Subgroups — C₃×C₅×#3138 #14276172

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 300 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, complemented
2	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#490	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#486	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#418	C₂	1	4	yes	maximal, complemented
6	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3138	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×D₄	—	2	2	no	—
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
12	C₃×C₅×#127	—	2	2	no	—
13	C₃×C₅×#127	C₂²	1	2	yes	—
14	C₃×C₅×#127	C₂²	1	2	yes	—
15	C₃×C₅×#125	C₂²	1	2	yes	complemented
16	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
17	C₃×C₄×C₅×Q₈	—	2	2	no	—
18	C₃×C₅×#117	C₂²	1	2	yes	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×#102	C₂²	1	16	yes	complemented
21	C₃×C₅×#97	C₂²	1	2	yes	complemented
22	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
23	C₃×#3138	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₅×#533	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
26	C₅×#490	C₂×C₃	1	4	yes	complemented
27	C₅×#486	C₂×C₃	1	4	yes	complemented
28	C₅×#418	C₂×C₃	1	4	yes	complemented
29	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂⁴×C₃×C₅	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×D₄	—	2	4	no	—
33	C₂×C₃×C₅×D₄	—	2	4	no	—
34	C₂×C₃×C₅×D₄	—	2	2	no	—
35	C₂×C₃×C₅×D₄	—	2	8	no	—
36	C₂×C₃×C₅×D₄	—	4	8	no	—
37	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₅×M₁₆	—	2	4	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₅×C₄⋊C₄	—	2	2	no	—
44	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
45	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
46	C₃×C₄²×C₅	—	2	2	no	—
47	C₃×C₄²×C₅	D₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₃×#533	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
52	C₃×#490	C₂×C₅	1	4	yes	complemented
53	C₃×#486	C₂×C₅	1	4	yes	complemented
54	C₃×#418	C₂×C₅	1	4	yes	complemented
55	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
56	C₂²×C₅×D₄	—	2	2	no	—
57	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
58	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
59	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	complemented
60	C₅×#127	—	2	2	no	—
61	C₅×#127	C₂²×C₃	1	2	yes	—
62	C₅×#127	C₂²×C₃	1	2	yes	—
63	C₅×#125	C₂²×C₃	1	2	yes	complemented
64	C₅×C₂²⋊Q₈	—	2	2	no	—
65	C₄×C₅×Q₈	—	2	2	no	—
66	C₅×#117	C₂²×C₃	1	2	yes	—
67	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
68	C₅×#102	C₂²×C₃	1	16	yes	complemented
69	C₅×#97	C₂²×C₃	1	2	yes	complemented
70	C₄×C₅×C₈	C₂²×C₃	1	2	yes	complemented
71	#3138	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃×C₅	—	2	4	no	—
73	C₂³×C₃×C₅	—	4	8	no	—
74	C₂³×C₃×C₅	—	4	4	no	—
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₅×Q₈	—	4	4	no	—
77	C₃×C₅×D₄	—	2	8	no	—
78	C₃×C₅×D₄	—	4	8	no	—
79	C₃×C₅×D₄	—	4	4	no	—
80	C₃×C₅×D₄	—	4	8	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	#41	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₃×C₅×C₈	—	2	4	no	—
89	C₂²×C₃×D₄	—	2	2	no	—
90	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
91	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
92	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
93	C₃×#127	—	2	2	no	—
94	C₃×#127	C₂²×C₅	1	2	yes	—
95	C₃×#127	C₂²×C₅	1	2	yes	—
96	C₃×#125	C₂²×C₅	1	2	yes	complemented
97	C₃×C₂²⋊Q₈	—	2	2	no	—
98	C₃×C₄×Q₈	—	2	2	no	—
99	C₃×#117	C₂²×C₅	1	2	yes	—
100	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
101	C₃×#102	C₂²×C₅	1	16	yes	complemented
102	C₃×#97	C₂²×C₅	1	2	yes	complemented
103	C₃×C₄×C₈	C₂²×C₅	1	2	yes	complemented
104	C₂⁴×C₅	—	2	2	no	—
105	C₂×C₅×Q₈	—	2	2	no	—
106	C₂×C₅×D₄	—	2	2	no	—
