Subgroups — C₂×C₃×C₅×#392 #14275418

Order 1920.

3624 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₄	C₂	1	3	yes	maximal, complemented
2	C₂²×C₃×C₅×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#392	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#392	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	3	yes	—
7	C₂²×C₃×C₅×D₄	—	2	24	no	—
8	C₂²×C₃×C₅×D₄	—	2	12	no	—
9	C₂²×C₃×C₅×D₄	C₂²	1	12	yes	complemented
10	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	—	2	12	no	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	complemented
14	C₃×C₅×#95	C₂²	1	4	yes	complemented
15	C₂×C₃×#392	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂³×C₅×D₄	C₂×C₃	1	3	yes	complemented
17	C₂²×C₅×C₂²⋊C₄	C₂×C₃	1	3	yes	complemented
18	C₅×#392	C₂×C₃	1	8	yes	direct product factor
19	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂⁴×C₃×C₅	—	2	6	no	—
21	C₂⁴×C₃×C₅	—	2	12	no	—
22	C₂⁴×C₃×C₅	—	2	12	no	—
23	C₂⁴×C₃×C₅	—	2	6	no	—
24	C₂⁴×C₃×C₅	D₄	1	12	yes	complemented
25	C₂⁴×C₃×C₅	C₂³	1	6	yes	—
26	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₅×D₄	—	2	96	no	—
28	C₂×C₃×C₅×D₄	—	2	48	no	—
29	C₂×C₃×C₅×D₄	—	2	24	no	—
30	C₂²×C₃×C₄×C₅	—	2	12	no	—
31	C₂²×C₃×C₄×C₅	—	2	6	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	12	yes	complemented
33	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
35	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
36	C₂³×C₃×D₄	C₂×C₅	1	3	yes	complemented
37	C₂²×C₃×C₂²⋊C₄	C₂×C₅	1	3	yes	complemented
38	C₃×#392	C₂×C₅	1	8	yes	direct product factor
39	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂⁵×C₅	C₂²×C₃	1	3	yes	—
41	C₂²×C₅×D₄	—	2	24	no	—
42	C₂²×C₅×D₄	—	2	12	no	—
43	C₂²×C₅×D₄	C₂²×C₃	1	12	yes	complemented
44	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
46	C₂×C₅×C₂²⋊C₄	—	2	12	no	—
47	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	complemented
48	C₅×#95	C₂²×C₃	1	4	yes	complemented
49	C₂×#392	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	2	48	no	—
51	C₂³×C₃×C₅	—	2	24	no	—
52	C₂³×C₃×C₅	—	2	6	no	—
53	C₂³×C₃×C₅	—	2	12	no	—
54	C₂³×C₃×C₅	—	2	12	no	—
55	C₂³×C₃×C₅	—	2	6	no	—
56	C₂³×C₃×C₅	—	4	12	no	—
57	C₂³×C₃×C₅	—	4	24	no	—
58	C₂³×C₃×C₅	—	4	12	no	—
59	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
60	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
61	C₂³×C₃×C₅	C₂×D₄	1	12	yes	complemented
62	C₂³×C₃×C₅	#41	1	2	yes	—
63	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
64	C₃×C₅×D₄	—	2	96	no	—
65	C₂×C₃×C₄×C₅	—	2	24	no	—
66	C₂×C₃×C₄×C₅	—	2	12	no	—
67	C₂×C₃×C₄×C₅	—	2	6	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	complemented
70	C₂⁵×C₃	C₂²×C₅	1	3	yes	—
71	C₂²×C₃×D₄	—	2	24	no	—
72	C₂²×C₃×D₄	—	2	12	no	—
73	C₂²×C₃×D₄	C₂²×C₅	1	12	yes	complemented
74	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
75	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
76	C₂×C₃×C₂²⋊C₄	—	2	12	no	—
77	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
