Subgroups — C₂×C₃×C₅×#381 #14275313

Order 1920.

2288 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
3	C₃×C₅×#381	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#95	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#381	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂³×C₃×C₄×C₅	—	2	8	no	—
8	C₂³×C₃×C₄×C₅	C₄	1	8	yes	complemented
9	C₂³×C₃×C₄×C₅	C₂²	1	4	yes	—
10	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
11	C₂³×C₃×C₄×C₅	C₂²	1	4	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	8	no	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
14	C₃×C₅×#95	C₂²	1	16	yes	complemented
15	C₂×C₃×#381	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂⁴×C₄×C₅	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	2	yes	—
18	C₅×#381	C₂×C₃	1	8	yes	direct product factor
19	C₂×C₅×#95	C₂×C₃	1	4	yes	complemented
20	C₂⁴×C₃×C₅	—	2	8	no	—
21	C₂⁴×C₃×C₅	C₂×C₄	1	4	yes	—
22	C₂⁴×C₃×C₅	D₄	1	8	yes	—
23	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	32	no	—
26	C₂²×C₃×C₄×C₅	—	2	8	no	—
27	C₂²×C₃×C₄×C₅	—	2	16	no	—
28	C₂²×C₃×C₄×C₅	—	2	8	no	—
29	C₂²×C₃×C₄×C₅	—	2	16	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	16	yes	complemented
32	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₃×C₅×C₂²⋊C₄	—	2	16	no	—
35	C₂⁴×C₃×C₄	C₂×C₅	1	1	yes	characteristic
36	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	2	yes	—
37	C₃×#381	C₂×C₅	1	8	yes	direct product factor
38	C₂×C₃×#95	C₂×C₅	1	4	yes	complemented
39	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
40	C₂³×C₄×C₅	—	2	8	no	—
41	C₂³×C₄×C₅	C₃×C₄	1	8	yes	complemented
42	C₂³×C₄×C₅	C₂²×C₃	1	4	yes	—
43	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
44	C₂³×C₄×C₅	C₂²×C₃	1	4	yes	—
45	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
46	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
47	C₅×#95	C₂²×C₃	1	16	yes	complemented
48	C₂×#381	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
49	C₂³×C₃×C₅	—	2	16	no	—
50	C₂³×C₃×C₅	—	2	32	no	—
51	C₂³×C₃×C₅	—	2	8	no	—
52	C₂³×C₃×C₅	C₂²⋊C₄	1	16	yes	—
53	C₂³×C₃×C₅	C₂²×C₄	1	4	yes	—
54	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
55	C₂³×C₃×C₅	C₂×D₄	1	8	yes	—
56	C₂³×C₃×C₅	C₂×D₄	1	4	yes	—
57	C₂³×C₃×C₅	#41	1	8	yes	—
58	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	16	no	—
60	C₂×C₃×C₄×C₅	—	2	32	no	—
61	C₂×C₃×C₄×C₅	—	2	8	no	—
62	C₂×C₃×C₄×C₅	—	2	8	no	—
63	C₂×C₃×C₄×C₅	—	2	16	no	—
64	C₂×C₃×C₄×C₅	—	4	16	no	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
66	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
67	C₂³×C₃×C₄	—	2	8	no	—
68	C₂³×C₃×C₄	C₄×C₅	1	8	yes	complemented
69	C₂³×C₃×C₄	C₂²×C₅	1	4	yes	—
70	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
71	C₂³×C₃×C₄	C₂²×C₅	1	4	yes	—
72	C₂×C₃×C₂²⋊C₄	—	2	8	no	—
73	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
74	C₃×#95	C₂²×C₅	1	16	yes	complemented
75	C₂⁴×C₅	—	2	8	no	—
76	C₂⁴×C₅	C₂×C₃×C₄	1	4	yes	—
77	C₂⁴×C₅	C₃×D₄	1	8	yes	—
78	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
79	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
80	C₂²×C₄×C₅	—	2	32	no	—
81	C₂²×C₄×C₅	—	2	8	no	—
82	C₂²×C₄×C₅	—	2	16	no	—
83	C₂²×C₄×C₅	—	2	8	no	—
84	C₂²×C₄×C₅	—	2	16	no	—
85	C₂²×C₄×C₅	C₂×C₃×C₄	1	8	yes	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	16	yes	complemented
87	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
88	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
89	C₅×C₂²⋊C₄	—	2	16	no	—
90	C₂⁴×C₄	C₂×C₃×C₅	1	1	yes	characteristic
91	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	2	yes	—
92	#381	C₂×C₃×C₅	1	8	yes	direct product factor
93	C₂×#95	C₂×C₃×C₅	1	4	yes	complemented
94	C₂²×C₃×C₅	—	2	8	no	—
