Subgroups — C₃×C₅×#1961 #14274995

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×C₃×C₅×#102	C₂	1	4	yes	maximal, complemented
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1961	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	—	2	2	no	—
7	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₅×#127	—	2	2	no	—
10	C₃×C₅×#127	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#102	—	2	8	no	—
14	C₃×C₅×#97	—	2	2	no	—
15	C₃×C₅×#95	—	2	2	no	—
16	C₃×#1961	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
18	C₂×C₅×#102	C₂×C₃	1	4	yes	complemented
19	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#382	C₂×C₃	1	1	yes	characteristic, complemented
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	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	D₄	1	4	yes	complemented
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
38	C₂×C₃×#102	C₂×C₅	1	4	yes	complemented
39	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#382	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂²×C₅×D₄	—	2	2	no	—
42	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
43	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
44	C₅×#127	—	2	2	no	—
45	C₅×#127	C₃×C₄	1	4	yes	complemented
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₅×#102	—	2	8	no	—
49	C₅×#97	—	2	2	no	—
50	C₅×#95	—	2	2	no	—
51	#1961	C₃×C₅	1	1	yes	characteristic, direct product factor, 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	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₃×C₅×C₈	—	2	4	no	—
69	C₂²×C₃×D₄	—	2	2	no	—
70	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
71	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
72	C₃×#127	—	2	2	no	—
73	C₃×#127	C₄×C₅	1	4	yes	complemented
74	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₃×#102	—	2	8	no	—
77	C₃×#97	—	2	2	no	—
78	C₃×#95	—	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	2	no	—
83	C₂×C₅×D₄	—	2	4	no	—
84	C₂×C₅×D₄	—	4	8	no	—
85	C₂×C₅×D₄	C₂×C₃×C₄	1	4	yes	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	C₃×D₄	1	4	yes	complemented
92	C₅×C₄⋊C₄	—	2	4	no	—
93	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
94	C₄²×C₅	C₃×D₄	1	2	yes	—
95	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
96	C₂×#102	C₂×C₃×C₅	1	4	yes	complemented
97	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic, complemented
98	#382	C₂×C₃×C₅	1	1	yes	characteristic, complemented
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	—	8	2	no	—
109	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
110	C₃×C₄×C₅	#121	1	2	yes	—
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	—	2	2	no	—
119	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
120	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₅×D₄	1	4	yes	complemented
124	C₃×C₄⋊C₄	—	2	4	no	—
125	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
126	C₃×C₄²	C₅×D₄	1	2	yes	—
127	C₂³×C₅	—	2	4	no	—
128	C₂³×C₅	—	4	2	no	—
129	C₂³×C₅	—	4	2	no	—
130	C₂³×C₅	—	4	8	no	—
131	C₂³×C₅	C₂×C₃×D₄	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	—	4	2	no	—
137	C₂×C₄×C₅	—	4	4	no	—
138	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
139	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₃×#41	1	2	yes	—
143	C₅×C₈	—	2	4	no	—
144	C₂²×D₄	—	2	2	no	—
145	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
146	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
147	#127	—	2	2	no	—
148	#127	C₃×C₄×C₅	1	4	yes	complemented
149	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
150	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
151	#102	—	2	8	no	—
