Subgroups — C₂×C₄×C₅×Dic₁₂ #14416151

Order 1920.

976 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	C₂²×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×Dic₆	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#818	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#816	C₂	1	2	yes	maximal
6	C₄×C₅×Dic₁₂	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×C₅×Q₁₆	—	3	1	no	maximal, Hall
8	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
14	C₂×C₄×C₅×Dic₃	—	2	2	no	—
15	C₂×C₅×Dic₁₂	C₄	1	8	yes	direct product factor
16	C₂×C₅×Dic₁₂	C₂²	1	4	yes	—
17	C₄×C₅×Dic₆	—	2	4	no	—
18	C₄×C₅×Dic₆	C₂²	1	8	yes	—
19	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
20	C₅×#818	C₂²	1	4	yes	—
21	C₅×#816	C₂²	1	8	yes	—
22	C₂×C₄×Dic₁₂	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂²×C₅×Q₁₆	—	3	1	no	—
24	C₂×C₄×C₅×Q₈	—	3	2	no	—
25	C₄×C₅×Q₁₆	—	3	8	no	—
26	C₂×C₅×#107	—	3	1	no	—
27	C₂×C₅×#103	—	3	2	no	—
28	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	—	2	2	no	—
33	C₂²×C₅×Dic₃	—	2	2	no	—
34	C₂×C₅×Dic₆	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	4	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	8	yes	—
37	C₂×C₅×Dic₆	C₂³	1	4	yes	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₄²×C₅	D₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	2	yes	—
44	C₅×C₄⋊Dic₃	—	2	2	no	—
45	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
46	C₅×Dic₃⋊C₄	—	2	8	no	—
47	C₄×C₅×Dic₃	—	2	8	no	—
48	C₅×Dic₁₂	C₂×C₄	1	16	yes	direct product factor
49	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic
50	C₂×C₄×Dic₆	C₂×C₅	1	2	yes	—
51	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
52	C₂×#818	C₂×C₅	1	1	yes	characteristic
53	C₂×#816	C₂×C₅	1	2	yes	—
54	C₄×Dic₁₂	C₂×C₅	1	8	yes	direct product factor
55	C₂²×C₅×Q₈	—	3	2	no	—
56	C₂×C₅×Q₁₆	—	3	8	no	—
57	C₂×C₅×Q₁₆	—	3	4	no	—
58	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	C₄×C₅×Q₈	—	3	8	no	—
61	C₄×C₅×Q₈	—	6	4	no	—
62	C₂×C₅×C₄⋊C₄	—	3	2	no	—
63	C₂×C₅×C₄⋊C₄	—	6	2	no	—
64	C₂×C₄²×C₅	—	6	2	no	—
65	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
66	C₅×#107	—	3	4	no	—
67	C₅×#103	—	3	8	no	—
68	C₄×C₅×C₈	C₂×D₃	1	4	yes	—
69	C₂×C₄×Q₁₆	—	3	1	no	Sylow
70	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	complemented
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂×C₃×C₄×C₅	#41	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×C₅×Dic₃	—	2	8	no	—
80	C₂×C₅×Dic₃	—	2	4	no	—
81	C₂×C₅×Dic₃	—	2	4	no	—
82	C₂×C₅×Dic₃	—	4	4	no	—
83	C₅×Dic₆	—	2	8	no	—
84	C₅×Dic₆	—	2	4	no	—
85	C₅×Dic₆	C₂²×C₄	1	8	yes	—
86	C₃×C₅×C₈	—	2	2	no	—
87	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
88	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
89	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
93	C₂×Dic₃⋊C₄	—	2	2	no	—
94	C₂×C₄×Dic₃	—	2	2	no	—
95	C₂×Dic₁₂	C₄×C₅	1	8	yes	direct product factor
96	C₂×Dic₁₂	C₂²×C₅	1	4	yes	—
97	C₄×Dic₆	—	2	4	no	—
98	C₄×Dic₆	C₂²×C₅	1	8	yes	—
99	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
100	#818	C₂²×C₅	1	4	yes	—
101	#816	C₂²×C₅	1	8	yes	—
102	C₂×C₅×Q₈	—	3	8	no	—
103	C₂×C₅×Q₈	—	3	4	no	—
104	C₂×C₅×Q₈	—	6	2	no	—
105	C₂×C₅×Q₈	—	6	4	no	—
106	C₂²×C₄×C₅	—	6	2	no	—
107	C₂²×C₄×C₅	—	6	2	no	—
108	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
109	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
110	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₅×Q₁₆	—	3	16	no	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	C₄×D₃	1	4	yes	—
114	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
115	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
116	C₅×C₄⋊C₄	—	3	4	no	—
117	C₅×C₄⋊C₄	—	6	2	no	—
118	C₅×C₄⋊C₄	—	6	8	no	—
119	C₄²×C₅	—	6	8	no	—
120	C₄²×C₅	D₁₂	1	2	yes	—
