Subgroups — C₅×#69855 #14282285

Order 1920.

632 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4926	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4459	C₂	1	4	yes	maximal, complemented
5	C₅×#1490	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅×D₃	—	2	1	no	—
8	C₂×C₃×C₅×M₁₆	—	2	1	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
11	C₂×C₅×C₄⋊Dic₃	C₄	1	1	yes	characteristic
12	C₄×C₅×D₁₂	—	2	1	no	—
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₄×C₅×D₁₂	C₄	1	2	yes	—
15	C₄×C₅×D₁₂	C₂²	1	2	yes	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
18	C₅×#830	—	2	1	no	—
19	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	complemented
20	C₅×#803	—	2	1	no	—
21	#69855	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₄×C₅×D₄	—	3	1	no	—
23	C₅×#432	—	3	1	no	—
24	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
25	C₅×D₄⋊C₈	—	3	4	no	—
26	C₂³×C₅×D₃	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₅×Dic₃	—	2	1	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	4	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₄×C₅×D₃	—	2	2	no	—
36	C₂×C₄×C₅×D₃	—	4	4	no	—
37	C₃×C₅×M₁₆	—	2	4	no	—
38	C₂×C₃×C₅×C₈	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
49	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
50	#4926	C₂×C₅	1	1	yes	characteristic, complemented
51	#4459	C₂×C₅	1	4	yes	complemented
52	C₂²×C₅×D₄	—	3	1	no	—
53	C₂³×C₄×C₅	—	6	1	no	—
54	C₂×C₅×M₁₆	—	2	1	no	—
55	C₄×C₅×D₄	—	3	2	no	—
56	C₄×C₅×D₄	—	3	2	no	—
57	C₄×C₅×D₄	—	6	1	no	—
58	C₄×C₅×D₄	—	6	1	no	—
59	C₂×C₅×C₄⋊C₄	—	3	1	no	—
60	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₅×C₄⋊C₈	—	3	2	no	—
63	C₅×C₂²⋊C₈	—	6	1	no	—
64	C₅×#97	—	2	1	no	—
65	C₅×#97	—	6	1	no	—
66	C₄×C₅×C₈	C₂×D₃	1	2	yes	complemented
67	#1490	—	3	1	no	Sylow
68	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₅×D₃	—	2	4	no	—
70	C₂²×C₅×D₃	—	2	2	no	—
71	C₂²×C₅×D₃	—	4	2	no	—
72	C₂²×C₅×D₃	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₅×Dic₃	—	2	2	no	—
83	C₂×C₅×Dic₃	—	4	2	no	—
84	C₅×D₁₂	—	2	4	no	—
85	C₅×D₁₂	—	2	2	no	—
86	C₅×D₁₂	M₁₆	1	4	yes	complemented
87	C₄×C₅×D₃	—	4	4	no	—
88	C₄×C₅×D₃	—	4	4	no	—
89	C₃×C₅×C₈	—	2	4	no	—
90	C₅×C₃⋊C₈	—	4	2	no	—
91	C₂²×D₁₂	C₄×C₅	1	1	yes	characteristic
92	C₂²×C₄×D₃	—	2	1	no	—
93	C₂×C₃×M₁₆	—	2	1	no	—
94	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
95	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
96	C₂×C₄⋊Dic₃	C₄×C₅	1	1	yes	characteristic
97	C₄×D₁₂	—	2	1	no	—
98	C₄×D₁₂	—	2	1	no	—
99	C₄×D₁₂	C₄×C₅	1	2	yes	—
100	C₄×D₁₂	C₂²×C₅	1	2	yes	—
101	C₃×#97	—	2	1	no	—
102	C₃×C₄×C₈	C₂²×C₅	1	2	yes	complemented
103	#830	—	2	1	no	—
104	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	complemented
105	#803	—	2	1	no	—
106	C₂⁴×C₅	—	6	1	no	—
107	C₂×C₅×D₄	—	3	2	no	—
108	C₂×C₅×D₄	—	6	4	no	—
109	C₂×C₅×D₄	—	6	1	no	—
110	C₂²×C₄×C₅	—	6	1	no	—
111	C₂²×C₄×C₅	—	6	2	no	—
112	C₂²×C₄×C₅	—	6	2	no	—
113	C₂²×C₄×C₅	—	12	4	no	—
114	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₅×M₁₆	—	2	4	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	—	6	2	no	—
120	C₅×C₄⋊C₄	—	3	2	no	—
121	C₅×C₄⋊C₄	—	6	1	no	—
122	C₅×C₂²⋊C₄	—	6	2	no	—
123	C₅×C₂²⋊C₄	—	6	2	no	—
124	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
125	C₄²×C₅	D₁₂	1	1	yes	characteristic
126	C₄²×C₅	#67	1	1	yes	characteristic
127	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₂×C₄×D₄	—	3	1	no	—
