Subgroups — C₅×#70662 #14305229

Order 1920.

416 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#819	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₅×#4999	C₂	1	2	yes	maximal
5	C₅×#4532	C₂	1	2	yes	maximal
6	C₅×#1945	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₅×#208	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	C₄	1	2	yes	complemented
13	C₃×C₅×#106	C₄	1	2	yes	complemented
14	C₅×#819	—	2	2	no	—
15	C₅×C₈×Dic₃	—	2	2	no	—
16	C₅×C₄⋊(C₃⋊C₈)	—	2	4	no	—
17	#70662	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#108	—	3	1	no	—
19	C₅×#427	D₃	1	1	yes	characteristic
20	C₅×#424	—	3	2	no	—
21	C₂×C₄×C₅×C₈	—	3	1	no	—
22	C₅×#339	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×Dic₃	—	2	1	no	—
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
29	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
31	C₃×C₄²×C₅	—	2	2	no	—
32	C₄×C₅×Dic₃	—	2	1	no	—
33	C₄×C₅×Dic₃	Q₈	1	2	yes	—
34	C₅×#208	—	2	2	no	—
35	C₅×#208	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	1	no	—
37	C₂×C₅×C₃⋊C₈	—	2	1	no	—
38	C₂×C₅×C₃⋊C₈	—	2	2	no	—
39	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
40	C₃×#427	C₂×C₅	1	1	yes	characteristic
41	C₂×#819	C₂×C₅	1	1	yes	characteristic
42	C₂×C₈×Dic₃	C₂×C₅	1	1	yes	characteristic
43	#4999	C₂×C₅	1	2	yes	—
44	#4532	C₂×C₅	1	2	yes	—
45	C₂×C₅×M₁₆	—	3	2	no	—
46	C₂²×C₅×C₈	—	3	1	no	—
47	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
48	C₅×#117	C₂×D₃	1	2	yes	—
49	C₂×C₄²×C₅	—	3	1	no	—
50	C₅×#108	—	6	2	no	—
51	C₅×#107	Dic₃	1	2	yes	complemented
52	C₅×#106	Dic₃	1	2	yes	complemented
53	C₅×C₄⋊C₈	—	6	4	no	—
54	C₄×C₅×C₈	—	6	2	no	—
55	#1945	—	3	1	no	Sylow
56	C₂³×C₃×C₅	#41	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
64	C₂×C₅×Dic₃	—	2	4	no	—
65	C₂×C₅×Dic₃	—	2	2	no	—
66	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
67	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
68	C₅×C₃⋊C₈	—	2	2	no	—
69	C₅×C₃⋊C₈	—	4	2	no	—
70	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₃×#117	C₂²×C₅	1	2	yes	—
72	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
73	C₂×#208	C₂²×C₅	1	2	yes	—
74	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×#107	C₄×C₅	1	2	yes	complemented
76	C₃×#106	C₄×C₅	1	2	yes	complemented
77	#819	—	2	2	no	—
78	C₈×Dic₃	—	2	2	no	—
79	C₄⋊(C₃⋊C₈)	—	2	4	no	—
80	C₂²×C₄×C₅	—	6	1	no	—
81	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
82	C₅×M₁₆	—	6	2	no	—
83	C₅×M₁₆	—	6	2	no	—
84	C₂×C₅×C₈	—	3	2	no	—
85	C₂×C₅×C₈	—	6	1	no	—
86	C₂×C₅×C₈	—	6	1	no	—
87	C₂×C₅×C₈	—	6	2	no	—
88	C₂×C₅×C₈	Dic₆	1	2	yes	—
89	C₂×C₅×C₈	D₁₂	1	2	yes	—
90	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
92	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₄²×C₅	—	2	2	no	—
95	C₄²×C₅	—	3	2	no	—
96	C₄²×C₅	—	6	1	no	—
97	C₂×#108	—	3	1	no	—
98	#427	C₅×D₃	1	1	yes	characteristic
99	#424	—	3	2	no	—
100	C₂×C₄×C₈	—	3	1	no	—
101	#339	—	3	2	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂²×C₃×C₅	#125	1	1	yes	characteristic
106	C₃×C₄×C₅	—	4	4	no	—
107	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
109	C₃×C₄×C₅	#121	1	1	yes	characteristic
110	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
111	C₅×Dic₃	—	2	4	no	—
112	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂²×Dic₃	—	2	1	no	—
114	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
117	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
118	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
119	C₃×C₂²⋊C₄	—	2	2	no	—
120	C₃×C₄²	—	2	2	no	—
121	C₄×Dic₃	—	2	1	no	—
122	C₄×Dic₃	C₅×Q₈	1	2	yes	—
123	#208	—	2	2	no	—
124	#208	—	2	2	no	—
125	C₂×C₃⋊C₈	—	2	1	no	—
126	C₂×C₃⋊C₈	—	2	1	no	—
127	C₂×C₃⋊C₈	—	2	2	no	—
128	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
129	C₂³×C₅	#237	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	6	4	no	—
132	C₂×C₄×C₅	—	6	2	no	—
133	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
135	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₅	#239	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
139	C₅×C₈	—	6	2	no	—
140	C₅×C₈	—	12	2	no	—
141	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
142	C₅×C₈	C₄⋊Dic₃	1	2	yes	—
143	C₂×M₁₆	—	3	2	no	—
144	C₂²×C₈	—	3	1	no	—
