Subgroups — C₂×C₃×#65470 #14438797

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#4090	C₂	1	2	yes	maximal
3	C₂²×C₃×#689	C₂	1	2	yes	maximal
4	C₂²×C₃×Dic₂₀	C₂	1	1	yes	maximal, characteristic
5	C₃×#65470	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₃×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
7	C₂×#65470	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
10	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×#4090	—	2	4	no	—
14	C₃×#4090	C₂²	1	8	yes	—
15	C₂×C₃×#689	—	2	8	no	—
16	C₂×C₃×#689	C₂²	1	8	yes	—
17	C₂×C₃×#683	—	2	4	no	—
18	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
19	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
20	C₂×C₃×Dic₂₀	—	2	4	no	—
21	C₂×C₃×Dic₂₀	C₂²	1	4	yes	—
22	C₃×C₈×Dic₅	C₂²	1	4	yes	—
23	C₂×C₃×#494	—	5	1	no	maximal, Hall
24	C₂²×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
25	C₂×#4090	C₂×C₃	1	2	yes	—
26	C₂²×#689	C₂×C₃	1	2	yes	—
27	C₂²×Dic₂₀	C₂×C₃	1	1	yes	characteristic
28	#65470	C₂×C₃	1	8	yes	direct product factor
29	C₂×C₈×Dic₅	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×C₅×Q₈	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	—	2	4	no	—
32	C₂×C₃×C₅×Q₈	—	2	4	no	—
33	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	—	2	2	no	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₃×Dic₅	—	2	2	no	—
37	C₂²×C₃×Dic₅	D₄	1	2	yes	—
38	C₂×C₃×Dic₁₀	—	2	4	no	—
39	C₂×C₃×Dic₁₀	—	2	4	no	—
40	C₂×C₃×Dic₁₀	—	2	2	no	—
41	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
42	C₃×C₅×Q₁₆	—	2	8	no	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₃×#689	—	2	16	no	—
46	C₃×#683	—	2	16	no	—
47	C₃×C₄×Dic₅	D₄	1	2	yes	—
48	C₃×C₄×Dic₅	C₂³	1	2	yes	—
49	C₂×C₃×#168	D₄	1	4	yes	—
50	C₂×C₃×#168	C₂³	1	2	yes	—
51	C₃×Dic₂₀	—	2	8	no	—
52	C₂²×C₃×Q₁₆	—	5	2	no	—
53	C₂²×C₃×Q₁₆	—	5	1	no	—
54	C₂²×C₃×Q₁₆	D₅	1	1	yes	characteristic
55	C₂×C₃×C₄⋊Q₈	—	5	2	no	—
56	C₃×#494	—	5	8	no	—
57	C₂×C₃×C₄×C₈	—	5	1	no	—
58	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
59	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
60	C₂×C₅×Q₁₆	—	2	4	no	—
61	C₂×C₅×Q₁₆	C₂²×C₃	1	4	yes	—
62	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
63	#4090	—	2	4	no	—
64	#4090	C₂²×C₃	1	8	yes	—
65	C₂×#689	—	2	8	no	—
66	C₂×#689	C₂²×C₃	1	8	yes	—
67	C₂×#683	—	2	4	no	—
68	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
70	C₂×Dic₂₀	—	2	4	no	—
71	C₂×Dic₂₀	C₂²×C₃	1	4	yes	—
72	C₈×Dic₅	C₂²×C₃	1	4	yes	—
73	C₂×#494	—	5	1	no	Sylow
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₃×C₅×Q₈	—	2	4	no	—
76	C₃×C₅×Q₈	—	2	4	no	—
77	C₃×C₅×Q₈	—	4	4	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	—	4	4	no	—