129	#432	—	3	1	no	—
130	C₄×M₁₆	C₅×D₃	1	1	yes	characteristic, complemented
131	D₄⋊C₈	—	3	4	no	—
132	C₂²×C₃×C₅	—	2	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#104	1	2	yes	—
135	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₂×C₅×D₃	—	2	4	no	—
137	C₂×C₅×D₃	—	4	8	no	—
138	C₂×C₅×D₃	—	4	2	no	—
139	C₂×C₅×D₃	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	—	2	1	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
145	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
146	C₅×Dic₃	—	4	2	no	—
147	C₂³×D₃	—	2	1	no	—
148	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
151	C₂²×Dic₃	—	2	1	no	—
152	C₂×D₁₂	—	2	1	no	—
153	C₂×D₁₂	—	2	4	no	—
154	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
155	C₂×C₄×D₃	—	2	2	no	—
156	C₂×C₄×D₃	—	2	2	no	—
157	C₂×C₄×D₃	—	4	4	no	—
158	C₃×M₁₆	—	2	4	no	—
159	C₂×C₃×C₈	—	2	2	no	—
160	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
161	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
162	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
163	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
164	(C₂×D₃)⋊C₄	—	2	2	no	—
165	(C₂×D₃)⋊C₄	—	2	2	no	—
166	C₄⋊Dic₃	—	2	1	no	—
167	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
168	C₂×C₃⋊C₈	—	2	2	no	—
169	C₂³×C₅	—	6	4	no	—
170	C₂³×C₅	—	6	2	no	—
171	C₂³×C₅	—	12	2	no	—
172	C₂³×C₅	—	12	2	no	—
173	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
174	C₅×D₄	—	3	4	no	—
175	C₅×D₄	—	6	4	no	—
176	C₅×D₄	—	6	2	no	—
177	C₂×C₄×C₅	—	2	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	—	12	4	no	—
182	C₂×C₄×C₅	—	12	2	no	—
183	C₂×C₄×C₅	—	12	4	no	—
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
190	C₅×C₈	—	2	4	no	—
191	C₅×C₈	—	12	2	no	—
192	C₂²×D₄	—	3	1	no	—
193	C₂³×C₄	—	6	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₄×D₄	—	3	2	no	—
196	C₄×D₄	—	3	2	no	—
197	C₄×D₄	—	6	1	no	—
198	C₄×D₄	—	6	1	no	—
199	C₂×C₄⋊C₄	—	3	1	no	—
200	C₂×C₂²⋊C₄	—	6	1	no	—
201	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
202	C₄⋊C₈	—	3	2	no	—
203	C₂²⋊C₈	—	6	1	no	—
204	#97	—	2	1	no	—
205	#97	—	6	1	no	—
206	C₄×C₈	C₂×C₅×D₃	1	2	yes	complemented
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	#407	1	1	yes	characteristic
209	C₂×C₃×C₅	#418	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
211	C₅×D₃	—	4	4	no	—
212	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×D₃	—	2	4	no	—
214	C₂²×D₃	—	2	2	no	—
215	C₂²×D₃	—	4	2	no	—
216	C₂²×D₃	—	4	2	no	—
217	C₂×C₃×C₄	—	2	2	no	—
218	C₂×C₃×C₄	—	2	2	no	—
219	C₂×C₃×C₄	—	2	2	no	—
220	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
226	C₂×Dic₃	—	2	2	no	—
227	C₂×Dic₃	—	4	2	no	—
228	D₁₂	—	2	4	no	—
229	D₁₂	—	2	2	no	—
230	D₁₂	C₅×M₁₆	1	4	yes	complemented
231	C₄×D₃	—	4	4	no	—
232	C₄×D₃	—	4	4	no	—
233	C₃×C₈	—	2	4	no	—
234	C₃⋊C₈	—	4	2	no	—
235	C₂²×C₅	—	2	1	no	—
236	C₂²×C₅	—	2	1	no	—
237	C₂²×C₅	—	6	4	no	—
238	C₂²×C₅	—	12	8	no	—
239	C₂²×C₅	—	12	2	no	—
240	C₂²×C₅	—	12	2	no	—
241	C₂²×C₅	#805	1	2	yes	—
242	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	2	1	no	—
245	C₄×C₅	—	2	1	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	12	2	no	—
248	C₄×C₅	D₃×M₁₆	1	2	yes	—
249	C₄×C₅	#908	1	2	yes	—
250	C₂⁴	—	6	1	no	—
251	C₂×D₄	—	3	2	no	—
252	C₂×D₄	—	6	4	no	—
253	C₂×D₄	—	6	1	no	—
254	C₂²×C₄	—	6	1	no	—
255	C₂²×C₄	—	6	2	no	—
256	C₂²×C₄	—	6	2	no	—
257	C₂²×C₄	—	12	4	no	—
258	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
