Subgroups — C₅×#71660 #14301483

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#519	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#944	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#836	C₂	1	1	yes	maximal, characteristic
4	C₅×#5004	C₂	1	2	yes	maximal
5	C₅×#4532	C₂	1	2	yes	maximal
6	C₅×#1886	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₅×Q₈	—	2	4	no	—
10	C₃×C₅×#117	—	2	2	no	—
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₅×#944	—	2	2	no	—
13	C₅×#944	C₄	1	4	yes	complemented
14	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
15	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₅×#208	C₂²	1	2	yes	—
17	C₅×#836	—	2	2	no	—
18	C₅×#810	—	2	4	no	—
19	#71660	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
21	C₅×#519	D₃	1	1	yes	characteristic
22	C₅×#419	—	3	2	no	—
23	C₂×C₅×#101	—	3	1	no	—
24	C₅×#339	—	3	2	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	2	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×Dic₃	—	2	1	no	—
32	C₂²×C₅×Dic₃	—	2	1	no	—
33	C₂×C₅×Dic₆	—	2	2	no	—
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
40	C₃×C₄²×C₅	—	2	4	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₅×Dic₃⋊C₄	—	2	8	no	—
43	C₄×C₅×Dic₃	—	2	1	no	—
44	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
45	C₅×#208	—	2	2	no	—
46	C₅×#208	—	2	2	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₃×#519	C₂×C₅	1	1	yes	characteristic
49	C₂×#944	C₂×C₅	1	1	yes	characteristic
50	C₂×#836	C₂×C₅	1	1	yes	characteristic
51	#5004	C₂×C₅	1	2	yes	—
52	#4532	C₂×C₅	1	2	yes	—
53	C₂²×C₅×Q₈	—	3	1	no	—
54	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₅×M₁₆	—	3	2	no	—
56	C₅×C₄⋊Q₈	—	3	4	no	—
57	C₅×C₄⋊Q₈	—	6	2	no	—
58	C₄×C₅×Q₈	—	2	4	no	—
59	C₅×#117	—	2	2	no	—
60	C₅×#117	C₂×D₃	1	2	yes	—
61	C₂×C₅×C₄⋊C₄	—	6	2	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	C₅×#103	—	6	4	no	—
64	C₅×#101	—	6	2	no	—
65	#1886	—	3	1	no	Sylow
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	#41	1	2	yes	—
77	C₂×C₅×Dic₃	—	2	4	no	—
78	C₂×C₅×Dic₃	—	2	2	no	—
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₂×C₅×Dic₃	—	4	1	no	—
81	C₂×C₅×Dic₃	—	4	1	no	—
82	C₅×Dic₆	—	2	4	no	—
83	C₅×Dic₆	—	4	2	no	—
84	C₅×C₃⋊C₈	—	4	2	no	—
85	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
87	C₃×C₄×Q₈	—	2	4	no	—
88	C₃×#117	—	2	2	no	—
89	C₃×#117	C₂²×C₅	1	2	yes	—
90	#944	—	2	2	no	—
91	#944	C₄×C₅	1	4	yes	complemented
92	C₂×Dic₃⋊C₄	—	2	2	no	—
93	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
94	C₂×#208	C₂²×C₅	1	2	yes	—
95	#836	—	2	2	no	—
96	#810	—	2	4	no	—
97	C₂×C₅×Q₈	—	2	2	no	—
98	C₂×C₅×Q₈	—	2	2	no	—
99	C₂×C₅×Q₈	—	3	2	no	—
100	C₂×C₅×Q₈	—	6	1	no	—
101	C₂×C₅×Q₈	—	6	2	no	—
102	C₂×C₅×Q₈	—	6	2	no	—
103	C₂×C₅×Q₈	C₄×D₃	1	2	yes	—
104	C₂×C₅×Q₈	D₁₂	1	2	yes	—
105	C₂²×C₄×C₅	—	6	1	no	—
106	C₂²×C₄×C₅	—	6	1	no	—
107	C₂²×C₄×C₅	#67	1	2	yes	—
108	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
109	C₅×M₁₆	—	6	2	no	—
