Subgroups — C₅×#71453 #14315242

Order 1920.

288 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5144	C₂	1	1	yes	maximal, characteristic
3	C₅×#4901	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4561	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4515	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4511	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4502	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2171	—	3	1	no	maximal, Hall
9	C₃×C₅×#131	C₂²	1	1	yes	characteristic
10	C₅×#950	C₄	1	1	yes	characteristic
11	C₅×#901	C₂²	1	1	yes	characteristic
12	C₃×C₅×M₃₂	—	2	1	no	—
13	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₆	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#837	C₄	1	1	yes	characteristic
17	C₅×#825	C₄	1	1	yes	characteristic
18	C₅×#822	C₄	1	1	yes	characteristic
19	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
20	C₅×#812	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₅×#798	—	2	1	no	—
23	C₅×D₃×C₁₆	—	2	1	no	—
24	#71453	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#504	—	3	1	no	—
26	C₅×#504	D₃	1	1	yes	characteristic, complemented
27	C₅×#443	—	3	1	no	—
28	C₅×#364	—	3	1	no	—
29	C₅×#350	—	3	1	no	—
30	C₅×#350	—	3	1	no	—
31	C₅×#346	—	3	1	no	—
32	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	C₅×#235	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×#216	C₈	1	1	yes	characteristic
39	C₅×#215	C₈	1	1	yes	characteristic
40	C₅×#214	C₈	1	1	yes	characteristic
41	C₅×#213	C₈	1	1	yes	characteristic
42	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
43	C₅×#208	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
46	C₅×D₃×C₈	—	2	1	no	—
47	C₃×C₅×C₁₆	—	2	1	no	—
48	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
49	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
50	C₅×C₃⋊C₁₆	—	2	1	no	—
51	C₅×C₃⋊C₁₆	—	2	1	no	—
52	C₃×#504	C₂×C₅	1	1	yes	characteristic, complemented
53	#5144	C₂×C₅	1	1	yes	characteristic
54	#4901	C₂×C₅	1	1	yes	characteristic, complemented
55	#4561	C₂×C₅	1	1	yes	characteristic, complemented
56	#4515	C₂×C₅	1	1	yes	characteristic, complemented
57	#4511	C₂×C₅	1	1	yes	characteristic, complemented
58	#4502	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#135	—	3	1	no	—
60	C₅×#131	—	3	1	no	—
61	C₅×#131	C₂×D₃	1	1	yes	characteristic
62	C₅×M₃₂	—	2	1	no	—
63	C₅×M₃₂	—	3	1	no	—
64	C₅×M₃₂	—	6	1	no	—
65	C₅×M₃₂	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×C₁₆	—	2	1	no	—
67	C₂×C₅×C₁₆	—	3	1	no	—
68	C₂×C₅×C₁₆	—	6	1	no	—
69	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
70	C₅×#108	—	3	1	no	—
71	C₅×#104	—	3	1	no	—
72	C₅×#104	—	3	1	no	—
73	C₄×C₅×C₈	—	3	1	no	—
74	#2171	—	3	1	no	Sylow
75	C₃×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₅×#67	—	2	1	no	—
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₅×D₁₂	C₂×C₈	1	1	yes	characteristic
83	C₄×C₅×D₃	—	2	1	no	—
84	C₅×Dic₆	C₂×C₈	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×C₈	#41	1	1	yes	characteristic
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₅×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
90	C₅×C₃⋊C₈	C₂×C₈	1	1	yes	characteristic
91	C₃×#131	C₂²×C₅	1	1	yes	characteristic
92	#950	C₄×C₅	1	1	yes	characteristic
93	#901	C₂²×C₅	1	1	yes	characteristic
94	C₃×M₃₂	—	2	1	no	—
95	C₃×M₃₂	C₂²×C₅	1	1	yes	characteristic
96	C₂×C₃×C₁₆	—	2	1	no	—
97	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
