Subgroups — C₃×C₅×#2736 #14275770

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 256 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	3	yes	maximal, complemented
2	C₃×C₅×#393	C₂	1	3	yes	maximal, complemented
3	C₃×C₅×#388	C₂	1	6	yes	maximal, complemented
4	C₃×C₅×#379	C₂	1	3	yes	maximal
5	C₅×#2736	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂⁵×C₃×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	complemented
9	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
10	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	—
12	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	3	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	complemented
14	C₃×C₅×#95	C₂²	1	3	yes	complemented
15	C₃×C₅×#95	C₂²	1	6	yes	complemented
16	C₃×C₅×#95	C₂²	1	1	yes	characteristic, complemented
17	C₃×#2736	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#397	C₂×C₃	1	3	yes	complemented
19	C₅×#393	C₂×C₃	1	3	yes	complemented
20	C₅×#388	C₂×C₃	1	6	yes	complemented
21	C₅×#379	C₂×C₃	1	3	yes	—
22	C₂⁴×C₃×C₅	—	2	6	no	—
23	C₂⁴×C₃×C₅	—	4	3	no	—
24	C₂⁴×C₃×C₅	—	4	1	no	—
25	C₂⁴×C₃×C₅	C₂³	1	3	yes	—
26	C₂×C₃×C₅×Q₈	—	4	3	no	—
27	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
30	C₃×C₅×C₄⋊C₄	—	4	3	no	—
31	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
32	C₃×C₅×C₂²⋊C₄	—	4	3	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	6	no	—
34	C₃×C₅×C₂²⋊C₄	—	4	3	no	—
35	C₃×C₄²×C₅	—	4	2	no	—
36	C₃×#397	C₂×C₅	1	3	yes	complemented
37	C₃×#393	C₂×C₅	1	3	yes	complemented
38	C₃×#388	C₂×C₅	1	6	yes	complemented
39	C₃×#379	C₂×C₅	1	3	yes	—
40	C₂⁵×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic, complemented
42	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	complemented
43	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
44	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
45	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	—
46	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	3	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	complemented
48	C₅×#95	C₂²×C₃	1	3	yes	complemented
49	C₅×#95	C₂²×C₃	1	6	yes	complemented
50	C₅×#95	C₂²×C₃	1	1	yes	characteristic, complemented
51	#2736	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	—	2	6	no	—
53	C₂³×C₃×C₅	—	2	3	no	—
54	C₂³×C₃×C₅	—	2	3	no	—
55	C₂³×C₃×C₅	—	2	6	no	—
56	C₂³×C₃×C₅	—	4	12	no	—
57	C₂³×C₃×C₅	—	4	12	no	—
58	C₂³×C₃×C₅	—	4	4	no	—
59	C₂³×C₃×C₅	#41	1	6	yes	—
60	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
61	C₃×C₅×Q₈	—	4	4	no	—
62	C₂×C₃×C₄×C₅	—	2	6	no	—
63	C₂×C₃×C₄×C₅	—	2	6	no	—
64	C₂×C₃×C₄×C₅	—	2	6	no	—
65	C₂×C₃×C₄×C₅	—	2	6	no	—
66	C₂×C₃×C₄×C₅	—	2	3	no	—
67	C₂×C₃×C₄×C₅	—	2	6	no	—
68	C₂×C₃×C₄×C₅	—	2	3	no	—
69	C₂⁵×C₃	C₂²×C₅	1	1	yes	characteristic
70	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic, complemented
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	complemented
72	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
73	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
74	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	—
75	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	3	yes	—
76	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	complemented
77	C₃×#95	C₂²×C₅	1	3	yes	complemented
78	C₃×#95	C₂²×C₅	1	6	yes	complemented
79	C₃×#95	C₂²×C₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	—	2	6	no	—
81	C₂⁴×C₅	—	4	3	no	—
82	C₂⁴×C₅	—	4	1	no	—
83	C₂⁴×C₅	C₂³×C₃	1	3	yes	—
84	C₂×C₅×Q₈	—	4	3	no	—
