Subgroups — C₅×#78538 #14439213

Order 1920.

1104 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	C₂×C₅×#909	C₂	1	4	yes	maximal, complemented
2	C₅×#5477	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#869	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4715	C₂	1	4	yes	maximal, complemented
6	C₅×#4699	C₂	1	2	yes	maximal, complemented
7	C₅×#4693	C₂	1	2	yes	maximal, complemented
8	C₅×#3152	—	3	1	no	maximal, Hall
9	C₂×C₅×#235	C₂²	1	2	yes	—
10	C₂²×C₅×Dic₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₅×#930	—	2	2	no	—
14	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
15	C₅×#909	—	2	16	no	—
16	C₂×C₅×Dic₁₂	C₂²	1	8	yes	—
17	C₂×C₅×#204	C₂²	1	8	yes	—
18	C₅×#875	C₂²	1	2	yes	—
19	C₅×#874	C₂²	1	1	yes	characteristic
20	C₄×C₅×D₁₂	—	2	2	no	—
21	C₅×#869	—	2	2	no	—
22	C₅×#869	C₂²	1	2	yes	—
23	C₅×#869	C₂²	1	1	yes	characteristic
24	C₅×#869	C₂²	1	2	yes	—
25	C₃×C₅×#97	C₂²	1	2	yes	—
26	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
27	#78538	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
28	C₂×C₅×#137	—	3	4	no	—
29	C₅×#532	—	3	1	no	—
30	C₂×C₅×C₄⋊Q₈	—	3	1	no	—
31	C₅×#497	—	3	4	no	—
32	C₅×#494	—	3	2	no	—
33	C₅×#492	—	3	2	no	—
34	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂×C₅×#67	—	2	2	no	—
38	C₂²×C₅×Dic₃	—	2	2	no	—
39	C₅×#235	—	2	8	no	—
40	C₂×C₅×D₁₂	—	2	2	no	—
41	C₂×C₅×D₁₂	C₂³	1	2	yes	—
42	C₂×C₄×C₅×D₃	—	2	2	no	—
43	C₂×C₅×Dic₆	—	2	8	no	—
44	C₂×C₅×Dic₆	—	2	2	no	—
45	C₂×C₅×Dic₆	C₂³	1	4	yes	—
46	C₂×C₅×Dic₆	C₂³	1	2	yes	—
47	C₃×C₅×M₁₆	D₄	1	8	yes	complemented
48	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
53	C₅×C₄⋊Dic₃	—	2	4	no	—
54	C₅×C₄⋊Dic₃	—	2	4	no	—
55	C₅×C₄⋊Dic₃	—	2	2	no	—
56	C₅×Dic₁₂	—	2	16	no	—
57	C₅×#204	—	2	16	no	—
58	C₂×#909	C₂×C₅	1	4	yes	complemented
59	#5477	C₂×C₅	1	1	yes	characteristic
60	C₂×#869	C₂×C₅	1	1	yes	characteristic, complemented
61	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
62	#4715	C₂×C₅	1	4	yes	complemented
63	#4699	C₂×C₅	1	2	yes	complemented
64	#4693	C₂×C₅	1	2	yes	complemented
65	C₂×C₅×#41	—	3	2	no	—
66	C₂²×C₅×Q₈	—	3	2	no	—
67	C₅×#137	—	6	16	no	—
68	C₂×C₅×Q₁₆	—	3	8	no	—
69	C₂×C₅×SD₁₆	—	3	8	no	—
70	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
71	C₅×C₄⋊Q₈	—	3	2	no	—
72	C₅×C₄⋊Q₈	—	3	1	no	—
73	C₅×C₄⋊Q₈	—	3	2	no	—
74	C₅×C₄⋊Q₈	—	6	2	no	—
75	C₅×#127	—	3	1	no	—
76	C₅×#124	—	3	2	no	—
77	C₅×#121	—	6	2	no	—
78	C₄×C₅×D₄	—	6	2	no	—
79	C₂×C₅×C₄⋊C₄	—	6	2	no	—
80	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
81	C₅×#97	C₂×D₃	1	2	yes	—
82	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
83	#3152	—	3	1	no	Sylow
84	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₅×D₃	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	4	no	—
87	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
90	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
