Subgroups — C₅×#69580 #14305801

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#809	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1957	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	2	no	—
7	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	C₅×#874	—	2	2	no	—
13	C₅×#874	C₄	1	4	yes	complemented
14	C₅×#809	—	2	8	no	—
15	C₄×C₅×C₃⋊C₈	—	2	2	no	—
16	#69580	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#127	—	3	1	no	—
18	C₂×C₅×#102	—	3	4	no	—
19	C₂×C₄×C₅×C₈	—	3	1	no	—
20	C₅×#384	D₃	1	1	yes	characteristic, complemented
21	C₂³×C₅×D₃	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×D₁₂	—	2	4	no	—
26	C₂×C₅×D₁₂	—	2	4	no	—
27	C₂×C₅×D₁₂	—	2	8	no	—
28	C₂×C₅×D₁₂	—	2	2	no	—
29	C₂×C₅×D₁₂	—	4	4	no	—
30	C₂×C₅×D₁₂	—	4	4	no	—
31	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₄⋊C₄	—	4	4	no	—
34	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
39	C₂×#874	C₂×C₅	1	1	yes	characteristic
40	C₃×#384	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×#809	C₂×C₅	1	4	yes	complemented
42	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂²×C₅×D₄	—	3	2	no	—
44	C₂²×C₅×D₄	—	6	2	no	—
45	C₂²×C₅×C₈	—	3	2	no	—
46	C₅×#127	—	3	4	no	—
47	C₅×#127	—	6	2	no	—
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₅×#102	—	6	8	no	—
52	C₄×C₅×C₈	—	6	2	no	—
53	#1957	—	3	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₅×D₃	—	2	4	no	—
56	C₂²×C₅×D₃	—	4	2	no	—
57	C₂²×C₅×D₃	—	4	2	no	—
58	C₂²×C₅×D₃	—	4	4	no	—
59	C₂²×C₅×D₃	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	4	4	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
64	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	2	yes	—
69	C₅×D₁₂	—	2	8	no	—
70	C₅×D₁₂	—	4	16	no	—
71	C₅×D₁₂	—	4	4	no	—
72	C₅×C₃⋊C₈	—	2	4	no	—
73	C₂²×D₁₂	—	2	2	no	—
74	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
75	C₂×C₃×C₄⋊C₄	—	2	2	no	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
77	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
78	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
79	#874	—	2	2	no	—
80	#874	C₄×C₅	1	4	yes	complemented
81	#809	—	2	8	no	—
82	C₄×C₃⋊C₈	—	2	2	no	—
83	C₂⁴×C₅	—	6	2	no	—
84	C₂×C₅×D₄	—	3	4	no	—
85	C₂×C₅×D₄	—	6	8	no	—
86	C₂×C₅×D₄	—	6	2	no	—
87	C₂×C₅×D₄	—	6	4	no	—
88	C₂×C₅×D₄	—	6	4	no	—
89	C₂×C₅×D₄	—	12	4	no	—
90	C₂×C₅×D₄	—	12	4	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	#67	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂×C₅×C₈	—	3	4	no	—
95	C₂×C₅×C₈	—	6	2	no	—
96	C₂×C₅×C₈	—	6	2	no	—
97	C₅×C₄⋊C₄	—	2	4	no	—
98	C₅×C₄⋊C₄	—	4	4	no	—
99	C₄²×C₅	C₄×D₃	1	2	yes	—
100	C₄²×C₅	D₁₂	1	2	yes	—
101	C₂×#127	—	3	1	no	—
102	C₂×#102	—	3	4	no	—
103	C₂×C₄×C₈	—	3	1	no	—
104	#384	C₅×D₃	1	1	yes	characteristic, complemented
105	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂²×C₃×C₅	#124	1	1	yes	characteristic
107	C₂²×C₃×C₅	#127	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
110	C₂×C₅×D₃	—	4	4	no	—
111	C₂×C₅×D₃	—	4	8	no	—
112	C₂×C₅×D₃	—	8	4	no	—
113	C₂×C₅×D₃	—	8	2	no	—
114	C₂×C₅×D₃	—	8	2	no	—
115	C₃×C₄×C₅	—	8	2	no	—
116	C₃×C₄×C₅	#102	1	8	yes	complemented
117	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₂³×D₃	—	2	2	no	—
120	C₂²×C₃×C₄	—	2	2	no	—
