Subgroups — C₅×#71765 #14305961

Order 1920.

824 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#832	C₂	1	4	yes	maximal, complemented
3	C₅×#4936	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1961	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×D₄	—	2	2	no	—
7	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
8	C₃×C₅×#127	—	2	2	no	—
9	C₃×C₅×#127	C₄	1	4	yes	complemented
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
13	C₅×#832	—	2	8	no	—
14	C₅×C₃⋊#95	—	2	2	no	—
15	C₅×#803	—	2	2	no	—
16	#71765	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#127	D₃	1	1	yes	characteristic
18	C₂×C₅×#102	—	3	4	no	—
19	C₂×C₅×#97	—	3	1	no	—
20	C₅×#382	—	3	1	no	—
21	C₂⁴×C₃×C₅	—	2	2	no	—
22	C₂×C₃×C₅×D₄	—	2	8	no	—
23	C₂×C₃×C₅×D₄	—	2	4	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	2	4	no	—
26	C₂×C₃×C₅×D₄	—	4	8	no	—
27	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	—
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂²×C₅×Dic₃	—	2	2	no	—
31	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₃×C₄²×C₅	D₄	1	2	yes	—
33	C₅×C₄⋊Dic₃	—	2	4	no	—
34	C₂×C₅×C₃⋊C₈	—	2	2	no	—
35	C₂×C₅×C₃⋊C₈	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
37	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
38	C₂×#832	C₂×C₅	1	4	yes	complemented
39	#4936	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂×#803	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂²×C₅×D₄	—	2	2	no	—
42	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
43	C₂²×C₅×C₈	—	3	2	no	—
44	C₅×#127	—	2	2	no	—
45	C₅×#127	Dic₃	1	4	yes	complemented
46	C₂×C₅×C₄⋊C₄	—	3	2	no	—
47	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
48	C₅×#102	—	6	8	no	—
49	C₅×#97	—	6	2	no	—
50	C₅×#95	—	6	2	no	—
51	#1961	—	3	1	no	Sylow
52	C₂³×C₃×C₅	—	2	4	no	—
53	C₂³×C₃×C₅	—	4	2	no	—
54	C₂³×C₃×C₅	—	4	2	no	—
55	C₂³×C₃×C₅	—	4	8	no	—
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₃×C₅×D₄	—	2	8	no	—
58	C₃×C₅×D₄	—	4	16	no	—
59	C₃×C₅×D₄	—	4	4	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	2	yes	—
66	C₂×C₅×Dic₃	—	4	2	no	—
67	C₂×C₅×Dic₃	—	4	4	no	—
68	C₅×C₃⋊C₈	—	2	4	no	—
69	C₂²×C₃×D₄	—	2	2	no	—
70	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
71	C₃×#127	—	2	2	no	—
72	C₃×#127	C₄×C₅	1	4	yes	complemented
73	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
75	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
76	#832	—	2	8	no	—
77	C₃⋊#95	—	2	2	no	—
78	#803	—	2	2	no	—
79	C₂⁴×C₅	—	2	2	no	—
80	C₂×C₅×D₄	—	2	8	no	—
81	C₂×C₅×D₄	—	2	4	no	—
82	C₂×C₅×D₄	—	2	4	no	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂×C₅×D₄	—	4	8	no	—
85	C₂×C₅×D₄	C₂×Dic₃	1	4	yes	—
86	C₂²×C₄×C₅	—	6	2	no	—
87	C₂²×C₄×C₅	#67	1	2	yes	—
88	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
89	C₂×C₅×C₈	—	3	4	no	—
90	C₂×C₅×C₈	—	6	2	no	—
91	C₂×C₅×C₈	—	6	2	no	—
92	C₅×C₄⋊C₄	—	6	4	no	—
93	C₄²×C₅	C₂×Dic₃	1	2	yes	—
94	C₄²×C₅	#67	1	2	yes	—
95	C₂×#127	C₅×D₃	1	1	yes	characteristic
