Subgroups — C₅×#72139 #14296862

Order 1920.

1224 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#834	C₂	1	4	yes	maximal, complemented
3	C₅×#5211	C₂	1	2	yes	maximal, complemented
4	C₅×#1782	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×#41	—	2	8	no	—
6	C₂×C₃×C₅×#41	—	2	4	no	—
7	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic, complemented
8	C₂²×C₃×C₅×D₄	C₄	1	1	yes	characteristic, complemented
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂³×C₃×C₄×C₅	C₄	1	2	yes	complemented
11	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#217	C₂²	1	4	yes	—
13	C₅×#834	—	2	8	no	—
14	C₅×C₃⋊#95	—	2	4	no	—
15	#72139	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₅×#41	D₃	1	1	yes	characteristic
17	C₂×C₅×#99	—	3	4	no	—
18	C₅×#381	—	3	2	no	—
19	C₂⁴×C₃×C₅	C₂×C₄	1	2	yes	—
20	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
21	C₃×C₅×#41	—	2	32	no	—
22	C₂×C₃×C₅×Q₈	—	2	4	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	4	no	—
25	C₂×C₃×C₅×D₄	—	2	4	no	—
26	C₂×C₃×C₅×D₄	—	2	4	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	D₄	1	8	yes	—
29	C₂²×C₃×C₄×C₅	—	2	4	no	—
30	C₂²×C₃×C₄×C₅	—	2	4	no	—
31	C₂²×C₃×C₄×C₅	—	2	4	no	—
32	C₂²×C₃×C₄×C₅	—	2	4	no	—
33	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
37	C₂²×C₅×Dic₃	—	2	4	no	—
38	C₅×#217	—	2	8	no	—
39	C₂²×C₃×#41	C₂×C₅	1	1	yes	characteristic
40	C₂×#834	C₂×C₅	1	4	yes	complemented
41	#5211	C₂×C₅	1	2	yes	complemented
42	C₂×C₅×#41	—	2	8	no	—
43	C₂×C₅×#41	—	2	4	no	—
44	C₂²×C₅×Q₈	Dic₃	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	Dic₃	1	1	yes	characteristic, complemented
46	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
47	C₂³×C₄×C₅	Dic₃	1	2	yes	complemented
48	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
50	C₅×#99	—	6	8	no	—
51	C₅×#95	—	6	4	no	—
52	#1782	—	3	1	no	Sylow
53	C₂³×C₃×C₅	—	2	2	no	—
54	C₂³×C₃×C₅	—	2	2	no	—
55	C₂³×C₃×C₅	—	4	2	no	—
56	C₂³×C₃×C₅	—	4	4	no	—
57	C₂³×C₃×C₅	C₂²⋊C₄	1	4	yes	—
58	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
60	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
61	C₃×C₅×Q₈	—	2	8	no	—
62	C₃×C₅×D₄	—	2	8	no	—
63	C₃×C₅×D₄	—	2	16	no	—
64	C₂×C₃×C₄×C₅	—	2	16	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	2	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	8	no	—
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂×C₅×Dic₃	—	4	8	no	—
74	C₂×C₅×Dic₃	—	4	4	no	—
75	C₂×C₃×#41	—	2	8	no	—
76	C₂×C₃×#41	—	2	4	no	—
77	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic, complemented
78	C₂²×C₃×D₄	C₄×C₅	1	1	yes	characteristic, complemented
79	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
80	C₂³×C₃×C₄	C₄×C₅	1	2	yes	complemented
81	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
82	C₂×#217	C₂²×C₅	1	4	yes	—
83	#834	—	2	8	no	—
84	C₃⋊#95	—	2	4	no	—
85	C₂⁴×C₅	C₂×Dic₃	1	2	yes	—
86	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₅×#41	—	2	32	no	—
88	C₂×C₅×Q₈	—	2	4	no	—
89	C₂×C₅×Q₈	—	2	2	no	—
90	C₂×C₅×D₄	—	2	4	no	—
91	C₂×C₅×D₄	—	2	4	no	—
92	C₂×C₅×D₄	—	2	4	no	—
93	C₂×C₅×D₄	—	2	2	no	—
94	C₂×C₅×D₄	#67	1	8	yes	—
95	C₂²×C₄×C₅	—	2	4	no	—
96	C₂²×C₄×C₅	—	2	4	no	—
97	C₂²×C₄×C₅	—	2	4	no	—
98	C₂²×C₄×C₅	—	2	4	no	—
99	C₂²×C₄×C₅	—	6	4	no	—
100	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
101	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