121	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
122	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
123	C₂×D₁₂	—	2	4	no	—
124	C₂×D₁₂	—	2	4	no	—
125	C₂×D₁₂	—	2	8	no	—
126	C₂×D₁₂	—	2	2	no	—
127	C₂×D₁₂	—	4	4	no	—
128	C₂×D₁₂	—	4	4	no	—
129	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
130	C₃×C₄⋊C₄	—	2	4	no	—
131	C₃×C₄⋊C₄	—	4	4	no	—
132	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
133	C₃×C₄²	C₅×D₄	1	2	yes	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	C₂×C₃⋊C₈	—	2	2	no	—
136	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	complemented
137	C₂³×C₅	—	6	4	no	—
138	C₂³×C₅	—	12	4	no	—
139	C₂³×C₅	—	12	2	no	—
140	C₂³×C₅	—	12	4	no	—
141	C₂³×C₅	—	12	2	no	—
142	C₂³×C₅	C₂×#67	1	1	yes	characteristic
143	C₅×D₄	—	6	8	no	—
144	C₅×D₄	—	12	16	no	—
145	C₅×D₄	—	12	4	no	—
146	C₂×C₄×C₅	—	4	4	no	—
147	C₂×C₄×C₅	—	4	2	no	—
148	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	#213	1	4	yes	—
150	C₂×C₄×C₅	#215	1	4	yes	—
151	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
153	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
154	C₂×C₄×C₅	#239	1	2	yes	—
155	C₅×C₈	—	6	4	no	—
156	C₂²×D₄	—	3	2	no	—
157	C₂²×D₄	—	6	2	no	—
158	C₂²×C₈	—	3	2	no	—
159	#127	—	3	4	no	—
160	#127	—	6	2	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
163	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
164	#102	—	6	8	no	—
165	C₄×C₈	—	6	2	no	—
166	C₂×C₃×C₅	#390	1	1	yes	characteristic
167	C₂×C₃×C₅	C₂×#102	1	2	yes	—
168	C₂×C₃×C₅	#486	1	2	yes	—
169	C₂×C₃×C₅	#492	1	1	yes	characteristic
170	C₂×C₃×C₅	#493	1	1	yes	characteristic
171	C₅×D₃	—	8	4	no	—
172	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×D₃	—	2	4	no	—
174	C₂²×D₃	—	4	2	no	—
175	C₂²×D₃	—	4	2	no	—
176	C₂²×D₃	—	4	4	no	—
177	C₂²×D₃	—	4	4	no	—
178	C₂×C₃×C₄	—	4	2	no	—
179	C₂×C₃×C₄	—	4	4	no	—
180	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
181	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
182	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
183	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
186	C₂×C₃×C₄	C₅×#41	1	2	yes	—
187	D₁₂	—	2	8	no	—
188	D₁₂	—	4	4	no	—
189	D₁₂	—	4	16	no	—
190	C₃⋊C₈	—	2	4	no	—
191	C₂²×C₅	—	12	8	no	—
192	C₂²×C₅	—	12	4	no	—
193	C₂²×C₅	—	24	2	no	—
194	C₂²×C₅	—	24	2	no	—
195	C₂²×C₅	—	24	4	no	—
196	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂²×C₅	C₂×#213	1	2	yes	—
198	C₂²×C₅	#940	1	1	yes	characteristic
199	C₂²×C₅	C₂×#215	1	2	yes	—
200	C₂²×C₅	#947	1	1	yes	characteristic
201	C₄×C₅	—	8	2	no	—
202	C₄×C₅	#809	1	8	yes	complemented
203	C₄×C₅	#893	1	2	yes	—
204	C₄×C₅	#895	1	2	yes	—
205	C₂⁴	—	6	2	no	—
206	C₂×D₄	—	3	4	no	—
207	C₂×D₄	—	6	8	no	—
208	C₂×D₄	—	6	2	no	—
209	C₂×D₄	—	6	4	no	—
210	C₂×D₄	—	6	4	no	—
211	C₂×D₄	—	12	4	no	—
212	C₂×D₄	—	12	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₅×#67	1	2	yes	—
215	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
216	C₂×C₈	—	3	4	no	—
217	C₂×C₈	—	6	2	no	—
218	C₂×C₈	—	6	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄²	C₄×C₅×D₃	1	2	yes	—
222	C₄²	C₅×D₁₂	1	2	yes	—
223	C₃×C₅	#1957	1	1	yes	characteristic, complemented, Hall
224	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
225	C₂²×C₃	C₅×#124	1	1	yes	characteristic