92	C₅×#67	—	4	4	no	—
93	C₂×C₅×Dic₃	—	2	4	no	—
94	C₂×C₅×Dic₃	—	2	2	no	—
95	C₂×C₅×Dic₃	—	4	4	no	—
96	C₅×D₁₂	—	2	4	no	—
97	C₅×D₁₂	—	4	4	no	—
98	C₄×C₅×D₃	—	4	4	no	—
99	C₅×Dic₆	—	2	8	no	—
100	C₅×Dic₆	—	2	4	no	—
101	C₅×Dic₆	—	4	4	no	—
102	C₃×C₅×C₈	C₂×D₄	1	8	yes	—
103	C₂×#235	C₂²×C₅	1	2	yes	—
104	C₂²×Dic₆	C₂²×C₅	1	2	yes	—
105	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
106	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
107	#930	—	2	2	no	—
108	C₂×C₄⋊Dic₃	—	2	2	no	—
109	#909	—	2	16	no	—
110	C₂×Dic₁₂	C₂²×C₅	1	8	yes	—
111	C₂×#204	C₂²×C₅	1	8	yes	—
112	#875	C₂²×C₅	1	2	yes	—
113	#874	C₂²×C₅	1	1	yes	characteristic
114	C₄×D₁₂	—	2	2	no	—
115	#869	—	2	2	no	—
116	#869	C₂²×C₅	1	2	yes	—
117	#869	C₂²×C₅	1	1	yes	characteristic
118	#869	C₂²×C₅	1	2	yes	—
119	C₃×#97	C₂²×C₅	1	2	yes	—
120	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
121	C₅×#41	—	6	8	no	—
122	C₂×C₅×Q₈	—	3	2	no	—
123	C₂×C₅×Q₈	—	3	4	no	—
124	C₂×C₅×Q₈	—	6	2	no	—
125	C₂×C₅×Q₈	—	6	8	no	—
126	C₂×C₅×D₄	—	3	2	no	—
127	C₂×C₅×D₄	—	6	2	no	—
128	C₂×C₅×D₄	—	6	2	no	—
129	C₂²×C₄×C₅	—	6	2	no	—
130	C₂²×C₄×C₅	—	6	2	no	—
131	C₂²×C₄×C₅	D₁₂	1	2	yes	—
132	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
133	C₅×Q₁₆	—	6	16	no	—
134	C₅×SD₁₆	—	6	16	no	—
135	C₅×M₁₆	D₁₂	1	8	yes	complemented
136	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
137	C₅×C₄⋊C₄	—	6	4	no	—
138	C₅×C₄⋊C₄	—	6	4	no	—
139	C₅×C₄⋊C₄	—	6	2	no	—
140	C₅×C₂²⋊C₄	—	6	4	no	—
141	C₄²×C₅	D₁₂	1	2	yes	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂×#137	—	3	4	no	—
145	#532	—	3	1	no	—
146	C₂×C₄⋊Q₈	—	3	1	no	—
147	#497	—	3	4	no	—
148	#494	—	3	2	no	—
149	#492	—	3	2	no	—
150	C₄×M₁₆	C₅×D₃	1	1	yes	characteristic, complemented
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	#127	1	2	yes	—
153	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₅×D₃	—	4	2	no	—
155	C₂×C₅×D₃	—	4	4	no	—
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	#127	1	2	yes	—
158	C₃×C₄×C₅	#137	1	4	yes	complemented
159	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
160	C₅×Dic₃	—	4	4	no	—
161	C₅×Dic₃	—	4	2	no	—
162	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
163	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
164	C₂×#67	—	2	2	no	—
165	C₂²×Dic₃	—	2	2	no	—
166	#235	—	2	8	no	—
167	C₂×D₁₂	—	2	2	no	—
168	C₂×D₁₂	C₂³×C₅	1	2	yes	—
169	C₂×C₄×D₃	—	2	2	no	—
170	C₂×Dic₆	—	2	8	no	—
171	C₂×Dic₆	—	2	2	no	—
172	C₂×Dic₆	C₂³×C₅	1	4	yes	—
173	C₂×Dic₆	C₂³×C₅	1	2	yes	—
174	C₃×M₁₆	C₅×D₄	1	8	yes	complemented
175	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
176	C₃×C₄²	C₅×D₄	1	2	yes	—
177	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
178	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
179	(C₂×D₃)⋊C₄	—	2	4	no	—
180	C₄⋊Dic₃	—	2	4	no	—
181	C₄⋊Dic₃	—	2	4	no	—
182	C₄⋊Dic₃	—	2	2	no	—
183	Dic₁₂	—	2	16	no	—