152	#97	—	2	2	no	—
153	#95	—	2	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	—	4	2	no	—
169	C₂×C₃×C₄	—	4	4	no	—
170	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	4	yes	—
171	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₅×#41	1	2	yes	—
175	C₃×C₈	—	2	4	no	—
176	C₂²×C₅	—	4	4	no	—
177	C₂²×C₅	—	4	8	no	—
178	C₂²×C₅	—	8	4	no	—
179	C₂²×C₅	—	8	4	no	—
180	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₅	C₃×#124	1	1	yes	characteristic
182	C₂²×C₅	C₃×#127	1	1	yes	characteristic
183	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	8	2	no	—
186	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
187	C₄×C₅	C₃×#121	1	2	yes	—
188	C₂⁴	—	2	2	no	—
189	C₂×D₄	—	2	8	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	4	no	—
193	C₂×D₄	—	4	8	no	—
194	C₂×D₄	C₂×C₃×C₄×C₅	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
197	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₃×C₅×D₄	1	4	yes	complemented
201	C₄⋊C₄	—	2	4	no	—
202	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
203	C₄²	C₃×C₅×D₄	1	2	yes	—
204	C₃×C₅	#1961	1	1	yes	characteristic, direct product factor, 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	—	8	2	no	—
215	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
216	C₃×C₄	C₅×#121	1	2	yes	—
217	C₂×C₅	—	8	4	no	—
218	C₂×C₅	C₃×#390	1	1	yes	characteristic
219	C₂×C₅	C₃×#418	1	2	yes	—
220	C₂×C₅	C₃×#490	1	2	yes	—
221	C₂×C₅	C₃×#496	1	2	yes	—
222	C₂³	—	2	4	no	—
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	8	no	—
226	C₂³	C₂×C₃×C₅×D₄	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	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	4	yes	—
234	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
236	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
237	C₂×C₄	C₃×C₅×#41	1	2	yes	—
238	C₈	—	2	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	C₃×#1961	1	1	yes	characteristic, direct product factor, Sylow
245	C₂²	—	4	4	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	8	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
250	C₂²	C₃×C₅×#124	1	1	yes	characteristic
251	C₂²	C₃×C₅×#127	1	1	yes	characteristic
252	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	4	yes	—
253	C₄	—	2	4	no	—
254	C₄	—	8	2	no	—
255	C₄	C₃×C₄×C₅×D₄	1	2	yes	—
256	C₄	C₃×C₅×#121	1	2	yes	—
257	C₃	C₅×#1961	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	8	4	no	—
259	C₂	C₃×C₅×#390	1	1	yes	characteristic
260	C₂	C₃×C₅×#418	1	2	yes	—
261	C₂	C₃×C₅×#490	1	2	yes	—
262	C₂	C₃×C₅×#496	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#102	C₂	semidirect
3. C₂×C₃×C₅×#97	C₂	semidirect
4. C₃×C₅×#382	C₂	semidirect
5. C₅×#1961	C₃	direct (N × H)
10. C₃×C₅×#127	C₄	semidirect
16. C₃×#1961	C₅	direct (N × H)
18. C₂×C₅×#102	C₂×C₃	semidirect
19. C₂×C₅×#97	C₂×C₃	semidirect
20. C₅×#382	C₂×C₃	semidirect
33. C₂×C₃×C₅×C₈	D₄	semidirect
38. C₂×C₃×#102	C₂×C₅	semidirect
39. C₂×C₃×#97	C₂×C₅	semidirect
40. C₃×#382	C₂×C₅	semidirect
45. C₅×#127	C₃×C₄	semidirect
51. #1961	C₃×C₅	direct (N × H)
73. C₃×#127	C₄×C₅	semidirect
91. C₂×C₅×C₈	C₃×D₄	semidirect
96. C₂×#102	C₂×C₃×C₅	semidirect
97. C₂×#97	C₂×C₃×C₅	semidirect
98. #382	C₂×C₃×C₅	semidirect
123. C₂×C₃×C₈	C₅×D₄	semidirect
148. #127	C₃×C₄×C₅	semidirect
200. C₂×C₈	C₃×C₅×D₄	semidirect
204. C₃×C₅	#1961	direct (N × H)
244. C₅	C₃×#1961	direct (N × H)
257. C₃	C₅×#1961	direct (N × H)

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×C₃×C₅×#102	2
C₂×C₃×C₅×#97	2
C₃×C₅×#382	2
C₅×#1961	3
C₃×#1961	5

Subgroups — C3×C5×#1961 #14274995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1961 #14274995