Subgroups — C₅×#81782 #14438892

Order 1920.

1144 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×#912	C₂	1	4	yes	maximal, complemented
2	C₅×#5716	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5155	C₂	1	4	yes	maximal, complemented
5	C₅×#5076	C₂	1	2	yes	maximal, complemented
6	C₅×#4699	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3147	—	3	1	no	maximal, Hall
9	C₂×C₅×#237	C₂²	1	4	yes	—
10	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
11	C₃×C₅×#127	C₂²	1	2	yes	—
12	C₅×#938	—	2	4	no	—
13	C₅×#936	C₂²	1	4	yes	—
14	C₅×D₄×Dic₃	—	2	4	no	—
15	C₂×C₅×#214	C₂²	1	8	yes	—
16	C₅×#912	—	2	16	no	—
17	C₂×C₅×Dic₁₂	C₂²	1	4	yes	—
18	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
19	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
20	C₅×#869	C₂²	1	2	yes	—
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×C₈×Dic₃	C₂²	1	2	yes	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#81782	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#135	—	3	4	no	—
26	C₅×#532	—	3	2	no	—
27	C₅×#495	—	3	4	no	—
28	C₅×#494	—	3	1	no	—
29	C₅×#493	D₃	1	1	yes	characteristic, complemented
30	C₅×#492	—	3	2	no	—
31	C₂×C₄×C₅×C₈	—	3	1	no	—
32	C₂×C₃×C₅×D₄	—	2	4	no	—
33	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
34	C₂×C₅×#67	—	2	4	no	—
35	C₂²×C₅×Dic₃	—	2	4	no	—
36	C₅×#237	—	2	16	no	—
37	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
39	C₂×C₅×Dic₆	C₂³	1	4	yes	—
40	C₃×C₅×D₈	—	2	8	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#217	—	2	8	no	—
44	C₅×#214	—	2	16	no	—
45	C₅×C₄⋊Dic₃	—	2	4	no	—
46	C₄×C₅×Dic₃	D₄	1	2	yes	—
47	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
48	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
49	C₅×Dic₁₂	—	2	8	no	—
50	C₅×D₃×C₈	D₄	1	8	yes	complemented
51	C₂×#912	C₂×C₅	1	4	yes	complemented
52	#5716	C₂×C₅	1	2	yes	complemented
53	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
54	#5155	C₂×C₅	1	4	yes	complemented
55	#5076	C₂×C₅	1	2	yes	complemented
56	#4699	C₂×C₅	1	1	yes	characteristic, complemented
57	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×#41	—	3	4	no	—
59	C₅×#135	—	6	16	no	—
60	C₂×C₅×Q₁₆	—	3	4	no	—
61	C₂×C₅×SD₁₆	—	3	8	no	—
62	C₂×C₅×D₈	C₂×D₃	1	4	yes	—
63	C₂²×C₅×C₈	—	3	2	no	—
64	C₅×C₄⋊Q₈	—	3	2	no	—
65	C₅×#127	C₂×D₃	1	2	yes	—
66	C₅×#124	—	3	4	no	—
67	C₅×#121	—	6	4	no	—
68	C₄×C₅×D₄	—	6	4	no	—
69	C₂×C₄²×C₅	—	3	1	no	—
70	C₄×C₅×C₈	—	3	2	no	—
71	C₄×C₅×C₈	—	3	1	no	—
72	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
73	#3147	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	2	4	no	—
75	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	8	no	—
77	C₃×C₅×D₄	—	4	8	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₅×#67	—	4	8	no	—
81	C₂×C₅×Dic₃	—	2	4	no	—
82	C₂×C₅×Dic₃	—	4	8	no	—
83	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
84	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
85	C₄×C₅×D₃	—	2	4	no	—
86	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
87	C₅×Dic₆	—	2	8	no	—
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
90	C₂×#237	C₂²×C₅	1	4	yes	—
91	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
92	C₃×#127	C₂²×C₅	1	2	yes	—
93	#938	—	2	4	no	—
94	#936	C₂²×C₅	1	4	yes	—
95	D₄×Dic₃	—	2	4	no	—
96	C₂×#214	C₂²×C₅	1	8	yes	—
97	#912	—	2	16	no	—
98	C₂×Dic₁₂	C₂²×C₅	1	4	yes	—
99	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
100	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
101	#869	C₂²×C₅	1	2	yes	—
