Subgroups — C₂×C₃×#65157 #14428045

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×#676	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#4043	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#3942	C₂	1	2	yes	maximal, complemented
6	C₃×#65157	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65157	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×Dic₁₀	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×#685	—	2	2	no	—
13	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
14	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
15	C₂×C₃×Dic₂₀	—	2	4	no	—
16	C₂×C₃×Dic₂₀	C₂²	1	4	yes	complemented
17	C₂×C₃×#676	—	2	4	no	—
18	C₂×C₃×#676	C₂²	1	4	yes	—
19	C₃×#4043	—	2	4	no	—
20	C₃×#4043	C₂²	1	8	yes	—
21	C₃×C₅×#106	C₂²	1	4	yes	complemented
22	C₃×#3942	C₂²	1	8	yes	complemented
23	C₂×C₃×#470	—	5	1	no	maximal, Hall
24	C₂²×Dic₂₀	C₂×C₃	1	1	yes	characteristic, complemented
25	C₂²×#676	C₂×C₃	1	1	yes	characteristic
26	C₂×#4043	C₂×C₃	1	2	yes	—
27	C₂×C₅×#106	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂×#3942	C₂×C₃	1	2	yes	complemented
29	#65157	C₂×C₃	1	8	yes	direct product factor
30	C₂³×C₃×D₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	—	2	2	no	—
34	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
35	C₂×C₃×C₄×D₅	—	2	2	no	—
36	C₂×C₃×C₄×D₅	—	2	2	no	—
37	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
38	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
39	C₂×C₃×Dic₁₀	—	2	4	no	—
40	C₂×C₃×Dic₁₀	—	2	4	no	—
41	C₂×C₃×Dic₁₀	—	2	2	no	—
42	C₂×C₃×Dic₁₀	C₂³	1	4	yes	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₃×C₅×C₄⋊C₄	—	2	2	no	—
46	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
47	C₃×#685	—	2	8	no	—
48	C₃×C₄⋊Dic₅	—	2	8	no	—
49	C₂×C₃×#168	—	2	2	no	—
50	C₂×C₃×#168	C₂³	1	2	yes	—
51	C₃×Dic₂₀	—	2	8	no	—
52	C₃×#676	—	2	8	no	—
53	C₂²×C₃×Q₁₆	—	5	1	no	—
54	C₂²×C₃×M₁₆	—	5	1	no	—
55	C₂×C₃×C₂²⋊Q₈	—	5	2	no	—
56	C₃×#470	—	5	8	no	—
57	C₂×C₃×#106	D₅	1	1	yes	characteristic, complemented
58	C₂×C₃×#103	—	5	2	no	—
59	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
60	C₂²×Dic₁₀	C₂²×C₃	1	2	yes	—
61	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
63	C₂×#685	—	2	2	no	—
64	C₂×C₄⋊Dic₅	—	2	2	no	—
65	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
66	C₂×Dic₂₀	—	2	4	no	—
67	C₂×Dic₂₀	C₂²×C₃	1	4	yes	complemented
68	C₂×#676	—	2	4	no	—
69	C₂×#676	C₂²×C₃	1	4	yes	—
70	#4043	—	2	4	no	—
71	#4043	C₂²×C₃	1	8	yes	—
72	C₅×#106	C₂²×C₃	1	4	yes	complemented
73	#3942	C₂²×C₃	1	8	yes	complemented
74	C₂×#470	—	5	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₃×D₅	—	2	2	no	—
77	C₂²×C₃×D₅	—	2	4	no	—
78	C₂²×C₃×D₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	—	2	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	#41	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₂×C₃×Dic₅	—	2	2	no	—
86	C₂×C₃×Dic₅	—	2	4	no	—
87	C₂×C₃×Dic₅	—	4	4	no	—
