Subgroups — C₅×#71903 #14308899

Order 1920.

564 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	C₃×C₅×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5001	C₂	1	2	yes	maximal
3	C₅×#4988	C₂	1	1	yes	maximal, characteristic
4	C₅×#4532	C₂	1	2	yes	maximal, complemented
5	C₅×#4530	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2027	—	3	1	no	maximal, Hall
7	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
9	C₃×C₅×#125	—	2	2	no	—
10	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
11	C₃×C₅×#117	C₂²	1	2	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
14	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₅×#208	C₂²	1	2	yes	—
16	C₅×#809	—	2	2	no	—
17	C₅×#809	—	2	2	no	—
18	#71903	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#544	D₃	1	1	yes	characteristic, complemented
20	C₅×#418	—	3	2	no	—
21	C₅×#390	—	3	1	no	—
22	C₅×#343	—	3	1	no	—
23	C₅×#339	—	3	2	no	—
24	C₂³×C₅×D₃	—	2	1	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×C₅×Dic₃	—	2	1	no	—
29	C₂×C₅×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	2	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	Q₈	1	2	yes	complemented
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
38	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
39	C₃×C₄²×C₅	—	2	2	no	—
40	C₅×(C₂×D₃)⋊C₄	—	4	4	no	—
41	C₄×C₅×Dic₃	—	2	2	no	—
42	C₅×#208	—	2	2	no	—
43	C₅×#208	—	2	2	no	—
44	C₂×C₅×C₃⋊C₈	—	2	2	no	—
45	C₃×#544	C₂×C₅	1	1	yes	characteristic, complemented
46	#5001	C₂×C₅	1	2	yes	—
47	#4988	C₂×C₅	1	1	yes	characteristic
48	#4532	C₂×C₅	1	2	yes	complemented
49	#4530	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×C₅×D₄	—	3	1	no	—
51	C₂×C₅×M₁₆	—	3	2	no	—
52	C₅×C₄⋊Q₈	—	2	2	no	—
53	C₅×#125	—	2	2	no	—
54	C₅×C₂²⋊Q₈	—	2	2	no	—
55	C₅×#117	C₂×D₃	1	2	yes	—
56	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
58	C₂×C₄²×C₅	—	3	1	no	—
59	C₅×#102	—	6	2	no	—
60	C₅×#102	—	6	2	no	—
61	#2027	—	3	1	no	Sylow
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₅×D₃	—	2	2	no	—
64	C₂²×C₅×D₃	—	4	2	no	—
65	C₂²×C₅×D₃	—	4	2	no	—
66	C₂²×C₅×D₃	—	4	1	no	—
67	C₂²×C₅×D₃	—	4	1	no	—
68	C₃×C₅×Q₈	—	8	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	#41	1	2	yes	—
76	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
77	C₂×C₅×Dic₃	—	2	2	no	—
78	C₂×C₅×Dic₃	—	4	1	no	—
79	C₂×C₅×Dic₃	—	4	1	no	—
80	C₅×D₁₂	—	2	4	no	—
81	C₅×D₁₂	—	4	2	no	—
82	C₅×C₃⋊C₈	—	4	2	no	—
83	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊Q₈	—	2	2	no	—
85	C₃×#125	—	2	2	no	—
86	C₃×C₂²⋊Q₈	—	2	2	no	—
87	C₃×#117	C₂²×C₅	1	2	yes	—
88	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
89	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
90	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
91	C₂×#208	C₂²×C₅	1	2	yes	—
92	#809	—	2	2	no	—
93	#809	—	2	2	no	—
94	C₂⁴×C₅	—	6	1	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×D₄	—	3	2	no	—
97	C₂×C₅×D₄	—	6	2	no	—
98	C₂×C₅×D₄	—	6	1	no	—
99	C₂×C₅×D₄	—	6	2	no	—
100	C₂²×C₄×C₅	—	6	1	no	—
101	C₂²×C₄×C₅	#67	1	2	yes	—
102	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₅×M₁₆	—	6	2	no	—
104	C₅×M₁₆	—	6	2	no	—
105	C₂×C₅×C₈	—	6	2	no	—
106	C₅×C₄⋊C₄	—	2	2	no	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	#67	1	2	yes	—
110	C₅×C₄⋊C₄	#67	1	2	yes	—
111	C₅×C₂²⋊C₄	—	2	2	no	—
112	C₅×C₂²⋊C₄	—	12	4	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	6	2	no	—
115	#544	C₅×D₃	1	1	yes	characteristic, complemented
