Subgroups — C₅×#72030 #14303037

Order 1920.

680 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	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5830	C₂	1	1	yes	maximal, characteristic
3	C₅×#5237	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#823	C₂	1	2	yes	maximal, complemented
5	C₅×#4559	C₂	1	2	yes	maximal, complemented
6	C₅×#1914	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂²×C₅×D₁₂	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×#947	—	2	1	no	—
13	C₅×#940	—	2	1	no	—
14	C₅×#937	C₄	1	4	yes	—
15	C₅×#930	—	2	2	no	—
16	C₅×#926	C₂²	1	1	yes	characteristic
17	C₂×C₅×#208	C₂²	1	2	yes	—
18	C₅×#836	—	2	1	no	—
19	C₅×#833	—	2	1	no	—
20	C₅×#823	—	2	4	no	—
21	#72030	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#568	D₃	1	1	yes	characteristic, complemented
23	C₅×#535	—	3	1	no	—
24	C₅×#413	—	3	1	no	—
25	C₂×C₅×#100	—	3	2	no	—
26	C₅×#344	—	3	2	no	—
27	C₂³×C₅×D₃	C₂×C₄	1	2	yes	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₃×C₅×#41	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×#67	—	2	4	no	—
36	C₂×C₅×D₁₂	—	2	1	no	—
37	C₂×C₅×D₁₂	—	2	1	no	—
38	C₂×C₅×D₁₂	—	2	2	no	—
39	C₂×C₅×D₁₂	—	2	2	no	—
40	C₃×C₅×M₁₆	—	2	4	no	—
41	C₃×C₅×M₁₆	—	2	2	no	—
42	C₃×C₅×M₁₆	D₄	1	4	yes	complemented
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	—	2	2	no	—
45	C₅×#217	C₂×C₄	1	2	yes	—
46	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
47	C₅×C₄⋊Dic₃	—	2	1	no	—
48	C₄×C₅×Dic₃	—	2	1	no	—
49	C₅×#208	—	2	2	no	—
50	C₅×#208	—	2	2	no	—
51	C₂×C₅×C₃⋊C₈	—	2	2	no	—
52	C₃×#568	C₂×C₅	1	1	yes	characteristic, complemented
53	#5830	C₂×C₅	1	1	yes	characteristic
54	#5237	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×#823	C₂×C₅	1	2	yes	complemented
56	#4559	C₂×C₅	1	2	yes	complemented
57	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₅×D₄	—	3	1	no	—
59	C₅×#131	—	2	4	no	—
60	C₂×C₅×M₁₆	—	2	2	no	—
61	C₂×C₅×M₁₆	—	3	2	no	—
62	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
63	C₅×#127	—	6	1	no	—
64	C₅×#124	—	6	1	no	—
65	C₅×#121	—	6	2	no	—
66	C₅×#120	—	3	4	no	—
67	C₅×#117	—	3	1	no	—
68	C₅×#101	—	6	1	no	—
69	C₅×#100	—	6	1	no	—
70	C₅×#100	—	6	4	no	—
71	#1914	—	3	1	no	Sylow
72	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₅×D₃	—	2	2	no	—
75	C₂²×C₅×D₃	—	2	4	no	—
76	C₂²×C₅×D₃	—	8	2	no	—
77	C₃×C₅×Q₈	—	2	2	no	—
78	C₃×C₅×D₄	—	2	4	no	—
79	C₃×C₅×D₄	—	2	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	#41	1	2	yes	—
87	C₅×#67	—	8	4	no	—
88	C₂×C₅×Dic₃	—	2	2	no	—
89	C₅×D₁₂	—	4	1	no	—
90	C₅×D₁₂	—	4	2	no	—
91	C₅×D₁₂	—	4	1	no	—
92	C₃×C₅×C₈	—	2	4	no	—
93	C₅×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
95	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
