Subgroups — C₅×#70746 #14285597

Order 1920.

360 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	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic
4	C₅×#4480	C₂	1	2	yes	maximal, complemented
5	C₅×#4479	C₂	1	2	yes	maximal, complemented
6	C₅×#1565	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₅×#208	C₄	1	1	yes	characteristic
9	C₂×C₅×#208	C₄	1	1	yes	characteristic
10	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
14	C₃×C₄×C₅×C₈	—	2	1	no	—
15	C₅×C₄⋊(C₃⋊C₈)	—	2	1	no	—
16	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₅×#803	C₄	1	2	yes	—
18	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
19	#70746	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×#431	D₃	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₈	—	3	1	no	—
22	C₄×C₅×M₁₆	—	3	1	no	—
23	C₅×#335	—	3	2	no	—
24	C₅×#334	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
32	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×#208	—	2	2	no	—
35	C₅×#208	C₈	1	4	yes	complemented
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	—	4	2	no	—
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₃×#431	C₂×C₅	1	1	yes	characteristic
41	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	1	yes	characteristic
42	C₄×#208	C₂×C₅	1	1	yes	characteristic
43	#4480	C₂×C₅	1	2	yes	complemented
44	#4479	C₂×C₅	1	2	yes	complemented
45	C₂×C₅×M₁₆	—	3	1	no	—
46	C₂×C₅×M₁₆	—	3	1	no	—
47	C₂²×C₅×C₈	—	6	1	no	—
48	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	3	2	no	—
50	C₅×C₄⋊C₈	—	6	1	no	—
51	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
52	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₅×#97	—	3	2	no	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	—	3	2	no	—
57	#1565	—	3	1	no	Sylow
58	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₃×C₅×C₈	—	4	1	no	—
71	C₃×C₅×C₈	—	4	1	no	—
72	C₅×C₃⋊C₈	—	2	2	no	—
73	C₅×C₃⋊C₈	—	4	2	no	—
74	C₅×C₃⋊C₈	C₂×C₈	1	4	yes	complemented
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₂×#208	C₄×C₅	1	1	yes	characteristic
77	C₂×#208	C₄×C₅	1	1	yes	characteristic
78	C₂²×C₃⋊C₈	—	2	1	no	—
79	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₂²⋊C₈	—	2	1	no	—
82	C₃×C₄×C₈	—	2	1	no	—
83	C₄⋊(C₃⋊C₈)	—	2	1	no	—
84	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
85	#803	C₄×C₅	1	2	yes	—
86	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
87	C₂²×C₄×C₅	Dic₆	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
89	C₂²×C₄×C₅	#67	1	1	yes	characteristic
90	C₅×M₁₆	—	3	4	no	—
91	C₅×M₁₆	—	6	2	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	3	2	no	—
95	C₂×C₅×C₈	—	3	2	no	—
96	C₂×C₅×C₈	—	6	2	no	—
97	C₂×C₅×C₈	—	12	2	no	—
98	C₄²×C₅	Dic₆	1	1	yes	characteristic
99	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
100	C₄²×C₅	#67	1	1	yes	characteristic
101	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
102	#431	C₅×D₃	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	3	1	no	—
104	C₄×M₁₆	—	3	1	no	—
105	#335	—	3	2	no	—
106	#334	—	3	2	no	—
107	C₂²×C₃×C₅	—	2	1	no	—
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
110	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
115	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₃×C₄×C₅	#137	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
125	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
126	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
127	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
128	#208	—	2	2	no	—
129	#208	C₅×C₈	1	4	yes	complemented
130	C₂×C₃⋊C₈	—	2	2	no	—
131	C₂×C₃⋊C₈	—	4	2	no	—
132	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
133	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
134	C₂³×C₅	Dic₃⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
139	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	2	yes	—
140	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
146	C₅×C₈	—	3	4	no	—
147	C₅×C₈	—	4	1	no	—
148	C₅×C₈	—	4	1	no	—
149	C₅×C₈	—	6	2	no	—
150	C₅×C₈	—	12	2	no	—
151	C₂×M₁₆	—	3	1	no	—
152	C₂×M₁₆	—	3	1	no	—
153	C₂²×C₈	—	6	1	no	—
154	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
