Subgroups — C₄×C₅×#805 #14295574

Order 1920.

720 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	C₄×C₅×#235	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#805	C₂	1	2	yes	maximal
4	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4523	C₂	1	2	yes	maximal, complemented
6	C₄×C₅×#104	—	3	1	no	maximal, Hall
7	C₂×C₅×#235	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄²×C₅	—	2	2	no	—
9	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₄×C₅×#67	—	2	2	no	—
12	C₅×#926	C₂²	1	1	yes	characteristic
13	C₂×C₅×#208	C₂²	1	2	yes	—
14	C₄×C₅×D₁₂	C₄	1	2	yes	complemented
15	C₅×#872	—	2	1	no	—
16	C₄²×C₅×D₃	—	2	1	no	—
17	C₄×C₅×Dic₆	C₄	1	2	yes	complemented
18	C₅×#805	C₄	1	8	yes	direct product factor
19	C₅×#803	—	2	1	no	—
20	C₄×C₅×C₃⋊C₈	—	2	1	no	—
21	C₄×#805	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₄×C₅×#41	—	3	1	no	—
23	C₂×C₅×#104	—	3	2	no	—
24	C₄×C₅×M₁₆	—	3	1	no	—
25	C₄³×C₅	D₃	1	1	yes	characteristic, complemented
26	C₅×#339	—	3	2	no	—
27	C₂²×C₃×C₄×C₅	—	2	2	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₂×C₅×#67	—	2	1	no	—
31	C₅×#235	—	2	2	no	—
32	C₅×#235	C₂×C₄	1	4	yes	—
33	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	—	2	1	no	—
35	C₂×C₄×C₅×D₃	—	2	2	no	—
36	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	—	2	8	no	—
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₅×#217	—	2	1	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
46	C₅×Dic₃⋊C₄	—	2	2	no	—
47	C₄×C₅×Dic₃	—	2	2	no	—
48	C₄×C₅×Dic₃	—	2	1	no	—
49	C₅×#208	—	2	2	no	—
50	C₅×#208	C₂×C₄	1	4	yes	complemented
51	C₂×C₅×C₃⋊C₈	—	2	2	no	—
52	C₄×#235	C₂×C₅	1	1	yes	characteristic
53	C₃×C₄³	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#805	C₂×C₅	1	2	yes	—
55	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
56	#4523	C₂×C₅	1	2	yes	complemented
57	C₂×C₅×#41	—	3	1	no	—
58	C₂×C₅×M₁₆	—	3	2	no	—
59	C₄×C₅×Q₈	—	3	2	no	—
60	C₄×C₅×D₄	—	3	2	no	—
61	C₄×C₅×D₄	—	6	2	no	—
62	C₅×#117	—	3	1	no	—
63	C₅×#117	—	6	1	no	—
64	C₂×C₄²×C₅	—	2	2	no	—
65	C₂×C₄²×C₅	—	6	1	no	—
66	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	C₅×#104	—	3	8	no	—
69	C₅×#97	—	6	1	no	—
70	C₄×C₅×C₈	—	6	1	no	—
71	C₄×#104	—	3	1	no	Sylow
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₅×D₃	—	2	1	no	—
74	C₂×C₃×C₄×C₅	—	2	8	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	2	4	no	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₅×#67	—	2	4	no	—
84	C₂×C₅×Dic₃	—	2	1	no	—
85	C₂×C₅×Dic₃	—	2	2	no	—
86	C₅×D₁₂	—	2	1	no	—
87	C₅×D₁₂	C₄²	1	2	yes	complemented
88	C₄×C₅×D₃	—	2	4	no	—
89	C₄×C₅×D₃	—	4	4	no	—
90	C₅×Dic₆	—	2	1	no	—
91	C₅×Dic₆	C₄²	1	2	yes	complemented
92	C₅×C₃⋊C₈	—	2	4	no	—
93	C₂×#235	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₃×C₄²	—	2	2	no	—
95	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
96	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
97	C₄×#67	—	2	2	no	—
98	#926	C₂²×C₅	1	1	yes	characteristic
99	C₂×#208	C₂²×C₅	1	2	yes	—
