Subgroups — C₅×#70847 #14291412

Order 1920.

408 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5015	C₂	1	2	yes	maximal, complemented
3	C₅×#4695	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4485	C₂	1	2	yes	maximal, complemented
5	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1667	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	—	2	1	no	—
8	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
9	C₂×C₅×#213	—	2	2	no	—
10	C₂×C₅×D₂₄	—	2	1	no	—
11	C₂×C₅×D₂₄	—	2	1	no	—
12	C₅×#874	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#809	—	2	2	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	#70847	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#493	—	3	1	no	—
20	C₅×#459	—	3	2	no	—
21	C₅×C₈×D₄	D₃	1	1	yes	characteristic, complemented
22	C₅×#335	—	3	1	no	—
23	C₅×#331	—	3	2	no	—
24	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂×C₅×D₁₂	—	2	2	no	—
27	C₂×C₅×D₁₂	—	2	2	no	—
28	C₂×C₅×D₁₂	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	—	4	2	no	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
34	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₅×#213	—	4	4	no	—
37	C₂×C₅×C₃⋊C₈	—	2	2	no	—
38	C₅×D₂₄	—	4	2	no	—
39	C₅×D₂₄	—	4	2	no	—
40	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic, complemented
41	#5015	C₂×C₅	1	2	yes	complemented
42	#4695	C₂×C₅	1	1	yes	characteristic, complemented
43	#4485	C₂×C₅	1	2	yes	complemented
44	#4458	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂×C₅×D₈	—	6	2	no	—
46	C₂×C₅×D₈	—	6	1	no	—
47	C₂×C₅×D₈	—	6	1	no	—
48	C₂²×C₅×C₈	—	2	1	no	—
49	C₅×#127	—	3	2	no	—
50	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	—	3	2	no	—
52	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
53	C₅×#102	—	6	2	no	—
54	C₅×C₂²⋊C₈	—	2	1	no	—
55	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	#1667	—	3	1	no	Sylow
57	C₂³×C₃×C₅	—	2	1	no	—
58	C₂²×C₅×D₃	—	4	2	no	—
59	C₃×C₅×D₄	—	2	1	no	—
60	C₃×C₅×D₄	D₈	1	2	yes	complemented
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₅×D₁₂	—	4	2	no	—
68	C₅×D₁₂	—	4	2	no	—
69	C₅×D₁₂	—	4	2	no	—
70	C₅×D₁₂	—	4	2	no	—
71	C₅×D₁₂	—	8	2	no	—
72	C₃×C₅×C₈	—	2	1	no	—
73	C₃×C₅×C₈	—	4	1	no	—
74	C₃×C₅×C₈	D₈	1	2	yes	complemented
75	C₅×C₃⋊C₈	—	4	2	no	—
76	C₂²×C₃×C₈	—	2	1	no	—
77	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
78	C₂×#213	—	2	2	no	—
79	C₂×D₂₄	—	2	1	no	—
80	C₂×D₂₄	—	2	1	no	—
81	#874	C₂²×C₅	1	2	yes	—
82	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₂²⋊C₈	—	2	1	no	—
84	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
85	#809	—	2	2	no	—
86	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
87	C₂×C₅×D₄	—	6	2	no	—
88	C₂×C₅×D₄	—	6	2	no	—
89	C₂×C₅×D₄	—	6	2	no	—
90	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
91	C₂²×C₄×C₅	—	2	1	no	—
92	C₅×D₈	—	12	2	no	—
93	C₅×D₈	—	12	4	no	—
94	C₅×D₈	—	12	2	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	4	2	no	—
97	C₂×C₅×C₈	—	6	2	no	—
98	C₂×C₅×C₈	#67	1	1	yes	characteristic
99	C₂×C₅×C₈	#67	1	1	yes	characteristic
100	C₅×C₄⋊C₄	D₁₂	1	1	yes	characteristic
101	C₅×C₂²⋊C₄	—	2	1	no	—
