Subgroups — C₃×C₅×#2679 #14275713

Order 1920.

2000 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#120	C₂	1	6	yes	maximal, complemented
2	C₃×C₅×#394	C₂	1	6	yes	maximal, complemented
3	C₃×C₅×#390	C₂	1	3	yes	maximal, complemented
4	C₅×#2679	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂⁵×C₃×C₅	C₂²	1	2	yes	complemented
6	C₂²×C₃×C₅×D₄	C₂²	1	6	yes	—
7	C₂²×C₃×C₅×D₄	C₂²	1	3	yes	—
8	C₃×C₅×#120	—	2	24	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	complemented
10	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	12	yes	complemented
11	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	6	yes	complemented
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
13	C₃×C₅×#95	C₂²	1	2	yes	complemented
14	C₃×#2679	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₂×C₅×#120	C₂×C₃	1	6	yes	complemented
16	C₅×#394	C₂×C₃	1	6	yes	complemented
17	C₅×#390	C₂×C₃	1	3	yes	complemented
18	C₂⁴×C₃×C₅	—	2	12	no	—
19	C₂⁴×C₃×C₅	—	4	6	no	—
20	C₂⁴×C₃×C₅	—	4	2	no	—
21	C₂⁴×C₃×C₅	C₂³	1	6	yes	—
22	C₂×C₃×C₅×D₄	—	2	24	no	—
23	C₂×C₃×C₅×D₄	—	4	3	no	—
24	C₂×C₃×C₅×D₄	—	4	3	no	—
25	C₂×C₃×C₅×D₄	—	4	6	no	—
26	C₂×C₃×C₅×D₄	D₄	1	12	yes	complemented
27	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
29	C₃×C₅×C₄⋊C₄	—	4	3	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	12	no	—
31	C₃×C₅×C₂²⋊C₄	—	2	24	no	—
32	C₃×C₄²×C₅	—	4	1	no	—
33	C₂×C₃×#120	C₂×C₅	1	6	yes	complemented
34	C₃×#394	C₂×C₅	1	6	yes	complemented
35	C₃×#390	C₂×C₅	1	3	yes	complemented
36	C₂⁵×C₅	C₂²×C₃	1	2	yes	complemented
37	C₂²×C₅×D₄	C₂²×C₃	1	6	yes	—
38	C₂²×C₅×D₄	C₂²×C₃	1	3	yes	—
39	C₅×#120	—	2	24	no	—
40	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	complemented
41	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	12	yes	complemented
42	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	6	yes	complemented
43	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic, complemented
44	C₅×#95	C₂²×C₃	1	2	yes	complemented
45	#2679	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	—	2	12	no	—
47	C₂³×C₃×C₅	—	2	6	no	—
48	C₂³×C₃×C₅	—	2	6	no	—
49	C₂³×C₃×C₅	—	2	12	no	—
50	C₂³×C₃×C₅	—	4	24	no	—
51	C₂³×C₃×C₅	—	4	24	no	—
52	C₂³×C₃×C₅	—	4	8	no	—
53	C₂³×C₃×C₅	C₂×D₄	1	12	yes	—
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×D₄	—	4	12	no	—
56	C₃×C₅×D₄	—	4	24	no	—
57	C₂×C₃×C₄×C₅	—	2	3	no	—
58	C₂×C₃×C₄×C₅	—	2	3	no	—
59	C₂×C₃×C₄×C₅	—	2	12	no	—
60	C₂×C₃×C₄×C₅	—	2	6	no	—
61	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
62	C₂⁵×C₃	C₂²×C₅	1	2	yes	complemented