102	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₈×Dic₃	C₂²×C₅	1	2	yes	—
104	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
105	C₅×#41	—	6	16	no	—
106	C₂×C₅×Q₈	—	3	4	no	—
107	C₂×C₅×D₄	—	2	4	no	—
108	C₂×C₅×D₄	—	6	4	no	—
109	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
110	C₂²×C₄×C₅	—	3	1	no	—
111	C₂²×C₄×C₅	—	3	2	no	—
112	C₂²×C₄×C₅	—	6	4	no	—
113	C₅×Q₁₆	—	6	8	no	—
114	C₅×SD₁₆	—	6	16	no	—
115	C₅×D₈	—	2	8	no	—
116	C₂×C₅×C₈	—	3	2	no	—
117	C₂×C₅×C₈	—	3	8	no	—
118	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
119	C₅×C₄⋊C₄	—	6	4	no	—
120	C₅×C₂²⋊C₄	—	6	8	no	—
121	C₄²×C₅	—	3	2	no	—
122	C₄²×C₅	—	3	1	no	—
123	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
124	C₂×#135	—	3	4	no	—
125	#532	—	3	2	no	—
126	#495	—	3	4	no	—
127	#494	—	3	1	no	—
128	#493	C₅×D₃	1	1	yes	characteristic, complemented
129	#492	—	3	2	no	—
130	C₂×C₄×C₈	—	3	1	no	—
131	C₂²×C₃×C₅	—	4	4	no	—
132	C₂²×C₃×C₅	—	4	8	no	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₅×D₃	—	2	2	no	—
135	C₂×C₅×D₃	#127	1	2	yes	—
136	C₃×C₄×C₅	#135	1	4	yes	complemented
137	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
138	C₅×Dic₃	—	2	2	no	—
139	C₅×Dic₃	—	4	4	no	—
140	C₅×Dic₃	#127	1	2	yes	—
141	C₂×C₃×D₄	—	2	4	no	—
142	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
143	C₂×#67	—	2	4	no	—
144	C₂²×Dic₃	—	2	4	no	—
145	#237	—	2	16	no	—
146	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
147	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
148	C₂×Dic₆	C₂³×C₅	1	4	yes	—
149	C₃×D₈	—	2	8	no	—
150	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
151	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
152	#217	—	2	8	no	—
153	#214	—	2	16	no	—
154	C₄⋊Dic₃	—	2	4	no	—
155	C₄×Dic₃	C₅×D₄	1	2	yes	—
156	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
157	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
158	Dic₁₂	—	2	8	no	—
159	D₃×C₈	C₅×D₄	1	8	yes	complemented
160	C₂³×C₅	—	2	4	no	—
161	C₂³×C₅	—	3	1	no	—
162	C₅×Q₈	—	6	8	no	—
163	C₅×D₄	—	2	8	no	—
164	C₅×D₄	—	4	8	no	—
165	C₅×D₄	—	12	8	no	—
166	C₂×C₄×C₅	—	3	1	no	—
167	C₂×C₄×C₅	—	3	4	no	—
168	C₂×C₄×C₅	—	3	2	no	—
169	C₂×C₄×C₅	—	6	4	no	—
170	C₂×C₄×C₅	—	6	4	no	—
171	C₂×C₄×C₅	—	12	8	no	—
172	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
174	C₅×C₈	—	3	4	no	—
175	C₅×C₈	D₃×D₄	1	4	yes	—
176	C₂×#41	—	3	4	no	—
177	#135	—	6	16	no	—
178	C₂×Q₁₆	—	3	4	no	—
179	C₂×SD₁₆	—	3	8	no	—
180	C₂×D₈	C₂×C₅×D₃	1	4	yes	—
181	C₂²×C₈	—	3	2	no	—
182	C₄⋊Q₈	—	3	2	no	—
183	#127	C₂×C₅×D₃	1	2	yes	—
184	#124	—	3	4	no	—
185	#121	—	6	4	no	—
186	C₄×D₄	—	6	4	no	—
187	C₂×C₄²	—	3	1	no	—
188	C₄×C₈	—	3	2	no	—
189	C₄×C₈	—	3	1	no	—
190	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
191	C₂×C₃×C₅	—	8	4	no	—
192	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂×#135	1	2	yes	—
194	C₅×D₃	—	2	2	no	—
195	C₂³×C₃	—	2	4	no	—
196	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
197	C₃×D₄	—	2	8	no	—
198	C₃×D₄	—	4	8	no	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
201	#67	—	4	8	no	—
202	C₂×Dic₃	—	2	4	no	—
203	C₂×Dic₃	—	4	8	no	—
204	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
205	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
206	C₄×D₃	—	2	4	no	—
207	C₄×D₃	C₂×C₅×D₄	1	4	yes	—
208	Dic₆	—	2	8	no	—
209	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
210	C₃⋊C₈	C₂×C₅×D₄	1	4	yes	—
211	C₂²×C₅	—	3	2	no	—
212	C₂²×C₅	—	4	4	no	—
213	C₂²×C₅	—	4	8	no	—
214	C₂²×C₅	—	6	2	no	—
215	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
