Subgroups — C₅×#83172 #14324380

Order 1920.

1480 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×D₄×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₅×#5221	C₂	1	2	yes	maximal, complemented
4	C₅×#5220	C₂	1	8	yes	maximal, complemented
5	C₅×#4936	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4932	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2328	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
10	C₂³×C₅×Dic₃	C₂²	1	4	yes	—
11	C₃×C₅×#127	C₄	1	8	yes	complemented
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₅×#217	C₂²	1	4	yes	—
14	C₂×C₅×#217	C₂²	1	8	yes	—
15	C₅×D₄×Dic₃	—	2	8	no	—
16	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
19	C₅×C₃⋊#95	C₂²	1	4	yes	—
20	C₅×C₃⋊#95	C₂²	1	4	yes	—
21	#83172	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#127	D₃	1	1	yes	characteristic
23	C₂×C₄×C₅×D₄	—	3	2	no	—
24	C₅×#390	—	3	2	no	—
25	C₅×#386	—	3	8	no	—
26	C₅×#382	—	3	1	no	—
27	C₅×#376	—	3	1	no	—
28	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	2	4	no	—
31	C₂×C₃×C₅×D₄	—	4	8	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	complemented
33	C₂×C₃×C₅×D₄	C₂×C₄	1	16	yes	complemented
34	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×C₅×Dic₃	—	2	8	no	—
37	C₂²×C₅×Dic₃	—	4	8	no	—
38	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
39	C₂²×C₅×Dic₃	C₂³	1	4	yes	—
40	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
41	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
42	C₅×#217	—	2	8	no	—
43	C₅×#217	—	4	8	no	—
44	C₅×C₄⋊Dic₃	—	2	4	no	—
45	C₄×C₅×Dic₃	—	2	2	no	—
46	C₄×C₅×Dic₃	D₄	1	4	yes	complemented
47	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
48	C₂×D₄×Dic₃	C₂×C₅	1	2	yes	complemented
49	#5221	C₂×C₅	1	2	yes	complemented
50	#5220	C₂×C₅	1	8	yes	complemented
51	#4936	C₂×C₅	1	1	yes	characteristic, complemented
52	#4932	C₂×C₅	1	1	yes	characteristic, complemented
53	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
54	C₂²×C₅×D₄	C₂×D₃	1	4	yes	—
55	C₂³×C₄×C₅	—	3	4	no	—
56	C₅×#127	Dic₃	1	8	yes	complemented
57	C₄×C₅×D₄	—	6	8	no	—
58	C₂×C₅×C₄⋊C₄	—	3	2	no	—
59	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
60	C₂×C₅×C₂²⋊C₄	—	3	8	no	—
61	C₂×C₄²×C₅	—	3	1	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
64	C₅×#95	—	3	4	no	—
65	C₅×#95	—	3	4	no	—
66	#2328	—	3	1	no	Sylow
67	C₂³×C₃×C₅	—	2	4	no	—
68	C₂³×C₃×C₅	—	2	4	no	—
69	C₂³×C₃×C₅	—	2	8	no	—
70	C₂³×C₃×C₅	—	2	8	no	—
71	C₂³×C₃×C₅	C₂²×C₄	1	8	yes	—
72	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
73	C₃×C₅×D₄	—	2	16	no	—
74	C₃×C₅×D₄	—	4	16	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	#41	1	2	yes	—
80	C₂×C₅×Dic₃	—	2	4	no	—
81	C₂×C₅×Dic₃	—	2	4	no	—
82	C₂×C₅×Dic₃	—	2	2	no	—
83	C₂×C₅×Dic₃	—	2	4	no	—
84	C₂×C₅×Dic₃	—	2	8	no	—
85	C₂×C₅×Dic₃	—	4	16	no	—
86	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
87	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
