Subgroups — C₅×#69576 #14299055

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#808	C₂	1	4	yes	maximal
3	C₅×#4935	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅×#1844	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
12	C₅×#808	—	2	8	no	—
13	C₄×C₅×C₃⋊C₈	C₄	1	4	yes	complemented
14	#69576	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#106	—	3	4	no	—
16	C₂×C₄×C₅×C₈	—	3	1	no	—
17	C₅×#384	—	3	1	no	—
18	C₅×#384	D₃	1	1	yes	characteristic
19	C₂²×C₃×C₄×C₅	—	2	2	no	—
20	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₅×Dic₃	—	2	2	no	—
23	C₃×C₅×C₄⋊C₄	—	2	4	no	—
24	C₃×C₅×C₄⋊C₄	—	4	4	no	—
25	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
26	C₃×C₄²×C₅	Q₈	1	2	yes	—
27	C₅×C₄⋊Dic₃	—	2	4	no	—
28	C₅×C₄⋊Dic₃	—	4	4	no	—
29	C₂×C₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	—
31	C₂×C₅×C₃⋊C₈	Q₈	1	4	yes	—
32	C₃×#384	C₂×C₅	1	1	yes	characteristic
33	C₂×#808	C₂×C₅	1	4	yes	—
34	#4935	C₂×C₅	1	1	yes	characteristic
35	C₂×C₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	3	2	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	3	2	no	—
39	C₂×C₅×C₄⋊C₄	—	6	2	no	—
40	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
41	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
42	C₅×#106	—	6	8	no	—
43	C₄×C₅×C₈	—	3	4	no	—
44	#1844	—	3	1	no	Sylow
45	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄×C₅	—	4	2	no	—
47	C₂×C₃×C₄×C₅	—	4	4	no	—
48	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
49	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
50	C₂×C₃×C₄×C₅	SD₁₆	1	4	yes	—
51	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
53	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₅×Dic₃	—	4	2	no	—
56	C₂×C₅×Dic₃	—	4	4	no	—
57	C₅×C₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
58	C₂×C₃×C₄⋊C₄	—	2	2	no	—
59	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
60	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
61	C₂×C₄⋊Dic₃	—	2	2	no	—
62	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
63	C₂²×C₃⋊C₈	C₂²×C₅	1	2	yes	—
64	#808	—	2	8	no	—
65	C₄×C₃⋊C₈	C₄×C₅	1	4	yes	complemented
66	C₂²×C₄×C₅	—	2	2	no	—
67	C₂²×C₄×C₅	—	6	2	no	—
68	C₂²×C₄×C₅	#67	1	2	yes	—
69	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₅×C₈	—	3	4	no	—
71	C₂×C₅×C₈	—	3	4	no	—
72	C₂×C₅×C₈	—	3	4	no	—
73	C₅×C₄⋊C₄	—	2	4	no	—
74	C₅×C₄⋊C₄	—	4	4	no	—
75	C₅×C₄⋊C₄	—	6	4	no	—
76	C₅×C₄⋊C₄	—	12	4	no	—
77	C₄²×C₅	Dic₆	1	2	yes	—
78	C₄²×C₅	C₄×D₃	1	2	yes	—
79	C₂×#106	—	3	4	no	—
80	C₂×C₄×C₈	—	3	1	no	—
81	#384	—	3	1	no	—
82	#384	C₅×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₃×C₅	#127	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
86	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
87	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
88	C₃×C₄×C₅	—	8	2	no	—
89	C₃×C₄×C₅	#106	1	8	yes	complemented
90	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
91	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
92	C₅×Dic₃	—	8	2	no	—
93	C₂²×C₃×C₄	—	2	2	no	—
94	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
95	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
96	C₂²×Dic₃	—	2	2	no	—
97	C₃×C₄⋊C₄	—	2	4	no	—
98	C₃×C₄⋊C₄	—	4	4	no	—
99	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
100	C₃×C₄²	C₅×Q₈	1	2	yes	—
101	C₄⋊Dic₃	—	2	4	no	—
102	C₄⋊Dic₃	—	4	4	no	—
103	C₂×C₃⋊C₈	C₂×C₄×C₅	1	4	yes	—
104	C₂×C₃⋊C₈	C₅×D₄	1	4	yes	—
105	C₂×C₃⋊C₈	C₅×Q₈	1	4	yes	—
106	C₂³×C₅	C₂×#67	1	1	yes	characteristic
107	C₂×C₄×C₅	—	4	4	no	—
108	C₂×C₄×C₅	—	4	2	no	—
109	C₂×C₄×C₅	—	12	2	no	—
110	C₂×C₄×C₅	—	12	4	no	—
111	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
112	C₂×C₄×C₅	#214	1	4	yes	—
113	C₂×C₄×C₅	#215	1	4	yes	—
114	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
117	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
118	C₅×C₈	—	3	8	no	—
119	C₂²×C₈	—	3	2	no	—
120	C₂×C₄⋊C₄	—	2	2	no	—
121	C₂×C₄⋊C₄	—	3	2	no	—
122	C₂×C₄⋊C₄	—	6	2	no	—
123	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
124	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
125	#106	—	6	8	no	—
126	C₄×C₈	—	3	4	no	—
127	C₂×C₃×C₅	#384	1	1	yes	characteristic
