Subgroups — C₂×C₃×C₅×#357 #14275170

Order 1920.

544 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	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#357	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	—	2	1	no	—
7	C₂×C₃×C₅×D₈	—	2	1	no	—
8	C₂×C₃×C₅×D₈	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₅×C₁₆	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
15	C₃×C₅×#107	—	2	1	no	—
16	C₃×C₅×#107	C₂²	1	2	yes	complemented
17	C₂×C₃×#357	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×D₈	C₂×C₃	1	1	yes	characteristic
19	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#357	C₂×C₃	1	4	yes	direct product factor
22	C₂⁴×C₃×C₅	—	4	1	no	—
23	C₂×C₃×C₅×D₄	—	2	4	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	4	1	no	—
26	C₂×C₃×C₅×D₄	—	4	2	no	—
27	C₂²×C₃×C₄×C₅	—	4	1	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₃×C₅×D₈	—	2	4	no	—
30	C₃×C₅×D₈	—	2	2	no	—
31	C₃×C₅×D₈	C₂×C₄	1	4	yes	complemented
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	4	1	no	—
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂²×C₃×D₈	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#357	C₂×C₅	1	4	yes	direct product factor
43	C₂²×C₅×D₄	—	2	1	no	—
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	C₃×C₄	1	4	yes	complemented
47	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
48	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₁₆	—	2	2	no	—
51	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	complemented
52	C₅×#107	—	2	1	no	—
53	C₅×#107	C₂²×C₃	1	2	yes	complemented
54	C₂×#357	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	4	4	no	—
56	C₂³×C₃×C₅	—	4	2	no	—
57	C₂³×C₃×C₅	—	8	2	no	—
58	C₂³×C₃×C₅	—	8	2	no	—
59	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
60	C₃×C₅×D₄	—	2	4	no	—
61	C₃×C₅×D₄	—	4	4	no	—
62	C₃×C₅×D₄	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	—	8	2	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₃×D₄	—	2	1	no	—
73	C₂×C₃×D₈	—	2	1	no	—
74	C₂×C₃×D₈	—	2	2	no	—
75	C₂×C₃×D₈	C₄×C₅	1	4	yes	complemented
76	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
77	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×C₄⋊C₄	—	2	1	no	—
79	C₂×C₃×C₁₆	—	2	2	no	—
80	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
81	C₃×#107	—	2	1	no	—
82	C₃×#107	C₂²×C₅	1	2	yes	complemented
83	C₂⁴×C₅	—	4	1	no	—
84	C₂×C₅×D₄	—	2	4	no	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	4	1	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂²×C₄×C₅	—	4	1	no	—
89	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
90	C₅×D₈	—	2	4	no	—
91	C₅×D₈	—	2	2	no	—
92	C₅×D₈	C₂×C₃×C₄	1	4	yes	complemented
93	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₁₆	—	2	2	no	—
100	C₂²×D₈	C₂×C₃×C₅	1	1	yes	characteristic
101	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic, complemented
103	#357	C₂×C₃×C₅	1	4	yes	direct product factor
104	C₂²×C₃×C₅	—	4	4	no	—
105	C₂²×C₃×C₅	—	8	8	no	—
106	C₂²×C₃×C₅	—	8	2	no	—
107	C₂²×C₃×C₅	—	8	2	no	—
108	C₂²×C₃×C₅	#102	1	2	yes	—
109	C₂²×C₃×C₅	D₁₆	1	2	yes	—
110	C₂²×C₃×C₅	SD₃₂	1	2	yes	—
111	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
112	C₃×C₄×C₅	—	8	2	no	—
113	C₃×C₄×C₅	#102	1	2	yes	—
114	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂⁴×C₃	—	4	1	no	—
117	C₂×C₃×D₄	—	2	4	no	—
118	C₂×C₃×D₄	—	2	2	no	—
119	C₂×C₃×D₄	—	4	1	no	—
120	C₂×C₃×D₄	—	4	2	no	—
121	C₂²×C₃×C₄	—	4	1	no	—
122	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
123	C₃×D₈	—	2	4	no	—
124	C₃×D₈	—	2	2	no	—
125	C₃×D₈	C₂×C₄×C₅	1	4	yes	complemented
126	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
128	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
129	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄⋊C₄	—	4	1	no	—
132	C₃×C₁₆	—	2	2	no	—
133	C₂³×C₅	—	4	4	no	—
134	C₂³×C₅	—	4	2	no	—
135	C₂³×C₅	—	8	2	no	—
136	C₂³×C₅	—	8	2	no	—
137	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
138	C₅×D₄	—	2	4	no	—
139	C₅×D₄	—	4	4	no	—
140	C₅×D₄	—	4	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	8	2	no	—
143	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
148	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
149	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂²×D₄	—	2	1	no	—
151	C₂×D₈	—	2	1	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	C₃×C₄×C₅	1	4	yes	complemented
154	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
155	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	complemented
159	#107	—	2	1	no	—
160	#107	C₂²×C₃×C₅	1	2	yes	complemented
161	C₂×C₃×C₅	—	8	4	no	—
162	C₂×C₃×C₅	#357	1	4	yes	direct product factor
163	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
166	C₂³×C₃	—	4	2	no	—
167	C₂³×C₃	—	4	4	no	—
168	C₂³×C₃	—	8	2	no	—
169	C₂³×C₃	—	8	2	no	—
170	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
171	C₃×D₄	—	2	4	no	—
172	C₃×D₄	—	4	2	no	—
173	C₃×D₄	—	4	4	no	—