102	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	#67	1	4	yes	—
104	C₅×C₂²⋊C₄	—	6	8	no	—
105	C₂²×#41	C₅×D₃	1	1	yes	characteristic
106	C₂×#99	—	3	4	no	—
107	#381	—	3	2	no	—
108	C₂²×C₃×C₅	—	2	4	no	—
109	C₂²×C₃×C₅	—	2	2	no	—
110	C₂²×C₃×C₅	—	2	4	no	—
111	C₂²×C₃×C₅	—	4	4	no	—
112	C₂²×C₃×C₅	—	4	4	no	—
113	C₂²×C₃×C₅	—	4	4	no	—
114	C₂²×C₃×C₅	—	4	4	no	—
115	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#120	1	4	yes	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	4	no	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₅×Dic₃	—	8	4	no	—
122	C₂⁴×C₃	C₂×C₄×C₅	1	2	yes	—
123	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
124	C₃×#41	—	2	32	no	—
125	C₂×C₃×Q₈	—	2	4	no	—
126	C₂×C₃×Q₈	—	2	2	no	—
127	C₂×C₃×D₄	—	2	4	no	—
128	C₂×C₃×D₄	—	2	4	no	—
129	C₂×C₃×D₄	—	2	4	no	—
130	C₂×C₃×D₄	—	2	2	no	—
131	C₂×C₃×D₄	C₅×D₄	1	8	yes	—
132	C₂²×C₃×C₄	—	2	4	no	—
133	C₂²×C₃×C₄	—	2	4	no	—
134	C₂²×C₃×C₄	—	2	4	no	—
135	C₂²×C₃×C₄	—	2	4	no	—
136	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
138	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
140	C₂²×Dic₃	—	2	4	no	—
141	#217	—	2	8	no	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	2	2	no	—
144	C₂³×C₅	—	4	2	no	—
145	C₂³×C₅	—	4	4	no	—
146	C₂³×C₅	#217	1	4	yes	—
147	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
148	C₂³×C₅	C₂×#67	1	2	yes	—
149	C₂³×C₅	C₂×#67	1	4	yes	—
150	C₅×Q₈	—	2	8	no	—
151	C₅×D₄	—	2	8	no	—
152	C₅×D₄	—	2	16	no	—
153	C₂×C₄×C₅	—	2	16	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	8	no	—
159	C₂×C₄×C₅	—	12	8	no	—
160	C₂×C₄×C₅	—	12	4	no	—
161	C₂×C₄×C₅	#217	1	4	yes	—
162	C₂×C₄×C₅	#217	1	2	yes	—
163	C₂×C₄×C₅	#217	1	2	yes	—
164	C₂×#41	—	2	8	no	—
165	C₂×#41	—	2	4	no	—
166	C₂²×Q₈	C₅×Dic₃	1	1	yes	characteristic, complemented
167	C₂²×D₄	C₅×Dic₃	1	1	yes	characteristic, complemented
168	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
169	C₂³×C₄	C₅×Dic₃	1	2	yes	complemented
170	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	3	4	no	—
172	#99	—	6	8	no	—
173	#95	—	6	4	no	—
174	C₂×C₃×C₅	—	2	4	no	—
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	—	4	4	no	—
177	C₂×C₃×C₅	#379	1	1	yes	characteristic
178	C₂×C₃×C₅	#410	1	2	yes	—
179	C₂³×C₃	—	2	2	no	—
180	C₂³×C₃	—	2	2	no	—
181	C₂³×C₃	—	4	2	no	—
182	C₂³×C₃	—	4	4	no	—
183	C₂³×C₃	C₅×C₂²⋊C₄	1	4	yes	—
184	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
185	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
186	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
187	C₃×Q₈	—	2	8	no	—
188	C₃×D₄	—	2	8	no	—
189	C₃×D₄	—	2	16	no	—
190	C₂×C₃×C₄	—	2	16	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	8	no	—
196	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
197	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
198	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
199	C₂×Dic₃	—	4	8	no	—
200	C₂×Dic₃	—	4	4	no	—
201	C₂²×C₅	—	2	4	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	4	no	—
204	C₂²×C₅	—	4	4	no	—
205	C₂²×C₅	—	4	4	no	—
206	C₂²×C₅	—	4	4	no	—
207	C₂²×C₅	—	4	4	no	—
208	C₂²×C₅	C₂×#217	1	2	yes	—
209	C₂²×C₅	C₂×#217	1	1	yes	characteristic
210	C₂²×C₅	#953	1	4	yes	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	4	no	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	24	4	no	—
215	C₂⁴	C₂×C₅×Dic₃	1	2	yes	—