100	C₄×D₁₂	C₄×C₅	1	2	yes	complemented
101	#872	—	2	1	no	—
102	C₄²×D₃	—	2	1	no	—
103	C₄×Dic₆	C₄×C₅	1	2	yes	complemented
104	#805	C₄×C₅	1	8	yes	direct product factor
105	#803	—	2	1	no	—
106	C₄×C₃⋊C₈	—	2	1	no	—
107	C₅×#41	—	3	4	no	—
108	C₅×#41	—	6	2	no	—
109	C₂×C₅×Q₈	—	3	1	no	—
110	C₂×C₅×D₄	—	3	1	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂²×C₄×C₅	—	2	2	no	—
113	C₂²×C₄×C₅	—	6	1	no	—
114	C₂²×C₄×C₅	—	6	2	no	—
115	C₂²×C₄×C₅	#67	1	2	yes	—
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₅×M₁₆	—	3	4	no	—
118	C₅×M₁₆	—	6	2	no	—
119	C₂×C₅×C₈	—	6	2	no	—
120	C₅×C₄⋊C₄	—	3	2	no	—
121	C₅×C₄⋊C₄	—	6	2	no	—
122	C₅×C₂²⋊C₄	—	6	1	no	—
123	C₅×C₂²⋊C₄	—	6	2	no	—
124	C₄²×C₅	—	2	8	no	—
125	C₄²×C₅	—	2	2	no	—
126	C₄²×C₅	—	6	2	no	—
127	C₄²×C₅	—	6	1	no	—
128	C₄²×C₅	C₄×D₃	1	2	yes	—
129	C₄²×C₅	C₄×D₃	1	4	yes	complemented
130	C₄²×C₅	D₁₂	1	1	yes	characteristic
131	C₄²×C₅	D₁₂	1	1	yes	characteristic
132	C₄×#41	—	3	1	no	—
133	C₂×#104	—	3	2	no	—
134	C₄×M₁₆	—	3	1	no	—
135	C₄³	C₅×D₃	1	1	yes	characteristic, complemented
136	#339	—	3	2	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
140	C₂×C₅×D₃	—	2	2	no	—
141	C₂×C₅×D₃	—	4	2	no	—
142	C₃×C₄×C₅	—	2	8	no	—
143	C₃×C₄×C₅	—	2	2	no	—
144	C₃×C₄×C₅	#104	1	4	yes	complemented
145	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
146	C₃×C₄×C₅	#117	1	1	yes	characteristic
147	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
148	C₅×Dic₃	—	2	2	no	—
149	C₅×Dic₃	—	4	2	no	—
150	C₂²×C₃×C₄	—	2	2	no	—
151	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
152	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
153	C₂×#67	—	2	1	no	—
154	#235	—	2	2	no	—
155	#235	C₂×C₄×C₅	1	4	yes	—
156	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₄×D₃	—	2	1	no	—
158	C₂×C₄×D₃	—	2	2	no	—
159	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
160	C₃×C₄²	—	2	8	no	—
161	C₃×C₄²	—	2	2	no	—
162	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
163	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
164	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
165	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
166	#217	—	2	1	no	—
167	(C₂×D₃)⋊C₄	—	2	2	no	—
168	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
169	Dic₃⋊C₄	—	2	2	no	—
170	C₄×Dic₃	—	2	2	no	—
171	C₄×Dic₃	—	2	1	no	—
172	#208	—	2	2	no	—
173	#208	C₂×C₄×C₅	1	4	yes	complemented
174	C₂×C₃⋊C₈	—	2	2	no	—
175	C₂³×C₅	—	6	1	no	—
176	C₂³×C₅	C₂×#67	1	1	yes	characteristic
177	C₅×Q₈	—	3	2	no	—
178	C₅×Q₈	—	6	1	no	—
179	C₅×D₄	—	3	2	no	—
180	C₅×D₄	—	6	4	no	—
181	C₅×D₄	—	6	1	no	—
182	C₂×C₄×C₅	—	2	8	no	—
183	C₂×C₄×C₅	—	2	4	no	—
184	C₂×C₄×C₅	—	2	4	no	—
185	C₂×C₄×C₅	—	6	1	no	—
186	C₂×C₄×C₅	—	6	4	no	—
187	C₂×C₄×C₅	—	6	2	no	—
188	C₂×C₄×C₅	—	12	4	no	—
189	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
190	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
191	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
