Subgroups — C₃×C₅×#2644 #14275678

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#397	C₂	1	6	yes	maximal, complemented
2	C₃×C₅×#396	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#390	C₂	1	3	yes	maximal
4	C₃×C₅×#389	C₂	1	3	yes	maximal, complemented
5	C₃×C₅×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2644	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
13	C₃×C₅×#95	C₂²	1	1	yes	characteristic
14	C₃×C₅×#95	C₂²	1	3	yes	—
15	C₃×C₅×#95	C₂²	1	6	yes	—
16	C₃×#2644	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#397	C₂×C₃	1	6	yes	complemented
18	C₅×#396	C₂×C₃	1	2	yes	complemented
19	C₅×#390	C₂×C₃	1	3	yes	—
20	C₅×#389	C₂×C₃	1	3	yes	complemented
21	C₅×#376	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
23	C₂×C₃×C₅×D₄	—	2	6	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₃×C₅×C₄⋊C₄	—	2	6	no	—
29	C₃×C₅×C₄⋊C₄	—	4	6	no	—
30	C₃×C₅×C₂²⋊C₄	—	4	6	no	—
31	C₃×C₅×C₂²⋊C₄	—	4	6	no	—
32	C₃×C₄²×C₅	—	2	6	no	—
33	C₃×#397	C₂×C₅	1	6	yes	complemented
34	C₃×#396	C₂×C₅	1	2	yes	complemented
35	C₃×#390	C₂×C₅	1	3	yes	—
36	C₃×#389	C₂×C₅	1	3	yes	complemented
37	C₃×#376	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂²×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
39	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
40	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
41	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
42	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
43	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
44	C₅×#95	C₂²×C₃	1	1	yes	characteristic
45	C₅×#95	C₂²×C₃	1	3	yes	—
46	C₅×#95	C₂²×C₃	1	6	yes	—
47	#2644	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂³×C₃×C₅	—	2	6	no	—
49	C₂³×C₃×C₅	—	2	6	no	—
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
52	C₃×C₅×D₄	—	4	4	no	—
53	C₂×C₃×C₄×C₅	—	2	3	no	—
54	C₂×C₃×C₄×C₅	—	2	3	no	—
55	C₂×C₃×C₄×C₅	—	2	6	no	—
56	C₂×C₃×C₄×C₅	—	2	3	no	—
57	C₂×C₃×C₄×C₅	—	2	3	no	—
58	C₂×C₃×C₄×C₅	—	2	6	no	—
59	C₂×C₃×C₄×C₅	—	2	6	no	—
60	C₂×C₃×C₄×C₅	—	2	3	no	—
61	C₂×C₃×C₄×C₅	—	2	3	no	—
62	C₂×C₃×C₄×C₅	#41	1	6	yes	—
63	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
65	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
66	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
67	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
68	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
69	C₃×#95	C₂²×C₅	1	1	yes	characteristic
70	C₃×#95	C₂²×C₅	1	3	yes	—
71	C₃×#95	C₂²×C₅	1	6	yes	—
72	C₂⁴×C₅	C₂³×C₃	1	2	yes	—
73	C₂×C₅×D₄	—	2	6	no	—
74	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
75	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
76	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
77	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
78	C₅×C₄⋊C₄	—	2	6	no	—
79	C₅×C₄⋊C₄	—	4	6	no	—
80	C₅×C₂²⋊C₄	—	4	6	no	—
81	C₅×C₂²⋊C₄	—	4	6	no	—
82	C₄²×C₅	—	2	6	no	—
83	#397	C₂×C₃×C₅	1	6	yes	complemented
84	#396	C₂×C₃×C₅	1	2	yes	complemented
85	#390	C₂×C₃×C₅	1	3	yes	—
86	#389	C₂×C₃×C₅	1	3	yes	complemented
87	#376	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	C₂²×C₃×C₅	—	4	2	no	—
89	C₂²×C₃×C₅	—	4	6	no	—
90	C₂²×C₃×C₅	—	4	6	no	—
91	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
92	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
93	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
94	C₃×C₄×C₅	—	4	3	no	—
95	C₃×C₄×C₅	—	4	6	no	—
96	C₃×C₄×C₅	—	4	3	no	—
97	C₃×C₄×C₅	#126	1	4	yes	—
98	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
99	C₂×C₃×D₄	—	2	6	no	—
100	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
101	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
102	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
103	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
104	C₃×C₄⋊C₄	—	2	6	no	—
105	C₃×C₄⋊C₄	—	4	6	no	—
106	C₃×C₂²⋊C₄	—	4	6	no	—
107	C₃×C₂²⋊C₄	—	4	6	no	—
108	C₃×C₄²	—	2	6	no	—
109	C₂³×C₅	—	2	6	no	—
110	C₂³×C₅	—	2	6	no	—
111	C₂³×C₅	—	2	2	no	—
112	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
113	C₅×D₄	—	4	4	no	—
114	C₂×C₄×C₅	—	2	3	no	—
115	C₂×C₄×C₅	—	2	3	no	—
116	C₂×C₄×C₅	—	2	6	no	—
117	C₂×C₄×C₅	—	2	3	no	—
118	C₂×C₄×C₅	—	2	3	no	—
119	C₂×C₄×C₅	—	2	6	no	—
120	C₂×C₄×C₅	—	2	6	no	—
