Subgroups — C₃×C₅×#3245 #14276279

Order 1920.

872 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#474	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#466	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#459	C₂	1	4	yes	maximal, complemented
5	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3245	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	2	no	—
8	C₂×C₃×C₅×D₈	C₂²	1	4	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₅×#127	C₂²	1	2	yes	—
12	C₃×C₅×#124	—	2	2	no	—
13	C₃×C₅×#121	—	2	2	no	—
14	C₃×C₅×#121	C₂²	1	4	yes	—
15	C₃×C₄×C₅×D₄	—	2	2	no	—
16	C₃×C₄×C₅×D₄	C₂²	1	4	yes	—
17	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
18	C₃×C₅×#107	C₂²	1	4	yes	—
19	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
20	C₃×C₅×#102	C₂²	1	8	yes	—
21	C₃×#3245	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	C₂×C₃	1	2	yes	complemented
23	C₅×#474	C₂×C₃	1	4	yes	complemented
24	C₅×#466	C₂×C₃	1	4	yes	complemented
25	C₅×#459	C₂×C₃	1	4	yes	complemented
26	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
27	C₃×C₅×#41	—	4	8	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	2	2	no	—
31	C₂×C₃×C₅×D₄	—	2	2	no	—
32	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
33	C₂²×C₃×C₄×C₅	—	2	4	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×D₈	—	4	4	no	—
38	C₂×C₃×C₅×C₈	—	4	2	no	—
39	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
40	C₃×C₅×C₄⋊C₄	—	2	2	no	—
41	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
42	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
43	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
44	C₃×C₄²×C₅	D₄	1	2	yes	—
45	C₃×C₄²×C₅	C₂³	1	2	yes	—
46	C₃×#532	C₂×C₅	1	2	yes	complemented
47	C₃×#474	C₂×C₅	1	4	yes	complemented
48	C₃×#466	C₂×C₅	1	4	yes	complemented
49	C₃×#459	C₂×C₅	1	4	yes	complemented
50	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×C₅×#41	—	2	2	no	—
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	—	2	2	no	—
57	C₅×#121	—	2	2	no	—
58	C₅×#121	C₂²×C₃	1	4	yes	—
59	C₄×C₅×D₄	—	2	2	no	—
60	C₄×C₅×D₄	C₂²×C₃	1	4	yes	—
61	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
62	C₅×#107	C₂²×C₃	1	4	yes	—
63	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
64	C₅×#102	C₂²×C₃	1	8	yes	—
65	#3245	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
66	C₂³×C₃×C₅	—	2	4	no	—
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×Q₈	—	4	4	no	—
69	C₃×C₅×D₄	—	2	4	no	—
70	C₃×C₅×D₄	—	2	4	no	—
71	C₃×C₅×D₄	—	4	4	no	—
72	C₃×C₅×D₄	—	4	8	no	—
73	C₃×C₅×D₄	—	4	4	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	8	no	—
79	C₂×C₃×C₄×C₅	D₈	1	4	yes	complemented
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	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
84	C₃×C₅×C₈	—	2	4	no	—
85	C₂×C₃×#41	—	2	2	no	—
86	C₂×C₃×D₈	C₂²×C₅	1	4	yes	—
87	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
88	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
89	C₃×#127	C₂²×C₅	1	2	yes	—
90	C₃×#124	—	2	2	no	—
91	C₃×#121	—	2	2	no	—
92	C₃×#121	C₂²×C₅	1	4	yes	—
93	C₃×C₄×D₄	—	2	2	no	—
94	C₃×C₄×D₄	C₂²×C₅	1	4	yes	—
95	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₃×#107	C₂²×C₅	1	4	yes	—
97	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
98	C₃×#102	C₂²×C₅	1	8	yes	—
99	C₅×#41	—	4	8	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×D₄	—	2	4	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	2	2	no	—
