Subgroups — C₅×#68570 #14285712

Order 1920.

336 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4835	C₂	1	2	yes	maximal
2	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4480	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1568	—	3	1	no	maximal, Hall
7	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
8	C₂×C₅×D₃×C₈	C₄	1	1	yes	characteristic
9	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×(C₂×D₃)⋊C₈	—	2	2	no	—
13	C₅×#815	—	2	2	no	—
14	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
15	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic
16	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	#68570	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#432	—	3	2	no	—
20	C₂×C₄×C₅×C₈	—	3	1	no	—
21	C₅×#335	—	3	1	no	—
22	C₅×#335	D₃	1	1	yes	characteristic, complemented
23	C₅×#334	—	3	2	no	—
24	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₅×D₃	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	—	2	2	no	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
31	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
33	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₅×D₃×C₈	—	2	2	no	—
38	C₅×D₃×C₈	—	2	2	no	—
39	#4835	C₂×C₅	1	2	yes	—
40	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
41	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
42	#4480	C₂×C₅	1	2	yes	complemented
43	#4458	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂²×C₅×C₈	—	3	1	no	—
45	C₂²×C₅×C₈	—	3	1	no	—
46	C₂×C₄²×C₅	—	3	1	no	—
47	C₅×C₄⋊C₈	—	3	2	no	—
48	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
49	C₅×C₂²⋊C₈	—	6	2	no	—
50	C₅×#97	—	6	2	no	—
51	C₄×C₅×C₈	—	3	1	no	—
52	C₄×C₅×C₈	—	3	1	no	—
53	C₄×C₅×C₈	—	3	1	no	—
54	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
55	#1568	—	3	1	no	Sylow
56	C₂²×C₅×D₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
63	C₄×C₅×D₃	—	2	2	no	—
64	C₄×C₅×D₃	—	2	1	no	—
65	C₄×C₅×D₃	—	2	1	no	—
66	C₄×C₅×D₃	—	2	1	no	—
67	C₄×C₅×D₃	—	2	1	no	—
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	—	4	2	no	—
70	C₃×C₅×C₈	M₁₆	1	2	yes	—
71	C₅×C₃⋊C₈	—	2	1	no	—
72	C₅×C₃⋊C₈	—	4	2	no	—
73	C₅×C₃⋊C₈	M₁₆	1	2	yes	—
74	C₂×D₃×C₈	C₄×C₅	1	1	yes	characteristic
75	C₂×D₃×C₈	C₄×C₅	1	1	yes	characteristic
76	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
77	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
79	(C₂×D₃)⋊C₈	—	2	2	no	—
80	#815	—	2	2	no	—
81	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
82	C₈×Dic₃	C₄×C₅	1	1	yes	characteristic
83	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
84	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	3	1	no	—
86	C₂²×C₄×C₅	—	3	1	no	—
87	C₂²×C₄×C₅	—	3	1	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	3	1	no	—
90	C₂×C₅×C₈	—	3	1	no	—
91	C₂×C₅×C₈	—	6	2	no	—
92	C₂×C₅×C₈	—	6	2	no	—
93	C₂×C₅×C₈	—	6	2	no	—
94	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
95	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
96	C₄²×C₅	—	3	1	no	—
97	C₄²×C₅	—	3	1	no	—
98	C₄²×C₅	—	3	1	no	—
99	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
100	#432	—	3	2	no	—
101	C₂×C₄×C₈	—	3	1	no	—
102	#335	—	3	1	no	—
103	#335	C₅×D₃	1	1	yes	characteristic, complemented
104	#334	—	3	2	no	—
105	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₅×D₃	—	2	1	no	—
107	C₂×C₅×D₃	—	2	1	no	—
108	C₂×C₅×D₃	#108	1	2	yes	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
111	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₃×C₄×C₅	#135	1	1	yes	characteristic
113	C₃×C₄×C₅	#135	1	1	yes	characteristic
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	2	1	no	—
116	C₅×Dic₃	—	2	1	no	—
117	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
118	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
119	C₂×C₄×D₃	C₅×Q₈	1	1	yes	characteristic
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
123	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
124	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
125	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
126	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
127	C₂×C₃⋊C₈	—	2	2	no	—
128	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
130	D₃×C₈	—	2	2	no	—
131	D₃×C₈	—	2	2	no	—
132	C₂³×C₅	—	3	1	no	—
133	C₂×C₄×C₅	—	3	1	no	—
134	C₂×C₄×C₅	—	3	1	no	—
135	C₂×C₄×C₅	—	3	1	no	—
136	C₂×C₄×C₅	—	6	2	no	—
137	C₂×C₄×C₅	—	6	1	no	—
138	C₂×C₄×C₅	—	6	1	no	—
139	C₂×C₄×C₅	—	6	1	no	—
140	C₂×C₄×C₅	—	6	1	no	—
141	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	—	3	2	no	—
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	—	6	1	no	—
148	C₅×C₈	—	12	2	no	—