184	#204	—	2	16	no	—
185	C₂³×C₅	—	6	2	no	—
186	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
187	C₅×Q₈	—	6	8	no	—
188	C₅×Q₈	—	6	4	no	—
189	C₅×Q₈	—	12	4	no	—
190	C₅×D₄	—	6	4	no	—
191	C₅×D₄	—	12	4	no	—
192	C₅×D₄	—	12	4	no	—
193	C₂×C₄×C₅	—	2	4	no	—
194	C₂×C₄×C₅	—	6	2	no	—
195	C₂×C₄×C₅	—	6	4	no	—
196	C₂×C₄×C₅	—	12	4	no	—
197	C₂×C₄×C₅	—	12	4	no	—
198	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
199	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
201	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
202	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
203	C₅×C₈	C₂×D₁₂	1	8	yes	—
204	C₂×#41	—	3	2	no	—
205	C₂²×Q₈	—	3	2	no	—
206	#137	—	6	16	no	—
207	C₂×Q₁₆	—	3	8	no	—
208	C₂×SD₁₆	—	3	8	no	—
209	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
210	C₄⋊Q₈	—	3	2	no	—
211	C₄⋊Q₈	—	3	2	no	—
212	C₄⋊Q₈	—	3	1	no	—
213	C₄⋊Q₈	—	6	2	no	—
214	#127	—	3	1	no	—
215	#124	—	3	2	no	—
216	#121	—	6	2	no	—
217	C₄×D₄	—	6	2	no	—
218	C₂×C₄⋊C₄	—	6	2	no	—
219	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
220	#97	C₂×C₅×D₃	1	2	yes	—
221	C₄×C₈	C₂×C₅×D₃	1	2	yes	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
224	C₂×C₃×C₅	C₂×#137	1	2	yes	—
225	C₅×D₃	—	8	2	no	—
226	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
227	C₂²×D₃	—	2	2	no	—
228	C₂×C₃×C₄	—	2	4	no	—
229	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
230	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
232	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
233	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
234	#67	—	4	4	no	—
235	C₂×Dic₃	—	2	4	no	—
236	C₂×Dic₃	—	2	2	no	—
237	C₂×Dic₃	—	4	4	no	—
238	D₁₂	—	2	4	no	—
239	D₁₂	—	4	4	no	—
240	C₄×D₃	—	4	4	no	—
241	Dic₆	—	2	8	no	—
242	Dic₆	—	2	4	no	—
243	Dic₆	—	4	4	no	—
244	C₃×C₈	C₂×C₅×D₄	1	8	yes	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	—	12	2	no	—
247	C₂²×C₅	—	12	4	no	—
248	C₂²×C₅	#874	1	2	yes	—
249	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	—	12	2	no	—
252	C₄×C₅	—	12	4	no	—
253	C₄×C₅	#874	1	2	yes	—
254	C₄×C₅	#909	1	4	yes	complemented
255	C₄×C₅	C₂²×D₁₂	1	2	yes	—
256	#41	—	6	8	no	—
257	C₂×Q₈	—	3	2	no	—
258	C₂×Q₈	—	3	4	no	—
259	C₂×Q₈	—	6	2	no	—
260	C₂×Q₈	—	6	8	no	—
261	C₂×D₄	—	3	2	no	—
262	C₂×D₄	—	6	2	no	—
263	C₂×D₄	—	6	2	no	—
264	C₂²×C₄	—	6	2	no	—
265	C₂²×C₄	—	6	2	no	—
266	C₂²×C₄	C₅×D₁₂	1	2	yes	—
267	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
268	Q₁₆	—	6	16	no	—
269	SD₁₆	—	6	16	no	—
270	M₁₆	C₅×D₁₂	1	8	yes	complemented
271	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
272	C₄⋊C₄	—	6	4	no	—
273	C₄⋊C₄	—	6	4	no	—
274	C₄⋊C₄	—	6	2	no	—
275	C₂²⋊C₄	—	6	4	no	—
276	C₄²	C₅×D₁₂	1	2	yes	—
277	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
278	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	C₅×#127	1	2	yes	—
282	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
