Subgroups — C₃×C₅×#2656 #14275690

Order 1920.

1608 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#121	C₂	1	6	yes	maximal, complemented
2	C₃×C₅×#399	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#392	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×C₅×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2656	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
7	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
8	C₂³×C₃×C₄×C₅	C₂²	1	3	yes	—
9	C₃×C₅×#121	—	2	24	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	—
12	C₃×C₅×#95	C₂²	1	4	yes	—
13	C₃×#2656	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₅×#121	C₂×C₃	1	6	yes	complemented
15	C₅×#399	C₂×C₃	1	4	yes	complemented
16	C₅×#392	C₂×C₃	1	4	yes	complemented
17	C₂×C₅×#95	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂⁴×C₃×C₅	C₂³	1	4	yes	—
19	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
20	C₂×C₃×C₅×D₄	—	2	24	no	—
21	C₂×C₃×C₅×D₄	—	2	12	no	—
22	C₂×C₃×C₅×D₄	—	4	12	no	—
23	C₂²×C₃×C₄×C₅	—	4	6	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	12	yes	complemented
25	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
26	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
27	C₃×C₅×C₄⋊C₄	—	2	12	no	—
28	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
29	C₂×C₃×#121	C₂×C₅	1	6	yes	complemented
30	C₃×#399	C₂×C₅	1	4	yes	complemented
31	C₃×#392	C₂×C₅	1	4	yes	complemented
32	C₂×C₃×#95	C₂×C₅	1	1	yes	characteristic, complemented
33	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
34	C₂²×C₅×D₄	C₂²×C₃	1	6	yes	—
35	C₂³×C₄×C₅	C₂²×C₃	1	3	yes	—
36	C₅×#121	—	2	24	no	—
37	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
38	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	—
39	C₅×#95	C₂²×C₃	1	4	yes	—
40	#2656	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
41	C₂³×C₃×C₅	—	2	3	no	—
42	C₂³×C₃×C₅	—	2	1	no	—
43	C₂³×C₃×C₅	—	2	12	no	—
44	C₂³×C₃×C₅	—	2	12	no	—
45	C₂³×C₃×C₅	—	2	4	no	—
46	C₂³×C₃×C₅	C₂×D₄	1	6	yes	—
47	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
48	C₃×C₅×D₄	—	4	24	no	—
49	C₃×C₅×D₄	—	4	16	no	—
50	C₂×C₃×C₄×C₅	—	2	12	no	—
51	C₂×C₃×C₄×C₅	—	2	12	no	—
52	C₂×C₃×C₄×C₅	—	4	12	no	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	12	yes	—
54	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
55	C₂²×C₃×D₄	C₂²×C₅	1	6	yes	—
56	C₂³×C₃×C₄	C₂²×C₅	1	3	yes	—
57	C₃×#121	—	2	24	no	—
58	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
59	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	—
60	C₃×#95	C₂²×C₅	1	4	yes	—
61	C₂⁴×C₅	C₂³×C₃	1	4	yes	—
62	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
63	C₂×C₅×D₄	—	2	24	no	—
64	C₂×C₅×D₄	—	2	12	no	—
65	C₂×C₅×D₄	—	4	12	no	—
66	C₂²×C₄×C₅	—	4	6	no	—
67	C₂²×C₄×C₅	C₃×D₄	1	12	yes	complemented
68	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
69	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
