Subgroups — C₂×C₅×#4709 #14413832

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×(C₂×D₃)⋊C₈	C₂	1	2	yes	maximal
4	C₂×C₅×Dic₃⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4709	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#433	—	3	1	no	maximal, Hall
8	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
12	C₂×C₅×Dic₃⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₅×#872	C₄	1	4	yes	—
16	C₅×#872	C₂²	1	4	yes	—
17	C₃×C₅×#97	C₂²	1	4	yes	complemented
18	C₅×(C₂×D₃)⋊C₈	C₂²	1	8	yes	—
19	C₅×Dic₃⋊C₈	C₂²	1	8	yes	complemented
20	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
21	C₂×#4709	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#117	—	3	1	no	—
23	C₅×#433	—	3	8	no	—
24	C₂×C₅×C₄⋊C₈	—	3	2	no	—
25	C₂×C₅×C₂²⋊C₈	—	3	2	no	—
26	C₂×C₅×#97	D₃	1	1	yes	characteristic, complemented
27	C₂×C₄×C₅×C₈	—	3	1	no	—
28	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
32	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	—	2	4	no	—
34	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
36	C₂×C₃×C₅×C₈	—	2	4	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
38	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	2	yes	—
40	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
41	C₅×Dic₃⋊C₄	C₂×C₄	1	8	yes	—
42	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
43	C₄×C₅×Dic₃	C₂³	1	2	yes	—
44	C₂×C₅×C₃⋊C₈	—	2	4	no	—
45	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
46	C₂×#872	C₂×C₅	1	1	yes	characteristic
47	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×(C₂×D₃)⋊C₈	C₂×C₅	1	2	yes	—
49	C₂×Dic₃⋊C₈	C₂×C₅	1	2	yes	complemented
50	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
51	#4709	C₂×C₅	1	8	yes	direct product factor
52	C₂³×C₄×C₅	—	3	1	no	—
53	C₂²×C₅×C₈	—	3	2	no	—
54	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
55	C₅×#117	—	3	4	no	—
56	C₅×#117	—	3	4	no	—
57	C₂×C₅×C₄⋊C₄	—	3	2	no	—
58	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
59	C₂×C₄²×C₅	—	3	1	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₅×C₄⋊C₈	—	3	8	no	—
62	C₅×C₂²⋊C₈	—	3	8	no	—
63	C₅×#97	C₂×D₃	1	4	yes	complemented
64	C₄×C₅×C₈	—	3	4	no	—
65	C₂×#433	—	3	1	no	Sylow
66	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₅×D₃	—	2	2	no	—
68	C₂²×C₅×D₃	—	2	4	no	—
69	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	4	yes	—
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₂×C₅×Dic₃	—	2	2	no	—
76	C₂×C₅×Dic₃	—	2	4	no	—
77	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
78	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
79	C₄×C₅×D₃	—	2	4	no	—
80	C₄×C₅×D₃	—	2	4	no	—
81	C₃×C₅×C₈	—	2	4	no	—
82	C₅×C₃⋊C₈	—	2	4	no	—
83	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
84	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
85	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
86	C₂×(C₂×D₃)⋊C₄	C₄×C₅	1	2	yes	—
87	C₂×Dic₃⋊C₄	C₄×C₅	1	2	yes	—
88	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
89	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
90	#872	C₄×C₅	1	4	yes	—
91	#872	C₂²×C₅	1	4	yes	—
92	C₃×#97	C₂²×C₅	1	4	yes	complemented
93	(C₂×D₃)⋊C₈	C₂²×C₅	1	8	yes	—
94	Dic₃⋊C₈	C₂²×C₅	1	8	yes	complemented
95	C₄×C₃⋊C₈	C₂²×C₅	1	4	yes	complemented
96	C₂⁴×C₅	—	3	1	no	—
97	C₂²×C₄×C₅	—	3	2	no	—
98	C₂²×C₄×C₅	—	3	1	no	—
99	C₂²×C₄×C₅	—	3	2	no	—
100	C₂²×C₄×C₅	—	3	2	no	—
101	C₂²×C₄×C₅	—	6	4	no	—
102	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