96	C₃×#131	—	2	4	no	—
97	C₂×C₃×M₁₆	—	2	2	no	—
98	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
99	#947	—	2	1	no	—
100	#940	—	2	1	no	—
101	#937	C₄×C₅	1	4	yes	—
102	#930	—	2	2	no	—
103	#926	C₂²×C₅	1	1	yes	characteristic
104	C₂×#208	C₂²×C₅	1	2	yes	—
105	#836	—	2	1	no	—
106	#833	—	2	1	no	—
107	#823	—	2	4	no	—
108	C₂⁴×C₅	—	3	2	no	—
109	C₅×#41	—	2	2	no	—
110	C₅×#41	—	2	2	no	—
111	C₂×C₅×Q₈	D₁₂	1	1	yes	characteristic
112	C₂×C₅×D₄	—	6	4	no	—
113	C₂×C₅×D₄	—	6	1	no	—
114	C₂×C₅×D₄	—	6	2	no	—
115	C₂×C₅×D₄	—	6	1	no	—
116	C₂×C₅×D₄	—	6	2	no	—
117	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
118	C₂×C₅×D₄	D₁₂	1	1	yes	characteristic
119	C₂²×C₄×C₅	#67	1	2	yes	—
120	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₅×M₁₆	—	2	4	no	—
122	C₅×M₁₆	—	2	2	no	—
123	C₅×M₁₆	—	6	2	no	—
124	C₅×M₁₆	—	6	2	no	—
125	C₅×M₁₆	#67	1	4	yes	complemented
126	C₂×C₅×C₈	—	2	4	no	—
127	C₂×C₅×C₈	—	2	2	no	—
128	C₂×C₅×C₈	—	6	2	no	—
129	C₅×C₄⋊C₄	—	6	1	no	—
130	C₅×C₂²⋊C₄	—	3	2	no	—
131	C₅×C₂²⋊C₄	—	6	4	no	—
132	C₄²×C₅	—	6	1	no	—
133	#568	C₅×D₃	1	1	yes	characteristic, complemented
134	#535	—	3	1	no	—
135	#413	—	3	1	no	—
136	C₂×#100	—	3	2	no	—
137	#344	—	3	2	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	1	no	—
140	C₂²×C₃×C₅	—	4	2	no	—
141	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
142	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
143	C₂×C₅×D₃	—	4	4	no	—
144	C₂×C₅×D₃	—	4	2	no	—
145	C₂×C₅×D₃	—	8	4	no	—
146	C₃×C₄×C₅	—	2	2	no	—
147	C₃×C₄×C₅	#120	1	1	yes	characteristic
148	C₃×C₄×C₅	#121	1	2	yes	—
149	C₃×C₄×C₅	#123	1	1	yes	characteristic
150	C₅×Dic₃	—	4	2	no	—
151	C₂³×D₃	C₂×C₄×C₅	1	2	yes	—
152	C₃×#41	—	2	2	no	—
153	C₃×#41	—	2	2	no	—
154	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
157	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
158	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
159	C₂×#67	—	2	4	no	—
160	C₂×D₁₂	—	2	2	no	—
161	C₂×D₁₂	—	2	1	no	—
162	C₂×D₁₂	—	2	1	no	—
163	C₂×D₁₂	—	2	2	no	—
164	C₃×M₁₆	—	2	4	no	—
165	C₃×M₁₆	—	2	2	no	—
166	C₃×M₁₆	C₅×D₄	1	4	yes	complemented
167	C₂×C₃×C₈	—	2	4	no	—
168	C₂×C₃×C₈	—	2	2	no	—
169	#217	C₂×C₄×C₅	1	2	yes	—
170	(C₂×D₃)⋊C₄	—	2	4	no	—
171	C₄⋊Dic₃	—	2	1	no	—
172	C₄×Dic₃	—	2	1	no	—
173	#208	—	2	2	no	—
174	#208	—	2	2	no	—
175	C₂×C₃⋊C₈	—	2	2	no	—
176	C₂³×C₅	—	6	2	no	—
177	C₂³×C₅	—	6	4	no	—
178	C₂³×C₅	—	24	2	no	—
179	C₂³×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
180	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
181	C₅×Q₈	—	2	2	no	—
182	C₅×D₄	—	2	4	no	—
183	C₅×D₄	—	2	2	no	—
184	C₅×D₄	—	12	1	no	—
185	C₅×D₄	—	12	2	no	—
186	C₅×D₄	—	12	1	no	—
187	C₅×D₄	—	24	4	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	6	2	no	—
