Subgroups — C₅×#81799 #14438898

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×#919	C₂	1	4	yes	maximal, complemented
2	C₅×#5725	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5191	C₂	1	4	yes	maximal, complemented
5	C₅×#5085	C₂	1	2	yes	maximal, complemented
6	C₅×#4695	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×#239	C₂²	1	4	yes	—
10	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₅×#947	C₂²	1	4	yes	—
13	C₂×C₅×#215	C₂²	1	8	yes	—
14	C₅×#919	—	2	16	no	—
15	C₂×C₅×D₂₄	C₂²	1	4	yes	—
16	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
17	C₅×#895	—	2	4	no	—
18	C₅×#893	—	2	4	no	—
19	C₅×#874	C₂²	1	2	yes	—
20	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
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	#81799	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	D₃	1	1	yes	characteristic, complemented
29	C₅×#493	—	3	1	no	—
30	C₅×#492	—	3	2	no	—
31	C₂×C₄×C₅×C₈	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
33	C₅×#239	—	2	16	no	—
34	C₂×C₅×D₁₂	—	2	4	no	—
35	C₂×C₅×D₁₂	—	2	4	no	—
36	C₂×C₅×D₁₂	C₂³	1	4	yes	—
37	C₂×C₄×C₅×D₃	—	2	4	no	—
38	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
39	C₂×C₄×C₅×D₃	C₂³	1	1	yes	characteristic
40	C₃×C₅×Q₁₆	—	2	8	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#215	—	2	16	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	8	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×D₂₄	—	2	8	no	—
50	C₅×D₃×C₈	D₄	1	8	yes	complemented
51	C₂×#919	C₂×C₅	1	4	yes	complemented
52	#5725	C₂×C₅	1	2	yes	complemented
53	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
54	#5191	C₂×C₅	1	4	yes	complemented
55	#5085	C₂×C₅	1	2	yes	complemented
56	#4695	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	C₂×D₃	1	4	yes	—
61	C₂×C₅×SD₁₆	—	3	8	no	—
62	C₂×C₅×D₈	—	3	4	no	—
63	C₂²×C₅×C₈	—	3	2	no	—
64	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
65	C₅×#127	—	3	2	no	—
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	C₂²×C₅×D₃	—	2	4	no	—
75	C₂²×C₅×D₃	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×Q₈	—	2	8	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
80	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
81	C₂×C₅×Dic₃	C₂×D₄	1	1	yes	characteristic
82	C₅×D₁₂	—	2	8	no	—
83	C₅×D₁₂	—	4	8	no	—
84	C₅×D₁₂	—	4	8	no	—
85	C₄×C₅×D₃	—	2	4	no	—
86	C₄×C₅×D₃	—	4	8	no	—
87	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
88	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
89	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
90	C₂×#239	C₂²×C₅	1	4	yes	—
91	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
92	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
93	#947	C₂²×C₅	1	4	yes	—
94	C₂×#215	C₂²×C₅	1	8	yes	—
95	#919	—	2	16	no	—
96	C₂×D₂₄	C₂²×C₅	1	4	yes	—
97	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
98	#895	—	2	4	no	—
99	#893	—	2	4	no	—
100	#874	C₂²×C₅	1	2	yes	—
101	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
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	C₂²×D₃	1	4	yes	—
107	C₂×C₅×D₄	—	3	4	no	—
108	C₂×C₅×D₄	—	6	4	no	—
109	C₂×C₅×D₄	—	6	4	no	—
110	C₂²×C₄×C₅	—	3	1	no	—
111	C₂²×C₄×C₅	—	3	2	no	—
112	C₂²×C₄×C₅	—	6	4	no	—
113	C₅×Q₁₆	—	2	8	no	—
114	C₅×SD₁₆	—	6	16	no	—
115	C₅×D₈	—	6	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	—	2	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	C₅×D₃	1	1	yes	characteristic, complemented
128	#493	—	3	1	no	—
129	#492	—	3	2	no	—
130	C₂×C₄×C₈	—	3	1	no	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₅×D₃	—	2	2	no	—
133	C₂×C₅×D₃	—	4	4	no	—
134	C₂×C₅×D₃	—	4	8	no	—
135	C₂×C₅×D₃	#127	1	2	yes	—
136	C₃×C₄×C₅	—	4	4	no	—
137	C₃×C₄×C₅	#135	1	4	yes	complemented
138	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
139	C₅×Dic₃	—	2	2	no	—
140	C₅×Dic₃	#127	1	2	yes	—
141	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
142	#239	—	2	16	no	—
143	C₂×D₁₂	—	2	4	no	—
144	C₂×D₁₂	—	2	4	no	—
145	C₂×D₁₂	C₂³×C₅	1	4	yes	—
146	C₂×C₄×D₃	—	2	4	no	—
147	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
148	C₂×C₄×D₃	C₂³×C₅	1	1	yes	characteristic
149	C₃×Q₁₆	—	2	8	no	—
150	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
153	#215	—	2	16	no	—
154	(C₂×D₃)⋊C₄	—	2	8	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	D₂₄	—	2	8	no	—
159	D₃×C₈	C₅×D₄	1	8	yes	complemented
160	C₂³×C₅	—	3	1	no	—
161	C₂³×C₅	—	6	4	no	—
162	C₅×Q₈	—	2	8	no	—
163	C₅×D₄	—	6	8	no	—
164	C₅×D₄	—	12	8	no	—
165	C₅×D₄	—	12	8	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	3	1	no	—
168	C₂×C₄×C₅	—	3	4	no	—
169	C₂×C₄×C₅	—	3	2	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	C₂×C₅×D₃	1	4	yes	—
179	C₂×SD₁₆	—	3	8	no	—
180	C₂×D₈	—	3	4	no	—
181	C₂²×C₈	—	3	2	no	—
182	C₄⋊Q₈	C₂×C₅×D₃	1	2	yes	—
183	#127	—	3	2	no	—
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	C₂×#127	1	1	yes	characteristic
192	C₂×C₃×C₅	C₂×#135	1	2	yes	—
193	C₅×D₃	—	2	2	no	—
194	C₅×D₃	—	8	4	no	—
195	C₂²×D₃	—	2	4	no	—
196	C₂²×D₃	C₂×C₅×D₄	1	1	yes	characteristic
197	C₃×Q₈	—	2	8	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
201	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
202	C₂×Dic₃	C₂×C₅×D₄	1	1	yes	characteristic
203	D₁₂	—	2	8	no	—
204	D₁₂	—	4	8	no	—
205	D₁₂	—	4	8	no	—
206	C₄×D₃	—	2	4	no	—
207	C₄×D₃	—	4	8	no	—
208	C₄×D₃	C₂×C₅×D₄	1	4	yes	—
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	—	6	2	no	—
213	C₂²×C₅	—	12	4	no	—
214	C₂²×C₅	—	12	8	no	—
215	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
216	C₄×C₅	—	3	2	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	—	6	2	no	—
219	C₄×C₅	#919	1	4	yes	complemented
220	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
221	#41	—	6	16	no	—
222	C₂×Q₈	C₂²×C₅×D₃	1	4	yes	—
223	C₂×D₄	—	3	4	no	—
224	C₂×D₄	—	6	4	no	—
225	C₂×D₄	—	6	4	no	—
226	C₂²×C₄	—	3	1	no	—
227	C₂²×C₄	—	3	2	no	—
228	C₂²×C₄	—	6	4	no	—
229	Q₁₆	—	2	8	no	—
230	SD₁₆	—	6	16	no	—
231	D₈	—	6	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	—	2	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	C₂²×C₅×D₄	1	1	yes	characteristic
242	C₂×D₃	—	2	2	no	—
243	C₂×D₃	—	4	4	no	—
244	C₂×D₃	—	4	8	no	—
245	C₂×D₃	C₅×#127	1	2	yes	—
246	C₃×C₄	—	4	4	no	—
247	C₃×C₄	C₅×#135	1	4	yes	complemented
248	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
249	Dic₃	—	2	2	no	—
250	Dic₃	C₅×#127	1	2	yes	—
251	C₂×C₅	—	6	2	no	—
252	C₂×C₅	—	24	4	no	—
253	C₂×C₅	D₃×#127	1	1	yes	characteristic
254	C₂×C₅	C₂×#919	1	2	yes	—
255	C₂³	—	3	1	no	—
256	C₂³	—	6	4	no	—
257	Q₈	—	2	8	no	—
258	D₄	—	6	8	no	—
259	D₄	—	12	8	no	—
260	D₄	—	12	8	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	3	1	no	—
263	C₂×C₄	—	3	4	no	—
264	C₂×C₄	—	3	2	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	C₂×C₅×#127	1	1	yes	characteristic
272	C₂×C₃	C₂×C₅×#135	1	2	yes	—
273	D₃	—	2	2	no	—
274	D₃	—	8	4	no	—
275	C₅	#81799	1	1	yes	characteristic, direct product factor, Sylow
276	C₂²	—	3	2	no	—
277	C₂²	—	6	2	no	—
278	C₂²	—	12	4	no	—
279	C₂²	—	12	8	no	—
280	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
281	C₄	—	3	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	6	2	no	—
284	C₄	C₅×#919	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	—	24	4	no	—
289	C₂	C₅×D₃×#127	1	1	yes	characteristic
290	C₂	C₂×C₅×#919	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×#494	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
hypercenter	C₅×#494	D₃
hypocenter	C₃	C₅×#3147
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#494

