Subgroups — C₅×#69502 #14298455

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#830	C₂	1	1	yes	maximal, characteristic
4	C₅×#4529	C₂	1	4	yes	maximal, complemented
5	C₅×#1824	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₅×Dic₃	—	2	1	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	—	2	1	no	—
10	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₅×C₄⋊Dic₃	—	2	1	no	—
12	C₂×C₅×C₄⋊Dic₃	C₄	1	4	yes	complemented
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×C₅×#95	—	2	1	no	—
16	C₅×#830	—	2	2	no	—
17	#69502	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×C₄⋊C₄	—	3	1	no	—
19	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
20	C₅×#380	D₃	1	1	yes	characteristic
21	C₅×#340	—	3	4	no	—
22	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₅×Dic₃	—	2	4	no	—
30	C₂²×C₅×Dic₃	—	2	2	no	—
31	C₂²×C₅×Dic₃	—	4	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	4	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
34	C₅×C₄⋊Dic₃	—	2	4	no	—
35	C₅×C₄⋊Dic₃	—	2	2	no	—
36	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₂×C₅×C₃⋊C₈	—	2	4	no	—
39	C₂²×C₄⋊Dic₃	C₂×C₅	1	1	yes	characteristic
40	C₃×#380	C₂×C₅	1	1	yes	characteristic
41	C₂×#830	C₂×C₅	1	1	yes	characteristic
42	#4529	C₂×C₅	1	4	yes	complemented
43	C₂³×C₄×C₅	—	6	1	no	—
44	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
45	C₂²×C₅×C₈	—	3	2	no	—
46	C₂×C₅×C₄⋊C₄	—	3	4	no	—
47	C₂×C₅×C₄⋊C₄	—	3	2	no	—
48	C₂×C₅×C₄⋊C₄	—	6	2	no	—
49	C₂×C₅×C₄⋊C₄	—	6	1	no	—
50	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
52	C₅×C₂²⋊C₈	—	6	2	no	—
53	C₅×#95	—	2	1	no	—
54	#1824	—	3	1	no	Sylow
55	C₂³×C₃×C₅	—	2	2	no	—
56	C₂³×C₃×C₅	—	2	2	no	—
57	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
58	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
59	C₂³×C₃×C₅	Q₁₆	1	2	yes	—
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	4	yes	—
71	C₂×C₅×Dic₃	—	2	4	no	—
72	C₂×C₅×Dic₃	—	4	8	no	—
73	C₂×C₅×Dic₃	—	4	4	no	—
74	C₅×C₃⋊C₈	—	4	2	no	—
75	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
76	C₂³×Dic₃	—	2	1	no	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
78	C₂×C₃×C₂²⋊C₄	—	2	1	no	—
79	C₂×C₄⋊Dic₃	—	2	2	no	—
80	C₂×C₄⋊Dic₃	—	2	1	no	—
81	C₂×C₄⋊Dic₃	C₄×C₅	1	4	yes	complemented
82	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
83	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
84	C₃×#95	—	2	1	no	—
85	#830	—	2	2	no	—
86	C₂⁴×C₅	#67	1	1	yes	characteristic
87	C₂²×C₄×C₅	—	2	4	no	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	—	6	4	no	—
90	C₂²×C₄×C₅	—	6	2	no	—
91	C₂²×C₄×C₅	—	12	4	no	—
92	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
93	C₂²×C₄×C₅	D₁₂	1	2	yes	—
94	C₂²×C₄×C₅	#67	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂×C₅×C₈	—	6	2	no	—
97	C₂×C₅×C₈	—	6	4	no	—
98	C₅×C₄⋊C₄	—	2	4	no	—
99	C₅×C₄⋊C₄	—	3	4	no	—
100	C₅×C₄⋊C₄	—	6	4	no	—
101	C₅×C₄⋊C₄	—	6	2	no	—
102	C₅×C₂²⋊C₄	—	4	2	no	—
103	C₂²×C₄⋊C₄	—	3	1	no	—
104	C₂×C₂²⋊C₈	—	3	1	no	—
105	#380	C₅×D₃	1	1	yes	characteristic
106	#340	—	3	4	no	—
107	C₂²×C₃×C₅	—	2	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	2	4	no	—
110	C₂²×C₃×C₅	—	2	4	no	—