226	C₂²×C₃	C₅×#127	1	1	yes	characteristic
227	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
228	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
229	C₂×D₃	—	4	4	no	—
230	C₂×D₃	—	4	8	no	—
231	C₂×D₃	—	8	4	no	—
232	C₂×D₃	—	8	2	no	—
233	C₂×D₃	—	8	2	no	—
234	C₃×C₄	—	8	2	no	—
235	C₃×C₄	C₅×#102	1	8	yes	complemented
236	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
237	C₃×C₄	C₅×#121	1	2	yes	—
238	C₂×C₅	—	24	4	no	—
239	C₂×C₅	C₂×#809	1	2	yes	—
240	C₂×C₅	#4988	1	1	yes	characteristic
241	C₂×C₅	#5073	1	1	yes	characteristic
242	C₂×C₅	#5085	1	1	yes	characteristic
243	C₂×C₅	#5088	1	2	yes	—
244	C₂³	—	6	4	no	—
245	C₂³	—	12	4	no	—
246	C₂³	—	12	2	no	—
247	C₂³	—	12	4	no	—
248	C₂³	—	12	2	no	—
249	C₂³	C₂×C₅×#67	1	1	yes	characteristic
250	D₄	—	6	8	no	—
251	D₄	—	12	4	no	—
252	D₄	—	12	16	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
256	C₂×C₄	C₅×#213	1	4	yes	—
257	C₂×C₄	C₅×#215	1	4	yes	—
258	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
260	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
261	C₂×C₄	C₅×#239	1	2	yes	—
262	C₈	—	6	4	no	—
263	C₂×C₃	C₅×#390	1	1	yes	characteristic
264	C₂×C₃	C₂×C₅×#102	1	2	yes	—
265	C₂×C₃	C₅×#486	1	2	yes	—
266	C₂×C₃	C₅×#492	1	1	yes	characteristic
267	C₂×C₃	C₅×#493	1	1	yes	characteristic
268	D₃	—	8	4	no	—
269	C₅	#69580	1	1	yes	characteristic, direct product factor, Sylow
270	C₂²	—	12	8	no	—
271	C₂²	—	12	4	no	—
272	C₂²	—	24	2	no	—
273	C₂²	—	24	2	no	—
274	C₂²	—	24	4	no	—
275	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
276	C₂²	C₂×C₅×#213	1	2	yes	—
277	C₂²	C₅×#940	1	1	yes	characteristic
278	C₂²	C₂×C₅×#215	1	2	yes	—
279	C₂²	C₅×#947	1	1	yes	characteristic
280	C₄	—	8	2	no	—
281	C₄	C₅×#809	1	8	yes	complemented
282	C₄	C₅×#893	1	2	yes	—
283	C₄	C₅×#895	1	2	yes	—
284	C₃	C₅×#1957	1	1	yes	characteristic, complemented, Sylow
285	C₂	—	24	4	no	—
286	C₂	C₂×C₅×#809	1	2	yes	—
287	C₂	C₅×#4988	1	1	yes	characteristic
288	C₂	C₅×#5073	1	1	yes	characteristic
289	C₂	C₅×#5085	1	1	yes	characteristic
290	C₂	C₅×#5088	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#384	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂×C₅×#874	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#384	D₃
hypocenter	C₃	C₅×#1957
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#384

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#384	C₂	semidirect
3. C₂×C₅×#809	C₂	semidirect
4. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
13. C₅×#874	C₄	semidirect
16. #69580	C₅	direct (N × H)
20. C₅×#384	D₃	semidirect
38. C₂×C₅×C₃⋊C₈	D₄	semidirect
40. C₃×#384	C₂×C₅	semidirect
41. C₂×#809	C₂×C₅	semidirect
42. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
80. #874	C₄×C₅	semidirect
104. #384	C₅×D₃	semidirect
116. C₃×C₄×C₅	#102	semidirect
136. C₂×C₃⋊C₈	C₅×D₄	semidirect
202. C₄×C₅	#809	semidirect
223. C₃×C₅	#1957	semidirect
235. C₃×C₄	C₅×#102	semidirect
269. C₅	#69580	direct (N × H)
281. C₄	C₅×#809	semidirect
284. C₃	C₅×#1957	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#874	2
C₃×C₅×#384	2
C₂×C₅×#809	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#1957	3
#69580	5

Subgroups — C5×#69580 #14305801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69580 #14305801