96	C₂×#102	—	3	4	no	—
97	C₂×#97	—	3	1	no	—
98	#382	—	3	1	no	—
99	C₂²×C₃×C₅	—	4	4	no	—
100	C₂²×C₃×C₅	—	4	8	no	—
101	C₂²×C₃×C₅	—	8	4	no	—
102	C₂²×C₃×C₅	—	8	4	no	—
103	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	#124	1	1	yes	characteristic
105	C₂²×C₃×C₅	#127	1	1	yes	characteristic
106	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	4	yes	—
107	C₃×C₄×C₅	—	2	4	no	—
108	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₃×C₄×C₅	#121	1	2	yes	—
110	C₅×Dic₃	—	8	2	no	—
111	C₂⁴×C₃	—	2	2	no	—
112	C₂×C₃×D₄	—	2	8	no	—
113	C₂×C₃×D₄	—	2	4	no	—
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	2	4	no	—
116	C₂×C₃×D₄	—	4	8	no	—
117	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	—
118	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
119	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
120	C₂²×Dic₃	—	2	2	no	—
121	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
122	C₃×C₄²	C₅×D₄	1	2	yes	—
123	C₄⋊Dic₃	—	2	4	no	—
124	C₂×C₃⋊C₈	—	2	2	no	—
125	C₂×C₃⋊C₈	—	2	2	no	—
126	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	complemented
127	C₂³×C₅	—	2	4	no	—
128	C₂³×C₅	—	4	8	no	—
129	C₂³×C₅	—	4	2	no	—
130	C₂³×C₅	—	4	2	no	—
131	C₂³×C₅	C₂×#67	1	1	yes	characteristic
132	C₅×D₄	—	2	8	no	—
133	C₅×D₄	—	4	16	no	—
134	C₅×D₄	—	4	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	12	4	no	—
137	C₂×C₄×C₅	—	12	2	no	—
138	C₂×C₄×C₅	#217	1	4	yes	—
139	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
140	C₂×C₄×C₅	#237	1	2	yes	—
141	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
143	C₅×C₈	—	6	4	no	—
144	C₂²×D₄	—	2	2	no	—
145	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
146	C₂²×C₈	—	3	2	no	—
147	#127	—	2	2	no	—
148	#127	C₅×Dic₃	1	4	yes	complemented
149	C₂×C₄⋊C₄	—	3	2	no	—
150	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
151	#102	—	6	8	no	—
152	#97	—	6	2	no	—
153	#95	—	6	2	no	—
154	C₂×C₃×C₅	—	8	4	no	—
155	C₂×C₃×C₅	#390	1	1	yes	characteristic
156	C₂×C₃×C₅	#418	1	2	yes	—
157	C₂×C₃×C₅	#490	1	2	yes	—
158	C₂×C₃×C₅	#496	1	2	yes	—
159	C₂³×C₃	—	2	4	no	—
160	C₂³×C₃	—	4	2	no	—
161	C₂³×C₃	—	4	2	no	—
162	C₂³×C₃	—	4	8	no	—
163	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
164	C₃×D₄	—	2	8	no	—
165	C₃×D₄	—	4	16	no	—
166	C₃×D₄	—	4	4	no	—
167	C₂×C₃×C₄	—	2	4	no	—
168	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
169	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₅×#41	1	2	yes	—
173	C₂×Dic₃	—	4	2	no	—
174	C₂×Dic₃	—	4	4	no	—
175	C₃⋊C₈	—	2	4	no	—
176	C₂²×C₅	—	4	8	no	—
177	C₂²×C₅	—	4	4	no	—
178	C₂²×C₅	—	8	4	no	—
179	C₂²×C₅	—	8	4	no	—
180	C₂²×C₅	#932	1	4	yes	—
181	C₂²×C₅	#936	1	1	yes	characteristic
182	C₂²×C₅	#940	1	1	yes	characteristic
183	C₂²×C₅	C₂×#217	1	1	yes	characteristic
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	24	2	no	—
186	C₄×C₅	D₄×Dic₃	1	2	yes	—
187	C₄×C₅	#938	1	2	yes	—
188	C₂⁴	—	2	2	no	—
189	C₂×D₄	—	2	8	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	C₂×C₅×Dic₃	1	4	yes	—
