Subgroups — C₃×C₅×#3185 #14276219

Order 1920.

716 conjugacy classes of subgroups.

The subgroup lattice has 276 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	2	yes	maximal
2	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#487	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#470	C₂	1	4	yes	maximal
6	C₃×C₅×#441	C₂	1	4	yes	maximal, complemented
7	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
8	C₅×#3185	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×#125	—	2	2	no	—
14	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
15	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₃×C₄×C₅×Q₈	—	2	2	no	—
17	C₃×C₄×C₅×Q₈	C₂²	1	4	yes	—
18	C₃×C₅×#117	—	2	2	no	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×#106	C₂²	1	4	yes	—
21	C₃×C₅×#103	C₂²	1	8	yes	—
22	C₃×C₅×#97	C₂²	1	2	yes	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
25	C₃×#3185	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₅×#533	C₂×C₃	1	2	yes	—
27	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
29	C₅×#487	C₂×C₃	1	2	yes	complemented
30	C₅×#470	C₂×C₃	1	4	yes	—
31	C₅×#441	C₂×C₃	1	4	yes	complemented
32	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×C₅×Q₈	—	2	2	no	—
34	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×Q₁₆	—	2	8	no	—
38	C₃×C₅×M₁₆	—	2	4	no	—
39	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	—	2	2	no	—
43	C₃×C₅×C₄⋊C₄	—	2	2	no	—
44	C₃×C₅×C₄⋊C₄	—	2	4	no	—
45	C₃×C₅×C₄⋊C₄	—	2	4	no	—
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
48	C₃×C₄²×C₅	—	2	4	no	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₃×#533	C₂×C₅	1	2	yes	—
53	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#487	C₂×C₅	1	2	yes	complemented
56	C₃×#470	C₂×C₅	1	4	yes	—
57	C₃×#441	C₂×C₅	1	4	yes	complemented
58	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
59	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
60	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
61	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
62	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
63	C₅×#125	—	2	2	no	—
64	C₅×C₂²⋊Q₈	—	2	2	no	—
65	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
66	C₄×C₅×Q₈	—	2	2	no	—
67	C₄×C₅×Q₈	C₂²×C₃	1	4	yes	—
68	C₅×#117	—	2	2	no	—
69	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
70	C₅×#106	C₂²×C₃	1	4	yes	—
71	C₅×#103	C₂²×C₃	1	8	yes	—
72	C₅×#97	C₂²×C₃	1	2	yes	—
73	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
74	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
75	#3185	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
76	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
77	C₃×C₅×Q₈	—	2	8	no	—
78	C₃×C₅×Q₈	—	4	4	no	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₃×C₅×C₈	—	2	2	no	—
88	C₃×C₅×C₈	#41	1	4	yes	—
89	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	—
90	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
91	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
92	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
93	C₃×#125	—	2	2	no	—
94	C₃×C₂²⋊Q₈	—	2	2	no	—
95	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
96	C₃×C₄×Q₈	—	2	2	no	—
97	C₃×C₄×Q₈	C₂²×C₅	1	4	yes	—
98	C₃×#117	—	2	2	no	—
99	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
100	C₃×#106	C₂²×C₅	1	4	yes	—
101	C₃×#103	C₂²×C₅	1	8	yes	—
102	C₃×#97	C₂²×C₅	1	2	yes	—
103	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
105	C₂×C₅×Q₈	—	2	2	no	—
106	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
107	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₅×Q₁₆	—	2	8	no	—
110	C₅×M₁₆	—	2	4	no	—
111	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
112	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	4	no	—
117	C₅×C₄⋊C₄	—	2	4	no	—
118	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
119	C₅×C₂²⋊C₄	—	2	4	no	—
120	C₄²×C₅	—	2	4	no	—
121	C₄²×C₅	C₃×D₄	1	2	yes	—
122	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
123	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
124	#533	C₂×C₃×C₅	1	2	yes	—
125	#498	C₂×C₃×C₅	1	1	yes	characteristic, complemented
126	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
127	#487	C₂×C₃×C₅	1	2	yes	complemented
128	#470	C₂×C₃×C₅	1	4	yes	—
129	#441	C₂×C₃×C₅	1	4	yes	complemented
130	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	—	2	2	no	—
133	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	2	no	—
136	C₃×C₄×C₅	—	4	4	no	—
137	C₃×C₄×C₅	—	4	4	no	—
138	C₃×C₄×C₅	#137	1	4	yes	—
139	C₃×C₄×C₅	C₂×#41	1	2	yes	—
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
142	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
143	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₃×Q₁₆	—	2	8	no	—