103	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
104	C₂×C₅×C₈	—	2	4	no	—
105	C₂×C₅×C₈	—	3	4	no	—
106	C₂×C₅×C₈	—	6	4	no	—
107	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
108	C₅×C₄⋊C₄	—	3	8	no	—
109	C₅×C₂²⋊C₄	—	3	8	no	—
110	C₄²×C₅	—	3	2	no	—
111	C₄²×C₅	—	3	2	no	—
112	C₄²×C₅	C₄×D₃	1	2	yes	—
113	C₄²×C₅	C₂²×D₃	1	2	yes	—
114	C₂×#117	—	3	1	no	—
115	#433	—	3	8	no	—
116	C₂×C₄⋊C₈	—	3	2	no	—
117	C₂×C₂²⋊C₈	—	3	2	no	—
118	C₂×#97	C₅×D₃	1	1	yes	characteristic, complemented
119	C₂×C₄×C₈	—	3	1	no	—
120	C₂²×C₃×C₅	#117	1	2	yes	—
121	C₂²×C₃×C₅	#131	1	4	yes	—
122	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₅×D₃	—	2	2	no	—
124	C₂×C₅×D₃	—	2	4	no	—
125	C₂×C₅×D₃	—	4	4	no	—
126	C₃×C₄×C₅	—	2	4	no	—
127	C₃×C₄×C₅	#117	1	2	yes	—
128	C₃×C₄×C₅	C₂×#41	1	2	yes	—
129	C₅×Dic₃	—	2	2	no	—
130	C₅×Dic₃	—	2	4	no	—
131	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
133	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
134	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
135	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
136	C₂×C₄×D₃	—	2	4	no	—
137	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
139	C₂×C₃×C₈	—	2	4	no	—
140	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
141	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
142	C₃×C₄²	C₂³×C₅	1	2	yes	—
143	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	—
144	Dic₃⋊C₄	C₂×C₄×C₅	1	8	yes	—
145	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
146	C₄×Dic₃	C₂³×C₅	1	2	yes	—
147	C₂×C₃⋊C₈	—	2	4	no	—
148	C₂×C₃⋊C₈	C₂³×C₅	1	4	yes	—
149	C₂³×C₅	—	3	2	no	—
150	C₂³×C₅	—	6	2	no	—
151	C₂³×C₅	—	6	4	no	—
152	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	3	4	no	—
155	C₂×C₄×C₅	—	3	2	no	—
156	C₂×C₄×C₅	—	6	2	no	—
157	C₂×C₄×C₅	—	6	4	no	—
158	C₂×C₄×C₅	—	6	4	no	—
159	C₂×C₄×C₅	—	6	4	no	—
160	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	#235	1	4	yes	—
163	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
164	C₅×C₈	—	2	4	no	—
165	C₅×C₈	—	6	4	no	—
166	C₂³×C₄	—	3	1	no	—
167	C₂²×C₈	—	3	2	no	—
168	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
169	#117	—	3	4	no	—
170	#117	—	3	4	no	—
171	C₂×C₄⋊C₄	—	3	2	no	—
172	C₂×C₂²⋊C₄	—	3	2	no	—
173	C₂×C₄²	—	3	1	no	—
174	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
175	C₄⋊C₈	—	3	8	no	—
176	C₂²⋊C₈	—	3	8	no	—
177	#97	C₂×C₅×D₃	1	4	yes	complemented
178	C₄×C₈	—	3	4	no	—
179	C₂×C₃×C₅	#433	1	4	yes	complemented
180	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
181	C₂×C₃×C₅	C₂×#131	1	2	yes	—
182	C₅×D₃	—	4	2	no	—
183	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₂²×D₃	—	2	2	no	—
185	C₂²×D₃	—	2	4	no	—
186	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
189	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₅×#41	1	4	yes	—
191	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
192	C₂×Dic₃	—	2	2	no	—
193	C₂×Dic₃	—	2	4	no	—
194	C₂×Dic₃	C₂²×C₄×C₅	1	4	yes	—
195	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
196	C₄×D₃	—	2	4	no	—
197	C₄×D₃	—	2	4	no	—
198	C₃×C₈	—	2	4	no	—
199	C₃⋊C₈	—	2	4	no	—
200	C₂²×C₅	—	6	2	no	—
201	C₂²×C₅	—	6	4	no	—
202	C₂²×C₅	—	12	4	no	—
203	C₂²×C₅	#872	1	2	yes	—
204	C₂²×C₅	#907	1	4	yes	—
205	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
206	C₄×C₅	—	2	4	no	—
207	C₄×C₅	—	6	2	no	—
208	C₄×C₅	—	6	4	no	—
209	C₄×C₅	#872	1	2	yes	—
210	C₄×C₅	C₂×#235	1	2	yes	—
211	C₂⁴	—	3	1	no	—
212	C₂²×C₄	—	3	2	no	—
213	C₂²×C₄	—	3	1	no	—
214	C₂²×C₄	—	3	2	no	—
