Subgroups — C₂×C₅×#4867 #14428044

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×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#897	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#810	C₂	1	2	yes	maximal, complemented
6	C₅×#4867	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₅×#470	—	3	1	no	maximal, Hall
8	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₆	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×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
13	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
14	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×Dic₁₂	—	2	4	no	—
16	C₂×C₅×Dic₁₂	C₂²	1	4	yes	complemented
17	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
18	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	—
19	C₅×#897	—	2	4	no	—
20	C₅×#897	C₂²	1	8	yes	—
21	C₃×C₅×#106	C₂²	1	4	yes	complemented
22	C₅×#810	C₂²	1	8	yes	complemented
23	C₂×#4867	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₅×Q₁₆	—	3	1	no	—
25	C₂²×C₅×M₁₆	—	3	1	no	—
26	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
27	C₅×#470	—	3	8	no	—
28	C₂×C₅×#106	D₃	1	1	yes	characteristic, complemented
29	C₂×C₅×#103	—	3	2	no	—
30	C₂³×C₅×D₃	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×C₅×Dic₃	—	2	2	no	—
34	C₂²×C₅×Dic₃	D₄	1	1	yes	characteristic
35	C₂×C₄×C₅×D₃	—	2	2	no	—
36	C₂×C₄×C₅×D₃	—	2	2	no	—
37	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
39	C₂×C₅×Dic₆	—	2	4	no	—
40	C₂×C₅×Dic₆	—	2	4	no	—
41	C₂×C₅×Dic₆	—	2	2	no	—
42	C₂×C₅×Dic₆	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	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
48	C₅×C₄⋊Dic₃	—	2	8	no	—
49	C₂×C₅×C₃⋊C₈	—	2	2	no	—
50	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
51	C₅×Dic₁₂	—	2	8	no	—
52	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
53	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂²×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic
55	C₂×#897	C₂×C₅	1	2	yes	—
56	C₂×C₃×#106	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×#810	C₂×C₅	1	2	yes	complemented
58	#4867	C₂×C₅	1	8	yes	direct product factor
59	C₂²×C₅×Q₈	—	3	2	no	—
60	C₂³×C₄×C₅	—	3	1	no	—
61	C₂×C₅×Q₁₆	—	3	4	no	—
62	C₂×C₅×Q₁₆	—	6	4	no	—
63	C₂×C₅×M₁₆	—	3	4	no	—
64	C₂×C₅×M₁₆	—	6	4	no	—
65	C₂²×C₅×C₈	—	3	1	no	—
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₅×C₂²⋊Q₈	—	3	8	no	—
68	C₅×C₂²⋊Q₈	—	6	4	no	—
69	C₂×C₅×C₄⋊C₄	—	6	2	no	—
70	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
71	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
72	C₅×#106	C₂×D₃	1	4	yes	complemented
73	C₅×#103	—	3	8	no	—
74	C₂×#470	—	3	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₅×D₃	—	2	2	no	—
77	C₂²×C₅×D₃	—	2	4	no	—
78	C₂²×C₅×D₃	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×C₅×Dic₃	—	2	2	no	—
86	C₂×C₅×Dic₃	—	2	4	no	—
87	C₂×C₅×Dic₃	—	4	4	no	—
88	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
89	C₄×C₅×D₃	—	2	4	no	—