145	C₃×M₁₆	—	2	4	no	—
146	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
147	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
148	C₃×C₄⋊C₄	—	2	2	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄⋊C₄	—	2	4	no	—
153	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
154	C₃×C₂²⋊C₄	—	2	4	no	—
155	C₃×C₄²	—	2	4	no	—
156	C₃×C₄²	C₅×D₄	1	2	yes	—
157	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
158	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
159	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₅×Q₈	—	2	8	no	—
161	C₅×Q₈	—	4	4	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
169	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	C₃×#41	1	4	yes	—
172	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	—
173	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
174	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
175	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
176	#125	—	2	2	no	—
177	C₂²⋊Q₈	—	2	2	no	—
178	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
179	C₄×Q₈	—	2	2	no	—
180	C₄×Q₈	C₂²×C₃×C₅	1	4	yes	—
181	#117	—	2	2	no	—
182	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
183	#106	C₂²×C₃×C₅	1	4	yes	—
184	#103	C₂²×C₃×C₅	1	8	yes	—
185	#97	C₂²×C₃×C₅	1	2	yes	—
186	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
187	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	#532	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×#137	1	2	yes	—
191	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
192	C₃×Q₈	—	2	8	no	—
193	C₃×Q₈	—	4	4	no	—
194	C₂×C₃×C₄	—	2	4	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	4	no	—
197	C₂×C₃×C₄	—	2	4	no	—
198	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
201	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
202	C₃×C₈	—	2	2	no	—
203	C₃×C₈	C₅×#41	1	4	yes	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	1	no	—
206	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
207	C₄×C₅	—	2	2	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	4	4	no	—
210	C₄×C₅	—	4	4	no	—
211	C₄×C₅	C₃×#137	1	4	yes	—
212	C₄×C₅	C₂×C₃×#41	1	2	yes	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
215	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
216	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
217	Q₁₆	—	2	8	no	—
218	M₁₆	—	2	4	no	—
219	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
220	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	C₃×C₅×D₄	1	2	yes	—
230	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
231	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
232	C₃×C₅	#3185	1	1	yes	characteristic, direct product factor, Hall
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	2	no	—
235	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	2	2	no	—
238	C₃×C₄	—	4	4	no	—
239	C₃×C₄	—	4	4	no	—
240	C₃×C₄	C₅×#137	1	4	yes	—
241	C₃×C₄	C₂×C₅×#41	1	2	yes	—
242	C₂×C₅	—	4	1	no	—
243	C₂×C₅	C₃×#532	1	1	yes	characteristic
244	C₂×C₅	C₂×C₃×#137	1	2	yes	—
245	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
246	Q₈	—	2	8	no	—
247	Q₈	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
253	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
255	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	C₃×C₅×#41	1	4	yes	—
258	C₂×C₃	—	4	1	no	—
259	C₂×C₃	C₅×#532	1	1	yes	characteristic
260	C₂×C₃	C₂×C₅×#137	1	2	yes	—
261	C₅	C₃×#3185	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²	—	2	1	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	4	no	—
268	C₄	—	4	4	no	—
269	C₄	C₃×C₅×#137	1	4	yes	—
270	C₄	C₂×C₃×C₅×#41	1	2	yes	—
271	C₃	C₅×#3185	1	1	yes	characteristic, direct product factor, Sylow
272	C₂	—	4	1	no	—
273	C₂	C₃×C₅×#532	1	1	yes	characteristic
274	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
2. C₃×C₅×#498	C₂	semidirect
3. C₃×C₅×#494	C₂	semidirect
4. C₃×C₅×#487	C₂	semidirect
6. C₃×C₅×#441	C₂	semidirect
8. C₅×#3185	C₃	direct (N × H)
25. C₃×#3185	C₅	direct (N × H)
27. C₅×#498	C₂×C₃	semidirect
28. C₅×#494	C₂×C₃	semidirect
29. C₅×#487	C₂×C₃	semidirect
31. C₅×#441	C₂×C₃	semidirect
53. C₃×#498	C₂×C₅	semidirect
54. C₃×#494	C₂×C₅	semidirect
55. C₃×#487	C₂×C₅	semidirect
57. C₃×#441	C₂×C₅	semidirect
75. #3185	C₃×C₅	direct (N × H)
125. #498	C₂×C₃×C₅	semidirect
126. #494	C₂×C₃×C₅	semidirect
127. #487	C₂×C₃×C₅	semidirect
129. #441	C₂×C₃×C₅	semidirect
232. C₃×C₅	#3185	direct (N × H)
261. C₅	C₃×#3185	direct (N × H)
271. C₃	C₅×#3185	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
C₃×C₅×#498	2
C₃×C₅×#494	2
C₃×C₅×#487	2
C₃×C₅×#470	2
C₃×C₅×#441	2
C₃×C₄×C₅×M₁₆	2
C₅×#3185	3
C₃×#3185	5

Subgroups — C3×C5×#3185 #14276219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3185 #14276219