Subgroups — C₅×#83813 #14412161

Order 1920.

1512 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	C₂×C₅×#1009	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5132	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#823	C₂	1	6	yes	maximal, complemented
6	C₅×#2888	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₅×#1009	—	2	4	no	—
9	C₅×#1009	—	2	4	no	—
10	C₂×C₅×#239	C₄	1	2	yes	—
11	C₂×C₅×D₃×D₄	C₄	1	6	yes	—
12	C₂×C₅×#235	C₂²	1	3	yes	—
13	C₂²×C₅×D₁₂	C₂²	1	3	yes	—
14	C₃×C₅×#131	—	2	4	no	—
15	C₂×C₃×C₅×M₁₆	C₂²	1	6	yes	complemented
16	C₅×#948	—	2	4	no	—
17	C₂×C₅×#208	C₂²	1	6	yes	complemented
18	C₅×#824	C₂²	1	4	yes	complemented
19	C₅×#823	C₂²	1	12	yes	complemented
20	#83813	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₅×H(5,𝔽₂)	—	3	1	no	—
22	C₅×#568	—	3	1	no	—
23	C₅×#568	D₃	1	1	yes	characteristic, complemented
24	C₅×#413	—	3	6	no	—
25	C₂×C₅×#100	—	3	6	no	—
26	C₂³×C₅×D₃	C₂×C₄	1	6	yes	—
27	C₃×C₅×#41	D₄	1	4	yes	—
28	C₃×C₅×#41	D₄	1	4	yes	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	3	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
32	C₂×C₅×#67	C₂×C₄	1	6	yes	—
33	C₅×#239	—	2	8	no	—
34	C₅×D₃×D₄	—	2	24	no	—
35	C₅×#235	—	2	6	no	—
36	C₅×#235	—	2	6	no	—
37	C₂×C₅×D₁₂	—	2	6	no	—
38	C₂×C₅×D₁₂	—	2	6	no	—
39	C₂×C₅×D₁₂	C₂×C₄	1	6	yes	—
40	C₂×C₅×D₁₂	C₂³	1	3	yes	—
41	C₂×C₄×C₅×D₃	C₂×C₄	1	6	yes	—
42	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
43	C₃×C₅×M₁₆	—	2	6	no	—
44	C₃×C₅×M₁₆	C₂³	1	4	yes	—
45	C₂×C₃×C₅×C₈	—	2	6	no	—
46	C₅×#208	—	2	6	no	—
47	C₅×#208	C₂³	1	4	yes	—
48	C₂×C₅×C₃⋊C₈	—	2	6	no	—
49	C₂×#1009	C₂×C₅	1	1	yes	characteristic
50	C₃×#568	C₂×C₅	1	1	yes	characteristic, complemented
51	#5819	C₂×C₅	1	1	yes	characteristic, complemented
52	#5132	C₂×C₅	1	6	yes	complemented
53	C₂×#823	C₂×C₅	1	6	yes	complemented
54	C₅×H(5,𝔽₂)	—	6	4	no	—
55	C₅×H(5,𝔽₂)	—	6	4	no	—
56	C₂×C₅×#41	—	3	2	no	—
57	C₂×C₅×#41	—	3	3	no	—
58	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
59	C₂²×C₅×D₄	—	3	6	no	—
60	C₂²×C₅×D₄	—	3	3	no	—
61	C₅×#131	—	2	4	no	—
62	C₅×#131	—	6	4	no	—
63	C₂×C₅×M₁₆	—	3	6	no	—
64	C₂×C₅×M₁₆	C₂×D₃	1	6	yes	complemented
65	C₅×#101	—	3	4	no	—
66	C₅×#100	—	3	12	no	—
67	#2888	—	3	1	no	Sylow
68	C₂³×C₃×C₅	C₂²×C₄	1	3	yes	—
69	C₂²×C₅×D₃	—	2	12	no	—
70	C₂²×C₅×D₃	—	8	6	no	—
71	C₂²×C₅×D₃	C₂²×C₄	1	6	yes	—
72	C₃×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
73	C₃×C₅×D₄	C₂²⋊C₄	1	12	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
77	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
78	C₅×#67	—	2	12	no	—
79	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
80	C₅×D₁₂	—	2	12	no	—
81	C₅×D₁₂	—	2	3	no	—
82	C₅×D₁₂	—	2	3	no	—
83	C₄×C₅×D₃	—	2	12	no	—
84	C₅×Dic₆	—	2	1	no	—
