Subgroups — C₃×C₅×#3195 #14276229

Order 1920.

820 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	C₂×C₃×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#483	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#470	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#451	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#407	C₂	1	1	yes	maximal, characteristic
6	C₅×#3195	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	complemented
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
12	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
13	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
14	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
15	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
16	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₂²⋊C₄	—	2	2	no	—
18	C₃×C₅×#106	C₂²	1	4	yes	complemented
19	C₃×C₅×#103	C₂²	1	8	yes	complemented
20	C₃×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₃×#3195	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×C₂²⋊Q₈	C₂×C₃	1	2	yes	—
23	C₅×#483	C₂×C₃	1	4	yes	complemented
24	C₅×#470	C₂×C₃	1	4	yes	complemented
25	C₅×#451	C₂×C₃	1	4	yes	complemented
26	C₅×#407	C₂×C₃	1	1	yes	characteristic
27	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	—	4	4	no	—
30	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	—	4	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
36	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
38	C₃×C₅×Q₁₆	—	4	4	no	—
39	C₃×C₅×M₁₆	—	4	2	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
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	4	no	—
46	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
47	C₂×C₃×C₂²⋊Q₈	C₂×C₅	1	2	yes	—
48	C₃×#483	C₂×C₅	1	4	yes	complemented
49	C₃×#470	C₂×C₅	1	4	yes	complemented
50	C₃×#451	C₂×C₅	1	4	yes	complemented
51	C₃×#407	C₂×C₅	1	1	yes	characteristic
52	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
53	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	complemented
55	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
56	C₅×C₂²⋊Q₈	—	2	2	no	—
57	C₅×C₂²⋊Q₈	—	2	2	no	—
58	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
59	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
60	C₂×C₅×C₄⋊C₄	—	2	2	no	—
61	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
62	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
63	C₅×#106	C₂²×C₃	1	4	yes	complemented
64	C₅×#103	C₂²×C₃	1	8	yes	complemented
65	C₅×C₂²⋊C₈	C₂²×C₃	1	4	yes	—
66	#3195	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
67	C₂³×C₃×C₅	—	2	4	no	—
68	C₂³×C₃×C₅	—	4	1	no	—
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
71	C₃×C₅×Q₈	—	2	8	no	—
72	C₃×C₅×Q₈	—	4	4	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	—	4	2	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	4	no	—
80	C₂×C₃×C₄×C₅	—	4	4	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
86	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₃×Q₁₆	C₂²×C₅	1	4	yes	complemented
88	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
89	C₃×C₂²⋊Q₈	—	2	2	no	—
90	C₃×C₂²⋊Q₈	—	2	2	no	—
91	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
92	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
93	C₂×C₃×C₄⋊C₄	—	2	2	no	—
94	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
96	C₃×#106	C₂²×C₅	1	4	yes	complemented
97	C₃×#103	C₂²×C₅	1	8	yes	complemented
98	C₃×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
99	C₂⁴×C₅	C₃×D₄	1	1	yes	characteristic
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	—	4	4	no	—
102	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
103	C₂²×C₄×C₅	—	2	2	no	—
104	C₂²×C₄×C₅	—	2	2	no	—
105	C₂²×C₄×C₅	—	4	2	no	—
106	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
107	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
109	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
110	C₅×Q₁₆	—	4	4	no	—
111	C₅×M₁₆	—	4	2	no	—
112	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
113	C₅×C₄⋊C₄	—	2	4	no	—
114	C₅×C₄⋊C₄	—	2	4	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
117	C₅×C₂²⋊C₄	—	2	4	no	—
118	C₅×C₂²⋊C₄	—	2	4	no	—
119	C₂×C₂²⋊Q₈	C₂×C₃×C₅	1	2	yes	—
120	#483	C₂×C₃×C₅	1	4	yes	complemented
121	#470	C₂×C₃×C₅	1	4	yes	complemented
122	#451	C₂×C₃×C₅	1	4	yes	complemented
123	#407	C₂×C₃×C₅	1	1	yes	characteristic
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	2	no	—
128	C₂²×C₃×C₅	—	4	4	no	—
129	C₂²×C₃×C₅	#137	1	4	yes	—
130	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	2	no	—
133	C₃×C₄×C₅	—	4	4	no	—
134	C₃×C₄×C₅	—	4	4	no	—
135	C₃×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
136	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
137	C₂×C₃×Q₈	—	2	2	no	—
138	C₂×C₃×Q₈	—	4	4	no	—
139	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
140	C₂²×C₃×C₄	—	2	2	no	—
141	C₂²×C₃×C₄	—	2	2	no	—
142	C₂²×C₃×C₄	—	4	2	no	—
143	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
144	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₃×Q₁₆	—	4	4	no	—
148	C₃×M₁₆	—	4	2	no	—
149	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
150	C₃×C₄⋊C₄	—	2	4	no	—
