Subgroups — C₅×#69859 #14283976

Order 1920.

540 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4464	C₂	1	4	yes	maximal, complemented
5	C₅×#1525	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₄	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₅×#930	—	2	1	no	—
10	C₅×#930	—	2	1	no	—
11	C₂×C₅×#208	—	2	1	no	—
12	C₅×#875	C₂²	1	2	yes	—
13	C₅×#874	C₄	1	1	yes	characteristic
14	C₄×C₅×D₁₂	—	2	1	no	—
15	C₄×C₅×D₁₂	—	2	1	no	—
16	C₅×#869	C₄	1	1	yes	characteristic
17	C₃×C₅×#97	C₂²	1	2	yes	—
18	C₃×C₄×C₅×C₈	—	2	1	no	—
19	C₅×#803	C₂²	1	2	yes	—
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	#69859	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	—	3	1	no	—
23	C₄×C₅×M₁₆	—	3	1	no	—
24	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
25	C₅×#329	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×#67	—	2	2	no	—
30	C₅×#235	—	4	4	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
36	C₃×C₅×M₁₆	—	2	4	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×#208	—	2	4	no	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	#5477	C₂×C₅	1	1	yes	characteristic
49	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
50	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
51	#4464	C₂×C₅	1	4	yes	complemented
52	C₂×C₅×#41	—	3	2	no	—
53	C₂×C₅×M₁₆	—	2	1	no	—
54	C₂×C₅×M₁₆	—	6	1	no	—
55	C₅×C₄⋊Q₈	—	3	1	no	—
56	C₅×#127	—	3	1	no	—
57	C₅×#124	—	3	2	no	—
58	C₅×#121	—	6	1	no	—
59	C₅×#121	—	6	1	no	—
60	C₄×C₅×D₄	—	6	1	no	—
61	C₄×C₅×D₄	—	6	1	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	C₅×#97	—	3	2	no	—
64	C₅×#97	C₂×D₃	1	2	yes	—
65	C₄×C₅×C₈	—	2	1	no	—
66	C₄×C₅×C₈	—	6	1	no	—
67	#1525	—	3	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₅×D₃	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₅×#67	—	4	4	no	—
80	C₂×C₅×Dic₃	—	2	2	no	—
81	C₅×D₁₂	—	4	2	no	—
82	C₅×D₁₂	—	4	2	no	—
83	C₅×D₁₂	—	4	2	no	—
84	C₄×C₅×D₃	—	4	4	no	—
85	C₅×Dic₆	—	4	2	no	—
86	C₃×C₅×C₈	—	2	4	no	—
87	C₅×C₃⋊C₈	—	2	4	no	—
88	C₂×#235	C₄×C₅	1	2	yes	—
89	C₂×C₃×M₁₆	—	2	1	no	—
90	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
91	#930	—	2	1	no	—
92	#930	—	2	1	no	—
93	C₂×#208	—	2	1	no	—
94	#875	C₂²×C₅	1	2	yes	—
95	#874	C₄×C₅	1	1	yes	characteristic
96	C₄×D₁₂	—	2	1	no	—
97	C₄×D₁₂	—	2	1	no	—
98	#869	C₄×C₅	1	1	yes	characteristic
99	C₃×#97	C₂²×C₅	1	2	yes	—
100	C₃×C₄×C₈	—	2	1	no	—
101	#803	C₂²×C₅	1	2	yes	—
102	C₄×C₃⋊C₈	—	2	1	no	—
103	C₅×#41	—	12	4	no	—
104	C₂×C₅×Q₈	—	3	2	no	—
105	C₂×C₅×D₄	—	3	2	no	—
106	C₂×C₅×D₄	—	6	2	no	—
107	C₂×C₅×D₄	—	6	1	no	—
108	C₂×C₅×D₄	—	6	1	no	—
109	C₂²×C₄×C₅	—	6	2	no	—
110	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
111	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
112	C₂²×C₄×C₅	#67	1	1	yes	characteristic
113	C₅×M₁₆	—	2	4	no	—
114	C₅×M₁₆	—	6	4	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	6	2	no	—
117	C₅×C₄⋊C₄	—	6	1	no	—
118	C₅×C₄⋊C₄	—	6	1	no	—
119	C₅×C₂²⋊C₄	—	6	2	no	—
120	C₅×C₂²⋊C₄	—	6	2	no	—