85	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
87	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
88	C₅×C₄⋊C₄	—	4	3	no	—
89	C₅×C₂²⋊C₄	—	2	12	no	—
90	C₅×C₂²⋊C₄	—	4	3	no	—
91	C₅×C₂²⋊C₄	—	4	6	no	—
92	C₅×C₂²⋊C₄	—	4	3	no	—
93	C₄²×C₅	—	4	2	no	—
94	#397	C₂×C₃×C₅	1	3	yes	complemented
95	#393	C₂×C₃×C₅	1	3	yes	complemented
96	#388	C₂×C₃×C₅	1	6	yes	complemented
97	#379	C₂×C₃×C₅	1	3	yes	—
98	C₂²×C₃×C₅	—	2	6	no	—
99	C₂²×C₃×C₅	—	2	12	no	—
100	C₂²×C₃×C₅	—	4	16	no	—
101	C₂²×C₃×C₅	—	4	6	no	—
102	C₂²×C₃×C₅	—	4	3	no	—
103	C₂²×C₃×C₅	—	4	3	no	—
104	C₂²×C₃×C₅	C₂×#41	1	3	yes	—
105	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
107	C₃×C₄×C₅	—	4	6	no	—
108	C₃×C₄×C₅	—	4	3	no	—
109	C₃×C₄×C₅	—	4	3	no	—
110	C₂⁴×C₃	—	2	6	no	—
111	C₂⁴×C₃	—	4	3	no	—
112	C₂⁴×C₃	—	4	1	no	—
113	C₂⁴×C₃	C₂³×C₅	1	3	yes	—
114	C₂×C₃×Q₈	—	4	3	no	—
115	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
116	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
117	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
118	C₃×C₄⋊C₄	—	4	3	no	—
119	C₃×C₂²⋊C₄	—	2	12	no	—
120	C₃×C₂²⋊C₄	—	4	3	no	—
121	C₃×C₂²⋊C₄	—	4	6	no	—
122	C₃×C₂²⋊C₄	—	4	3	no	—
123	C₃×C₄²	—	4	2	no	—
124	C₂³×C₅	—	2	6	no	—
125	C₂³×C₅	—	2	3	no	—
126	C₂³×C₅	—	2	3	no	—
127	C₂³×C₅	—	2	6	no	—
128	C₂³×C₅	—	4	12	no	—
129	C₂³×C₅	—	4	12	no	—
130	C₂³×C₅	—	4	4	no	—
131	C₂³×C₅	C₃×#41	1	6	yes	—
132	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
133	C₅×Q₈	—	4	4	no	—
134	C₂×C₄×C₅	—	2	6	no	—
135	C₂×C₄×C₅	—	2	6	no	—
136	C₂×C₄×C₅	—	2	6	no	—
137	C₂×C₄×C₅	—	2	6	no	—
138	C₂×C₄×C₅	—	2	3	no	—
139	C₂×C₄×C₅	—	2	6	no	—
140	C₂×C₄×C₅	—	2	3	no	—
141	C₂⁵	C₂²×C₃×C₅	1	1	yes	characteristic
142	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
143	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	complemented
144	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
145	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
146	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	—
147	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	3	yes	—
148	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	complemented
149	#95	C₂²×C₃×C₅	1	3	yes	complemented
150	#95	C₂²×C₃×C₅	1	6	yes	complemented
151	#95	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
152	C₂×C₃×C₅	—	4	6	no	—
153	C₂×C₃×C₅	#538	1	3	yes	—
154	C₂×C₃×C₅	#551	1	3	yes	—
155	C₂×C₃×C₅	H(3,𝔽₄)	1	1	yes	characteristic
156	C₂³×C₃	—	2	6	no	—
157	C₂³×C₃	—	2	3	no	—
158	C₂³×C₃	—	2	3	no	—
159	C₂³×C₃	—	2	6	no	—
160	C₂³×C₃	—	4	12	no	—
161	C₂³×C₃	—	4	12	no	—
162	C₂³×C₃	—	4	4	no	—
163	C₂³×C₃	C₅×#41	1	6	yes	—
164	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
165	C₃×Q₈	—	4	4	no	—
166	C₂×C₃×C₄	—	2	6	no	—
167	C₂×C₃×C₄	—	2	6	no	—
168	C₂×C₃×C₄	—	2	6	no	—
169	C₂×C₃×C₄	—	2	6	no	—
170	C₂×C₃×C₄	—	2	3	no	—
171	C₂×C₃×C₄	—	2	6	no	—
172	C₂×C₃×C₄	—	2	3	no	—
173	C₂²×C₅	—	2	6	no	—
174	C₂²×C₅	—	2	12	no	—
175	C₂²×C₅	—	4	16	no	—
176	C₂²×C₅	—	4	3	no	—
177	C₂²×C₅	—	4	6	no	—
178	C₂²×C₅	—	4	3	no	—
179	C₂²×C₅	C₂×C₃×#41	1	3	yes	—
180	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
181	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
182	C₄×C₅	—	4	3	no	—
183	C₄×C₅	—	4	6	no	—
184	C₄×C₅	—	4	3	no	—
185	C₂⁴	—	2	6	no	—
186	C₂⁴	—	4	3	no	—
187	C₂⁴	—	4	1	no	—
188	C₂⁴	C₂³×C₃×C₅	1	3	yes	—
189	C₂×Q₈	—	4	3	no	—
190	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
191	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
192	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