121	C₄²×C₅	C₂²×D₃	1	2	yes	—
122	C₂²×Q₁₆	—	3	1	no	—
123	C₂×C₄×Q₈	—	3	2	no	—
124	C₄×Q₁₆	—	3	8	no	—
125	C₂×#107	—	3	1	no	—
126	C₂×#103	—	3	2	no	—
127	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic
128	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
129	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
130	C₂²×C₃×C₅	#135	1	2	yes	—
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₃×C₄×C₅	—	2	2	no	—
133	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	complemented
135	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
137	C₅×Dic₃	—	2	8	no	—
138	C₅×Dic₃	—	4	4	no	—
139	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
140	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
142	C₂²×Dic₃	—	2	2	no	—
143	C₂²×Dic₃	—	2	2	no	—
144	C₂×Dic₆	—	2	2	no	—
145	C₂×Dic₆	—	2	4	no	—
146	C₂×Dic₆	C₂×C₄×C₅	1	8	yes	—
147	C₂×Dic₆	C₂³×C₅	1	4	yes	—
148	C₂×C₃×C₈	—	2	2	no	—
149	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
150	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
151	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
152	C₃×C₄²	C₅×D₄	1	2	yes	—
153	C₃×C₄²	C₂³×C₅	1	2	yes	—
154	C₄⋊Dic₃	—	2	2	no	—
155	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
156	Dic₃⋊C₄	—	2	8	no	—
157	C₄×Dic₃	—	2	8	no	—
158	Dic₁₂	C₂×C₄×C₅	1	16	yes	direct product factor
159	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
160	C₅×Q₈	—	3	8	no	—
161	C₅×Q₈	—	6	8	no	—
162	C₅×Q₈	—	6	4	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	6	8	no	—
165	C₂×C₄×C₅	—	6	4	no	—
166	C₂×C₄×C₅	—	6	4	no	—
167	C₂×C₄×C₅	—	12	4	no	—
168	C₂×C₄×C₅	Dic₁₂	1	4	yes	direct product factor
169	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
170	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
171	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
173	C₂×C₄×C₅	#235	1	2	yes	—
174	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
175	C₅×C₈	—	2	2	no	—
176	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
177	C₂²×Q₈	—	3	2	no	—
178	C₂×Q₁₆	—	3	8	no	—
179	C₂×Q₁₆	—	3	4	no	—
180	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
181	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
182	C₄×Q₈	—	3	8	no	—
183	C₄×Q₈	—	6	4	no	—
184	C₂×C₄⋊C₄	—	3	2	no	—
185	C₂×C₄⋊C₄	—	6	2	no	—
186	C₂×C₄²	—	6	2	no	—
187	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
188	#107	—	3	4	no	—
189	#103	—	3	8	no	—
190	C₄×C₈	C₂×C₅×D₃	1	4	yes	—
191	C₂×C₃×C₅	C₄×Q₁₆	1	4	yes	complemented
192	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
194	C₂×C₃×C₅	C₂×#135	1	1	yes	characteristic
195	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	complemented
198	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
202	C₂×C₃×C₄	C₅×#41	1	2	yes	—
203	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
204	C₂×Dic₃	—	2	8	no	—
205	C₂×Dic₃	—	2	4	no	—
206	C₂×Dic₃	—	2	4	no	—
207	C₂×Dic₃	—	4	4	no	—
208	Dic₆	—	2	8	no	—
209	Dic₆	—	2	4	no	—
210	Dic₆	C₂²×C₄×C₅	1	8	yes	—
211	C₃×C₈	—	2	2	no	—
212	C₃×C₈	C₂²×C₄×C₅	1	4	yes	—
213	C₂²×C₅	C₄×D₁₂	1	2	yes	—
214	C₂²×C₅	#904	1	2	yes	—
215	C₂²×C₅	C₂×Dic₁₂	1	2	yes	—
216	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	6	8	no	—
219	C₄×C₅	—	12	4	no	—
220	C₄×C₅	C₄×D₁₂	1	2	yes	—
221	C₄×C₅	C₂×Dic₁₂	1	4	yes	direct product factor
222	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
223	C₄×C₅	C₂×#235	1	1	yes	characteristic
224	C₂×Q₈	—	3	8	no	—
225	C₂×Q₈	—	3	4	no	—
226	C₂×Q₈	—	6	2	no	—
227	C₂×Q₈	—	6	4	no	—
228	C₂²×C₄	—	6	2	no	—
229	C₂²×C₄	—	6	2	no	—
230	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
231	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
233	Q₁₆	—	3	16	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₄×C₅×D₃	1	4	yes	—
