Subgroups — C₅×#78561 #14417342

Order 1920.

1056 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₅×#5521	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5131	C₂	1	2	yes	maximal, complemented
5	C₅×#5106	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4711	C₂	1	4	yes	maximal, complemented
7	C₄×C₅×D₂₄	C₂	1	4	yes	maximal, complemented
8	C₅×#2948	—	3	1	no	maximal, Hall
9	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₅×#926	C₂²	1	2	yes	—
14	C₂×C₅×D₂₄	C₄	1	8	yes	complemented
15	C₂×C₅×D₂₄	C₂²	1	2	yes	—
16	C₂×C₅×D₂₄	C₂²	1	2	yes	—
17	C₅×#893	—	2	4	no	—
18	C₅×D₃×C₂²⋊C₄	—	2	4	no	—
19	C₄×C₅×D₁₂	C₂²	1	8	yes	complemented
20	C₃×C₅×#97	C₂²	1	2	yes	complemented
21	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
22	C₅×#821	C₂²	1	8	yes	complemented
23	C₅×#818	C₂²	1	2	yes	complemented
24	C₅×#817	C₂²	1	2	yes	complemented
25	#78561	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂²×C₅×D₈	—	3	1	no	—
27	C₅×#518	—	3	2	no	—
28	C₅×#442	—	3	4	no	—
29	C₄×C₅×D₈	—	3	4	no	—
30	C₅×#427	—	3	1	no	—
31	C₅×#418	—	3	2	no	—
32	C₅×#405	D₃	1	1	yes	characteristic, complemented
33	C₂³×C₅×D₃	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×D₁₂	—	2	2	no	—
36	C₂×C₅×D₁₂	—	2	4	no	—
37	C₂×C₅×D₁₂	C₂×C₄	1	8	yes	complemented
38	C₂×C₅×D₁₂	C₂³	1	4	yes	—
39	C₂×C₄×C₅×D₃	—	2	8	no	—
40	C₃×C₅×M₁₆	—	2	2	no	—
41	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
42	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
46	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
47	C₃×C₄²×C₅	C₂³	1	2	yes	—
48	C₅×#217	—	2	2	no	—
49	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
50	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
51	C₄×C₅×Dic₃	—	2	2	no	—
52	C₅×D₂₄	C₂×C₄	1	8	yes	complemented
53	C₅×D₂₄	C₂×C₄	1	8	yes	complemented
54	C₂²×D₂₄	C₂×C₅	1	1	yes	characteristic
55	#5521	C₂×C₅	1	2	yes	complemented
56	C₃×#405	C₂×C₅	1	1	yes	characteristic, complemented
57	#5131	C₂×C₅	1	2	yes	complemented
58	#5106	C₂×C₅	1	1	yes	characteristic, complemented
59	#4711	C₂×C₅	1	4	yes	complemented
60	C₄×D₂₄	C₂×C₅	1	4	yes	complemented
61	C₂²×C₅×D₄	—	3	2	no	—
62	C₂×C₅×D₈	—	3	8	no	—
63	C₂×C₅×D₈	—	3	2	no	—
64	C₂×C₅×D₈	—	3	2	no	—
65	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₄×C₅×D₄	—	3	8	no	—
68	C₄×C₅×D₄	—	6	4	no	—
69	C₅×#117	—	3	2	no	—
70	C₅×#117	C₂×D₃	1	1	yes	characteristic
71	C₂×C₅×C₂²⋊C₄	—	6	4	no	—
72	C₅×#107	—	3	2	no	—
73	C₅×#106	—	3	2	no	—
74	C₅×#102	—	3	8	no	—
75	C₅×#97	C₂×D₃	1	2	yes	complemented
76	C₄×C₅×C₈	C₂×D₃	1	2	yes	complemented
77	#2948	—	3	1	no	Sylow
78	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₅×D₃	—	2	4	no	—
80	C₂²×C₅×D₃	—	2	8	no	—
81	C₂²×C₅×D₃	—	8	4	no	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₂×C₅×Dic₃	—	2	4	no	—
88	C₅×D₁₂	—	2	8	no	—
89	C₅×D₁₂	—	2	4	no	—
90	C₅×D₁₂	C₂²×C₄	1	8	yes	—
91	C₄×C₅×D₃	—	8	8	no	—
92	C₃×C₅×C₈	—	2	2	no	—
93	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
96	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