70	C₅×C₄⋊C₄	—	2	12	no	—
71	C₅×C₂²⋊C₄	—	2	24	no	—
72	C₂×#121	C₂×C₃×C₅	1	6	yes	complemented
73	#399	C₂×C₃×C₅	1	4	yes	complemented
74	#392	C₂×C₃×C₅	1	4	yes	complemented
75	C₂×#95	C₂×C₃×C₅	1	1	yes	characteristic, complemented
76	C₂²×C₃×C₅	—	2	6	no	—
77	C₂²×C₃×C₅	—	2	6	no	—
78	C₂²×C₃×C₅	—	4	4	no	—
79	C₂²×C₃×C₅	—	4	12	no	—
80	C₂²×C₃×C₅	—	4	12	no	—
81	C₂²×C₃×C₅	#127	1	4	yes	—
82	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
83	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
84	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
85	C₃×C₄×C₅	—	2	12	no	—
86	C₃×C₄×C₅	—	4	4	no	—
87	C₂⁴×C₃	C₂³×C₅	1	4	yes	—
88	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
89	C₂×C₃×D₄	—	2	24	no	—
90	C₂×C₃×D₄	—	2	12	no	—
91	C₂×C₃×D₄	—	4	12	no	—
92	C₂²×C₃×C₄	—	4	6	no	—
93	C₂²×C₃×C₄	C₅×D₄	1	12	yes	complemented
94	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
95	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
96	C₃×C₄⋊C₄	—	2	12	no	—
97	C₃×C₂²⋊C₄	—	2	24	no	—
98	C₂³×C₅	—	2	3	no	—
99	C₂³×C₅	—	2	1	no	—
100	C₂³×C₅	—	2	12	no	—
101	C₂³×C₅	—	2	12	no	—
102	C₂³×C₅	—	2	4	no	—
103	C₂³×C₅	C₂×C₃×D₄	1	6	yes	—
104	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
105	C₅×D₄	—	4	24	no	—
106	C₅×D₄	—	4	16	no	—
107	C₂×C₄×C₅	—	2	12	no	—
108	C₂×C₄×C₅	—	2	12	no	—
109	C₂×C₄×C₅	—	4	12	no	—
110	C₂×C₄×C₅	C₂×C₃×D₄	1	12	yes	—
111	C₂²×D₄	C₂²×C₃×C₅	1	4	yes	—
112	C₂²×D₄	C₂²×C₃×C₅	1	6	yes	—
113	C₂³×C₄	C₂²×C₃×C₅	1	3	yes	—
114	#121	—	2	24	no	—
115	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	—
117	#95	C₂²×C₃×C₅	1	4	yes	—
118	C₂×C₃×C₅	—	2	4	no	—
119	C₂×C₃×C₅	—	8	4	no	—
120	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
121	C₂×C₃×C₅	#534	1	3	yes	—
122	C₂×C₃×C₅	#537	1	3	yes	—
123	C₂³×C₃	—	2	3	no	—
124	C₂³×C₃	—	2	1	no	—
125	C₂³×C₃	—	2	12	no	—
126	C₂³×C₃	—	2	12	no	—
127	C₂³×C₃	—	2	4	no	—
128	C₂³×C₃	C₂×C₅×D₄	1	6	yes	—
129	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
130	C₃×D₄	—	4	24	no	—
131	C₃×D₄	—	4	16	no	—
132	C₂×C₃×C₄	—	2	12	no	—
133	C₂×C₃×C₄	—	2	12	no	—
134	C₂×C₃×C₄	—	4	12	no	—
135	C₂×C₃×C₄	C₂×C₅×D₄	1	12	yes	—
136	C₂²×C₅	—	2	6	no	—
137	C₂²×C₅	—	2	6	no	—
138	C₂²×C₅	—	4	4	no	—
139	C₂²×C₅	—	4	12	no	—
140	C₂²×C₅	—	4	12	no	—
141	C₂²×C₅	C₃×#127	1	4	yes	—
142	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
143	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
144	C₂²×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
145	C₄×C₅	—	2	12	no	—
146	C₄×C₅	—	4	4	no	—
147	C₂⁴	C₂³×C₃×C₅	1	4	yes	—
148	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
149	C₂×D₄	—	2	24	no	—
150	C₂×D₄	—	2	12	no	—
151	C₂×D₄	—	4	12	no	—
152	C₂²×C₄	—	4	6	no	—
153	C₂²×C₄	C₃×C₅×D₄	1	12	yes	complemented
154	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
155	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
156	C₄⋊C₄	—	2	12	no	—
157	C₂²⋊C₄	—	2	24	no	—
