Subgroups — C₂×C₃×C₅×#493 #14276178

Order 1920.

1520 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×D₈	C₂	1	4	yes	maximal, complemented
2	C₂×C₃×C₅×#127	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#493	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#493	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	—	2	4	no	—
7	C₂²×C₃×C₅×D₄	C₂²	1	4	yes	—
8	C₂×C₃×C₅×D₈	—	2	16	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	16	yes	complemented
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₃×C₅×#127	—	2	4	no	—
12	C₃×C₅×#127	C₂²	1	8	yes	complemented
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	complemented
15	C₂×C₃×#493	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂²×C₅×D₈	C₂×C₃	1	4	yes	complemented
17	C₂×C₅×#127	C₂×C₃	1	2	yes	complemented
18	C₅×#493	C₂×C₃	1	8	yes	direct product factor
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂⁴×C₃×C₅	—	2	4	no	—
21	C₂×C₃×C₅×D₄	—	2	16	no	—
22	C₂×C₃×C₅×D₄	—	2	16	no	—
23	C₂×C₃×C₅×D₄	—	2	4	no	—
24	C₂×C₃×C₅×D₄	—	4	16	no	—
25	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₃×C₅×D₈	—	2	32	no	—
29	C₂×C₃×C₅×C₈	D₄	1	8	yes	complemented
30	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂³	1	2	yes	—
33	C₂²×C₃×D₈	C₂×C₅	1	4	yes	complemented
34	C₂×C₃×#127	C₂×C₅	1	2	yes	complemented
35	C₃×#493	C₂×C₅	1	8	yes	direct product factor
36	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂²×C₅×D₄	—	2	4	no	—
38	C₂²×C₅×D₄	C₂²×C₃	1	4	yes	—
39	C₂×C₅×D₈	—	2	16	no	—
40	C₂×C₅×D₈	C₂²×C₃	1	16	yes	complemented
41	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
42	C₅×#127	—	2	4	no	—
43	C₅×#127	C₂²×C₃	1	8	yes	complemented
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂²×C₃	1	4	yes	complemented
46	C₂×#493	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₅	—	2	8	no	—
48	C₂³×C₃×C₅	—	4	8	no	—
49	C₂³×C₃×C₅	—	4	16	no	—
50	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
51	C₃×C₅×D₄	—	2	16	no	—
52	C₃×C₅×D₄	—	4	8	no	—
53	C₃×C₅×D₄	—	4	32	no	—
54	C₂×C₃×C₄×C₅	D₈	1	8	yes	complemented
55	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
57	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
58	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
59	C₃×C₅×C₈	C₂×D₄	1	8	yes	complemented
60	C₂²×C₃×D₄	—	2	4	no	—
61	C₂²×C₃×D₄	C₂²×C₅	1	4	yes	—
62	C₂×C₃×D₈	—	2	16	no	—
63	C₂×C₃×D₈	C₂²×C₅	1	16	yes	complemented
64	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
65	C₃×#127	—	2	4	no	—
66	C₃×#127	C₂²×C₅	1	8	yes	complemented
67	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
68	C₃×C₄×C₈	C₂²×C₅	1	4	yes	complemented
69	C₂⁴×C₅	—	2	4	no	—
70	C₂×C₅×D₄	—	2	16	no	—
71	C₂×C₅×D₄	—	2	4	no	—
72	C₂×C₅×D₄	—	2	16	no	—
73	C₂×C₅×D₄	—	4	16	no	—
74	C₂×C₅×D₄	C₂³×C₃	1	8	yes	—
75	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
76	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
77	C₅×D₈	—	2	32	no	—
78	C₂×C₅×C₈	C₃×D₄	1	8	yes	complemented
79	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
80	C₄²×C₅	C₃×D₄	1	2	yes	—
81	C₄²×C₅	C₂³×C₃	1	2	yes	—
82	C₂²×D₈	C₂×C₃×C₅	1	4	yes	complemented
83	C₂×#127	C₂×C₃×C₅	1	2	yes	complemented
84	#493	C₂×C₃×C₅	1	8	yes	direct product factor
85	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₂²×C₃×C₅	—	4	8	no	—
87	C₂²×C₃×C₅	—	4	16	no	—
88	C₂²×C₃×C₅	—	8	16	no	—
89	C₂²×C₃×C₅	#127	1	2	yes	—