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	C₂×C₃×Dic₅	—	2	4	no	—
85	C₂×C₃×Dic₅	—	4	4	no	—
86	C₂×C₃×Dic₅	Q₁₆	1	8	yes	complemented
87	C₂×C₃×Dic₅	C₂×D₄	1	4	yes	—
88	C₃×Dic₁₀	—	2	4	no	—
89	C₃×Dic₁₀	—	2	4	no	—
90	C₃×Dic₁₀	—	4	4	no	—
91	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
92	C₃×#168	C₂×D₄	1	4	yes	—
93	C₂²×C₃×Q₈	—	5	2	no	—
94	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
95	C₂×C₃×Q₁₆	—	2	4	no	—
96	C₂×C₃×Q₁₆	—	5	8	no	—
97	C₂×C₃×Q₁₆	—	5	4	no	—
98	C₂×C₃×Q₁₆	—	10	8	no	—
99	C₂×C₃×Q₁₆	—	10	4	no	—
100	C₂×C₃×Q₁₆	C₂×D₅	1	4	yes	—
101	C₂²×C₃×C₈	—	5	1	no	—
102	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
103	C₃×C₄⋊Q₈	—	5	8	no	—
104	C₃×C₄⋊Q₈	—	10	4	no	—
105	C₂×C₃×C₄⋊C₄	—	10	4	no	—
106	C₂×C₃×C₄²	—	5	1	no	—
107	C₃×C₄×C₈	—	5	4	no	—
108	C₂×C₅×Q₈	—	2	2	no	—
109	C₂×C₅×Q₈	—	2	4	no	—
110	C₂×C₅×Q₈	—	2	4	no	—
111	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
112	C₂²×C₄×C₅	—	2	2	no	—
113	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
114	C₂²×Dic₅	—	2	2	no	—
115	C₂²×Dic₅	C₃×D₄	1	2	yes	—
116	C₂×Dic₁₀	—	2	4	no	—
117	C₂×Dic₁₀	—	2	4	no	—
118	C₂×Dic₁₀	—	2	2	no	—
119	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
120	C₅×Q₁₆	—	2	8	no	—
121	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
122	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
123	#689	—	2	16	no	—
124	#683	—	2	16	no	—
125	C₄×Dic₅	C₃×D₄	1	2	yes	—
126	C₄×Dic₅	C₂³×C₃	1	2	yes	—
127	C₂×#168	C₃×D₄	1	4	yes	—
128	C₂×#168	C₂³×C₃	1	2	yes	—
129	Dic₂₀	—	2	8	no	—
130	C₂²×Q₁₆	—	5	2	no	—
131	C₂²×Q₁₆	—	5	1	no	—
132	C₂²×Q₁₆	C₃×D₅	1	1	yes	characteristic
133	C₂×C₄⋊Q₈	—	5	2	no	—
134	#494	—	5	8	no	—
135	C₂×C₄×C₈	—	5	1	no	—
136	C₂²×C₃×C₅	#127	1	2	yes	—
137	C₂²×C₃×C₅	C₂×Q₁₆	1	4	yes	—
138	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	—	4	4	no	—
140	C₃×C₄×C₅	#127	1	2	yes	—
141	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₃×Dic₅	—	4	4	no	—
144	C₃×Dic₅	C₂×Q₁₆	1	8	yes	complemented
145	C₂×C₃×Q₈	—	2	2	no	—
146	C₂×C₃×Q₈	—	2	4	no	—
147	C₂×C₃×Q₈	—	2	4	no	—
148	C₂×C₃×Q₈	—	5	4	no	—
149	C₂×C₃×Q₈	—	10	2	no	—
150	C₂×C₃×Q₈	—	10	4	no	—
151	C₂×C₃×Q₈	—	10	4	no	—
152	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
153	C₂²×C₃×C₄	—	2	2	no	—
154	C₂²×C₃×C₄	—	5	2	no	—
155	C₂²×C₃×C₄	—	10	2	no	—
156	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
157	C₃×Q₁₆	—	2	8	no	—
158	C₃×Q₁₆	—	10	16	no	—
159	C₃×Q₁₆	—	10	8	no	—
160	C₂×C₃×C₈	—	5	4	no	—
161	C₂×C₃×C₈	—	5	2	no	—
162	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
163	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
164	C₃×C₄⋊C₄	—	10	16	no	—
165	C₃×C₄²	—	5	2	no	—
166	C₃×C₄²	—	5	2	no	—
