Subgroups — C₅×#72029 #14301708

Order 1920.

684 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5828	C₂	1	1	yes	maximal, characteristic
3	C₅×#5237	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5126	C₂	1	2	yes	maximal, complemented
5	C₅×#4556	C₂	1	2	yes	maximal, complemented
6	C₅×#1891	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
9	C₃×C₅×#131	—	2	4	no	—
10	C₂×C₃×C₅×M₁₆	—	2	2	no	—
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₅×#946	—	2	1	no	—
13	C₅×#938	—	2	1	no	—
14	C₅×#937	C₄	1	2	yes	—
15	C₅×#935	C₄	1	2	yes	—
16	C₄×C₅×#67	—	2	2	no	—
17	C₅×#926	C₂²	1	1	yes	characteristic
18	C₂×C₅×#208	C₂²	1	2	yes	—
19	C₅×#836	—	2	1	no	—
20	C₅×#833	—	2	1	no	—
21	C₅×(C₂×D₃)⋊C₈	—	2	4	no	—
22	#72029	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#568	D₃	1	1	yes	characteristic, complemented
24	C₅×#525	—	3	1	no	—
25	C₅×#413	—	3	1	no	—
26	C₅×#407	—	3	2	no	—
27	C₅×#344	—	3	2	no	—
28	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
29	C₃×C₅×#41	—	2	2	no	—
30	C₃×C₅×#41	—	2	2	no	—
31	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×C₅×#67	—	2	2	no	—
37	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	—	2	1	no	—
39	C₂×C₄×C₅×D₃	—	2	1	no	—
40	C₂×C₄×C₅×D₃	—	2	2	no	—
41	C₂×C₄×C₅×D₃	—	2	2	no	—
42	C₃×C₅×M₁₆	—	2	4	no	—
43	C₃×C₅×M₁₆	—	2	2	no	—
44	C₃×C₅×M₁₆	—	2	2	no	—
45	C₂×C₃×C₅×C₈	—	2	4	no	—
46	C₂×C₃×C₅×C₈	D₄	1	4	yes	complemented
47	C₅×#217	—	2	2	no	—
48	C₅×#217	C₂×C₄	1	2	yes	—
49	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×Dic₃⋊C₄	—	2	2	no	—
52	C₄×C₅×Dic₃	—	2	1	no	—
53	C₅×#208	—	2	2	no	—
54	C₅×#208	—	2	2	no	—
55	C₂×C₅×C₃⋊C₈	—	2	2	no	—
56	C₃×#568	C₂×C₅	1	1	yes	characteristic, complemented
57	#5828	C₂×C₅	1	1	yes	characteristic
58	#5237	C₂×C₅	1	1	yes	characteristic, complemented
59	#5126	C₂×C₅	1	2	yes	complemented
60	#4556	C₂×C₅	1	2	yes	complemented
61	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
62	C₂³×C₄×C₅	—	3	1	no	—
63	C₅×#131	—	2	4	no	—
64	C₂×C₅×M₁₆	—	2	2	no	—
65	C₂×C₅×M₁₆	—	3	2	no	—
66	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
67	C₅×#123	—	3	2	no	—
68	C₅×C₂²⋊Q₈	—	6	1	no	—
69	C₅×#121	—	6	1	no	—
70	C₅×#120	—	3	2	no	—
71	C₄×C₅×D₄	—	6	2	no	—
72	C₅×#117	—	3	1	no	—
73	C₅×#101	—	6	1	no	—
74	C₅×#100	—	6	1	no	—
75	C₅×C₂²⋊C₈	—	6	4	no	—
76	#1891	—	3	1	no	Sylow
77	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
78	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₅×D₃	—	2	1	no	—
80	C₂²×C₅×D₃	—	2	2	no	—
81	C₂²×C₅×D₃	—	4	2	no	—
82	C₃×C₅×Q₈	—	2	2	no	—
83	C₃×C₅×D₄	—	2	4	no	—
84	C₃×C₅×D₄	—	2	2	no	—
85	C₂×C₃×C₄×C₅	—	2	2	no	—
86	C₂×C₃×C₄×C₅	—	2	2	no	—
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	#41	1	2	yes	—
92	C₅×#67	—	4	4	no	—
93	C₂×C₅×Dic₃	—	2	2	no	—
94	C₂×C₅×Dic₃	—	2	1	no	—
95	C₂×C₅×Dic₃	—	2	2	no	—
96	C₄×C₅×D₃	—	4	1	no	—
97	C₄×C₅×D₃	—	4	2	no	—
98	C₄×C₅×D₃	—	4	1	no	—
99	C₃×C₅×C₈	—	2	4	no	—
100	C₅×C₃⋊C₈	—	4	2	no	—
101	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
102	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
103	C₃×#131	—	2	4	no	—