88	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
89	C₃×C₄×D₅	—	2	4	no	—
90	C₃×C₄×D₅	—	2	2	no	—
91	C₃×C₄×D₅	—	2	2	no	—
92	C₃×Dic₁₀	—	2	4	no	—
93	C₃×Dic₁₀	—	2	4	no	—
94	C₃×Dic₁₀	—	4	4	no	—
95	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
96	C₃×#168	—	2	2	no	—
97	C₂²×C₃×Q₈	—	5	2	no	—
98	C₂³×C₃×C₄	—	5	1	no	—
99	C₂×C₃×Q₁₆	—	5	4	no	—
100	C₂×C₃×Q₁₆	—	10	4	no	—
101	C₂×C₃×M₁₆	—	5	4	no	—
102	C₂×C₃×M₁₆	—	10	4	no	—
103	C₂²×C₃×C₈	—	5	1	no	—
104	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
105	C₃×C₂²⋊Q₈	—	5	8	no	—
106	C₃×C₂²⋊Q₈	—	10	4	no	—
107	C₂×C₃×C₄⋊C₄	—	10	2	no	—
108	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
109	C₂×C₃×C₂²⋊C₄	—	10	2	no	—
110	C₃×#106	C₂×D₅	1	4	yes	complemented
111	C₃×#103	—	5	8	no	—
112	C₂³×D₅	C₃×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₅	—	2	2	no	—
114	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
115	C₂²×Dic₅	—	2	2	no	—
116	C₂²×Dic₅	C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×D₅	—	2	2	no	—
118	C₂×C₄×D₅	—	2	2	no	—
119	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
120	C₂×C₄×D₅	C₂³×C₃	1	2	yes	—
121	C₂×Dic₁₀	—	2	4	no	—
122	C₂×Dic₁₀	—	2	4	no	—
123	C₂×Dic₁₀	—	2	2	no	—
124	C₂×Dic₁₀	C₂³×C₃	1	4	yes	—
125	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
126	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
129	#685	—	2	8	no	—
130	C₄⋊Dic₅	—	2	8	no	—
131	C₂×#168	—	2	2	no	—
132	C₂×#168	C₂³×C₃	1	2	yes	—
133	Dic₂₀	—	2	8	no	—
134	#676	—	2	8	no	—
135	C₂²×Q₁₆	—	5	1	no	—
136	C₂²×M₁₆	—	5	1	no	—
137	C₂×C₂²⋊Q₈	—	5	2	no	—
138	#470	—	5	8	no	—
139	C₂×#106	C₃×D₅	1	1	yes	characteristic, complemented
140	C₂×#103	—	5	2	no	—
141	C₂²×C₃×C₅	#121	1	2	yes	—
142	C₂²×C₃×C₅	#137	1	4	yes	—
143	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₃×D₅	—	2	2	no	—
145	C₂×C₃×D₅	—	2	4	no	—
146	C₂×C₃×D₅	—	4	4	no	—
147	C₃×C₄×C₅	—	4	4	no	—
148	C₃×C₄×C₅	#121	1	2	yes	—
149	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
151	C₃×Dic₅	—	2	2	no	—
152	C₃×Dic₅	—	4	4	no	—
153	C₂⁴×C₃	—	5	1	no	—
154	C₂×C₃×Q₈	—	5	4	no	—
155	C₂×C₃×Q₈	—	10	2	no	—
156	C₂×C₃×Q₈	—	10	4	no	—
157	C₂×C₃×Q₈	—	10	4	no	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	—	5	1	no	—
160	C₂²×C₃×C₄	—	5	2	no	—
161	C₂²×C₃×C₄	—	5	2	no	—
162	C₂²×C₃×C₄	—	10	2	no	—
163	C₂²×C₃×C₄	—	10	2	no	—
164	C₂²×C₃×C₄	—	10	2	no	—
165	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
166	C₃×Q₁₆	—	10	8	no	—
167	C₃×M₁₆	—	10	8	no	—
168	C₂×C₃×C₈	—	5	2	no	—
169	C₂×C₃×C₈	—	10	2	no	—
170	C₂×C₃×C₈	D₂₀	1	4	yes	—
171	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
172	C₃×C₄⋊C₄	—	2	2	no	—
173	C₃×C₄⋊C₄	—	10	8	no	—
174	C₃×C₄⋊C₄	C₂²×D₅	1	4	yes	—
175	C₃×C₂²⋊C₄	—	10	8	no	—
176	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
177	C₂²×D₅	—	2	2	no	—
178	C₂²×D₅	—	2	4	no	—
179	C₂²×D₅	C₂×C₃×D₄	1	2	yes	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	—	4	4	no	—
182	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