151	C₃×C₄⋊C₄	—	2	4	no	—
152	C₃×C₄⋊C₄	—	2	2	no	—
153	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
154	C₃×C₂²⋊C₄	—	2	4	no	—
155	C₃×C₂²⋊C₄	—	2	4	no	—
156	C₂³×C₅	—	2	4	no	—
157	C₂³×C₅	—	4	1	no	—
158	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
160	C₅×Q₈	—	2	8	no	—
161	C₅×Q₈	—	4	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	4	no	—
169	C₂×C₄×C₅	—	4	4	no	—
170	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
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	C₂²×Q₈	C₂²×C₃×C₅	1	2	yes	—
175	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
176	C₂×Q₁₆	C₂²×C₃×C₅	1	4	yes	complemented
177	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
178	C₂²⋊Q₈	—	2	2	no	—
179	C₂²⋊Q₈	—	2	2	no	—
180	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
181	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
184	C₂×C₂²⋊C₄	—	2	2	no	—
185	#106	C₂²×C₃×C₅	1	4	yes	complemented
186	#103	C₂²×C₃×C₅	1	8	yes	complemented
187	C₂²⋊C₈	C₂²×C₃×C₅	1	4	yes	—
188	C₂×C₃×C₅	—	2	4	no	—
189	C₂×C₃×C₅	—	4	1	no	—
190	C₂×C₃×C₅	#534	1	1	yes	characteristic
191	C₂×C₃×C₅	C₂×#137	1	2	yes	—
192	C₂³×C₃	—	2	4	no	—
193	C₂³×C₃	—	4	1	no	—
194	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
195	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
196	C₃×Q₈	—	2	8	no	—
197	C₃×Q₈	—	4	4	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	4	2	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	—	4	4	no	—
205	C₂×C₃×C₄	—	4	4	no	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
209	C₃×C₈	—	2	4	no	—
210	C₂²×C₅	—	2	2	no	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	—	2	1	no	—
214	C₂²×C₅	—	4	4	no	—
215	C₂²×C₅	C₃×#137	1	4	yes	—
216	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	2	2	no	—
219	C₄×C₅	—	4	4	no	—
220	C₄×C₅	—	4	4	no	—
221	C₄×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
222	C₂⁴	C₃×C₅×D₄	1	1	yes	characteristic
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	4	4	no	—
225	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	4	2	no	—
229	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
230	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
232	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
233	Q₁₆	—	4	4	no	—
234	M₁₆	—	4	2	no	—
235	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
236	C₄⋊C₄	—	2	4	no	—
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	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₃×C₅	#3195	1	1	yes	characteristic, direct product factor, Hall
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	4	4	no	—
248	C₂²×C₃	C₅×#137	1	4	yes	—
249	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	—	2	2	no	—
252	C₃×C₄	—	4	4	no	—
253	C₃×C₄	—	4	4	no	—
254	C₃×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
255	C₂×C₅	—	2	4	no	—
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	C₃×#534	1	1	yes	characteristic
258	C₂×C₅	C₂×C₃×#137	1	2	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
262	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
263	Q₈	—	2	8	no	—
264	Q₈	—	4	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
274	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
275	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₂×C₃	—	2	4	no	—
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	C₅×#534	1	1	yes	characteristic
280	C₂×C₃	C₂×C₅×#137	1	2	yes	—
281	C₅	C₃×#3195	1	1	yes	characteristic, direct product factor, Sylow
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₃×C₅×#137	1	4	yes	—
288	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	4	no	—
293	C₄	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
294	C₃	C₅×#3195	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	4	no	—
296	C₂	—	4	1	no	—
297	C₂	C₃×C₅×#534	1	1	yes	characteristic
298	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	C₂⁴×C₃×C₅	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#483	C₂	semidirect
3. C₃×C₅×#470	C₂	semidirect
4. C₃×C₅×#451	C₂	semidirect
6. C₅×#3195	C₃	direct (N × H)
9. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
18. C₃×C₅×#106	C₂²	semidirect
19. C₃×C₅×#103	C₂²	semidirect
21. C₃×#3195	C₅	direct (N × H)
23. C₅×#483	C₂×C₃	semidirect
24. C₅×#470	C₂×C₃	semidirect
25. C₅×#451	C₂×C₃	semidirect
48. C₃×#483	C₂×C₅	semidirect
49. C₃×#470	C₂×C₅	semidirect
50. C₃×#451	C₂×C₅	semidirect
54. C₂×C₅×Q₁₆	C₂²×C₃	semidirect
63. C₅×#106	C₂²×C₃	semidirect
64. C₅×#103	C₂²×C₃	semidirect
66. #3195	C₃×C₅	direct (N × H)
87. C₂×C₃×Q₁₆	C₂²×C₅	semidirect
96. C₃×#106	C₂²×C₅	semidirect
97. C₃×#103	C₂²×C₅	semidirect
120. #483	C₂×C₃×C₅	semidirect
121. #470	C₂×C₃×C₅	semidirect
122. #451	C₂×C₃×C₅	semidirect
176. C₂×Q₁₆	C₂²×C₃×C₅	semidirect
185. #106	C₂²×C₃×C₅	semidirect
186. #103	C₂²×C₃×C₅	semidirect
242. C₃×C₅	#3195	direct (N × H)
281. C₅	C₃×#3195	direct (N × H)
294. C₃	C₅×#3195	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×C₂²⋊Q₈	2
C₃×C₅×#483	2
C₃×C₅×#470	2
C₃×C₅×#451	2
C₃×C₅×#407	2
C₅×#3195	3
C₃×#3195	5

Subgroups — C3×C5×#3195 #14276229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3195 #14276229