116	#418	—	3	2	no	—
117	#390	—	3	1	no	—
118	#343	—	3	1	no	—
119	#339	—	3	2	no	—
120	C₂²×C₃×C₅	—	2	1	no	—
121	C₂²×C₃×C₅	—	2	1	no	—
122	C₂²×C₃×C₅	#120	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
124	C₂²×C₃×C₅	#124	1	1	yes	characteristic
125	C₂×C₅×D₃	—	4	4	no	—
126	C₂×C₅×D₃	—	4	2	no	—
127	C₂×C₅×D₃	—	8	2	no	—
128	C₂×C₅×D₃	—	8	1	no	—
129	C₂×C₅×D₃	—	8	1	no	—
130	C₃×C₄×C₅	—	4	2	no	—
131	C₃×C₄×C₅	—	4	2	no	—
132	C₃×C₄×C₅	—	4	2	no	—
133	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
134	C₃×C₄×C₅	#124	1	2	yes	—
135	C₅×Dic₃	—	4	2	no	—
136	C₂³×D₃	—	2	1	no	—
137	C₂×C₃×Q₈	—	2	2	no	—
138	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
139	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
140	C₂²×Dic₃	—	2	1	no	—
141	C₂×D₁₂	—	2	2	no	—
142	C₂×D₁₂	—	2	2	no	—
143	C₂×D₁₂	—	2	1	no	—
144	C₂×D₁₂	C₅×Q₈	1	2	yes	complemented
145	C₃×C₄⋊C₄	—	2	2	no	—
146	C₃×C₄⋊C₄	—	2	4	no	—
147	C₃×C₄⋊C₄	—	2	2	no	—
148	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
149	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
150	C₃×C₂²⋊C₄	—	2	2	no	—
151	C₃×C₄²	—	2	2	no	—
152	(C₂×D₃)⋊C₄	—	4	4	no	—
153	C₄×Dic₃	—	2	2	no	—
154	#208	—	2	2	no	—
155	#208	—	2	2	no	—
156	C₂×C₃⋊C₈	—	2	2	no	—
157	C₂³×C₅	—	6	2	no	—
158	C₂³×C₅	—	12	2	no	—
159	C₂³×C₅	—	12	2	no	—
160	C₂³×C₅	—	12	1	no	—
161	C₂³×C₅	—	12	1	no	—
162	C₂³×C₅	C₂×#67	1	1	yes	characteristic
163	C₅×Q₈	—	8	2	no	—
164	C₅×D₄	—	6	4	no	—
165	C₅×D₄	—	12	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	2	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	6	2	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	—	12	1	no	—
173	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₅	#239	1	2	yes	—
175	C₂×C₄×C₅	#239	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×#67	1	2	yes	—
177	C₅×C₈	—	12	2	no	—
178	C₂²×D₄	—	3	1	no	—
179	C₂×M₁₆	—	3	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	#125	—	2	2	no	—
182	C₂²⋊Q₈	—	2	2	no	—
183	#117	C₂×C₅×D₃	1	2	yes	—
184	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
185	C₂×C₂²⋊C₄	—	6	2	no	—
186	C₂×C₄²	—	3	1	no	—
187	#102	—	6	2	no	—
188	#102	—	6	2	no	—
189	C₂×C₃×C₅	—	2	1	no	—
190	C₂×C₃×C₅	—	2	1	no	—
191	C₂×C₃×C₅	#393	1	1	yes	characteristic
192	C₂×C₃×C₅	#453	1	1	yes	characteristic
193	C₂×C₃×C₅	#454	1	1	yes	characteristic
194	C₅×D₃	—	8	2	no	—
195	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
196	C₂²×D₃	—	2	2	no	—
197	C₂²×D₃	—	4	2	no	—
198	C₂²×D₃	—	4	2	no	—
199	C₂²×D₃	—	4	1	no	—
200	C₂²×D₃	—	4	1	no	—
201	C₃×Q₈	—	8	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	4	2	no	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
208	C₂×C₃×C₄	C₅×#41	1	2	yes	—
209	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	4	1	no	—
212	C₂×Dic₃	—	4	1	no	—
213	D₁₂	—	2	4	no	—
214	D₁₂	—	4	2	no	—
215	C₃⋊C₈	—	4	2	no	—
216	C₂²×C₅	—	2	1	no	—
217	C₂²×C₅	—	2	1	no	—
218	C₂²×C₅	—	12	4	no	—
219	C₂²×C₅	—	12	2	no	—
220	C₂²×C₅	—	24	2	no	—
221	C₂²×C₅	—	24	1	no	—
222	C₂²×C₅	—	24	1	no	—
223	C₂²×C₅	#946	1	1	yes	characteristic
224	C₂²×C₅	#947	1	1	yes	characteristic
225	C₂²×C₅	#953	1	1	yes	characteristic
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	—	12	2	no	—
230	C₄×C₅	#946	1	2	yes	—
231	C₄×C₅	#947	1	2	yes	—
232	C₂⁴	—	6	1	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×D₄	—	3	2	no	—
235	C₂×D₄	—	6	2	no	—
236	C₂×D₄	—	6	1	no	—
237	C₂×D₄	—	6	2	no	—
238	C₂²×C₄	—	6	1	no	—
239	C₂²×C₄	C₅×#67	1	2	yes	—