215	C₂²×C₄	—	3	2	no	—
216	C₂²×C₄	—	6	4	no	—
217	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
218	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	3	4	no	—
221	C₂×C₈	—	6	4	no	—
222	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
223	C₄⋊C₄	—	3	8	no	—
224	C₂²⋊C₄	—	3	8	no	—
225	C₄²	—	3	2	no	—
226	C₄²	—	3	2	no	—
227	C₄²	C₄×C₅×D₃	1	2	yes	—
228	C₄²	C₂²×C₅×D₃	1	2	yes	—
229	C₃×C₅	C₂×#433	1	1	yes	characteristic, complemented, Hall
230	C₂²×C₃	C₅×#117	1	2	yes	—
231	C₂²×C₃	C₅×#131	1	4	yes	—
232	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
233	C₂×D₃	—	2	2	no	—
234	C₂×D₃	—	2	4	no	—
235	C₂×D₃	—	4	4	no	—
236	C₃×C₄	—	2	4	no	—
237	C₃×C₄	C₅×#117	1	2	yes	—
238	C₃×C₄	C₂×C₅×#41	1	2	yes	—
239	Dic₃	—	2	2	no	—
240	Dic₃	—	2	4	no	—
241	C₂×C₅	—	12	2	no	—
242	C₂×C₅	#4709	1	4	yes	direct product factor
243	C₂×C₅	C₂×#872	1	1	yes	characteristic
244	C₂×C₅	C₂×#907	1	2	yes	—
245	C₂³	—	3	2	no	—
246	C₂³	—	6	2	no	—
247	C₂³	—	6	4	no	—
248	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	3	4	no	—
251	C₂×C₄	—	3	2	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	6	4	no	—
254	C₂×C₄	—	6	4	no	—
255	C₂×C₄	—	6	4	no	—
256	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
257	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
258	C₂×C₄	C₅×#235	1	4	yes	—
259	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
260	C₈	—	2	4	no	—
261	C₈	—	6	4	no	—
262	C₂×C₃	C₅×#433	1	4	yes	complemented
263	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
264	C₂×C₃	C₂×C₅×#131	1	2	yes	—
265	D₃	—	4	2	no	—
266	C₅	C₂×#4709	1	1	yes	characteristic, direct product factor, Sylow
267	C₂²	—	6	2	no	—
268	C₂²	—	6	4	no	—
269	C₂²	—	12	4	no	—
270	C₂²	C₅×#872	1	2	yes	—
271	C₂²	C₅×#907	1	4	yes	—
272	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	6	2	no	—
275	C₄	—	6	4	no	—
276	C₄	C₅×#872	1	2	yes	—
277	C₄	C₂×C₅×#235	1	2	yes	—
278	C₃	C₂×C₅×#433	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	12	2	no	—
280	C₂	C₅×#4709	1	4	yes	direct product factor
281	C₂	C₂×C₅×#872	1	1	yes	characteristic
282	C₂	C₂×C₅×#907	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×C₅×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₂×C₅×#97	D₃
hypocenter	C₃	C₂×C₅×#433
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#97

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#97	C₂	semidirect
4. C₂×C₅×Dic₃⋊C₈	C₂	semidirect
5. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
6. C₅×#4709	C₂	direct (N × H)
17. C₃×C₅×#97	C₂²	semidirect
19. C₅×Dic₃⋊C₈	C₂²	semidirect
20. C₄×C₅×C₃⋊C₈	C₂²	semidirect
21. C₂×#4709	C₅	direct (N × H)
26. C₂×C₅×#97	D₃	semidirect
47. C₂×C₃×#97	C₂×C₅	semidirect
49. C₂×Dic₃⋊C₈	C₂×C₅	semidirect
50. C₂×C₄×C₃⋊C₈	C₂×C₅	semidirect
51. #4709	C₂×C₅	direct (N × H)
63. C₅×#97	C₂×D₃	semidirect
92. C₃×#97	C₂²×C₅	semidirect
94. Dic₃⋊C₈	C₂²×C₅	semidirect
95. C₄×C₃⋊C₈	C₂²×C₅	semidirect
118. C₂×#97	C₅×D₃	semidirect
177. #97	C₂×C₅×D₃	semidirect
179. C₂×C₃×C₅	#433	semidirect
229. C₃×C₅	C₂×#433	semidirect
242. C₂×C₅	#4709	direct (N × H)
262. C₂×C₃	C₅×#433	semidirect
266. C₅	C₂×#4709	direct (N × H)
278. C₃	C₂×C₅×#433	semidirect
280. C₂	C₅×#4709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#872	2
C₂×C₃×C₅×#97	2
C₂×C₅×(C₂×D₃)⋊C₈	2
C₂×C₅×Dic₃⋊C₈	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#4709	2
C₂×C₅×#433	3
C₂×#4709	5

Subgroups — C2×C5×#4709 #14413832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4709 #14413832