Subgroups — C₃×C₅×#2371 #14275405

Order 1920.

896 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	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#387	C₂	1	4	yes	maximal
3	C₃×C₅×#387	C₂	1	4	yes	maximal
4	C₃×C₄×C₅×C₄⋊C₄	C₂	1	4	yes	maximal
5	C₃×C₅×#376	C₂	1	2	yes	maximal
6	C₅×#2371	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#125	C₄	1	8	yes	complemented
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×#95	C₂²	1	2	yes	—
15	C₃×C₅×#95	C₂²	1	2	yes	—
16	C₃×C₅×#95	C₂²	1	4	yes	—
17	C₃×C₅×#95	C₂²	1	4	yes	—
18	C₃×C₅×#95	C₂²	1	4	yes	—
19	C₃×#2371	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
21	C₅×#387	C₂×C₃	1	4	yes	—
22	C₅×#387	C₂×C₃	1	4	yes	—
23	C₄×C₅×C₄⋊C₄	C₂×C₃	1	4	yes	—
24	C₅×#376	C₂×C₃	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	8	no	—
31	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
32	C₃×C₅×C₄⋊C₄	C₂×C₄	1	8	yes	—
33	C₃×C₄²×C₅	—	2	8	no	—
34	C₃×C₄²×C₅	—	2	8	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
36	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
37	C₃×#387	C₂×C₅	1	4	yes	—
38	C₃×#387	C₂×C₅	1	4	yes	—
39	C₃×C₄×C₄⋊C₄	C₂×C₅	1	4	yes	—
40	C₃×#376	C₂×C₅	1	2	yes	—
41	C₅×#125	C₃×C₄	1	8	yes	complemented
42	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
46	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₅×#95	C₂²×C₃	1	2	yes	—
49	C₅×#95	C₂²×C₃	1	2	yes	—
50	C₅×#95	C₂²×C₃	1	4	yes	—
51	C₅×#95	C₂²×C₃	1	4	yes	—
52	C₅×#95	C₂²×C₃	1	4	yes	—
53	#2371	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	4	no	—
56	C₂×C₃×C₄×C₅	—	2	4	no	—
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	—	2	4	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
67	C₂×C₃×C₄×C₅	#41	1	8	yes	—
68	C₃×#125	C₄×C₅	1	8	yes	complemented
69	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
70	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
72	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
73	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₃×#95	C₂²×C₅	1	2	yes	—
76	C₃×#95	C₂²×C₅	1	2	yes	—
77	C₃×#95	C₂²×C₅	1	4	yes	—
78	C₃×#95	C₂²×C₅	1	4	yes	—
79	C₃×#95	C₂²×C₅	1	4	yes	—
80	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
81	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
82	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
83	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
85	C₅×C₄⋊C₄	—	2	8	no	—
86	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
87	C₅×C₄⋊C₄	C₂×C₃×C₄	1	8	yes	—
88	C₄²×C₅	—	2	8	no	—
89	C₄²×C₅	—	2	8	no	—
90	C₄²×C₅	C₂×C₃×C₄	1	4	yes	—
91	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
92	#387	C₂×C₃×C₅	1	4	yes	—
93	#387	C₂×C₃×C₅	1	4	yes	—
94	C₄×C₄⋊C₄	C₂×C₃×C₅	1	4	yes	—
95	#376	C₂×C₃×C₅	1	2	yes	—
96	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
98	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
99	C₃×C₄×C₅	—	2	4	no	—
100	C₃×C₄×C₅	—	2	8	no	—
101	C₃×C₄×C₅	—	2	2	no	—
102	C₃×C₄×C₅	—	2	8	no	—
103	C₃×C₄×C₅	—	4	4	no	—
104	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
105	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
107	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
108	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
109	C₃×C₄⋊C₄	—	2	8	no	—
110	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
111	C₃×C₄⋊C₄	C₂×C₄×C₅	1	8	yes	—
112	C₃×C₄²	—	2	8	no	—
113	C₃×C₄²	—	2	8	no	—
114	C₃×C₄²	C₂×C₄×C₅	1	4	yes	—
115	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	4	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	4	no	—
120	C₂×C₄×C₅	—	2	2	no	—
121	C₂×C₄×C₅	—	2	4	no	—
122	C₂×C₄×C₅	—	2	4	no	—
123	C₂×C₄×C₅	—	2	4	no	—
124	C₂×C₄×C₅	—	2	4	no	—
125	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
126	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
