Subgroups — C₅×#68548 #14291916

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4850	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#4848	C₂	1	1	yes	maximal, characteristic
3	C₅×#4694	C₂	1	1	yes	maximal, characteristic
4	C₅×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic
7	C₅×#4481	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1678	—	3	1	no	maximal, Hall
9	C₂×C₅×#204	—	2	1	no	—
10	C₅×#874	C₂²	1	1	yes	characteristic
11	C₅×#869	C₂²	1	1	yes	characteristic
12	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#821	—	2	1	no	—
17	C₅×#821	—	2	1	no	—
18	C₅×#821	—	2	1	no	—
19	C₅×#818	—	2	1	no	—
20	C₅×#816	—	2	1	no	—
21	C₅×Dic₃⋊C₈	—	2	1	no	—
22	C₅×C₈×Dic₃	—	2	1	no	—
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#68548	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#486	—	3	1	no	—
26	C₅×#477	—	3	1	no	—
27	C₅×#460	—	3	1	no	—
28	C₅×C₈×Q₈	—	3	1	no	—
29	C₅×#334	D₃	1	1	yes	characteristic, complemented
30	C₅×#333	—	3	1	no	—
31	C₅×#331	—	3	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×C₄⋊Dic₃	—	2	1	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
45	C₅×Dic₃⋊C₄	—	2	1	no	—
46	C₄×C₅×Dic₃	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₅×#204	—	4	2	no	—
49	#4850	C₂×C₅	1	1	yes	characteristic, complemented
50	#4848	C₂×C₅	1	1	yes	characteristic
51	#4694	C₂×C₅	1	1	yes	characteristic
52	C₈×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
53	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
54	#4485	C₂×C₅	1	1	yes	characteristic
55	#4481	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂×C₅×SD₁₆	—	6	1	no	—
57	C₅×C₄⋊Q₈	—	3	1	no	—
58	C₅×#127	—	3	1	no	—
59	C₄×C₅×Q₈	—	3	1	no	—
60	C₅×#107	—	6	1	no	—
61	C₅×C₄⋊C₈	—	3	1	no	—
62	C₅×C₄⋊C₈	—	6	1	no	—
63	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
64	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₅×#103	—	6	1	no	—
66	C₅×#102	—	6	1	no	—
67	C₅×#102	—	6	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₄×C₅×C₈	—	6	1	no	—
70	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
71	#1678	—	3	1	no	Sylow
72	C₂²×C₅×D₃	—	4	1	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
76	C₂×C₅×Dic₃	—	2	1	no	—
77	C₂×C₅×Dic₃	—	2	1	no	—
78	C₂×C₅×Dic₃	—	4	1	no	—
79	C₅×D₁₂	—	4	1	no	—
80	C₅×D₁₂	—	4	1	no	—
81	C₅×D₁₂	—	4	1	no	—
82	C₅×D₁₂	—	4	1	no	—
83	C₅×D₁₂	—	8	1	no	—
84	C₅×Dic₆	—	2	1	no	—
85	C₅×Dic₆	—	8	1	no	—
86	C₅×Dic₆	SD₁₆	1	2	yes	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃×C₅×C₈	—	2	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₃×C₅×C₈	—	4	1	no	—
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₂×#204	—	2	1	no	—
93	#874	C₂²×C₅	1	1	yes	characteristic
94	#869	C₂²×C₅	1	1	yes	characteristic
95	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
99	#821	—	2	1	no	—
100	#821	—	2	1	no	—
101	#821	—	2	1	no	—
102	#818	—	2	1	no	—
103	#816	—	2	1	no	—
104	Dic₃⋊C₈	—	2	1	no	—
105	C₈×Dic₃	—	2	1	no	—
106	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
107	C₂×C₅×Q₈	—	3	1	no	—
108	C₂×C₅×Q₈	—	6	1	no	—
109	C₂×C₅×D₄	—	6	1	no	—
110	C₂×C₅×D₄	—	6	1	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₅×SD₁₆	—	12	2	no	—
113	C₂×C₅×C₈	—	2	1	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	6	1	no	—