97	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×#117	C₂²×C₅	1	1	yes	characteristic
99	#926	C₂²×C₅	1	2	yes	—
100	C₂×D₂₄	C₄×C₅	1	8	yes	complemented
101	C₂×D₂₄	C₂²×C₅	1	2	yes	—
102	C₂×D₂₄	C₂²×C₅	1	2	yes	—
103	#893	—	2	4	no	—
104	D₃×C₂²⋊C₄	—	2	4	no	—
105	C₄×D₁₂	C₂²×C₅	1	8	yes	complemented
106	C₃×#97	C₂²×C₅	1	2	yes	complemented
107	C₃×C₄×C₈	C₂²×C₅	1	2	yes	complemented
108	#821	C₂²×C₅	1	8	yes	complemented
109	#818	C₂²×C₅	1	2	yes	complemented
110	#817	C₂²×C₅	1	2	yes	complemented
111	C₂⁴×C₅	—	6	2	no	—
112	C₂×C₅×D₄	—	3	8	no	—
113	C₂×C₅×D₄	—	3	4	no	—
114	C₂×C₅×D₄	—	6	2	no	—
115	C₂×C₅×D₄	—	6	4	no	—
116	C₂²×C₄×C₅	—	6	8	no	—
117	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₅×D₈	—	3	8	no	—
119	C₅×D₈	—	3	8	no	—
120	C₅×M₁₆	—	2	2	no	—
121	C₂×C₅×C₈	C₄×D₃	1	4	yes	complemented
122	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
123	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₅×C₄⋊C₄	—	3	4	no	—
126	C₅×C₄⋊C₄	D₁₂	1	2	yes	—
127	C₅×C₂²⋊C₄	—	6	2	no	—
128	C₅×C₂²⋊C₄	—	6	8	no	—
129	C₅×C₂²⋊C₄	D₁₂	1	2	yes	—
130	C₄²×C₅	—	6	2	no	—
131	C₄²×C₅	C₂²×D₃	1	2	yes	—
132	C₂²×D₈	—	3	1	no	—
133	#518	—	3	2	no	—
134	#442	—	3	4	no	—
135	C₄×D₈	—	3	4	no	—
136	#427	—	3	1	no	—
137	#418	—	3	2	no	—
138	#405	C₅×D₃	1	1	yes	characteristic, complemented
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₅×D₃	—	2	8	no	—
143	C₂×C₅×D₃	—	4	8	no	—
144	C₂×C₅×D₃	—	8	8	no	—
145	C₃×C₄×C₅	—	2	4	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₅×Dic₃	—	4	4	no	—
150	C₂³×D₃	—	2	2	no	—
151	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂×D₁₂	—	2	2	no	—
153	C₂×D₁₂	—	2	4	no	—
154	C₂×D₁₂	C₂×C₄×C₅	1	8	yes	complemented
155	C₂×D₁₂	C₂³×C₅	1	4	yes	—
156	C₂×C₄×D₃	—	2	8	no	—
157	C₃×M₁₆	—	2	2	no	—
158	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	complemented
159	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
160	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
162	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
163	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
164	C₃×C₄²	C₂³×C₅	1	2	yes	—
165	#217	—	2	2	no	—
166	(C₂×D₃)⋊C₄	—	2	8	no	—
167	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
168	C₄×Dic₃	—	2	2	no	—
169	D₂₄	C₂×C₄×C₅	1	8	yes	complemented
170	D₂₄	C₂×C₄×C₅	1	8	yes	complemented
171	C₂³×C₅	—	6	4	no	—
172	C₂³×C₅	—	6	8	no	—
173	C₂³×C₅	—	24	4	no	—
174	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
175	C₅×D₄	—	3	8	no	—
176	C₅×D₄	—	6	8	no	—
177	C₅×D₄	—	6	4	no	—
178	C₂×C₄×C₅	—	6	4	no	—
179	C₂×C₄×C₅	—	24	8	no	—
180	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
181	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
182	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
183	C₂×C₄×C₅	#235	1	2	yes	—
184	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
185	C₅×C₈	—	2	2	no	—
186	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
187	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