236	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
237	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
238	C₄⋊C₄	—	3	4	no	—
239	C₄⋊C₄	—	6	2	no	—
240	C₄⋊C₄	—	6	8	no	—
241	C₄²	—	6	8	no	—
242	C₄²	C₅×D₁₂	1	2	yes	—
243	C₄²	C₂²×C₅×D₃	1	2	yes	—
244	C₃×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
246	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
247	C₂²×C₃	C₅×#135	1	2	yes	—
248	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
249	C₃×C₄	—	2	2	no	—
250	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
251	C₃×C₄	C₂×C₅×Q₁₆	1	4	yes	complemented
252	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
253	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
254	Dic₃	—	2	8	no	—
255	Dic₃	—	4	4	no	—
256	C₂×C₅	C₄×Dic₁₂	1	4	yes	direct product factor
257	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
258	C₂×C₅	C₂×#904	1	1	yes	characteristic
259	C₂×C₅	C₂²×Dic₁₂	1	1	yes	characteristic
260	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
261	Q₈	—	3	8	no	—
262	Q₈	—	6	8	no	—
263	Q₈	—	6	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	6	8	no	—
266	C₂×C₄	—	6	4	no	—
267	C₂×C₄	—	6	4	no	—
268	C₂×C₄	—	12	4	no	—
269	C₂×C₄	C₅×Dic₁₂	1	4	yes	direct product factor
270	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
271	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
272	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
273	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
274	C₂×C₄	C₅×#235	1	2	yes	—
275	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
276	C₈	—	2	2	no	—
277	C₈	C₂×C₄×C₅×D₃	1	4	yes	—
278	C₂×C₃	C₄×C₅×Q₁₆	1	4	yes	complemented
279	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
280	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
281	C₂×C₃	C₂×C₅×#135	1	1	yes	characteristic
282	C₅	C₂×C₄×Dic₁₂	1	1	yes	characteristic, direct product factor, Sylow
283	C₂²	C₄×C₅×D₁₂	1	2	yes	—
284	C₂²	C₅×#904	1	2	yes	—
285	C₂²	C₂×C₅×Dic₁₂	1	2	yes	—
286	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	6	8	no	—
289	C₄	—	12	4	no	—
290	C₄	C₄×C₅×D₁₂	1	2	yes	—
291	C₄	C₂×C₅×Dic₁₂	1	4	yes	direct product factor
292	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
293	C₄	C₂×C₅×#235	1	1	yes	characteristic
294	C₃	C₂×C₄×C₅×Q₁₆	1	1	yes	characteristic, complemented, Sylow
295	C₂	C₄×C₅×Dic₁₂	1	4	yes	direct product factor
296	C₂	C₂×C₄×C₅×D₁₂	1	1	yes	characteristic
297	C₂	C₂×C₅×#904	1	1	yes	characteristic
298	C₂	C₂²×C₅×Dic₁₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	D₁₂
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₂×C₄×C₅×Q₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
6. C₄×C₅×Dic₁₂	C₂	direct (N × H)
15. C₂×C₅×Dic₁₂	C₄	direct (N × H)
22. C₂×C₄×Dic₁₂	C₅	direct (N × H)
48. C₅×Dic₁₂	C₂×C₄	direct (N × H)
54. C₄×Dic₁₂	C₂×C₅	direct (N × H)
72. C₂×C₃×C₄×C₅	Q₁₆	semidirect
95. C₂×Dic₁₂	C₄×C₅	direct (N × H)
134. C₃×C₄×C₅	C₂×Q₁₆	semidirect
158. Dic₁₂	C₂×C₄×C₅	direct (N × H)
168. C₂×C₄×C₅	Dic₁₂	direct (N × H)
191. C₂×C₃×C₅	C₄×Q₁₆	semidirect
197. C₂×C₃×C₄	C₅×Q₁₆	semidirect
221. C₄×C₅	C₂×Dic₁₂	direct (N × H)
244. C₃×C₅	C₂×C₄×Q₁₆	semidirect
251. C₃×C₄	C₂×C₅×Q₁₆	semidirect
256. C₂×C₅	C₄×Dic₁₂	direct (N × H)
269. C₂×C₄	C₅×Dic₁₂	direct (N × H)
278. C₂×C₃	C₄×C₅×Q₁₆	semidirect
282. C₅	C₂×C₄×Dic₁₂	direct (N × H)
291. C₄	C₂×C₅×Dic₁₂	direct (N × H)
294. C₃	C₂×C₄×C₅×Q₁₆	semidirect
295. C₂	C₄×C₅×Dic₁₂	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×C₅×Dic₁₂	2
C₂×C₄×C₅×Dic₆	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#818	2
C₂×C₅×#816	2
C₄×C₅×Dic₁₂	2
C₂×C₄×C₅×Q₁₆	3
C₂×C₄×Dic₁₂	5

Subgroups — C2×C4×C5×Dic12 #14416151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×Dic₁₂ #14416151