111	C₂²×C₃×C₅	#103	1	4	yes	—
112	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₃×C₅	#123	1	2	yes	—
114	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
116	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
117	C₃×C₄×C₅	—	2	2	no	—
118	C₃×C₄×C₅	—	8	2	no	—
119	C₃×C₄×C₅	#117	1	2	yes	—
120	C₃×C₄×C₅	#123	1	2	yes	—
121	C₅×Dic₃	—	4	4	no	—
122	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
123	C₂²×C₃×C₄	—	2	4	no	—
124	C₂²×C₃×C₄	—	2	2	no	—
125	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
126	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
127	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
129	C₂²×Dic₃	—	2	4	no	—
130	C₂²×Dic₃	—	2	2	no	—
131	C₂²×Dic₃	—	4	4	no	—
132	C₃×C₄⋊C₄	—	2	4	no	—
133	C₃×C₂²⋊C₄	—	4	2	no	—
134	C₄⋊Dic₃	—	2	4	no	—
135	C₄⋊Dic₃	—	2	2	no	—
136	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
137	C₂×C₃⋊C₈	—	2	2	no	—
138	C₂×C₃⋊C₈	—	2	4	no	—
139	C₂³×C₅	—	2	2	no	—
140	C₂³×C₅	—	2	2	no	—
141	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
142	C₂³×C₅	#214	1	2	yes	—
143	C₂³×C₅	#216	1	2	yes	—
144	C₂³×C₅	C₂×#67	1	1	yes	characteristic
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	—	2	4	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	4	2	no	—
149	C₂×C₄×C₅	—	4	2	no	—
150	C₂×C₄×C₅	—	4	2	no	—
151	C₂×C₄×C₅	—	6	4	no	—
152	C₂×C₄×C₅	—	12	8	no	—
153	C₂×C₄×C₅	—	12	4	no	—
154	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
157	C₂×C₄×C₅	#237	1	4	yes	—
158	C₅×C₈	—	12	2	no	—
159	C₂³×C₄	—	6	1	no	—
160	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
161	C₂²×C₈	—	3	2	no	—
162	C₂×C₄⋊C₄	—	3	4	no	—
163	C₂×C₄⋊C₄	—	3	2	no	—
164	C₂×C₄⋊C₄	—	6	2	no	—
165	C₂×C₄⋊C₄	—	6	1	no	—
166	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
167	C₂×C₂²⋊C₄	—	2	1	no	—
168	C₂²⋊C₈	—	6	2	no	—
169	#95	—	2	1	no	—
170	C₂×C₃×C₅	—	2	4	no	—
171	C₂×C₃×C₅	#381	1	1	yes	characteristic
172	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
173	C₂×C₃×C₅	#418	1	1	yes	characteristic
174	C₂×C₃×C₅	#481	1	2	yes	—
175	C₂×C₃×C₅	#484	1	2	yes	—
176	C₂³×C₃	—	2	2	no	—
177	C₂³×C₃	—	2	2	no	—
178	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
179	C₂³×C₃	C₅×SD₁₆	1	2	yes	—
180	C₂³×C₃	C₅×Q₁₆	1	2	yes	—
181	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	4	2	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
189	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₅×#41	1	4	yes	—
192	C₂×Dic₃	—	2	4	no	—
193	C₂×Dic₃	—	4	8	no	—
194	C₂×Dic₃	—	4	4	no	—
195	C₃⋊C₈	—	4	2	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	#810	1	4	yes	—
201	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
202	C₂²×C₅	C₂×#214	1	1	yes	characteristic
203	C₂²×C₅	#935	1	2	yes	—
204	C₂²×C₅	C₂×#216	1	1	yes	characteristic
205	C₂²×C₅	#949	1	2	yes	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	8	2	no	—
208	C₄×C₅	—	12	4	no	—
209	C₄×C₅	#877	1	2	yes	—
210	C₄×C₅	#886	1	2	yes	—
211	C₂⁴	C₅×#67	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	6	4	no	—
215	C₂²×C₄	—	6	2	no	—
216	C₂²×C₄	—	12	4	no	—
217	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
218	C₂²×C₄	C₅×D₁₂	1	2	yes	—