78	C₃×#95	C₂²×C₅	1	4	yes	complemented
79	C₂⁴×C₅	—	2	6	no	—
80	C₂⁴×C₅	—	2	12	no	—
81	C₂⁴×C₅	—	2	12	no	—
82	C₂⁴×C₅	—	2	6	no	—
83	C₂⁴×C₅	C₃×D₄	1	12	yes	complemented
84	C₂⁴×C₅	C₂³×C₃	1	6	yes	—
85	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₅×D₄	—	2	96	no	—
87	C₂×C₅×D₄	—	2	48	no	—
88	C₂×C₅×D₄	—	2	24	no	—
89	C₂²×C₄×C₅	—	2	12	no	—
90	C₂²×C₄×C₅	—	2	6	no	—
91	C₂²×C₄×C₅	C₃×D₄	1	12	yes	complemented
92	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
94	C₅×C₂²⋊C₄	—	2	24	no	—
95	C₂³×D₄	C₂×C₃×C₅	1	3	yes	complemented
96	C₂²×C₂²⋊C₄	C₂×C₃×C₅	1	3	yes	complemented
97	#392	C₂×C₃×C₅	1	8	yes	direct product factor
98	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	C₂²×C₃×C₅	—	2	24	no	—
100	C₂²×C₃×C₅	—	2	12	no	—
101	C₂²×C₃×C₅	—	4	6	no	—
102	C₂²×C₃×C₅	—	4	12	no	—
103	C₂²×C₃×C₅	—	4	6	no	—
104	C₂²×C₃×C₅	—	4	48	no	—
105	C₂²×C₃×C₅	#120	1	12	yes	—
106	C₂²×C₃×C₅	#121	1	12	yes	—
107	C₂²×C₃×C₅	#127	1	4	yes	—
108	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
109	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
110	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
111	C₃×C₄×C₅	—	2	12	no	—
112	C₃×C₄×C₅	—	4	2	no	—
113	C₂⁴×C₃	—	2	6	no	—
114	C₂⁴×C₃	—	2	12	no	—
115	C₂⁴×C₃	—	2	12	no	—
116	C₂⁴×C₃	—	2	6	no	—
117	C₂⁴×C₃	C₅×D₄	1	12	yes	complemented
118	C₂⁴×C₃	C₂³×C₅	1	6	yes	—
119	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
120	C₂×C₃×D₄	—	2	96	no	—
121	C₂×C₃×D₄	—	2	48	no	—
122	C₂×C₃×D₄	—	2	24	no	—
123	C₂²×C₃×C₄	—	2	12	no	—
124	C₂²×C₃×C₄	—	2	6	no	—
125	C₂²×C₃×C₄	C₅×D₄	1	12	yes	complemented
126	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
127	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
128	C₃×C₂²⋊C₄	—	2	24	no	—
129	C₂³×C₅	—	2	48	no	—
130	C₂³×C₅	—	2	24	no	—
131	C₂³×C₅	—	2	6	no	—
132	C₂³×C₅	—	2	12	no	—
133	C₂³×C₅	—	2	12	no	—
134	C₂³×C₅	—	2	6	no	—
135	C₂³×C₅	—	4	12	no	—
136	C₂³×C₅	—	4	24	no	—
137	C₂³×C₅	—	4	12	no	—
138	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
139	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
140	C₂³×C₅	C₂×C₃×D₄	1	12	yes	complemented
141	C₂³×C₅	C₃×#41	1	2	yes	—
142	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
143	C₅×D₄	—	2	96	no	—
144	C₂×C₄×C₅	—	2	24	no	—
145	C₂×C₄×C₅	—	2	12	no	—
146	C₂×C₄×C₅	—	2	6	no	—
147	C₂×C₄×C₅	—	4	4	no	—
148	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	complemented
149	C₂⁵	C₂²×C₃×C₅	1	3	yes	—
150	C₂²×D₄	—	2	24	no	—
151	C₂²×D₄	—	2	12	no	—
152	C₂²×D₄	C₂²×C₃×C₅	1	12	yes	complemented
153	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
155	C₂×C₂²⋊C₄	—	2	12	no	—
156	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	complemented
157	#95	C₂²×C₃×C₅	1	4	yes	complemented
158	C₂×C₃×C₅	—	4	12	no	—
159	C₂×C₃×C₅	#392	1	8	yes	direct product factor
160	C₂×C₃×C₅	C₂×#120	1	3	yes	—
161	C₂×C₃×C₅	C₂×#121	1	3	yes	—
162	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
163	C₂³×C₃	—	2	48	no	—
164	C₂³×C₃	—	2	24	no	—