127	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
128	C₂×C₄×C₅	C₃×#41	1	8	yes	—
129	#125	C₃×C₄×C₅	1	8	yes	complemented
130	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
131	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
132	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
133	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
134	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
135	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
136	#95	C₂²×C₃×C₅	1	2	yes	—
137	#95	C₂²×C₃×C₅	1	2	yes	—
138	#95	C₂²×C₃×C₅	1	4	yes	—
139	#95	C₂²×C₃×C₅	1	4	yes	—
140	#95	C₂²×C₃×C₅	1	4	yes	—
141	C₂×C₃×C₅	C₄×#41	1	2	yes	—
142	C₂×C₃×C₅	#519	1	1	yes	characteristic
143	C₂×C₃×C₅	#552	1	4	yes	—
144	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
145	C₂×C₃×C₄	—	2	4	no	—
146	C₂×C₃×C₄	—	2	4	no	—
147	C₂×C₃×C₄	—	2	4	no	—
148	C₂×C₃×C₄	—	2	4	no	—
149	C₂×C₃×C₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	4	no	—
151	C₂×C₃×C₄	—	2	4	no	—
152	C₂×C₃×C₄	—	2	4	no	—
153	C₂×C₃×C₄	—	2	4	no	—
154	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
155	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
156	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
157	C₂×C₃×C₄	C₅×#41	1	8	yes	—
158	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
159	C₂²×C₅	C₂×C₃×#41	1	4	yes	—
160	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
161	C₄×C₅	—	2	4	no	—
162	C₄×C₅	—	2	8	no	—
163	C₄×C₅	—	2	2	no	—
164	C₄×C₅	—	2	8	no	—
165	C₄×C₅	—	4	4	no	—
166	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
167	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
169	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
170	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
171	C₄⋊C₄	—	2	8	no	—
172	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
173	C₄⋊C₄	C₂×C₃×C₄×C₅	1	8	yes	—
174	C₄²	—	2	8	no	—
175	C₄²	—	2	8	no	—
176	C₄²	C₂×C₃×C₄×C₅	1	4	yes	—
177	C₃×C₅	#2371	1	1	yes	characteristic, direct product factor, Hall
178	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
179	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
180	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
181	C₃×C₄	—	2	4	no	—
182	C₃×C₄	—	2	8	no	—
183	C₃×C₄	—	2	2	no	—
184	C₃×C₄	—	2	8	no	—
185	C₃×C₄	—	4	4	no	—
186	C₂×C₅	C₃×C₄×#41	1	2	yes	—
187	C₂×C₅	C₃×#519	1	1	yes	characteristic
188	C₂×C₅	C₃×#552	1	4	yes	—
189	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
200	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
201	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
202	C₂×C₄	C₃×C₅×#41	1	8	yes	—
203	C₂×C₃	C₄×C₅×#41	1	2	yes	—
204	C₂×C₃	C₅×#519	1	1	yes	characteristic
205	C₂×C₃	C₅×#552	1	4	yes	—
206	C₅	C₃×#2371	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
208	C₂²	C₂×C₃×C₅×#41	1	4	yes	—
209	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
210	C₄	—	2	4	no	—
211	C₄	—	2	8	no	—
212	C₄	—	2	2	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	4	no	—
215	C₃	C₅×#2371	1	1	yes	characteristic, direct product factor, Sylow
216	C₂	C₃×C₄×C₅×#41	1	2	yes	—
217	C₂	C₃×C₅×#519	1	1	yes	characteristic
218	C₂	C₃×C₅×#552	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2371	C₃	direct (N × H)
7. C₃×C₅×#125	C₄	semidirect
19. C₃×#2371	C₅	direct (N × H)
41. C₅×#125	C₃×C₄	semidirect
53. #2371	C₃×C₅	direct (N × H)
68. C₃×#125	C₄×C₅	semidirect
129. #125	C₃×C₄×C₅	semidirect
177. C₃×C₅	#2371	direct (N × H)
206. C₅	C₃×#2371	direct (N × H)
215. C₃	C₅×#2371	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#125	2
C₃×C₅×#387	2
C₃×C₄×C₅×C₄⋊C₄	2
C₃×C₅×#376	2
C₅×#2371	3
C₃×#2371	5

Subgroups — C3×C5×#2371 #14275405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2371 #14275405