Subgroups — C₂×C₅×#4696 #14438835

Order 1920.

1184 conjugacy classes of subgroups.

The subgroup lattice has 320 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, complemented
2	C₂²×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#204	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#875	C₂	1	2	yes	maximal, complemented
5	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4696	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#495	—	3	1	no	maximal, Hall
8	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
9	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
14	C₂×C₅×Dic₁₂	—	2	4	no	—
15	C₂×C₅×Dic₁₂	C₂²	1	4	yes	complemented
16	C₂×C₅×D₂₄	—	2	4	no	—
17	C₂×C₅×D₂₄	C₂²	1	4	yes	—
18	C₂×C₅×#204	—	2	8	no	—
19	C₂×C₅×#204	C₂²	1	8	yes	complemented
20	C₅×#875	—	2	4	no	—
21	C₅×#875	C₂²	1	8	yes	complemented
22	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
23	C₂×#4696	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₅×Q₁₆	—	3	1	no	—
25	C₂²×C₅×SD₁₆	—	3	2	no	—
26	C₂²×C₅×D₈	—	3	1	no	—
27	C₂×C₅×#124	—	3	2	no	—
28	C₅×#495	—	3	8	no	—
29	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
30	C₂³×C₅×D₃	—	2	2	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂²×C₅×Dic₃	—	2	2	no	—
34	C₂×C₅×D₁₂	—	2	4	no	—
35	C₂×C₅×D₁₂	—	2	4	no	—
36	C₂×C₅×D₁₂	—	2	2	no	—
37	C₂×C₅×D₁₂	C₂³	1	4	yes	—
38	C₂×C₅×Dic₆	—	2	4	no	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	2	2	no	—
41	C₂×C₅×Dic₆	C₂³	1	4	yes	—
42	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
43	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
44	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	C₃×C₄²×C₅	D₄	1	2	yes	—
47	C₃×C₄²×C₅	C₂³	1	2	yes	—
48	C₅×(C₂×D₃)⋊C₄	—	2	16	no	—
49	C₅×Dic₁₂	—	2	8	no	—
50	C₅×D₂₄	—	2	8	no	—
51	C₅×#204	—	2	16	no	—
52	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂²×D₂₄	C₂×C₅	1	1	yes	characteristic
54	C₂²×#204	C₂×C₅	1	2	yes	complemented
55	C₂×#875	C₂×C₅	1	2	yes	complemented
56	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
57	#4696	C₂×C₅	1	8	yes	direct product factor
58	C₂²×C₅×Q₈	—	3	2	no	—
59	C₂²×C₅×D₄	—	3	2	no	—
60	C₂×C₅×Q₁₆	—	3	4	no	—
61	C₂×C₅×Q₁₆	—	6	4	no	—
62	C₂×C₅×SD₁₆	—	3	8	no	—
63	C₂×C₅×SD₁₆	—	6	8	no	—
64	C₂×C₅×D₈	—	3	4	no	—
65	C₂×C₅×D₈	—	6	4	no	—
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
68	C₅×#124	—	3	8	no	—
69	C₅×#124	—	6	4	no	—
70	C₂×C₅×C₂²⋊C₄	—	6	4	no	—
71	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
72	C₄×C₅×C₈	C₂×D₃	1	4	yes	complemented
73	C₂×#495	—	3	1	no	Sylow
74	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₅×D₃	—	2	4	no	—
76	C₂²×C₅×D₃	—	4	4	no	—
77	C₂²×C₅×D₃	—	4	8	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
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×C₅×Dic₃	—	2	4	no	—
85	C₂×C₅×Dic₃	—	4	4	no	—
86	C₅×D₁₂	—	2	4	no	—
87	C₅×D₁₂	—	2	4	no	—
88	C₅×D₁₂	—	4	4	no	—
89	C₅×Dic₆	—	2	4	no	—
90	C₅×Dic₆	—	2	4	no	—
91	C₅×Dic₆	—	4	4	no	—