121	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
122	C₄²×C₅	D₁₂	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
125	#532	—	3	1	no	—
126	C₄×M₁₆	—	3	1	no	—
127	C₄×M₁₆	C₅×D₃	1	1	yes	characteristic, complemented
128	#329	—	3	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₅×D₃	—	4	2	no	—
134	C₂×C₅×D₃	—	4	2	no	—
135	C₂×C₅×D₃	—	4	2	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	#100	1	2	yes	—
140	C₃×C₄×C₅	#104	1	2	yes	—
141	C₃×C₄×C₅	#104	1	2	yes	—
142	C₅×Dic₃	—	4	2	no	—
143	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
146	C₂×#67	—	2	2	no	—
147	#235	—	4	4	no	—
148	C₂×D₁₂	—	2	1	no	—
149	C₂×D₁₂	—	2	1	no	—
150	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
151	C₂×C₄×D₃	—	2	2	no	—
152	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
153	C₃×M₁₆	—	2	4	no	—
154	C₂×C₃×C₈	—	2	2	no	—
155	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
156	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
157	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
158	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
159	(C₂×D₃)⋊C₄	—	2	2	no	—
160	(C₂×D₃)⋊C₄	—	2	2	no	—
161	C₄⋊Dic₃	—	2	1	no	—
162	C₄⋊Dic₃	—	2	1	no	—
163	#208	—	2	4	no	—
164	C₂×C₃⋊C₈	—	2	2	no	—
165	C₂³×C₅	—	6	2	no	—
166	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
167	C₅×Q₈	—	12	2	no	—
168	C₅×D₄	—	12	2	no	—
169	C₅×D₄	—	12	4	no	—
170	C₅×D₄	—	12	2	no	—
171	C₅×D₄	—	12	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	2	2	no	—
175	C₂×C₄×C₅	—	6	2	no	—
176	C₂×C₄×C₅	—	12	4	no	—
177	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
183	C₅×C₈	—	2	4	no	—
184	C₅×C₈	—	6	4	no	—
185	C₂×#41	—	3	2	no	—
186	C₂×M₁₆	—	2	1	no	—
187	C₂×M₁₆	—	6	1	no	—
188	C₄⋊Q₈	—	3	1	no	—
189	#127	—	3	1	no	—
190	#124	—	3	2	no	—
191	#121	—	6	1	no	—
192	#121	—	6	1	no	—
193	C₄×D₄	—	6	1	no	—
194	C₄×D₄	—	6	1	no	—
195	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
196	#97	—	3	2	no	—
197	#97	C₂×C₅×D₃	1	2	yes	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	—	6	1	no	—
200	C₂×C₃×C₅	—	4	1	no	—
201	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
204	C₅×D₃	—	8	2	no	—
205	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
206	C₂²×D₃	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	#67	—	4	4	no	—
217	C₂×Dic₃	—	2	2	no	—
218	D₁₂	—	4	2	no	—
219	D₁₂	—	4	2	no	—
220	D₁₂	—	4	2	no	—
221	C₄×D₃	—	4	4	no	—
222	Dic₆	—	4	2	no	—
223	C₃×C₈	—	2	4	no	—
224	C₃⋊C₈	—	2	4	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	12	2	no	—
229	C₂²×C₅	—	12	2	no	—
230	C₂²×C₅	—	12	2	no	—
231	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	12	2	no	—
236	C₄×C₅	#805	1	2	yes	—
237	C₄×C₅	#823	1	2	yes	—
238	C₄×C₅	#825	1	2	yes	—
239	#41	—	12	4	no	—
240	C₂×Q₈	—	3	2	no	—
241	C₂×D₄	—	3	2	no	—
242	C₂×D₄	—	6	2	no	—
243	C₂×D₄	—	6	1	no	—
244	C₂×D₄	—	6	1	no	—
245	C₂²×C₄	—	6	2	no	—
246	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
247	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
248	C₂²×C₄	C₅×#67	1	1	yes	characteristic