174	C₂×C₃×C₄	—	4	2	no	—
175	C₂×C₃×C₄	—	8	2	no	—
176	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
177	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
178	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
181	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
182	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂²×C₅	—	4	4	no	—
184	C₂²×C₅	—	8	2	no	—
185	C₂²×C₅	—	8	2	no	—
186	C₂²×C₅	—	8	8	no	—
187	C₂²×C₅	C₃×#102	1	2	yes	—
188	C₂²×C₅	C₃×D₁₆	1	2	yes	—
189	C₂²×C₅	C₃×SD₃₂	1	2	yes	—
190	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
191	C₄×C₅	—	8	2	no	—
192	C₄×C₅	C₃×#102	1	2	yes	—
193	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
194	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
195	C₂⁴	—	4	1	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
202	D₈	—	2	4	no	—
203	D₈	—	2	2	no	—
204	D₈	C₂×C₃×C₄×C₅	1	4	yes	complemented
205	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
206	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
207	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
208	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₁₆	—	2	2	no	—
212	C₃×C₅	C₂×#357	1	1	yes	characteristic, direct product factor, Hall
213	C₂²×C₃	—	4	4	no	—
214	C₂²×C₃	—	8	8	no	—
215	C₂²×C₃	—	8	2	no	—
216	C₂²×C₃	—	8	2	no	—
217	C₂²×C₃	C₅×#102	1	2	yes	—
218	C₂²×C₃	C₅×D₁₆	1	2	yes	—
219	C₂²×C₃	C₅×SD₃₂	1	2	yes	—
220	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
221	C₃×C₄	—	8	2	no	—
222	C₃×C₄	C₅×#102	1	2	yes	—
223	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
225	C₂×C₅	—	8	4	no	—
226	C₂×C₅	C₃×#357	1	4	yes	direct product factor
227	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
228	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
229	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
230	C₂³	—	4	2	no	—
231	C₂³	—	4	4	no	—
232	C₂³	—	8	2	no	—
233	C₂³	—	8	2	no	—
234	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
235	D₄	—	2	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
241	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
242	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
243	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
244	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
245	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
246	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
247	C₂×C₃	—	8	4	no	—
248	C₂×C₃	C₅×#357	1	4	yes	direct product factor
249	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
250	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
251	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
252	C₅	C₂×C₃×#357	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	4	4	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	8	no	—
257	C₂²	C₃×C₅×#102	1	2	yes	—
258	C₂²	C₃×C₅×D₁₆	1	2	yes	—
259	C₂²	C₃×C₅×SD₃₂	1	2	yes	—
260	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
261	C₄	—	8	2	no	—
262	C₄	C₃×C₅×#102	1	2	yes	—
263	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
265	C₃	C₂×C₅×#357	1	1	yes	characteristic, direct product factor, Sylow
266	C₂	—	8	4	no	—
267	C₂	C₃×C₅×#357	1	4	yes	direct product factor
268	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
269	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
270	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂²×C₃×C₅×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₂×C₃×C₅×#107	C₂	semidirect
4. C₃×C₅×#357	C₂	direct (N × H)
5. C₂×C₅×#357	C₃	direct (N × H)
9. C₂×C₃×C₅×D₈	C₄	semidirect
14. C₂×C₃×C₅×C₁₆	C₂²	semidirect
16. C₃×C₅×#107	C₂²	semidirect
17. C₂×C₃×#357	C₅	direct (N × H)
19. C₂²×C₅×C₁₆	C₂×C₃	semidirect
20. C₂×C₅×#107	C₂×C₃	semidirect
21. C₅×#357	C₂×C₃	direct (N × H)
31. C₃×C₅×D₈	C₂×C₄	semidirect
40. C₂²×C₃×C₁₆	C₂×C₅	semidirect
41. C₂×C₃×#107	C₂×C₅	semidirect
42. C₃×#357	C₂×C₅	direct (N × H)
46. C₂×C₅×D₈	C₃×C₄	semidirect
51. C₂×C₅×C₁₆	C₂²×C₃	semidirect
53. C₅×#107	C₂²×C₃	semidirect
54. C₂×#357	C₃×C₅	direct (N × H)
75. C₂×C₃×D₈	C₄×C₅	semidirect
80. C₂×C₃×C₁₆	C₂²×C₅	semidirect
82. C₃×#107	C₂²×C₅	semidirect
92. C₅×D₈	C₂×C₃×C₄	semidirect
101. C₂²×C₁₆	C₂×C₃×C₅	semidirect
102. C₂×#107	C₂×C₃×C₅	semidirect
103. #357	C₂×C₃×C₅	direct (N × H)
125. C₃×D₈	C₂×C₄×C₅	semidirect
153. C₂×D₈	C₃×C₄×C₅	semidirect
158. C₂×C₁₆	C₂²×C₃×C₅	semidirect
160. #107	C₂²×C₃×C₅	semidirect
162. C₂×C₃×C₅	#357	direct (N × H)
204. D₈	C₂×C₃×C₄×C₅	semidirect
212. C₃×C₅	C₂×#357	direct (N × H)
226. C₂×C₅	C₃×#357	direct (N × H)
248. C₂×C₃	C₅×#357	direct (N × H)
252. C₅	C₂×C₃×#357	direct (N × H)
265. C₃	C₂×C₅×#357	direct (N × H)
267. C₂	C₃×C₅×#357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×D₈	2
C₂²×C₃×C₅×C₁₆	2
C₂×C₃×C₅×#107	2
C₃×C₅×#357	2
C₂×C₅×#357	3
C₂×C₃×#357	5

Subgroups — C2×C3×C5×#357 #14275170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#357 #14275170