191	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
192	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
193	C₂×C₄×C₅	#235	1	2	yes	—
194	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×#67	1	2	yes	—
196	C₅×C₈	—	2	4	no	—
197	C₅×C₈	—	12	2	no	—
198	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
199	C₂²×D₄	—	3	1	no	—
200	#131	—	2	4	no	—
201	C₂×M₁₆	—	2	2	no	—
202	C₂×M₁₆	—	3	2	no	—
203	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
204	#127	—	6	1	no	—
205	#124	—	6	1	no	—
206	#121	—	6	2	no	—
207	#120	—	3	4	no	—
208	#117	—	3	1	no	—
209	#101	—	6	1	no	—
210	#100	—	6	1	no	—
211	#100	—	6	4	no	—
212	C₂×C₃×C₅	—	2	1	no	—
213	C₂×C₃×C₅	—	2	2	no	—
214	C₂×C₃×C₅	—	4	2	no	—
215	C₂×C₃×C₅	#386	1	1	yes	characteristic
216	C₅×D₃	—	8	2	no	—
217	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
218	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂²×D₃	—	2	2	no	—
220	C₂²×D₃	—	2	4	no	—
221	C₂²×D₃	—	8	2	no	—
222	C₃×Q₈	—	2	2	no	—
223	C₃×D₄	—	2	4	no	—
224	C₃×D₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	2	2	no	—
227	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
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	C₅×#41	1	2	yes	—
232	#67	—	8	4	no	—
233	C₂×Dic₃	—	2	2	no	—
234	D₁₂	—	4	1	no	—
235	D₁₂	—	4	2	no	—
236	D₁₂	—	4	1	no	—
237	C₃×C₈	—	2	4	no	—
238	C₃⋊C₈	—	4	2	no	—
239	C₂²×C₅	—	2	2	no	—
240	C₂²×C₅	—	4	1	no	—
241	C₂²×C₅	—	4	2	no	—
242	C₂²×C₅	—	12	4	no	—
243	C₂²×C₅	—	12	2	no	—
244	C₂²×C₅	—	24	4	no	—
245	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
246	C₂²×C₅	C₄×#67	1	2	yes	—
247	C₄×C₅	—	2	2	no	—
248	C₄×C₅	—	12	2	no	—
249	C₄×C₅	#929	1	1	yes	characteristic
250	C₄×C₅	#930	1	2	yes	—
251	C₄×C₅	#953	1	1	yes	characteristic
252	C₂⁴	—	3	2	no	—
253	#41	—	2	2	no	—
254	#41	—	2	2	no	—
255	C₂×Q₈	C₅×D₁₂	1	1	yes	characteristic
256	C₂×D₄	—	6	4	no	—
257	C₂×D₄	—	6	1	no	—
258	C₂×D₄	—	6	2	no	—
259	C₂×D₄	—	6	2	no	—
260	C₂×D₄	—	6	1	no	—
261	C₂×D₄	C₄×C₅×D₃	1	2	yes	—
262	C₂×D₄	C₅×D₁₂	1	1	yes	characteristic
263	C₂²×C₄	C₅×#67	1	2	yes	—
264	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
265	M₁₆	—	2	4	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	6	2	no	—
268	M₁₆	—	6	2	no	—
269	M₁₆	C₅×#67	1	4	yes	complemented
270	C₂×C₈	—	2	4	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	6	2	no	—
273	C₄⋊C₄	—	6	1	no	—
274	C₂²⋊C₄	—	3	2	no	—
275	C₂²⋊C₄	—	6	4	no	—
276	C₄²	—	6	1	no	—
277	C₃×C₅	#1914	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	2	2	no	—
279	C₂²×C₃	—	4	1	no	—
280	C₂²×C₃	—	4	2	no	—
281	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
282	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
283	C₂×D₃	—	4	4	no	—
284	C₂×D₃	—	4	2	no	—
