Subgroups — C₅×#79042 #14415339

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#907	C₂	1	2	yes	maximal, complemented
2	C₅×#5477	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5129	C₂	1	2	yes	maximal, complemented
5	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4836	C₂	1	8	yes	maximal, complemented
7	C₅×#2932	—	3	1	no	maximal, Hall
8	C₂×C₅×#235	C₂²	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₅×#930	C₄	1	4	yes	—
12	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₅×#907	—	2	8	no	—
15	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	4	yes	—
16	C₂×C₅×D₃×C₈	C₂²	1	4	yes	—
17	C₅×#875	C₄	1	2	yes	—
18	C₅×#874	C₄	1	1	yes	characteristic
19	C₄×C₅×D₁₂	C₂²	1	4	yes	—
20	C₅×#869	C₄	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
22	C₅×(C₂×D₃)⋊C₈	C₂²	1	8	yes	—
23	C₅×#803	C₂²	1	4	yes	—
24	#79042	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₅×#131	—	3	2	no	—
26	C₅×#532	—	3	1	no	—
27	C₅×#435	—	3	8	no	—
28	C₅×#423	D₃	1	1	yes	characteristic, complemented
29	C₅×#408	—	3	2	no	—
30	C₂×C₅×#97	—	3	1	no	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×#67	C₂×C₄	1	4	yes	—
34	C₅×#235	—	2	4	no	—
35	C₅×#235	—	2	4	no	—
36	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
37	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
38	C₂×C₄×C₅×D₃	C₂³	1	4	yes	—
39	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
40	C₃×C₅×M₁₆	—	2	4	no	—
41	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
42	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂³	1	2	yes	—
45	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	—
46	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
47	C₂×C₅×C₃⋊C₈	—	2	2	no	—
48	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
49	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
50	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
51	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
52	C₅×D₃×C₈	—	2	8	no	—
53	C₂×#907	C₂×C₅	1	2	yes	complemented
54	#5477	C₂×C₅	1	1	yes	characteristic
55	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
56	#5129	C₂×C₅	1	2	yes	complemented
57	C₂×#803	C₂×C₅	1	1	yes	characteristic, complemented
58	#4836	C₂×C₅	1	8	yes	complemented
59	C₂×C₅×#41	—	3	2	no	—
60	C₅×#131	—	6	8	no	—
61	C₂×C₅×M₁₆	—	3	4	no	—
62	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
63	C₂²×C₅×C₈	—	3	1	no	—
64	C₂²×C₅×C₈	—	3	1	no	—
65	C₂²×C₅×C₈	—	3	4	no	—
66	C₅×C₄⋊Q₈	—	3	1	no	—
67	C₅×#127	—	3	1	no	—
68	C₅×#124	—	3	2	no	—
69	C₅×#121	—	3	4	no	—
70	C₄×C₅×D₄	—	3	4	no	—
71	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
72	C₅×C₄⋊C₈	C₂×D₃	1	4	yes	—
73	C₅×C₂²⋊C₈	—	3	8	no	—
74	C₅×#97	—	3	4	no	—
75	#2932	—	3	1	no	Sylow
76	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	—	4	2	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₅×#67	—	2	8	no	—
86	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
87	C₅×D₁₂	—	2	4	no	—
88	C₅×D₁₂	—	4	4	no	—
89	C₄×C₅×D₃	—	2	4	no	—
90	C₄×C₅×D₃	—	2	4	no	—
91	C₅×Dic₆	—	2	4	no	—
92	C₃×C₅×C₈	—	2	4	no	—
93	C₅×C₃⋊C₈	—	2	2	no	—
94	C₅×C₃⋊C₈	C₂×D₄	1	4	yes	—
95	C₂×#235	C₂²×C₅	1	2	yes	—
96	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
97	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
98	#930	C₄×C₅	1	4	yes	—
99	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
101	#907	—	2	8	no	—
102	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	4	yes	—
103	C₂×D₃×C₈	C₂²×C₅	1	4	yes	—
104	#875	C₄×C₅	1	2	yes	—
105	#874	C₄×C₅	1	1	yes	characteristic
106	C₄×D₁₂	C₂²×C₅	1	4	yes	—
107	#869	C₄×C₅	1	1	yes	characteristic
108	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
109	(C₂×D₃)⋊C₈	C₂²×C₅	1	8	yes	—
110	#803	C₂²×C₅	1	4	yes	—
111	C₅×#41	—	6	4	no	—
112	C₅×#41	—	6	4	no	—
113	C₂×C₅×Q₈	—	3	2	no	—
114	C₂×C₅×D₄	—	3	4	no	—
115	C₂×C₅×D₄	—	3	2	no	—
