Subgroups — C₅×#71553 #14298429

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#5212	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#830	C₂	1	1	yes	maximal, characteristic
4	C₅×#4529	C₂	1	4	yes	maximal, complemented
5	C₅×#1823	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	—	2	1	no	—
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₅×#217	—	2	1	no	—
13	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
15	C₅×C₃⋊#95	—	2	1	no	—
16	C₅×#830	—	2	2	no	—
17	#71553	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic
19	C₂×C₅×C₂²⋊C₈	—	3	1	no	—
20	C₅×#380	—	3	1	no	—
21	C₅×#340	—	3	4	no	—
22	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	—	2	4	no	—
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	4	2	no	—
27	C₂²×C₃×C₄×C₅	—	4	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×C₅×Dic₃	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	4	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₅×#217	—	4	2	no	—
37	C₅×C₄⋊Dic₃	—	2	4	no	—
38	C₂×C₅×C₃⋊C₈	—	2	2	no	—
39	C₂×C₅×C₃⋊C₈	—	2	4	no	—
40	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
41	#5212	C₂×C₅	1	1	yes	characteristic
42	C₂×#830	C₂×C₅	1	1	yes	characteristic
43	#4529	C₂×C₅	1	4	yes	complemented
44	C₂³×C₄×C₅	—	2	1	no	—
45	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂²×C₅×C₈	—	3	2	no	—
47	C₂×C₅×C₄⋊C₄	—	2	1	no	—
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	—	3	2	no	—
50	C₂×C₅×C₄⋊C₄	Dic₃	1	4	yes	complemented
51	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
53	C₅×C₂²⋊C₈	—	6	2	no	—
54	C₅×#95	—	6	1	no	—
55	#1823	—	3	1	no	Sylow
56	C₂³×C₃×C₅	—	2	2	no	—
57	C₂³×C₃×C₅	—	2	2	no	—
58	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
59	C₂³×C₃×C₅	D₈	1	2	yes	—
60	C₂³×C₃×C₅	SD₁₆	1	2	yes	—
61	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	—	4	8	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	—	4	2	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	4	yes	—
73	C₂×C₅×Dic₃	—	4	2	no	—
74	C₂×C₅×Dic₃	—	4	2	no	—
75	C₂×C₅×Dic₃	—	4	2	no	—
76	C₅×C₃⋊C₈	—	4	2	no	—
77	C₂³×C₃×C₄	—	2	1	no	—
78	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	—	2	1	no	—
80	C₂×C₃×C₄⋊C₄	—	2	2	no	—
81	C₂×C₃×C₄⋊C₄	C₄×C₅	1	4	yes	complemented
82	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
83	C₂×#217	—	2	1	no	—
84	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
85	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
86	C₃⋊#95	—	2	1	no	—
87	#830	—	2	2	no	—
88	C₂⁴×C₅	#67	1	1	yes	characteristic
89	C₂²×C₄×C₅	—	2	4	no	—
90	C₂²×C₄×C₅	—	2	4	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂²×C₄×C₅	—	4	2	no	—
94	C₂²×C₄×C₅	—	6	2	no	—
95	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
96	C₂²×C₄×C₅	#67	1	1	yes	characteristic
97	C₂²×C₄×C₅	#67	1	2	yes	—
98	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
99	C₂×C₅×C₈	—	6	4	no	—
100	C₂×C₅×C₈	—	6	2	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	6	4	no	—
104	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
105	C₅×C₂²⋊C₄	—	12	2	no	—
106	C₂²×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic
107	C₂×C₂²⋊C₈	—	3	1	no	—
108	#380	—	3	1	no	—
109	#340	—	3	4	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	—	2	4	no	—