188	C₂²×D₄	—	3	2	no	—
189	C₂×D₈	—	3	8	no	—
190	C₂×D₈	—	3	2	no	—
191	C₂×D₈	—	3	2	no	—
192	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
193	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
194	C₄×D₄	—	3	8	no	—
195	C₄×D₄	—	6	4	no	—
196	#117	—	3	2	no	—
197	#117	C₂×C₅×D₃	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	6	4	no	—
199	#107	—	3	2	no	—
200	#106	—	3	2	no	—
201	#102	—	3	8	no	—
202	#97	C₂×C₅×D₃	1	2	yes	complemented
203	C₄×C₈	C₂×C₅×D₃	1	2	yes	complemented
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂×C₃×C₅	#576	1	2	yes	—
207	C₅×D₃	—	4	8	no	—
208	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
209	C₂²×D₃	—	2	4	no	—
210	C₂²×D₃	—	2	8	no	—
211	C₂²×D₃	—	8	4	no	—
212	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
213	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×#41	1	2	yes	—
216	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
217	C₂×Dic₃	—	2	4	no	—
218	D₁₂	—	2	8	no	—
219	D₁₂	—	2	4	no	—
220	D₁₂	C₂²×C₄×C₅	1	8	yes	—
221	C₄×D₃	—	8	8	no	—
222	C₃×C₈	—	2	2	no	—
223	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
224	C₃×C₈	C₂²×C₄×C₅	1	2	yes	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	—	6	8	no	—
227	C₂²×C₅	—	12	8	no	—
228	C₂²×C₅	—	24	8	no	—
229	C₂²×C₅	C₄×D₁₂	1	2	yes	—
230	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
231	C₄×C₅	—	2	4	no	—
232	C₄×C₅	—	12	4	no	—
233	C₄×C₅	C₄×D₁₂	1	2	yes	—
234	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
235	C₄×C₅	C₂×#235	1	1	yes	characteristic
236	C₂⁴	—	6	2	no	—
237	C₂×D₄	—	3	8	no	—
238	C₂×D₄	—	3	4	no	—
239	C₂×D₄	—	6	2	no	—
240	C₂×D₄	—	6	4	no	—
241	C₂²×C₄	—	6	8	no	—
242	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
243	D₈	—	3	8	no	—
244	D₈	—	3	8	no	—
245	M₁₆	—	2	2	no	—
246	C₂×C₈	C₄×C₅×D₃	1	4	yes	complemented
247	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
248	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
250	C₄⋊C₄	—	3	4	no	—
251	C₄⋊C₄	C₅×D₁₂	1	2	yes	—
252	C₂²⋊C₄	—	6	2	no	—
253	C₂²⋊C₄	—	6	8	no	—
254	C₂²⋊C₄	C₅×D₁₂	1	2	yes	—
255	C₄²	—	6	2	no	—
256	C₄²	C₂²×C₅×D₃	1	2	yes	—
257	C₃×C₅	#2948	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
260	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
261	C₂×D₃	—	2	8	no	—
262	C₂×D₃	—	4	8	no	—
263	C₂×D₃	—	8	8	no	—
264	C₃×C₄	—	2	4	no	—
265	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
266	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
267	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
268	Dic₃	—	4	4	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	12	8	no	—
271	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
272	C₂×C₅	#5752	1	2	yes	—
273	C₂³	—	6	4	no	—
274	C₂³	—	6	8	no	—
275	C₂³	—	24	4	no	—
276	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
277	D₄	—	3	8	no	—
278	D₄	—	6	8	no	—
279	D₄	—	6	4	no	—
280	C₂×C₄	—	6	4	no	—
281	C₂×C₄	—	24	8	no	—
282	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
283	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