116	C₂×C₅×D₄	—	3	4	no	—
117	C₂²×C₄×C₅	—	3	4	no	—
118	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
119	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
120	C₅×M₁₆	—	2	4	no	—
121	C₅×M₁₆	—	6	8	no	—
122	C₂×C₅×C₈	—	3	4	no	—
123	C₂×C₅×C₈	—	3	2	no	—
124	C₂×C₅×C₈	—	3	2	no	—
125	C₂×C₅×C₈	—	6	8	no	—
126	C₂×C₅×C₈	—	6	2	no	—
127	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
128	C₅×C₄⋊C₄	—	3	4	no	—
129	C₅×C₂²⋊C₄	—	3	8	no	—
130	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
131	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
132	C₄²×C₅	C₂²×D₃	1	2	yes	—
133	C₂×#131	—	3	2	no	—
134	#532	—	3	1	no	—
135	#435	—	3	8	no	—
136	#423	C₅×D₃	1	1	yes	characteristic, complemented
137	#408	—	3	2	no	—
138	C₂×#97	—	3	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
141	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₅×D₃	—	2	4	no	—
143	C₂×C₅×D₃	—	2	8	no	—
144	C₃×C₄×C₅	—	2	2	no	—
145	C₃×C₄×C₅	—	2	2	no	—
146	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
147	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
149	C₅×Dic₃	—	2	4	no	—
150	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
151	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
152	C₂×#67	C₂×C₄×C₅	1	4	yes	—
153	#235	—	2	4	no	—
154	#235	—	2	4	no	—
155	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
156	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
157	C₂×C₄×D₃	C₂³×C₅	1	4	yes	—
158	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
159	C₃×M₁₆	—	2	4	no	—
160	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
161	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
162	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
163	C₃×C₄²	C₂³×C₅	1	2	yes	—
164	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	—
165	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
166	C₂×C₃⋊C₈	—	2	2	no	—
167	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	complemented
168	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
169	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
170	ΓL(1,𝔽₂₅)	—	2	8	no	—
171	D₃×C₈	—	2	8	no	—
172	C₂³×C₅	—	3	4	no	—
173	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₅×Q₈	—	6	4	no	—
175	C₅×D₄	—	6	8	no	—
176	C₅×D₄	—	6	4	no	—
177	C₅×D₄	—	12	4	no	—
178	C₂×C₄×C₅	—	3	4	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	6	4	no	—
181	C₂×C₄×C₅	—	6	4	no	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
183	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
184	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
185	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
186	C₂×C₄×C₅	#239	1	2	yes	—
187	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
188	C₅×C₈	—	2	4	no	—
189	C₅×C₈	—	3	4	no	—
190	C₅×C₈	—	6	2	no	—
191	C₂×#41	—	3	2	no	—
192	#131	—	6	8	no	—
193	C₂×M₁₆	—	3	4	no	—
194	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
195	C₂²×C₈	—	3	1	no	—
196	C₂²×C₈	—	3	1	no	—
197	C₂²×C₈	—	3	4	no	—
198	C₄⋊Q₈	—	3	1	no	—
199	#127	—	3	1	no	—
200	#124	—	3	2	no	—
201	#121	—	3	4	no	—
202	C₄×D₄	—	3	4	no	—
203	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
204	C₄⋊C₈	C₂×C₅×D₃	1	4	yes	—
205	C₂²⋊C₈	—	3	8	no	—
206	#97	—	3	4	no	—
207	C₂×C₃×C₅	—	2	2	no	—
208	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂×C₃×C₅	#568	1	2	yes	—
210	C₅×D₃	—	4	4	no	—
211	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
212	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
215	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
216	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
218	C₂×C₃×C₄	C₅×#41	1	2	yes	—
219	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
220	#67	—	2	8	no	—
221	C₂×Dic₃	C₂²×C₄×C₅	1	4	yes	—
222	D₁₂	—	2	4	no	—
223	D₁₂	—	4	4	no	—
224	C₄×D₃	—	2	4	no	—