116	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
117	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
118	C₅×C₄⋊C₄	—	3	1	no	—
119	C₅×C₄⋊C₄	—	6	1	no	—
120	C₅×C₄⋊C₄	—	6	1	no	—
121	C₅×C₄⋊C₄	—	6	1	no	—
122	C₄²×C₅	—	6	1	no	—
123	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
124	#486	—	3	1	no	—
125	#477	—	3	1	no	—
126	#460	—	3	1	no	—
127	C₈×Q₈	—	3	1	no	—
128	#334	C₅×D₃	1	1	yes	characteristic, complemented
129	#333	—	3	1	no	—
130	#331	—	3	1	no	—
131	C₂²×C₃×C₅	#121	1	1	yes	characteristic
132	C₂×C₅×D₃	—	8	1	no	—
133	C₂×C₅×D₃	—	8	1	no	—
134	C₂×C₅×D₃	—	8	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
137	C₃×C₄×C₅	#135	1	1	yes	characteristic
138	C₃×C₄×C₅	#135	1	1	yes	characteristic
139	C₃×C₄×C₅	#137	1	1	yes	characteristic
140	C₅×Dic₃	—	2	2	no	—
141	C₅×Dic₃	—	4	1	no	—
142	C₅×Dic₃	—	8	1	no	—
143	C₂×D₁₂	—	2	1	no	—
144	C₂×D₁₂	—	2	1	no	—
145	C₂×D₁₂	—	2	1	no	—
146	C₂×Dic₆	—	2	1	no	—
147	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
151	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
152	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
153	C₄⋊Dic₃	—	2	1	no	—
154	C₄⋊Dic₃	—	2	1	no	—
155	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
156	Dic₃⋊C₄	—	2	1	no	—
157	C₄×Dic₃	—	2	1	no	—
158	C₂×C₃⋊C₈	—	2	1	no	—
159	#204	—	4	2	no	—
160	C₂³×C₅	—	12	1	no	—
161	C₅×Q₈	—	3	2	no	—
162	C₅×Q₈	—	6	1	no	—
163	C₅×Q₈	—	24	1	no	—
164	C₅×D₄	—	12	1	no	—
165	C₅×D₄	—	12	1	no	—
166	C₅×D₄	—	12	1	no	—
167	C₅×D₄	—	12	1	no	—
168	C₅×D₄	—	24	1	no	—
169	C₂×C₄×C₅	—	6	1	no	—
170	C₂×C₄×C₅	—	6	1	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	#239	1	1	yes	characteristic
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	—	4	1	no	—
179	C₅×C₈	—	12	1	no	—
180	C₂×SD₁₆	—	6	1	no	—
181	C₄⋊Q₈	—	3	1	no	—
182	#127	—	3	1	no	—
183	C₄×Q₈	—	3	1	no	—
184	#107	—	6	1	no	—
185	C₄⋊C₈	—	3	1	no	—
186	C₄⋊C₈	—	6	1	no	—
187	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
188	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
189	#103	—	6	1	no	—
190	#102	—	6	1	no	—
191	#102	—	6	1	no	—
192	#102	—	6	1	no	—
193	C₄×C₈	—	6	1	no	—
194	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
195	C₂×C₃×C₅	#464	1	1	yes	characteristic
196	C₂×C₃×C₅	#465	1	1	yes	characteristic
197	C₂×C₃×C₅	#472	1	1	yes	characteristic
198	C₅×D₃	—	16	1	no	—
199	C₂²×D₃	—	4	1	no	—
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
203	C₂×Dic₃	—	2	1	no	—
204	C₂×Dic₃	—	2	1	no	—
205	C₂×Dic₃	—	4	1	no	—
206	D₁₂	—	4	1	no	—
207	D₁₂	—	4	1	no	—
208	D₁₂	—	4	1	no	—
209	D₁₂	—	4	1	no	—
210	D₁₂	—	8	1	no	—
211	Dic₆	—	2	1	no	—
212	Dic₆	—	8	1	no	—
213	Dic₆	C₅×SD₁₆	1	2	yes	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃×C₈	—	4	1	no	—
218	C₃⋊C₈	—	4	1	no	—
219	C₂²×C₅	—	24	1	no	—
220	C₂²×C₅	—	24	1	no	—
221	C₂²×C₅	—	24	1	no	—
222	C₂²×C₅	#895	1	1	yes	characteristic
223	C₄×C₅	—	2	1	no	—
224	C₄×C₅	—	6	2	no	—
225	C₄×C₅	—	12	1	no	—
226	C₄×C₅	—	24	1	no	—
227	C₄×C₅	#904	1	1	yes	characteristic
228	C₄×C₅	#909	1	1	yes	characteristic
229	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
230	C₄×C₅	#916	1	1	yes	characteristic
231	C₂×Q₈	—	3	1	no	—
232	C₂×Q₈	—	6	1	no	—
233	C₂×D₄	—	6	1	no	—
