Subgroups — C₃×C₅×#3139 #14276173

Order 1920.

760 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	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#491	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#487	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#419	C₂	1	4	yes	maximal
6	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₅×#3139	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₃×C₅×#125	C₂²	1	2	yes	—
15	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
16	C₃×C₄×C₅×Q₈	—	2	2	no	—
17	C₃×C₅×#117	C₂²	1	2	yes	—
18	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	C₂²	1	16	yes	—
21	C₃×C₅×#97	C₂²	1	2	yes	—
22	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
23	C₃×#3139	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₅×#533	C₂×C₃	1	1	yes	characteristic
25	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
26	C₅×#491	C₂×C₃	1	4	yes	complemented
27	C₅×#487	C₂×C₃	1	4	yes	complemented
28	C₅×#419	C₂×C₃	1	4	yes	—
29	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×Q₈	—	2	8	no	—
33	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	4	no	—
38	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	4	no	—
40	C₃×C₅×C₄⋊C₄	—	2	4	no	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₃×C₄²×C₅	—	2	2	no	—
46	C₃×C₄²×C₅	D₄	1	2	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×#533	C₂×C₅	1	1	yes	characteristic
50	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
51	C₃×#491	C₂×C₅	1	4	yes	complemented
52	C₃×#487	C₂×C₅	1	4	yes	complemented
53	C₃×#419	C₂×C₅	1	4	yes	—
54	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
55	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
56	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
57	C₅×C₄⋊Q₈	—	2	2	no	—
58	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
59	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
60	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
61	C₅×#125	C₂²×C₃	1	2	yes	—
62	C₅×C₂²⋊Q₈	—	2	2	no	—
63	C₄×C₅×Q₈	—	2	2	no	—
64	C₅×#117	C₂²×C₃	1	2	yes	—
65	C₂×C₅×C₄⋊C₄	—	2	2	no	—
66	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₅×#103	C₂²×C₃	1	16	yes	—
68	C₅×#97	C₂²×C₃	1	2	yes	—
69	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
70	#3139	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
71	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×Q₈	—	2	8	no	—
73	C₃×C₅×Q₈	—	4	4	no	—
74	C₃×C₅×Q₈	—	4	4	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	4	4	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	#41	1	4	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₃×C₅×C₈	—	2	4	no	—
85	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
86	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
87	C₃×C₄⋊Q₈	—	2	2	no	—
88	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
89	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
90	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
91	C₃×#125	C₂²×C₅	1	2	yes	—
92	C₃×C₂²⋊Q₈	—	2	2	no	—
93	C₃×C₄×Q₈	—	2	2	no	—
94	C₃×#117	C₂²×C₅	1	2	yes	—
95	C₂×C₃×C₄⋊C₄	—	2	2	no	—
96	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	C₃×#103	C₂²×C₅	1	16	yes	—
98	C₃×#97	C₂²×C₅	1	2	yes	—
99	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	—	2	2	no	—
102	C₂×C₅×Q₈	—	2	8	no	—
103	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
106	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
107	C₅×M₁₆	—	2	4	no	—
108	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	C₃×D₄	1	2	yes	—
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
119	#533	C₂×C₃×C₅	1	1	yes	characteristic
120	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
121	#491	C₂×C₃×C₅	1	4	yes	complemented
122	#487	C₂×C₃×C₅	1	4	yes	complemented
123	#419	C₂×C₃×C₅	1	4	yes	—
124	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	#124	1	2	yes	—
127	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₃×C₄×C₅	—	2	2	no	—
129	C₃×C₄×C₅	—	4	4	no	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	#124	1	2	yes	—
132	C₃×C₄×C₅	#137	1	4	yes	—
133	C₃×C₄×C₅	C₂×#41	1	2	yes	—
134	C₂×C₃×Q₈	—	2	2	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×Q₈	—	2	8	no	—
137	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
138	C₂²×C₃×C₄	—	2	2	no	—
