Subgroups — C₅×#70512 #14305987

Order 1920.

720 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	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#825	C₂	1	4	yes	maximal, complemented
4	C₄²×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1962	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₂²	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#930	—	2	2	no	—
10	C₂×C₄×C₅×Dic₃	—	2	2	no	—
11	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
12	C₅×#875	C₄	1	2	yes	complemented
13	C₅×#874	C₄	1	1	yes	characteristic, complemented
14	C₄×C₅×D₁₂	—	2	2	no	—
15	C₅×#869	C₄	1	1	yes	characteristic, complemented
16	C₃×C₅×C₄⋊C₈	—	2	2	no	—
17	C₅×#825	—	2	8	no	—
18	#70512	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#532	—	3	1	no	—
20	C₅×#423	D₃	1	1	yes	characteristic, complemented
21	C₂×C₅×#104	—	3	4	no	—
22	C₄³×C₅	—	3	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×C₅×#67	—	2	2	no	—
26	C₂²×C₅×Dic₃	—	2	2	no	—
27	C₅×#235	—	2	4	no	—
28	C₅×#235	—	2	4	no	—
29	C₂×C₅×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
31	C₂×C₄×C₅×D₃	—	2	2	no	—
32	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
33	C₃×C₅×M₁₆	—	2	4	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
39	C₅×C₄⋊Dic₃	—	2	2	no	—
40	C₄×C₅×Dic₃	—	2	8	no	—
41	C₄×C₅×Dic₃	—	2	2	no	—
42	C₄×C₅×Dic₃	D₄	1	4	yes	complemented
43	#5477	C₂×C₅	1	1	yes	characteristic
44	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂×#825	C₂×C₅	1	4	yes	complemented
46	C₄²×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
47	C₂×C₅×#41	—	3	2	no	—
48	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
49	C₅×C₄⋊Q₈	—	3	1	no	—
50	C₅×#127	—	3	1	no	—
51	C₅×#124	—	3	2	no	—
52	C₅×#121	—	6	2	no	—
53	C₄×C₅×D₄	—	6	2	no	—
54	C₂×C₄²×C₅	—	3	2	no	—
55	C₂×C₄²×C₅	—	6	2	no	—
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	2	no	—
58	C₅×#104	—	6	8	no	—
59	#1962	—	3	1	no	Sylow
60	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₅×D₃	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	2	yes	—
69	C₅×#67	—	4	4	no	—
70	C₂×C₅×Dic₃	—	2	8	no	—
71	C₂×C₅×Dic₃	—	2	4	no	—
72	C₂×C₅×Dic₃	—	2	2	no	—
73	C₅×D₁₂	—	2	4	no	—
74	C₅×D₁₂	—	4	4	no	—
75	C₄×C₅×D₃	—	4	2	no	—
76	C₄×C₅×D₃	—	4	2	no	—
77	C₅×Dic₆	—	2	4	no	—
78	C₃×C₅×C₈	—	4	2	no	—
79	C₂×#235	C₂²×C₅	1	2	yes	—
80	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
81	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
82	#930	—	2	2	no	—
83	C₂×C₄×Dic₃	—	2	2	no	—
84	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
85	#875	C₄×C₅	1	2	yes	complemented
86	#874	C₄×C₅	1	1	yes	characteristic, complemented
87	C₄×D₁₂	—	2	2	no	—
88	#869	C₄×C₅	1	1	yes	characteristic, complemented
89	C₃×C₄⋊C₈	—	2	2	no	—
90	#825	—	2	8	no	—
91	C₅×#41	—	6	4	no	—
92	C₅×#41	—	6	4	no	—
93	C₂×C₅×Q₈	—	3	2	no	—
94	C₂×C₅×D₄	—	3	2	no	—
95	C₂×C₅×D₄	—	6	2	no	—
96	C₂×C₅×D₄	—	6	2	no	—
97	C₂²×C₄×C₅	—	6	2	no	—
98	C₂²×C₄×C₅	—	6	2	no	—
99	C₂²×C₄×C₅	D₁₂	1	2	yes	—
100	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₅×M₁₆	—	2	4	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₅×C₄⋊C₄	—	6	2	no	—
104	C₅×C₂²⋊C₄	—	6	4	no	—
105	C₄²×C₅	—	3	4	no	—
106	C₄²×C₅	—	6	8	no	—
107	C₄²×C₅	—	6	2	no	—
108	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