216	C₄×C₅	—	3	2	no	—
217	C₄×C₅	—	6	2	no	—
218	C₄×C₅	—	12	4	no	—
219	C₄×C₅	#912	1	4	yes	complemented
220	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
221	#41	—	6	16	no	—
222	C₂×Q₈	—	3	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	6	4	no	—
225	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	—	6	4	no	—
229	Q₁₆	—	6	8	no	—
230	SD₁₆	—	6	16	no	—
231	D₈	—	2	8	no	—
232	C₂×C₈	—	3	2	no	—
233	C₂×C₈	—	3	8	no	—
234	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
235	C₄⋊C₄	—	6	4	no	—
236	C₂²⋊C₄	—	6	8	no	—
237	C₄²	—	3	2	no	—
238	C₄²	—	3	1	no	—
239	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
240	C₃×C₅	#3147	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	4	4	no	—
242	C₂²×C₃	—	4	8	no	—
243	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
244	C₂×D₃	—	2	2	no	—
245	C₂×D₃	C₅×#127	1	2	yes	—
246	C₃×C₄	C₅×#135	1	4	yes	complemented
247	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
248	Dic₃	—	2	2	no	—
249	Dic₃	—	4	4	no	—
250	Dic₃	C₅×#127	1	2	yes	—
251	C₂×C₅	—	6	2	no	—
252	C₂×C₅	—	8	4	no	—
253	C₂×C₅	D₃×#127	1	1	yes	characteristic
254	C₂×C₅	C₂×#912	1	2	yes	—
255	C₂³	—	2	4	no	—
256	C₂³	—	3	1	no	—
257	Q₈	—	6	8	no	—
258	D₄	—	2	8	no	—
259	D₄	—	4	8	no	—
260	D₄	—	12	8	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	3	4	no	—
263	C₂×C₄	—	3	2	no	—
264	C₂×C₄	—	6	4	no	—
265	C₂×C₄	—	6	4	no	—
266	C₂×C₄	—	12	8	no	—
267	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
268	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
269	C₈	—	3	4	no	—
270	C₈	C₅×D₃×D₄	1	4	yes	—
271	C₂×C₃	—	8	4	no	—
272	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
273	C₂×C₃	C₂×C₅×#135	1	2	yes	—
274	D₃	—	2	2	no	—
275	C₅	#81782	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	3	2	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	8	no	—
279	C₂²	—	6	2	no	—
280	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
281	C₄	—	3	2	no	—
282	C₄	—	6	2	no	—
283	C₄	—	12	4	no	—
284	C₄	C₅×#912	1	4	yes	complemented
285	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
286	C₃	C₅×#3147	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	6	2	no	—
288	C₂	—	8	4	no	—
289	C₂	C₅×D₃×#127	1	1	yes	characteristic
290	C₂	C₂×C₅×#912	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#493	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#493	D₃
hypocenter	C₃	C₅×#3147
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#493

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#912	C₂	semidirect
2. C₅×#5716	C₂	semidirect
3. C₃×C₅×#493	C₂	semidirect
4. C₅×#5155	C₂	semidirect
5. C₅×#5076	C₂	semidirect
6. C₅×#4699	C₂	semidirect
7. C₄×C₅×D₃×C₈	C₂	semidirect
24. #81782	C₅	direct (N × H)
29. C₅×#493	D₃	semidirect
50. C₅×D₃×C₈	D₄	semidirect
51. C₂×#912	C₂×C₅	semidirect
52. #5716	C₂×C₅	semidirect
53. C₃×#493	C₂×C₅	semidirect
54. #5155	C₂×C₅	semidirect
55. #5076	C₂×C₅	semidirect
56. #4699	C₂×C₅	semidirect
57. C₄×D₃×C₈	C₂×C₅	semidirect
128. #493	C₅×D₃	semidirect
136. C₃×C₄×C₅	#135	semidirect
159. D₃×C₈	C₅×D₄	semidirect
219. C₄×C₅	#912	semidirect
240. C₃×C₅	#3147	semidirect
246. C₃×C₄	C₅×#135	semidirect
275. C₅	#81782	direct (N × H)
284. C₄	C₅×#912	semidirect
286. C₃	C₅×#3147	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#912	2
C₅×#5716	2
C₃×C₅×#493	2
C₅×#5155	2
C₅×#5076	2
C₅×#4699	2
C₄×C₅×D₃×C₈	2
C₅×#3147	3
#81782	5

Subgroups — C5×#81782 #14438892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81782 #14438892