98	#837	C₄×C₅	1	1	yes	characteristic
99	#825	C₄×C₅	1	1	yes	characteristic
100	#822	C₄×C₅	1	1	yes	characteristic
101	C₈×Dic₃	C₂²×C₅	1	1	yes	characteristic
102	#812	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
104	#798	—	2	1	no	—
105	D₃×C₁₆	—	2	1	no	—
106	C₅×#41	—	3	1	no	—
107	C₅×#41	C₄×D₃	1	1	yes	characteristic
108	C₅×Q₁₆	—	3	1	no	—
109	C₅×SD₁₆	—	3	1	no	—
110	C₅×SD₁₆	—	3	1	no	—
111	C₅×D₈	—	3	1	no	—
112	C₅×M₁₆	—	2	1	no	—
113	C₅×M₁₆	—	3	1	no	—
114	C₅×M₁₆	—	6	1	no	—
115	C₅×M₁₆	C₄×D₃	1	1	yes	characteristic
116	C₂×C₅×C₈	—	2	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	6	1	no	—
119	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
120	C₄²×C₅	—	3	1	no	—
121	C₅×C₁₆	—	2	1	no	—
122	C₅×C₁₆	—	6	1	no	—
123	C₅×C₁₆	—	6	1	no	—
124	C₅×C₁₆	#67	1	1	yes	characteristic
125	C₅×C₁₆	#67	1	1	yes	characteristic
126	#504	—	3	1	no	—
127	#504	C₅×D₃	1	1	yes	characteristic, complemented
128	#443	—	3	1	no	—
129	#364	—	3	1	no	—
130	#350	—	3	1	no	—
131	#350	—	3	1	no	—
132	#346	—	3	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#131	1	1	yes	characteristic
135	C₂×C₅×D₃	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
138	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
139	C₅×Dic₃	—	2	1	no	—
140	C₅×Dic₃	—	4	1	no	—
141	C₃×#41	C₂×C₄×C₅	1	1	yes	characteristic
142	#235	C₂×C₄×C₅	1	1	yes	characteristic
143	C₃×M₁₆	—	2	1	no	—
144	C₃×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
145	C₂×C₃×C₈	—	2	1	no	—
146	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
147	#216	C₅×C₈	1	1	yes	characteristic
148	#215	C₅×C₈	1	1	yes	characteristic
149	#214	C₅×C₈	1	1	yes	characteristic
150	#213	C₅×C₈	1	1	yes	characteristic
151	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
152	#208	C₂×C₄×C₅	1	1	yes	characteristic
153	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
154	ΓL(1,𝔽₂₅)	—	2	1	no	—
155	D₃×C₈	—	2	1	no	—
156	C₃×C₁₆	—	2	1	no	—
157	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
158	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
159	C₃⋊C₁₆	—	2	1	no	—
160	C₃⋊C₁₆	—	2	1	no	—
161	C₅×Q₈	—	3	1	no	—
162	C₅×Q₈	D₃×C₈	1	1	yes	characteristic
163	C₅×D₄	—	2	1	no	—
164	C₅×D₄	—	3	1	no	—
165	C₅×D₄	—	6	1	no	—
166	C₅×D₄	D₃×C₈	1	1	yes	characteristic
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	6	1	no	—
169	C₂×C₄×C₅	—	6	1	no	—
170	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	C₅×C₈	—	3	1	no	—
173	C₅×C₈	—	3	1	no	—
174	C₅×C₈	—	6	1	no	—
175	C₅×C₈	#235	1	1	yes	characteristic
176	C₅×C₈	C₂×#67	1	1	yes	characteristic
177	#135	—	3	1	no	—
178	#131	—	3	1	no	—
179	#131	C₂×C₅×D₃	1	1	yes	characteristic
180	M₃₂	—	2	1	no	—
181	M₃₂	—	3	1	no	—
182	M₃₂	—	6	1	no	—
183	M₃₂	C₂×C₅×D₃	1	1	yes	characteristic
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	—	3	1	no	—
186	C₂×C₁₆	—	6	1	no	—
187	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
188	#108	—	3	1	no	—
189	#104	—	3	1	no	—
190	#104	—	3	1	no	—
191	C₄×C₈	—	3	1	no	—
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	4	1	no	—
194	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
195	C₅×D₃	—	4	1	no	—
196	C₃×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	C₂×C₅×C₈	1	1	yes	characteristic
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
201	#67	—	2	1	no	—
202	C₂×Dic₃	—	2	1	no	—
203	D₁₂	C₂×C₅×C₈	1	1	yes	characteristic
204	C₄×D₃	—	2	1	no	—
205	Dic₆	C₂×C₅×C₈	1	1	yes	characteristic
