Subgroups — C₂×C₅×#5137 #14428041

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×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#924	C₂	1	2	yes	maximal
3	C₂²×C₅×Dic₁₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5137	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#817	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#816	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#470	—	3	1	no	maximal, Hall
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	—	2	4	no	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	4	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₅×#217	—	2	2	no	—
15	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₅×#924	—	2	4	no	—
17	C₅×#924	C₂²	1	8	yes	—
18	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
19	C₂×C₅×Dic₁₂	—	2	4	no	—
20	C₂×C₅×Dic₁₂	C₂²	1	4	yes	complemented
21	C₅×#817	C₂²	1	4	yes	complemented
22	C₅×#816	C₂²	1	8	yes	complemented
23	C₂×#5137	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₂²×C₅×Q₁₆	—	3	1	no	—
25	C₂²×C₅×M₁₆	D₃	1	1	yes	characteristic
26	C₂×C₅×C₂²⋊Q₈	—	3	2	no	—
27	C₅×#470	—	3	8	no	—
28	C₂×C₅×#106	—	3	1	no	—
29	C₂×C₅×#103	—	3	2	no	—
30	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×C₅×Dic₃	—	2	2	no	—
38	C₂²×C₅×Dic₃	—	2	2	no	—
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	C₃×C₅×M₁₆	—	2	8	no	—
44	C₂×C₃×C₅×C₈	—	2	2	no	—
45	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
46	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
47	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
48	C₅×#217	—	2	8	no	—
49	C₅×C₄⋊Dic₃	—	2	2	no	—
50	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
51	C₅×Dic₃⋊C₄	—	2	8	no	—
52	C₅×Dic₁₂	—	2	8	no	—
53	C₂²×C₃×M₁₆	C₂×C₅	1	1	yes	characteristic
54	C₂×#924	C₂×C₅	1	2	yes	—
55	C₂²×Dic₁₂	C₂×C₅	1	1	yes	characteristic, complemented
56	#5137	C₂×C₅	1	8	yes	direct product factor
57	C₂×#817	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×#816	C₂×C₅	1	2	yes	complemented
59	C₂²×C₅×Q₈	—	3	2	no	—
60	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
61	C₂×C₅×Q₁₆	—	3	4	no	—
62	C₂×C₅×Q₁₆	—	6	4	no	—
63	C₂×C₅×M₁₆	—	2	4	no	—
64	C₂×C₅×M₁₆	C₂×D₃	1	4	yes	—
65	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
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	—	3	2	no	—
70	C₂×C₅×C₄⋊C₄	—	6	2	no	—
71	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
72	C₅×#106	—	3	4	no	—
73	C₅×#103	—	3	8	no	—
74	C₂×#470	—	3	1	no	Sylow
75	C₂³×C₃×C₅	—	2	2	no	—
76	C₂³×C₃×C₅	—	2	4	no	—
77	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×C₅×Dic₃	—	2	4	no	—
89	C₂×C₅×Dic₃	—	2	4	no	—
90	C₂×C₅×Dic₃	—	4	4	no	—
91	C₂×C₅×Dic₃	—	4	4	no	—
92	C₅×Dic₆	—	2	4	no	—
93	C₅×Dic₆	—	2	4	no	—
94	C₅×Dic₆	—	4	4	no	—
95	C₃×C₅×C₈	—	2	2	no	—
96	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
97	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
98	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
99	C₂×C₃×M₁₆	—	2	4	no	—
100	C₂×C₃×M₁₆	C₂²×C₅	1	4	yes	—
101	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
102	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
103	C₂×#217	—	2	2	no	—
104	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
105	#924	—	2	4	no	—
106	#924	C₂²×C₅	1	8	yes	—
107	C₂×Dic₃⋊C₄	—	2	2	no	—
108	C₂×Dic₁₂	—	2	4	no	—
109	C₂×Dic₁₂	C₂²×C₅	1	4	yes	complemented
110	#817	C₂²×C₅	1	4	yes	complemented
111	#816	C₂²×C₅	1	8	yes	complemented
112	C₂⁴×C₅	D₁₂	1	1	yes	characteristic
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	—	2	2	no	—
119	C₂²×C₄×C₅	—	6	2	no	—
120	C₂²×C₄×C₅	—	6	2	no	—
121	C₂²×C₄×C₅	D₁₂	1	2	yes	—
122	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
124	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
