Subgroups — C₅×#70521 #14306116

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#874	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#823	C₂	1	4	yes	maximal, complemented
4	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1965	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	C₄	1	4	yes	—
7	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	C₂²	1	2	yes	—
11	C₅×#874	—	2	1	no	—
12	C₅×#874	—	2	1	no	—
13	C₅×#874	—	2	2	no	—
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	C₅×#823	—	2	8	no	—
16	C₅×#803	—	2	1	no	—
17	C₄×C₅×C₃⋊C₈	—	2	1	no	—
18	#70521	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#127	—	3	1	no	—
20	C₅×#423	D₃	1	1	yes	characteristic, complemented
21	C₂×C₅×#100	—	3	4	no	—
22	C₄×C₅×M₁₆	—	3	1	no	—
23	C₂³×C₅×D₃	C₂×C₄	1	4	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₅×D₁₂	—	2	4	no	—
27	C₂×C₅×D₁₂	—	2	4	no	—
28	C₂×C₅×D₁₂	—	2	2	no	—
29	C₂×C₅×D₁₂	—	2	2	no	—
30	C₂×C₅×D₁₂	—	2	4	no	—
31	C₂×C₅×D₁₂	—	2	4	no	—
32	C₂×C₅×D₁₂	—	4	8	no	—
33	C₃×C₅×M₁₆	—	2	4	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	C₅×#208	—	2	2	no	—
39	C₅×#208	D₄	1	4	yes	complemented
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×#874	C₂×C₅	1	1	yes	characteristic
42	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×#823	C₂×C₅	1	4	yes	complemented
44	C₄×#208	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	—	3	4	no	—
46	C₂²×C₅×D₄	—	3	2	no	—
47	C₂×C₅×M₁₆	—	3	2	no	—
48	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
49	C₅×#127	—	6	1	no	—
50	C₅×#127	—	6	1	no	—
51	C₅×#127	—	6	2	no	—
52	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×#100	—	6	8	no	—
55	C₅×#97	—	6	1	no	—
56	C₄×C₅×C₈	—	6	1	no	—
57	#1965	—	3	1	no	Sylow
58	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₅×D₃	—	2	4	no	—
60	C₂²×C₅×D₃	—	2	8	no	—
61	C₂²×C₅×D₃	—	4	8	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	2	yes	—
69	C₅×D₁₂	—	4	8	no	—
70	C₅×D₁₂	—	4	8	no	—
71	C₅×D₁₂	—	4	4	no	—
72	C₅×D₁₂	—	4	2	no	—
73	C₅×D₁₂	—	4	2	no	—
74	C₃×C₅×C₈	—	4	2	no	—
75	C₅×C₃⋊C₈	—	2	4	no	—
76	C₂²×D₁₂	C₄×C₅	1	4	yes	—
77	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
78	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
80	C₂×#208	C₂²×C₅	1	2	yes	—
81	#874	—	2	1	no	—
82	#874	—	2	1	no	—
83	#874	—	2	2	no	—
84	C₃×C₄⋊C₈	—	2	2	no	—
85	#823	—	2	8	no	—
86	#803	—	2	1	no	—
87	C₄×C₃⋊C₈	—	2	1	no	—
88	C₂⁴×C₅	—	3	4	no	—
89	C₂×C₅×D₄	—	6	4	no	—
90	C₂×C₅×D₄	—	6	4	no	—
91	C₂×C₅×D₄	—	6	2	no	—
92	C₂×C₅×D₄	—	6	4	no	—
93	C₂×C₅×D₄	—	6	4	no	—
94	C₂×C₅×D₄	—	6	2	no	—
95	C₂×C₅×D₄	—	12	8	no	—
96	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
97	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
98	C₅×M₁₆	—	2	4	no	—
99	C₅×M₁₆	—	3	4	no	—
100	C₅×M₁₆	—	6	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	6	2	no	—
103	C₄²×C₅	D₁₂	1	1	yes	characteristic
104	C₄²×C₅	D₁₂	1	1	yes	characteristic
105	C₄²×C₅	#67	1	2	yes	—
106	C₂×#127	—	3	1	no	—
107	#423	C₅×D₃	1	1	yes	characteristic, complemented
108	C₂×#100	—	3	4	no	—
109	C₄×M₁₆	—	3	1	no	—
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
113	C₂×C₅×D₃	—	4	4	no	—
114	C₂×C₅×D₃	—	4	8	no	—
115	C₂×C₅×D₃	—	8	8	no	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	#100	1	4	yes	complemented
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₃×C₄×C₅	#124	1	1	yes	characteristic
120	C₃×C₄×C₅	#127	1	1	yes	characteristic
121	C₂³×D₃	C₂×C₄×C₅	1	4	yes	—
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
123	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂×D₁₂	—	2	4	no	—
125	C₂×D₁₂	—	2	4	no	—
126	C₂×D₁₂	—	2	4	no	—
127	C₂×D₁₂	—	2	2	no	—
128	C₂×D₁₂	—	2	2	no	—
129	C₂×D₁₂	—	2	4	no	—
130	C₂×D₁₂	—	4	8	no	—
131	C₃×M₁₆	—	2	4	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
134	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
135	C₃×C₄²	C₅×D₄	1	2	yes	—
136	#208	—	2	2	no	—
137	#208	C₅×D₄	1	4	yes	complemented
138	C₂×C₃⋊C₈	—	2	2	no	—
139	C₂³×C₅	—	6	4	no	—
140	C₂³×C₅	—	6	8	no	—
141	C₂³×C₅	—	12	8	no	—
142	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
143	C₅×D₄	—	12	8	no	—
144	C₅×D₄	—	12	8	no	—
145	C₅×D₄	—	12	4	no	—
146	C₅×D₄	—	12	2	no	—
147	C₅×D₄	—	12	2	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
152	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
153	C₂×C₄×C₅	#239	1	2	yes	—
154	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
155	C₅×C₈	—	4	2	no	—
156	C₅×C₈	—	6	4	no	—