167	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₅×Q₈	—	2	4	no	—
169	C₅×Q₈	—	2	4	no	—
170	C₅×Q₈	—	4	4	no	—
171	C₂×C₄×C₅	—	2	4	no	—
172	C₂×C₄×C₅	—	4	4	no	—
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	C₂×C₃×D₄	1	2	yes	—
176	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
177	C₂×Dic₅	—	2	4	no	—
178	C₂×Dic₅	—	4	4	no	—
179	C₂×Dic₅	C₃×Q₁₆	1	8	yes	complemented
180	C₂×Dic₅	C₂×C₃×D₄	1	4	yes	—
181	Dic₁₀	—	2	4	no	—
182	Dic₁₀	—	2	4	no	—
183	Dic₁₀	—	4	4	no	—
184	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
185	#168	C₂×C₃×D₄	1	4	yes	—
186	C₂²×Q₈	—	5	2	no	—
187	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
188	C₂×Q₁₆	—	2	4	no	—
189	C₂×Q₁₆	—	5	8	no	—
190	C₂×Q₁₆	—	5	4	no	—
191	C₂×Q₁₆	—	10	8	no	—
192	C₂×Q₁₆	—	10	4	no	—
193	C₂×Q₁₆	C₂×C₃×D₅	1	4	yes	—
194	C₂²×C₈	—	5	1	no	—
195	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
196	C₄⋊Q₈	—	5	8	no	—
197	C₄⋊Q₈	—	10	4	no	—
198	C₂×C₄⋊C₄	—	10	4	no	—
199	C₂×C₄²	—	5	1	no	—
200	C₄×C₈	—	5	4	no	—
201	C₂×C₃×C₅	#494	1	4	yes	complemented
202	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂²×Q₁₆	1	2	yes	—
204	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
205	C₃×Q₈	—	2	4	no	—
206	C₃×Q₈	—	2	4	no	—
207	C₃×Q₈	—	4	4	no	—
208	C₃×Q₈	—	10	4	no	—
209	C₃×Q₈	—	10	4	no	—
210	C₃×Q₈	—	20	4	no	—
211	C₂×C₃×C₄	—	2	4	no	—
212	C₂×C₃×C₄	—	4	4	no	—
213	C₂×C₃×C₄	—	5	4	no	—
214	C₂×C₃×C₄	—	5	8	no	—
215	C₂×C₃×C₄	—	10	4	no	—
216	C₂×C₃×C₄	—	20	4	no	—
217	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
218	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
220	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
221	C₃×C₈	—	5	4	no	—
222	C₃×C₈	C₂×C₂²⋊D₅	1	4	yes	—
223	C₂²×C₅	C₃×#127	1	2	yes	—
224	C₂²×C₅	C₂×C₃×Q₁₆	1	4	yes	—
225	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
226	C₄×C₅	—	4	4	no	—
227	C₄×C₅	C₃×#127	1	2	yes	—
228	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
229	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
230	Dic₅	—	4	4	no	—
231	Dic₅	C₂×C₃×Q₁₆	1	8	yes	complemented
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	5	4	no	—
236	C₂×Q₈	—	10	2	no	—
237	C₂×Q₈	—	10	4	no	—
238	C₂×Q₈	—	10	4	no	—
239	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	5	2	no	—
242	C₂²×C₄	—	10	2	no	—
243	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
244	Q₁₆	—	2	8	no	—
245	Q₁₆	—	10	16	no	—
246	Q₁₆	—	10	8	no	—
247	C₂×C₈	—	5	4	no	—
248	C₂×C₈	—	5	2	no	—
249	C₂×C₈	C₃×C₂²⋊D₅	1	4	yes	—
250	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
251	C₄⋊C₄	—	10	16	no	—
252	C₄²	—	5	2	no	—
253	C₄²	—	5	2	no	—
254	C₃×C₅	C₂×#494	1	1	yes	characteristic, complemented, Hall