184	C₂×C₄×C₅	C₃×#41	1	2	yes	—
185	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
186	C₂×Dic₅	—	2	2	no	—
187	C₂×Dic₅	—	2	4	no	—
188	C₂×Dic₅	—	4	4	no	—
189	C₂×Dic₅	C₂×C₃×D₄	1	2	yes	—
190	C₄×D₅	—	2	4	no	—
191	C₄×D₅	—	2	2	no	—
192	C₄×D₅	—	2	2	no	—
193	Dic₁₀	—	2	4	no	—
194	Dic₁₀	—	2	4	no	—
195	Dic₁₀	—	4	4	no	—
196	C₅×C₈	C₂×C₃×D₄	1	4	yes	—
197	#168	—	2	2	no	—
198	C₂²×Q₈	—	5	2	no	—
199	C₂³×C₄	—	5	1	no	—
200	C₂×Q₁₆	—	5	4	no	—
201	C₂×Q₁₆	—	10	4	no	—
202	C₂×M₁₆	—	5	4	no	—
203	C₂×M₁₆	—	10	4	no	—
204	C₂²×C₈	—	5	1	no	—
205	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
206	C₂²⋊Q₈	—	5	8	no	—
207	C₂²⋊Q₈	—	10	4	no	—
208	C₂×C₄⋊C₄	—	10	2	no	—
209	C₂×C₄⋊C₄	C₂×C₃×D₅	1	2	yes	—
210	C₂×C₂²⋊C₄	—	10	2	no	—
211	#106	C₂×C₃×D₅	1	4	yes	complemented
212	#103	—	5	8	no	—
213	C₂×C₃×C₅	#470	1	4	yes	complemented
214	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂×#137	1	2	yes	—
216	C₃×D₅	—	4	2	no	—
217	C₂³×C₃	—	5	2	no	—
218	C₂³×C₃	—	10	2	no	—
219	C₂³×C₃	—	10	4	no	—
220	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
221	C₃×Q₈	—	10	4	no	—
222	C₃×Q₈	—	10	4	no	—
223	C₃×Q₈	—	20	4	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	4	4	no	—
226	C₂×C₃×C₄	—	5	2	no	—
227	C₂×C₃×C₄	—	10	2	no	—
228	C₂×C₃×C₄	—	10	2	no	—
229	C₂×C₃×C₄	—	10	2	no	—
230	C₂×C₃×C₄	—	10	4	no	—
231	C₂×C₃×C₄	—	10	4	no	—
232	C₂×C₃×C₄	—	20	4	no	—
233	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
234	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	#713	1	2	yes	—
236	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
237	C₃×C₈	—	10	2	no	—
238	C₃×C₈	C₂×D₂₀	1	4	yes	—
239	C₂²×C₅	C₃×#121	1	2	yes	—
240	C₂²×C₅	C₃×#137	1	4	yes	—
241	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
242	C₂×D₅	—	2	4	no	—
243	C₂×D₅	—	2	2	no	—
244	C₂×D₅	—	4	4	no	—
245	C₄×C₅	—	4	4	no	—
246	C₄×C₅	C₃×#121	1	2	yes	—
247	C₄×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
248	C₄×C₅	C₂×C₃×#41	1	1	yes	characteristic
249	Dic₅	—	2	2	no	—
250	Dic₅	—	4	4	no	—
251	C₂⁴	—	5	1	no	—
252	C₂×Q₈	—	5	4	no	—
253	C₂×Q₈	—	10	2	no	—
254	C₂×Q₈	—	10	4	no	—
255	C₂×Q₈	—	10	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	5	1	no	—
258	C₂²×C₄	—	5	2	no	—
259	C₂²×C₄	—	5	2	no	—
260	C₂²×C₄	—	10	2	no	—
261	C₂²×C₄	—	10	2	no	—
262	C₂²×C₄	—	10	2	no	—
263	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
264	Q₁₆	—	10	8	no	—
265	M₁₆	—	10	8	no	—
266	C₂×C₈	—	5	2	no	—
267	C₂×C₈	—	10	2	no	—
268	C₂×C₈	C₃×D₂₀	1	4	yes	—
269	C₂×C₈	C₂²×C₃×D₅	1	2	yes	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	10	8	no	—
272	C₄⋊C₄	C₂²×C₃×D₅	1	4	yes	—
273	C₂²⋊C₄	—	10	8	no	—
274	C₃×C₅	C₂×#470	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	—	10	4	no	—
276	C₂²×C₃	—	10	2	no	—
277	C₂²×C₃	—	20	4	no	—
278	C₂²×C₃	#4041	1	2	yes	—
279	C₂²×C₃	#4061	1	4	yes	—
280	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	—	10	2	no	—
283	C₃×C₄	—	20	4	no	—