102	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
103	#493	—	3	1	no	—
104	#459	—	3	2	no	—
105	C₈×D₄	C₅×D₃	1	1	yes	characteristic, complemented
106	#335	—	3	1	no	—
107	#331	—	3	2	no	—
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	1	no	—
111	C₂²×C₃×C₅	#121	1	1	yes	characteristic
112	C₂×C₅×D₃	—	8	2	no	—
113	C₂×C₅×D₃	—	8	2	no	—
114	C₂×C₅×D₃	—	8	2	no	—
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	—	4	1	no	—
117	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
118	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
119	C₃×C₄×C₅	#135	1	1	yes	characteristic
120	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
122	C₂²×C₃×C₄	—	2	1	no	—
123	C₂×D₁₂	—	2	2	no	—
124	C₂×D₁₂	—	2	2	no	—
125	C₂×D₁₂	—	2	2	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	4	2	no	—
128	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
129	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
130	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
131	C₃×C₂²⋊C₄	—	2	1	no	—
132	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
133	#213	—	4	4	no	—
134	C₂×C₃⋊C₈	—	2	2	no	—
135	D₂₄	—	4	2	no	—
136	D₂₄	—	4	2	no	—
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	—	12	2	no	—
139	C₅×D₄	—	2	1	no	—
140	C₅×D₄	—	12	2	no	—
141	C₅×D₄	—	12	2	no	—
142	C₅×D₄	—	12	2	no	—
143	C₅×D₄	—	12	2	no	—
144	C₅×D₄	—	24	2	no	—
145	C₅×D₄	D₂₄	1	2	yes	complemented
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₄×C₅	#235	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	4	1	no	—
154	C₅×C₈	—	12	2	no	—
155	C₅×C₈	#213	1	2	yes	complemented
156	C₂×D₈	—	6	2	no	—
157	C₂×D₈	—	6	1	no	—
158	C₂×D₈	—	6	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	#127	—	3	2	no	—
161	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
162	C₄⋊C₈	—	3	2	no	—
163	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
164	#102	—	6	2	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
167	C₂×C₃×C₅	—	4	2	no	—
168	C₂×C₃×C₅	#459	1	1	yes	characteristic
169	C₂×C₃×C₅	#466	1	1	yes	characteristic
170	C₂×C₃×C₅	#472	1	1	yes	characteristic
171	C₅×D₃	—	16	2	no	—
172	C₂³×C₃	—	2	1	no	—
173	C₂²×D₃	—	4	2	no	—
174	C₃×D₄	—	2	1	no	—
175	C₃×D₄	C₅×D₈	1	2	yes	complemented
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
182	D₁₂	—	4	2	no	—
183	D₁₂	—	4	2	no	—
184	D₁₂	—	4	2	no	—
185	D₁₂	—	4	2	no	—
186	D₁₂	—	8	2	no	—
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	4	1	no	—
189	C₃×C₈	C₅×D₈	1	2	yes	complemented
190	C₃⋊C₈	—	4	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	24	2	no	—
195	C₂²×C₅	—	24	2	no	—
196	C₂²×C₅	—	24	2	no	—
197	C₂²×C₅	#930	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
201	C₄×C₅	#904	1	1	yes	characteristic
202	C₄×C₅	C₂×#213	1	1	yes	characteristic
203	C₄×C₅	#949	1	1	yes	characteristic
204	C₂×D₄	—	6	2	no	—
205	C₂×D₄	—	6	2	no	—
206	C₂×D₄	—	6	2	no	—
207	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
208	C₂²×C₄	—	2	1	no	—
209	D₈	—	12	2	no	—
210	D₈	—	12	4	no	—
211	D₈	—	12	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	6	2	no	—
215	C₂×C₈	C₅×#67	1	1	yes	characteristic