125	C₅×Q₁₆	—	6	8	no	—
126	C₅×M₁₆	—	2	8	no	—
127	C₂×C₅×C₈	—	2	2	no	—
128	C₂×C₅×C₈	#67	1	4	yes	—
129	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
130	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
131	C₅×C₄⋊C₄	—	3	4	no	—
132	C₅×C₄⋊C₄	—	6	2	no	—
133	C₅×C₄⋊C₄	—	6	8	no	—
134	C₅×C₂²⋊C₄	—	6	8	no	—
135	C₂²×Q₁₆	—	3	1	no	—
136	C₂²×M₁₆	C₅×D₃	1	1	yes	characteristic
137	C₂×C₂²⋊Q₈	—	3	2	no	—
138	#470	—	3	8	no	—
139	C₂×#106	—	3	1	no	—
140	C₂×#103	—	3	2	no	—
141	C₂²×C₃×C₅	—	2	2	no	—
142	C₂²×C₃×C₅	—	2	4	no	—
143	C₂²×C₃×C₅	—	4	4	no	—
144	C₂²×C₃×C₅	#121	1	2	yes	—
145	C₂²×C₃×C₅	#137	1	4	yes	—
146	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
147	C₃×C₄×C₅	—	2	2	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	—	4	4	no	—
152	C₅×Dic₃	—	4	4	no	—
153	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
154	C₂²×C₃×C₄	—	2	2	no	—
155	C₂²×C₃×C₄	—	2	2	no	—
156	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
158	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
159	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
160	C₂²×Dic₃	—	2	2	no	—
161	C₂²×Dic₃	—	2	2	no	—
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	C₃×M₁₆	—	2	8	no	—
167	C₂×C₃×C₈	—	2	2	no	—
168	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
169	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
170	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
171	#217	—	2	8	no	—
172	C₄⋊Dic₃	—	2	2	no	—
173	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
174	Dic₃⋊C₄	—	2	8	no	—
175	Dic₁₂	—	2	8	no	—
176	C₂³×C₅	—	2	4	no	—
177	C₂³×C₅	—	2	2	no	—
178	C₂³×C₅	C₂×D₁₂	1	2	yes	—
179	C₂³×C₅	C₂×D₁₂	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	2	no	—
184	C₂×C₄×C₅	—	2	4	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	2	2	no	—
187	C₂×C₄×C₅	—	6	4	no	—
188	C₂×C₄×C₅	—	6	4	no	—
189	C₂×C₄×C₅	—	12	4	no	—
190	C₂×C₄×C₅	—	12	4	no	—
191	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
192	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
193	C₂×C₄×C₅	#235	1	2	yes	—
194	C₂×C₄×C₅	C₂×#67	1	2	yes	—
195	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
196	C₅×C₈	—	2	2	no	—
197	C₅×C₈	C₂×#67	1	4	yes	—
198	C₂²×Q₈	—	3	2	no	—
199	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
200	C₂×Q₁₆	—	3	4	no	—
201	C₂×Q₁₆	—	6	4	no	—
202	C₂×M₁₆	—	2	4	no	—
203	C₂×M₁₆	C₂×C₅×D₃	1	4	yes	—
204	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
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	—	3	2	no	—
209	C₂×C₄⋊C₄	—	6	2	no	—
210	C₂×C₂²⋊C₄	—	6	2	no	—
211	#106	—	3	4	no	—
212	#103	—	3	8	no	—
213	C₂×C₃×C₅	—	4	2	no	—
214	C₂×C₃×C₅	#470	1	4	yes	complemented
215	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
216	C₂×C₃×C₅	C₂×#137	1	2	yes	—
217	C₂³×C₃	—	2	2	no	—
218	C₂³×C₃	—	2	4	no	—
219	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
220	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
221	C₂×C₃×C₄	—	2	2	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	2	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
226	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
228	C₂×C₃×C₄	C₅×#41	1	2	yes	—
229	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
230	C₂×Dic₃	—	2	4	no	—
231	C₂×Dic₃	—	2	4	no	—
232	C₂×Dic₃	—	4	4	no	—
233	C₂×Dic₃	—	4	4	no	—
234	Dic₆	—	2	4	no	—
235	Dic₆	—	2	4	no	—
236	Dic₆	—	4	4	no	—
237	C₃×C₈	—	2	2	no	—
238	C₃×C₈	C₂×C₅×D₄	1	4	yes	—
239	C₂²×C₅	—	2	4	no	—
240	C₂²×C₅	—	2	2	no	—
241	C₂²×C₅	—	4	4	no	—
242	C₂²×C₅	#909	1	4	yes	—
243	C₂²×C₅	#930	1	2	yes	—
244	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
245	C₄×C₅	—	2	2	no	—
246	C₄×C₅	—	12	4	no	—
247	C₄×C₅	—	12	4	no	—
248	C₄×C₅	#930	1	2	yes	—
249	C₄×C₅	C₂×#235	1	1	yes	characteristic