128	C₂×C₃×C₅	C₂×#106	1	2	yes	—
129	C₂×C₃×C₅	#492	1	1	yes	characteristic
130	C₂×C₃×C₅	#492	1	1	yes	characteristic
131	C₂×C₃×C₅	#498	1	2	yes	—
132	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
133	C₂×C₃×C₄	—	4	2	no	—
134	C₂×C₃×C₄	—	4	4	no	—
135	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
136	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
137	C₂×C₃×C₄	C₅×SD₁₆	1	4	yes	—
138	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
142	C₂×Dic₃	—	4	2	no	—
143	C₂×Dic₃	—	4	4	no	—
144	C₃⋊C₈	C₅×C₄⋊C₄	1	8	yes	complemented
145	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
146	C₂²×C₅	C₂×#214	1	2	yes	—
147	C₂²×C₅	#940	1	1	yes	characteristic
148	C₂²×C₅	C₂×#215	1	2	yes	—
149	C₂²×C₅	#944	1	1	yes	characteristic
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	—	24	2	no	—
152	C₄×C₅	#808	1	8	yes	complemented
153	C₄×C₅	#888	1	2	yes	—
154	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	—	6	2	no	—
157	C₂²×C₄	C₅×#67	1	2	yes	—
158	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
159	C₂×C₈	—	3	4	no	—
160	C₂×C₈	—	3	4	no	—
161	C₂×C₈	—	3	4	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	6	4	no	—
165	C₄⋊C₄	—	12	4	no	—
166	C₄²	C₅×Dic₆	1	2	yes	—
167	C₄²	C₄×C₅×D₃	1	2	yes	—
168	C₃×C₅	#1844	1	1	yes	characteristic, complemented, Hall
169	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
170	C₂²×C₃	C₅×#127	1	1	yes	characteristic
171	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
172	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
173	C₂²×C₃	C₂×C₅×SD₁₆	1	2	yes	—
174	C₃×C₄	—	8	2	no	—
175	C₃×C₄	C₅×#106	1	8	yes	complemented
176	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
177	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
178	Dic₃	—	8	2	no	—
179	C₂×C₅	C₂×#808	1	2	yes	—
180	C₂×C₅	#4975	1	1	yes	characteristic
181	C₂×C₅	#5076	1	1	yes	characteristic
182	C₂×C₅	#5080	1	2	yes	—
183	C₂×C₅	#5085	1	1	yes	characteristic
184	C₂³	C₂×C₅×#67	1	1	yes	characteristic
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	12	2	no	—
188	C₂×C₄	—	12	4	no	—
189	C₂×C₄	C₅×Dic₃⋊C₄	1	4	yes	—
190	C₂×C₄	C₅×#214	1	4	yes	—
191	C₂×C₄	C₅×#215	1	4	yes	—
192	C₂×C₄	C₂×C₅×Dic₆	1	1	yes	characteristic
193	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
194	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
195	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
196	C₈	—	3	8	no	—
197	C₂×C₃	C₅×#384	1	1	yes	characteristic
198	C₂×C₃	C₂×C₅×#106	1	2	yes	—
199	C₂×C₃	C₅×#492	1	1	yes	characteristic
200	C₂×C₃	C₅×#492	1	1	yes	characteristic
201	C₂×C₃	C₅×#498	1	2	yes	—
202	C₅	#69576	1	1	yes	characteristic, direct product factor, Sylow
203	C₂²	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
204	C₂²	C₂×C₅×#214	1	2	yes	—
205	C₂²	C₅×#940	1	1	yes	characteristic
206	C₂²	C₂×C₅×#215	1	2	yes	—
207	C₂²	C₅×#944	1	1	yes	characteristic
208	C₄	—	8	2	no	—
209	C₄	—	24	2	no	—
210	C₄	C₅×#808	1	8	yes	complemented
211	C₄	C₅×#888	1	2	yes	—
212	C₄	C₅×D₃×C₄⋊C₄	1	2	yes	—
213	C₃	C₅×#1844	1	1	yes	characteristic, complemented, Sylow
214	C₂	C₂×C₅×#808	1	2	yes	—
215	C₂	C₅×#4975	1	1	yes	characteristic
216	C₂	C₅×#5076	1	1	yes	characteristic
217	C₂	C₅×#5080	1	2	yes	—
218	C₂	C₅×#5085	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#384	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#384	D₃
hypocenter	C₃	C₅×#1844
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₅×#384

Complements

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

N	H ≅ G/N	type
13. C₄×C₅×C₃⋊C₈	C₄	semidirect
14. #69576	C₅	direct (N × H)
57. C₅×C₃⋊C₈	C₄⋊C₄	semidirect
65. C₄×C₃⋊C₈	C₄×C₅	semidirect
89. C₃×C₄×C₅	#106	semidirect
144. C₃⋊C₈	C₅×C₄⋊C₄	semidirect
152. C₄×C₅	#808	semidirect
168. C₃×C₅	#1844	semidirect
175. C₃×C₄	C₅×#106	semidirect
202. C₅	#69576	direct (N × H)
210. C₄	C₅×#808	semidirect
213. C₃	C₅×#1844	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#384	2
C₂×C₅×#808	2
C₅×#4935	2
C₂×C₄×C₅×C₃⋊C₈	2
C₅×#1844	3
#69576	5

Subgroups — C5×#69576 #14299055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69576 #14299055