240	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
241	M₁₆	—	6	2	no	—
242	M₁₆	—	6	2	no	—
243	C₂×C₈	—	6	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	C₅×#67	1	2	yes	—
248	C₄⋊C₄	C₅×#67	1	2	yes	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	12	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	6	2	no	—
253	C₃×C₅	#2027	1	1	yes	characteristic, complemented, Hall
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	C₅×#120	1	1	yes	characteristic
257	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂²×C₃	C₅×#124	1	1	yes	characteristic
259	C₂×D₃	—	4	4	no	—
260	C₂×D₃	—	4	2	no	—
261	C₂×D₃	—	8	2	no	—
262	C₂×D₃	—	8	1	no	—
263	C₂×D₃	—	8	1	no	—
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	—	4	2	no	—
266	C₃×C₄	—	4	2	no	—
267	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
268	C₃×C₄	C₅×#124	1	2	yes	—
269	Dic₃	—	4	2	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	—	2	1	no	—
272	C₂×C₅	—	24	2	no	—
273	C₂×C₅	#5231	1	1	yes	characteristic
274	C₂×C₅	#5241	1	1	yes	characteristic
275	C₂×C₅	#5245	1	1	yes	characteristic
276	C₂³	—	6	2	no	—
277	C₂³	—	12	2	no	—
278	C₂³	—	12	2	no	—
279	C₂³	—	12	1	no	—
280	C₂³	—	12	1	no	—
281	C₂³	C₂×C₅×#67	1	1	yes	characteristic
282	Q₈	—	8	2	no	—
283	D₄	—	6	4	no	—
284	D₄	—	12	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	6	2	no	—
290	C₂×C₄	—	12	1	no	—
291	C₂×C₄	—	12	1	no	—
292	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
293	C₂×C₄	C₅×#239	1	2	yes	—
294	C₂×C₄	C₅×#239	1	1	yes	characteristic
295	C₂×C₄	C₂×C₅×#67	1	2	yes	—
296	C₈	—	12	2	no	—
297	C₂×C₃	—	2	1	no	—
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	C₅×#393	1	1	yes	characteristic
300	C₂×C₃	C₅×#453	1	1	yes	characteristic
301	C₂×C₃	C₅×#454	1	1	yes	characteristic
302	D₃	—	8	2	no	—
303	C₅	#71903	1	1	yes	characteristic, direct product factor, Sylow
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	12	4	no	—
307	C₂²	—	12	2	no	—
308	C₂²	—	24	2	no	—
309	C₂²	—	24	1	no	—
310	C₂²	—	24	1	no	—
311	C₂²	C₅×#946	1	1	yes	characteristic
312	C₂²	C₅×#947	1	1	yes	characteristic
313	C₂²	C₅×#953	1	1	yes	characteristic
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	12	2	no	—
318	C₄	C₅×#946	1	2	yes	—
319	C₄	C₅×#947	1	2	yes	—
320	C₃	C₅×#2027	1	1	yes	characteristic, complemented, Sylow
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	24	2	no	—
324	C₂	C₅×#5231	1	1	yes	characteristic
325	C₂	C₅×#5241	1	1	yes	characteristic
326	C₂	C₅×#5245	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#953
commutator, cosocle	C₂²×C₃×C₄	C₂³×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#544	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂²×C₅×D₁₂	C₂²
hypercenter	C₅×#544	D₃
hypocenter	C₃	C₅×#2027
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#544

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#544	C₂	semidirect
4. C₅×#4532	C₂	semidirect
5. C₅×#4530	C₂	semidirect
18. #71903	C₅	direct (N × H)
19. C₅×#544	D₃	semidirect
32. C₂×C₅×D₁₂	Q₈	semidirect
45. C₃×#544	C₂×C₅	semidirect
48. #4532	C₂×C₅	semidirect
49. #4530	C₂×C₅	semidirect
115. #544	C₅×D₃	semidirect
144. C₂×D₁₂	C₅×Q₈	semidirect
253. C₃×C₅	#2027	semidirect
303. C₅	#71903	direct (N × H)
320. C₃	C₅×#2027	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#544	2
C₅×#5001	2
C₅×#4988	2
C₅×#4532	2
C₅×#4530	2
C₅×#2027	3
#71903	5

Subgroups — C5×#71903 #14308899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71903 #14308899