Subgroups — C₅×#70761 #14281981

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4483	C₂	1	4	yes	maximal, complemented
5	C₅×#1484	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#930	—	2	1	no	—
9	C₅×#930	C₄	1	2	yes	—
10	C₂×C₅×#208	—	2	1	no	—
11	C₅×#875	—	2	1	no	—
12	C₅×#874	C₄	1	1	yes	characteristic
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₄×C₅×D₁₂	C₂²	1	2	yes	—
15	C₅×#869	C₄	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₄×C₅×C₈	—	2	1	no	—
19	C₅×#803	—	2	1	no	—
20	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
21	#70761	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₅×#532	—	3	1	no	—
23	C₅×#431	D₃	1	1	yes	characteristic, complemented
24	C₄×C₅×M₁₆	—	3	1	no	—
25	C₅×D₄⋊C₈	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×#67	—	2	2	no	—
30	C₅×#235	—	4	4	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
45	C₅×#208	—	2	4	no	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	#5477	C₂×C₅	1	1	yes	characteristic
48	C₃×#431	C₂×C₅	1	1	yes	characteristic, complemented
49	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
50	#4483	C₂×C₅	1	4	yes	complemented
51	C₂×C₅×#41	—	6	1	no	—
52	C₂×C₅×M₁₆	—	6	1	no	—
53	C₅×C₄⋊Q₈	—	3	1	no	—
54	C₅×#127	—	3	1	no	—
55	C₅×#124	—	6	1	no	—
56	C₅×#121	—	3	2	no	—
57	C₅×#121	—	6	1	no	—
58	C₄×C₅×D₄	—	3	2	no	—
59	C₄×C₅×D₄	—	6	1	no	—
60	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
61	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
62	C₅×C₂²⋊C₈	—	2	1	no	—
63	C₅×#97	—	6	1	no	—
64	C₄×C₅×C₈	—	2	1	no	—
65	C₄×C₅×C₈	—	3	2	no	—
66	#1484	—	3	1	no	Sylow
67	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₅×D₃	—	2	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	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₅×#67	—	4	4	no	—
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₅×D₁₂	—	2	4	no	—
81	C₅×D₁₂	—	4	2	no	—
82	C₅×D₁₂	—	4	2	no	—
83	C₄×C₅×D₃	—	4	4	no	—
84	C₅×Dic₆	—	4	2	no	—
85	C₃×C₅×C₈	—	4	2	no	—
86	C₅×C₃⋊C₈	—	2	4	no	—
87	C₂×#235	—	2	1	no	—
88	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
89	#930	—	2	1	no	—
90	#930	C₄×C₅	1	2	yes	—
91	C₂×#208	—	2	1	no	—
92	#875	—	2	1	no	—
93	#874	C₄×C₅	1	1	yes	characteristic
94	C₄×D₁₂	—	2	1	no	—
95	C₄×D₁₂	C₂²×C₅	1	2	yes	—
96	#869	C₄×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
98	C₃×C₂²⋊C₈	—	2	1	no	—
99	C₃×C₄×C₈	—	2	1	no	—
100	#803	—	2	1	no	—
101	C₄×C₃⋊C₈	C₂²×C₅	1	2	yes	—
102	C₅×#41	—	12	4	no	—
103	C₂×C₅×Q₈	—	6	1	no	—
104	C₂×C₅×D₄	—	3	2	no	—
105	C₂×C₅×D₄	—	6	2	no	—
106	C₂×C₅×D₄	—	6	1	no	—
107	C₂×C₅×D₄	—	6	1	no	—
108	C₂²×C₄×C₅	—	6	2	no	—
109	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₅×M₁₆	—	6	4	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	6	2	no	—
115	C₅×C₄⋊C₄	—	3	2	no	—
116	C₅×C₄⋊C₄	—	6	1	no	—
117	C₅×C₂²⋊C₄	—	6	2	no	—
118	C₅×C₂²⋊C₄	—	6	2	no	—
119	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
120	C₄²×C₅	D₁₂	1	1	yes	characteristic
121	C₄²×C₅	#67	1	1	yes	characteristic
122	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
123	#532	—	3	1	no	—
124	#431	C₅×D₃	1	1	yes	characteristic, complemented
125	C₄×M₁₆	—	3	1	no	—
126	D₄⋊C₈	—	3	4	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₅×D₃	—	4	2	no	—
132	C₂×C₅×D₃	—	4	2	no	—
133	C₂×C₅×D₃	—	4	2	no	—
134	C₃×C₄×C₅	—	2	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	#104	1	2	yes	—
138	C₃×C₄×C₅	#131	1	2	yes	—
139	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
140	C₅×Dic₃	—	4	2	no	—
141	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
144	C₂×#67	—	2	2	no	—
145	#235	—	4	4	no	—
146	C₂×D₁₂	—	2	1	no	—
147	C₂×D₁₂	—	2	1	no	—
148	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
149	C₂×C₄×D₃	—	2	2	no	—
150	C₂×Dic₆	—	2	1	no	—
151	C₂×C₃×C₈	—	2	2	no	—
152	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
153	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
154	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
155	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
156	(C₂×D₃)⋊C₄	—	2	2	no	—
157	(C₂×D₃)⋊C₄	—	2	2	no	—
158	C₄⋊Dic₃	—	2	1	no	—
159	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
160	#208	—	2	4	no	—
161	C₂×C₃⋊C₈	—	2	2	no	—
162	C₂³×C₅	—	6	2	no	—
163	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
164	C₅×Q₈	—	12	2	no	—
165	C₅×D₄	—	6	4	no	—
166	C₅×D₄	—	12	2	no	—
167	C₅×D₄	—	12	4	no	—
168	C₅×D₄	—	12	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	6	2	no	—
173	C₂×C₄×C₅	—	12	4	no	—