219	C₂²×C₄	C₅×#67	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
221	C₂×C₈	—	6	2	no	—
222	C₂×C₈	—	6	4	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	3	4	no	—
225	C₄⋊C₄	—	6	4	no	—
226	C₄⋊C₄	—	6	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₃×C₅	#1824	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	2	2	no	—
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	2	4	no	—
233	C₂²×C₃	C₅×#103	1	4	yes	—
234	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
235	C₂²×C₃	C₅×#123	1	2	yes	—
236	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
237	C₂²×C₃	C₂×C₅×Q₁₆	1	1	yes	characteristic
238	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	—	8	2	no	—
241	C₃×C₄	C₅×#117	1	2	yes	—
242	C₃×C₄	C₅×#123	1	2	yes	—
243	Dic₃	—	4	4	no	—
244	C₂×C₅	—	2	4	no	—
245	C₂×C₅	#4943	1	1	yes	characteristic
246	C₂×C₅	C₂×#810	1	1	yes	characteristic
247	C₂×C₅	#5001	1	1	yes	characteristic
248	C₂×C₅	#5034	1	2	yes	—
249	C₂×C₅	#5044	1	2	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	2	no	—
252	C₂³	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
253	C₂³	C₅×#214	1	2	yes	—
254	C₂³	C₅×#216	1	2	yes	—
255	C₂³	C₂×C₅×#67	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	6	4	no	—
263	C₂×C₄	—	12	8	no	—
264	C₂×C₄	—	12	4	no	—
265	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
266	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
268	C₂×C₄	C₅×#237	1	4	yes	—
269	C₈	—	12	2	no	—
270	C₂×C₃	—	2	4	no	—
271	C₂×C₃	C₅×#381	1	1	yes	characteristic
272	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
273	C₂×C₃	C₅×#418	1	1	yes	characteristic
274	C₂×C₃	C₅×#481	1	2	yes	—
275	C₂×C₃	C₅×#484	1	2	yes	—
276	C₅	#69502	1	1	yes	characteristic, direct product factor, Sylow
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	C₅×#810	1	4	yes	—
282	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
283	C₂²	C₂×C₅×#214	1	1	yes	characteristic
284	C₂²	C₅×#935	1	2	yes	—
285	C₂²	C₂×C₅×#216	1	1	yes	characteristic
286	C₂²	C₅×#949	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	12	4	no	—
290	C₄	C₅×#877	1	2	yes	—
291	C₄	C₅×#886	1	2	yes	—
292	C₃	C₅×#1824	1	1	yes	characteristic, complemented, Sylow
293	C₂	—	2	4	no	—
294	C₂	C₅×#4943	1	1	yes	characteristic
295	C₂	C₂×C₅×#810	1	1	yes	characteristic
296	C₂	C₅×#5001	1	1	yes	characteristic
297	C₂	C₅×#5034	1	2	yes	—
298	C₂	C₅×#5044	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×#380	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#380	D₃
hypocenter	C₃	C₅×#1824
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₅×#380

Complements

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

N	H ≅ G/N	type
4. C₅×#4529	C₂	semidirect
12. C₂×C₅×C₄⋊Dic₃	C₄	semidirect
17. #69502	C₅	direct (N × H)
42. #4529	C₂×C₅	semidirect
81. C₂×C₄⋊Dic₃	C₄×C₅	semidirect
228. C₃×C₅	#1824	semidirect
276. C₅	#69502	direct (N × H)
292. C₃	C₅×#1824	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₄⋊Dic₃	2
C₃×C₅×#380	2
C₂×C₅×#830	2
C₅×#4529	2
C₅×#1824	3
#69502	5

Subgroups — C5×#69502 #14298455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69502 #14298455