Subgroups — C₃×C₅×#3150 #14276184

Order 1920.

1128 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	C₂×C₃×C₅×#135	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#532	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#497	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#496	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3150	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	4	yes	—
8	C₃×C₅×#135	—	2	16	no	—
9	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
10	C₂×C₃×C₅×SD₁₆	C₂²	1	8	yes	—
11	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₃×C₅×#127	C₂²	1	2	yes	—
15	C₃×C₅×#124	C₂²	1	4	yes	—
16	C₃×C₅×#121	—	2	4	no	—
17	C₃×C₄×C₅×D₄	—	2	4	no	—
18	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
19	C₃×C₅×#97	C₂²	1	4	yes	—
20	C₃×#3150	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×#135	C₂×C₃	1	4	yes	complemented
22	C₅×#532	C₂×C₃	1	2	yes	complemented
23	C₅×#497	C₂×C₃	1	4	yes	complemented
24	C₅×#496	C₂×C₃	1	4	yes	complemented
25	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×C₅×#41	—	2	16	no	—
27	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
28	C₂×C₃×C₅×D₄	—	2	4	no	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄×C₅	—	2	4	no	—
32	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×Q₁₆	—	2	8	no	—
35	C₃×C₅×SD₁₆	—	2	16	no	—
36	C₃×C₅×D₈	—	2	8	no	—
37	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
38	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	4	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
41	C₃×C₄²×C₅	D₄	1	2	yes	—
42	C₃×C₄²×C₅	C₂³	1	2	yes	—
43	C₂×C₃×#135	C₂×C₅	1	4	yes	complemented
44	C₃×#532	C₂×C₅	1	2	yes	complemented
45	C₃×#497	C₂×C₅	1	4	yes	complemented
46	C₃×#496	C₂×C₅	1	4	yes	complemented
47	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×C₅×#41	C₂²×C₃	1	4	yes	—
49	C₅×#135	—	2	16	no	—
50	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
51	C₂×C₅×SD₁₆	C₂²×C₃	1	8	yes	—
52	C₂×C₅×D₈	C₂²×C₃	1	4	yes	—
53	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
54	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
55	C₅×#127	C₂²×C₃	1	2	yes	—
56	C₅×#124	C₂²×C₃	1	4	yes	—
57	C₅×#121	—	2	4	no	—
58	C₄×C₅×D₄	—	2	4	no	—
59	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
60	C₅×#97	C₂²×C₃	1	4	yes	—
61	#3150	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	—	2	4	no	—
63	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
64	C₃×C₅×Q₈	—	2	8	no	—
65	C₃×C₅×D₄	—	2	8	no	—
66	C₃×C₅×D₄	—	4	8	no	—
67	C₃×C₅×D₄	—	4	8	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	4	2	no	—
70	C₂×C₃×C₄×C₅	—	4	8	no	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
76	C₂×C₃×#41	C₂²×C₅	1	4	yes	—
77	C₃×#135	—	2	16	no	—
78	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
79	C₂×C₃×SD₁₆	C₂²×C₅	1	8	yes	—
80	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
81	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
82	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
83	C₃×#127	C₂²×C₅	1	2	yes	—
84	C₃×#124	C₂²×C₅	1	4	yes	—
85	C₃×#121	—	2	4	no	—
86	C₃×C₄×D₄	—	2	4	no	—
87	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
88	C₃×#97	C₂²×C₅	1	4	yes	—
89	C₅×#41	—	2	16	no	—
90	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
91	C₂×C₅×D₄	—	2	4	no	—
92	C₂×C₅×D₄	—	2	4	no	—
93	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
94	C₂²×C₄×C₅	—	2	4	no	—
95	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
97	C₅×Q₁₆	—	2	8	no	—
98	C₅×SD₁₆	—	2	16	no	—
99	C₅×D₈	—	2	8	no	—
100	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
101	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₂²⋊C₄	—	2	8	no	—
104	C₄²×C₅	C₃×D₄	1	2	yes	—
105	C₄²×C₅	C₂³×C₃	1	2	yes	—
106	C₂×#135	C₂×C₃×C₅	1	4	yes	complemented
107	#532	C₂×C₃×C₅	1	2	yes	complemented
108	#497	C₂×C₃×C₅	1	4	yes	complemented
109	#496	C₂×C₃×C₅	1	4	yes	complemented
110	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	—	4	4	no	—
113	C₂²×C₃×C₅	—	4	8	no	—
114	C₂²×C₃×C₅	#127	1	2	yes	—
115	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
116	C₃×C₄×C₅	—	2	4	no	—
117	C₃×C₄×C₅	—	4	4	no	—
118	C₃×C₄×C₅	#127	1	2	yes	—
119	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
120	C₃×#41	—	2	16	no	—
121	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
122	C₂×C₃×D₄	—	2	4	no	—
123	C₂×C₃×D₄	—	2	4	no	—
124	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
125	C₂²×C₃×C₄	—	2	4	no	—
126	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
127	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
128	C₃×Q₁₆	—	2	8	no	—
129	C₃×SD₁₆	—	2	16	no	—
130	C₃×D₈	—	2	8	no	—
131	C₂×C₃×C₈	C₅×D₄	1	8	yes	complemented
132	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
133	C₃×C₄⋊C₄	—	2	4	no	—
134	C₃×C₂²⋊C₄	—	2	8	no	—
135	C₃×C₄²	C₅×D₄	1	2	yes	—
136	C₃×C₄²	C₂³×C₅	1	2	yes	—
137	C₂³×C₅	—	2	4	no	—
138	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₅×Q₈	—	2	8	no	—