158	C₃×C₅	#2656	1	1	yes	characteristic, direct product factor, Hall
159	C₂²×C₃	—	2	6	no	—
160	C₂²×C₃	—	2	6	no	—
161	C₂²×C₃	—	4	4	no	—
162	C₂²×C₃	—	4	12	no	—
163	C₂²×C₃	—	4	12	no	—
164	C₂²×C₃	C₅×#127	1	4	yes	—
165	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
166	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
167	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
168	C₃×C₄	—	2	12	no	—
169	C₃×C₄	—	4	4	no	—
170	C₂×C₅	—	2	4	no	—
171	C₂×C₅	—	8	4	no	—
172	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
173	C₂×C₅	C₃×#534	1	3	yes	—
174	C₂×C₅	C₃×#537	1	3	yes	—
175	C₂³	—	2	3	no	—
176	C₂³	—	2	1	no	—
177	C₂³	—	2	12	no	—
178	C₂³	—	2	12	no	—
179	C₂³	—	2	4	no	—
180	C₂³	C₂×C₃×C₅×D₄	1	6	yes	—
181	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
182	D₄	—	4	24	no	—
183	D₄	—	4	16	no	—
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	12	no	—
186	C₂×C₄	—	4	12	no	—
187	C₂×C₄	C₂×C₃×C₅×D₄	1	12	yes	—
188	C₂×C₃	—	2	4	no	—
189	C₂×C₃	—	8	4	no	—
190	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
191	C₂×C₃	C₅×#534	1	3	yes	—
192	C₂×C₃	C₅×#537	1	3	yes	—
193	C₅	C₃×#2656	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²	—	2	6	no	—
195	C₂²	—	2	6	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	12	no	—
198	C₂²	—	4	12	no	—
199	C₂²	C₃×C₅×#127	1	4	yes	—
200	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
201	C₂²	C₃×C₅×H(5,𝔽₂)	1	3	yes	—
202	C₂²	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
203	C₄	—	2	12	no	—
204	C₄	—	4	4	no	—
205	C₃	C₅×#2656	1	1	yes	characteristic, direct product factor, Sylow
206	C₂	—	2	4	no	—
207	C₂	—	8	4	no	—
208	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
209	C₂	C₃×C₅×#534	1	3	yes	—
210	C₂	C₃×C₅×#537	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
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. C₂×C₃×C₅×#121	C₂	semidirect
2. C₃×C₅×#399	C₂	semidirect
3. C₃×C₅×#392	C₂	semidirect
4. C₂×C₃×C₅×#95	C₂	semidirect
5. C₅×#2656	C₃	direct (N × H)
13. C₃×#2656	C₅	direct (N × H)
14. C₂×C₅×#121	C₂×C₃	semidirect
15. C₅×#399	C₂×C₃	semidirect
16. C₅×#392	C₂×C₃	semidirect
17. C₂×C₅×#95	C₂×C₃	semidirect
24. C₂²×C₃×C₄×C₅	D₄	semidirect
29. C₂×C₃×#121	C₂×C₅	semidirect
30. C₃×#399	C₂×C₅	semidirect
31. C₃×#392	C₂×C₅	semidirect
32. C₂×C₃×#95	C₂×C₅	semidirect
40. #2656	C₃×C₅	direct (N × H)
67. C₂²×C₄×C₅	C₃×D₄	semidirect
72. C₂×#121	C₂×C₃×C₅	semidirect
73. #399	C₂×C₃×C₅	semidirect
74. #392	C₂×C₃×C₅	semidirect
75. C₂×#95	C₂×C₃×C₅	semidirect
93. C₂²×C₃×C₄	C₅×D₄	semidirect
153. C₂²×C₄	C₃×C₅×D₄	semidirect
158. C₃×C₅	#2656	direct (N × H)
193. C₅	C₃×#2656	direct (N × H)
205. C₃	C₅×#2656	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×#121	2
C₃×C₅×#399	2
C₃×C₅×#392	2
C₂×C₃×C₅×#95	2
C₅×#2656	3
C₃×#2656	5

Subgroups — C3×C5×#2656 #14275690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2656 #14275690