113	C₂²×C₃×C₅	—	2	4	no	—
114	C₂²×C₃×C₅	#102	1	4	yes	—
115	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	#123	1	2	yes	—
117	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×C₃×C₅	#137	1	2	yes	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	4	4	no	—
122	C₃×C₄×C₅	#117	1	2	yes	—
123	C₃×C₄×C₅	#123	1	2	yes	—
124	C₅×Dic₃	—	8	2	no	—
125	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
126	C₂²×C₃×C₄	—	2	4	no	—
127	C₂²×C₃×C₄	—	2	4	no	—
128	C₂²×C₃×C₄	—	2	2	no	—
129	C₂²×C₃×C₄	—	4	2	no	—
130	C₂²×C₃×C₄	—	4	2	no	—
131	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
132	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
134	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₂²×Dic₃	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	4	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
139	#217	—	4	2	no	—
140	C₄⋊Dic₃	—	2	4	no	—
141	C₂×C₃⋊C₈	—	2	2	no	—
142	C₂×C₃⋊C₈	—	2	4	no	—
143	C₂³×C₅	—	2	2	no	—
144	C₂³×C₅	—	2	2	no	—
145	C₂³×C₅	#213	1	2	yes	—
146	C₂³×C₅	#214	1	2	yes	—
147	C₂³×C₅	#217	1	2	yes	—
148	C₂³×C₅	C₂×#67	1	1	yes	characteristic
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	2	4	no	—
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	4	8	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	12	2	no	—
157	C₂×C₄×C₅	—	12	2	no	—
158	C₂×C₄×C₅	—	12	2	no	—
159	C₂×C₄×C₅	#217	1	2	yes	—
160	C₂×C₄×C₅	#235	1	4	yes	—
161	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
163	C₅×C₈	—	12	2	no	—
164	C₂³×C₄	—	2	1	no	—
165	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
166	C₂²×C₈	—	3	2	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	3	2	no	—
170	C₂×C₄⋊C₄	C₅×Dic₃	1	4	yes	complemented
171	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
172	C₂×C₂²⋊C₄	—	6	1	no	—
173	C₂²⋊C₈	—	6	2	no	—
174	#95	—	6	1	no	—
175	C₂×C₃×C₅	—	2	4	no	—
176	C₂×C₃×C₅	#381	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
178	C₂×C₃×C₅	#419	1	1	yes	characteristic
179	C₂×C₃×C₅	#480	1	2	yes	—
180	C₂×C₃×C₅	#483	1	2	yes	—
181	C₂³×C₃	—	2	2	no	—
182	C₂³×C₃	—	2	2	no	—
183	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
184	C₂³×C₃	C₅×D₈	1	2	yes	—
185	C₂³×C₃	C₅×SD₁₆	1	2	yes	—
186	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	8	no	—
194	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
195	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
196	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂×C₃×C₄	C₅×#41	1	4	yes	—
198	C₂×Dic₃	—	4	2	no	—
199	C₂×Dic₃	—	4	2	no	—
200	C₂×Dic₃	—	4	2	no	—
201	C₃⋊C₈	—	4	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	4	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	4	no	—
206	C₂²×C₅	#832	1	4	yes	—
207	C₂²×C₅	#929	1	2	yes	—
208	C₂²×C₅	C₂×#213	1	1	yes	characteristic
209	C₂²×C₅	C₂×#214	1	1	yes	characteristic
210	C₂²×C₅	#942	1	2	yes	—
211	C₂²×C₅	C₂×#217	1	1	yes	characteristic
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	4	4	no	—
214	C₄×C₅	—	24	2	no	—
215	C₄×C₅	#926	1	2	yes	—
216	C₄×C₅	#929	1	2	yes	—
217	C₂⁴	C₅×#67	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	4	2	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	—	6	2	no	—
