Subgroups — C₅×#70046 #14302776

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5825	C₂	1	1	yes	maximal, characteristic
2	C₅×#5819	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic
4	C₅×#5004	C₂	1	2	yes	maximal, complemented
5	C₅×#4531	C₂	1	2	yes	maximal, complemented
6	C₅×#1909	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
9	C₃×C₅×#117	C₂²	1	2	yes	—
10	C₅×#948	—	2	4	no	—
11	C₅×#944	C₄	1	2	yes	complemented
12	C₅×#936	C₄	1	2	yes	complemented
13	C₅×#935	—	2	2	no	—
14	C₅×#926	C₂²	1	1	yes	characteristic
15	C₅×#924	—	2	2	no	—
16	C₂×C₅×#208	—	2	2	no	—
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₃×C₅×#99	—	2	2	no	—
19	C₅×#810	—	2	4	no	—
20	#70046	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#568	—	3	1	no	—
22	C₅×#536	—	3	1	no	—
23	C₅×#419	—	3	2	no	—
24	C₅×#410	D₃	1	1	yes	characteristic
25	C₅×H(3,ℤ/4)	—	3	2	no	—
26	C₃×C₅×#41	—	2	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×Dic₃	—	2	1	no	—
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	2	no	—
36	C₂×C₅×Dic₆	—	2	1	no	—
37	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₄²×C₅	—	2	2	no	—
41	C₅×#217	—	2	4	no	—
42	C₅×#217	—	2	1	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×Dic₃⋊C₄	—	2	4	no	—
45	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
46	C₅×#208	—	2	4	no	—
47	C₅×#208	—	2	2	no	—
48	C₅×#208	—	2	2	no	—
49	C₂×C₅×C₃⋊C₈	—	2	4	no	—
50	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
51	#5825	C₂×C₅	1	1	yes	characteristic
52	#5819	C₂×C₅	1	1	yes	characteristic
53	C₃×#410	C₂×C₅	1	1	yes	characteristic
54	#5004	C₂×C₅	1	2	yes	complemented
55	#4531	C₂×C₅	1	2	yes	complemented
56	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
57	C₂²×C₅×Q₈	—	3	1	no	—
58	C₅×#131	—	6	4	no	—
59	C₂×C₅×M₁₆	—	3	2	no	—
60	C₂×C₅×M₁₆	—	6	2	no	—
61	C₅×C₄⋊Q₈	—	3	2	no	—
62	C₅×#124	—	3	2	no	—
63	C₅×#123	—	6	2	no	—
64	C₅×C₂²⋊Q₈	—	6	2	no	—
65	C₅×#117	—	3	1	no	—
66	C₅×#117	C₂×D₃	1	2	yes	—
67	C₅×#103	—	6	4	no	—
68	C₅×#99	—	2	2	no	—
69	#1909	—	3	1	no	Sylow
70	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	2	no	—
73	C₃×C₅×D₄	—	2	4	no	—
74	C₃×C₅×D₄	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	2	2	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₅×Dic₃	—	2	2	no	—
84	C₂×C₅×Dic₃	—	2	2	no	—
85	C₂×C₅×Dic₃	—	4	2	no	—
86	C₅×Dic₆	—	2	4	no	—
87	C₅×Dic₆	—	4	1	no	—
88	C₅×Dic₆	—	4	1	no	—
89	C₅×C₃⋊C₈	—	2	4	no	—
90	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
91	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
92	C₃×#117	C₂²×C₅	1	2	yes	—
93	#948	—	2	4	no	—
94	#944	C₄×C₅	1	2	yes	complemented
95	#936	C₄×C₅	1	2	yes	complemented
96	#935	—	2	2	no	—
97	#926	C₂²×C₅	1	1	yes	characteristic
98	#924	—	2	2	no	—
99	C₂×#208	—	2	2	no	—
100	C₂×#208	C₂²×C₅	1	2	yes	—
101	C₃×#99	—	2	2	no	—
102	#810	—	2	4	no	—
103	C₅×#41	—	2	2	no	—
104	C₅×#41	—	2	2	no	—
105	C₂×C₅×Q₈	—	3	2	no	—
106	C₂×C₅×Q₈	—	6	1	no	—
107	C₂×C₅×Q₈	—	6	2	no	—
108	C₂×C₅×Q₈	—	6	2	no	—
109	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
110	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
111	C₂×C₅×D₄	#67	1	2	yes	—
112	C₂²×C₄×C₅	—	6	1	no	—
113	C₂²×C₄×C₅	D₁₂	1	2	yes	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₅×M₁₆	—	6	4	no	—
116	C₅×M₁₆	—	6	2	no	—
117	C₅×M₁₆	—	6	2	no	—
118	C₂×C₅×C₈	—	3	4	no	—