145	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
146	#117	C₂×C₅×D₃	1	2	yes	—
147	C₂×C₄²	—	3	1	no	—
148	#108	—	6	2	no	—
149	#107	C₅×Dic₃	1	2	yes	complemented
150	#106	C₅×Dic₃	1	2	yes	complemented
151	C₄⋊C₈	—	6	4	no	—
152	C₄×C₈	—	6	2	no	—
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	#389	1	1	yes	characteristic
156	C₂×C₃×C₅	#443	1	2	yes	—
157	C₂³×C₃	C₅×#41	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
160	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
165	C₂×Dic₃	—	2	4	no	—
166	C₂×Dic₃	—	2	2	no	—
167	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
168	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
169	C₃⋊C₈	—	2	2	no	—
170	C₃⋊C₈	—	4	2	no	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	#890	1	1	yes	characteristic
173	C₂²×C₅	#897	1	1	yes	characteristic
174	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
175	C₄×C₅	—	4	4	no	—
176	C₄×C₅	—	6	4	no	—
177	C₄×C₅	#888	1	1	yes	characteristic
178	C₄×C₅	#895	1	1	yes	characteristic
179	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
180	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
181	C₂²×C₄	—	6	1	no	—
182	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
183	M₁₆	—	6	2	no	—
184	M₁₆	—	6	2	no	—
185	C₂×C₈	—	3	2	no	—
186	C₂×C₈	—	6	1	no	—
187	C₂×C₈	—	6	1	no	—
188	C₂×C₈	—	6	2	no	—
189	C₂×C₈	C₅×Dic₆	1	2	yes	—
190	C₂×C₈	C₅×D₁₂	1	2	yes	—
191	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
192	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
193	C₄⋊C₄	C₂×C₅×Dic₃	1	4	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	3	2	no	—
197	C₄²	—	6	1	no	—
198	C₃×C₅	#1945	1	1	yes	characteristic, complemented, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂²×C₃	C₅×#125	1	1	yes	characteristic
203	C₃×C₄	—	4	4	no	—
204	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
205	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
206	C₃×C₄	C₅×#121	1	1	yes	characteristic
207	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
208	Dic₃	—	2	4	no	—
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	#4984	1	1	yes	characteristic
212	C₂×C₅	#5196	1	2	yes	—
213	C₂³	C₅×#237	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	6	4	no	—
216	C₂×C₄	—	6	2	no	—
217	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
218	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
219	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
220	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
221	C₂×C₄	C₅×#239	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
223	C₈	—	6	2	no	—
224	C₈	—	12	2	no	—
225	C₈	C₅×C₄⋊Dic₃	1	2	yes	—
226	C₈	C₅×C₄⋊Dic₃	1	2	yes	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	C₅×#389	1	1	yes	characteristic
230	C₂×C₃	C₅×#443	1	2	yes	—
231	C₅	#70662	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	C₅×#890	1	1	yes	characteristic
234	C₂²	C₅×#897	1	1	yes	characteristic
235	C₂²	C₅×D₄×Dic₃	1	1	yes	characteristic
236	C₄	—	4	4	no	—
237	C₄	—	6	4	no	—
238	C₄	C₅×#888	1	1	yes	characteristic
239	C₄	C₅×#895	1	1	yes	characteristic
240	C₄	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
241	C₄	C₅×Q₈×Dic₃	1	1	yes	characteristic
242	C₃	C₅×#1945	1	1	yes	characteristic, complemented, Sylow
243	C₂	—	2	1	no	—
244	C₂	—	2	1	no	—
245	C₂	C₅×#4984	1	1	yes	characteristic
246	C₂	C₅×#5196	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#427	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#427	D₃
hypocenter	C₃	C₅×#1945
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#427

Complements

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

N	H ≅ G/N	type
12. C₃×C₅×#107	C₄	semidirect
13. C₃×C₅×#106	C₄	semidirect
17. #70662	C₅	direct (N × H)
51. C₅×#107	Dic₃	semidirect
52. C₅×#106	Dic₃	semidirect
75. C₃×#107	C₄×C₅	semidirect
76. C₃×#106	C₄×C₅	semidirect
149. #107	C₅×Dic₃	semidirect
150. #106	C₅×Dic₃	semidirect
198. C₃×C₅	#1945	semidirect
231. C₅	#70662	direct (N × H)
242. C₃	C₅×#1945	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#427	2
C₂×C₅×#819	2
C₂×C₅×C₈×Dic₃	2
C₅×#4999	2
C₅×#4532	2
C₅×#1945	3
#70662	5

Subgroups — C5×#70662 #14305229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70662 #14305229