234	C₂×D₄	—	6	1	no	—
235	C₂×D₄	—	6	1	no	—
236	SD₁₆	—	12	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	6	1	no	—
240	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
241	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
242	C₄⋊C₄	—	3	1	no	—
243	C₄⋊C₄	—	6	1	no	—
244	C₄⋊C₄	—	6	1	no	—
245	C₄⋊C₄	—	6	1	no	—
246	C₄²	—	6	1	no	—
247	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
248	C₃×C₅	#1678	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	C₅×#121	1	1	yes	characteristic
250	C₂×D₃	—	8	1	no	—
251	C₂×D₃	—	8	1	no	—
252	C₂×D₃	—	8	1	no	—
253	C₃×C₄	—	2	1	no	—
254	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
255	C₃×C₄	C₅×#135	1	1	yes	characteristic
256	C₃×C₄	C₅×#135	1	1	yes	characteristic
257	C₃×C₄	C₅×#137	1	1	yes	characteristic
258	Dic₃	—	2	2	no	—
259	Dic₃	—	4	1	no	—
260	Dic₃	—	8	1	no	—
261	C₂×C₅	—	48	1	no	—
262	C₂×C₅	#4845	1	1	yes	characteristic
263	C₂×C₅	#4864	1	1	yes	characteristic
264	C₂×C₅	#4897	1	1	yes	characteristic
265	C₂³	—	12	1	no	—
266	Q₈	—	3	2	no	—
267	Q₈	—	6	1	no	—
268	Q₈	—	24	1	no	—
269	D₄	—	12	1	no	—
270	D₄	—	12	1	no	—
271	D₄	—	12	1	no	—
272	D₄	—	12	1	no	—
273	D₄	—	24	1	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	12	1	no	—
277	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
278	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
279	C₂×C₄	C₅×#239	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	12	1	no	—
285	C₂×C₃	C₅×#464	1	1	yes	characteristic
286	C₂×C₃	C₅×#465	1	1	yes	characteristic
287	C₂×C₃	C₅×#472	1	1	yes	characteristic
288	D₃	—	16	1	no	—
289	C₅	#68548	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	24	1	no	—
291	C₂²	—	24	1	no	—
292	C₂²	—	24	1	no	—
293	C₂²	C₅×#895	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	6	2	no	—
296	C₄	—	12	1	no	—
297	C₄	—	24	1	no	—
298	C₄	C₅×#904	1	1	yes	characteristic
299	C₄	C₅×#909	1	1	yes	characteristic
300	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
301	C₄	C₅×#916	1	1	yes	characteristic
302	C₃	C₅×#1678	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	48	1	no	—
304	C₂	C₅×#4845	1	1	yes	characteristic
305	C₂	C₅×#4864	1	1	yes	characteristic
306	C₂	C₅×#4897	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#895
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#334	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#874	C₂²
hypercenter	C₅×#334	D₃
hypocenter	C₃	C₅×#1678
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#334

Complements

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

N	H ≅ G/N	type
1. C₅×#4850	C₂	semidirect
4. C₅×C₈×Dic₆	C₂	semidirect
5. C₃×C₅×#334	C₂	semidirect
7. C₅×#4481	C₂	semidirect
24. #68548	C₅	direct (N × H)
29. C₅×#334	D₃	semidirect
49. #4850	C₂×C₅	semidirect
52. C₈×Dic₆	C₂×C₅	semidirect
53. C₃×#334	C₂×C₅	semidirect
55. #4481	C₂×C₅	semidirect
128. #334	C₅×D₃	semidirect
248. C₃×C₅	#1678	semidirect
289. C₅	#68548	direct (N × H)
302. C₃	C₅×#1678	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4850	2
C₅×#4848	2
C₅×#4694	2
C₅×C₈×Dic₆	2
C₃×C₅×#334	2
C₅×#4485	2
C₅×#4481	2
C₅×#1678	3
#68548	5

Subgroups — C5×#68548 #14291916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68548 #14291916