284	C₃×C₄	#4041	1	2	yes	—
285	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
286	C₃×C₄	C₂×#713	1	1	yes	characteristic
287	C₂×C₅	C₃×#470	1	4	yes	complemented
288	C₂×C₅	C₂×C₃×#121	1	1	yes	characteristic
289	C₂×C₅	C₂×C₃×#137	1	2	yes	—
290	D₅	—	4	2	no	—
291	C₂³	—	5	2	no	—
292	C₂³	—	10	2	no	—
293	C₂³	—	10	4	no	—
294	C₂³	C₃×D₄×D₅	1	1	yes	characteristic
295	Q₈	—	10	4	no	—
296	Q₈	—	10	4	no	—
297	Q₈	—	20	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	10	2	no	—
302	C₂×C₄	—	10	2	no	—
303	C₂×C₄	—	10	2	no	—
304	C₂×C₄	—	10	4	no	—
305	C₂×C₄	—	10	4	no	—
306	C₂×C₄	—	20	4	no	—
307	C₂×C₄	C₂×C₃×D₂₀	1	2	yes	—
308	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
309	C₂×C₄	C₃×#713	1	2	yes	—
310	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
311	C₈	—	10	2	no	—
312	C₈	C₂×C₃×D₂₀	1	4	yes	—
313	C₂×C₃	—	20	2	no	—
314	C₂×C₃	#65157	1	4	yes	direct product factor
315	C₂×C₃	C₂×#4041	1	1	yes	characteristic
316	C₂×C₃	C₂×#4061	1	2	yes	—
317	C₅	C₂×C₃×#470	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	10	4	no	—
319	C₂²	—	10	2	no	—
320	C₂²	—	20	4	no	—
321	C₂²	C₃×#4041	1	2	yes	—
322	C₂²	C₃×#4061	1	4	yes	—
323	C₂²	C₂×C₃×D₄×D₅	1	1	yes	characteristic
324	C₄	—	4	4	no	—
325	C₄	—	10	2	no	—
326	C₄	—	20	4	no	—
327	C₄	C₃×#4041	1	2	yes	—
328	C₄	C₂²×C₃×D₂₀	1	1	yes	characteristic
329	C₄	C₂×C₃×#713	1	1	yes	characteristic
330	C₃	C₂×#65157	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	20	2	no	—
332	C₂	C₃×#65157	1	4	yes	direct product factor
333	C₂	C₂×C₃×#4041	1	1	yes	characteristic
334	C₂	C₂×C₃×#4061	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×#106	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×C₃×D₅	D₄
hypercenter	C₂×C₃×#106	D₅
hypocenter	C₅	C₂×C₃×#470
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#106

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×Dic₂₀	C₂	semidirect
4. C₂×C₃×C₅×#106	C₂	semidirect
5. C₂×C₃×#3942	C₂	semidirect
6. C₃×#65157	C₂	direct (N × H)
7. C₂×#65157	C₃	direct (N × H)
16. C₂×C₃×Dic₂₀	C₂²	semidirect
21. C₃×C₅×#106	C₂²	semidirect
22. C₃×#3942	C₂²	semidirect
24. C₂²×Dic₂₀	C₂×C₃	semidirect
27. C₂×C₅×#106	C₂×C₃	semidirect
28. C₂×#3942	C₂×C₃	semidirect
29. #65157	C₂×C₃	direct (N × H)
57. C₂×C₃×#106	D₅	semidirect
67. C₂×Dic₂₀	C₂²×C₃	semidirect
72. C₅×#106	C₂²×C₃	semidirect
73. #3942	C₂²×C₃	semidirect
110. C₃×#106	C₂×D₅	semidirect
139. C₂×#106	C₃×D₅	semidirect
211. #106	C₂×C₃×D₅	semidirect
213. C₂×C₃×C₅	#470	semidirect
274. C₃×C₅	C₂×#470	semidirect
287. C₂×C₅	C₃×#470	semidirect
314. C₂×C₃	#65157	direct (N × H)
317. C₅	C₂×C₃×#470	semidirect
330. C₃	C₂×#65157	direct (N × H)
332. C₂	C₃×#65157	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×Dic₂₀	2
C₂²×C₃×#676	2
C₂×C₃×#4043	2
C₂×C₃×C₅×#106	2
C₂×C₃×#3942	2
C₃×#65157	2
C₂×#65157	3
C₂×C₃×#470	5

Subgroups — C2×C3×#65157 #14428045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#65157 #14428045