Subgroups — C₅×#63964 #13087251

Order 1440.

330 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Dic₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#627	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3757	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#217	C₂	1	1	yes	maximal, characteristic
5	C₅×#3724	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×Dic₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×Dic₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#924	—	3	1	no	maximal
9	C₅×#878	—	3	1	no	maximal
10	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×Dic₃	—	2	2	no	—
14	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₅×#627	—	2	2	no	—
19	#63964	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂²×C₃×C₄×C₅	—	3	1	no	—
21	C₂²×C₅×Dic₃	D₃	1	1	yes	characteristic
22	C₂×C₅×Dic₆	—	3	1	no	—
23	C₂×C₅×Dic₆	—	3	1	no	—
24	C₃×C₅×C₂²⋊C₄	—	3	1	no	—
25	C₅×#217	—	3	1	no	—
26	C₅×#217	—	3	1	no	—
27	C₅×#217	—	6	1	no	—
28	C₅×#217	D₃	1	1	yes	characteristic
29	C₅×C₄⋊Dic₃	—	3	1	no	—
30	C₅×C₄⋊Dic₃	—	3	1	no	—
31	C₅×Dic₃⋊C₄	—	3	1	no	—
32	C₅×Dic₃⋊C₄	—	3	1	no	—
33	C₅×Dic₃⋊C₄	—	3	1	no	—
34	C₅×Dic₃⋊C₄	—	3	1	no	—
35	C₂²×C₃²×C₅	—	2	1	no	—
36	C₂²×C₃²×C₅	—	2	1	no	—
37	C₂²×C₃²×C₅	Q₈	1	2	yes	complemented
38	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
39	C₅×C₃⋊Dic₃	—	2	1	no	—
40	C₅×C₃⋊Dic₃	—	2	1	no	—
41	C₃×C₅×Dic₃	—	2	1	no	—
42	C₃×C₅×Dic₃	—	2	1	no	—
43	C₃×C₅×Dic₃	—	2	1	no	—
44	C₃×C₅×Dic₃	D₄	1	2	yes	—
45	C₅×C₂²⋊Q₈	—	9	1	no	Hall
46	C₂²×C₃×Dic₃	C₂×C₅	1	1	yes	characteristic
47	C₂×#627	C₂×C₅	1	1	yes	characteristic, complemented
48	#3757	C₂×C₅	1	1	yes	characteristic
49	C₃×#217	C₂×C₅	1	1	yes	characteristic
50	#3724	C₂×C₅	1	1	yes	characteristic, complemented
51	Dic₃⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
52	Dic₃⋊Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂³×C₃×C₅	—	2	1	no	—
54	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	3	1	no	—
57	C₂×C₃×C₄×C₅	—	3	1	no	—
58	C₂×C₃×C₄×C₅	—	3	1	no	—
59	C₂×C₃×C₄×C₅	—	3	1	no	—
60	C₂×C₃×C₄×C₅	—	6	2	no	—
61	C₂×C₅×Dic₃	—	2	2	no	—
62	C₂×C₅×Dic₃	—	3	1	no	—
63	C₂×C₅×Dic₃	—	3	1	no	—
64	C₂×C₅×Dic₃	—	3	1	no	—
65	C₂×C₅×Dic₃	—	3	1	no	—
66	C₂×C₅×Dic₃	—	6	1	no	—
67	C₂×C₅×Dic₃	—	6	1	no	—
68	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
69	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
71	C₂×C₅×Dic₃	C₂×D₃	1	1	yes	characteristic
72	C₅×Dic₆	—	6	2	no	—
73	C₅×Dic₆	—	6	2	no	—
74	#924	—	3	1	no	—
75	#878	—	3	1	no	—
76	C₂×C₃²×C₅	—	2	2	no	—
77	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃²×C₅	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃²×C₅	#41	1	1	yes	characteristic
80	C₂×C₅×Q₈	—	9	1	no	—
81	C₂²×C₄×C₅	—	3	1	no	—
82	C₅×C₄⋊C₄	—	9	1	no	—
83	C₅×C₄⋊C₄	—	9	1	no	—
84	C₅×C₄⋊C₄	—	9	1	no	—
85	C₅×C₂²⋊C₄	—	3	1	no	—
86	C₅×C₂²⋊C₄	—	9	1	no	—
87	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
89	C₂×C₃⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
93	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
95	#627	—	2	2	no	—
96	C₂²×C₃×C₅	—	2	1	no	—
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	1	no	—
99	C₂²×C₃×C₅	—	2	1	no	—
100	C₂²×C₃×C₅	—	2	2	no	—
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	4	1	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	Dic₆	1	2	yes	complemented
