Subgroups — C₅×#68074 #14280670

Order 1920.

356 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4685	C₂	1	1	yes	maximal, characteristic
2	C₅×#4685	C₂	1	1	yes	maximal, characteristic
3	C₅×#4685	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1454	—	3	1	no	maximal, Hall
9	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
14	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
15	C₅×(C₂×D₃)⋊C₈	C₄	1	2	yes	—
16	C₅×Dic₃⋊C₈	C₄	1	2	yes	—
17	C₅×Dic₃⋊C₈	C₄	1	2	yes	—
18	C₅×Dic₃⋊C₈	C₄	1	2	yes	—
19	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#68074	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#431	—	3	1	no	—
24	C₅×#431	—	3	1	no	—
25	C₅×#431	—	3	1	no	—
26	C₅×#322	—	3	1	no	—
27	C₅×#322	—	3	1	no	—
28	C₅×#322	D₃	1	1	yes	characteristic, complemented
29	C₅×C₈²	—	3	1	no	—
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	#4685	C₂×C₅	1	1	yes	characteristic
44	#4685	C₂×C₅	1	1	yes	characteristic
45	#4685	C₂×C₅	1	1	yes	characteristic
46	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
47	#4455	C₂×C₅	1	1	yes	characteristic, complemented
48	#4454	C₂×C₅	1	1	yes	characteristic, complemented
49	C₈×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×C₄²×C₅	—	3	1	no	—
51	C₅×C₄⋊C₈	—	3	2	no	—
52	C₅×C₄⋊C₈	—	3	2	no	—
53	C₅×C₄⋊C₈	—	3	2	no	—
54	C₅×C₂²⋊C₈	—	3	2	no	—
55	C₅×C₂²⋊C₈	—	3	2	no	—
56	C₅×C₂²⋊C₈	—	3	2	no	—
57	C₄×C₅×C₈	—	3	1	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₄×C₅×C₈	—	3	1	no	—
60	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
61	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	#1454	—	3	1	no	Sylow
64	C₂²×C₅×D₃	C₄²	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
69	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
71	C₄×C₅×D₃	—	2	2	no	—
72	C₄×C₅×D₃	—	2	2	no	—
73	C₄×C₅×D₃	—	2	2	no	—
74	C₃×C₅×C₈	—	2	2	no	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₅×C₃⋊C₈	—	2	2	no	—
78	C₅×C₃⋊C₈	—	2	2	no	—
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	(C₂×D₃)⋊C₈	C₄×C₅	1	2	yes	—
85	(C₂×D₃)⋊C₈	C₄×C₅	1	2	yes	—
86	(C₂×D₃)⋊C₈	C₄×C₅	1	2	yes	—
87	Dic₃⋊C₈	C₄×C₅	1	2	yes	—
88	Dic₃⋊C₈	C₄×C₅	1	2	yes	—
89	Dic₃⋊C₈	C₄×C₅	1	2	yes	—
90	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
91	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	—	3	1	no	—
94	C₂²×C₄×C₅	—	3	1	no	—
95	C₂²×C₄×C₅	—	3	1	no	—
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
100	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
101	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
102	C₄²×C₅	—	3	1	no	—
103	C₄²×C₅	—	3	1	no	—
104	C₄²×C₅	—	3	1	no	—
105	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
106	#431	—	3	1	no	—
107	#431	—	3	1	no	—
108	#431	—	3	1	no	—
109	#322	—	3	1	no	—
110	#322	—	3	1	no	—
111	#322	C₅×D₃	1	1	yes	characteristic, complemented
112	C₈²	—	3	1	no	—
113	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₅×D₃	—	2	1	no	—
115	C₂×C₅×D₃	—	2	1	no	—
116	C₂×C₅×D₃	—	2	1	no	—
117	C₃×C₄×C₅	#131	1	2	yes	—
118	C₃×C₄×C₅	#131	1	2	yes	—
119	C₃×C₄×C₅	#131	1	2	yes	—
120	C₅×Dic₃	—	2	1	no	—
121	C₅×Dic₃	—	2	1	no	—
122	C₅×Dic₃	—	2	1	no	—
123	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
125	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
128	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
129	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
130	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
131	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
132	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
133	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
134	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
135	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
136	C₂³×C₅	—	3	1	no	—
137	C₂×C₄×C₅	—	3	1	no	—
138	C₂×C₄×C₅	—	3	1	no	—
139	C₂×C₄×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	6	2	no	—
141	C₂×C₄×C₅	—	6	2	no	—
142	C₂×C₄×C₅	—	6	2	no	—
143	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	2	2	no	—
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	—	6	2	no	—