107	C₂×C₅×D₄	—	2	4	no	—
108	C₂×C₅×D₄	—	2	4	no	—
109	C₂×C₅×D₄	—	2	8	no	—
110	C₂×C₅×D₄	—	4	8	no	—
111	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
112	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
113	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
114	C₅×M₁₆	—	2	4	no	—
115	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
116	C₅×C₄⋊C₄	—	2	4	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
119	C₅×C₂²⋊C₄	—	2	2	no	—
120	C₄²×C₅	—	2	2	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	1	yes	characteristic, complemented
125	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
126	#490	C₂×C₃×C₅	1	4	yes	complemented
127	#486	C₂×C₃×C₅	1	4	yes	complemented
128	#418	C₂×C₃×C₅	1	4	yes	complemented
129	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
130	C₂²×C₃×C₅	—	2	2	no	—
131	C₂²×C₃×C₅	—	4	4	no	—
132	C₂²×C₃×C₅	—	4	8	no	—
133	C₂²×C₃×C₅	—	8	8	no	—
134	C₂²×C₃×C₅	#124	1	2	yes	—
135	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	—	4	4	no	—
138	C₃×C₄×C₅	#124	1	2	yes	—
139	C₃×C₄×C₅	AGL(1,ℤ/8)	1	4	yes	—
140	C₃×C₄×C₅	C₂×#41	1	2	yes	—
141	C₂⁴×C₃	—	2	2	no	—
142	C₂×C₃×Q₈	—	2	2	no	—
143	C₂×C₃×D₄	—	2	2	no	—
144	C₂×C₃×D₄	—	2	4	no	—
145	C₂×C₃×D₄	—	2	4	no	—
146	C₂×C₃×D₄	—	2	8	no	—
147	C₂×C₃×D₄	—	4	8	no	—
148	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
149	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
150	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
151	C₃×M₁₆	—	2	4	no	—
152	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
153	C₃×C₄⋊C₄	—	2	4	no	—
154	C₃×C₄⋊C₄	—	2	2	no	—
155	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₄²	—	2	2	no	—
158	C₃×C₄²	C₅×D₄	1	2	yes	—
159	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
160	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
161	C₂³×C₅	—	2	4	no	—
162	C₂³×C₅	—	4	8	no	—
163	C₂³×C₅	—	4	4	no	—
164	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₅×Q₈	—	4	4	no	—
166	C₅×D₄	—	2	8	no	—
167	C₅×D₄	—	4	8	no	—
168	C₅×D₄	—	4	4	no	—
169	C₅×D₄	—	4	8	no	—
170	C₂×C₄×C₅	—	2	4	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	C₃×#41	1	4	yes	—
176	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
177	C₅×C₈	—	2	4	no	—
178	C₂²×D₄	—	2	2	no	—
179	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
180	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
181	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	complemented
182	#127	—	2	2	no	—
183	#127	C₂²×C₃×C₅	1	2	yes	—
184	#127	C₂²×C₃×C₅	1	2	yes	—
185	#125	C₂²×C₃×C₅	1	2	yes	complemented
186	C₂²⋊Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	#117	C₂²×C₃×C₅	1	2	yes	—
189	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
190	#102	C₂²×C₃×C₅	1	16	yes	complemented
191	#97	C₂²×C₃×C₅	1	2	yes	complemented
192	C₄×C₈	C₂²×C₃×C₅	1	2	yes	complemented
193	C₂×C₃×C₅	—	2	2	no	—
194	C₂×C₃×C₅	—	8	4	no	—
195	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
196	C₂×C₃×C₅	C₂×AGL(1,ℤ/8)	1	2	yes	—
197	C₂³×C₃	—	2	4	no	—
198	C₂³×C₃	—	4	8	no	—
199	C₂³×C₃	—	4	4	no	—
200	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
201	C₃×Q₈	—	4	4	no	—
202	C₃×D₄	—	2	8	no	—
203	C₃×D₄	—	4	8	no	—
204	C₃×D₄	—	4	4	no	—
205	C₃×D₄	—	4	8	no	—
206	C₂×C₃×C₄	—	2	4	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
211	C₂×C₃×C₄	C₅×#41	1	4	yes	—
212	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
213	C₃×C₈	—	2	4	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	4	8	no	—
216	C₂²×C₅	—	4	4	no	—
217	C₂²×C₅	—	8	8	no	—
218	C₂²×C₅	C₃×#124	1	2	yes	—
219	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	4	4	no	—
222	C₄×C₅	C₃×#124	1	2	yes	—
223	C₄×C₅	C₃×AGL(1,ℤ/8)	1	4	yes	—
224	C₄×C₅	C₂×C₃×#41	1	2	yes	—
225	C₂⁴	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	8	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