193	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
194	C₂×C₄×C₅	#235	1	2	yes	—
195	C₅×C₈	—	6	4	no	—
196	C₂×#41	—	3	1	no	—
197	C₂×M₁₆	—	3	2	no	—
198	C₄×Q₈	—	3	2	no	—
199	C₄×D₄	—	3	2	no	—
200	C₄×D₄	—	6	2	no	—
201	#117	—	3	1	no	—
202	#117	—	6	1	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	6	1	no	—
205	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
206	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
207	#104	—	3	8	no	—
208	#97	—	6	1	no	—
209	C₄×C₈	—	6	1	no	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₃×C₅	C₂×#104	1	2	yes	—
213	C₅×D₃	—	4	2	no	—
214	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂²×D₃	—	2	1	no	—
216	C₂×C₃×C₄	—	2	8	no	—
217	C₂×C₃×C₄	—	2	4	no	—
218	C₂×C₃×C₄	—	2	4	no	—
219	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
220	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
221	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
223	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₅×#41	1	2	yes	—
225	#67	—	2	4	no	—
226	C₂×Dic₃	—	2	1	no	—
227	C₂×Dic₃	—	2	2	no	—
228	D₁₂	—	2	1	no	—
229	D₁₂	C₄²×C₅	1	2	yes	complemented
230	C₄×D₃	—	2	4	no	—
231	C₄×D₃	—	4	4	no	—
232	Dic₆	—	2	1	no	—
233	Dic₆	C₄²×C₅	1	2	yes	complemented
234	C₃⋊C₈	—	2	4	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	—	6	2	no	—
237	C₂²×C₅	—	12	2	no	—
238	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
239	C₂²×C₅	C₄×#67	1	2	yes	—
240	C₄×C₅	—	2	8	no	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	6	2	no	—
243	C₄×C₅	—	12	2	no	—
244	C₄×C₅	#805	1	4	yes	direct product factor
245	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
246	C₄×C₅	#872	1	1	yes	characteristic
247	C₄×C₅	C₄×D₁₂	1	2	yes	—
248	#41	—	3	4	no	—
249	#41	—	6	2	no	—
250	C₂×Q₈	—	3	1	no	—
251	C₂×D₄	—	3	1	no	—
252	C₂×D₄	—	6	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	6	1	no	—
255	C₂²×C₄	—	6	2	no	—
256	C₂²×C₄	C₅×#67	1	2	yes	—
257	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
258	M₁₆	—	3	4	no	—
259	M₁₆	—	6	2	no	—
260	C₂×C₈	—	6	2	no	—
261	C₄⋊C₄	—	3	2	no	—
262	C₄⋊C₄	—	6	2	no	—
263	C₂²⋊C₄	—	6	1	no	—
264	C₂²⋊C₄	—	6	2	no	—
265	C₄²	—	2	8	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	6	2	no	—
268	C₄²	—	6	1	no	—
269	C₄²	C₄×C₅×D₃	1	2	yes	—
270	C₄²	C₄×C₅×D₃	1	4	yes	complemented
271	C₄²	C₅×D₁₂	1	1	yes	characteristic
272	C₄²	C₅×D₁₂	1	1	yes	characteristic
273	C₃×C₅	C₄×#104	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
276	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
277	C₂×D₃	—	2	2	no	—
278	C₂×D₃	—	4	2	no	—
279	C₃×C₄	—	2	8	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	C₅×#104	1	4	yes	complemented
282	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
283	C₃×C₄	C₅×#117	1	1	yes	characteristic
284	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
285	Dic₃	—	2	2	no	—
286	Dic₃	—	4	2	no	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	12	2	no	—
289	C₂×C₅	C₂×#805	1	2	yes	—