250	C₄×C₅	C₂²×#67	1	1	yes	characteristic
251	C₂⁴	C₅×D₁₂	1	1	yes	characteristic
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	—	2	2	no	—
258	C₂²×C₄	—	6	2	no	—
259	C₂²×C₄	—	6	2	no	—
260	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
261	C₂²×C₄	C₅×D₁₂	1	2	yes	—
262	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
263	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
264	Q₁₆	—	6	8	no	—
265	M₁₆	—	2	8	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₅×#67	1	4	yes	—
268	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
269	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
270	C₄⋊C₄	—	3	4	no	—
271	C₄⋊C₄	—	6	2	no	—
272	C₄⋊C₄	—	6	8	no	—
273	C₂²⋊C₄	—	6	8	no	—
274	C₃×C₅	C₂×#470	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	2	4	no	—
277	C₂²×C₃	—	4	4	no	—
278	C₂²×C₃	C₅×#121	1	2	yes	—
279	C₂²×C₃	C₅×#137	1	4	yes	—
280	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
281	C₃×C₄	—	2	2	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	—	4	4	no	—
286	Dic₃	—	4	4	no	—
287	C₂×C₅	—	4	2	no	—
288	C₂×C₅	#5137	1	4	yes	direct product factor
289	C₂×C₅	C₂×#909	1	2	yes	—
290	C₂×C₅	C₂×#930	1	1	yes	characteristic
291	C₂³	—	2	4	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂×C₅×D₁₂	1	2	yes	—
294	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
295	Q₈	—	6	4	no	—
296	Q₈	—	6	4	no	—
297	Q₈	—	12	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	6	4	no	—
303	C₂×C₄	—	6	4	no	—
304	C₂×C₄	—	12	4	no	—
305	C₂×C₄	—	12	4	no	—
306	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
307	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
308	C₂×C₄	C₅×#235	1	2	yes	—
309	C₂×C₄	C₂×C₅×#67	1	2	yes	—
310	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	C₂×C₅×#67	1	4	yes	—
313	C₂×C₃	—	4	2	no	—
314	C₂×C₃	C₅×#470	1	4	yes	complemented
315	C₂×C₃	C₂×C₅×#121	1	1	yes	characteristic
316	C₂×C₃	C₂×C₅×#137	1	2	yes	—
317	C₅	C₂×#5137	1	1	yes	characteristic, direct product factor, Sylow
318	C₂²	—	2	4	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	4	4	no	—
321	C₂²	C₅×#909	1	4	yes	—
322	C₂²	C₅×#930	1	2	yes	—
323	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	12	4	no	—
326	C₄	—	12	4	no	—
327	C₄	C₅×#930	1	2	yes	—
328	C₄	C₂×C₅×#235	1	1	yes	characteristic
329	C₄	C₂²×C₅×#67	1	1	yes	characteristic
330	C₃	C₂×C₅×#470	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	4	2	no	—
332	C₂	C₅×#5137	1	4	yes	direct product factor
333	C₂	C₂×C₅×#909	1	2	yes	—
334	C₂	C₂×C₅×#930	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₂²×C₃×C₅×M₁₆	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
hypercenter	C₂²×C₅×M₁₆	D₃
hypocenter	C₃	C₂×C₅×#470
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂²×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₂²×C₅×Dic₁₂	C₂	semidirect
4. C₅×#5137	C₂	direct (N × H)
5. C₂×C₅×#817	C₂	semidirect
6. C₂×C₅×#816	C₂	semidirect
20. C₂×C₅×Dic₁₂	C₂²	semidirect
21. C₅×#817	C₂²	semidirect
22. C₅×#816	C₂²	semidirect
23. C₂×#5137	C₅	direct (N × H)
55. C₂²×Dic₁₂	C₂×C₅	semidirect
56. #5137	C₂×C₅	direct (N × H)
57. C₂×#817	C₂×C₅	semidirect
58. C₂×#816	C₂×C₅	semidirect
109. C₂×Dic₁₂	C₂²×C₅	semidirect
110. #817	C₂²×C₅	semidirect
111. #816	C₂²×C₅	semidirect
214. C₂×C₃×C₅	#470	semidirect
274. C₃×C₅	C₂×#470	semidirect
288. C₂×C₅	#5137	direct (N × H)
314. C₂×C₃	C₅×#470	semidirect
317. C₅	C₂×#5137	direct (N × H)
330. C₃	C₂×C₅×#470	semidirect
332. C₂	C₅×#5137	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×M₁₆	2
C₂×C₅×#924	2
C₂²×C₅×Dic₁₂	2
C₅×#5137	2
C₂×C₅×#817	2
C₂×C₅×#816	2
C₂×C₅×#470	3
C₂×#5137	5

Subgroups — C2×C5×#5137 #14428041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5137 #14428041