92	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
93	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
94	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
95	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
96	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
97	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
99	C₂×(C₂×D₃)⋊C₄	—	2	4	no	—
100	C₂×Dic₁₂	—	2	4	no	—
101	C₂×Dic₁₂	C₂²×C₅	1	4	yes	complemented
102	C₂×D₂₄	—	2	4	no	—
103	C₂×D₂₄	C₂²×C₅	1	4	yes	—
104	C₂×#204	—	2	8	no	—
105	C₂×#204	C₂²×C₅	1	8	yes	complemented
106	#875	—	2	4	no	—
107	#875	C₂²×C₅	1	8	yes	complemented
108	C₃×C₄×C₈	C₂²×C₅	1	4	yes	complemented
109	C₂⁴×C₅	—	6	2	no	—
110	C₂×C₅×Q₈	—	3	4	no	—
111	C₂×C₅×Q₈	—	6	2	no	—
112	C₂×C₅×Q₈	—	6	4	no	—
113	C₂×C₅×Q₈	—	6	4	no	—
114	C₂×C₅×D₄	—	3	4	no	—
115	C₂×C₅×D₄	—	6	4	no	—
116	C₂×C₅×D₄	—	6	4	no	—
117	C₂×C₅×D₄	—	6	2	no	—
118	C₂²×C₄×C₅	—	6	2	no	—
119	C₂²×C₄×C₅	D₁₂	1	2	yes	—
120	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₅×Q₁₆	—	6	8	no	—
122	C₅×SD₁₆	—	6	16	no	—
123	C₅×D₈	—	6	8	no	—
124	C₂×C₅×C₈	D₁₂	1	4	yes	—
125	C₂×C₅×C₈	D₁₂	1	4	yes	—
126	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
127	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
128	C₅×C₂²⋊C₄	—	6	16	no	—
129	C₄²×C₅	D₁₂	1	2	yes	—
130	C₄²×C₅	C₂²×D₃	1	2	yes	—
131	C₂²×Q₁₆	—	3	1	no	—
132	C₂²×SD₁₆	—	3	2	no	—
133	C₂²×D₈	—	3	1	no	—
134	C₂×#124	—	3	2	no	—
135	#495	—	3	8	no	—
136	C₂×C₄×C₈	C₅×D₃	1	1	yes	characteristic, complemented
137	C₂²×C₃×C₅	#127	1	2	yes	—
138	C₂²×C₃×C₅	#135	1	4	yes	—
139	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₅×D₃	—	4	4	no	—
141	C₂×C₅×D₃	—	4	8	no	—
142	C₂×C₅×D₃	—	8	8	no	—
143	C₃×C₄×C₅	—	2	4	no	—
144	C₃×C₄×C₅	#127	1	2	yes	—
145	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₅×Dic₃	—	4	4	no	—
148	C₂³×D₃	—	2	2	no	—
149	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
150	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
151	C₂²×Dic₃	—	2	2	no	—
152	C₂×D₁₂	—	2	4	no	—
153	C₂×D₁₂	—	2	4	no	—
154	C₂×D₁₂	—	2	2	no	—
155	C₂×D₁₂	C₂³×C₅	1	4	yes	—
156	C₂×Dic₆	—	2	4	no	—
157	C₂×Dic₆	—	2	4	no	—
158	C₂×Dic₆	—	2	2	no	—
159	C₂×Dic₆	C₂³×C₅	1	4	yes	—
160	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
161	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
162	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
163	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
164	C₃×C₄²	C₅×D₄	1	2	yes	—
165	C₃×C₄²	C₂³×C₅	1	2	yes	—
166	(C₂×D₃)⋊C₄	—	2	16	no	—
167	Dic₁₂	—	2	8	no	—
168	D₂₄	—	2	8	no	—
169	#204	—	2	16	no	—
170	C₂³×C₅	—	6	4	no	—
171	C₂³×C₅	—	12	4	no	—
172	C₂³×C₅	—	12	8	no	—
173	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
174	C₅×Q₈	—	6	4	no	—
175	C₅×Q₈	—	6	4	no	—
176	C₅×Q₈	—	12	4	no	—
177	C₅×D₄	—	6	4	no	—
178	C₅×D₄	—	6	4	no	—
179	C₅×D₄	—	12	4	no	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	—	6	4	no	—
182	C₂×C₄×C₅	—	12	4	no	—
183	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