85	C₅×Dic₆	—	2	1	no	—
86	C₃×C₅×C₈	—	2	4	no	—
87	C₅×C₃⋊C₈	—	2	4	no	—
88	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
89	#1009	—	2	4	no	—
90	#1009	—	2	4	no	—
91	C₂×#239	C₄×C₅	1	2	yes	—
92	C₂×D₃×D₄	C₄×C₅	1	6	yes	—
93	C₂×#235	C₂²×C₅	1	3	yes	—
94	C₂²×D₁₂	C₂²×C₅	1	3	yes	—
95	C₃×#131	—	2	4	no	—
96	C₂×C₃×M₁₆	C₂²×C₅	1	6	yes	complemented
97	#948	—	2	4	no	—
98	C₂×#208	C₂²×C₅	1	6	yes	complemented
99	#824	C₂²×C₅	1	4	yes	complemented
100	#823	C₂²×C₅	1	12	yes	complemented
101	C₂⁴×C₅	—	3	6	no	—
102	C₅×#41	—	6	8	no	—
103	C₅×#41	—	6	6	no	—
104	C₅×#41	—	6	6	no	—
105	C₅×#41	D₁₂	1	4	yes	—
106	C₅×#41	#67	1	4	yes	—
107	C₂×C₅×Q₈	—	3	1	no	—
108	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
109	C₂×C₅×D₄	—	3	6	no	—
110	C₂×C₅×D₄	—	3	6	no	—
111	C₂×C₅×D₄	—	3	3	no	—
112	C₂×C₅×D₄	—	6	24	no	—
113	C₂×C₅×D₄	—	6	6	no	—
114	C₂×C₅×D₄	—	6	6	no	—
115	C₂×C₅×D₄	C₄×D₃	1	3	yes	—
116	C₂²×C₄×C₅	—	3	6	no	—
117	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
118	C₅×M₁₆	—	2	6	no	—
119	C₅×M₁₆	—	3	4	no	—
120	C₅×M₁₆	—	6	6	no	—
121	C₅×M₁₆	C₂²×D₃	1	4	yes	—
122	C₂×C₅×C₈	—	2	6	no	—
123	C₂×C₅×C₈	—	6	6	no	—
124	C₂×H(5,𝔽₂)	—	3	1	no	—
125	#568	—	3	1	no	—
126	#568	C₅×D₃	1	1	yes	characteristic, complemented
127	#413	—	3	6	no	—
128	C₂×#100	—	3	6	no	—
129	C₂²×C₃×C₅	—	4	3	no	—
130	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
131	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
132	C₂×C₅×D₃	—	2	6	no	—
133	C₂×C₅×D₃	—	4	6	no	—
134	C₂×C₅×D₃	—	8	12	no	—
135	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
136	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
137	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
138	C₅×Dic₃	—	2	2	no	—
139	C₂³×D₃	C₂×C₄×C₅	1	6	yes	—
140	C₃×#41	C₅×D₄	1	4	yes	—
141	C₃×#41	C₅×D₄	1	4	yes	—
142	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂×C₃×D₄	C₂×C₄×C₅	1	3	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
145	C₂×#67	C₂×C₄×C₅	1	6	yes	—
146	#239	—	2	8	no	—
147	D₃×D₄	—	2	24	no	—
148	#235	—	2	6	no	—
149	#235	—	2	6	no	—
150	C₂×D₁₂	—	2	6	no	—
151	C₂×D₁₂	—	2	6	no	—
152	C₂×D₁₂	C₂×C₄×C₅	1	6	yes	—
153	C₂×D₁₂	C₂³×C₅	1	3	yes	—
154	C₂×C₄×D₃	C₂×C₄×C₅	1	6	yes	—
155	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
156	C₃×M₁₆	—	2	6	no	—
157	C₃×M₁₆	C₂³×C₅	1	4	yes	—
158	C₂×C₃×C₈	—	2	6	no	—
159	#208	—	2	6	no	—
160	#208	C₂³×C₅	1	4	yes	—
161	C₂×C₃⋊C₈	—	2	6	no	—
162	C₂³×C₅	—	3	6	no	—
163	C₂³×C₅	—	6	12	no	—
164	C₂³×C₅	—	24	6	no	—
165	C₂³×C₅	C₂×C₄×D₃	1	3	yes	—
166	C₅×Q₈	—	6	1	no	—
167	C₅×Q₈	—	6	1	no	—
168	C₅×Q₈	(C₂×D₃)⋊C₄	1	4	yes	—
169	C₅×D₄	—	6	12	no	—
170	C₅×D₄	—	6	12	no	—
171	C₅×D₄	—	6	3	no	—
172	C₅×D₄	—	6	3	no	—
173	C₅×D₄	(C₂×D₃)⋊C₄	1	12	yes	—
174	C₂×C₄×C₅	—	3	2	no	—
175	C₂×C₄×C₅	—	6	12	no	—