119	C₂×C₅×C₈	—	6	4	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	6	4	no	—
122	C₅×C₄⋊C₄	—	6	1	no	—
123	C₅×C₂²⋊C₄	—	2	2	no	—
124	C₅×C₂²⋊C₄	—	6	4	no	—
125	C₅×C₂²⋊C₄	—	6	1	no	—
126	C₄²×C₅	—	2	2	no	—
127	C₄²×C₅	—	3	2	no	—
128	#568	—	3	1	no	—
129	#536	—	3	1	no	—
130	#419	—	3	2	no	—
131	#410	C₅×D₃	1	1	yes	characteristic
132	H(3,ℤ/4)	—	3	2	no	—
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	2	no	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₃×C₅	#121	1	2	yes	—
138	C₃×C₄×C₅	—	2	2	no	—
139	C₃×C₄×C₅	—	8	2	no	—
140	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
141	C₃×C₄×C₅	#120	1	1	yes	characteristic
142	C₃×C₄×C₅	#123	1	1	yes	characteristic
143	C₅×Dic₃	—	4	2	no	—
144	C₅×Dic₃	—	4	2	no	—
145	C₃×#41	—	2	2	no	—
146	C₃×#41	—	2	2	no	—
147	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
148	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
150	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
151	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂²×Dic₃	—	2	1	no	—
153	C₂×Dic₆	—	2	2	no	—
154	C₂×Dic₆	—	2	2	no	—
155	C₂×Dic₆	—	2	1	no	—
156	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₂²⋊C₄	—	2	2	no	—
159	C₃×C₄²	—	2	2	no	—
160	#217	—	2	4	no	—
161	#217	—	2	1	no	—
162	C₄⋊Dic₃	—	2	1	no	—
163	Dic₃⋊C₄	—	2	4	no	—
164	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
165	#208	—	2	4	no	—
166	#208	—	2	2	no	—
167	#208	—	2	2	no	—
168	C₂×C₃⋊C₈	—	2	4	no	—
169	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
170	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
171	C₂³×C₅	C₂×#67	1	1	yes	characteristic
172	C₅×Q₈	—	2	2	no	—
173	C₅×Q₈	—	6	4	no	—
174	C₅×Q₈	—	12	1	no	—
175	C₅×Q₈	—	12	1	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	—	6	2	no	—
182	C₂×C₄×C₅	—	6	2	no	—
183	C₂×C₄×C₅	—	12	2	no	—
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
185	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
187	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
188	C₂×C₄×C₅	#237	1	2	yes	—
189	C₅×C₈	—	6	4	no	—
190	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
191	C₂²×Q₈	—	3	1	no	—
192	#131	—	6	4	no	—
193	C₂×M₁₆	—	3	2	no	—
194	C₂×M₁₆	—	6	2	no	—
195	C₄⋊Q₈	—	3	2	no	—
196	#124	—	3	2	no	—
197	#123	—	6	2	no	—
198	C₂²⋊Q₈	—	6	2	no	—
199	#117	—	3	1	no	—
200	#117	C₂×C₅×D₃	1	2	yes	—
201	#103	—	6	4	no	—
202	#99	—	2	2	no	—
203	C₂×C₃×C₅	—	2	1	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	—	4	2	no	—
206	C₂×C₃×C₅	#386	1	1	yes	characteristic
207	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
208	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
209	C₃×Q₈	—	2	2	no	—
210	C₃×D₄	—	2	4	no	—
211	C₃×D₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	4	2	no	—
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
216	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
219	C₂×C₃×C₄	C₅×#41	1	2	yes	—
220	C₂×Dic₃	—	2	2	no	—
221	C₂×Dic₃	—	2	2	no	—
222	C₂×Dic₃	—	4	2	no	—
223	Dic₆	—	2	4	no	—
224	Dic₆	—	4	1	no	—
225	Dic₆	—	4	1	no	—
226	C₃⋊C₈	—	2	4	no	—
227	C₂²×C₅	—	2	2	no	—
228	C₂²×C₅	—	4	1	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
231	C₂²×C₅	#939	1	2	yes	—
232	C₄×C₅	—	2	2	no	—
233	C₄×C₅	—	8	2	no	—
234	C₄×C₅	—	12	2	no	—
235	C₄×C₅	—	12	2	no	—
236	C₄×C₅	#67⋊C₄	1	2	yes	—
237	C₄×C₅	#882	1	1	yes	characteristic
238	C₄×C₅	#886	1	1	yes	characteristic