225	C₄×D₃	—	2	4	no	—
226	Dic₆	—	2	4	no	—
227	C₃×C₈	—	2	4	no	—
228	C₃⋊C₈	—	2	2	no	—
229	C₃⋊C₈	C₂×C₅×D₄	1	4	yes	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	6	8	no	—
232	C₂²×C₅	—	6	4	no	—
233	C₂²×C₅	#893	1	2	yes	—
234	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	6	4	no	—
238	C₄×C₅	#893	1	2	yes	—
239	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
240	C₄×C₅	C₂×#239	1	1	yes	characteristic
241	#41	—	6	4	no	—
242	#41	—	6	4	no	—
243	C₂×Q₈	—	3	2	no	—
244	C₂×D₄	—	3	4	no	—
245	C₂×D₄	—	3	2	no	—
246	C₂×D₄	—	3	4	no	—
247	C₂²×C₄	—	3	4	no	—
248	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
249	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
250	M₁₆	—	2	4	no	—
251	M₁₆	—	6	8	no	—
252	C₂×C₈	—	3	4	no	—
253	C₂×C₈	—	3	2	no	—
254	C₂×C₈	—	3	2	no	—
255	C₂×C₈	—	6	8	no	—
256	C₂×C₈	—	6	2	no	—
257	C₂×C₈	C₂²×C₅×D₃	1	4	yes	—
258	C₄⋊C₄	—	3	4	no	—
259	C₂²⋊C₄	—	3	8	no	—
260	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
261	C₄²	C₄×C₅×D₃	1	1	yes	characteristic
262	C₄²	C₂²×C₅×D₃	1	2	yes	—
263	C₃×C₅	#2932	1	1	yes	characteristic, complemented, Hall
264	C₂²×C₃	—	2	2	no	—
265	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
266	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
267	C₂×D₃	—	2	4	no	—
268	C₂×D₃	—	2	8	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	2	no	—
271	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
272	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
273	C₃×C₄	C₂×C₅×#41	1	1	yes	characteristic
274	Dic₃	—	2	4	no	—
275	C₂×C₅	—	2	2	no	—
276	C₂×C₅	—	12	4	no	—
277	C₂×C₅	C₂×#893	1	1	yes	characteristic
278	C₂×C₅	#5745	1	2	yes	—
279	C₂³	—	3	4	no	—
280	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
281	Q₈	—	6	4	no	—
282	D₄	—	6	8	no	—
283	D₄	—	6	4	no	—
284	D₄	—	12	4	no	—
285	C₂×C₄	—	3	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	6	4	no	—
288	C₂×C₄	—	6	4	no	—
289	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
290	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
291	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
292	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
293	C₂×C₄	C₅×#239	1	2	yes	—
294	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	3	4	no	—
297	C₈	—	6	2	no	—
298	C₂×C₃	—	2	2	no	—
299	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
300	C₂×C₃	C₅×#568	1	2	yes	—
301	D₃	—	4	4	no	—
302	C₅	#79042	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	2	no	—
304	C₂²	—	6	8	no	—
305	C₂²	—	6	4	no	—
306	C₂²	C₅×#893	1	2	yes	—
307	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	6	4	no	—
311	C₄	C₅×#893	1	2	yes	—
312	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
313	C₄	C₂×C₅×#239	1	1	yes	characteristic
314	C₃	C₅×#2932	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	2	2	no	—
316	C₂	—	12	4	no	—
317	C₂	C₂×C₅×#893	1	1	yes	characteristic
318	C₂	C₅×#5745	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₅×#5477	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#2932
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#907	C₂	semidirect
3. C₃×C₅×#423	C₂	semidirect
4. C₅×#5129	C₂	semidirect
5. C₂×C₅×#803	C₂	semidirect
6. C₅×#4836	C₂	semidirect
24. #79042	C₅	direct (N × H)
28. C₅×#423	D₃	semidirect
48. C₂×C₅×C₃⋊C₈	D₄	semidirect
53. C₂×#907	C₂×C₅	semidirect
55. C₃×#423	C₂×C₅	semidirect
56. #5129	C₂×C₅	semidirect
57. C₂×#803	C₂×C₅	semidirect
58. #4836	C₂×C₅	semidirect
136. #423	C₅×D₃	semidirect
167. C₂×C₃⋊C₈	C₅×D₄	semidirect
263. C₃×C₅	#2932	semidirect
302. C₅	#79042	direct (N × H)
314. C₃	C₅×#2932	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#907	2
C₅×#5477	2
C₃×C₅×#423	2
C₅×#5129	2
C₂×C₅×#803	2
C₅×#4836	2
C₅×#2932	3
#79042	5

Subgroups — C5×#79042 #14415339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#79042 #14415339