90	C₄×C₅×D₃	—	2	2	no	—
91	C₄×C₅×D₃	—	2	2	no	—
92	C₅×Dic₆	—	2	4	no	—
93	C₅×Dic₆	—	2	4	no	—
94	C₅×Dic₆	—	4	4	no	—
95	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
96	C₅×C₃⋊C₈	—	2	2	no	—
97	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
98	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
99	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
101	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
102	C₂×C₄⋊Dic₃	—	2	2	no	—
103	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×Dic₁₂	—	2	4	no	—
105	C₂×Dic₁₂	C₂²×C₅	1	4	yes	complemented
106	C₂×ΓL(1,𝔽₂₅)	—	2	4	no	—
107	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	4	yes	—
108	#897	—	2	4	no	—
109	#897	C₂²×C₅	1	8	yes	—
110	C₃×#106	C₂²×C₅	1	4	yes	complemented
111	#810	C₂²×C₅	1	8	yes	complemented
112	C₂⁴×C₅	—	3	1	no	—
113	C₂×C₅×Q₈	—	3	4	no	—
114	C₂×C₅×Q₈	—	6	2	no	—
115	C₂×C₅×Q₈	—	6	4	no	—
116	C₂×C₅×Q₈	—	6	4	no	—
117	C₂²×C₄×C₅	—	2	2	no	—
118	C₂²×C₄×C₅	—	3	1	no	—
119	C₂²×C₄×C₅	—	3	2	no	—
120	C₂²×C₄×C₅	—	3	2	no	—
121	C₂²×C₄×C₅	—	6	2	no	—
122	C₂²×C₄×C₅	—	6	2	no	—
123	C₂²×C₄×C₅	—	6	2	no	—
124	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₅×Q₁₆	—	6	8	no	—
126	C₅×M₁₆	—	6	8	no	—
127	C₂×C₅×C₈	—	3	2	no	—
128	C₂×C₅×C₈	—	6	2	no	—
129	C₂×C₅×C₈	D₁₂	1	4	yes	—
130	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
131	C₅×C₄⋊C₄	—	2	2	no	—
132	C₅×C₄⋊C₄	—	6	8	no	—
133	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	—
134	C₅×C₂²⋊C₄	—	6	8	no	—
135	C₂²×Q₁₆	—	3	1	no	—
136	C₂²×M₁₆	—	3	1	no	—
137	C₂×C₂²⋊Q₈	—	3	2	no	—
138	#470	—	3	8	no	—
139	C₂×#106	C₅×D₃	1	1	yes	characteristic, complemented
140	C₂×#103	—	3	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×C₅×D₃	—	2	2	no	—
145	C₂×C₅×D₃	—	2	4	no	—
146	C₂×C₅×D₃	—	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	C₅×Dic₃	—	2	2	no	—
152	C₅×Dic₃	—	4	4	no	—
153	C₂³×D₃	C₅×D₄	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
156	C₂²×Dic₃	—	2	2	no	—
157	C₂²×Dic₃	C₅×D₄	1	1	yes	characteristic
158	C₂×C₄×D₃	—	2	2	no	—
159	C₂×C₄×D₃	—	2	2	no	—
160	C₂×C₄×D₃	C₅×D₄	1	2	yes	—
161	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
162	C₂×Dic₆	—	2	4	no	—
163	C₂×Dic₆	—	2	4	no	—
164	C₂×Dic₆	—	2	2	no	—
165	C₂×Dic₆	C₂³×C₅	1	4	yes	—
166	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
167	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
168	C₃×C₄⋊C₄	—	2	2	no	—
169	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
170	(C₂×D₃)⋊C₄	—	2	8	no	—
171	C₄⋊Dic₃	—	2	8	no	—
172	C₂×C₃⋊C₈	—	2	2	no	—
173	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
174	Dic₁₂	—	2	8	no	—
175	ΓL(1,𝔽₂₅)	—	2	8	no	—
176	C₂³×C₅	—	3	2	no	—
177	C₂³×C₅	—	6	2	no	—
178	C₂³×C₅	—	6	4	no	—
179	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
180	C₅×Q₈	—	6	4	no	—
181	C₅×Q₈	—	6	4	no	—
182	C₅×Q₈	—	12	4	no	—
183	C₂×C₄×C₅	—	2	4	no	—
184	C₂×C₄×C₅	—	3	2	no	—
185	C₂×C₄×C₅	—	4	4	no	—
186	C₂×C₄×C₅	—	6	2	no	—
187	C₂×C₄×C₅	—	6	4	no	—
