Subgroups — C₅×#70133 #14295511

Order 1920.

568 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#948	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#413	C₂	1	1	yes	maximal, characteristic
3	C₅×#5240	C₂	1	1	yes	maximal, characteristic
4	C₅×#5122	C₂	1	2	yes	maximal, complemented
5	C₅×#4559	C₂	1	2	yes	maximal, complemented
6	C₅×#1757	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₅×#948	—	2	2	no	—
10	C₅×#948	—	2	2	no	—
11	C₅×#926	C₂²	1	2	yes	—
12	C₂×C₅×#208	C₄	1	2	yes	complemented
13	C₂×C₅×#208	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₃⋊C₈	C₄	1	2	yes	complemented
15	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#101	C₄	1	2	yes	complemented
17	C₃×C₅×#100	C₄	1	2	yes	complemented
18	C₅×#834	C₄	1	4	yes	—
19	C₅×#815	—	2	2	no	—
20	C₅×C₈×Dic₃	—	2	2	no	—
21	#70133	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#131	—	3	1	no	—
23	C₅×#413	D₃	1	1	yes	characteristic
24	C₅×#410	—	3	1	no	—
25	C₅×#405	—	3	2	no	—
26	C₅×#344	—	3	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×M₁₆	—	2	2	no	—
35	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₅×#217	C₂×C₄	1	4	yes	—
38	C₅×C₄⋊Dic₃	—	2	2	no	—
39	C₄×C₅×Dic₃	—	2	2	no	—
40	C₅×#208	—	2	2	no	—
41	C₅×#208	—	2	4	no	—
42	C₂×C₅×C₃⋊C₈	—	2	2	no	—
43	C₂×C₅×C₃⋊C₈	—	2	2	no	—
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
48	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
49	C₂×#948	C₂×C₅	1	1	yes	characteristic
50	C₃×#413	C₂×C₅	1	1	yes	characteristic
51	#5240	C₂×C₅	1	1	yes	characteristic
52	#5122	C₂×C₅	1	2	yes	complemented
53	#4559	C₂×C₅	1	2	yes	complemented
54	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
55	C₅×#131	—	6	2	no	—
56	C₅×#131	—	6	2	no	—
57	C₂×C₅×M₁₆	—	3	2	no	—
58	C₂×C₅×M₁₆	—	3	1	no	—
59	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
60	C₂²×C₅×C₈	—	3	1	no	—
61	C₂²×C₅×C₈	—	3	2	no	—
62	C₅×#117	—	3	2	no	—
63	C₅×#101	Dic₃	1	2	yes	complemented
64	C₅×#100	Dic₃	1	2	yes	complemented
65	C₅×#99	—	3	4	no	—
66	C₅×#97	—	6	2	no	—
67	C₄×C₅×C₈	—	6	2	no	—
68	#1757	—	3	1	no	Sylow
69	C₂³×C₃×C₅	C₄²	1	2	yes	—
70	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×Q₈	—	2	2	no	—
72	C₃×C₅×D₄	—	2	4	no	—
73	C₃×C₅×D₄	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	#41	1	2	yes	—
81	C₂×C₅×Dic₃	—	2	4	no	—
82	C₃×C₅×C₈	—	2	4	no	—
83	C₅×C₃⋊C₈	—	2	2	no	—
84	C₅×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₅×C₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
86	C₅×C₃⋊C₈	C₂²⋊C₄	1	2	yes	—
87	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
89	#948	—	2	2	no	—
90	#948	—	2	2	no	—
91	#926	C₂²×C₅	1	2	yes	—
92	C₂×#208	C₄×C₅	1	2	yes	complemented
93	C₂×#208	C₂²×C₅	1	1	yes	characteristic
94	C₂²×C₃⋊C₈	C₄×C₅	1	2	yes	complemented
95	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×#101	C₄×C₅	1	2	yes	complemented
97	C₃×#100	C₄×C₅	1	2	yes	complemented
98	#834	C₄×C₅	1	4	yes	—
99	#815	—	2	2	no	—
100	C₈×Dic₃	—	2	2	no	—
101	C₅×#41	—	2	2	no	—
102	C₅×#41	—	2	2	no	—
103	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
105	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
106	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
107	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₅×M₁₆	—	2	2	no	—
109	C₅×M₁₆	—	6	4	no	—
110	C₅×M₁₆	—	6	2	no	—
111	C₅×M₁₆	C₂×Dic₃	1	4	yes	—
112	C₂×C₅×C₈	—	2	2	no	—
113	C₂×C₅×C₈	—	3	1	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	C₂×C₅×C₈	—	3	2	no	—
116	C₂×C₅×C₈	—	3	2	no	—
117	C₂×C₅×C₈	—	3	2	no	—
118	C₂×C₅×C₈	—	6	2	no	—
119	C₂×C₅×C₈	—	6	2	no	—
120	C₅×C₄⋊C₄	—	6	2	no	—
121	C₅×C₂²⋊C₄	—	3	4	no	—
122	C₄²×C₅	—	6	2	no	—
123	C₂×#131	—	3	1	no	—
124	#413	C₅×D₃	1	1	yes	characteristic
125	#410	—	3	1	no	—
126	#405	—	3	2	no	—
127	#344	—	3	2	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	4	1	no	—
130	C₂²×C₃×C₅	—	4	2	no	—
131	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
133	C₃×C₄×C₅	—	2	2	no	—
134	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#117	1	1	yes	characteristic
136	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
137	C₅×Dic₃	—	4	4	no	—
138	C₃×#41	—	2	2	no	—
139	C₃×#41	—	2	2	no	—
140	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
141	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
142	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
144	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
145	C₃×M₁₆	—	2	2	no	—
146	C₃×M₁₆	C₂×C₄×C₅	1	4	yes	—
147	C₂×C₃×C₈	—	2	2	no	—
148	#217	C₂×C₄×C₅	1	4	yes	—
149	C₄⋊Dic₃	—	2	2	no	—
150	C₄×Dic₃	—	2	2	no	—
151	#208	—	2	2	no	—
152	#208	—	2	4	no	—
153	C₂×C₃⋊C₈	—	2	2	no	—
154	C₂×C₃⋊C₈	—	2	2	no	—