216	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
217	#41	—	2	32	no	—
218	C₂×Q₈	—	2	4	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×D₄	—	2	4	no	—
221	C₂×D₄	—	2	4	no	—
222	C₂×D₄	—	2	4	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₅×#67	1	8	yes	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	6	4	no	—
230	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
232	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
233	C₂²×C₄	C₅×#67	1	4	yes	—
234	C₂²⋊C₄	—	6	8	no	—
235	C₃×C₅	#1782	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	4	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	4	no	—
239	C₂²×C₃	—	4	4	no	—
240	C₂²×C₃	—	4	4	no	—
241	C₂²×C₃	—	4	4	no	—
242	C₂²×C₃	—	4	4	no	—
243	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
244	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₂²×C₃	C₅×#120	1	4	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	2	4	no	—
248	C₃×C₄	—	2	2	no	—
249	Dic₃	—	8	4	no	—
250	C₂×C₅	—	2	4	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	4	4	no	—
253	C₂×C₅	#5240	1	2	yes	—
254	C₂×C₅	#5247	1	1	yes	characteristic
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	4	2	no	—
258	C₂³	—	4	4	no	—
259	C₂³	C₅×#217	1	4	yes	—
260	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
261	C₂³	C₂×C₅×#67	1	2	yes	—
262	C₂³	C₂×C₅×#67	1	4	yes	—
263	Q₈	—	2	8	no	—
264	D₄	—	2	8	no	—
265	D₄	—	2	16	no	—
266	C₂×C₄	—	2	16	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	12	8	no	—
273	C₂×C₄	—	12	4	no	—
274	C₂×C₄	C₅×#217	1	2	yes	—
275	C₂×C₄	C₅×#217	1	4	yes	—
276	C₂×C₄	C₅×#217	1	2	yes	—
277	C₂×C₃	—	2	4	no	—
278	C₂×C₃	—	2	2	no	—
279	C₂×C₃	—	4	4	no	—
280	C₂×C₃	C₅×#379	1	1	yes	characteristic
281	C₂×C₃	C₅×#410	1	2	yes	—
282	C₅	#72139	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	—	4	4	no	—
289	C₂²	—	4	4	no	—
290	C₂²	C₂×C₅×#217	1	2	yes	—
291	C₂²	C₂×C₅×#217	1	1	yes	characteristic
292	C₂²	C₅×#953	1	4	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	24	4	no	—
297	C₃	C₅×#1782	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	2	4	no	—
299	C₂	—	2	2	no	—
300	C₂	—	4	4	no	—
301	C₂	C₅×#5240	1	2	yes	—
302	C₂	C₅×#5247	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×#217
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂⁴	C₂²×C₅×D₃
Fitting, prime radical	C₂²×C₃×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂⁴×C₃	C₂³×C₅
hypercenter	C₂²×C₅×#41	D₃
hypocenter	C₃	C₅×#1782
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₂²×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#834	C₂	semidirect
3. C₅×#5211	C₂	semidirect
7. C₂²×C₃×C₅×Q₈	C₄	semidirect
8. C₂²×C₃×C₅×D₄	C₄	semidirect
10. C₂³×C₃×C₄×C₅	C₄	semidirect
15. #72139	C₅	direct (N × H)
40. C₂×#834	C₂×C₅	semidirect
41. #5211	C₂×C₅	semidirect
44. C₂²×C₅×Q₈	Dic₃	semidirect
45. C₂²×C₅×D₄	Dic₃	semidirect
47. C₂³×C₄×C₅	Dic₃	semidirect
77. C₂²×C₃×Q₈	C₄×C₅	semidirect
78. C₂²×C₃×D₄	C₄×C₅	semidirect
80. C₂³×C₃×C₄	C₄×C₅	semidirect
166. C₂²×Q₈	C₅×Dic₃	semidirect
167. C₂²×D₄	C₅×Dic₃	semidirect
169. C₂³×C₄	C₅×Dic₃	semidirect
235. C₃×C₅	#1782	semidirect
282. C₅	#72139	direct (N × H)
297. C₃	C₅×#1782	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×#41	2
C₂×C₅×#834	2
C₅×#5211	2
C₅×#1782	3
#72139	5

Subgroups — C5×#72139 #14296862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72139 #14296862