149	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
150	C₂²×C₈	—	3	1	no	—
151	C₂²×C₈	—	3	1	no	—
152	C₂×C₄²	—	3	1	no	—
153	C₄⋊C₈	—	3	2	no	—
154	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
155	C₂²⋊C₈	—	6	2	no	—
156	#97	—	6	2	no	—
157	C₄×C₈	—	3	1	no	—
158	C₄×C₈	—	3	1	no	—
159	C₄×C₈	—	3	1	no	—
160	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
161	C₂×C₃×C₅	#423	1	1	yes	characteristic
162	C₂×C₃×C₅	#427	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×#108	1	1	yes	characteristic
164	C₅×D₃	—	2	2	no	—
165	C₂²×D₃	C₅×C₄⋊C₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
169	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
171	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₄×D₃	—	2	2	no	—
173	C₄×D₃	—	2	1	no	—
174	C₄×D₃	—	2	1	no	—
175	C₄×D₃	—	2	1	no	—
176	C₄×D₃	—	2	1	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	4	2	no	—
179	C₃×C₈	C₅×M₁₆	1	2	yes	—
180	C₃⋊C₈	—	2	1	no	—
181	C₃⋊C₈	—	4	2	no	—
182	C₃⋊C₈	C₅×M₁₆	1	2	yes	—
183	C₂²×C₅	—	3	2	no	—
184	C₂²×C₅	—	6	1	no	—
185	C₂²×C₅	—	6	1	no	—
186	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
187	C₄×C₅	—	2	1	no	—
188	C₄×C₅	—	6	1	no	—
189	C₄×C₅	—	6	1	no	—
190	C₄×C₅	—	6	1	no	—
191	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
192	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
193	C₄×C₅	#912	1	1	yes	characteristic
194	C₄×C₅	#919	1	1	yes	characteristic
195	C₂²×C₄	—	3	1	no	—
196	C₂²×C₄	—	3	1	no	—
197	C₂²×C₄	—	3	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	3	1	no	—
200	C₂×C₈	—	3	1	no	—
201	C₂×C₈	—	6	2	no	—
202	C₂×C₈	—	6	2	no	—
203	C₂×C₈	—	6	2	no	—
204	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
205	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
206	C₄²	—	3	1	no	—
207	C₄²	—	3	1	no	—
208	C₄²	—	3	1	no	—
209	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
210	C₃×C₅	#1568	1	1	yes	characteristic, complemented, Hall
211	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
212	C₂×D₃	—	2	1	no	—
213	C₂×D₃	—	2	1	no	—
214	C₂×D₃	C₅×#108	1	2	yes	—
215	C₃×C₄	—	2	1	no	—
216	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
217	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
218	C₃×C₄	C₅×#135	1	1	yes	characteristic
219	C₃×C₄	C₅×#135	1	1	yes	characteristic
220	Dic₃	—	2	1	no	—
221	Dic₃	—	2	1	no	—
222	Dic₃	—	2	1	no	—
223	C₂×C₅	—	6	2	no	—
224	C₂×C₅	#4837	1	1	yes	characteristic
225	C₂×C₅	#4880	1	1	yes	characteristic
226	C₂×C₅	D₃×#108	1	1	yes	characteristic
227	C₂³	—	3	1	no	—
228	C₂×C₄	—	3	1	no	—
229	C₂×C₄	—	3	1	no	—
230	C₂×C₄	—	3	1	no	—
231	C₂×C₄	—	6	2	no	—
232	C₂×C₄	—	6	1	no	—
233	C₂×C₄	—	6	1	no	—
234	C₂×C₄	—	6	1	no	—
235	C₂×C₄	—	6	1	no	—
236	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
237	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
238	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
239	C₈	—	2	1	no	—
240	C₈	—	3	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	6	1	no	—
243	C₈	—	12	2	no	—
244	C₈	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
245	C₂×C₃	C₅×#423	1	1	yes	characteristic
246	C₂×C₃	C₅×#427	1	1	yes	characteristic
247	C₂×C₃	C₂×C₅×#108	1	1	yes	characteristic
248	D₃	—	2	2	no	—
249	C₅	#68570	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	3	2	no	—
251	C₂²	—	6	1	no	—
252	C₂²	—	6	1	no	—
253	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	6	1	no	—
256	C₄	—	6	1	no	—
257	C₄	—	6	1	no	—
258	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
259	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
260	C₄	C₅×#912	1	1	yes	characteristic
261	C₄	C₅×#919	1	1	yes	characteristic
262	C₃	C₅×#1568	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	6	2	no	—
264	C₂	C₅×#4837	1	1	yes	characteristic
265	C₂	C₅×#4880	1	1	yes	characteristic
266	C₂	C₅×D₃×#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₅×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#335	D₃
hypocenter	C₃	C₅×#1568
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#335

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#335	C₂	semidirect
4. C₅×#4480	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #68570	C₅	direct (N × H)
22. C₅×#335	D₃	semidirect
41. C₃×#335	C₂×C₅	semidirect
42. #4480	C₂×C₅	semidirect
43. #4458	C₂×C₅	semidirect
103. #335	C₅×D₃	semidirect
210. C₃×C₅	#1568	semidirect
249. C₅	#68570	direct (N × H)
262. C₃	C₅×#1568	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4835	2
C₄×C₅×D₃×C₈	2
C₃×C₅×#335	2
C₅×#4480	2
C₅×#4458	2
C₅×#1568	3
#68570	5

Subgroups — C5×#68570 #14285712

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68570 #14285712