285	C₂×D₃	—	8	4	no	—
286	C₃×C₄	—	2	2	no	—
287	C₃×C₄	C₅×#120	1	1	yes	characteristic
288	C₃×C₄	C₅×#121	1	2	yes	—
289	C₃×C₄	C₅×#123	1	1	yes	characteristic
290	Dic₃	—	4	2	no	—
291	C₂×C₅	—	2	1	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	4	2	no	—
294	C₂×C₅	—	24	2	no	—
295	C₂×C₅	#5213	1	1	yes	characteristic
296	C₂³	—	6	2	no	—
297	C₂³	—	6	4	no	—
298	C₂³	—	24	2	no	—
299	C₂³	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
300	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
301	Q₈	—	2	2	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	2	no	—
304	D₄	—	12	1	no	—
305	D₄	—	12	2	no	—
306	D₄	—	12	1	no	—
307	D₄	—	24	4	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	6	2	no	—
311	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
312	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
313	C₂×C₄	C₅×#235	1	2	yes	—
314	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#67	1	2	yes	—
316	C₈	—	2	4	no	—
317	C₈	—	12	2	no	—
318	C₂×C₃	—	2	1	no	—
319	C₂×C₃	—	2	2	no	—
320	C₂×C₃	—	4	2	no	—
321	C₂×C₃	C₅×#386	1	1	yes	characteristic
322	D₃	—	8	2	no	—
323	C₅	#72030	1	1	yes	characteristic, direct product factor, Sylow
324	C₂²	—	2	2	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	2	no	—
327	C₂²	—	12	4	no	—
328	C₂²	—	12	2	no	—
329	C₂²	—	24	4	no	—
330	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
331	C₂²	C₄×C₅×#67	1	2	yes	—
332	C₄	—	2	2	no	—
333	C₄	—	12	2	no	—
334	C₄	C₅×#929	1	1	yes	characteristic
335	C₄	C₅×#930	1	2	yes	—
336	C₄	C₅×#953	1	1	yes	characteristic
337	C₃	C₅×#1914	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	1	no	—
339	C₂	—	2	2	no	—
340	C₂	—	4	2	no	—
341	C₂	—	24	2	no	—
342	C₂	C₅×#5213	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	#5213
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×#5830	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#568	D₃
hypocenter	C₃	C₅×#1914
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×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. C₂×C₅×#823	C₂	semidirect
5. C₅×#4559	C₂	semidirect
21. #72030	C₅	direct (N × H)
22. C₅×#568	D₃	semidirect
42. C₃×C₅×M₁₆	D₄	semidirect
52. C₃×#568	C₂×C₅	semidirect
54. #5237	C₂×C₅	semidirect
55. C₂×#823	C₂×C₅	semidirect
56. #4559	C₂×C₅	semidirect
125. C₅×M₁₆	#67	semidirect
133. #568	C₅×D₃	semidirect
166. C₃×M₁₆	C₅×D₄	semidirect
269. M₁₆	C₅×#67	semidirect
277. C₃×C₅	#1914	semidirect
323. C₅	#72030	direct (N × H)
337. C₃	C₅×#1914	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×#5830	2
C₅×#5237	2
C₂×C₅×#823	2
C₅×#4559	2
C₅×#1914	3
#72030	5

Subgroups — C5×#72030 #14303037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72030 #14303037