249	M₁₆	—	2	4	no	—
250	M₁₆	—	6	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	6	2	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₄⋊C₄	—	6	1	no	—
255	C₂²⋊C₄	—	6	2	no	—
256	C₂²⋊C₄	—	6	2	no	—
257	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
258	C₄²	C₅×D₁₂	1	1	yes	characteristic
259	C₄²	C₅×#67	1	1	yes	characteristic
260	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
261	C₃×C₅	#1525	1	1	yes	characteristic, complemented, Hall
262	C₂²×C₃	—	2	1	no	—
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₂×D₃	—	4	2	no	—
267	C₂×D₃	—	4	2	no	—
268	C₂×D₃	—	4	2	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	C₅×#100	1	2	yes	—
273	C₃×C₄	C₅×#104	1	2	yes	—
274	C₃×C₄	C₅×#104	1	2	yes	—
275	Dic₃	—	4	2	no	—
276	C₂×C₅	—	4	1	no	—
277	C₂×C₅	—	24	2	no	—
278	C₂×C₅	C₂×#805	1	1	yes	characteristic
279	C₂×C₅	C₂×#823	1	1	yes	characteristic
280	C₂×C₅	C₂×#825	1	1	yes	characteristic
281	C₂³	—	6	2	no	—
282	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
283	Q₈	—	12	2	no	—
284	D₄	—	12	2	no	—
285	D₄	—	12	4	no	—
286	D₄	—	12	2	no	—
287	D₄	—	12	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	6	2	no	—
292	C₂×C₄	—	12	4	no	—
293	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
294	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
295	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
296	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
298	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
299	C₈	—	2	4	no	—
300	C₈	—	6	4	no	—
301	C₂×C₃	—	4	1	no	—
302	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
303	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
304	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
305	D₃	—	8	2	no	—
306	C₅	#69859	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	12	2	no	—
311	C₂²	—	12	2	no	—
312	C₂²	—	12	2	no	—
313	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	12	2	no	—
318	C₄	C₅×#805	1	2	yes	—
319	C₄	C₅×#823	1	2	yes	—
320	C₄	C₅×#825	1	2	yes	—
321	C₃	C₅×#1525	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	4	1	no	—
323	C₂	—	24	2	no	—
324	C₂	C₂×C₅×#805	1	1	yes	characteristic
325	C₂	C₂×C₅×#823	1	1	yes	characteristic
326	C₂	C₂×C₅×#825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₄×C₅×M₁₆	D₃
hypocenter	C₃	C₅×#1525
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₄×C₅×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×M₁₆	C₂	semidirect
3. C₄×C₅×#208	C₂	semidirect
4. C₅×#4464	C₂	semidirect
21. #69859	C₅	direct (N × H)
24. C₄×C₅×M₁₆	D₃	semidirect
49. C₃×C₄×M₁₆	C₂×C₅	semidirect
50. C₄×#208	C₂×C₅	semidirect
51. #4464	C₂×C₅	semidirect
127. C₄×M₁₆	C₅×D₃	semidirect
261. C₃×C₅	#1525	semidirect
306. C₅	#69859	direct (N × H)
321. C₃	C₅×#1525	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₃×C₄×C₅×M₁₆	2
C₄×C₅×#208	2
C₅×#4464	2
C₅×#1525	3
#69859	5

Subgroups — C5×#69859 #14283976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69859 #14283976