88	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
89	C₂³×Dic₃	C₂²×C₅	1	4	yes	—
90	C₃×#127	C₄×C₅	1	8	yes	complemented
91	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
92	C₂×#217	C₂²×C₅	1	4	yes	—
93	C₂×#217	C₂²×C₅	1	8	yes	—
94	D₄×Dic₃	—	2	8	no	—
95	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
96	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
97	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
98	C₃⋊#95	C₂²×C₅	1	4	yes	—
99	C₃⋊#95	C₂²×C₅	1	4	yes	—
100	C₂⁴×C₅	C₂²×D₃	1	4	yes	—
101	C₂×C₅×D₄	—	2	4	no	—
102	C₂×C₅×D₄	—	2	4	no	—
103	C₂×C₅×D₄	—	4	8	no	—
104	C₂×C₅×D₄	C₂×Dic₃	1	8	yes	complemented
105	C₂×C₅×D₄	C₂×Dic₃	1	16	yes	complemented
106	C₂²×C₄×C₅	—	3	2	no	—
107	C₂²×C₄×C₅	—	3	2	no	—
108	C₂²×C₄×C₅	—	3	4	no	—
109	C₂²×C₄×C₅	—	6	8	no	—
110	C₂²×C₄×C₅	—	12	8	no	—
111	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
112	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	6	4	no	—
114	C₅×C₂²⋊C₄	—	6	8	no	—
115	C₅×C₂²⋊C₄	—	12	8	no	—
116	C₄²×C₅	—	3	4	no	—
117	C₄²×C₅	—	6	2	no	—
118	C₄²×C₅	C₂×Dic₃	1	4	yes	complemented
119	C₂×#127	C₅×D₃	1	1	yes	characteristic
120	C₂×C₄×D₄	—	3	2	no	—
121	#390	—	3	2	no	—
122	#386	—	3	8	no	—
123	#382	—	3	1	no	—
124	#376	—	3	1	no	—
125	C₂²×C₃×C₅	—	2	8	no	—
126	C₂²×C₃×C₅	—	2	16	no	—
127	C₂²×C₃×C₅	—	4	4	no	—
128	C₂²×C₃×C₅	—	4	4	no	—
129	C₂²×C₃×C₅	—	4	8	no	—
130	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
133	C₂²×C₃×C₅	H(5,𝔽₂)	1	4	yes	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
136	C₅×Dic₃	—	2	4	no	—
137	C₅×Dic₃	—	4	4	no	—
138	C₅×Dic₃	—	4	2	no	—
139	C₂⁴×C₃	C₂³×C₅	1	4	yes	—
140	C₂×C₃×D₄	—	2	4	no	—
141	C₂×C₃×D₄	—	2	4	no	—
142	C₂×C₃×D₄	—	4	8	no	—
143	C₂×C₃×D₄	C₂×C₄×C₅	1	8	yes	complemented
144	C₂×C₃×D₄	C₂×C₄×C₅	1	16	yes	complemented
145	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
147	C₂²×Dic₃	—	2	8	no	—
148	C₂²×Dic₃	—	4	8	no	—
149	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
150	C₂²×Dic₃	C₂³×C₅	1	4	yes	—
151	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
152	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
153	#217	—	2	8	no	—
154	#217	—	4	8	no	—
155	C₄⋊Dic₃	—	2	4	no	—
156	C₄×Dic₃	—	2	2	no	—
157	C₄×Dic₃	C₅×D₄	1	4	yes	complemented
158	C₂³×C₅	—	2	4	no	—
159	C₂³×C₅	—	2	4	no	—
160	C₂³×C₅	—	2	8	no	—
161	C₂³×C₅	—	2	8	no	—
162	C₂³×C₅	C₂²×Dic₃	1	8	yes	—
163	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
164	C₅×D₄	—	2	16	no	—
165	C₅×D₄	—	4	16	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	3	4	no	—
168	C₂×C₄×C₅	—	6	4	no	—
169	C₂×C₄×C₅	—	6	4	no	—
170	C₂×C₄×C₅	—	6	2	no	—
171	C₂×C₄×C₅	—	6	4	no	—
172	C₂×C₄×C₅	—	6	8	no	—
173	C₂×C₄×C₅	—	12	16	no	—
174	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
175	C₂×C₄×C₅	#237	1	2	yes	—
176	C₂×C₄×C₅	C₂²×Dic₃	1	4	yes	—
177	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
178	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