188	C₂×C₄×C₅	—	6	2	no	—
189	C₂×C₄×C₅	—	6	2	no	—
190	C₂×C₄×C₅	—	6	4	no	—
191	C₂×C₄×C₅	—	12	4	no	—
192	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
193	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
194	C₂×C₄×C₅	#239	1	2	yes	—
195	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₅×C₈	—	6	2	no	—
197	C₅×C₈	C₂×D₁₂	1	4	yes	—
198	C₂²×Q₈	—	3	2	no	—
199	C₂³×C₄	—	3	1	no	—
200	C₂×Q₁₆	—	3	4	no	—
201	C₂×Q₁₆	—	6	4	no	—
202	C₂×M₁₆	—	3	4	no	—
203	C₂×M₁₆	—	6	4	no	—
204	C₂²×C₈	—	3	1	no	—
205	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
206	C₂²⋊Q₈	—	3	8	no	—
207	C₂²⋊Q₈	—	6	4	no	—
208	C₂×C₄⋊C₄	—	6	2	no	—
209	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
210	C₂×C₂²⋊C₄	—	6	2	no	—
211	#106	C₂×C₅×D₃	1	4	yes	complemented
212	#103	—	3	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	C₅×D₃	—	4	2	no	—
217	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
218	C₂²×D₃	—	2	2	no	—
219	C₂²×D₃	—	2	4	no	—
220	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₃×C₄	—	2	4	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
224	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₅×#41	1	2	yes	—
226	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
227	C₂×Dic₃	—	2	2	no	—
228	C₂×Dic₃	—	2	4	no	—
229	C₂×Dic₃	—	4	4	no	—
230	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
231	C₄×D₃	—	2	4	no	—
232	C₄×D₃	—	2	2	no	—
233	C₄×D₃	—	2	2	no	—
234	Dic₆	—	2	4	no	—
235	Dic₆	—	2	4	no	—
236	Dic₆	—	4	4	no	—
237	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
238	C₃⋊C₈	—	2	2	no	—
239	C₂²×C₅	—	6	4	no	—
240	C₂²×C₅	—	6	2	no	—
241	C₂²×C₅	—	12	4	no	—
242	C₂²×C₅	#895	1	2	yes	—
243	C₂²×C₅	#915	1	4	yes	—
244	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
245	C₄×C₅	—	4	4	no	—
246	C₄×C₅	—	6	2	no	—
247	C₄×C₅	—	12	4	no	—
248	C₄×C₅	#895	1	2	yes	—
249	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
250	C₄×C₅	C₂×#239	1	1	yes	characteristic
251	C₂⁴	—	3	1	no	—
252	C₂×Q₈	—	3	4	no	—
253	C₂×Q₈	—	6	2	no	—
254	C₂×Q₈	—	6	4	no	—
255	C₂×Q₈	—	6	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	3	1	no	—
258	C₂²×C₄	—	3	2	no	—
259	C₂²×C₄	—	3	2	no	—
260	C₂²×C₄	—	6	2	no	—
261	C₂²×C₄	—	6	2	no	—
262	C₂²×C₄	—	6	2	no	—
263	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
264	Q₁₆	—	6	8	no	—
265	M₁₆	—	6	8	no	—
266	C₂×C₈	—	3	2	no	—
267	C₂×C₈	—	6	2	no	—
268	C₂×C₈	C₅×D₁₂	1	4	yes	—
269	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	6	8	no	—
272	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	—
273	C₂²⋊C₄	—	6	8	no	—
274	C₃×C₅	C₂×#470	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	C₅×#121	1	2	yes	—
276	C₂²×C₃	C₅×#137	1	4	yes	—
277	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
278	C₂×D₃	—	2	2	no	—
279	C₂×D₃	—	2	4	no	—
280	C₂×D₃	—	4	4	no	—
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	C₅×#121	1	2	yes	—
283	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
284	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