155	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
156	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
157	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
158	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
159	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
160	C₂³×C₅	C₄×Dic₃	1	2	yes	—
161	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₅×Q₈	—	2	2	no	—
163	C₅×D₄	—	2	4	no	—
164	C₅×D₄	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	6	4	no	—
168	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
169	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
171	C₂×C₄×C₅	#237	1	2	yes	—
172	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
173	C₅×C₈	—	2	4	no	—
174	C₅×C₈	—	3	1	no	—
175	C₅×C₈	—	3	1	no	—
176	C₅×C₈	—	3	2	no	—
177	C₅×C₈	—	6	2	no	—
178	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
179	#131	—	6	2	no	—
180	#131	—	6	2	no	—
181	C₂×M₁₆	—	3	2	no	—
182	C₂×M₁₆	—	3	1	no	—
183	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
184	C₂²×C₈	—	3	1	no	—
185	C₂²×C₈	—	3	2	no	—
186	#117	—	3	2	no	—
187	#101	C₅×Dic₃	1	2	yes	complemented
188	#100	C₅×Dic₃	1	2	yes	complemented
189	#99	—	3	4	no	—
190	#97	—	6	2	no	—
191	C₄×C₈	—	6	2	no	—
192	C₂×C₃×C₅	—	2	2	no	—
193	C₂×C₃×C₅	—	2	1	no	—
194	C₂×C₃×C₅	—	4	2	no	—
195	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
196	C₂³×C₃	C₄²×C₅	1	2	yes	—
197	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
198	C₃×Q₈	—	2	2	no	—
199	C₃×D₄	—	2	4	no	—
200	C₃×D₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	C₄²×C₅	1	2	yes	—
204	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
207	C₂×C₃×C₄	C₅×#41	1	2	yes	—
208	C₂×Dic₃	—	2	4	no	—
209	C₃×C₈	—	2	4	no	—
210	C₃⋊C₈	—	2	2	no	—
211	C₃⋊C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
212	C₃⋊C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
213	C₃⋊C₈	C₅×C₂²⋊C₄	1	2	yes	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	4	1	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	#67⋊C₄	1	2	yes	—
218	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	12	4	no	—
221	C₄×C₅	#877	1	1	yes	characteristic
222	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
223	C₄×C₅	D₄×Dic₃	1	2	yes	—
224	#41	—	2	2	no	—
225	#41	—	2	2	no	—
226	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic
227	C₂×D₄	C₄×C₅×D₃	1	2	yes	—
228	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
229	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
230	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
231	M₁₆	—	2	2	no	—
232	M₁₆	—	6	4	no	—
233	M₁₆	—	6	2	no	—
234	M₁₆	C₂×C₅×Dic₃	1	4	yes	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	3	1	no	—
238	C₂×C₈	—	3	2	no	—
239	C₂×C₈	—	3	2	no	—
240	C₂×C₈	—	3	2	no	—
241	C₂×C₈	—	6	2	no	—
242	C₂×C₈	—	6	2	no	—
243	C₄⋊C₄	—	6	2	no	—
244	C₂²⋊C₄	—	3	4	no	—
245	C₄²	—	6	2	no	—
246	C₃×C₅	#1757	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	4	1	no	—
249	C₂²×C₃	—	4	2	no	—
250	C₂²×C₃	C₂×C₄²×C₅	1	1	yes	characteristic
251	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
252	C₃×C₄	—	2	2	no	—
253	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
254	C₃×C₄	C₅×#117	1	1	yes	characteristic
255	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
256	Dic₃	—	4	4	no	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	2	1	no	—
259	C₂×C₅	—	4	2	no	—
260	C₂×C₅	Dic₃×C₂²⋊C₄	1	1	yes	characteristic
261	C₂³	C₄×C₅×Dic₃	1	2	yes	—
262	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
263	Q₈	—	2	2	no	—
264	D₄	—	2	4	no	—
265	D₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	C₄×C₅×Dic₃	1	2	yes	—
270	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
271	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
272	C₂×C₄	C₅×#237	1	2	yes	—
273	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	3	1	no	—
276	C₈	—	3	1	no	—
277	C₈	—	3	2	no	—
278	C₈	—	6	2	no	—
279	C₂×C₃	—	2	2	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
283	C₅	#70133	1	1	yes	characteristic, direct product factor, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₅×#67⋊C₄	1	2	yes	—
288	C₂²	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	12	4	no	—
291	C₄	C₅×#877	1	1	yes	characteristic
292	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
293	C₄	C₅×D₄×Dic₃	1	2	yes	—
294	C₃	C₅×#1757	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	4	2	no	—
298	C₂	C₅×Dic₃×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#413	C₂
socle	C₂×C₃×C₅	C₄×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#413	D₃
hypocenter	C₃	C₅×#1757
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#413