121	C₂×C₄×C₅	—	2	3	no	—
122	C₂×C₄×C₅	—	2	3	no	—
123	C₂×C₄×C₅	C₃×#41	1	6	yes	—
124	C₂²×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
125	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
126	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
129	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
130	#95	C₂²×C₃×C₅	1	1	yes	characteristic
131	#95	C₂²×C₃×C₅	1	3	yes	—
132	#95	C₂²×C₃×C₅	1	6	yes	—
133	C₂×C₃×C₅	—	8	2	no	—
134	C₂×C₃×C₅	C₂×#126	1	1	yes	characteristic
135	C₂×C₃×C₅	#540	1	3	yes	—
136	C₂×C₃×C₅	#542	1	3	yes	—
137	C₂³×C₃	—	2	6	no	—
138	C₂³×C₃	—	2	6	no	—
139	C₂³×C₃	—	2	2	no	—
140	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
141	C₃×D₄	—	4	4	no	—
142	C₂×C₃×C₄	—	2	3	no	—
143	C₂×C₃×C₄	—	2	3	no	—
144	C₂×C₃×C₄	—	2	6	no	—
145	C₂×C₃×C₄	—	2	3	no	—
146	C₂×C₃×C₄	—	2	3	no	—
147	C₂×C₃×C₄	—	2	6	no	—
148	C₂×C₃×C₄	—	2	6	no	—
149	C₂×C₃×C₄	—	2	3	no	—
150	C₂×C₃×C₄	—	2	3	no	—
151	C₂×C₃×C₄	C₅×#41	1	6	yes	—
152	C₂²×C₅	—	4	2	no	—
153	C₂²×C₅	—	4	6	no	—
154	C₂²×C₅	—	4	6	no	—
155	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
156	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
157	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
158	C₄×C₅	—	4	3	no	—
159	C₄×C₅	—	4	6	no	—
160	C₄×C₅	—	4	3	no	—
161	C₄×C₅	C₃×#126	1	4	yes	—
162	C₂⁴	C₂³×C₃×C₅	1	2	yes	—
163	C₂×D₄	—	2	6	no	—
164	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
165	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
166	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
167	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
168	C₄⋊C₄	—	2	6	no	—
169	C₄⋊C₄	—	4	6	no	—
170	C₂²⋊C₄	—	4	6	no	—
171	C₂²⋊C₄	—	4	6	no	—
172	C₄²	—	2	6	no	—
173	C₃×C₅	#2644	1	1	yes	characteristic, direct product factor, Hall
174	C₂²×C₃	—	4	2	no	—
175	C₂²×C₃	—	4	6	no	—
176	C₂²×C₃	—	4	6	no	—
177	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
178	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
179	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
180	C₃×C₄	—	4	3	no	—
181	C₃×C₄	—	4	6	no	—
182	C₃×C₄	—	4	3	no	—
183	C₃×C₄	C₅×#126	1	4	yes	—
184	C₂×C₅	—	8	2	no	—
185	C₂×C₅	C₂×C₃×#126	1	1	yes	characteristic
186	C₂×C₅	C₃×#540	1	3	yes	—
187	C₂×C₅	C₃×#542	1	3	yes	—
188	C₂³	—	2	6	no	—
189	C₂³	—	2	6	no	—
190	C₂³	—	2	2	no	—
191	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
192	D₄	—	4	4	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	6	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	6	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	C₃×C₅×#41	1	6	yes	—
203	C₂×C₃	—	8	2	no	—
204	C₂×C₃	C₂×C₅×#126	1	1	yes	characteristic
205	C₂×C₃	C₅×#540	1	3	yes	—
206	C₂×C₃	C₅×#542	1	3	yes	—
207	C₅	C₃×#2644	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	4	2	no	—
209	C₂²	—	4	6	no	—
210	C₂²	—	4	6	no	—
211	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
212	C₂²	C₃×C₅×H(5,𝔽₂)	1	3	yes	—
213	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
214	C₄	—	4	3	no	—
215	C₄	—	4	6	no	—
216	C₄	—	4	3	no	—
217	C₄	C₃×C₅×#126	1	4	yes	—
218	C₃	C₅×#2644	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	8	2	no	—
220	C₂	C₂×C₃×C₅×#126	1	1	yes	characteristic
221	C₂	C₃×C₅×#540	1	3	yes	—
222	C₂	C₃×C₅×#542	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂²×C₃×C₅×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#397	C₂	semidirect
2. C₃×C₅×#396	C₂	semidirect
4. C₃×C₅×#389	C₂	semidirect
5. C₃×C₅×#376	C₂	semidirect
6. C₅×#2644	C₃	direct (N × H)
16. C₃×#2644	C₅	direct (N × H)
17. C₅×#397	C₂×C₃	semidirect
18. C₅×#396	C₂×C₃	semidirect
20. C₅×#389	C₂×C₃	semidirect
21. C₅×#376	C₂×C₃	semidirect
33. C₃×#397	C₂×C₅	semidirect
34. C₃×#396	C₂×C₅	semidirect
36. C₃×#389	C₂×C₅	semidirect
37. C₃×#376	C₂×C₅	semidirect
47. #2644	C₃×C₅	direct (N × H)
83. #397	C₂×C₃×C₅	semidirect
84. #396	C₂×C₃×C₅	semidirect
86. #389	C₂×C₃×C₅	semidirect
87. #376	C₂×C₃×C₅	semidirect
173. C₃×C₅	#2644	direct (N × H)
207. C₅	C₃×#2644	direct (N × H)
218. C₃	C₅×#2644	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#397	2
C₃×C₅×#396	2
C₃×C₅×#390	2
C₃×C₅×#389	2
C₃×C₅×#376	2
C₅×#2644	3
C₃×#2644	5

Subgroups — C3×C5×#2644 #14275678

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2644 #14275678