195	C₂²×C₄	—	6	2	no	—
196	C₂²×C₄	C₅×#67	1	2	yes	—
197	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
198	C₂×C₈	—	3	4	no	—
199	C₂×C₈	—	6	2	no	—
200	C₂×C₈	—	6	2	no	—
201	C₄⋊C₄	—	6	4	no	—
202	C₄²	C₂×C₅×Dic₃	1	2	yes	—
203	C₄²	C₅×#67	1	2	yes	—
204	C₃×C₅	#1961	1	1	yes	characteristic, complemented, Hall
205	C₂²×C₃	—	4	4	no	—
206	C₂²×C₃	—	4	8	no	—
207	C₂²×C₃	—	8	4	no	—
208	C₂²×C₃	—	8	4	no	—
209	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₃	C₅×#124	1	1	yes	characteristic
211	C₂²×C₃	C₅×#127	1	1	yes	characteristic
212	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	4	yes	—
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
215	C₃×C₄	C₅×#121	1	2	yes	—
216	Dic₃	—	8	2	no	—
217	C₂×C₅	—	8	4	no	—
218	C₂×C₅	#5071	1	2	yes	—
219	C₂×C₅	#5074	1	2	yes	—
220	C₂×C₅	#5215	1	2	yes	—
221	C₂×C₅	#5221	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	4	8	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	C₂×C₅×#67	1	1	yes	characteristic
227	D₄	—	2	8	no	—
228	D₄	—	4	16	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	12	4	no	—
232	C₂×C₄	—	12	2	no	—
233	C₂×C₄	C₅×#217	1	4	yes	—
234	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
235	C₂×C₄	C₅×#237	1	2	yes	—
236	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
237	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
238	C₈	—	6	4	no	—
239	C₂×C₃	—	8	4	no	—
240	C₂×C₃	C₅×#390	1	1	yes	characteristic
241	C₂×C₃	C₅×#418	1	2	yes	—
242	C₂×C₃	C₅×#490	1	2	yes	—
243	C₂×C₃	C₅×#496	1	2	yes	—
244	C₅	#71765	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	4	8	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	C₅×#932	1	4	yes	—
250	C₂²	C₅×#936	1	1	yes	characteristic
251	C₂²	C₅×#940	1	1	yes	characteristic
252	C₂²	C₂×C₅×#217	1	1	yes	characteristic
253	C₄	—	2	4	no	—
254	C₄	—	24	2	no	—
255	C₄	C₅×D₄×Dic₃	1	2	yes	—
256	C₄	C₅×#938	1	2	yes	—
257	C₃	C₅×#1961	1	1	yes	characteristic, complemented, Sylow
258	C₂	—	8	4	no	—
259	C₂	C₅×#5071	1	2	yes	—
260	C₂	C₅×#5074	1	2	yes	—
261	C₂	C₅×#5215	1	2	yes	—
262	C₂	C₅×#5221	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#127	D₃
hypocenter	C₃	C₅×#1961
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₅×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#832	C₂	semidirect
3. C₅×#4936	C₂	semidirect
4. C₂×C₅×#803	C₂	semidirect
9. C₃×C₅×#127	C₄	semidirect
16. #71765	C₅	direct (N × H)
36. C₂×C₅×C₃⋊C₈	D₄	semidirect
38. C₂×#832	C₂×C₅	semidirect
39. #4936	C₂×C₅	semidirect
40. C₂×#803	C₂×C₅	semidirect
45. C₅×#127	Dic₃	semidirect
72. C₃×#127	C₄×C₅	semidirect
126. C₂×C₃⋊C₈	C₅×D₄	semidirect
148. #127	C₅×Dic₃	semidirect
204. C₃×C₅	#1961	semidirect
244. C₅	#71765	direct (N × H)
257. C₃	C₅×#1961	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×#127	2
C₂×C₅×#832	2
C₅×#4936	2
C₂×C₅×#803	2
C₅×#1961	3
#71765	5

Subgroups — C5×#71765 #14305961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71765 #14305961