104	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
105	C₂²×C₄×C₅	—	2	4	no	—
106	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₅×D₈	—	4	4	no	—
110	C₂×C₅×C₈	—	4	2	no	—
111	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
114	C₅×C₂²⋊C₄	—	2	4	no	—
115	C₅×C₂²⋊C₄	—	2	4	no	—
116	C₄²×C₅	C₃×D₄	1	2	yes	—
117	C₄²×C₅	C₂³×C₃	1	2	yes	—
118	#532	C₂×C₃×C₅	1	2	yes	complemented
119	#474	C₂×C₃×C₅	1	4	yes	complemented
120	#466	C₂×C₃×C₅	1	4	yes	complemented
121	#459	C₂×C₃×C₅	1	4	yes	complemented
122	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	C₂²×C₃×C₅	—	2	1	no	—
124	C₂²×C₃×C₅	—	2	1	no	—
125	C₂²×C₃×C₅	—	4	4	no	—
126	C₂²×C₃×C₅	—	4	4	no	—
127	C₂²×C₃×C₅	—	4	4	no	—
128	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
129	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
130	C₃×C₄×C₅	—	2	1	no	—
131	C₃×C₄×C₅	—	2	2	no	—
132	C₃×C₄×C₅	—	4	4	no	—
133	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
134	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
135	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
136	C₃×#41	—	4	8	no	—
137	C₂×C₃×Q₈	—	2	2	no	—
138	C₂×C₃×D₄	—	2	4	no	—
139	C₂×C₃×D₄	—	2	2	no	—
140	C₂×C₃×D₄	—	2	2	no	—
141	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
142	C₂²×C₃×C₄	—	2	4	no	—
143	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₃×D₈	—	4	4	no	—
147	C₂×C₃×C₈	—	4	2	no	—
148	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
151	C₃×C₂²⋊C₄	—	2	4	no	—
152	C₃×C₂²⋊C₄	—	2	4	no	—
153	C₃×C₄²	C₅×D₄	1	2	yes	—
154	C₃×C₄²	C₂³×C₅	1	2	yes	—
155	C₂³×C₅	—	2	4	no	—
156	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
157	C₅×Q₈	—	4	4	no	—
158	C₅×D₄	—	2	4	no	—
159	C₅×D₄	—	2	4	no	—
160	C₅×D₄	—	4	4	no	—
161	C₅×D₄	—	4	8	no	—
162	C₅×D₄	—	4	4	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	8	no	—
168	C₂×C₄×C₅	C₃×D₈	1	4	yes	complemented
169	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
171	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
172	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
173	C₅×C₈	—	2	4	no	—
174	C₂×#41	—	2	2	no	—
175	C₂×D₈	C₂²×C₃×C₅	1	4	yes	—
176	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
177	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
178	#127	C₂²×C₃×C₅	1	2	yes	—
179	#124	—	2	2	no	—
180	#121	—	2	2	no	—
181	#121	C₂²×C₃×C₅	1	4	yes	—
182	C₄×D₄	—	2	2	no	—
183	C₄×D₄	C₂²×C₃×C₅	1	4	yes	—
184	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
185	#107	C₂²×C₃×C₅	1	4	yes	—
186	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
187	#102	C₂²×C₃×C₅	1	8	yes	—
188	C₂×C₃×C₅	—	2	2	no	—
189	C₂×C₃×C₅	—	8	4	no	—
190	C₂×C₃×C₅	#537	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
192	C₂×C₃×C₅	#575	1	1	yes	characteristic
193	C₂³×C₃	—	2	4	no	—
194	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
195	C₃×Q₈	—	4	4	no	—
196	C₃×D₄	—	2	4	no	—
197	C₃×D₄	—	2	4	no	—
198	C₃×D₄	—	4	4	no	—
199	C₃×D₄	—	4	8	no	—
200	C₃×D₄	—	4	4	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	4	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	4	8	no	—
206	C₂×C₃×C₄	C₅×D₈	1	4	yes	complemented
207	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
210	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
211	C₃×C₈	—	2	4	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	4	4	no	—
215	C₂²×C₅	—	4	4	no	—
216	C₂²×C₅	—	4	4	no	—
217	C₂²×C₅	C₂×C₃×D₈	1	2	yes	—