165	C₂³×C₃	—	2	6	no	—
166	C₂³×C₃	—	2	12	no	—
167	C₂³×C₃	—	2	12	no	—
168	C₂³×C₃	—	2	6	no	—
169	C₂³×C₃	—	4	12	no	—
170	C₂³×C₃	—	4	24	no	—
171	C₂³×C₃	—	4	12	no	—
172	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
173	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
174	C₂³×C₃	C₂×C₅×D₄	1	12	yes	complemented
175	C₂³×C₃	C₅×#41	1	2	yes	—
176	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
177	C₃×D₄	—	2	96	no	—
178	C₂×C₃×C₄	—	2	24	no	—
179	C₂×C₃×C₄	—	2	12	no	—
180	C₂×C₃×C₄	—	2	6	no	—
181	C₂×C₃×C₄	—	4	4	no	—
182	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	complemented
183	C₂²×C₅	—	2	24	no	—
184	C₂²×C₅	—	2	12	no	—
185	C₂²×C₅	—	4	48	no	—
186	C₂²×C₅	—	4	6	no	—
187	C₂²×C₅	—	4	12	no	—
188	C₂²×C₅	—	4	6	no	—
189	C₂²×C₅	C₃×#120	1	12	yes	—
190	C₂²×C₅	C₃×#121	1	12	yes	—
191	C₂²×C₅	C₃×#127	1	4	yes	—
192	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
193	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
194	C₂²×C₅	C₂×C₃×#41	1	1	yes	characteristic
195	C₄×C₅	—	2	12	no	—
196	C₄×C₅	—	4	2	no	—
197	C₂⁴	—	2	6	no	—
198	C₂⁴	—	2	12	no	—
199	C₂⁴	—	2	12	no	—
200	C₂⁴	—	2	6	no	—
201	C₂⁴	C₃×C₅×D₄	1	12	yes	complemented
202	C₂⁴	C₂³×C₃×C₅	1	6	yes	—
203	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
204	C₂×D₄	—	2	96	no	—
205	C₂×D₄	—	2	48	no	—
206	C₂×D₄	—	2	24	no	—
207	C₂²×C₄	—	2	12	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	C₃×C₅×D₄	1	12	yes	complemented
210	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
211	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
212	C₂²⋊C₄	—	2	24	no	—
213	C₃×C₅	C₂×#392	1	1	yes	characteristic, direct product factor, Hall
214	C₂²×C₃	—	2	24	no	—
215	C₂²×C₃	—	2	12	no	—
216	C₂²×C₃	—	4	48	no	—
217	C₂²×C₃	—	4	6	no	—
218	C₂²×C₃	—	4	12	no	—
219	C₂²×C₃	—	4	6	no	—
220	C₂²×C₃	C₅×#120	1	12	yes	—
221	C₂²×C₃	C₅×#121	1	12	yes	—
222	C₂²×C₃	C₅×#127	1	4	yes	—
223	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
224	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
225	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
226	C₃×C₄	—	2	12	no	—
227	C₃×C₄	—	4	2	no	—
228	C₂×C₅	—	4	12	no	—
229	C₂×C₅	C₃×#392	1	8	yes	direct product factor
230	C₂×C₅	C₂×C₃×#120	1	3	yes	—
231	C₂×C₅	C₂×C₃×#121	1	3	yes	—
232	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
233	C₂³	—	2	48	no	—
234	C₂³	—	2	6	no	—
235	C₂³	—	2	12	no	—
236	C₂³	—	2	12	no	—
237	C₂³	—	2	24	no	—
238	C₂³	—	2	6	no	—
239	C₂³	—	4	12	no	—
240	C₂³	—	4	24	no	—
241	C₂³	—	4	12	no	—
242	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
243	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
244	C₂³	C₂×C₃×C₅×D₄	1	12	yes	complemented
245	C₂³	C₃×C₅×#41	1	2	yes	—
246	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
247	D₄	—	2	96	no	—
248	C₂×C₄	—	2	24	no	—
249	C₂×C₄	—	2	12	no	—
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	complemented
253	C₂×C₃	—	4	12	no	—
254	C₂×C₃	C₅×#392	1	8	yes	direct product factor