90	C₂²×C₃×C₅	C₂×D₈	1	4	yes	—
91	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	#127	1	2	yes	—
93	C₃×C₄×C₅	C₂×D₈	1	8	yes	complemented
94	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
95	C₂⁴×C₃	—	2	4	no	—
96	C₂×C₃×D₄	—	2	16	no	—
97	C₂×C₃×D₄	—	2	4	no	—
98	C₂×C₃×D₄	—	2	16	no	—
99	C₂×C₃×D₄	—	4	16	no	—
100	C₂×C₃×D₄	C₂³×C₅	1	8	yes	—
101	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
102	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
103	C₃×D₈	—	2	32	no	—
104	C₂×C₃×C₈	C₅×D₄	1	8	yes	complemented
105	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
106	C₃×C₄²	C₅×D₄	1	2	yes	—
107	C₃×C₄²	C₂³×C₅	1	2	yes	—
108	C₂³×C₅	—	2	8	no	—
109	C₂³×C₅	—	4	8	no	—
110	C₂³×C₅	—	4	16	no	—
111	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
112	C₅×D₄	—	2	16	no	—
113	C₅×D₄	—	4	8	no	—
114	C₅×D₄	—	4	32	no	—
115	C₂×C₄×C₅	C₃×D₈	1	8	yes	complemented
116	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
118	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
119	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
120	C₅×C₈	C₂×C₃×D₄	1	8	yes	complemented
121	C₂²×D₄	—	2	4	no	—
122	C₂²×D₄	C₂²×C₃×C₅	1	4	yes	—
123	C₂×D₈	—	2	16	no	—
124	C₂×D₈	C₂²×C₃×C₅	1	16	yes	complemented
125	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
126	#127	—	2	4	no	—
127	#127	C₂²×C₃×C₅	1	8	yes	complemented
128	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₄×C₈	C₂²×C₃×C₅	1	4	yes	complemented
130	C₂×C₃×C₅	—	8	8	no	—
131	C₂×C₃×C₅	#493	1	4	yes	direct product factor
132	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
133	C₂×C₃×C₅	C₂²×D₈	1	2	yes	—
134	C₂³×C₃	—	2	8	no	—
135	C₂³×C₃	—	4	8	no	—
136	C₂³×C₃	—	4	16	no	—
137	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
138	C₃×D₄	—	2	16	no	—
139	C₃×D₄	—	4	8	no	—
140	C₃×D₄	—	4	32	no	—
141	C₂×C₃×C₄	C₅×D₈	1	8	yes	complemented
142	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
144	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
145	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
146	C₃×C₈	C₂×C₅×D₄	1	8	yes	complemented
147	C₂²×C₅	—	4	16	no	—
148	C₂²×C₅	—	4	8	no	—
149	C₂²×C₅	—	8	16	no	—
150	C₂²×C₅	C₃×#127	1	2	yes	—
151	C₂²×C₅	C₂×C₃×D₈	1	4	yes	—
152	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
153	C₄×C₅	C₃×#127	1	2	yes	—
154	C₄×C₅	C₂×C₃×D₈	1	8	yes	complemented
155	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
156	C₂⁴	—	2	4	no	—
157	C₂×D₄	—	2	16	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	2	16	no	—
160	C₂×D₄	—	4	16	no	—
161	C₂×D₄	C₂³×C₃×C₅	1	8	yes	—
162	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
163	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
164	D₈	—	2	32	no	—
165	C₂×C₈	C₃×C₅×D₄	1	8	yes	complemented
166	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
167	C₄²	C₃×C₅×D₄	1	2	yes	—
168	C₄²	C₂³×C₃×C₅	1	2	yes	—
169	C₃×C₅	C₂×#493	1	1	yes	characteristic, direct product factor, Hall
170	C₂²×C₃	—	4	8	no	—
171	C₂²×C₃	—	4	16	no	—
172	C₂²×C₃	—	8	16	no	—
173	C₂²×C₃	C₅×#127	1	2	yes	—
174	C₂²×C₃	C₂×C₅×D₈	1	4	yes	—
175	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
176	C₃×C₄	C₅×#127	1	2	yes	—
177	C₃×C₄	C₂×C₅×D₈	1	8	yes	complemented
178	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
179	C₂×C₅	—	8	8	no	—
180	C₂×C₅	C₃×#493	1	4	yes	direct product factor
181	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