255	C₂²×C₃	D₅×Q₁₆	1	4	yes	—
256	C₂²×C₃	#4086	1	2	yes	—
257	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
258	C₃×C₄	—	4	4	no	—
259	C₃×C₄	—	5	8	no	—
260	C₃×C₄	—	20	4	no	—
261	C₃×C₄	#4086	1	2	yes	—
262	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
263	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
264	C₂×C₅	C₃×#494	1	4	yes	complemented
265	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
266	C₂×C₅	C₂²×C₃×Q₁₆	1	2	yes	—
267	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
268	Q₈	—	2	4	no	—
269	Q₈	—	2	4	no	—
270	Q₈	—	4	4	no	—
271	Q₈	—	10	4	no	—
272	Q₈	—	10	4	no	—
273	Q₈	—	20	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	5	4	no	—
277	C₂×C₄	—	5	8	no	—
278	C₂×C₄	—	10	4	no	—
279	C₂×C₄	—	20	4	no	—
280	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
281	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
282	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
283	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
284	C₈	—	5	4	no	—
285	C₈	C₂×C₃×C₂²⋊D₅	1	4	yes	—
286	C₂×C₃	#65470	1	4	yes	direct product factor
287	C₂×C₃	C₂×D₅×Q₁₆	1	2	yes	—
288	C₂×C₃	C₂×#4086	1	1	yes	characteristic
289	C₅	C₂×C₃×#494	1	1	yes	characteristic, complemented, Sylow
290	C₂²	C₃×D₅×Q₁₆	1	4	yes	—
291	C₂²	C₃×#4086	1	2	yes	—
292	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
293	C₄	—	4	4	no	—
294	C₄	—	5	8	no	—
295	C₄	—	20	4	no	—
296	C₄	C₃×#4086	1	2	yes	—
297	C₄	C₂×C₃×D₄×D₅	1	1	yes	characteristic
298	C₄	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
299	C₃	C₂×#65470	1	1	yes	characteristic, direct product factor, Sylow
300	C₂	C₃×#65470	1	4	yes	direct product factor
301	C₂	C₂×C₃×D₅×Q₁₆	1	2	yes	—
302	C₂	C₂×C₃×#4086	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂²×C₃×C₅×Q₁₆	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂²×C₃×Q₁₆	D₅
hypocenter	C₅	C₂×C₃×#494
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
5. C₃×#65470	C₂	direct (N × H)
7. C₂×#65470	C₃	direct (N × H)
28. #65470	C₂×C₃	direct (N × H)
86. C₂×C₃×Dic₅	Q₁₆	semidirect
144. C₃×Dic₅	C₂×Q₁₆	semidirect
179. C₂×Dic₅	C₃×Q₁₆	semidirect
201. C₂×C₃×C₅	#494	semidirect
231. Dic₅	C₂×C₃×Q₁₆	semidirect
254. C₃×C₅	C₂×#494	semidirect
264. C₂×C₅	C₃×#494	semidirect
286. C₂×C₃	#65470	direct (N × H)
289. C₅	C₂×C₃×#494	semidirect
299. C₃	C₂×#65470	direct (N × H)
300. C₂	C₃×#65470	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×Q₁₆	2
C₂×C₃×#4090	2
C₂²×C₃×#689	2
C₂²×C₃×Dic₂₀	2
C₃×#65470	2
C₂×C₃×C₈×Dic₅	2
C₂×#65470	3
C₂×C₃×#494	5

Subgroups — C2×C3×#65470 #14438797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65470 #14438797