233	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
234	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	C₃×C₅×D₄	1	2	yes	—
243	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
244	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
245	C₃×C₅	#3138	1	1	yes	characteristic, direct product factor, Hall
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	4	4	no	—
248	C₂²×C₃	—	4	8	no	—
249	C₂²×C₃	—	8	8	no	—
250	C₂²×C₃	C₅×#124	1	2	yes	—
251	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	—	4	4	no	—
254	C₃×C₄	C₅×#124	1	2	yes	—
255	C₃×C₄	C₅×AGL(1,ℤ/8)	1	4	yes	—
256	C₃×C₄	C₂×C₅×#41	1	2	yes	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	8	4	no	—
259	C₂×C₅	C₂×C₃×#124	1	1	yes	characteristic
260	C₂×C₅	C₂×C₃×AGL(1,ℤ/8)	1	2	yes	—
261	C₂³	—	2	4	no	—
262	C₂³	—	4	8	no	—
263	C₂³	—	4	4	no	—
264	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
265	Q₈	—	4	4	no	—
266	D₄	—	2	8	no	—
267	D₄	—	4	8	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	8	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
273	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
275	C₂×C₄	C₃×C₅×#41	1	4	yes	—
276	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
277	C₈	—	2	4	no	—
278	C₂×C₃	—	2	2	no	—
279	C₂×C₃	—	8	4	no	—
280	C₂×C₃	C₂×C₅×#124	1	1	yes	characteristic
281	C₂×C₃	C₂×C₅×AGL(1,ℤ/8)	1	2	yes	—
282	C₅	C₃×#3138	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	—	2	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	—	4	8	no	—
286	C₂²	—	8	8	no	—
287	C₂²	C₃×C₅×#124	1	2	yes	—
288	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	4	4	no	—
291	C₄	C₃×C₅×#124	1	2	yes	—
292	C₄	C₃×C₅×AGL(1,ℤ/8)	1	4	yes	—
293	C₄	C₂×C₃×C₅×#41	1	2	yes	—
294	C₃	C₅×#3138	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	2	no	—
296	C₂	—	8	4	no	—
297	C₂	C₂×C₃×C₅×#124	1	1	yes	characteristic
298	C₂	C₂×C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂×C₃×C₅×#127	C₂
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
1. C₃×C₅×#533	C₂	semidirect
3. C₃×C₅×#490	C₂	semidirect
4. C₃×C₅×#486	C₂	semidirect
5. C₃×C₅×#418	C₂	semidirect
6. C₃×C₄×C₅×M₁₆	C₂	semidirect
7. C₅×#3138	C₃	direct (N × H)
11. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
15. C₃×C₅×#125	C₂²	semidirect
20. C₃×C₅×#102	C₂²	semidirect
21. C₃×C₅×#97	C₂²	semidirect
22. C₃×C₄×C₅×C₈	C₂²	semidirect
23. C₃×#3138	C₅	direct (N × H)
24. C₅×#533	C₂×C₃	semidirect
26. C₅×#490	C₂×C₃	semidirect
27. C₅×#486	C₂×C₃	semidirect
28. C₅×#418	C₂×C₃	semidirect
29. C₄×C₅×M₁₆	C₂×C₃	semidirect
50. C₃×#533	C₂×C₅	semidirect
52. C₃×#490	C₂×C₅	semidirect
53. C₃×#486	C₂×C₅	semidirect
54. C₃×#418	C₂×C₅	semidirect
55. C₃×C₄×M₁₆	C₂×C₅	semidirect
59. C₅×C₄⋊Q₈	C₂²×C₃	semidirect
63. C₅×#125	C₂²×C₃	semidirect
68. C₅×#102	C₂²×C₃	semidirect
69. C₅×#97	C₂²×C₃	semidirect
70. C₄×C₅×C₈	C₂²×C₃	semidirect
71. #3138	C₃×C₅	direct (N × H)
92. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
96. C₃×#125	C₂²×C₅	semidirect
101. C₃×#102	C₂²×C₅	semidirect
102. C₃×#97	C₂²×C₅	semidirect
103. C₃×C₄×C₈	C₂²×C₅	semidirect
124. #533	C₂×C₃×C₅	semidirect
126. #490	C₂×C₃×C₅	semidirect
127. #486	C₂×C₃×C₅	semidirect
128. #418	C₂×C₃×C₅	semidirect
129. C₄×M₁₆	C₂×C₃×C₅	semidirect
181. C₄⋊Q₈	C₂²×C₃×C₅	semidirect
185. #125	C₂²×C₃×C₅	semidirect
190. #102	C₂²×C₃×C₅	semidirect
191. #97	C₂²×C₃×C₅	semidirect
192. C₄×C₈	C₂²×C₃×C₅	semidirect
245. C₃×C₅	#3138	direct (N × H)
282. C₅	C₃×#3138	direct (N × H)
294. C₃	C₅×#3138	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×#127	2
C₃×C₅×#490	2
C₃×C₅×#486	2
C₃×C₅×#418	2
C₃×C₄×C₅×M₁₆	2
C₅×#3138	3
C₃×#3138	5

Subgroups — C3×C5×#3138 #14276172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3138 #14276172