176	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
177	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
178	C₂×C₄×C₅	C₂×#67	1	6	yes	—
179	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
180	C₅×C₈	—	2	4	no	—
181	C₅×C₈	—	6	4	no	—
182	H(5,𝔽₂)	—	6	4	no	—
183	H(5,𝔽₂)	—	6	4	no	—
184	C₂×#41	—	3	2	no	—
185	C₂×#41	—	3	3	no	—
186	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂²×D₄	—	3	6	no	—
188	C₂²×D₄	—	3	3	no	—
189	#131	—	2	4	no	—
190	#131	—	6	4	no	—
191	C₂×M₁₆	—	3	6	no	—
192	C₂×M₁₆	C₂×C₅×D₃	1	6	yes	complemented
193	#101	—	3	4	no	—
194	#100	—	3	12	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	2	6	no	—
197	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
198	C₅×D₃	—	4	6	no	—
199	C₂³×C₃	C₂²×C₄×C₅	1	3	yes	—
200	C₂²×D₃	—	2	12	no	—
201	C₂²×D₃	—	8	6	no	—
202	C₂²×D₃	C₂²×C₄×C₅	1	6	yes	—
203	C₃×Q₈	C₅×C₂²⋊C₄	1	4	yes	—
204	C₃×D₄	C₅×C₂²⋊C₄	1	12	yes	—
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	3	yes	—
206	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
207	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
208	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
209	#67	—	2	12	no	—
210	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
211	D₁₂	—	2	12	no	—
212	D₁₂	—	2	3	no	—
213	D₁₂	—	2	3	no	—
214	C₄×D₃	—	2	12	no	—
215	Dic₆	—	2	1	no	—
216	Dic₆	—	2	1	no	—
217	C₃×C₈	—	2	4	no	—
218	C₃⋊C₈	—	2	4	no	—
219	C₂²×C₅	—	4	3	no	—
220	C₂²×C₅	—	6	6	no	—
221	C₂²×C₅	—	12	6	no	—
222	C₂²×C₅	—	24	12	no	—
223	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
224	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
225	C₄×C₅	—	6	2	no	—
226	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
227	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
228	C₄×C₅	C₂²×#67	1	1	yes	characteristic
229	C₂⁴	—	3	6	no	—
230	#41	—	6	8	no	—
231	#41	—	6	6	no	—
232	#41	—	6	6	no	—
233	#41	C₅×D₁₂	1	4	yes	—
234	#41	C₅×#67	1	4	yes	—
235	C₂×Q₈	—	3	1	no	—
236	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
237	C₂×D₄	—	3	6	no	—
238	C₂×D₄	—	3	6	no	—
239	C₂×D₄	—	3	3	no	—
240	C₂×D₄	—	6	24	no	—
241	C₂×D₄	—	6	6	no	—
242	C₂×D₄	—	6	6	no	—
243	C₂×D₄	C₄×C₅×D₃	1	3	yes	—
244	C₂²×C₄	—	3	6	no	—
245	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
246	M₁₆	—	2	6	no	—
247	M₁₆	—	3	4	no	—
248	M₁₆	—	6	6	no	—
249	M₁₆	C₂²×C₅×D₃	1	4	yes	—
250	C₂×C₈	—	2	6	no	—
251	C₂×C₈	—	6	6	no	—
252	C₃×C₅	#2888	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	4	3	no	—
254	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	6	yes	—
255	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
256	C₂×D₃	—	2	6	no	—
257	C₂×D₃	—	4	6	no	—
258	C₂×D₃	—	8	12	no	—
259	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