63	C₂²×C₃×D₄	C₂²×C₅	1	6	yes	—
64	C₂²×C₃×D₄	C₂²×C₅	1	3	yes	—
65	C₃×#120	—	2	24	no	—
66	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	complemented
67	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	12	yes	complemented
68	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	6	yes	complemented
69	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic, complemented
70	C₃×#95	C₂²×C₅	1	2	yes	complemented
71	C₂⁴×C₅	—	2	12	no	—
72	C₂⁴×C₅	—	4	6	no	—
73	C₂⁴×C₅	—	4	2	no	—
74	C₂⁴×C₅	C₂³×C₃	1	6	yes	—
75	C₂×C₅×D₄	—	2	24	no	—
76	C₂×C₅×D₄	—	4	3	no	—
77	C₂×C₅×D₄	—	4	3	no	—
78	C₂×C₅×D₄	—	4	6	no	—
79	C₂×C₅×D₄	C₃×D₄	1	12	yes	complemented
80	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
81	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
82	C₅×C₄⋊C₄	—	4	3	no	—
83	C₅×C₂²⋊C₄	—	2	12	no	—
84	C₅×C₂²⋊C₄	—	2	24	no	—
85	C₄²×C₅	—	4	1	no	—
86	C₂×#120	C₂×C₃×C₅	1	6	yes	complemented
87	#394	C₂×C₃×C₅	1	6	yes	complemented
88	#390	C₂×C₃×C₅	1	3	yes	complemented
89	C₂²×C₃×C₅	—	2	12	no	—
90	C₂²×C₃×C₅	—	2	24	no	—
91	C₂²×C₃×C₅	—	4	32	no	—
92	C₂²×C₃×C₅	—	4	12	no	—
93	C₂²×C₃×C₅	—	4	6	no	—
94	C₂²×C₃×C₅	—	4	6	no	—
95	C₂²×C₃×C₅	C₂²×D₄	1	3	yes	—
96	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
97	C₂²×C₃×C₅	H(5,𝔽₂)	1	3	yes	—
98	C₃×C₄×C₅	—	2	6	no	—
99	C₃×C₄×C₅	—	4	6	no	—
100	C₂⁴×C₃	—	2	12	no	—
101	C₂⁴×C₃	—	4	6	no	—
102	C₂⁴×C₃	—	4	2	no	—
103	C₂⁴×C₃	C₂³×C₅	1	6	yes	—
104	C₂×C₃×D₄	—	2	24	no	—
105	C₂×C₃×D₄	—	4	3	no	—
106	C₂×C₃×D₄	—	4	3	no	—
107	C₂×C₃×D₄	—	4	6	no	—
108	C₂×C₃×D₄	C₅×D₄	1	12	yes	complemented
109	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
110	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
111	C₃×C₄⋊C₄	—	4	3	no	—
112	C₃×C₂²⋊C₄	—	2	12	no	—
113	C₃×C₂²⋊C₄	—	2	24	no	—
114	C₃×C₄²	—	4	1	no	—
115	C₂³×C₅	—	2	12	no	—
116	C₂³×C₅	—	2	6	no	—
117	C₂³×C₅	—	2	6	no	—
118	C₂³×C₅	—	2	12	no	—
119	C₂³×C₅	—	4	24	no	—
120	C₂³×C₅	—	4	24	no	—
121	C₂³×C₅	—	4	8	no	—
122	C₂³×C₅	C₂×C₃×D₄	1	12	yes	—
123	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
124	C₅×D₄	—	4	12	no	—
125	C₅×D₄	—	4	24	no	—
126	C₂×C₄×C₅	—	2	3	no	—
127	C₂×C₄×C₅	—	2	3	no	—
128	C₂×C₄×C₅	—	2	12	no	—
129	C₂×C₄×C₅	—	2	6	no	—
130	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
131	C₂⁵	C₂²×C₃×C₅	1	2	yes	complemented
132	C₂²×D₄	C₂²×C₃×C₅	1	6	yes	—
133	C₂²×D₄	C₂²×C₃×C₅	1	3	yes	—
134	#120	—	2	24	no	—
135	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	complemented
136	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	12	yes	complemented