140	C₅×D₄	—	2	8	no	—
141	C₅×D₄	—	4	8	no	—
142	C₅×D₄	—	4	8	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	4	2	no	—
145	C₂×C₄×C₅	—	4	8	no	—
146	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
147	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
149	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
150	C₅×C₈	C₂×C₃×D₄	1	8	yes	—
151	C₂×#41	C₂²×C₃×C₅	1	4	yes	—
152	#135	—	2	16	no	—
153	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	—
154	C₂×SD₁₆	C₂²×C₃×C₅	1	8	yes	—
155	C₂×D₈	C₂²×C₃×C₅	1	4	yes	—
156	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
157	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
158	#127	C₂²×C₃×C₅	1	2	yes	—
159	#124	C₂²×C₃×C₅	1	4	yes	—
160	#121	—	2	4	no	—
161	C₄×D₄	—	2	4	no	—
162	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
163	#97	C₂²×C₃×C₅	1	4	yes	—
164	C₂×C₃×C₅	—	2	2	no	—
165	C₂×C₃×C₅	—	8	4	no	—
166	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
167	C₂×C₃×C₅	#575	1	2	yes	—
168	C₂³×C₃	—	2	4	no	—
169	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
170	C₃×Q₈	—	2	8	no	—
171	C₃×D₄	—	2	8	no	—
172	C₃×D₄	—	4	8	no	—
173	C₃×D₄	—	4	8	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	—	4	2	no	—
176	C₂×C₃×C₄	—	4	8	no	—
177	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
178	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
180	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
181	C₃×C₈	C₂×C₅×D₄	1	8	yes	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	4	8	no	—
184	C₂²×C₅	—	4	4	no	—
185	C₂²×C₅	C₃×#127	1	2	yes	—
186	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
187	C₄×C₅	—	2	4	no	—
188	C₄×C₅	—	4	4	no	—
189	C₄×C₅	C₃×#127	1	2	yes	—
190	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
191	#41	—	2	16	no	—
192	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	2	4	no	—
195	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
198	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	16	no	—
201	D₈	—	2	8	no	—
202	C₂×C₈	C₃×C₅×D₄	1	8	yes	complemented
203	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
204	C₄⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	8	no	—
206	C₄²	C₃×C₅×D₄	1	2	yes	—
207	C₄²	C₂³×C₃×C₅	1	2	yes	—
208	C₃×C₅	#3150	1	1	yes	characteristic, direct product factor, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	4	4	no	—
211	C₂²×C₃	—	4	8	no	—
212	C₂²×C₃	C₅×#127	1	2	yes	—
213	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
214	C₃×C₄	—	2	4	no	—
215	C₃×C₄	—	4	4	no	—
216	C₃×C₄	C₅×#127	1	2	yes	—
217	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
218	C₂×C₅	—	2	2	no	—
219	C₂×C₅	—	8	4	no	—
220	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
221	C₂×C₅	C₃×#575	1	2	yes	—
222	C₂³	—	2	4	no	—
223	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
224	Q₈	—	2	8	no	—
225	D₄	—	2	8	no	—
226	D₄	—	4	8	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	8	no	—
231	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
232	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
233	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
235	C₈	C₂×C₃×C₅×D₄	1	8	yes	—
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	8	4	no	—
238	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
239	C₂×C₃	C₅×#575	1	2	yes	—
240	C₅	C₃×#3150	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	4	8	no	—
244	C₂²	C₃×C₅×#127	1	2	yes	—
245	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	C₃×C₅×#127	1	2	yes	—
249	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
250	C₃	C₅×#3150	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	8	4	no	—
253	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
254	C₂	C₃×C₅×#575	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#135	C₂	semidirect
2. C₃×C₅×#532	C₂	semidirect
3. C₃×C₅×#497	C₂	semidirect
4. C₃×C₅×#496	C₂	semidirect
5. C₂×C₃×C₅×#97	C₂	semidirect
6. C₅×#3150	C₃	direct (N × H)
20. C₃×#3150	C₅	direct (N × H)
21. C₂×C₅×#135	C₂×C₃	semidirect
22. C₅×#532	C₂×C₃	semidirect
23. C₅×#497	C₂×C₃	semidirect
24. C₅×#496	C₂×C₃	semidirect
25. C₂×C₅×#97	C₂×C₃	semidirect
37. C₂×C₃×C₅×C₈	D₄	semidirect
43. C₂×C₃×#135	C₂×C₅	semidirect
44. C₃×#532	C₂×C₅	semidirect
45. C₃×#497	C₂×C₅	semidirect
46. C₃×#496	C₂×C₅	semidirect
47. C₂×C₃×#97	C₂×C₅	semidirect
61. #3150	C₃×C₅	direct (N × H)
100. C₂×C₅×C₈	C₃×D₄	semidirect
106. C₂×#135	C₂×C₃×C₅	semidirect
107. #532	C₂×C₃×C₅	semidirect
108. #497	C₂×C₃×C₅	semidirect
109. #496	C₂×C₃×C₅	semidirect
110. C₂×#97	C₂×C₃×C₅	semidirect
131. C₂×C₃×C₈	C₅×D₄	semidirect
202. C₂×C₈	C₃×C₅×D₄	semidirect
208. C₃×C₅	#3150	direct (N × H)
240. C₅	C₃×#3150	direct (N × H)
250. C₃	C₅×#3150	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×#135	2
C₃×C₅×#532	2
C₃×C₅×#497	2
C₃×C₅×#496	2
C₂×C₃×C₅×#97	2
C₅×#3150	3
C₃×#3150	5

Subgroups — C3×C5×#3150 #14276184

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3150 #14276184