104	C₂×C₃×M₁₆	—	2	2	no	—
105	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
106	#946	—	2	1	no	—
107	#938	—	2	1	no	—
108	#937	C₄×C₅	1	2	yes	—
109	#935	C₄×C₅	1	2	yes	—
110	C₄×#67	—	2	2	no	—
111	#926	C₂²×C₅	1	1	yes	characteristic
112	C₂×#208	C₂²×C₅	1	2	yes	—
113	#836	—	2	1	no	—
114	#833	—	2	1	no	—
115	(C₂×D₃)⋊C₈	—	2	4	no	—
116	C₂⁴×C₅	—	3	1	no	—
117	C₅×#41	—	2	2	no	—
118	C₅×#41	—	2	2	no	—
119	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
120	C₂×C₅×D₄	—	6	2	no	—
121	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
122	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
123	C₂²×C₄×C₅	—	3	1	no	—
124	C₂²×C₄×C₅	—	6	1	no	—
125	C₂²×C₄×C₅	—	6	2	no	—
126	C₂²×C₄×C₅	—	6	1	no	—
127	C₂²×C₄×C₅	—	6	2	no	—
128	C₂²×C₄×C₅	#67	1	2	yes	—
129	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
130	C₅×M₁₆	—	2	4	no	—
131	C₅×M₁₆	—	2	2	no	—
132	C₅×M₁₆	—	2	2	no	—
133	C₅×M₁₆	—	6	2	no	—
134	C₅×M₁₆	—	6	2	no	—
135	C₂×C₅×C₈	—	2	4	no	—
136	C₂×C₅×C₈	—	6	2	no	—
137	C₂×C₅×C₈	#67	1	4	yes	complemented
138	C₅×C₄⋊C₄	—	6	2	no	—
139	C₅×C₄⋊C₄	—	6	1	no	—
140	C₅×C₂²⋊C₄	—	3	2	no	—
141	C₅×C₂²⋊C₄	—	6	2	no	—
142	C₅×C₂²⋊C₄	—	6	2	no	—
143	C₄²×C₅	—	6	1	no	—
144	#568	C₅×D₃	1	1	yes	characteristic, complemented
145	#525	—	3	1	no	—
146	#413	—	3	1	no	—
147	#407	—	3	2	no	—
148	#344	—	3	2	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	—	4	1	no	—
151	C₂²×C₃×C₅	—	4	2	no	—
152	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
154	C₂×C₅×D₃	—	2	2	no	—
155	C₂×C₅×D₃	—	2	4	no	—
156	C₂×C₅×D₃	—	4	1	no	—
157	C₂×C₅×D₃	—	4	2	no	—
158	C₂×C₅×D₃	—	4	1	no	—
159	C₃×C₄×C₅	—	2	2	no	—
160	C₃×C₄×C₅	#120	1	1	yes	characteristic
161	C₃×C₄×C₅	#121	1	2	yes	—
162	C₃×C₄×C₅	#123	1	1	yes	characteristic
163	C₅×Dic₃	—	4	2	no	—
164	C₅×Dic₃	—	4	1	no	—
165	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic
166	C₃×#41	—	2	2	no	—
167	C₃×#41	—	2	2	no	—
168	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
169	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
170	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
171	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
172	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
173	C₂×#67	—	2	2	no	—
174	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×C₄×D₃	—	2	2	no	—
176	C₂×C₄×D₃	—	2	1	no	—
177	C₂×C₄×D₃	—	2	1	no	—
178	C₂×C₄×D₃	—	2	2	no	—
179	C₃×M₁₆	—	2	4	no	—
180	C₃×M₁₆	—	2	2	no	—
181	C₃×M₁₆	—	2	2	no	—
182	C₂×C₃×C₈	—	2	4	no	—
183	C₂×C₃×C₈	C₅×D₄	1	4	yes	complemented
184	#217	—	2	2	no	—
185	#217	C₂×C₄×C₅	1	2	yes	—
186	(C₂×D₃)⋊C₄	—	2	2	no	—
187	C₄⋊Dic₃	—	2	1	no	—
188	Dic₃⋊C₄	—	2	2	no	—
189	C₄×Dic₃	—	2	1	no	—
190	#208	—	2	2	no	—
191	#208	—	2	2	no	—
192	C₂×C₃⋊C₈	—	2	2	no	—
193	C₂³×C₅	—	6	1	no	—
194	C₂³×C₅	—	6	2	no	—
195	C₂³×C₅	—	12	2	no	—
196	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
197	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
198	C₅×Q₈	—	2	2	no	—
199	C₅×D₄	—	2	4	no	—
200	C₅×D₄	—	2	2	no	—
201	C₅×D₄	—	12	4	no	—
202	C₂×C₄×C₅	—	2	2	no	—
203	C₂×C₄×C₅	—	2	2	no	—
204	C₂×C₄×C₅	—	6	2	no	—