105	C₂²×C₃×C₅	Dic₆	1	2	yes	complemented
106	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₃×C₄×C₅	—	3	2	no	—
109	C₃×C₄×C₅	—	6	1	no	—
110	C₃×C₄×C₅	—	6	1	no	—
111	C₃×C₄×C₅	—	6	1	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₅×Dic₃	—	2	1	no	—
114	C₅×Dic₃	—	2	1	no	—
115	C₅×Dic₃	—	6	1	no	—
116	C₅×Dic₃	—	6	1	no	—
117	C₅×Dic₃	—	6	1	no	—
118	C₅×Dic₃	—	6	1	no	—
119	C₅×Dic₃	—	12	1	no	—
120	C₅×Dic₃	—	12	1	no	—
121	C₅×Dic₃	#67	1	2	yes	—
122	C₂²×C₃×C₄	—	3	1	no	—
123	C₂²×Dic₃	C₅×D₃	1	1	yes	characteristic
124	C₂×Dic₆	—	3	1	no	—
125	C₂×Dic₆	—	3	1	no	—
126	C₃×C₂²⋊C₄	—	3	1	no	—
127	#217	—	3	1	no	—
128	#217	—	3	1	no	—
129	#217	—	6	1	no	—
130	#217	C₅×D₃	1	1	yes	characteristic
131	C₄⋊Dic₃	—	3	1	no	—
132	C₄⋊Dic₃	—	3	1	no	—
133	Dic₃⋊C₄	—	3	1	no	—
134	Dic₃⋊C₄	—	3	1	no	—
135	Dic₃⋊C₄	—	3	1	no	—
136	Dic₃⋊C₄	—	3	1	no	—
137	C₃²×C₅	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
138	C₂³×C₅	D₃²	1	1	yes	characteristic
139	C₅×Q₈	—	18	2	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	1	no	—
142	C₂×C₄×C₅	—	3	1	no	—
143	C₂×C₄×C₅	—	3	1	no	—
144	C₂×C₄×C₅	—	6	2	no	—
145	C₂×C₄×C₅	—	9	1	no	—
146	C₂×C₄×C₅	—	9	1	no	—
147	C₂²×C₃²	—	2	1	no	—
148	C₂²×C₃²	—	2	1	no	—
149	C₂²×C₃²	C₅×Q₈	1	2	yes	complemented
150	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
151	C₃⋊Dic₃	—	2	1	no	—
152	C₃⋊Dic₃	—	2	1	no	—
153	C₃×Dic₃	—	2	1	no	—
154	C₃×Dic₃	—	2	1	no	—
155	C₃×Dic₃	—	2	1	no	—
156	C₃×Dic₃	C₅×D₄	1	2	yes	—
157	C₂²⋊Q₈	—	9	1	no	Sylow
158	C₂×C₃×C₅	—	2	2	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	—	2	1	no	—
161	C₂×C₃×C₅	—	2	1	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	—	4	2	no	—
164	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×Dic₆	1	1	yes	characteristic
166	C₂×C₃×C₅	#235	1	1	yes	characteristic
167	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
168	C₂×C₃×C₅	#237	1	1	yes	characteristic
169	C₂×C₃×C₅	C₂×#67	1	1	yes	characteristic
170	C₂³×C₃	—	2	1	no	—
171	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
172	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	—	3	1	no	—
175	C₂×C₃×C₄	—	3	1	no	—
176	C₂×C₃×C₄	—	3	1	no	—
177	C₂×C₃×C₄	—	6	2	no	—
178	C₂×Dic₃	—	2	2	no	—
179	C₂×Dic₃	—	3	1	no	—
180	C₂×Dic₃	—	3	1	no	—
181	C₂×Dic₃	—	3	1	no	—
182	C₂×Dic₃	—	3	1	no	—
183	C₂×Dic₃	—	6	1	no	—
184	C₂×Dic₃	—	6	1	no	—
185	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
186	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
188	C₂×Dic₃	C₂×C₅×D₃	1	1	yes	characteristic
189	Dic₆	—	6	2	no	—
190	Dic₆	—	6	2	no	—
191	C₂²×C₅	—	2	1	no	—
192	C₂²×C₅	—	2	1	no	—
193	C₂²×C₅	#627	1	2	yes	complemented
194	C₂²×C₅	C₂×D₃²	1	1	yes	characteristic
195	C₄×C₅	—	3	2	no	—
196	C₄×C₅	—	6	1	no	—
197	C₄×C₅	—	6	1	no	—
198	C₄×C₅	—	6	1	no	—
199	C₄×C₅	—	18	1	no	—
200	C₄×C₅	—	18	1	no	—
201	C₂×C₃²	—	2	2	no	—
202	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃²	C₂×C₅×Q₈	1	1	yes	characteristic
204	C₂×C₃²	C₅×#41	1	1	yes	characteristic
205	C₂×Q₈	—	9	1	no	—
206	C₂²×C₄	—	3	1	no	—
207	C₄⋊C₄	—	9	1	no	—
208	C₄⋊C₄	—	9	1	no	—
209	C₄⋊C₄	—	9	1	no	—
210	C₂²⋊C₄	—	3	1	no	—
211	C₂²⋊C₄	—	9	1	no	—
212	C₃×C₅	—	2	1	no	—
213	C₃×C₅	#878	1	1	yes	characteristic, complemented
214	C₃×C₅	#924	1	1	yes	characteristic, complemented
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	1	no	—