260	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
261	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
262	Dic₃	—	2	2	no	—
263	C₂×C₅	—	2	1	no	—
264	C₂×C₅	—	2	6	no	—
265	C₂×C₅	—	12	6	no	—
266	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
267	C₂³	—	3	6	no	—
268	C₂³	—	6	12	no	—
269	C₂³	—	24	6	no	—
270	C₂³	C₂×C₄×C₅×D₃	1	3	yes	—
271	Q₈	—	6	1	no	—
272	Q₈	—	6	1	no	—
273	Q₈	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
274	D₄	—	6	12	no	—
275	D₄	—	6	12	no	—
276	D₄	—	6	3	no	—
277	D₄	—	6	3	no	—
278	D₄	C₅×(C₂×D₃)⋊C₄	1	12	yes	—
279	C₂×C₄	—	3	2	no	—
280	C₂×C₄	—	6	12	no	—
281	C₂×C₄	C₂×C₄×C₅×D₃	1	3	yes	—
282	C₂×C₄	C₂×C₅×D₁₂	1	6	yes	—
283	C₂×C₄	C₂×C₅×#67	1	6	yes	—
284	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
285	C₈	—	2	4	no	—
286	C₈	—	6	4	no	—
287	C₂×C₃	—	2	1	no	—
288	C₂×C₃	—	2	6	no	—
289	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
290	D₃	—	4	6	no	—
291	C₅	#83813	1	1	yes	characteristic, direct product factor, Sylow
292	C₂²	—	4	3	no	—
293	C₂²	—	6	6	no	—
294	C₂²	—	12	6	no	—
295	C₂²	—	24	12	no	—
296	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	6	yes	—
297	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
298	C₄	—	6	2	no	—
299	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	6	yes	—
300	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
301	C₄	C₂²×C₅×#67	1	1	yes	characteristic
302	C₃	C₅×#2888	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	2	1	no	—
304	C₂	—	2	6	no	—
305	C₂	—	12	6	no	—
306	C₂	C₂²×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1009	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#568	D₃
hypocenter	C₃	C₅×#2888
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₅×#41 ◁ C₅×#568

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#568	C₂	semidirect
3. C₅×#5819	C₂	semidirect
4. C₅×#5132	C₂	semidirect
5. C₂×C₅×#823	C₂	semidirect
15. C₂×C₃×C₅×M₁₆	C₂²	semidirect
17. C₂×C₅×#208	C₂²	semidirect
18. C₅×#824	C₂²	semidirect
19. C₅×#823	C₂²	semidirect
20. #83813	C₅	direct (N × H)
23. C₅×#568	D₃	semidirect
50. C₃×#568	C₂×C₅	semidirect
51. #5819	C₂×C₅	semidirect
52. #5132	C₂×C₅	semidirect
53. C₂×#823	C₂×C₅	semidirect
64. C₂×C₅×M₁₆	C₂×D₃	semidirect
96. C₂×C₃×M₁₆	C₂²×C₅	semidirect
98. C₂×#208	C₂²×C₅	semidirect
99. #824	C₂²×C₅	semidirect
100. #823	C₂²×C₅	semidirect
126. #568	C₅×D₃	semidirect
192. C₂×M₁₆	C₂×C₅×D₃	semidirect
252. C₃×C₅	#2888	semidirect
291. C₅	#83813	direct (N × H)
302. C₃	C₅×#2888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1009	2
C₃×C₅×#568	2
C₅×#5819	2
C₅×#5132	2
C₂×C₅×#823	2
C₅×#2888	3
#83813	5

Subgroups — C5×#83813 #14412161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#83813 #14412161