205	C₂×C₄×C₅	—	6	1	no	—
206	C₂×C₄×C₅	—	6	2	no	—
207	C₂×C₄×C₅	—	12	1	no	—
208	C₂×C₄×C₅	—	12	2	no	—
209	C₂×C₄×C₅	—	12	1	no	—
210	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
211	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
212	C₂×C₄×C₅	#235	1	2	yes	—
213	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
214	C₂×C₄×C₅	C₂×#67	1	2	yes	—
215	C₅×C₈	—	2	4	no	—
216	C₅×C₈	—	12	2	no	—
217	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
218	C₂³×C₄	—	3	1	no	—
219	#131	—	2	4	no	—
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	3	2	no	—
222	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
223	#123	—	3	2	no	—
224	C₂²⋊Q₈	—	6	1	no	—
225	#121	—	6	1	no	—
226	#120	—	3	2	no	—
227	C₄×D₄	—	6	2	no	—
228	#117	—	3	1	no	—
229	#101	—	6	1	no	—
230	#100	—	6	1	no	—
231	C₂²⋊C₈	—	6	4	no	—
232	C₂×C₃×C₅	—	2	1	no	—
233	C₂×C₃×C₅	—	2	2	no	—
234	C₂×C₃×C₅	—	4	2	no	—
235	C₂×C₃×C₅	#386	1	1	yes	characteristic
236	C₅×D₃	—	4	2	no	—
237	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
238	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
239	C₂²×D₃	—	2	1	no	—
240	C₂²×D₃	—	2	2	no	—
241	C₂²×D₃	—	4	2	no	—
242	C₃×Q₈	—	2	2	no	—
243	C₃×D₄	—	2	4	no	—
244	C₃×D₄	—	2	2	no	—
245	C₂×C₃×C₄	—	2	2	no	—
246	C₂×C₃×C₄	—	2	2	no	—
247	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
248	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
250	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂×C₃×C₄	C₅×#41	1	2	yes	—
252	#67	—	4	4	no	—
253	C₂×Dic₃	—	2	2	no	—
254	C₂×Dic₃	—	2	1	no	—
255	C₂×Dic₃	—	2	2	no	—
256	C₄×D₃	—	4	1	no	—
257	C₄×D₃	—	4	2	no	—
258	C₄×D₃	—	4	1	no	—
259	C₃×C₈	—	2	4	no	—
260	C₃⋊C₈	—	4	2	no	—
261	C₂²×C₅	—	2	2	no	—
262	C₂²×C₅	—	4	1	no	—
263	C₂²×C₅	—	4	2	no	—
264	C₂²×C₅	—	6	2	no	—
265	C₂²×C₅	—	6	4	no	—
266	C₂²×C₅	—	12	1	no	—
267	C₂²×C₅	—	12	2	no	—
268	C₂²×C₅	—	12	1	no	—
269	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
270	C₂²×C₅	C₄×#67	1	2	yes	—
271	C₄×C₅	—	2	2	no	—
272	C₄×C₅	—	12	2	no	—
273	C₄×C₅	—	12	1	no	—
274	C₄×C₅	#929	1	1	yes	characteristic
275	C₄×C₅	#930	1	2	yes	—
276	C₄×C₅	#953	1	1	yes	characteristic
277	C₂⁴	—	3	1	no	—
278	#41	—	2	2	no	—
279	#41	—	2	2	no	—
280	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
281	C₂×D₄	—	6	2	no	—
282	C₂×D₄	C₄×C₅×D₃	1	2	yes	—
283	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
284	C₂²×C₄	—	3	1	no	—
285	C₂²×C₄	—	6	1	no	—
286	C₂²×C₄	—	6	2	no	—
287	C₂²×C₄	—	6	2	no	—
288	C₂²×C₄	—	6	1	no	—
289	C₂²×C₄	C₅×#67	1	2	yes	—
290	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
291	M₁₆	—	2	4	no	—
292	M₁₆	—	2	2	no	—
293	M₁₆	—	2	2	no	—
294	M₁₆	—	6	2	no	—
295	M₁₆	—	6	2	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	6	2	no	—
298	C₂×C₈	C₅×#67	1	4	yes	complemented
299	C₄⋊C₄	—	6	2	no	—
300	C₄⋊C₄	—	6	1	no	—
301	C₂²⋊C₄	—	3	2	no	—
302	C₂²⋊C₄	—	6	2	no	—
303	C₂²⋊C₄	—	6	2	no	—
304	C₄²	—	6	1	no	—
305	C₃×C₅	#1891	1	1	yes	characteristic, complemented, Hall
306	C₂²×C₃	—	2	2	no	—
307	C₂²×C₃	—	4	1	no	—
308	C₂²×C₃	—	4	2	no	—