95	C₂²×C₃×C₅	—	2	16	no	—
96	C₂²×C₃×C₅	—	2	32	no	—
97	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	8	yes	—
98	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	4	yes	—
99	C₂²×C₃×C₅	#117	1	8	yes	—
100	C₂²×C₃×C₅	#123	1	16	yes	—
101	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
103	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
104	C₃×C₄×C₅	—	2	8	no	—
105	C₃×C₄×C₅	—	4	8	no	—
106	C₂⁴×C₃	—	2	8	no	—
107	C₂⁴×C₃	C₂×C₄×C₅	1	4	yes	—
108	C₂⁴×C₃	C₅×D₄	1	8	yes	—
109	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
110	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
111	C₂²×C₃×C₄	—	2	32	no	—
112	C₂²×C₃×C₄	—	2	8	no	—
113	C₂²×C₃×C₄	—	2	16	no	—
114	C₂²×C₃×C₄	—	2	8	no	—
115	C₂²×C₃×C₄	—	2	16	no	—
116	C₂²×C₃×C₄	C₂×C₄×C₅	1	8	yes	—
117	C₂²×C₃×C₄	C₂×C₄×C₅	1	16	yes	complemented
118	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
119	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
120	C₃×C₂²⋊C₄	—	2	16	no	—
121	C₂³×C₅	—	2	16	no	—
122	C₂³×C₅	—	2	32	no	—
123	C₂³×C₅	—	2	8	no	—
124	C₂³×C₅	C₃×C₂²⋊C₄	1	16	yes	—
125	C₂³×C₅	C₂²×C₃×C₄	1	4	yes	—
126	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
127	C₂³×C₅	C₂×C₃×D₄	1	8	yes	—
128	C₂³×C₅	C₂×C₃×D₄	1	4	yes	—
129	C₂³×C₅	C₃×#41	1	8	yes	—
130	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
131	C₂×C₄×C₅	—	2	16	no	—
132	C₂×C₄×C₅	—	2	32	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	2	8	no	—
135	C₂×C₄×C₅	—	2	16	no	—
136	C₂×C₄×C₅	—	4	16	no	—
137	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
138	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₂³×C₄	—	2	8	no	—
140	C₂³×C₄	C₃×C₄×C₅	1	8	yes	complemented
141	C₂³×C₄	C₂²×C₃×C₅	1	4	yes	—
142	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
143	C₂³×C₄	C₂²×C₃×C₅	1	4	yes	—
144	C₂×C₂²⋊C₄	—	2	8	no	—
145	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	8	yes	—
146	#95	C₂²×C₃×C₅	1	16	yes	complemented
147	C₂×C₃×C₅	—	2	8	no	—
148	C₂×C₃×C₅	#381	1	8	yes	direct product factor
149	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₃×C₅	C₂×#117	1	2	yes	—
151	C₂×C₃×C₅	C₂×#123	1	4	yes	—
152	C₂³×C₃	—	2	16	no	—
153	C₂³×C₃	—	2	32	no	—
154	C₂³×C₃	—	2	8	no	—
155	C₂³×C₃	C₅×C₂²⋊C₄	1	16	yes	—
156	C₂³×C₃	C₂²×C₄×C₅	1	4	yes	—
157	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
158	C₂³×C₃	C₂×C₅×D₄	1	8	yes	—
159	C₂³×C₃	C₂×C₅×D₄	1	4	yes	—
160	C₂³×C₃	C₅×#41	1	8	yes	—
161	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	16	no	—
163	C₂×C₃×C₄	—	2	32	no	—
164	C₂×C₃×C₄	—	2	8	no	—
165	C₂×C₃×C₄	—	2	8	no	—
166	C₂×C₃×C₄	—	2	16	no	—
167	C₂×C₃×C₄	—	4	16	no	—
168	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
169	C₂²×C₅	—	2	32	no	—
170	C₂²×C₅	—	2	8	no	—
171	C₂²×C₅	—	2	16	no	—
172	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	8	yes	—
173	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	4	yes	—
174	C₂²×C₅	C₃×#117	1	8	yes	—
175	C₂²×C₅	C₃×#123	1	16	yes	—
176	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
177	C₂²×C₅	C₂²×C₃×D₄	1	2	yes	—
178	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
179	C₄×C₅	—	2	8	no	—
180	C₄×C₅	—	4	8	no	—
181	C₂⁴	—	2	8	no	—
182	C₂⁴	C₂×C₃×C₄×C₅	1	4	yes	—
183	C₂⁴	C₃×C₅×D₄	1	8	yes	—
184	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
185	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
186	C₂²×C₄	—	2	32	no	—
187	C₂²×C₄	—	2	8	no	—
188	C₂²×C₄	—	2	16	no	—
189	C₂²×C₄	—	2	8	no	—
190	C₂²×C₄	—	2	16	no	—
191	C₂²×C₄	C₂×C₃×C₄×C₅	1	8	yes	—
192	C₂²×C₄	C₂×C₃×C₄×C₅	1	16	yes	complemented
193	C₂²×C₄	C₂³×C₃×C₅	1	8	yes	—
194	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
195	C₂²⋊C₄	—	2	16	no	—