109	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
110	C₄²×C₅	#67	1	2	yes	—
111	#532	—	3	1	no	—
112	#423	C₅×D₃	1	1	yes	characteristic, complemented
113	C₂×#104	—	3	4	no	—
114	C₄³	—	3	1	no	—
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	#121	1	2	yes	—
118	C₂×C₅×D₃	—	4	2	no	—
119	C₂×C₅×D₃	—	4	4	no	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	#104	1	4	yes	complemented
122	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
123	C₃×C₄×C₅	#124	1	1	yes	characteristic
124	C₃×C₄×C₅	#127	1	1	yes	characteristic
125	C₅×Dic₃	—	2	8	no	—
126	C₅×Dic₃	—	4	2	no	—
127	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
128	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
129	C₂×#67	—	2	2	no	—
130	C₂²×Dic₃	—	2	2	no	—
131	#235	—	2	4	no	—
132	#235	—	2	4	no	—
133	C₂×D₁₂	—	2	2	no	—
134	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₄×D₃	—	2	2	no	—
136	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
137	C₃×M₁₆	—	2	4	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
140	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
141	C₃×C₄²	C₅×D₄	1	2	yes	—
142	(C₂×D₃)⋊C₄	—	2	4	no	—
143	C₄⋊Dic₃	—	2	2	no	—
144	C₄×Dic₃	—	2	8	no	—
145	C₄×Dic₃	—	2	2	no	—
146	C₄×Dic₃	C₅×D₄	1	4	yes	complemented
147	C₂³×C₅	—	6	2	no	—
148	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
149	C₅×Q₈	—	6	4	no	—
150	C₅×D₄	—	6	4	no	—
151	C₅×D₄	—	12	4	no	—
152	C₅×D₄	—	12	4	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	6	8	no	—
155	C₂×C₄×C₅	—	6	4	no	—
156	C₂×C₄×C₅	—	6	2	no	—
157	C₂×C₄×C₅	—	12	2	no	—
158	C₂×C₄×C₅	—	12	2	no	—
159	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
160	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
163	C₂×C₄×C₅	#239	1	2	yes	—
164	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
165	C₅×C₈	—	4	2	no	—
166	C₂×#41	—	3	2	no	—
167	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
168	C₄⋊Q₈	—	3	1	no	—
169	#127	—	3	1	no	—
170	#124	—	3	2	no	—
171	#121	—	6	2	no	—
172	C₄×D₄	—	6	2	no	—
173	C₂×C₄²	—	3	2	no	—
174	C₂×C₄²	—	6	2	no	—
175	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
176	C₄⋊C₈	—	2	2	no	—
177	#104	—	6	8	no	—
178	C₂×C₃×C₅	—	2	2	no	—
179	C₂×C₃×C₅	#390	1	1	yes	characteristic
180	C₂×C₃×C₅	C₂×#104	1	2	yes	—
181	C₅×D₃	—	8	2	no	—
182	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂²×D₃	—	2	2	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
186	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
187	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₅×#41	1	2	yes	—
191	#67	—	4	4	no	—
192	C₂×Dic₃	—	2	8	no	—
193	C₂×Dic₃	—	2	4	no	—
194	C₂×Dic₃	—	2	2	no	—
195	D₁₂	—	2	4	no	—
196	D₁₂	—	4	4	no	—
197	C₄×D₃	—	4	2	no	—
198	C₄×D₃	—	4	2	no	—
199	Dic₆	—	2	4	no	—
200	C₃×C₈	—	4	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	12	4	no	—
203	C₂²×C₅	—	12	2	no	—
204	C₂²×C₅	#895	1	2	yes	—
205	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	—	6	8	no	—
208	C₄×C₅	—	12	2	no	—
209	C₄×C₅	#825	1	4	yes	complemented
210	C₄×C₅	#893	1	2	yes	—
211	C₄×C₅	#940	1	1	yes	characteristic
212	C₄×C₅	#947	1	1	yes	characteristic
213	#41	—	6	4	no	—
214	#41	—	6	4	no	—
215	C₂×Q₈	—	3	2	no	—
216	C₂×D₄	—	3	2	no	—
217	C₂×D₄	—	6	2	no	—