184	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
185	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
186	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
187	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
188	C₅×C₈	C₂×D₁₂	1	4	yes	—
189	C₅×C₈	C₂×D₁₂	1	4	yes	—
190	C₂²×Q₈	—	3	2	no	—
191	C₂²×D₄	—	3	2	no	—
192	C₂×Q₁₆	—	3	4	no	—
193	C₂×Q₁₆	—	6	4	no	—
194	C₂×SD₁₆	—	3	8	no	—
195	C₂×SD₁₆	—	6	8	no	—
196	C₂×D₈	—	3	4	no	—
197	C₂×D₈	—	6	4	no	—
198	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
199	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
200	#124	—	3	8	no	—
201	#124	—	6	4	no	—
202	C₂×C₂²⋊C₄	—	6	4	no	—
203	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
204	C₄×C₈	C₂×C₅×D₃	1	4	yes	complemented
205	C₂×C₃×C₅	#495	1	4	yes	complemented
206	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
207	C₂×C₃×C₅	C₂×#135	1	2	yes	—
208	C₅×D₃	—	8	4	no	—
209	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
210	C₂²×D₃	—	2	4	no	—
211	C₂²×D₃	—	4	4	no	—
212	C₂²×D₃	—	4	8	no	—
213	C₂×C₃×C₄	—	2	4	no	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
215	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
217	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
219	C₂×Dic₃	—	2	4	no	—
220	C₂×Dic₃	—	4	4	no	—
221	D₁₂	—	2	4	no	—
222	D₁₂	—	2	4	no	—
223	D₁₂	—	4	4	no	—
224	Dic₆	—	2	4	no	—
225	Dic₆	—	2	4	no	—
226	Dic₆	—	4	4	no	—
227	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
228	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
229	C₂²×C₅	—	12	4	no	—
230	C₂²×C₅	—	12	8	no	—
231	C₂²×C₅	—	24	8	no	—
232	C₂²×C₅	#874	1	2	yes	—
233	C₂²×C₅	#904	1	4	yes	—
234	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
235	C₄×C₅	—	2	4	no	—
236	C₄×C₅	—	12	4	no	—
237	C₄×C₅	#874	1	2	yes	—
238	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
239	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
240	C₂⁴	—	6	2	no	—
241	C₂×Q₈	—	3	4	no	—
242	C₂×Q₈	—	6	2	no	—
243	C₂×Q₈	—	6	4	no	—
244	C₂×Q₈	—	6	4	no	—
245	C₂×D₄	—	3	4	no	—
246	C₂×D₄	—	6	4	no	—
247	C₂×D₄	—	6	4	no	—
248	C₂×D₄	—	6	2	no	—
249	C₂²×C₄	—	6	2	no	—
250	C₂²×C₄	C₅×D₁₂	1	2	yes	—
251	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
252	Q₁₆	—	6	8	no	—
253	SD₁₆	—	6	16	no	—
254	D₈	—	6	8	no	—
255	C₂×C₈	C₅×D₁₂	1	4	yes	—
256	C₂×C₈	C₅×D₁₂	1	4	yes	—
257	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
258	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
259	C₂²⋊C₄	—	6	16	no	—
260	C₄²	C₅×D₁₂	1	2	yes	—
261	C₄²	C₂²×C₅×D₃	1	2	yes	—
262	C₃×C₅	C₂×#495	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	C₅×#127	1	2	yes	—
264	C₂²×C₃	C₅×#135	1	4	yes	—
265	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
266	C₂×D₃	—	4	4	no	—
267	C₂×D₃	—	4	8	no	—
268	C₂×D₃	—	8	8	no	—
269	C₃×C₄	—	2	4	no	—
270	C₃×C₄	C₅×#127	1	2	yes	—
271	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
272	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