174	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
180	C₅×C₈	—	4	2	no	—
181	C₅×C₈	—	6	4	no	—
182	C₂×#41	—	6	1	no	—
183	C₂×M₁₆	—	6	1	no	—
184	C₄⋊Q₈	—	3	1	no	—
185	#127	—	3	1	no	—
186	#124	—	6	1	no	—
187	#121	—	3	2	no	—
188	#121	—	6	1	no	—
189	C₄×D₄	—	3	2	no	—
190	C₄×D₄	—	6	1	no	—
191	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
192	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
193	C₂²⋊C₈	—	2	1	no	—
194	#97	—	6	1	no	—
195	C₄×C₈	—	2	1	no	—
196	C₄×C₈	—	3	2	no	—
197	C₂×C₃×C₅	—	4	1	no	—
198	C₂×C₃×C₅	#408	1	1	yes	characteristic
199	C₂×C₃×C₅	#418	1	1	yes	characteristic
200	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
201	C₅×D₃	—	8	2	no	—
202	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₂²×D₃	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
213	#67	—	4	4	no	—
214	C₂×Dic₃	—	2	2	no	—
215	D₁₂	—	2	4	no	—
216	D₁₂	—	4	2	no	—
217	D₁₂	—	4	2	no	—
218	C₄×D₃	—	4	4	no	—
219	Dic₆	—	4	2	no	—
220	C₃×C₈	—	4	2	no	—
221	C₃⋊C₈	—	2	4	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	12	2	no	—
226	C₂²×C₅	—	12	2	no	—
227	C₂²×C₅	—	12	2	no	—
228	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	12	2	no	—
233	C₄×C₅	#825	1	2	yes	—
234	C₄×C₅	#901	1	2	yes	—
235	C₄×C₅	#949	1	2	yes	—
236	#41	—	12	4	no	—
237	C₂×Q₈	—	6	1	no	—
238	C₂×D₄	—	3	2	no	—
239	C₂×D₄	—	6	2	no	—
240	C₂×D₄	—	6	1	no	—
241	C₂×D₄	—	6	1	no	—
242	C₂²×C₄	—	6	2	no	—
243	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
244	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
245	C₂²×C₄	C₅×#67	1	1	yes	characteristic
246	M₁₆	—	6	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	6	2	no	—
249	C₄⋊C₄	—	3	2	no	—
250	C₄⋊C₄	—	6	1	no	—
251	C₂²⋊C₄	—	6	2	no	—
252	C₂²⋊C₄	—	6	2	no	—
253	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
254	C₄²	C₅×D₁₂	1	1	yes	characteristic
255	C₄²	C₅×#67	1	1	yes	characteristic
256	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
257	C₃×C₅	#1484	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
262	C₂×D₃	—	4	2	no	—
263	C₂×D₃	—	4	2	no	—
264	C₂×D₃	—	4	2	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	C₅×#104	1	2	yes	—
269	C₃×C₄	C₅×#131	1	2	yes	—
270	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
271	Dic₃	—	4	2	no	—
272	C₂×C₅	—	4	1	no	—
273	C₂×C₅	—	24	2	no	—
274	C₂×C₅	#5001	1	1	yes	characteristic
275	C₂×C₅	#5110	1	1	yes	characteristic
276	C₂×C₅	C₂×#825	1	1	yes	characteristic
277	C₂³	—	6	2	no	—
278	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
279	Q₈	—	12	2	no	—
280	D₄	—	6	4	no	—
281	D₄	—	12	2	no	—
282	D₄	—	12	4	no	—
283	D₄	—	12	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	6	2	no	—
288	C₂×C₄	—	12	4	no	—
289	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
290	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
291	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
292	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
293	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
295	C₈	—	4	2	no	—
296	C₈	—	6	4	no	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	C₅×#408	1	1	yes	characteristic
299	C₂×C₃	C₅×#418	1	1	yes	characteristic
300	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
301	D₃	—	8	2	no	—
302	C₅	#70761	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	12	2	no	—
307	C₂²	—	12	2	no	—
308	C₂²	—	12	2	no	—
309	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	12	2	no	—
314	C₄	C₅×#825	1	2	yes	—
315	C₄	C₅×#901	1	2	yes	—
316	C₄	C₅×#949	1	2	yes	—
317	C₃	C₅×#1484	1	1	yes	characteristic, complemented, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	24	2	no	—
320	C₂	C₅×#5001	1	1	yes	characteristic
321	C₂	C₅×#5110	1	1	yes	characteristic
322	C₂	C₂×C₅×#825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#431	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#431	D₃
hypocenter	C₃	C₅×#1484
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#431

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#431	C₂	semidirect
3. C₄×C₅×#208	C₂	semidirect
4. C₅×#4483	C₂	semidirect
21. #70761	C₅	direct (N × H)
23. C₅×#431	D₃	semidirect
48. C₃×#431	C₂×C₅	semidirect
49. C₄×#208	C₂×C₅	semidirect
50. #4483	C₂×C₅	semidirect
124. #431	C₅×D₃	semidirect
257. C₃×C₅	#1484	semidirect
302. C₅	#70761	direct (N × H)
317. C₃	C₅×#1484	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
C₃×C₅×#431	2
C₄×C₅×#208	2
C₅×#4483	2
C₅×#1484	3
#70761	5

Subgroups — C5×#70761 #14281981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70761 #14281981