179	C₂²×D₄	C₂×C₅×D₃	1	4	yes	—
180	C₂³×C₄	—	3	4	no	—
181	#127	C₅×Dic₃	1	8	yes	complemented
182	C₄×D₄	—	6	8	no	—
183	C₂×C₄⋊C₄	—	3	2	no	—
184	C₂×C₂²⋊C₄	—	3	4	no	—
185	C₂×C₂²⋊C₄	—	3	8	no	—
186	C₂×C₄²	—	3	1	no	—
187	C₂×C₄²	—	3	1	no	—
188	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
189	#95	—	3	4	no	—
190	#95	—	3	4	no	—
191	C₂×C₃×C₅	—	4	8	no	—
192	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
193	C₂×C₃×C₅	#518	1	2	yes	—
194	C₂×C₃×C₅	#535	1	2	yes	—
195	C₂×C₃×C₅	#540	1	2	yes	—
196	C₂³×C₃	—	2	4	no	—
197	C₂³×C₃	—	2	4	no	—
198	C₂³×C₃	—	2	8	no	—
199	C₂³×C₃	—	2	8	no	—
200	C₂³×C₃	C₂²×C₄×C₅	1	8	yes	—
201	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
202	C₃×D₄	—	2	16	no	—
203	C₃×D₄	—	4	16	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
206	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
208	C₂×C₃×C₄	C₅×#41	1	2	yes	—
209	C₂×Dic₃	—	2	4	no	—
210	C₂×Dic₃	—	2	4	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	—	2	4	no	—
213	C₂×Dic₃	—	2	8	no	—
214	C₂×Dic₃	—	4	16	no	—
215	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
216	C₂²×C₅	—	2	8	no	—
217	C₂²×C₅	—	2	16	no	—
218	C₂²×C₅	—	4	4	no	—
219	C₂²×C₅	—	4	4	no	—
220	C₂²×C₅	—	4	8	no	—
221	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
222	C₂²×C₅	C₂×#237	1	1	yes	characteristic
223	C₂²×C₅	#1004	1	4	yes	—
224	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
225	C₄×C₅	—	2	4	no	—
226	C₄×C₅	—	6	4	no	—
227	C₄×C₅	—	12	4	no	—
228	C₄×C₅	—	12	2	no	—
229	C₄×C₅	D₄×Dic₃	1	4	yes	—
230	C₂⁴	C₂²×C₅×D₃	1	4	yes	—
231	C₂×D₄	—	2	4	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	8	no	—
234	C₂×D₄	C₂×C₅×Dic₃	1	8	yes	complemented
235	C₂×D₄	C₂×C₅×Dic₃	1	16	yes	complemented
236	C₂²×C₄	—	3	2	no	—
237	C₂²×C₄	—	3	2	no	—
238	C₂²×C₄	—	3	4	no	—
239	C₂²×C₄	—	6	8	no	—
240	C₂²×C₄	—	12	8	no	—
241	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
242	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
243	C₄⋊C₄	—	6	4	no	—
244	C₂²⋊C₄	—	6	8	no	—
245	C₂²⋊C₄	—	12	8	no	—
246	C₄²	—	3	4	no	—
247	C₄²	—	6	2	no	—
248	C₄²	C₂×C₅×Dic₃	1	4	yes	complemented
249	C₃×C₅	#2328	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	8	no	—
251	C₂²×C₃	—	2	16	no	—
252	C₂²×C₃	—	4	4	no	—
253	C₂²×C₃	—	4	4	no	—
254	C₂²×C₃	—	4	8	no	—
255	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
256	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
257	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
258	C₂²×C₃	C₅×H(5,𝔽₂)	1	4	yes	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
261	Dic₃	—	2	4	no	—
262	Dic₃	—	4	4	no	—
263	Dic₃	—	4	2	no	—
264	C₂×C₅	—	4	8	no	—
265	C₂×C₅	#5715	1	2	yes	—
266	C₂×C₅	#5719	1	2	yes	—
267	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
268	C₂×C₅	#5798	1	2	yes	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	4	no	—
271	C₂³	—	2	8	no	—
272	C₂³	—	2	8	no	—
273	C₂³	C₂²×C₅×Dic₃	1	8	yes	—
