×C₅	—	4	4	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	4	yes	—
88	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₅×#67	—	4	4	no	—
90	C₂×C₅×Dic₃	—	2	2	no	—
91	C₅×D₁₂	—	2	4	no	—
92	C₅×D₁₂	—	4	4	no	—
93	C₄×C₅×D₃	—	4	4	no	—
94	C₅×Dic₆	—	2	4	no	—
95	C₅×C₃⋊C₈	—	2	4	no	—
96	C₂×#235	C₂²×C₅	1	2	yes	—
97	C₃×#125	—	2	2	no	—
98	C₃×#125	C₂²×C₅	1	2	yes	—
99	C₃×#125	C₂²×C₅	1	2	yes	—
100	C₂×C₃×C₄⋊C₄	—	2	2	no	—
101	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
102	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
103	#930	—	2	2	no	—
104	C₂×#208	C₂²×C₅	1	2	yes	—
105	#875	C₂²×C₅	1	2	yes	—
106	#874	C₂²×C₅	1	1	yes	characteristic
107	C₄×D₁₂	—	2	2	no	—
108	#869	C₂²×C₅	1	1	yes	characteristic
109	#810	C₂²×C₅	1	8	yes	—
110	#809	C₂²×C₅	1	8	yes	—
111	#803	C₂²×C₅	1	2	yes	—
112	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
113	C₅×#41	—	6	8	no	—
114	C₂×C₅×Q₈	—	3	2	no	—
115	C₂×C₅×D₄	—	3	2	no	—
116	C₂×C₅×D₄	—	6	2	no	—
117	C₂×C₅×D₄	—	6	2	no	—
118	C₂²×C₄×C₅	—	2	2	no	—
119	C₂²×C₄×C₅	—	6	2	no	—
120	C₂²×C₄×C₅	#67	1	2	yes	—
121	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
122	C₅×M₁₆	—	6	4	no	—
123	C₂×C₅×C₈	—	3	4	no	—
124	C₅×C₄⋊C₄	—	2	4	no	—
125	C₅×C₄⋊C₄	—	2	4	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	6	2	no	—
128	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
129	C₅×C₂²⋊C₄	—	6	4	no	—
130	C₄²×C₅	#67	1	2	yes	—
131	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
133	#532	—	3	1	no	—
134	C₂×#125	C₅×D₃	1	1	yes	characteristic, complemented
135	#491	—	3	2	no	—
136	#490	—	3	2	no	—
137	#488	—	3	4	no	—
138	#417	—	3	4	no	—
139	C₄×M₁₆	—	3	1	no	—
140	C₂²×C₃×C₅	—	2	2	no	—
141	C₂²×C₃×C₅	#124	1	2	yes	—
142	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₅×D₃	—	4	2	no	—
144	C₂×C₅×D₃	—	4	4	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	—	4	4	no	—
148	C₃×C₄×C₅	#124	1	2	yes	—
149	C₃×C₄×C₅	C₂×#41	1	2	yes	—
150	C₅×Dic₃	—	4	2	no	—
151	C₂²×C₃×C₄	—	2	2	no	—
152	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
153	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂×#67	—	2	2	no	—
155	#235	—	2	8	no	—
156	C₂×D₁₂	—	2	2	no	—
157	C₂×D₁₂	C₂³×C₅	1	2	yes	—
158	C₂×C₄×D₃	—	2	2	no	—
159	C₂×Dic₆	C₂³×C₅	1	2	yes	—
160	C₃×C₄⋊C₄	—	2	4	no	—
161	C₃×C₄⋊C₄	—	2	4	no	—
162	C₃×C₄⋊C₄	—	2	2	no	—
163	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
164	C₃×C₄²	C₅×D₄	1	2	yes	—
165	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
166	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
167	(C₂×D₃)⋊C₄	—	2	4	no	—
168	C₄⋊Dic₃	—	2	2	no	—
169	#208	—	2	4	no	—
170	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
171	C₂³×C₅	—	6	2	no	—
172	C₂³×C₅	C₂×#67	1	1	yes	characteristic
173	C₅×Q₈	—	6	4	no	—
174	C₅×D₄	—	6	4	no	—
175	C₅×D₄	—	12	4	no	—
176	C₅×D₄	—	12	4	no	—
177	C₂×C₄×C₅	—	2	4	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	4	4	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	—	12	4	no	—
182	C₂×C₄×C₅	#239	1	4	yes	—
183	C₂×C₄×C₅	C₂×#67	1	2	yes	—
184	C₂×C₄×C₅	C₂×#67	1	2	yes	—
185	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
187	C₅×C₈	—	6	4	no	—
188	C₂×#41	—	3	2	no	—
189	C₂×M₁₆	—	3	2	no	—
190	C₄⋊Q₈	—	3	1	no	—
191	#127	—	3	1	no	—
192	#125	—	2	2	no	—
193	#125	C₂×C₅×D₃	1	2	yes	—
194	#125	C₂×C₅×D₃	1	2	yes	—
195	#124	—	3	2	no	—
196	#121	—	6	2	no	—
197	C₄×D₄	—	6	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
200	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
201	#103	—	3	8	no	—
202	#102	—	3	8	no	—
203	#97	—	3	2	no	—
204	C₄×C₈	—	3	2	no	—
205	C₂×C₃×C₅	—	2	2	no	—
206	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
207	C₂×C₃×C₅	#575	1	2	yes	—
208	C₅×D₃	—	8	2	no	—
209	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×D₃	—	2	2	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	4	no	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