137	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	6	yes	complemented
138	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
139	#95	C₂²×C₃×C₅	1	2	yes	complemented
140	C₂×C₃×C₅	—	4	12	no	—
141	C₂×C₃×C₅	#534	1	3	yes	—
142	C₂×C₃×C₅	D₄²	1	3	yes	—
143	C₂×C₃×C₅	H(3,𝔽₄)	1	1	yes	characteristic
144	C₂³×C₃	—	2	12	no	—
145	C₂³×C₃	—	2	6	no	—
146	C₂³×C₃	—	2	6	no	—
147	C₂³×C₃	—	2	12	no	—
148	C₂³×C₃	—	4	24	no	—
149	C₂³×C₃	—	4	24	no	—
150	C₂³×C₃	—	4	8	no	—
151	C₂³×C₃	C₂×C₅×D₄	1	12	yes	—
152	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
153	C₃×D₄	—	4	12	no	—
154	C₃×D₄	—	4	24	no	—
155	C₂×C₃×C₄	—	2	3	no	—
156	C₂×C₃×C₄	—	2	3	no	—
157	C₂×C₃×C₄	—	2	12	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
160	C₂²×C₅	—	2	12	no	—
161	C₂²×C₅	—	2	24	no	—
162	C₂²×C₅	—	4	32	no	—
163	C₂²×C₅	—	4	6	no	—
164	C₂²×C₅	—	4	12	no	—
165	C₂²×C₅	—	4	6	no	—
166	C₂²×C₅	C₂²×C₃×D₄	1	3	yes	—
167	C₂²×C₅	C₃×H(5,𝔽₂)	1	3	yes	—
168	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
169	C₄×C₅	—	2	6	no	—
170	C₄×C₅	—	4	6	no	—
171	C₂⁴	—	2	12	no	—
172	C₂⁴	—	4	6	no	—
173	C₂⁴	—	4	2	no	—
174	C₂⁴	C₂³×C₃×C₅	1	6	yes	—
175	C₂×D₄	—	2	24	no	—
176	C₂×D₄	—	4	3	no	—
177	C₂×D₄	—	4	3	no	—
178	C₂×D₄	—	4	6	no	—
179	C₂×D₄	C₃×C₅×D₄	1	12	yes	complemented
180	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
181	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
182	C₄⋊C₄	—	4	3	no	—
183	C₂²⋊C₄	—	2	12	no	—
184	C₂²⋊C₄	—	2	24	no	—
185	C₄²	—	4	1	no	—
186	C₃×C₅	#2679	1	1	yes	characteristic, direct product factor, Hall
187	C₂²×C₃	—	2	12	no	—
188	C₂²×C₃	—	2	24	no	—
189	C₂²×C₃	—	4	32	no	—
190	C₂²×C₃	—	4	12	no	—
191	C₂²×C₃	—	4	6	no	—
192	C₂²×C₃	—	4	6	no	—
193	C₂²×C₃	C₂²×C₅×D₄	1	3	yes	—
194	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
195	C₂²×C₃	C₅×H(5,𝔽₂)	1	3	yes	—
196	C₃×C₄	—	2	6	no	—
197	C₃×C₄	—	4	6	no	—
198	C₂×C₅	—	4	12	no	—
199	C₂×C₅	C₃×#534	1	3	yes	—
200	C₂×C₅	C₃×D₄²	1	3	yes	—
201	C₂×C₅	C₃×H(3,𝔽₄)	1	1	yes	characteristic
202	C₂³	—	2	12	no	—
203	C₂³	—	2	6	no	—
204	C₂³	—	2	6	no	—
205	C₂³	—	2	12	no	—
206	C₂³	—	4	24	no	—
207	C₂³	—	4	24	no	—
208	C₂³	—	4	8	no	—
209	C₂³	C₂×C₃×C₅×D₄	1	12	yes	—
210	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
211	D₄	—	4	12	no	—
212	D₄	—	4	24	no	—
213	C₂×C₄	—	2	3	no	—
214	C₂×C₄	—	2	3	no	—
215	C₂×C₄	—	2	12	no	—
216	C₂×C₄	—	2	6	no	—
217	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
218	C₂×C₃	—	4	12	no	—
219	C₂×C₃	C₅×#534	1	3	yes	—
220	C₂×C₃	C₅×D₄²	1	3	yes	—
221	C₂×C₃	C₅×H(3,𝔽₄)	1	1	yes	characteristic