283	C₂×D₃	—	4	2	no	—
284	C₂×D₃	—	4	4	no	—
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	C₅×#127	1	2	yes	—
287	C₃×C₄	C₅×#137	1	4	yes	complemented
288	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
289	Dic₃	—	4	4	no	—
290	Dic₃	—	4	2	no	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	—	24	2	no	—
293	C₂×C₅	C₂×#874	1	1	yes	characteristic
294	C₂×C₅	C₂×#909	1	2	yes	—
295	C₂³	—	6	2	no	—
296	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
297	Q₈	—	6	8	no	—
298	Q₈	—	6	4	no	—
299	Q₈	—	12	4	no	—
300	D₄	—	6	4	no	—
301	D₄	—	12	4	no	—
302	D₄	—	12	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	6	2	no	—
305	C₂×C₄	—	6	4	no	—
306	C₂×C₄	—	12	4	no	—
307	C₂×C₄	—	12	4	no	—
308	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
309	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
310	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
311	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
312	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
313	C₈	C₂×C₅×D₁₂	1	8	yes	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
316	C₂×C₃	C₂×C₅×#137	1	2	yes	—
317	D₃	—	8	2	no	—
318	C₅	#78538	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	12	2	no	—
321	C₂²	—	12	4	no	—
322	C₂²	C₅×#874	1	2	yes	—
323	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	12	2	no	—
326	C₄	—	12	4	no	—
327	C₄	C₅×#874	1	2	yes	—
328	C₄	C₅×#909	1	4	yes	complemented
329	C₄	C₂²×C₅×D₁₂	1	2	yes	—
330	C₃	C₅×#3152	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	24	2	no	—
333	C₂	C₂×C₅×#874	1	1	yes	characteristic
334	C₂	C₂×C₅×#909	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₅×#5477	C₂
hypercenter	C₄×C₅×M₁₆	D₃
hypocenter	C₃	C₅×#3152
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₄×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#909	C₂	semidirect
3. C₂×C₅×#869	C₂	semidirect
4. C₃×C₄×C₅×M₁₆	C₂	semidirect
5. C₅×#4715	C₂	semidirect
6. C₅×#4699	C₂	semidirect
7. C₅×#4693	C₂	semidirect
27. #78538	C₅	direct (N × H)
34. C₄×C₅×M₁₆	D₃	semidirect
47. C₃×C₅×M₁₆	D₄	semidirect
58. C₂×#909	C₂×C₅	semidirect
60. C₂×#869	C₂×C₅	semidirect
61. C₃×C₄×M₁₆	C₂×C₅	semidirect
62. #4715	C₂×C₅	semidirect
63. #4699	C₂×C₅	semidirect
64. #4693	C₂×C₅	semidirect
135. C₅×M₁₆	D₁₂	semidirect
150. C₄×M₁₆	C₅×D₃	semidirect
158. C₃×C₄×C₅	#137	semidirect
174. C₃×M₁₆	C₅×D₄	semidirect
254. C₄×C₅	#909	semidirect
270. M₁₆	C₅×D₁₂	semidirect
279. C₃×C₅	#3152	semidirect
287. C₃×C₄	C₅×#137	semidirect
318. C₅	#78538	direct (N × H)
328. C₄	C₅×#909	semidirect
330. C₃	C₅×#3152	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#909	2
C₅×#5477	2
C₂×C₅×#869	2
C₃×C₄×C₅×M₁₆	2
C₅×#4715	2
C₅×#4699	2
C₅×#4693	2
C₅×#3152	3
#78538	5

Subgroups — C5×#78538 #14439213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78538 #14439213