Complements

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

N	H ≅ G/N	type
4. C₅×#5122	C₂	semidirect
5. C₅×#4559	C₂	semidirect
12. C₂×C₅×#208	C₄	semidirect
14. C₂²×C₅×C₃⋊C₈	C₄	semidirect
16. C₃×C₅×#101	C₄	semidirect
17. C₃×C₅×#100	C₄	semidirect
21. #70133	C₅	direct (N × H)
52. #5122	C₂×C₅	semidirect
53. #4559	C₂×C₅	semidirect
63. C₅×#101	Dic₃	semidirect
64. C₅×#100	Dic₃	semidirect
92. C₂×#208	C₄×C₅	semidirect
94. C₂²×C₃⋊C₈	C₄×C₅	semidirect
96. C₃×#101	C₄×C₅	semidirect
97. C₃×#100	C₄×C₅	semidirect
187. #101	C₅×Dic₃	semidirect
188. #100	C₅×Dic₃	semidirect
246. C₃×C₅	#1757	semidirect
283. C₅	#70133	direct (N × H)
294. C₃	C₅×#1757	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#948	2
C₃×C₅×#413	2
C₅×#5240	2
C₅×#5122	2
C₅×#4559	2
C₅×#1757	3
#70133	5

Subgroups — C5×#70133 #14295511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70133 #14295511