139	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
140	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
141	C₃×M₁₆	—	2	4	no	—
142	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₄⋊C₄	—	2	4	no	—
146	C₃×C₄⋊C₄	—	2	2	no	—
147	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
148	C₃×C₂²⋊C₄	—	2	2	no	—
149	C₃×C₄²	—	2	2	no	—
150	C₃×C₄²	C₅×D₄	1	2	yes	—
151	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
152	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
153	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₅×Q₈	—	2	8	no	—
155	C₅×Q₈	—	4	4	no	—
156	C₅×Q₈	—	4	4	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	4	no	—
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
164	C₂×C₄×C₅	C₃×#41	1	4	yes	—
165	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
166	C₅×C₈	—	2	4	no	—
167	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
168	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
171	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
172	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
173	#125	C₂²×C₃×C₅	1	2	yes	—
174	C₂²⋊Q₈	—	2	2	no	—
175	C₄×Q₈	—	2	2	no	—
176	#117	C₂²×C₃×C₅	1	2	yes	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
179	#103	C₂²×C₃×C₅	1	16	yes	—
180	#97	C₂²×C₃×C₅	1	2	yes	—
181	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
182	C₂×C₃×C₅	—	2	2	no	—
183	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
184	C₂×C₃×C₅	C₂×#137	1	2	yes	—
185	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
186	C₃×Q₈	—	2	8	no	—
187	C₃×Q₈	—	4	4	no	—
188	C₃×Q₈	—	4	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	4	no	—
192	C₂×C₃×C₄	—	4	4	no	—
193	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
196	C₂×C₃×C₄	C₅×#41	1	4	yes	—
197	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
198	C₃×C₈	—	2	4	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	C₃×#124	1	2	yes	—
201	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	C₃×#124	1	2	yes	—
206	C₄×C₅	C₃×#137	1	4	yes	—
207	C₄×C₅	C₂×C₃×#41	1	2	yes	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
214	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	C₃×C₅×D₄	1	2	yes	—
225	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
226	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
227	C₃×C₅	#3139	1	1	yes	characteristic, direct product factor, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	C₅×#124	1	2	yes	—
230	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
231	C₃×C₄	—	2	2	no	—
232	C₃×C₄	—	4	4	no	—
233	C₃×C₄	—	4	4	no	—
234	C₃×C₄	C₅×#124	1	2	yes	—
235	C₃×C₄	C₅×#137	1	4	yes	—
236	C₃×C₄	C₂×C₅×#41	1	2	yes	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
239	C₂×C₅	C₂×C₃×#137	1	2	yes	—
240	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
241	Q₈	—	2	8	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
249	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
250	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
251	C₂×C₄	C₃×C₅×#41	1	4	yes	—
252	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
256	C₂×C₃	C₂×C₅×#137	1	2	yes	—
257	C₅	C₃×#3139	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	C₃×C₅×#124	1	2	yes	—
260	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	4	no	—
264	C₄	C₃×C₅×#124	1	2	yes	—
265	C₄	C₃×C₅×#137	1	4	yes	—
266	C₄	C₂×C₃×C₅×#41	1	2	yes	—
267	C₃	C₅×#3139	1	1	yes	characteristic, direct product factor, Sylow
268	C₂	—	2	2	no	—
269	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
270	C₂	C₂×C₃×C₅×#137	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#491	C₂	semidirect
4. C₃×C₅×#487	C₂	semidirect
7. C₅×#3139	C₃	direct (N × H)
23. C₃×#3139	C₅	direct (N × H)
26. C₅×#491	C₂×C₃	semidirect
27. C₅×#487	C₂×C₃	semidirect
51. C₃×#491	C₂×C₅	semidirect
52. C₃×#487	C₂×C₅	semidirect
70. #3139	C₃×C₅	direct (N × H)
121. #491	C₂×C₃×C₅	semidirect
122. #487	C₂×C₃×C₅	semidirect
227. C₃×C₅	#3139	direct (N × H)
257. C₅	C₃×#3139	direct (N × H)
267. C₃	C₅×#3139	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#533	2
C₂×C₃×C₅×C₄⋊Q₈	2
C₃×C₅×#491	2
C₃×C₅×#487	2
C₃×C₅×#419	2
C₃×C₄×C₅×M₁₆	2
C₅×#3139	3
C₃×#3139	5

Subgroups — C3×C5×#3139 #14276173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3139 #14276173