217	C₂²×C₃	—	2	1	no	—
218	C₂²×C₃	—	2	1	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	4	1	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	C₅×Dic₆	1	2	yes	complemented
224	C₂²×C₃	C₅×Dic₆	1	2	yes	complemented
225	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
226	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
227	C₃×C₄	—	3	2	no	—
228	C₃×C₄	—	6	1	no	—
229	C₃×C₄	—	6	1	no	—
230	C₃×C₄	—	6	1	no	—
231	Dic₃	—	2	1	no	—
232	Dic₃	—	2	1	no	—
233	Dic₃	—	2	1	no	—
234	Dic₃	—	6	1	no	—
235	Dic₃	—	6	1	no	—
236	Dic₃	—	6	1	no	—
237	Dic₃	—	6	1	no	—
238	Dic₃	—	12	1	no	—
239	Dic₃	—	12	1	no	—
240	Dic₃	C₅×#67	1	2	yes	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	#3804	1	1	yes	characteristic
243	C₂×C₅	C₂×#627	1	1	yes	characteristic
244	C₂×C₅	D₃×#67	1	1	yes	characteristic
245	C₃²	C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₅×D₃²	1	1	yes	characteristic
247	Q₈	—	18	2	no	—
248	C₂×C₄	—	3	1	no	—
249	C₂×C₄	—	3	1	no	—
250	C₂×C₄	—	3	1	no	—
251	C₂×C₄	—	3	1	no	—
252	C₂×C₄	—	6	2	no	—
253	C₂×C₄	—	9	1	no	—
254	C₂×C₄	—	9	1	no	—
255	C₂×C₃	—	2	2	no	—
256	C₂×C₃	—	2	2	no	—
257	C₂×C₃	—	2	1	no	—
258	C₂×C₃	—	2	1	no	—
259	C₂×C₃	—	2	1	no	—
260	C₂×C₃	—	4	2	no	—
261	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
262	C₂×C₃	C₂×C₅×Dic₆	1	1	yes	characteristic
263	C₂×C₃	C₅×#235	1	1	yes	characteristic
264	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
265	C₂×C₃	C₅×#237	1	1	yes	characteristic
266	C₂×C₃	C₂×C₅×#67	1	1	yes	characteristic
267	C₅	#63964	1	1	yes	characteristic, direct product factor, Sylow
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	C₅×#627	1	2	yes	complemented
271	C₂²	C₂×C₅×D₃²	1	1	yes	characteristic
272	C₄	—	3	2	no	—
273	C₄	—	6	1	no	—
274	C₄	—	6	1	no	—
275	C₄	—	6	1	no	—
276	C₄	—	18	1	no	—
277	C₄	—	18	1	no	—
278	C₃	—	2	1	no	—
279	C₃	C₅×#878	1	1	yes	characteristic, complemented
280	C₃	C₅×#924	1	1	yes	characteristic, complemented
281	C₂	—	2	2	no	—
282	C₂	C₅×#3804	1	1	yes	characteristic
283	C₂	C₂×C₅×#627	1	1	yes	characteristic
284	C₂	C₅×D₃×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃²
commutator, cosocle	C₂²×C₃²	C₂³×C₅
Frattini	C₂²	C₂×C₅×D₃²
Fitting, prime radical	C₂³×C₃²×C₅	C₂²
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₂³×C₅	D₃²
hypocenter	C₃²	C₅×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#627	C₂	semidirect
5. C₅×#3724	C₂	semidirect
6. C₅×Dic₃⋊Dic₃	C₂	semidirect
7. C₅×Dic₃⋊Dic₃	C₂	semidirect
19. #63964	C₅	direct (N × H)
37. C₂²×C₃²×C₅	Q₈	semidirect
47. C₂×#627	C₂×C₅	semidirect
50. #3724	C₂×C₅	semidirect
51. Dic₃⋊Dic₃	C₂×C₅	semidirect
52. Dic₃⋊Dic₃	C₂×C₅	semidirect
104. C₂²×C₃×C₅	Dic₆	semidirect
105. C₂²×C₃×C₅	Dic₆	semidirect
137. C₃²×C₅	C₂²⋊Q₈	semidirect
149. C₂²×C₃²	C₅×Q₈	semidirect
193. C₂²×C₅	#627	semidirect
213. C₃×C₅	#878	semidirect
214. C₃×C₅	#924	semidirect
223. C₂²×C₃	C₅×Dic₆	semidirect
224. C₂²×C₃	C₅×Dic₆	semidirect
245. C₃²	C₅×C₂²⋊Q₈	semidirect
267. C₅	#63964	direct (N × H)
270. C₂²	C₅×#627	semidirect
279. C₃	C₅×#878	semidirect
280. C₃	C₅×#924	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×Dic₃	2
C₂×C₅×#627	2
C₅×#3757	2
C₃×C₅×#217	2
C₅×#3724	2
C₅×Dic₃⋊Dic₃	2
C₅×#924	3
C₅×#878	3
#63964	5

Subgroups — C5×#63964 #13087251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#63964 #13087251