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
208	C₃×C₈	C₅×#41	1	1	yes	characteristic
209	C₃⋊C₈	—	2	1	no	—
210	C₃⋊C₈	C₂×C₅×C₈	1	1	yes	characteristic
211	C₃⋊C₈	C₂×C₅×C₈	1	1	yes	characteristic
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	6	1	no	—
214	C₂²×C₅	#901	1	1	yes	characteristic
215	C₄×C₅	—	2	1	no	—
216	C₄×C₅	—	6	1	no	—
217	C₄×C₅	—	12	1	no	—
218	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
219	C₄×C₅	C₄×#67	1	1	yes	characteristic
220	#41	—	3	1	no	—
221	#41	C₄×C₅×D₃	1	1	yes	characteristic
222	Q₁₆	—	3	1	no	—
223	SD₁₆	—	3	1	no	—
224	SD₁₆	—	3	1	no	—
225	D₈	—	3	1	no	—
226	M₁₆	—	2	1	no	—
227	M₁₆	—	3	1	no	—
228	M₁₆	—	6	1	no	—
229	M₁₆	C₄×C₅×D₃	1	1	yes	characteristic
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	3	1	no	—
232	C₂×C₈	—	6	1	no	—
233	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
234	C₄²	—	3	1	no	—
235	C₁₆	—	2	1	no	—
236	C₁₆	—	6	1	no	—
237	C₁₆	—	6	1	no	—
238	C₁₆	C₅×#67	1	1	yes	characteristic
239	C₁₆	C₅×#67	1	1	yes	characteristic
240	C₃×C₅	#2171	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	C₅×#131	1	1	yes	characteristic
243	C₂×D₃	—	2	1	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
246	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
247	Dic₃	—	2	1	no	—
248	Dic₃	—	4	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	—	12	1	no	—
252	C₂×C₅	C₈×#67	1	1	yes	characteristic
253	Q₈	—	3	1	no	—
254	Q₈	C₅×D₃×C₈	1	1	yes	characteristic
255	D₄	—	2	1	no	—
256	D₄	—	3	1	no	—
257	D₄	—	6	1	no	—
258	D₄	C₅×D₃×C₈	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	6	1	no	—
261	C₂×C₄	—	6	1	no	—
262	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	3	1	no	—
265	C₈	—	3	1	no	—
266	C₈	—	6	1	no	—
267	C₈	C₅×#235	1	1	yes	characteristic
268	C₈	C₂×C₅×#67	1	1	yes	characteristic
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
272	D₃	—	4	1	no	—
273	C₅	#71453	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	6	1	no	—
276	C₂²	C₅×#901	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	6	1	no	—
279	C₄	—	12	1	no	—
280	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
281	C₄	C₄×C₅×#67	1	1	yes	characteristic
282	C₃	C₅×#2171	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	4	1	no	—
285	C₂	—	12	1	no	—
286	C₂	C₅×C₈×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂×#67
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#504	C₂
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₅×#950	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#504	D₃
hypocenter	C₃	C₅×#2171
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂×C₅×C₁₆ ◁ C₅×#504

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#504	C₂	semidirect
3. C₅×#4901	C₂	semidirect
4. C₅×#4561	C₂	semidirect
5. C₅×#4515	C₂	semidirect
6. C₅×#4511	C₂	semidirect
7. C₅×#4502	C₂	semidirect
24. #71453	C₅	direct (N × H)
26. C₅×#504	D₃	semidirect
52. C₃×#504	C₂×C₅	semidirect
54. #4901	C₂×C₅	semidirect
55. #4561	C₂×C₅	semidirect
56. #4515	C₂×C₅	semidirect
57. #4511	C₂×C₅	semidirect
58. #4502	C₂×C₅	semidirect
127. #504	C₅×D₃	semidirect
240. C₃×C₅	#2171	semidirect
273. C₅	#71453	direct (N × H)
282. C₃	C₅×#2171	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#504	2
C₅×#5144	2
C₅×#4901	2
C₅×#4561	2
C₅×#4515	2
C₅×#4511	2
C₅×#4502	2
C₅×#2171	3
#71453	5

Subgroups — C5×#71453 #14315242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71453 #14315242