284	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
285	C₂×C₄	C₅×#235	1	2	yes	—
286	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
287	C₈	—	2	2	no	—
288	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
289	C₈	C₂×C₄×C₅×D₃	1	2	yes	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
292	C₂×C₃	C₅×#576	1	2	yes	—
293	D₃	—	4	8	no	—
294	C₅	#78561	1	1	yes	characteristic, direct product factor, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	6	8	no	—
297	C₂²	—	12	8	no	—
298	C₂²	—	24	8	no	—
299	C₂²	C₄×C₅×D₁₂	1	2	yes	—
300	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	12	4	no	—
303	C₄	C₄×C₅×D₁₂	1	2	yes	—
304	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
305	C₄	C₂×C₅×#235	1	1	yes	characteristic
306	C₃	C₅×#2948	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	12	8	no	—
309	C₂	C₂×C₄×C₅×D₁₂	1	1	yes	characteristic
310	C₂	C₅×#5752	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂²×D₁₂
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#405	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₅×D₂₄	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#405	D₃
hypocenter	C₃	C₅×#2948
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405

Complements

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

N	H ≅ G/N	type
2. C₅×#5521	C₂	semidirect
3. C₃×C₅×#405	C₂	semidirect
4. C₅×#5131	C₂	semidirect
5. C₅×#5106	C₂	semidirect
6. C₅×#4711	C₂	semidirect
7. C₄×C₅×D₂₄	C₂	semidirect
14. C₂×C₅×D₂₄	C₄	semidirect
19. C₄×C₅×D₁₂	C₂²	semidirect
20. C₃×C₅×#97	C₂²	semidirect
21. C₃×C₄×C₅×C₈	C₂²	semidirect
22. C₅×#821	C₂²	semidirect
23. C₅×#818	C₂²	semidirect
24. C₅×#817	C₂²	semidirect
25. #78561	C₅	direct (N × H)
32. C₅×#405	D₃	semidirect
37. C₂×C₅×D₁₂	C₂×C₄	semidirect
41. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
52. C₅×D₂₄	C₂×C₄	semidirect
53. C₅×D₂₄	C₂×C₄	semidirect
55. #5521	C₂×C₅	semidirect
56. C₃×#405	C₂×C₅	semidirect
57. #5131	C₂×C₅	semidirect
58. #5106	C₂×C₅	semidirect
59. #4711	C₂×C₅	semidirect
60. C₄×D₂₄	C₂×C₅	semidirect
75. C₅×#97	C₂×D₃	semidirect
76. C₄×C₅×C₈	C₂×D₃	semidirect
100. C₂×D₂₄	C₄×C₅	semidirect
105. C₄×D₁₂	C₂²×C₅	semidirect
106. C₃×#97	C₂²×C₅	semidirect
107. C₃×C₄×C₈	C₂²×C₅	semidirect
108. #821	C₂²×C₅	semidirect
109. #818	C₂²×C₅	semidirect
110. #817	C₂²×C₅	semidirect
121. C₂×C₅×C₈	C₄×D₃	semidirect
138. #405	C₅×D₃	semidirect
154. C₂×D₁₂	C₂×C₄×C₅	semidirect
158. C₂×C₃×C₈	C₂×C₄×C₅	semidirect
169. D₂₄	C₂×C₄×C₅	semidirect
170. D₂₄	C₂×C₄×C₅	semidirect
202. #97	C₂×C₅×D₃	semidirect
203. C₄×C₈	C₂×C₅×D₃	semidirect
246. C₂×C₈	C₄×C₅×D₃	semidirect
257. C₃×C₅	#2948	semidirect
294. C₅	#78561	direct (N × H)
306. C₃	C₅×#2948	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×D₂₄	2
C₅×#5521	2
C₃×C₅×#405	2
C₅×#5131	2
C₅×#5106	2
C₅×#4711	2
C₄×C₅×D₂₄	2
C₅×#2948	3
#78561	5

Subgroups — C5×#78561 #14417342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78561 #14417342