155	C₄⋊C₈	—	3	2	no	—
156	C₄⋊C₈	—	6	1	no	—
157	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
158	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
159	C₂²⋊C₈	—	2	1	no	—
160	#97	—	3	2	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	3	2	no	—
163	C₂×C₃×C₅	—	2	2	no	—
164	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	#428	1	1	yes	characteristic
166	C₂×C₃×C₅	#430	1	1	yes	characteristic
167	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	4	1	no	—
171	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
175	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
179	C₃×C₈	—	4	1	no	—
180	C₃×C₈	—	4	1	no	—
181	C₃⋊C₈	—	2	2	no	—
182	C₃⋊C₈	—	4	2	no	—
183	C₃⋊C₈	C₂×C₅×C₈	1	4	yes	complemented
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	Dic₃⋊C₈	1	2	yes	—
187	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	Dic₃⋊C₈	1	2	yes	—
192	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
193	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
194	C₄×C₅	#949	1	1	yes	characteristic
195	C₄×C₅	#951	1	1	yes	characteristic
196	C₂²×C₄	C₅×Dic₆	1	1	yes	characteristic
197	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
198	C₂²×C₄	C₅×#67	1	1	yes	characteristic
199	M₁₆	—	3	4	no	—
200	M₁₆	—	6	2	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	3	2	no	—
204	C₂×C₈	—	3	2	no	—
205	C₂×C₈	—	6	2	no	—
206	C₂×C₈	—	12	2	no	—
207	C₄²	C₅×Dic₆	1	1	yes	characteristic
208	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
209	C₄²	C₅×#67	1	1	yes	characteristic
210	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
211	C₃×C₅	#1565	1	1	yes	characteristic, complemented, Hall
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
215	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
220	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
221	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
222	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₃×C₄	C₅×#137	1	1	yes	characteristic
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	#4995	1	1	yes	characteristic
226	C₂×C₅	C₂×Dic₃⋊C₈	1	1	yes	characteristic
227	C₂×C₅	#5124	1	1	yes	characteristic
228	C₂³	C₅×Dic₃⋊C₄	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	C₅×D₃×C₈	1	2	yes	—
233	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
234	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
236	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
239	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
240	C₈	—	3	4	no	—
241	C₈	—	4	1	no	—
242	C₈	—	4	1	no	—
243	C₈	—	6	2	no	—
244	C₈	—	12	2	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
247	C₂×C₃	C₅×#428	1	1	yes	characteristic
248	C₂×C₃	C₅×#430	1	1	yes	characteristic
249	C₅	#70746	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₅×Dic₃⋊C₈	1	2	yes	—
253	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	C₅×Dic₃⋊C₈	1	2	yes	—
258	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
259	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
260	C₄	C₅×#949	1	1	yes	characteristic
261	C₄	C₅×#951	1	1	yes	characteristic
262	C₃	C₅×#1565	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	2	2	no	—
264	C₂	C₅×#4995	1	1	yes	characteristic
265	C₂	C₂×C₅×Dic₃⋊C₈	1	1	yes	characteristic
266	C₂	C₅×#5124	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#431	D₃
hypocenter	C₃	C₅×#1565
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#431

Complements

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

N	H ≅ G/N	type
4. C₅×#4480	C₂	semidirect
5. C₅×#4479	C₂	semidirect
19. #70746	C₅	direct (N × H)
35. C₅×#208	C₈	semidirect
43. #4480	C₂×C₅	semidirect
44. #4479	C₂×C₅	semidirect
74. C₅×C₃⋊C₈	C₂×C₈	semidirect
129. #208	C₅×C₈	semidirect
183. C₃⋊C₈	C₂×C₅×C₈	semidirect
211. C₃×C₅	#1565	semidirect
249. C₅	#70746	direct (N × H)
262. C₃	C₅×#1565	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#431	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₄×C₅×#208	2
C₅×#4480	2
C₅×#4479	2
C₅×#1565	3
#70746	5

Subgroups — C5×#70746 #14285597

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70746 #14285597