150	C₅×C₈	—	6	2	no	—
151	C₅×C₈	—	6	2	no	—
152	C₂×C₄²	—	3	1	no	—
153	C₄⋊C₈	—	3	2	no	—
154	C₄⋊C₈	—	3	2	no	—
155	C₄⋊C₈	—	3	2	no	—
156	C₂²⋊C₈	—	3	2	no	—
157	C₂²⋊C₈	—	3	2	no	—
158	C₂²⋊C₈	—	3	2	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	—	3	1	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
163	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
164	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
165	C₂×C₃×C₅	#405	1	1	yes	characteristic
166	C₂×C₃×C₅	#405	1	1	yes	characteristic
167	C₂×C₃×C₅	#405	1	1	yes	characteristic
168	C₅×D₃	—	4	1	no	—
169	C₂²×D₃	C₄²×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
174	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
175	C₂×Dic₃	C₄²×C₅	1	1	yes	characteristic
176	C₄×D₃	—	2	2	no	—
177	C₄×D₃	—	2	2	no	—
178	C₄×D₃	—	2	2	no	—
179	C₃×C₈	—	2	2	no	—
180	C₃×C₈	—	2	2	no	—
181	C₃×C₈	—	2	2	no	—
182	C₃⋊C₈	—	2	2	no	—
183	C₃⋊C₈	—	2	2	no	—
184	C₃⋊C₈	—	2	2	no	—
185	C₂²×C₅	—	6	1	no	—
186	C₂²×C₅	—	6	1	no	—
187	C₂²×C₅	—	6	1	no	—
188	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
189	C₄×C₅	—	6	1	no	—
190	C₄×C₅	—	6	1	no	—
191	C₄×C₅	—	6	1	no	—
192	C₄×C₅	#901	1	2	yes	—
193	C₄×C₅	#907	1	2	yes	—
194	C₄×C₅	#907	1	2	yes	—
195	C₂²×C₄	—	3	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂×C₈	—	3	2	no	—
199	C₂×C₈	—	3	2	no	—
200	C₂×C₈	—	3	2	no	—
201	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
202	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
203	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
204	C₄²	—	3	1	no	—
205	C₄²	—	3	1	no	—
206	C₄²	—	3	1	no	—
207	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
208	C₃×C₅	#1454	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
210	C₂×D₃	—	2	1	no	—
211	C₂×D₃	—	2	1	no	—
212	C₂×D₃	—	2	1	no	—
213	C₃×C₄	C₅×#131	1	2	yes	—
214	C₃×C₄	C₅×#131	1	2	yes	—
215	C₃×C₄	C₅×#131	1	2	yes	—
216	Dic₃	—	2	1	no	—
217	Dic₃	—	2	1	no	—
218	Dic₃	—	2	1	no	—
219	C₂×C₅	—	12	1	no	—
220	C₂×C₅	#4689	1	1	yes	characteristic
221	C₂×C₅	#4708	1	1	yes	characteristic
222	C₂×C₅	#4708	1	1	yes	characteristic
223	C₂³	—	3	1	no	—
224	C₂×C₄	—	3	1	no	—
225	C₂×C₄	—	3	1	no	—
226	C₂×C₄	—	3	1	no	—
227	C₂×C₄	—	6	2	no	—
228	C₂×C₄	—	6	2	no	—
229	C₂×C₄	—	6	2	no	—
230	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
231	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
232	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	6	2	no	—
237	C₈	—	6	2	no	—
238	C₈	—	6	2	no	—
239	C₂×C₃	C₅×#405	1	1	yes	characteristic
240	C₂×C₃	C₅×#405	1	1	yes	characteristic
241	C₂×C₃	C₅×#405	1	1	yes	characteristic
242	D₃	—	4	1	no	—
243	C₅	#68074	1	1	yes	characteristic, direct product factor, Sylow
244	C₂²	—	6	1	no	—
245	C₂²	—	6	1	no	—
246	C₂²	—	6	1	no	—
247	C₂²	C₄²×C₅×D₃	1	1	yes	characteristic
248	C₄	—	6	1	no	—
249	C₄	—	6	1	no	—
250	C₄	—	6	1	no	—
251	C₄	C₅×#901	1	2	yes	—
252	C₄	C₅×#907	1	2	yes	—
253	C₄	C₅×#907	1	2	yes	—
254	C₃	C₅×#1454	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	12	1	no	—
256	C₂	C₅×#4689	1	1	yes	characteristic
257	C₂	C₅×#4708	1	1	yes	characteristic
258	C₂	C₅×#4708	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#322	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₅×D₃	C₄²
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#322	D₃
hypocenter	C₃	C₅×#1454
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₅×#322

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#322	C₂	semidirect
5. C₅×#4455	C₂	semidirect
6. C₅×#4454	C₂	semidirect
7. C₅×C₈×C₃⋊C₈	C₂	semidirect
22. #68074	C₅	direct (N × H)
28. C₅×#322	D₃	semidirect
46. C₃×#322	C₂×C₅	semidirect
47. #4455	C₂×C₅	semidirect
48. #4454	C₂×C₅	semidirect
49. C₈×C₃⋊C₈	C₂×C₅	semidirect
111. #322	C₅×D₃	semidirect
208. C₃×C₅	#1454	semidirect
243. C₅	#68074	direct (N × H)
254. C₃	C₅×#1454	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4685	2
C₃×C₅×#322	2
C₅×#4455	2
C₅×#4454	2
C₅×C₈×C₃⋊C₈	2
C₅×#1454	3
#68074	5

Subgroups — C5×#68074 #14280670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68074 #14280670