309	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
310	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
311	C₂×D₃	—	2	2	no	—
312	C₂×D₃	—	2	4	no	—
313	C₂×D₃	—	4	1	no	—
314	C₂×D₃	—	4	2	no	—
315	C₂×D₃	—	4	1	no	—
316	C₃×C₄	—	2	2	no	—
317	C₃×C₄	C₅×#120	1	1	yes	characteristic
318	C₃×C₄	C₅×#121	1	2	yes	—
319	C₃×C₄	C₅×#123	1	1	yes	characteristic
320	Dic₃	—	4	2	no	—
321	Dic₃	—	4	1	no	—
322	C₂×C₅	—	2	1	no	—
323	C₂×C₅	—	2	2	no	—
324	C₂×C₅	—	4	2	no	—
325	C₂×C₅	—	12	2	no	—
326	C₂×C₅	#5213	1	1	yes	characteristic
327	C₂³	—	6	1	no	—
328	C₂³	—	6	2	no	—
329	C₂³	—	12	2	no	—
330	C₂³	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
331	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
332	Q₈	—	2	2	no	—
333	D₄	—	2	4	no	—
334	D₄	—	2	2	no	—
335	D₄	—	12	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	6	2	no	—
339	C₂×C₄	—	6	1	no	—
340	C₂×C₄	—	6	2	no	—
341	C₂×C₄	—	12	1	no	—
342	C₂×C₄	—	12	2	no	—
343	C₂×C₄	—	12	1	no	—
344	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
345	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
346	C₂×C₄	C₅×#235	1	2	yes	—
347	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
348	C₂×C₄	C₂×C₅×#67	1	2	yes	—
349	C₈	—	2	4	no	—
350	C₈	—	12	2	no	—
351	C₂×C₃	—	2	1	no	—
352	C₂×C₃	—	2	2	no	—
353	C₂×C₃	—	4	2	no	—
354	C₂×C₃	C₅×#386	1	1	yes	characteristic
355	D₃	—	4	2	no	—
356	C₅	#72029	1	1	yes	characteristic, direct product factor, Sylow
357	C₂²	—	2	2	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	6	2	no	—
361	C₂²	—	6	4	no	—
362	C₂²	—	12	1	no	—
363	C₂²	—	12	2	no	—
364	C₂²	—	12	1	no	—
365	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
366	C₂²	C₄×C₅×#67	1	2	yes	—
367	C₄	—	2	2	no	—
368	C₄	—	12	2	no	—
369	C₄	—	12	1	no	—
370	C₄	C₅×#929	1	1	yes	characteristic
371	C₄	C₅×#930	1	2	yes	—
372	C₄	C₅×#953	1	1	yes	characteristic
373	C₃	C₅×#1891	1	1	yes	characteristic, complemented, Sylow
374	C₂	—	2	1	no	—
375	C₂	—	2	2	no	—
376	C₂	—	4	2	no	—
377	C₂	—	12	2	no	—
378	C₂	C₅×#5213	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#953
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	#386
prime radical	C₅×#5828	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#568	D₃
hypocenter	C₃	C₅×#1891
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#568

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#568	C₂	semidirect
3. C₅×#5237	C₂	semidirect
4. C₅×#5126	C₂	semidirect
5. C₅×#4556	C₂	semidirect
22. #72029	C₅	direct (N × H)
23. C₅×#568	D₃	semidirect
46. C₂×C₃×C₅×C₈	D₄	semidirect
56. C₃×#568	C₂×C₅	semidirect
58. #5237	C₂×C₅	semidirect
59. #5126	C₂×C₅	semidirect
60. #4556	C₂×C₅	semidirect
137. C₂×C₅×C₈	#67	semidirect
144. #568	C₅×D₃	semidirect
183. C₂×C₃×C₈	C₅×D₄	semidirect
298. C₂×C₈	C₅×#67	semidirect
305. C₃×C₅	#1891	semidirect
356. C₅	#72029	direct (N × H)
373. C₃	C₅×#1891	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#568	2
C₅×#5828	2
C₅×#5237	2
C₅×#5126	2
C₅×#4556	2
C₅×#1891	3
#72029	5

Subgroups — C5×#72029 #14301708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72029 #14301708