Subgroups — C₅×#71583 #14297473

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#830	C₂	1	2	yes	maximal
3	C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal, complemented
4	C₅×#4524	C₂	1	2	yes	maximal, complemented
5	C₅×#1798	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₃×C₅×#117	—	2	4	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂²×C₅×C₃⋊C₈	C₂²	1	4	yes	—
13	C₅×#830	—	2	4	no	—
14	C₅×C₄⋊(C₃⋊C₈)	—	2	4	no	—
15	#71583	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×#117	D₃	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	—	3	2	no	—
18	C₂×C₅×C₂²⋊C₈	—	3	2	no	—
19	C₅×#336	—	3	2	no	—
20	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₃×C₄²×C₅	D₄	1	4	yes	—
33	C₂×C₅×C₃⋊C₈	—	2	4	no	—
34	C₂×C₅×C₃⋊C₈	—	2	4	no	—
35	C₂×C₅×C₃⋊C₈	—	2	4	no	—
36	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic
37	C₂×#830	C₂×C₅	1	2	yes	—
38	C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	1	2	yes	complemented
39	#4524	C₂×C₅	1	2	yes	complemented
40	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	3	4	no	—
42	C₅×#117	—	2	4	no	—
43	C₂×C₅×C₄⋊C₄	Dic₃	1	2	yes	—
44	C₂×C₅×C₂²⋊C₄	Dic₃	1	2	yes	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
47	C₅×C₄⋊C₈	—	6	4	no	—
48	C₅×C₂²⋊C₈	—	6	4	no	—
49	#1798	—	3	1	no	Sylow
50	C₂³×C₃×C₅	—	2	1	no	—
51	C₂³×C₃×C₅	—	2	1	no	—
52	C₂³×C₃×C₅	—	2	4	no	—
53	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	4	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₅×C₃⋊C₈	—	4	4	no	—
69	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
70	C₃×#117	—	2	4	no	—
71	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
72	C₂×C₃×C₂²⋊C₄	C₄×C₅	1	2	yes	—
73	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₂²×C₃⋊C₈	C₂²×C₅	1	4	yes	—
76	#830	—	2	4	no	—
77	C₄⋊(C₃⋊C₈)	—	2	4	no	—
78	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	Dic₆	1	2	yes	—
82	C₂²×C₄×C₅	D₁₂	1	2	yes	—
83	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
84	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
86	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
87	C₂×C₅×C₈	—	6	4	no	—
88	C₂×C₅×C₈	—	6	4	no	—
89	C₂×C₅×C₈	—	6	4	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	#67	1	4	yes	—
94	C₂×#117	C₅×D₃	1	1	yes	characteristic
95	C₂×C₄⋊C₈	—	3	2	no	—
96	C₂×C₂²⋊C₈	—	3	2	no	—
97	#336	—	3	2	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	2	4	no	—
100	C₂²×C₃×C₅	—	4	2	no	—
101	C₂²×C₃×C₅	—	4	2	no	—
102	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	#117	1	1	yes	characteristic
105	C₂²×C₃×C₅	#131	1	4	yes	—
106	C₃×C₄×C₅	—	2	2	no	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	—	4	2	no	—
109	C₃×C₄×C₅	#121	1	2	yes	—
110	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
111	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
115	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
116	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
118	C₂²×C₃×C₄	C₅×Q₈	1	2	yes	—
119	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
120	C₃×C₄⋊C₄	—	2	4	no	—
121	C₃×C₂²⋊C₄	—	2	4	no	—
122	C₃×C₄²	—	2	2	no	—
123	C₃×C₄²	C₅×D₄	1	4	yes	—
124	C₂×C₃⋊C₈	—	2	4	no	—
125	C₂×C₃⋊C₈	—	2	4	no	—
126	C₂×C₃⋊C₈	—	2	4	no	—
127	C₂³×C₅	—	2	1	no	—
128	C₂³×C₅	—	2	4	no	—
129	C₂³×C₅	—	2	1	no	—
130	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
131	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	4	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
140	C₂×C₄×C₅	#217	1	4	yes	—
141	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
143	C₂×C₄×C₅	#235	1	2	yes	—
144	C₂×C₄×C₅	C₂×#67	1	2	yes	—
145	C₅×C₈	—	12	4	no	—
146	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
147	C₂²×C₈	—	3	4	no	—
148	#117	—	2	4	no	—
149	C₂×C₄⋊C₄	C₅×Dic₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₅×Dic₃	1	2	yes	—
151	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