157	C₂²×D₄	—	3	4	no	—
158	C₂²×D₄	—	3	2	no	—
159	C₂×M₁₆	—	3	2	no	—
160	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
161	#127	—	6	1	no	—
162	#127	—	6	1	no	—
163	#127	—	6	2	no	—
164	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	#100	—	6	8	no	—
167	#97	—	6	1	no	—
168	C₄×C₈	—	6	1	no	—
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	#390	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×#100	1	2	yes	—
172	C₅×D₃	—	8	4	no	—
173	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
174	C₂²×D₃	—	2	4	no	—
175	C₂²×D₃	—	2	8	no	—
176	C₂²×D₃	—	4	8	no	—
177	C₂×C₃×C₄	—	2	4	no	—
178	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
179	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
180	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₅×#41	1	2	yes	—
184	D₁₂	—	4	8	no	—
185	D₁₂	—	4	4	no	—
186	D₁₂	—	4	2	no	—
187	D₁₂	—	4	2	no	—
188	D₁₂	—	4	8	no	—
189	C₃×C₈	—	4	2	no	—
190	C₃⋊C₈	—	2	4	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	12	8	no	—
193	C₂²×C₅	—	12	4	no	—
194	C₂²×C₅	—	24	8	no	—
195	C₂²×C₅	#893	1	2	yes	—
196	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	#823	1	4	yes	complemented
199	C₄×C₅	#895	1	2	yes	—
200	C₄×C₅	#940	1	1	yes	characteristic
201	C₄×C₅	#947	1	1	yes	characteristic
202	C₂⁴	—	3	4	no	—
203	C₂×D₄	—	6	4	no	—
204	C₂×D₄	—	6	4	no	—
205	C₂×D₄	—	6	2	no	—
206	C₂×D₄	—	6	4	no	—
207	C₂×D₄	—	6	4	no	—
208	C₂×D₄	—	6	2	no	—
209	C₂×D₄	—	12	8	no	—
210	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
211	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	3	4	no	—
214	M₁₆	—	6	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	6	2	no	—
217	C₄²	C₅×D₁₂	1	1	yes	characteristic
218	C₄²	C₅×D₁₂	1	1	yes	characteristic
219	C₄²	C₅×#67	1	2	yes	—
220	C₃×C₅	#1965	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
224	C₂×D₃	—	4	4	no	—
225	C₂×D₃	—	4	8	no	—
226	C₂×D₃	—	8	8	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	C₅×#100	1	4	yes	complemented
229	C₃×C₄	C₅×#121	1	2	yes	—
230	C₃×C₄	C₅×#124	1	1	yes	characteristic
231	C₃×C₄	C₅×#127	1	1	yes	characteristic
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	24	4	no	—
234	C₂×C₅	#4988	1	1	yes	characteristic
235	C₂×C₅	C₂×#823	1	2	yes	—
236	C₂³	—	6	4	no	—
237	C₂³	—	6	8	no	—
238	C₂³	—	12	8	no	—
239	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
240	D₄	—	12	8	no	—
241	D₄	—	12	4	no	—
242	D₄	—	12	2	no	—
243	D₄	—	12	2	no	—
244	D₄	—	12	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
247	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
249	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
250	C₂×C₄	C₅×#239	1	2	yes	—
251	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
252	C₈	—	4	2	no	—
253	C₈	—	6	4	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	C₅×#390	1	1	yes	characteristic
256	C₂×C₃	C₂×C₅×#100	1	2	yes	—
257	D₃	—	8	4	no	—
258	C₅	#70521	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	12	4	no	—
261	C₂²	—	12	8	no	—
262	C₂²	—	24	8	no	—
263	C₂²	C₅×#893	1	2	yes	—
264	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	C₅×#823	1	4	yes	complemented
267	C₄	C₅×#895	1	2	yes	—
268	C₄	C₅×#940	1	1	yes	characteristic
269	C₄	C₅×#947	1	1	yes	characteristic
270	C₃	C₅×#1965	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	2	2	no	—
272	C₂	—	24	4	no	—
273	C₂	C₅×#4988	1	1	yes	characteristic
274	C₂	C₂×C₅×#823	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅	C₂×(C₂×D₃)⋊C₄
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#874	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#423	D₃
hypocenter	C₃	C₅×#1965
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#423

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₂×C₅×#823	C₂	semidirect
4. C₄×C₅×#208	C₂	semidirect
18. #70521	C₅	direct (N × H)
20. C₅×#423	D₃	semidirect
39. C₅×#208	D₄	semidirect
42. C₃×#423	C₂×C₅	semidirect
43. C₂×#823	C₂×C₅	semidirect
44. C₄×#208	C₂×C₅	semidirect
107. #423	C₅×D₃	semidirect
117. C₃×C₄×C₅	#100	semidirect
137. #208	C₅×D₄	semidirect
198. C₄×C₅	#823	semidirect
220. C₃×C₅	#1965	semidirect
228. C₃×C₄	C₅×#100	semidirect
258. C₅	#70521	direct (N × H)
266. C₄	C₅×#823	semidirect
270. C₃	C₅×#1965	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#874	2
C₃×C₅×#423	2
C₂×C₅×#823	2
C₄×C₅×#208	2
C₅×#1965	3
#70521	5

Subgroups — C5×#70521 #14306116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70521 #14306116