255	C₂×C₃	C₂×C₅×#120	1	3	yes	—
256	C₂×C₃	C₂×C₅×#121	1	3	yes	—
257	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
258	C₅	C₂×C₃×#392	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	24	no	—
260	C₂²	—	2	12	no	—
261	C₂²	—	4	6	no	—
262	C₂²	—	4	12	no	—
263	C₂²	—	4	6	no	—
264	C₂²	—	4	48	no	—
265	C₂²	C₃×C₅×#120	1	12	yes	—
266	C₂²	C₃×C₅×#121	1	12	yes	—
267	C₂²	C₃×C₅×#127	1	4	yes	—
268	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
269	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
270	C₂²	C₂×C₃×C₅×#41	1	1	yes	characteristic
271	C₄	—	2	12	no	—
272	C₄	—	4	2	no	—
273	C₃	C₂×C₅×#392	1	1	yes	characteristic, direct product factor, Sylow
274	C₂	—	4	12	no	—
275	C₂	C₃×C₅×#392	1	8	yes	direct product factor
276	C₂	C₂×C₃×C₅×#120	1	3	yes	—
277	C₂	C₂×C₃×C₅×#121	1	3	yes	—
278	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₄	C₂	semidirect
2. C₂²×C₃×C₅×C₂²⋊C₄	C₂	semidirect
3. C₃×C₅×#392	C₂	direct (N × H)
4. C₂×C₃×C₅×#95	C₂	semidirect
5. C₂×C₅×#392	C₃	direct (N × H)
9. C₂²×C₃×C₅×D₄	C₂²	semidirect
13. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
14. C₃×C₅×#95	C₂²	semidirect
15. C₂×C₃×#392	C₅	direct (N × H)
16. C₂³×C₅×D₄	C₂×C₃	semidirect
17. C₂²×C₅×C₂²⋊C₄	C₂×C₃	semidirect
18. C₅×#392	C₂×C₃	direct (N × H)
19. C₂×C₅×#95	C₂×C₃	semidirect
24. C₂⁴×C₃×C₅	D₄	semidirect
32. C₂²×C₃×C₄×C₅	D₄	semidirect
36. C₂³×C₃×D₄	C₂×C₅	semidirect
37. C₂²×C₃×C₂²⋊C₄	C₂×C₅	semidirect
38. C₃×#392	C₂×C₅	direct (N × H)
39. C₂×C₃×#95	C₂×C₅	semidirect
43. C₂²×C₅×D₄	C₂²×C₃	semidirect
47. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
48. C₅×#95	C₂²×C₃	semidirect
49. C₂×#392	C₃×C₅	direct (N × H)
61. C₂³×C₃×C₅	C₂×D₄	semidirect
69. C₂×C₃×C₄×C₅	C₂×D₄	semidirect
73. C₂²×C₃×D₄	C₂²×C₅	semidirect
77. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
78. C₃×#95	C₂²×C₅	semidirect
83. C₂⁴×C₅	C₃×D₄	semidirect
91. C₂²×C₄×C₅	C₃×D₄	semidirect
95. C₂³×D₄	C₂×C₃×C₅	semidirect
96. C₂²×C₂²⋊C₄	C₂×C₃×C₅	semidirect
97. #392	C₂×C₃×C₅	direct (N × H)
98. C₂×#95	C₂×C₃×C₅	semidirect
117. C₂⁴×C₃	C₅×D₄	semidirect
125. C₂²×C₃×C₄	C₅×D₄	semidirect
140. C₂³×C₅	C₂×C₃×D₄	semidirect
148. C₂×C₄×C₅	C₂×C₃×D₄	semidirect
152. C₂²×D₄	C₂²×C₃×C₅	semidirect
156. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
157. #95	C₂²×C₃×C₅	semidirect
159. C₂×C₃×C₅	#392	direct (N × H)
174. C₂³×C₃	C₂×C₅×D₄	semidirect
182. C₂×C₃×C₄	C₂×C₅×D₄	semidirect
201. C₂⁴	C₃×C₅×D₄	semidirect
209. C₂²×C₄	C₃×C₅×D₄	semidirect
213. C₃×C₅	C₂×#392	direct (N × H)
229. C₂×C₅	C₃×#392	direct (N × H)
244. C₂³	C₂×C₃×C₅×D₄	semidirect
252. C₂×C₄	C₂×C₃×C₅×D₄	semidirect
254. C₂×C₃	C₅×#392	direct (N × H)
258. C₅	C₂×C₃×#392	direct (N × H)
273. C₃	C₂×C₅×#392	direct (N × H)
275. C₂	C₃×C₅×#392	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₄	2
C₂²×C₃×C₅×C₂²⋊C₄	2
C₃×C₅×#392	2
C₂×C₃×C₅×#95	2
C₂×C₅×#392	3
C₂×C₃×#392	5

Subgroups — C2×C3×C5×#392 #14275418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#392 #14275418