Subgroups — C₃×C₅×#2249 #14275283

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×SD₃₂	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#495	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2249	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
10	C₃×C₅×#124	—	2	2	no	—
11	C₃×C₅×Q₃₂	—	2	2	no	—
12	C₃×C₅×SD₃₂	—	2	4	no	—
13	C₃×C₅×D₁₆	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#2249	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	complemented
20	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic
21	C₅×#495	C₂×C₃	1	2	yes	complemented
22	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₃×C₅×Q₁₆	—	2	2	no	—
26	C₃×C₅×Q₁₆	—	2	2	no	—
27	C₃×C₅×SD₁₆	—	4	4	no	—
28	C₃×C₅×D₈	—	2	2	no	—
29	C₃×C₅×D₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₅×C₂²⋊C₄	—	4	4	no	—
33	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₃×C₅×C₁₆	D₄	1	2	yes	—
35	C₃×C₅×C₁₆	D₄	1	2	yes	—
36	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂×C₃×SD₃₂	C₂×C₅	1	2	yes	complemented
38	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic
39	C₃×#495	C₂×C₅	1	2	yes	complemented
40	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
42	C₂×C₅×SD₁₆	—	2	2	no	—
43	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
44	C₅×#124	—	2	2	no	—
45	C₅×Q₃₂	—	2	2	no	—
46	C₅×SD₃₂	—	2	4	no	—
47	C₅×D₁₆	—	2	2	no	—
48	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	#2249	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	—	4	2	no	—
53	C₃×C₅×Q₈	—	4	2	no	—
54	C₃×C₅×Q₈	—	4	2	no	—
55	C₃×C₅×D₄	—	4	2	no	—
56	C₃×C₅×D₄	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₃×C₅×C₈	D₈	1	2	yes	—
61	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
64	C₂×C₃×SD₁₆	—	2	2	no	—
65	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
66	C₃×#124	—	2	2	no	—
67	C₃×Q₃₂	—	2	2	no	—
68	C₃×SD₃₂	—	2	4	no	—
69	C₃×D₁₆	—	2	2	no	—
70	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×D₄	—	2	2	no	—
75	C₅×Q₁₆	—	2	2	no	—
76	C₅×Q₁₆	—	2	2	no	—
77	C₅×SD₁₆	—	4	4	no	—
78	C₅×D₈	—	2	2	no	—
79	C₅×D₈	—	2	2	no	—
80	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
81	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
82	C₅×C₂²⋊C₄	—	4	4	no	—
83	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
84	C₅×C₁₆	C₃×D₄	1	2	yes	—
85	C₅×C₁₆	C₃×D₄	1	2	yes	—
86	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	complemented
88	C₂×D₁₆	C₂×C₃×C₅	1	1	yes	characteristic
89	#495	C₂×C₃×C₅	1	2	yes	complemented
90	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
91	C₂²×C₃×C₅	—	8	2	no	—
92	C₂²×C₃×C₅	—	8	4	no	—
93	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
94	C₃×C₄×C₅	—	2	2	no	—
95	C₃×C₄×C₅	—	8	2	no	—
96	C₃×C₄×C₅	#127	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
98	C₂×C₃×Q₈	—	2	2	no	—
99	C₂×C₃×D₄	—	2	2	no	—
100	C₃×Q₁₆	—	2	2	no	—
101	C₃×Q₁₆	—	2	2	no	—
102	C₃×SD₁₆	—	4	4	no	—
103	C₃×D₈	—	2	2	no	—
104	C₃×D₈	—	2	2	no	—
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
107	C₃×C₂²⋊C₄	—	4	4	no	—
108	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
109	C₃×C₁₆	C₅×D₄	1	2	yes	—
110	C₃×C₁₆	C₅×D₄	1	2	yes	—
111	C₂³×C₅	—	4	2	no	—
112	C₅×Q₈	—	4	2	no	—
113	C₅×Q₈	—	4	2	no	—
114	C₅×D₄	—	4	2	no	—
115	C₅×D₄	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
118	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
119	C₅×C₈	C₃×D₈	1	2	yes	—
120	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
121	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
122	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
123	C₂×SD₁₆	—	2	2	no	—