153	C₄⋊C₈	—	6	4	no	—
154	C₂²⋊C₈	—	6	4	no	—
155	C₂×C₃×C₅	—	4	2	no	—
156	C₂×C₃×C₅	#380	1	1	yes	characteristic
157	C₂×C₃×C₅	#408	1	2	yes	—
158	C₂×C₃×C₅	#424	1	2	yes	—
159	C₂×C₃×C₅	#433	1	2	yes	—
160	C₂³×C₃	—	2	1	no	—
161	C₂³×C₃	—	2	1	no	—
162	C₂³×C₃	—	2	4	no	—
163	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
164	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	—	2	4	no	—
169	C₂×C₃×C₄	—	2	4	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
173	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
177	C₂×C₃×C₄	C₅×#41	1	2	yes	—
178	C₃⋊C₈	—	4	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	4	2	no	—
182	C₂²×C₅	—	4	2	no	—
183	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
184	C₂²×C₅	#926	1	1	yes	characteristic
185	C₂²×C₅	#948	1	4	yes	—
186	C₂²×C₅	C₂×#217	1	1	yes	characteristic
187	C₄×C₅	—	2	2	no	—
188	C₄×C₅	—	2	4	no	—
189	C₄×C₅	—	4	2	no	—
190	C₄×C₅	#924	1	2	yes	—
191	C₄×C₅	#930	1	2	yes	—
192	C₂⁴	C₂×C₅×Dic₃	1	1	yes	characteristic
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	C₅×Dic₆	1	2	yes	—
196	C₂²×C₄	C₅×D₁₂	1	2	yes	—
197	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
198	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
200	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
201	C₂×C₈	—	6	4	no	—
202	C₂×C₈	—	6	4	no	—
203	C₂×C₈	—	6	4	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₄²	C₅×#67	1	4	yes	—
208	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	4	no	—
211	C₂²×C₃	—	4	2	no	—
212	C₂²×C₃	—	4	2	no	—
213	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	C₅×#117	1	1	yes	characteristic
216	C₂²×C₃	C₅×#131	1	4	yes	—
217	C₃×C₄	—	2	2	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	4	2	no	—
220	C₃×C₄	C₅×#121	1	2	yes	—
221	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
222	C₂×C₅	—	4	2	no	—
223	C₂×C₅	#4999	1	2	yes	—
224	C₂×C₅	#5000	1	2	yes	—
225	C₂×C₅	#5212	1	1	yes	characteristic
226	C₂×C₅	#5234	1	2	yes	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	1	no	—
230	C₂³	C₅×C₄⋊Dic₃	1	2	yes	—
231	C₂³	C₂²×C₅×Dic₃	1	1	yes	characteristic
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
240	C₂×C₄	C₅×#217	1	4	yes	—
241	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
242	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
243	C₂×C₄	C₅×#235	1	2	yes	—
244	C₂×C₄	C₂×C₅×#67	1	2	yes	—
245	C₈	—	12	4	no	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	C₅×#380	1	1	yes	characteristic
248	C₂×C₃	C₅×#408	1	2	yes	—
249	C₂×C₃	C₅×#424	1	2	yes	—
250	C₂×C₃	C₅×#433	1	2	yes	—
251	C₅	#71583	1	1	yes	characteristic, direct product factor, Sylow
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
257	C₂²	C₅×#926	1	1	yes	characteristic
258	C₂²	C₅×#948	1	4	yes	—
259	C₂²	C₂×C₅×#217	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	2	no	—
263	C₄	C₅×#924	1	2	yes	—
264	C₄	C₅×#930	1	2	yes	—
265	C₃	C₅×#1798	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	4	2	no	—
267	C₂	C₅×#4999	1	2	yes	—
268	C₂	C₅×#5000	1	2	yes	—
269	C₂	C₅×#5212	1	1	yes	characteristic
270	C₂	C₅×#5234	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²×D₃
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#117	D₃
hypocenter	C₃	C₅×#1798
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#117

Complements

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

N	H ≅ G/N	type
3. C₂×C₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. C₅×#4524	C₂	semidirect
15. #71583	C₅	direct (N × H)
38. C₂×C₄⋊(C₃⋊C₈)	C₂×C₅	semidirect
39. #4524	C₂×C₅	semidirect
208. C₃×C₅	#1798	semidirect
251. C₅	#71583	direct (N × H)
265. C₃	C₅×#1798	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#117	2
C₂×C₅×#830	2
C₂×C₅×C₄⋊(C₃⋊C₈)	2
C₅×#4524	2
C₅×#1798	3
#71583	5

Subgroups — C5×#71583 #14297473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71583 #14297473