273	Dic₃	—	4	4	no	—
274	C₂×C₅	—	24	4	no	—
275	C₂×C₅	#4696	1	4	yes	direct product factor
276	C₂×C₅	C₂×#874	1	1	yes	characteristic
277	C₂×C₅	C₂×#904	1	2	yes	—
278	C₂³	—	6	4	no	—
279	C₂³	—	12	4	no	—
280	C₂³	—	12	8	no	—
281	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
282	Q₈	—	6	4	no	—
283	Q₈	—	6	4	no	—
284	Q₈	—	12	4	no	—
285	D₄	—	6	4	no	—
286	D₄	—	6	4	no	—
287	D₄	—	12	4	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	6	4	no	—
290	C₂×C₄	—	12	4	no	—
291	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
292	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
293	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
294	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
295	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
296	C₈	C₂×C₅×D₁₂	1	4	yes	—
297	C₈	C₂×C₅×D₁₂	1	4	yes	—
298	C₂×C₃	C₅×#495	1	4	yes	complemented
299	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
300	C₂×C₃	C₂×C₅×#135	1	2	yes	—
301	D₃	—	8	4	no	—
302	C₅	C₂×#4696	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	12	4	no	—
304	C₂²	—	12	8	no	—
305	C₂²	—	24	8	no	—
306	C₂²	C₅×#874	1	2	yes	—
307	C₂²	C₅×#904	1	4	yes	—
308	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	12	4	no	—
311	C₄	C₅×#874	1	2	yes	—
312	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
313	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
314	C₃	C₂×C₅×#495	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	24	4	no	—
316	C₂	C₅×#4696	1	4	yes	direct product factor
317	C₂	C₂×C₅×#874	1	1	yes	characteristic
318	C₂	C₂×C₅×#904	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₅×D₂₄	C₂
hypercenter	C₂×C₄×C₅×C₈	D₃
hypocenter	C₃	C₂×C₅×#495
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Dic₁₂	C₂	semidirect
3. C₂²×C₅×#204	C₂	semidirect
4. C₂×C₅×#875	C₂	semidirect
5. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
6. C₅×#4696	C₂	direct (N × H)
15. C₂×C₅×Dic₁₂	C₂²	semidirect
19. C₂×C₅×#204	C₂²	semidirect
21. C₅×#875	C₂²	semidirect
22. C₃×C₄×C₅×C₈	C₂²	semidirect
23. C₂×#4696	C₅	direct (N × H)
29. C₂×C₄×C₅×C₈	D₃	semidirect
52. C₂²×Dic₁₂	C₂×C₅	semidirect
54. C₂²×#204	C₂×C₅	semidirect
55. C₂×#875	C₂×C₅	semidirect
56. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
57. #4696	C₂×C₅	direct (N × H)
72. C₄×C₅×C₈	C₂×D₃	semidirect
101. C₂×Dic₁₂	C₂²×C₅	semidirect
105. C₂×#204	C₂²×C₅	semidirect
107. #875	C₂²×C₅	semidirect
108. C₃×C₄×C₈	C₂²×C₅	semidirect
136. C₂×C₄×C₈	C₅×D₃	semidirect
204. C₄×C₈	C₂×C₅×D₃	semidirect
205. C₂×C₃×C₅	#495	semidirect
262. C₃×C₅	C₂×#495	semidirect
275. C₂×C₅	#4696	direct (N × H)
298. C₂×C₃	C₅×#495	semidirect
302. C₅	C₂×#4696	direct (N × H)
314. C₃	C₂×C₅×#495	semidirect
316. C₂	C₅×#4696	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
C₂²×C₅×D₂₄	2
C₂²×C₅×#204	2
C₂×C₅×#875	2
C₂×C₃×C₄×C₅×C₈	2
C₅×#4696	2
C₂×C₅×#495	3
C₂×#4696	5

Subgroups — C2×C5×#4696 #14438835

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4696 #14438835