218	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	4	4	no	—
222	C₄×C₅	C₂×C₃×D₈	1	4	yes	—
223	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
224	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
225	#41	—	4	8	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
235	D₈	—	4	4	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₄²	C₃×C₅×D₄	1	2	yes	—
243	C₄²	C₂³×C₃×C₅	1	2	yes	—
244	C₃×C₅	#3245	1	1	yes	characteristic, direct product factor, Hall
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	1	no	—
247	C₂²×C₃	—	4	4	no	—
248	C₂²×C₃	—	4	4	no	—
249	C₂²×C₃	—	4	4	no	—
250	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
251	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
252	C₃×C₄	—	2	1	no	—
253	C₃×C₄	—	2	2	no	—
254	C₃×C₄	—	4	4	no	—
255	C₃×C₄	C₂×C₅×D₈	1	4	yes	—
256	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
257	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
258	C₂×C₅	—	2	2	no	—
259	C₂×C₅	—	8	4	no	—
260	C₂×C₅	C₃×#537	1	1	yes	characteristic
261	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
262	C₂×C₅	C₃×#575	1	1	yes	characteristic
263	C₂³	—	2	4	no	—
264	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
265	Q₈	—	4	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	C₃×C₅×D₈	1	4	yes	complemented
277	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
279	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
280	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₂×C₃	—	2	2	no	—
283	C₂×C₃	—	8	4	no	—
284	C₂×C₃	C₅×#537	1	1	yes	characteristic
285	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
286	C₂×C₃	C₅×#575	1	1	yes	characteristic
287	C₅	C₃×#3245	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	C₂×C₃×C₅×D₈	1	2	yes	—
294	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	C₂×C₃×C₅×D₈	1	4	yes	—
299	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
300	C₄	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
301	C₃	C₅×#3245	1	1	yes	characteristic, direct product factor, Sylow
302	C₂	—	2	2	no	—
303	C₂	—	8	4	no	—
304	C₂	C₃×C₅×#537	1	1	yes	characteristic
305	C₂	C₂²×C₃×C₅×D₈	1	1	yes	characteristic
306	C₂	C₃×C₅×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
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×#532	C₂	semidirect
2. C₃×C₅×#474	C₂	semidirect
3. C₃×C₅×#466	C₂	semidirect
4. C₃×C₅×#459	C₂	semidirect
5. C₂×C₃×C₅×C₄⋊C₈	C₂	semidirect
6. C₅×#3245	C₃	direct (N × H)
21. C₃×#3245	C₅	direct (N × H)
22. C₅×#532	C₂×C₃	semidirect
23. C₅×#474	C₂×C₃	semidirect
24. C₅×#466	C₂×C₃	semidirect
25. C₅×#459	C₂×C₃	semidirect
26. C₂×C₅×C₄⋊C₈	C₂×C₃	semidirect
46. C₃×#532	C₂×C₅	semidirect
47. C₃×#474	C₂×C₅	semidirect
48. C₃×#466	C₂×C₅	semidirect
49. C₃×#459	C₂×C₅	semidirect
50. C₂×C₃×C₄⋊C₈	C₂×C₅	semidirect
65. #3245	C₃×C₅	direct (N × H)
79. C₂×C₃×C₄×C₅	D₈	semidirect
118. #532	C₂×C₃×C₅	semidirect
119. #474	C₂×C₃×C₅	semidirect
120. #466	C₂×C₃×C₅	semidirect
121. #459	C₂×C₃×C₅	semidirect
122. C₂×C₄⋊C₈	C₂×C₃×C₅	semidirect
168. C₂×C₄×C₅	C₃×D₈	semidirect
206. C₂×C₃×C₄	C₅×D₈	semidirect
244. C₃×C₅	#3245	direct (N × H)
276. C₂×C₄	C₃×C₅×D₈	semidirect
287. C₅	C₃×#3245	direct (N × H)
301. C₃	C₅×#3245	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×#532	2
C₃×C₅×#474	2
C₃×C₅×#466	2
C₃×C₅×#459	2
C₂×C₃×C₅×C₄⋊C₈	2
C₅×#3245	3
C₃×#3245	5

Subgroups — C3×C5×#3245 #14276279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3245 #14276279