224	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
225	C₂²×C₄	C₅×#67	1	1	yes	characteristic
226	C₂²×C₄	C₅×#67	1	2	yes	—
227	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
228	C₂×C₈	—	6	4	no	—
229	C₂×C₈	—	6	2	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	6	4	no	—
233	C₄⋊C₄	C₂×C₅×Dic₃	1	4	yes	—
234	C₂²⋊C₄	—	12	2	no	—
235	C₃×C₅	#1823	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	2	4	no	—
239	C₂²×C₃	—	2	4	no	—
240	C₂²×C₃	C₅×#102	1	4	yes	—
241	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
242	C₂²×C₃	C₅×#123	1	2	yes	—
243	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
244	C₂²×C₃	C₂×C₅×SD₁₆	1	1	yes	characteristic
245	C₂²×C₃	C₅×#137	1	2	yes	—
246	C₃×C₄	—	2	2	no	—
247	C₃×C₄	—	4	4	no	—
248	C₃×C₄	C₅×#117	1	2	yes	—
249	C₃×C₄	C₅×#123	1	2	yes	—
250	Dic₃	—	8	2	no	—
251	C₂×C₅	—	2	4	no	—
252	C₂×C₅	#4967	1	2	yes	—
253	C₂×C₅	#4971	1	2	yes	—
254	C₂×C₅	#5211	1	1	yes	characteristic
255	C₂×C₅	C₂×#832	1	1	yes	characteristic
256	C₂×C₅	#5225	1	1	yes	characteristic
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	C₅×#213	1	2	yes	—
260	C₂³	C₅×#214	1	2	yes	—
261	C₂³	C₅×#217	1	2	yes	—
262	C₂³	C₂×C₅×#67	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	—	12	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	—	12	2	no	—
273	C₂×C₄	C₅×#217	1	2	yes	—
274	C₂×C₄	C₅×#235	1	4	yes	—
275	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
277	C₈	—	12	2	no	—
278	C₂×C₃	—	2	4	no	—
279	C₂×C₃	C₅×#381	1	1	yes	characteristic
280	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
281	C₂×C₃	C₅×#419	1	1	yes	characteristic
282	C₂×C₃	C₅×#480	1	2	yes	—
283	C₂×C₃	C₅×#483	1	2	yes	—
284	C₅	#71553	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	4	no	—
289	C₂²	C₅×#832	1	4	yes	—
290	C₂²	C₅×#929	1	2	yes	—
291	C₂²	C₂×C₅×#213	1	1	yes	characteristic
292	C₂²	C₂×C₅×#214	1	1	yes	characteristic
293	C₂²	C₅×#942	1	2	yes	—
294	C₂²	C₂×C₅×#217	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	4	4	no	—
297	C₄	—	24	2	no	—
298	C₄	C₅×#926	1	2	yes	—
299	C₄	C₅×#929	1	2	yes	—
300	C₃	C₅×#1823	1	1	yes	characteristic, complemented, Sylow
301	C₂	—	2	4	no	—
302	C₂	C₅×#4967	1	2	yes	—
303	C₂	C₅×#4971	1	2	yes	—
304	C₂	C₅×#5211	1	1	yes	characteristic
305	C₂	C₂×C₅×#832	1	1	yes	characteristic
306	C₂	C₅×#5225	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂²×C₃×C₅×C₄⋊C₄	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂⁴×C₃×C₅	D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂²×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#1823
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂²×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₅×#4529	C₂	semidirect
10. C₂×C₃×C₅×C₄⋊C₄	C₄	semidirect
17. #71553	C₅	direct (N × H)
43. #4529	C₂×C₅	semidirect
50. C₂×C₅×C₄⋊C₄	Dic₃	semidirect
81. C₂×C₃×C₄⋊C₄	C₄×C₅	semidirect
170. C₂×C₄⋊C₄	C₅×Dic₃	semidirect
235. C₃×C₅	#1823	semidirect
284. C₅	#71553	direct (N × H)
300. C₃	C₅×#1823	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×C₄⋊C₄	2
C₅×#5212	2
C₂×C₅×#830	2
C₅×#4529	2
C₅×#1823	3
#71553	5

Subgroups — C5×#71553 #14298429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71553 #14298429