239	#41	—	2	2	no	—
240	#41	—	2	2	no	—
241	C₂×Q₈	—	3	2	no	—
242	C₂×Q₈	—	6	1	no	—
243	C₂×Q₈	—	6	2	no	—
244	C₂×Q₈	—	6	2	no	—
245	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
246	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
247	C₂×D₄	C₅×#67	1	2	yes	—
248	C₂²×C₄	—	6	1	no	—
249	C₂²×C₄	C₅×D₁₂	1	2	yes	—
250	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
251	M₁₆	—	6	4	no	—
252	M₁₆	—	6	2	no	—
253	M₁₆	—	6	2	no	—
254	C₂×C₈	—	3	4	no	—
255	C₂×C₈	—	6	4	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	6	4	no	—
258	C₄⋊C₄	—	6	1	no	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	6	4	no	—
261	C₂²⋊C₄	—	6	1	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	3	2	no	—
264	C₃×C₅	#1909	1	1	yes	characteristic, complemented, Hall
265	C₂²×C₃	—	2	2	no	—
266	C₂²×C₃	—	4	1	no	—
267	C₂²×C₃	—	4	2	no	—
268	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
269	C₂²×C₃	C₅×#121	1	2	yes	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	—	8	2	no	—
272	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
273	C₃×C₄	C₅×#120	1	1	yes	characteristic
274	C₃×C₄	C₅×#123	1	1	yes	characteristic
275	Dic₃	—	4	2	no	—
276	Dic₃	—	4	2	no	—
277	C₂×C₅	—	2	2	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	—	4	2	no	—
280	C₂×C₅	#4958	1	1	yes	characteristic
281	C₂³	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
282	C₂³	C₂×C₅×#67	1	1	yes	characteristic
283	Q₈	—	2	2	no	—
284	Q₈	—	6	4	no	—
285	Q₈	—	12	1	no	—
286	Q₈	—	12	1	no	—
287	D₄	—	2	4	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	—	6	2	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
296	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
298	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
299	C₂×C₄	C₅×#237	1	2	yes	—
300	C₈	—	6	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	4	2	no	—
304	C₂×C₃	C₅×#386	1	1	yes	characteristic
305	C₅	#70046	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	—	2	2	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	2	no	—
309	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₂²	C₅×#939	1	2	yes	—
311	C₄	—	2	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	12	2	no	—
314	C₄	—	12	2	no	—
315	C₄	C₅×#67⋊C₄	1	2	yes	—
316	C₄	C₅×#882	1	1	yes	characteristic
317	C₄	C₅×#886	1	1	yes	characteristic
318	C₃	C₅×#1909	1	1	yes	characteristic, complemented, Sylow
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	4	2	no	—
322	C₂	C₅×#4958	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#4958
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#410	D₃
hypocenter	C₃	C₅×#1909
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#410

Complements

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

N	H ≅ G/N	type
4. C₅×#5004	C₂	semidirect
5. C₅×#4531	C₂	semidirect
11. C₅×#944	C₄	semidirect
12. C₅×#936	C₄	semidirect
20. #70046	C₅	direct (N × H)
54. #5004	C₂×C₅	semidirect
55. #4531	C₂×C₅	semidirect
94. #944	C₄×C₅	semidirect
95. #936	C₄×C₅	semidirect
264. C₃×C₅	#1909	semidirect
305. C₅	#70046	direct (N × H)
318. C₃	C₅×#1909	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5825	2
C₅×#5819	2
C₃×C₅×#410	2
C₅×#5004	2
C₅×#4531	2
C₅×#1909	3
#70046	5

Subgroups — C5×#70046 #14302776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70046 #14302776