259	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
260	C₂²×C₄	C₅×#67	1	1	yes	characteristic
261	M₁₆	—	2	4	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	6	2	no	—
264	C₄⋊C₄	—	3	2	no	—
265	C₄⋊C₄	—	6	1	no	—
266	C₂²⋊C₄	—	6	2	no	—
267	C₂²⋊C₄	—	6	2	no	—
268	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
269	C₄²	C₅×D₁₂	1	1	yes	characteristic
270	C₄²	C₅×#67	1	1	yes	characteristic
271	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
272	C₃×C₅	#1490	1	1	yes	characteristic, complemented, Hall
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	C₅×#104	1	2	yes	—
276	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×D₃	—	2	4	no	—
278	C₂×D₃	—	4	8	no	—
279	C₂×D₃	—	4	2	no	—
280	C₂×D₃	—	4	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
286	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
287	Dic₃	—	4	2	no	—
288	C₂×C₅	—	2	2	no	—
289	C₂×C₅	—	12	4	no	—
290	C₂×C₅	C₂×#805	1	1	yes	characteristic
291	C₂×C₅	#5126	1	1	yes	characteristic
292	C₂×C₅	#5131	1	1	yes	characteristic
293	C₂³	—	6	4	no	—
294	C₂³	—	6	2	no	—
295	C₂³	—	12	2	no	—
296	C₂³	—	12	2	no	—
297	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
298	D₄	—	3	4	no	—
299	D₄	—	6	4	no	—
300	D₄	—	6	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	6	2	no	—
305	C₂×C₄	—	12	4	no	—
306	C₂×C₄	—	12	2	no	—
307	C₂×C₄	—	12	4	no	—
308	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
313	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
314	C₈	—	2	4	no	—
315	C₈	—	12	2	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	C₅×#407	1	1	yes	characteristic
318	C₂×C₃	C₅×#418	1	1	yes	characteristic
319	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
320	D₃	—	4	4	no	—
321	C₅	#69855	1	1	yes	characteristic, direct product factor, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	6	4	no	—
325	C₂²	—	12	8	no	—
326	C₂²	—	12	2	no	—
327	C₂²	—	12	2	no	—
328	C₂²	C₅×#805	1	2	yes	—
329	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	12	2	no	—
335	C₄	C₅×D₃×M₁₆	1	2	yes	—
336	C₄	C₅×#908	1	2	yes	—
337	C₃	C₅×#1490	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	2	no	—
339	C₂	—	12	4	no	—
340	C₂	C₂×C₅×#805	1	1	yes	characteristic
341	C₂	C₅×#5126	1	1	yes	characteristic
342	C₂	C₅×#5131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₁₂
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	C₂²×C₅×D₁₂	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₄×C₅×M₁₆	D₃
hypocenter	C₃	C₅×#1490
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×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. C₅×#4926	C₂	semidirect
4. C₅×#4459	C₂	semidirect
17. C₃×C₄×C₅×C₈	C₂²	semidirect
19. C₅×C₄⋊(C₃⋊C₈)	C₂²	semidirect
21. #69855	C₅	direct (N × H)
24. C₄×C₅×M₁₆	D₃	semidirect
49. C₃×C₄×M₁₆	C₂×C₅	semidirect
50. #4926	C₂×C₅	semidirect
51. #4459	C₂×C₅	semidirect
66. C₄×C₅×C₈	C₂×D₃	semidirect
86. C₅×D₁₂	M₁₆	semidirect
102. C₃×C₄×C₈	C₂²×C₅	semidirect
104. C₄⋊(C₃⋊C₈)	C₂²×C₅	semidirect
130. C₄×M₁₆	C₅×D₃	semidirect
206. C₄×C₈	C₂×C₅×D₃	semidirect
230. D₁₂	C₅×M₁₆	semidirect
272. C₃×C₅	#1490	semidirect
321. C₅	#69855	direct (N × H)
337. C₃	C₅×#1490	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×D₁₂	2
C₃×C₄×C₅×M₁₆	2
C₅×#4926	2
C₅×#4459	2
C₅×#1490	3
#69855	5

Subgroups — C5×#69855 #14282285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69855 #14282285