124	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
125	#124	—	2	2	no	—
126	Q₃₂	—	2	2	no	—
127	SD₃₂	—	2	4	no	—
128	D₁₆	—	2	2	no	—
129	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
130	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
131	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₂×C₃×C₅	—	16	2	no	—
133	C₂×C₃×C₅	#493	1	1	yes	characteristic
134	C₂×C₃×C₅	#508	1	2	yes	—
135	C₂³×C₃	—	4	2	no	—
136	C₃×Q₈	—	4	2	no	—
137	C₃×Q₈	—	4	2	no	—
138	C₃×D₄	—	4	2	no	—
139	C₃×D₄	—	4	2	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₃×C₈	C₅×D₈	1	2	yes	—
144	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
145	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
146	C₂²×C₅	—	8	4	no	—
147	C₂²×C₅	—	8	2	no	—
148	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
149	C₄×C₅	—	2	2	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	C₃×#127	1	1	yes	characteristic
152	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
153	C₂×Q₈	—	2	2	no	—
154	C₂×D₄	—	2	2	no	—
155	Q₁₆	—	2	2	no	—
156	Q₁₆	—	2	2	no	—
157	SD₁₆	—	4	4	no	—
158	D₈	—	2	2	no	—
159	D₈	—	2	2	no	—
160	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
162	C₂²⋊C₄	—	4	4	no	—
163	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
164	C₁₆	C₃×C₅×D₄	1	2	yes	—
165	C₁₆	C₃×C₅×D₄	1	2	yes	—
166	C₃×C₅	#2249	1	1	yes	characteristic, direct product factor, Hall
167	C₂²×C₃	—	8	2	no	—
168	C₂²×C₃	—	8	4	no	—
169	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
170	C₃×C₄	—	2	2	no	—
171	C₃×C₄	—	8	2	no	—
172	C₃×C₄	C₅×#127	1	1	yes	characteristic
173	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
174	C₂×C₅	—	16	2	no	—
175	C₂×C₅	C₃×#493	1	1	yes	characteristic
176	C₂×C₅	C₃×#508	1	2	yes	—
177	C₂³	—	4	2	no	—
178	Q₈	—	4	2	no	—
179	Q₈	—	4	2	no	—
180	D₄	—	4	2	no	—
181	D₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
184	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
185	C₈	C₃×C₅×D₈	1	2	yes	—
186	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
187	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
188	C₂×C₃	—	16	2	no	—
189	C₂×C₃	C₅×#493	1	1	yes	characteristic
190	C₂×C₃	C₅×#508	1	2	yes	—
191	C₅	C₃×#2249	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	8	2	no	—
193	C₂²	—	8	4	no	—
194	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	8	2	no	—
197	C₄	C₃×C₅×#127	1	1	yes	characteristic
198	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
199	C₃	C₅×#2249	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	16	2	no	—
201	C₂	C₃×C₅×#493	1	1	yes	characteristic
202	C₂	C₃×C₅×#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×Q₃₂	C₂	semidirect
2. C₂×C₃×C₅×SD₃₂	C₂	semidirect
4. C₃×C₅×#495	C₂	semidirect
5. C₃×C₄×C₅×C₁₆	C₂	semidirect
6. C₅×#2249	C₃	direct (N × H)
17. C₃×#2249	C₅	direct (N × H)
18. C₂×C₅×Q₃₂	C₂×C₃	semidirect
19. C₂×C₅×SD₃₂	C₂×C₃	semidirect
21. C₅×#495	C₂×C₃	semidirect
22. C₄×C₅×C₁₆	C₂×C₃	semidirect
36. C₂×C₃×Q₃₂	C₂×C₅	semidirect
37. C₂×C₃×SD₃₂	C₂×C₅	semidirect
39. C₃×#495	C₂×C₅	semidirect
40. C₃×C₄×C₁₆	C₂×C₅	semidirect
51. #2249	C₃×C₅	direct (N × H)
86. C₂×Q₃₂	C₂×C₃×C₅	semidirect
87. C₂×SD₃₂	C₂×C₃×C₅	semidirect
89. #495	C₂×C₃×C₅	semidirect
90. C₄×C₁₆	C₂×C₃×C₅	semidirect
166. C₃×C₅	#2249	direct (N × H)
191. C₅	C₃×#2249	direct (N × H)
199. C₃	C₅×#2249	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×Q₃₂	2
C₂×C₃×C₅×SD₃₂	2
C₂×C₃×C₅×D₁₆	2
C₃×C₅×#495	2
C₃×C₄×C₅×C₁₆	2
C₅×#2249	3
C₃×#2249	5

Subgroups — C3×C5×#2249 #14275283

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2249 #14275283