182	C₂×C₅	C₂²×C₃×D₈	1	2	yes	—
183	C₂³	—	2	8	no	—
184	C₂³	—	4	8	no	—
185	C₂³	—	4	16	no	—
186	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
187	D₄	—	2	16	no	—
188	D₄	—	4	8	no	—
189	D₄	—	4	32	no	—
190	C₂×C₄	C₃×C₅×D₈	1	8	yes	complemented
191	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
192	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
193	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
194	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
195	C₈	C₂×C₃×C₅×D₄	1	8	yes	complemented
196	C₂×C₃	—	8	8	no	—
197	C₂×C₃	C₅×#493	1	4	yes	direct product factor
198	C₂×C₃	C₂×C₅×#127	1	1	yes	characteristic
199	C₂×C₃	C₂²×C₅×D₈	1	2	yes	—
200	C₅	C₂×C₃×#493	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	4	8	no	—
202	C₂²	—	4	16	no	—
203	C₂²	—	8	16	no	—
204	C₂²	C₃×C₅×#127	1	2	yes	—
205	C₂²	C₂×C₃×C₅×D₈	1	4	yes	—
206	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
207	C₄	C₃×C₅×#127	1	2	yes	—
208	C₄	C₂×C₃×C₅×D₈	1	8	yes	complemented
209	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
210	C₃	C₂×C₅×#493	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	8	8	no	—
212	C₂	C₃×C₅×#493	1	4	yes	direct product factor
213	C₂	C₂×C₃×C₅×#127	1	1	yes	characteristic
214	C₂	C₂²×C₃×C₅×D₈	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×D₈	C₂	semidirect
2. C₂×C₃×C₅×#127	C₂	semidirect
3. C₃×C₅×#493	C₂	direct (N × H)
4. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
5. C₂×C₅×#493	C₃	direct (N × H)
9. C₂×C₃×C₅×D₈	C₂²	semidirect
12. C₃×C₅×#127	C₂²	semidirect
14. C₃×C₄×C₅×C₈	C₂²	semidirect
15. C₂×C₃×#493	C₅	direct (N × H)
16. C₂²×C₅×D₈	C₂×C₃	semidirect
17. C₂×C₅×#127	C₂×C₃	semidirect
18. C₅×#493	C₂×C₃	direct (N × H)
19. C₂×C₄×C₅×C₈	C₂×C₃	semidirect
29. C₂×C₃×C₅×C₈	D₄	semidirect
33. C₂²×C₃×D₈	C₂×C₅	semidirect
34. C₂×C₃×#127	C₂×C₅	semidirect
35. C₃×#493	C₂×C₅	direct (N × H)
36. C₂×C₃×C₄×C₈	C₂×C₅	semidirect
40. C₂×C₅×D₈	C₂²×C₃	semidirect
43. C₅×#127	C₂²×C₃	semidirect
45. C₄×C₅×C₈	C₂²×C₃	semidirect
46. C₂×#493	C₃×C₅	direct (N × H)
54. C₂×C₃×C₄×C₅	D₈	semidirect
59. C₃×C₅×C₈	C₂×D₄	semidirect
63. C₂×C₃×D₈	C₂²×C₅	semidirect
66. C₃×#127	C₂²×C₅	semidirect
68. C₃×C₄×C₈	C₂²×C₅	semidirect
78. C₂×C₅×C₈	C₃×D₄	semidirect
82. C₂²×D₈	C₂×C₃×C₅	semidirect
83. C₂×#127	C₂×C₃×C₅	semidirect
84. #493	C₂×C₃×C₅	direct (N × H)
85. C₂×C₄×C₈	C₂×C₃×C₅	semidirect
93. C₃×C₄×C₅	C₂×D₈	semidirect
104. C₂×C₃×C₈	C₅×D₄	semidirect
115. C₂×C₄×C₅	C₃×D₈	semidirect
120. C₅×C₈	C₂×C₃×D₄	semidirect
124. C₂×D₈	C₂²×C₃×C₅	semidirect
127. #127	C₂²×C₃×C₅	semidirect
129. C₄×C₈	C₂²×C₃×C₅	semidirect
131. C₂×C₃×C₅	#493	direct (N × H)
141. C₂×C₃×C₄	C₅×D₈	semidirect
146. C₃×C₈	C₂×C₅×D₄	semidirect
154. C₄×C₅	C₂×C₃×D₈	semidirect
165. C₂×C₈	C₃×C₅×D₄	semidirect
169. C₃×C₅	C₂×#493	direct (N × H)
177. C₃×C₄	C₂×C₅×D₈	semidirect
180. C₂×C₅	C₃×#493	direct (N × H)
190. C₂×C₄	C₃×C₅×D₈	semidirect
195. C₈	C₂×C₃×C₅×D₄	semidirect
197. C₂×C₃	C₅×#493	direct (N × H)
200. C₅	C₂×C₃×#493	direct (N × H)
208. C₄	C₂×C₃×C₅×D₈	semidirect
210. C₃	C₂×C₅×#493	direct (N × H)
212. C₂	C₃×C₅×#493	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×D₈	2
C₂×C₃×C₅×#127	2
C₃×C₅×#493	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#493	3
C₂×C₃×#493	5

Subgroups — C2×C3×C5×#493 #14276178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#493 #14276178