218	C₂×D₄	—	6	2	no	—
219	C₂²×C₄	—	6	2	no	—
220	C₂²×C₄	—	6	2	no	—
221	C₂²×C₄	C₅×D₁₂	1	2	yes	—
222	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₄⋊C₄	—	6	2	no	—
226	C₂²⋊C₄	—	6	4	no	—
227	C₄²	—	3	4	no	—
228	C₄²	—	6	8	no	—
229	C₄²	—	6	2	no	—
230	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
231	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
232	C₄²	C₅×#67	1	2	yes	—
233	C₃×C₅	#1962	1	1	yes	characteristic, complemented, Hall
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂²×C₃	C₅×#121	1	2	yes	—
237	C₂×D₃	—	4	2	no	—
238	C₂×D₃	—	4	4	no	—
239	C₃×C₄	—	2	2	no	—
240	C₃×C₄	C₅×#104	1	4	yes	complemented
241	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
242	C₃×C₄	C₅×#124	1	1	yes	characteristic
243	C₃×C₄	C₅×#127	1	1	yes	characteristic
244	Dic₃	—	2	8	no	—
245	Dic₃	—	4	2	no	—
246	C₂×C₅	—	2	2	no	—
247	C₂×C₅	—	24	2	no	—
248	C₂×C₅	#4988	1	1	yes	characteristic
249	C₂×C₅	C₂×#825	1	2	yes	—
250	C₂³	—	6	2	no	—
251	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
252	Q₈	—	6	4	no	—
253	D₄	—	6	4	no	—
254	D₄	—	12	4	no	—
255	D₄	—	12	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	6	8	no	—
258	C₂×C₄	—	6	4	no	—
259	C₂×C₄	—	6	2	no	—
260	C₂×C₄	—	12	2	no	—
261	C₂×C₄	—	12	2	no	—
262	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
263	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
264	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
265	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
266	C₂×C₄	C₅×#239	1	2	yes	—
267	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
268	C₈	—	4	2	no	—
269	C₂×C₃	—	2	2	no	—
270	C₂×C₃	C₅×#390	1	1	yes	characteristic
271	C₂×C₃	C₂×C₅×#104	1	2	yes	—
272	D₃	—	8	2	no	—
273	C₅	#70512	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	2	no	—
275	C₂²	—	12	2	no	—
276	C₂²	—	12	4	no	—
277	C₂²	C₅×#895	1	2	yes	—
278	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	6	8	no	—
281	C₄	—	12	2	no	—
282	C₄	C₅×#825	1	4	yes	complemented
283	C₄	C₅×#893	1	2	yes	—
284	C₄	C₅×#940	1	1	yes	characteristic
285	C₄	C₅×#947	1	1	yes	characteristic
286	C₃	C₅×#1962	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	2	2	no	—
288	C₂	—	24	2	no	—
289	C₂	C₅×#4988	1	1	yes	characteristic
290	C₂	C₂×C₅×#825	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#1962
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₂×C₅×#825	C₂	semidirect
4. C₄²×C₅×Dic₃	C₂	semidirect
12. C₅×#875	C₄	semidirect
13. C₅×#874	C₄	semidirect
15. C₅×#869	C₄	semidirect
18. #70512	C₅	direct (N × H)
20. C₅×#423	D₃	semidirect
42. C₄×C₅×Dic₃	D₄	semidirect
44. C₃×#423	C₂×C₅	semidirect
45. C₂×#825	C₂×C₅	semidirect
46. C₄²×Dic₃	C₂×C₅	semidirect
85. #875	C₄×C₅	semidirect
86. #874	C₄×C₅	semidirect
88. #869	C₄×C₅	semidirect
112. #423	C₅×D₃	semidirect
121. C₃×C₄×C₅	#104	semidirect
146. C₄×Dic₃	C₅×D₄	semidirect
209. C₄×C₅	#825	semidirect
233. C₃×C₅	#1962	semidirect
240. C₃×C₄	C₅×#104	semidirect
273. C₅	#70512	direct (N × H)
282. C₄	C₅×#825	semidirect
286. C₃	C₅×#1962	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₃×C₅×#423	2
C₂×C₅×#825	2
C₄²×C₅×Dic₃	2
C₅×#1962	3
#70512	5

Subgroups — C5×#70512 #14305987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70512 #14305987