110	C₅×M₁₆	—	6	2	no	—
111	C₂×C₅×C₈	—	6	2	no	—
112	C₅×C₄⋊C₄	—	2	4	no	—
113	C₅×C₄⋊C₄	—	6	8	no	—
114	C₅×C₄⋊C₄	#67	1	4	yes	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₄²×C₅	—	2	4	no	—
117	C₄²×C₅	—	2	2	no	—
118	C₄²×C₅	—	3	2	no	—
119	C₄²×C₅	—	6	1	no	—
120	C₂×C₄⋊Q₈	—	3	1	no	—
121	#519	C₅×D₃	1	1	yes	characteristic
122	#419	—	3	2	no	—
123	C₂×#101	—	3	1	no	—
124	#339	—	3	2	no	—
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₃×C₅	#120	1	1	yes	characteristic
128	C₂²×C₃×C₅	#123	1	1	yes	characteristic
129	C₃×C₄×C₅	—	2	4	no	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
132	C₃×C₄×C₅	#121	1	2	yes	—
133	C₅×Dic₃	—	2	4	no	—
134	C₅×Dic₃	—	4	2	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×Q₈	—	2	2	no	—
137	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
138	C₂×C₃×Q₈	C₅×D₄	1	2	yes	—
139	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
140	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₂²×Dic₃	—	2	1	no	—
142	C₂²×Dic₃	—	2	1	no	—
143	C₂×Dic₆	—	2	2	no	—
144	C₂×Dic₆	—	2	2	no	—
145	C₂×Dic₆	—	2	1	no	—
146	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
147	C₃×C₄⋊C₄	—	2	4	no	—
148	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	—
149	C₃×C₂²⋊C₄	—	2	2	no	—
150	C₃×C₄²	—	2	4	no	—
151	C₃×C₄²	—	2	2	no	—
152	Dic₃⋊C₄	—	2	8	no	—
153	C₄×Dic₃	—	2	1	no	—
154	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
155	#208	—	2	2	no	—
156	#208	—	2	2	no	—
157	C₂×C₃⋊C₈	—	2	2	no	—
158	C₂³×C₅	C₂×#67	1	1	yes	characteristic
159	C₅×Q₈	—	2	4	no	—
160	C₅×Q₈	—	2	4	no	—
161	C₅×Q₈	—	6	4	no	—
162	C₅×Q₈	—	12	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	6	4	no	—
167	C₂×C₄×C₅	—	6	2	no	—
168	C₂×C₄×C₅	—	6	2	no	—
169	C₂×C₄×C₅	—	12	1	no	—
170	C₂×C₄×C₅	—	12	1	no	—
171	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	4	yes	—
172	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
174	C₂×C₄×C₅	#235	1	2	yes	—
175	C₂×C₄×C₅	C₂×#67	1	2	yes	—
176	C₅×C₈	—	12	2	no	—
177	C₂²×Q₈	—	3	1	no	—
178	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
179	C₂×M₁₆	—	3	2	no	—
180	C₄⋊Q₈	—	3	4	no	—
181	C₄⋊Q₈	—	6	2	no	—
182	C₄×Q₈	—	2	4	no	—
183	#117	—	2	2	no	—
184	#117	C₂×C₅×D₃	1	2	yes	—
185	C₂×C₄⋊C₄	—	6	2	no	—
186	C₂×C₄²	—	3	1	no	—
187	#103	—	6	4	no	—
188	#101	—	6	2	no	—
189	C₂×C₃×C₅	—	2	1	no	—
190	C₂×C₃×C₅	—	2	1	no	—
191	C₂×C₃×C₅	#386	1	1	yes	characteristic
192	C₂×C₃×C₅	#456	1	2	yes	—
193	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
194	C₃×Q₈	—	2	4	no	—
195	C₃×Q₈	—	2	4	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
200	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
203	C₂×C₃×C₄	C₅×#41	1	2	yes	—
204	C₂×Dic₃	—	2	4	no	—
205	C₂×Dic₃	—	2	2	no	—
206	C₂×Dic₃	—	2	2	no	—
207	C₂×Dic₃	—	4	1	no	—
208	C₂×Dic₃	—	4	1	no	—
209	Dic₆	—	2	4	no	—
210	Dic₆	—	4	2	no	—
211	C₃⋊C₈	—	4	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₂²×C₅	#929	1	1	yes	characteristic
215	C₂²×C₅	#953	1	1	yes	characteristic