290	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
291	C₂³	—	6	1	no	—
292	C₂³	C₂×C₅×#67	1	1	yes	characteristic
293	Q₈	—	3	2	no	—
294	Q₈	—	6	1	no	—
295	D₄	—	3	2	no	—
296	D₄	—	6	4	no	—
297	D₄	—	6	1	no	—
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	6	1	no	—
302	C₂×C₄	—	6	4	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	12	4	no	—
305	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
306	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
307	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
308	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
309	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
310	C₂×C₄	C₅×#235	1	2	yes	—
311	C₈	—	6	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₃	C₂×C₅×#104	1	2	yes	—
315	D₃	—	4	2	no	—
316	C₅	C₄×#805	1	1	yes	characteristic, direct product factor, Sylow
317	C₂²	—	2	2	no	—
318	C₂²	—	6	2	no	—
319	C₂²	—	12	2	no	—
320	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
321	C₂²	C₄×C₅×#67	1	2	yes	—
322	C₄	—	2	8	no	—
323	C₄	—	2	2	no	—
324	C₄	—	6	2	no	—
325	C₄	—	12	2	no	—
326	C₄	C₅×#805	1	4	yes	direct product factor
327	C₄	C₄²×C₅×D₃	1	1	yes	characteristic
328	C₄	C₅×#872	1	1	yes	characteristic
329	C₄	C₄×C₅×D₁₂	1	2	yes	—
330	C₃	C₄×C₅×#104	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	12	2	no	—
333	C₂	C₂×C₅×#805	1	2	yes	—
334	C₂	C₄×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₅	D₁₂
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₄³×C₅	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₄³×C₅	D₃
hypocenter	C₃	C₄×C₅×#104
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄²×C₅ ◁ C₂×C₄²×C₅ ◁ C₄³×C₅

Complements

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

N	H ≅ G/N	type
2. C₃×C₄³×C₅	C₂	semidirect
4. C₄×C₅×#208	C₂	semidirect
5. C₅×#4523	C₂	semidirect
14. C₄×C₅×D₁₂	C₄	semidirect
17. C₄×C₅×Dic₆	C₄	semidirect
18. C₅×#805	C₄	direct (N × H)
21. C₄×#805	C₅	direct (N × H)
25. C₄³×C₅	D₃	semidirect
40. C₃×C₄²×C₅	C₂×C₄	semidirect
50. C₅×#208	C₂×C₄	semidirect
53. C₃×C₄³	C₂×C₅	semidirect
55. C₄×#208	C₂×C₅	semidirect
56. #4523	C₂×C₅	semidirect
87. C₅×D₁₂	C₄²	semidirect
91. C₅×Dic₆	C₄²	semidirect
100. C₄×D₁₂	C₄×C₅	semidirect
103. C₄×Dic₆	C₄×C₅	semidirect
104. #805	C₄×C₅	direct (N × H)
129. C₄²×C₅	C₄×D₃	semidirect
135. C₄³	C₅×D₃	semidirect
144. C₃×C₄×C₅	#104	semidirect
163. C₃×C₄²	C₂×C₄×C₅	semidirect
173. #208	C₂×C₄×C₅	semidirect
229. D₁₂	C₄²×C₅	semidirect
233. Dic₆	C₄²×C₅	semidirect
244. C₄×C₅	#805	direct (N × H)
270. C₄²	C₄×C₅×D₃	semidirect
273. C₃×C₅	C₄×#104	semidirect
281. C₃×C₄	C₅×#104	semidirect
316. C₅	C₄×#805	direct (N × H)
326. C₄	C₅×#805	direct (N × H)
330. C₃	C₄×C₅×#104	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#235	2
C₃×C₄³×C₅	2
C₂×C₅×#805	2
C₄×C₅×#208	2
C₅×#4523	2
C₄×C₅×#104	3
C₄×#805	5

Subgroups — C4×C5×#805 #14295574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#805 #14295574