285	Dic₃	—	2	2	no	—
286	Dic₃	—	4	4	no	—
287	C₂×C₅	—	12	2	no	—
288	C₂×C₅	#4867	1	4	yes	direct product factor
289	C₂×C₅	C₂×#895	1	1	yes	characteristic
290	C₂×C₅	C₂×#915	1	2	yes	—
291	C₂³	—	3	2	no	—
292	C₂³	—	6	2	no	—
293	C₂³	—	6	4	no	—
294	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
295	Q₈	—	6	4	no	—
296	Q₈	—	6	4	no	—
297	Q₈	—	12	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	3	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	6	2	no	—
302	C₂×C₄	—	6	4	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	6	2	no	—
305	C₂×C₄	—	6	4	no	—
306	C₂×C₄	—	12	4	no	—
307	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
308	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
309	C₂×C₄	C₅×#239	1	2	yes	—
310	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
311	C₈	—	6	2	no	—
312	C₈	C₂×C₅×D₁₂	1	4	yes	—
313	C₂×C₃	C₅×#470	1	4	yes	complemented
314	C₂×C₃	C₂×C₅×#121	1	1	yes	characteristic
315	C₂×C₃	C₂×C₅×#137	1	2	yes	—
316	D₃	—	4	2	no	—
317	C₅	C₂×#4867	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	6	4	no	—
319	C₂²	—	6	2	no	—
320	C₂²	—	12	4	no	—
321	C₂²	C₅×#895	1	2	yes	—
322	C₂²	C₅×#915	1	4	yes	—
323	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
324	C₄	—	4	4	no	—
325	C₄	—	6	2	no	—
326	C₄	—	12	4	no	—
327	C₄	C₅×#895	1	2	yes	—
328	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
329	C₄	C₂×C₅×#239	1	1	yes	characteristic
330	C₃	C₂×C₅×#470	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	12	2	no	—
332	C₂	C₅×#4867	1	4	yes	direct product factor
333	C₂	C₂×C₅×#895	1	1	yes	characteristic
334	C₂	C₂×C₅×#915	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#106	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×C₅×D₃	D₄
hypercenter	C₂×C₅×#106	D₃
hypocenter	C₃	C₂×C₅×#470
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#106

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Dic₁₂	C₂	semidirect
4. C₂×C₃×C₅×#106	C₂	semidirect
5. C₂×C₅×#810	C₂	semidirect
6. C₅×#4867	C₂	direct (N × H)
16. C₂×C₅×Dic₁₂	C₂²	semidirect
21. C₃×C₅×#106	C₂²	semidirect
22. C₅×#810	C₂²	semidirect
23. C₂×#4867	C₅	direct (N × H)
28. C₂×C₅×#106	D₃	semidirect
53. C₂²×Dic₁₂	C₂×C₅	semidirect
56. C₂×C₃×#106	C₂×C₅	semidirect
57. C₂×#810	C₂×C₅	semidirect
58. #4867	C₂×C₅	direct (N × H)
72. C₅×#106	C₂×D₃	semidirect
105. C₂×Dic₁₂	C₂²×C₅	semidirect
110. C₃×#106	C₂²×C₅	semidirect
111. #810	C₂²×C₅	semidirect
139. C₂×#106	C₅×D₃	semidirect
211. #106	C₂×C₅×D₃	semidirect
213. C₂×C₃×C₅	#470	semidirect
274. C₃×C₅	C₂×#470	semidirect
288. C₂×C₅	#4867	direct (N × H)
313. C₂×C₃	C₅×#470	semidirect
317. C₅	C₂×#4867	direct (N × H)
330. C₃	C₂×C₅×#470	semidirect
332. C₂	C₅×#4867	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×ΓL(1,𝔽₂₅)	2
C₂×C₅×#897	2
C₂×C₃×C₅×#106	2
C₂×C₅×#810	2
C₅×#4867	2
C₂×C₅×#470	3
C₂×#4867	5

Subgroups — C2×C5×#4867 #14428044

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4867 #14428044