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	4	4	no	—
218	C₄×C₅	—	6	4	no	—
219	C₄×C₅	—	12	2	no	—
220	C₄×C₅	C₄×#67	1	2	yes	—
221	C₄×C₅	#930	1	2	yes	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	3	2	no	—
225	C₂×Q₈	—	6	1	no	—
226	C₂×Q₈	—	6	2	no	—
227	C₂×Q₈	—	6	2	no	—
228	C₂×Q₈	C₄×C₅×D₃	1	2	yes	—
229	C₂×Q₈	C₅×D₁₂	1	2	yes	—
230	C₂²×C₄	—	6	1	no	—
231	C₂²×C₄	—	6	1	no	—
232	C₂²×C₄	C₅×#67	1	2	yes	—
233	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
234	M₁₆	—	6	2	no	—
235	M₁₆	—	6	2	no	—
236	C₂×C₈	—	6	2	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	6	8	no	—
239	C₄⋊C₄	C₅×#67	1	4	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	3	2	no	—
244	C₄²	—	6	1	no	—
245	C₃×C₅	#1886	1	1	yes	characteristic, complemented, Hall
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₂²×C₃	C₅×#120	1	1	yes	characteristic
249	C₂²×C₃	C₅×#123	1	1	yes	characteristic
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	4	4	no	—
252	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
253	C₃×C₄	C₅×#121	1	2	yes	—
254	Dic₃	—	2	4	no	—
255	Dic₃	—	4	2	no	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	#5213	1	1	yes	characteristic
259	C₂×C₅	#5246	1	2	yes	—
260	C₂³	C₂×C₅×#67	1	1	yes	characteristic
261	Q₈	—	2	4	no	—
262	Q₈	—	2	4	no	—
263	Q₈	—	6	4	no	—
264	Q₈	—	12	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	—	6	2	no	—
270	C₂×C₄	—	6	2	no	—
271	C₂×C₄	—	12	1	no	—
272	C₂×C₄	—	12	1	no	—
273	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
274	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
276	C₂×C₄	C₅×#235	1	2	yes	—
277	C₂×C₄	C₂×C₅×#67	1	2	yes	—
278	C₈	—	12	2	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	C₅×#386	1	1	yes	characteristic
282	C₂×C₃	C₅×#456	1	2	yes	—
283	C₅	#71660	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
286	C₂²	C₅×#929	1	1	yes	characteristic
287	C₂²	C₅×#953	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	4	4	no	—
290	C₄	—	6	4	no	—
291	C₄	—	12	2	no	—
292	C₄	C₄×C₅×#67	1	2	yes	—
293	C₄	C₅×#930	1	2	yes	—
294	C₃	C₅×#1886	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	C₅×#5213	1	1	yes	characteristic
298	C₂	C₅×#5246	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	#953
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#519	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#519	D₃
hypocenter	C₃	C₅×#1886
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#519

Complements

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

N	H ≅ G/N	type
13. C₅×#944	C₄	semidirect
19. #71660	C₅	direct (N × H)
91. #944	C₄×C₅	semidirect
245. C₃×C₅	#1886	semidirect
283. C₅	#71660	direct (N × H)
294. C₃	C₅×#1886	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#519	2
C₂×C₅×#944	2
C₂×C₅×#836	2
C₅×#5004	2
C₅×#4532	2
C₅×#1886	3
#71660	5

Subgroups — C5×#71660 #14301483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71660 #14301483