Subgroups — C₃×C₅×#1548 #14274582

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#333	C₂	1	4	yes	maximal, complemented
5	C₅×#1548	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	1	no	—
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
19	C₃×C₄×C₅×C₈	—	2	1	no	—
20	C₃×#1548	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
22	C₅×#431	C₂×C₃	1	1	yes	characteristic
23	C₅×#423	C₂×C₃	1	1	yes	characteristic
24	C₅×#333	C₂×C₃	1	4	yes	complemented
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	4	4	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×M₁₆	—	4	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic
44	C₃×#431	C₂×C₅	1	1	yes	characteristic
45	C₃×#423	C₂×C₅	1	1	yes	characteristic
46	C₃×#333	C₂×C₅	1	4	yes	complemented
47	C₂²×C₅×Q₈	C₃×C₄	1	2	yes	—
48	C₂×C₅×M₁₆	—	2	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	C₃×C₄	1	2	yes	—
52	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₅×C₄⋊C₄	—	2	1	no	—
55	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
56	C₅×C₄⋊C₈	—	2	1	no	—
57	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
58	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
59	C₅×C₂²⋊C₈	—	2	1	no	—
60	C₄×C₅×C₈	—	2	1	no	—
61	#1548	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₃×C₅×Q₈	—	4	4	no	—
64	C₃×C₅×Q₈	—	4	4	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
73	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₃×C₅×C₈	—	4	2	no	—
78	C₃×C₅×C₈	—	4	2	no	—
79	C₂²×C₃×Q₈	C₄×C₅	1	2	yes	—
80	C₂×C₃×M₁₆	—	2	1	no	—
81	C₃×C₄⋊Q₈	—	2	1	no	—
82	C₃×C₄⋊Q₈	—	2	1	no	—
83	C₃×C₄⋊Q₈	C₄×C₅	1	2	yes	—
84	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
85	C₂×C₃×C₄⋊C₄	—	2	1	no	—
86	C₂×C₃×C₄⋊C₄	—	2	1	no	—
87	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄⋊C₈	—	2	1	no	—
89	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
90	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
91	C₃×C₂²⋊C₈	—	2	1	no	—
92	C₃×C₄×C₈	—	2	1	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	—	4	4	no	—
95	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
100	C₅×M₁₆	—	4	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
108	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
109	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
110	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
111	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	1	yes	characteristic
112	#431	C₂×C₃×C₅	1	1	yes	characteristic
113	#423	C₂×C₃×C₅	1	1	yes	characteristic
114	#333	C₂×C₃×C₅	1	4	yes	complemented
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	—	2	1	no	—
117	C₂²×C₃×C₅	#103	1	2	yes	—
118	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	4	4	no	—
122	C₃×C₄×C₅	#100	1	2	yes	—
123	C₃×C₄×C₅	#103	1	2	yes	—
124	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₃×C₄×C₅	#137	1	2	yes	—
127	C₂×C₃×Q₈	—	2	2	no	—
128	C₂×C₃×Q₈	—	4	4	no	—
129	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₃×M₁₆	—	4	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
142	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
144	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
145	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
146	C₅×Q₈	—	4	4	no	—
147	C₅×Q₈	—	4	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	4	4	no	—
152	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
156	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
157	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₅×C₈	—	4	2	no	—
161	C₅×C₈	—	4	2	no	—
162	C₂²×Q₈	C₃×C₄×C₅	1	2	yes	—
163	C₂×M₁₆	—	2	1	no	—
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	—	2	1	no	—
166	C₄⋊Q₈	C₃×C₄×C₅	1	2	yes	—
167	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₄⋊C₈	—	2	1	no	—
172	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
173	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
174	C₂²⋊C₈	—	2	1	no	—
175	C₄×C₈	—	2	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
179	C₂×C₃×C₅	#419	1	1	yes	characteristic
180	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₃×Q₈	—	4	4	no	—
182	C₃×Q₈	—	4	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	4	4	no	—
187	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
189	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
191	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
192	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
195	C₃×C₈	—	4	2	no	—
196	C₃×C₈	—	4	2	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	C₃×#103	1	2	yes	—
200	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	C₃×#100	1	2	yes	—
205	C₄×C₅	C₃×#103	1	2	yes	—
206	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
207	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
208	C₄×C₅	C₃×#137	1	2	yes	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
214	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
215	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
216	M₁₆	—	4	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
224	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
225	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
226	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
227	C₃×C₅	#1548	1	1	yes	characteristic, direct product factor, Hall
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	C₅×#103	1	2	yes	—
231	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	4	no	—
235	C₃×C₄	C₅×#100	1	2	yes	—
236	C₃×C₄	C₅×#103	1	2	yes	—
237	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
238	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
239	C₃×C₄	C₅×#137	1	2	yes	—
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
242	C₂×C₅	C₂×C₃×#103	1	1	yes	characteristic
243	C₂×C₅	C₃×#419	1	1	yes	characteristic
244	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
255	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
256	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
257	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
263	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
264	C₂×C₃	C₅×#419	1	1	yes	characteristic
265	C₅	C₃×#1548	1	1	yes	characteristic, direct product factor, Sylow
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	C₃×C₅×#103	1	2	yes	—
269	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	4	no	—
273	C₄	C₃×C₅×#100	1	2	yes	—
274	C₄	C₃×C₅×#103	1	2	yes	—
275	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
276	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
277	C₄	C₃×C₅×#137	1	2	yes	—
278	C₃	C₅×#1548	1	1	yes	characteristic, direct product factor, Sylow
279	C₂	—	2	2	no	—
280	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
281	C₂	C₂×C₃×C₅×#103	1	1	yes	characteristic
282	C₂	C₃×C₅×#419	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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
4. C₃×C₅×#333	C₂	semidirect
5. C₅×#1548	C₃	direct (N × H)
20. C₃×#1548	C₅	direct (N × H)
24. C₅×#333	C₂×C₃	semidirect
46. C₃×#333	C₂×C₅	semidirect
61. #1548	C₃×C₅	direct (N × H)
114. #333	C₂×C₃×C₅	semidirect
227. C₃×C₅	#1548	direct (N × H)
265. C₅	C₃×#1548	direct (N × H)
278. C₃	C₅×#1548	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×C₄⋊Q₈	2
C₃×C₅×#431	2
C₃×C₅×#423	2
C₃×C₅×#333	2
C₅×#1548	3
C₃×#1548	5

Subgroups — C3×C5×#1548 #14274582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1548 #14274582