215	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
216	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
217	C₂×C₃×C₄	C₅×#41	1	4	yes	—
218	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
219	#67	—	4	4	no	—
220	C₂×Dic₃	—	2	2	no	—
221	D₁₂	—	2	4	no	—
222	D₁₂	—	4	4	no	—
223	C₄×D₃	—	4	4	no	—
224	Dic₆	—	2	4	no	—
225	C₃⋊C₈	—	2	4	no	—
226	C₂²×C₅	—	2	2	no	—
227	C₂²×C₅	—	12	2	no	—
228	C₂²×C₅	—	12	4	no	—
229	C₂²×C₅	#947	1	2	yes	—
230	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
231	C₄×C₅	—	2	2	no	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	4	4	no	—
234	C₄×C₅	—	12	2	no	—
235	C₄×C₅	#947	1	2	yes	—
236	C₄×C₅	C₂×#239	1	2	yes	—
237	#41	—	6	8	no	—
238	C₂×Q₈	—	3	2	no	—
239	C₂×D₄	—	3	2	no	—
240	C₂×D₄	—	6	2	no	—
241	C₂×D₄	—	6	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	6	2	no	—
244	C₂²×C₄	C₅×#67	1	2	yes	—
245	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
246	M₁₆	—	6	4	no	—
247	C₂×C₈	—	3	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	6	2	no	—
252	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
253	C₂²⋊C₄	—	6	4	no	—
254	C₄²	C₅×#67	1	2	yes	—
255	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
256	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
257	C₃×C₅	#3142	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	C₅×#124	1	2	yes	—
260	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
261	C₂×D₃	—	4	2	no	—
262	C₂×D₃	—	4	4	no	—
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	4	4	no	—
266	C₃×C₄	C₅×#124	1	2	yes	—
267	C₃×C₄	C₂×C₅×#41	1	2	yes	—
268	Dic₃	—	4	2	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	24	2	no	—
271	C₂×C₅	C₂×#947	1	1	yes	characteristic
272	C₂×C₅	#5822	1	2	yes	—
273	C₂³	—	6	2	no	—
274	C₂³	C₂×C₅×#67	1	1	yes	characteristic
275	Q₈	—	6	4	no	—
276	D₄	—	6	4	no	—
277	D₄	—	12	4	no	—
278	D₄	—	12	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	6	2	no	—
283	C₂×C₄	—	12	4	no	—
284	C₂×C₄	C₅×#239	1	4	yes	—
285	C₂×C₄	C₂×C₅×#67	1	2	yes	—
286	C₂×C₄	C₂×C₅×#67	1	2	yes	—
287	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
288	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
289	C₈	—	6	4	no	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
292	C₂×C₃	C₅×#575	1	2	yes	—
293	D₃	—	8	2	no	—
294	C₅	#83012	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	12	2	no	—
297	C₂²	—	12	4	no	—
298	C₂²	C₅×#947	1	2	yes	—
299	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	4	no	—
303	C₄	—	12	2	no	—
304	C₄	C₅×#947	1	2	yes	—
305	C₄	C₂×C₅×#239	1	2	yes	—
306	C₃	C₅×#3142	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	24	2	no	—
309	C₂	C₂×C₅×#947	1	1	yes	characteristic
310	C₂	C₅×#5822	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#125	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5477	C₂
hypercenter	C₂×C₅×#125	D₃
hypocenter	C₃	C₅×#3142
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#125

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#125	C₂	semidirect
3. C₅×#5069	C₂	semidirect
4. C₅×#5068	C₂	semidirect
5. C₅×#5067	C₂	semidirect
6. C₅×#4966	C₂	semidirect
7. C₄×C₅×#208	C₂	semidirect
26. #83012	C₅	direct (N × H)
28. C₂×C₅×#125	D₃	semidirect
54. C₂×C₃×#125	C₂×C₅	semidirect
56. #5069	C₂×C₅	semidirect
57. #5068	C₂×C₅	semidirect
58. #5067	C₂×C₅	semidirect
59. #4966	C₂×C₅	semidirect
60. C₄×#208	C₂×C₅	semidirect
134. C₂×#125	C₅×D₃	semidirect
257. C₃×C₅	#3142	semidirect
294. C₅	#83012	direct (N × H)
306. C₃	C₅×#3142	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#125	2
C₅×#5477	2
C₅×#5069	2
C₅×#5068	2
C₅×#5067	2
C₅×#4966	2
C₄×C₅×#208	2
C₅×#3142	3
#83012	5

Subgroups — C5×#83012 #14438664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83012 #14438664