196	C₃×C₅	C₂×#381	1	1	yes	characteristic, direct product factor, Hall
197	C₂²×C₃	—	2	32	no	—
198	C₂²×C₃	—	2	8	no	—
199	C₂²×C₃	—	2	16	no	—
200	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	8	yes	—
201	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
202	C₂²×C₃	C₅×#117	1	8	yes	—
203	C₂²×C₃	C₅×#123	1	16	yes	—
204	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
205	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
206	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
207	C₃×C₄	—	2	8	no	—
208	C₃×C₄	—	4	8	no	—
209	C₂×C₅	—	2	8	no	—
210	C₂×C₅	C₃×#381	1	8	yes	direct product factor
211	C₂×C₅	C₂²×C₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₅	C₂×C₃×#117	1	2	yes	—
213	C₂×C₅	C₂×C₃×#123	1	4	yes	—
214	C₂³	—	2	16	no	—
215	C₂³	—	2	8	no	—
216	C₂³	—	2	32	no	—
217	C₂³	C₃×C₅×C₂²⋊C₄	1	16	yes	—
218	C₂³	C₂²×C₃×C₄×C₅	1	4	yes	—
219	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
220	C₂³	C₂×C₃×C₅×D₄	1	8	yes	—
221	C₂³	C₂×C₃×C₅×D₄	1	4	yes	—
222	C₂³	C₃×C₅×#41	1	8	yes	—
223	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	32	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	8	no	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	4	16	no	—
230	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
231	C₂×C₃	—	2	8	no	—
232	C₂×C₃	C₅×#381	1	8	yes	direct product factor
233	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₃	C₂×C₅×#117	1	2	yes	—
235	C₂×C₃	C₂×C₅×#123	1	4	yes	—
236	C₅	C₂×C₃×#381	1	1	yes	characteristic, direct product factor, Sylow
237	C₂²	—	2	8	no	—
238	C₂²	—	2	16	no	—
239	C₂²	—	2	32	no	—
240	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	8	yes	—
241	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	4	yes	—
242	C₂²	C₃×C₅×#117	1	8	yes	—
243	C₂²	C₃×C₅×#123	1	16	yes	—
244	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
245	C₂²	C₂²×C₃×C₅×D₄	1	2	yes	—
246	C₂²	C₂×C₃×C₅×#41	1	4	yes	—
247	C₄	—	2	8	no	—
248	C₄	—	4	8	no	—
249	C₃	C₂×C₅×#381	1	1	yes	characteristic, direct product factor, Sylow
250	C₂	—	2	8	no	—
251	C₂	C₃×C₅×#381	1	8	yes	direct product factor
252	C₂	C₂²×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂	C₂×C₃×C₅×#117	1	2	yes	—
254	C₂	C₂×C₃×C₅×#123	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃×C₅	C₂³
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁵×C₃×C₅	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#381	C₂	direct (N × H)
4. C₂×C₃×C₅×#95	C₂	semidirect
5. C₂×C₅×#381	C₃	direct (N × H)
8. C₂³×C₃×C₄×C₅	C₄	semidirect
14. C₃×C₅×#95	C₂²	semidirect
15. C₂×C₃×#381	C₅	direct (N × H)
18. C₅×#381	C₂×C₃	direct (N × H)
19. C₂×C₅×#95	C₂×C₃	semidirect
31. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
37. C₃×#381	C₂×C₅	direct (N × H)
38. C₂×C₃×#95	C₂×C₅	semidirect
41. C₂³×C₄×C₅	C₃×C₄	semidirect
47. C₅×#95	C₂²×C₃	semidirect
48. C₂×#381	C₃×C₅	direct (N × H)
68. C₂³×C₃×C₄	C₄×C₅	semidirect
74. C₃×#95	C₂²×C₅	semidirect
86. C₂²×C₄×C₅	C₂×C₃×C₄	semidirect
92. #381	C₂×C₃×C₅	direct (N × H)
93. C₂×#95	C₂×C₃×C₅	semidirect
117. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
140. C₂³×C₄	C₃×C₄×C₅	semidirect
146. #95	C₂²×C₃×C₅	semidirect
148. C₂×C₃×C₅	#381	direct (N × H)
192. C₂²×C₄	C₂×C₃×C₄×C₅	semidirect
196. C₃×C₅	C₂×#381	direct (N × H)
210. C₂×C₅	C₃×#381	direct (N × H)
232. C₂×C₃	C₅×#381	direct (N × H)
236. C₅	C₂×C₃×#381	direct (N × H)
249. C₃	C₂×C₅×#381	direct (N × H)
251. C₂	C₃×C₅×#381	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₄×C₅	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#381	2
C₂×C₃×C₅×#95	2
C₂×C₅×#381	3
C₂×C₃×#381	5

Subgroups — C2×C3×C5×#381 #14275313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#381 #14275313