274	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
275	D₄	—	2	16	no	—
276	D₄	—	4	16	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	3	4	no	—
279	C₂×C₄	—	6	4	no	—
280	C₂×C₄	—	6	4	no	—
281	C₂×C₄	—	6	2	no	—
282	C₂×C₄	—	6	4	no	—
283	C₂×C₄	—	6	8	no	—
284	C₂×C₄	—	12	16	no	—
285	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
286	C₂×C₄	C₅×#237	1	2	yes	—
287	C₂×C₄	C₂²×C₅×Dic₃	1	4	yes	—
288	C₂×C₄	C₂²×C₅×Dic₃	1	2	yes	—
289	C₂×C₃	—	4	8	no	—
290	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
291	C₂×C₃	C₅×#518	1	2	yes	—
292	C₂×C₃	C₅×#535	1	2	yes	—
293	C₂×C₃	C₅×#540	1	2	yes	—
294	C₅	#83172	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	8	no	—
296	C₂²	—	2	16	no	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	8	no	—
300	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
301	C₂²	C₂×C₅×#237	1	1	yes	characteristic
302	C₂²	C₅×#1004	1	4	yes	—
303	C₂²	C₂³×C₅×Dic₃	1	1	yes	characteristic
304	C₄	—	2	4	no	—
305	C₄	—	6	4	no	—
306	C₄	—	12	4	no	—
307	C₄	—	12	2	no	—
308	C₄	C₅×D₄×Dic₃	1	4	yes	—
309	C₃	C₅×#2328	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	4	8	no	—
311	C₂	C₅×#5715	1	2	yes	—
312	C₂	C₅×#5719	1	2	yes	—
313	C₂	C₂×C₅×D₄×Dic₃	1	1	yes	characteristic
314	C₂	C₅×#5798	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting, prime radical	C₂×C₃×C₅×#127	C₂
socle	C₂³×C₃×C₅	C₂⁴
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₂×C₅×#127	D₃
hypocenter	C₃	C₅×#2328
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#127

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×D₄×Dic₃	C₂	semidirect
3. C₅×#5221	C₂	semidirect
4. C₅×#5220	C₂	semidirect
5. C₅×#4936	C₂	semidirect
6. C₅×#4932	C₂	semidirect
11. C₃×C₅×#127	C₄	semidirect
21. #83172	C₅	direct (N × H)
32. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
33. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
41. C₃×C₄²×C₅	C₂×C₄	semidirect
46. C₄×C₅×Dic₃	D₄	semidirect
48. C₂×D₄×Dic₃	C₂×C₅	semidirect
49. #5221	C₂×C₅	semidirect
50. #5220	C₂×C₅	semidirect
51. #4936	C₂×C₅	semidirect
52. #4932	C₂×C₅	semidirect
56. C₅×#127	Dic₃	semidirect
90. C₃×#127	C₄×C₅	semidirect
104. C₂×C₅×D₄	C₂×Dic₃	semidirect
105. C₂×C₅×D₄	C₂×Dic₃	semidirect
118. C₄²×C₅	C₂×Dic₃	semidirect
143. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
144. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
152. C₃×C₄²	C₂×C₄×C₅	semidirect
157. C₄×Dic₃	C₅×D₄	semidirect
181. #127	C₅×Dic₃	semidirect
234. C₂×D₄	C₂×C₅×Dic₃	semidirect
235. C₂×D₄	C₂×C₅×Dic₃	semidirect
248. C₄²	C₂×C₅×Dic₃	semidirect
249. C₃×C₅	#2328	semidirect
294. C₅	#83172	direct (N × H)
309. C₃	C₅×#2328	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#127	2
C₂×C₅×D₄×Dic₃	2
C₅×#5221	2
C₅×#5220	2
C₅×#4936	2
C₅×#4932	2
C₅×#2328	3
#83172	5

Subgroups — C5×#83172 #14324380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83172 #14324380