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#919	C₂	semidirect
2. C₅×#5725	C₂	semidirect
3. C₃×C₅×#494	C₂	semidirect
4. C₅×#5191	C₂	semidirect
5. C₅×#5085	C₂	semidirect
6. C₅×#4695	C₂	semidirect
7. C₄×C₅×D₃×C₈	C₂	semidirect
24. #81799	C₅	direct (N × H)
28. C₅×#494	D₃	semidirect
50. C₅×D₃×C₈	D₄	semidirect
51. C₂×#919	C₂×C₅	semidirect
52. #5725	C₂×C₅	semidirect
53. C₃×#494	C₂×C₅	semidirect
54. #5191	C₂×C₅	semidirect
55. #5085	C₂×C₅	semidirect
56. #4695	C₂×C₅	semidirect
57. C₄×D₃×C₈	C₂×C₅	semidirect
127. #494	C₅×D₃	semidirect
137. C₃×C₄×C₅	#135	semidirect
159. D₃×C₈	C₅×D₄	semidirect
219. C₄×C₅	#919	semidirect
240. C₃×C₅	#3147	semidirect
247. C₃×C₄	C₅×#135	semidirect
275. C₅	#81799	direct (N × H)
284. C₄	C₅×#919	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×#919	2
C₅×#5725	2
C₃×C₅×#494	2
C₅×#5191	2
C₅×#5085	2
C₅×#4695	2
C₄×C₅×D₃×C₈	2
C₅×#3147	3
#81799	5

Subgroups — C5×#81799 #14438898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#81799 #14438898