216	C₂×C₈	C₅×#67	1	1	yes	characteristic
217	C₄⋊C₄	C₅×D₁₂	1	1	yes	characteristic
218	C₂²⋊C₄	—	2	1	no	—
219	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
220	C₃×C₅	#1667	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	4	1	no	—
224	C₂²×C₃	C₅×#121	1	1	yes	characteristic
225	C₂×D₃	—	8	2	no	—
226	C₂×D₃	—	8	2	no	—
227	C₂×D₃	—	8	2	no	—
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	1	no	—
230	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
231	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
232	C₃×C₄	C₅×#135	1	1	yes	characteristic
233	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂×C₅	—	4	2	no	—
235	C₂×C₅	—	48	2	no	—
236	C₂×C₅	#5015	1	1	yes	characteristic
237	C₂×C₅	#5115	1	1	yes	characteristic
238	C₂×C₅	#5142	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	12	2	no	—
241	D₄	—	2	1	no	—
242	D₄	—	12	2	no	—
243	D₄	—	12	2	no	—
244	D₄	—	12	2	no	—
245	D₄	—	12	2	no	—
246	D₄	—	24	2	no	—
247	D₄	C₅×D₂₄	1	2	yes	complemented
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
252	C₂×C₄	C₅×#235	1	1	yes	characteristic
253	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	4	1	no	—
256	C₈	—	12	2	no	—
257	C₈	C₅×#213	1	2	yes	complemented
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	C₅×#459	1	1	yes	characteristic
260	C₂×C₃	C₅×#466	1	1	yes	characteristic
261	C₂×C₃	C₅×#472	1	1	yes	characteristic
262	D₃	—	16	2	no	—
263	C₅	#70847	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	24	2	no	—
268	C₂²	—	24	2	no	—
269	C₂²	—	24	2	no	—
270	C₂²	C₅×#930	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	C₂×C₅×D₂₄	1	1	yes	characteristic
274	C₄	C₅×#904	1	1	yes	characteristic
275	C₄	C₂×C₅×#213	1	1	yes	characteristic
276	C₄	C₅×#949	1	1	yes	characteristic
277	C₃	C₅×#1667	1	1	yes	characteristic, complemented, Sylow
278	C₂	—	4	2	no	—
279	C₂	—	48	2	no	—
280	C₂	C₅×#5015	1	1	yes	characteristic
281	C₂	C₅×#5115	1	1	yes	characteristic
282	C₂	C₅×#5142	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#930
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×C₈×D₄	C₂
socle	C₂²×C₃×C₅	#121
hypercenter	C₅×C₈×D₄	D₃
hypocenter	C₃	C₅×#1667
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×C₈×D₄	C₂	semidirect
2. C₅×#5015	C₂	semidirect
3. C₅×#4695	C₂	semidirect
4. C₅×#4485	C₂	semidirect
5. C₅×#4458	C₂	semidirect
18. #70847	C₅	direct (N × H)
21. C₅×C₈×D₄	D₃	semidirect
40. C₃×C₈×D₄	C₂×C₅	semidirect
41. #5015	C₂×C₅	semidirect
42. #4695	C₂×C₅	semidirect
43. #4485	C₂×C₅	semidirect
44. #4458	C₂×C₅	semidirect
60. C₃×C₅×D₄	D₈	semidirect
74. C₃×C₅×C₈	D₈	semidirect
105. C₈×D₄	C₅×D₃	semidirect
145. C₅×D₄	D₂₄	semidirect
155. C₅×C₈	#213	semidirect
175. C₃×D₄	C₅×D₈	semidirect
189. C₃×C₈	C₅×D₈	semidirect
220. C₃×C₅	#1667	semidirect
247. D₄	C₅×D₂₄	semidirect
257. C₈	C₅×#213	semidirect
263. C₅	#70847	direct (N × H)
277. C₃	C₅×#1667	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×C₈×D₄	2
C₅×#5015	2
C₅×#4695	2
C₅×#4485	2
C₅×#4458	2
C₅×#1667	3
#70847	5

Subgroups — C5×#70847 #14291412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70847 #14291412