222	C₅	C₃×#2679	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	2	12	no	—
224	C₂²	—	2	24	no	—
225	C₂²	—	4	32	no	—
226	C₂²	—	4	12	no	—
227	C₂²	—	4	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	C₂²×C₃×C₅×D₄	1	3	yes	—
230	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
231	C₂²	C₃×C₅×H(5,𝔽₂)	1	3	yes	—
232	C₄	—	2	6	no	—
233	C₄	—	4	6	no	—
234	C₃	C₅×#2679	1	1	yes	characteristic, direct product factor, Sylow
235	C₂	—	4	12	no	—
236	C₂	C₃×C₅×#534	1	3	yes	—
237	C₂	C₃×C₅×D₄²	1	3	yes	—
238	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
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×#120	C₂	semidirect
2. C₃×C₅×#394	C₂	semidirect
3. C₃×C₅×#390	C₂	semidirect
4. C₅×#2679	C₃	direct (N × H)
5. C₂⁵×C₃×C₅	C₂²	semidirect
9. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
10. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
11. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
12. C₂×C₃×C₄²×C₅	C₂²	semidirect
13. C₃×C₅×#95	C₂²	semidirect
14. C₃×#2679	C₅	direct (N × H)
15. C₂×C₅×#120	C₂×C₃	semidirect
16. C₅×#394	C₂×C₃	semidirect
17. C₅×#390	C₂×C₃	semidirect
26. C₂×C₃×C₅×D₄	D₄	semidirect
33. C₂×C₃×#120	C₂×C₅	semidirect
34. C₃×#394	C₂×C₅	semidirect
35. C₃×#390	C₂×C₅	semidirect
36. C₂⁵×C₅	C₂²×C₃	semidirect
40. C₂×C₅×C₄⋊C₄	C₂²×C₃	semidirect
41. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
42. C₂×C₅×C₂²⋊C₄	C₂²×C₃	semidirect
43. C₂×C₄²×C₅	C₂²×C₃	semidirect
44. C₅×#95	C₂²×C₃	semidirect
45. #2679	C₃×C₅	direct (N × H)
62. C₂⁵×C₃	C₂²×C₅	semidirect
66. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
67. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
68. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
69. C₂×C₃×C₄²	C₂²×C₅	semidirect
70. C₃×#95	C₂²×C₅	semidirect
79. C₂×C₅×D₄	C₃×D₄	semidirect
86. C₂×#120	C₂×C₃×C₅	semidirect
87. #394	C₂×C₃×C₅	semidirect
88. #390	C₂×C₃×C₅	semidirect
108. C₂×C₃×D₄	C₅×D₄	semidirect
131. C₂⁵	C₂²×C₃×C₅	semidirect
135. C₂×C₄⋊C₄	C₂²×C₃×C₅	semidirect
136. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
137. C₂×C₂²⋊C₄	C₂²×C₃×C₅	semidirect
138. C₂×C₄²	C₂²×C₃×C₅	semidirect
139. #95	C₂²×C₃×C₅	semidirect
179. C₂×D₄	C₃×C₅×D₄	semidirect
186. C₃×C₅	#2679	direct (N × H)
222. C₅	C₃×#2679	direct (N × H)
234. C₃	C₅×#2679	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×#120	2
C₃×C₅×#394	2
C₃×C₅×#390	2
C₅×#2679	3
C₃×#2679	5

Subgroups — C3×C5×#2679 #14275713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2679 #14275713