193	C₄⋊C₄	—	4	3	no	—
194	C₂²⋊C₄	—	2	12	no	—
195	C₂²⋊C₄	—	4	3	no	—
196	C₂²⋊C₄	—	4	6	no	—
197	C₂²⋊C₄	—	4	3	no	—
198	C₄²	—	4	2	no	—
199	C₃×C₅	#2736	1	1	yes	characteristic, direct product factor, Hall
200	C₂²×C₃	—	2	6	no	—
201	C₂²×C₃	—	2	12	no	—
202	C₂²×C₃	—	4	16	no	—
203	C₂²×C₃	—	4	6	no	—
204	C₂²×C₃	—	4	3	no	—
205	C₂²×C₃	—	4	3	no	—
206	C₂²×C₃	C₂×C₅×#41	1	3	yes	—
207	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
208	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
209	C₃×C₄	—	4	6	no	—
210	C₃×C₄	—	4	3	no	—
211	C₃×C₄	—	4	3	no	—
212	C₂×C₅	—	4	6	no	—
213	C₂×C₅	C₃×#538	1	3	yes	—
214	C₂×C₅	C₃×#551	1	3	yes	—
215	C₂×C₅	C₃×H(3,𝔽₄)	1	1	yes	characteristic
216	C₂³	—	2	6	no	—
217	C₂³	—	2	3	no	—
218	C₂³	—	2	3	no	—
219	C₂³	—	2	6	no	—
220	C₂³	—	4	12	no	—
221	C₂³	—	4	12	no	—
222	C₂³	—	4	4	no	—
223	C₂³	C₃×C₅×#41	1	6	yes	—
224	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
225	Q₈	—	4	4	no	—
226	C₂×C₄	—	2	6	no	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	6	no	—
229	C₂×C₄	—	2	6	no	—
230	C₂×C₄	—	2	3	no	—
231	C₂×C₄	—	2	6	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₃	—	4	6	no	—
234	C₂×C₃	C₅×#538	1	3	yes	—
235	C₂×C₃	C₅×#551	1	3	yes	—
236	C₂×C₃	C₅×H(3,𝔽₄)	1	1	yes	characteristic
237	C₅	C₃×#2736	1	1	yes	characteristic, direct product factor, Sylow
238	C₂²	—	2	6	no	—
239	C₂²	—	2	12	no	—
240	C₂²	—	4	16	no	—
241	C₂²	—	4	6	no	—
242	C₂²	—	4	3	no	—
243	C₂²	—	4	3	no	—
244	C₂²	C₂×C₃×C₅×#41	1	3	yes	—
245	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
246	C₂²	C₃×C₅×H(5,𝔽₂)	1	3	yes	—
247	C₄	—	4	6	no	—
248	C₄	—	4	3	no	—
249	C₄	—	4	3	no	—
250	C₃	C₅×#2736	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	6	no	—
252	C₂	C₃×C₅×#538	1	3	yes	—
253	C₂	C₃×C₅×#551	1	3	yes	—
254	C₂	C₃×C₅×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁵×C₃×C₅	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×#393	C₂	semidirect
3. C₃×C₅×#388	C₂	semidirect
5. C₅×#2736	C₃	direct (N × H)
7. C₂²×C₃×C₅×Q₈	C₂²	semidirect
8. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
13. C₂×C₃×C₄²×C₅	C₂²	semidirect
14. C₃×C₅×#95	C₂²	semidirect
15. C₃×C₅×#95	C₂²	semidirect
16. C₃×C₅×#95	C₂²	semidirect
17. C₃×#2736	C₅	direct (N × H)
18. C₅×#397	C₂×C₃	semidirect
19. C₅×#393	C₂×C₃	semidirect
20. C₅×#388	C₂×C₃	semidirect
36. C₃×#397	C₂×C₅	semidirect
37. C₃×#393	C₂×C₅	semidirect
38. C₃×#388	C₂×C₅	semidirect
41. C₂²×C₅×Q₈	C₂²×C₃	semidirect
42. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
47. C₂×C₄²×C₅	C₂²×C₃	semidirect
48. C₅×#95	C₂²×C₃	semidirect
49. C₅×#95	C₂²×C₃	semidirect
50. C₅×#95	C₂²×C₃	semidirect
51. #2736	C₃×C₅	direct (N × H)
70. C₂²×C₃×Q₈	C₂²×C₅	semidirect
71. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
76. C₂×C₃×C₄²	C₂²×C₅	semidirect
77. C₃×#95	C₂²×C₅	semidirect
78. C₃×#95	C₂²×C₅	semidirect
79. C₃×#95	C₂²×C₅	semidirect
94. #397	C₂×C₃×C₅	semidirect
95. #393	C₂×C₃×C₅	semidirect
96. #388	C₂×C₃×C₅	semidirect
142. C₂²×Q₈	C₂²×C₃×C₅	semidirect
143. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
148. C₂×C₄²	C₂²×C₃×C₅	semidirect
149. #95	C₂²×C₃×C₅	semidirect
150. #95	C₂²×C₃×C₅	semidirect
151. #95	C₂²×C₃×C₅	semidirect
199. C₃×C₅	#2736	direct (N × H)
237. C₅	C₃×#2736	direct (N × H)
250. C₃	C₅×#2736	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×#393	2
C₃×C₅×#388	2
C₃×C₅×#379	2
C₅×#2736	3
C₃×#2736	5

Subgroups — C3×C5×#2736 #14275770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2736 #14275770