Subgroups — C₃×C₅×#2230 #14275264

Order 1920.

252 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	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#443	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#365	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#360	C₂	1	2	yes	maximal
6	C₅×#2230	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#135	C₂²	1	1	yes	characteristic
8	C₃×C₅×#131	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₃×C₅×#108	C₂²	1	1	yes	characteristic
12	C₃×C₅×#106	—	2	1	no	—
13	C₃×C₅×#106	C₂²	1	2	yes	—
14	C₃×C₅×#104	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#2230	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#498	C₂×C₃	1	1	yes	characteristic, complemented
18	C₅×#443	C₂×C₃	1	1	yes	characteristic
19	C₅×#365	C₂×C₃	1	2	yes	complemented
20	C₅×#364	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#360	C₂×C₃	1	2	yes	—
22	C₃×C₅×#41	—	2	1	no	—
23	C₂×C₃×C₅×Q₈	—	4	1	no	—
24	C₃×C₅×Q₁₆	Q₈	1	1	yes	characteristic, complemented
25	C₃×C₅×SD₁₆	—	2	1	no	—
26	C₃×C₅×D₈	Q₈	1	1	yes	characteristic, complemented
27	C₃×C₅×M₁₆	—	2	1	no	—
28	C₃×C₅×M₁₆	—	4	1	no	—
29	C₂×C₃×C₅×C₈	—	4	1	no	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	4	1	no	—
34	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
35	C₃×C₅×C₁₆	—	2	2	no	—
36	C₃×#498	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×#443	C₂×C₅	1	1	yes	characteristic
38	C₃×#365	C₂×C₅	1	2	yes	complemented
39	C₃×#364	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#360	C₂×C₅	1	2	yes	—
41	C₅×#135	C₂²×C₃	1	1	yes	characteristic
42	C₅×#131	—	2	1	no	—
43	C₅×C₄⋊Q₈	—	2	1	no	—
44	C₅×M₃₂	C₂²×C₃	1	2	yes	—
45	C₅×#108	C₂²×C₃	1	1	yes	characteristic
46	C₅×#106	—	2	1	no	—
47	C₅×#106	C₂²×C₃	1	2	yes	—
48	C₅×#104	—	2	1	no	—
49	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	#2230	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₃×C₅×Q₈	—	2	1	no	—
52	C₃×C₅×Q₈	—	8	2	no	—
53	C₃×C₅×D₄	—	2	1	no	—
54	C₃×C₅×D₄	—	4	1	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	4	1	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	4	1	no	—
60	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
61	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
62	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₃×C₅×C₈	#41	1	1	yes	characteristic
64	C₃×#135	C₂²×C₅	1	1	yes	characteristic
65	C₃×#131	—	2	1	no	—
66	C₃×C₄⋊Q₈	—	2	1	no	—
67	C₃×M₃₂	C₂²×C₅	1	2	yes	—
68	C₃×#108	C₂²×C₅	1	1	yes	characteristic
69	C₃×#106	—	2	1	no	—
70	C₃×#106	C₂²×C₅	1	2	yes	—
71	C₃×#104	—	2	1	no	—
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₅×#41	—	2	1	no	—
74	C₂×C₅×Q₈	—	4	1	no	—
75	C₅×Q₁₆	C₃×Q₈	1	1	yes	characteristic, complemented
76	C₅×SD₁₆	—	2	1	no	—
77	C₅×D₈	C₃×Q₈	1	1	yes	characteristic, complemented
78	C₅×M₁₆	—	2	1	no	—
79	C₅×M₁₆	—	4	1	no	—
80	C₂×C₅×C₈	—	4	1	no	—
81	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
83	C₅×C₄⋊C₄	—	2	2	no	—
84	C₅×C₄⋊C₄	—	4	1	no	—
85	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
86	C₅×C₁₆	—	2	2	no	—
87	#498	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	#443	C₂×C₃×C₅	1	1	yes	characteristic
89	#365	C₂×C₃×C₅	1	2	yes	complemented
90	#364	C₂×C₃×C₅	1	1	yes	characteristic, complemented
91	#360	C₂×C₃×C₅	1	2	yes	—
92	C₂²×C₃×C₅	—	4	1	no	—
93	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₃×C₄×C₅	—	2	2	no	—
95	C₃×C₄×C₅	—	4	1	no	—
96	C₃×C₄×C₅	—	8	2	no	—
97	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
98	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
99	C₃×#41	—	2	1	no	—
100	C₂×C₃×Q₈	—	4	1	no	—
101	C₃×Q₁₆	C₅×Q₈	1	1	yes	characteristic, complemented
102	C₃×SD₁₆	—	2	1	no	—
103	C₃×D₈	C₅×Q₈	1	1	yes	characteristic, complemented
104	C₃×M₁₆	—	2	1	no	—
105	C₃×M₁₆	—	4	1	no	—
106	C₂×C₃×C₈	—	4	1	no	—
107	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₄⋊C₄	—	4	1	no	—
111	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
112	C₃×C₁₆	—	2	2	no	—
113	C₅×Q₈	—	2	1	no	—
114	C₅×Q₈	—	8	2	no	—
115	C₅×D₄	—	2	1	no	—
116	C₅×D₄	—	4	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	4	1	no	—
119	C₂×C₄×C₅	—	4	2	no	—
120	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
121	C₅×C₈	—	4	1	no	—
122	C₅×C₈	C₃×SD₁₆	1	1	yes	characteristic
123	C₅×C₈	C₃×SD₁₆	1	1	yes	characteristic
124	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
125	C₅×C₈	C₃×#41	1	1	yes	characteristic
126	#135	C₂²×C₃×C₅	1	1	yes	characteristic
127	#131	—	2	1	no	—
128	C₄⋊Q₈	—	2	1	no	—
129	M₃₂	C₂²×C₃×C₅	1	2	yes	—
130	#108	C₂²×C₃×C₅	1	1	yes	characteristic
131	#106	—	2	1	no	—
132	#106	C₂²×C₃×C₅	1	2	yes	—
133	#104	—	2	1	no	—
134	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₂×C₃×C₅	—	2	1	no	—
136	C₂×C₃×C₅	—	8	1	no	—
137	C₂×C₃×C₅	#476	1	1	yes	characteristic
138	C₃×Q₈	—	2	1	no	—
139	C₃×Q₈	—	8	2	no	—
140	C₃×D₄	—	2	1	no	—
141	C₃×D₄	—	4	1	no	—
142	C₂×C₃×C₄	—	2	1	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
146	C₃×C₈	—	4	1	no	—
147	C₃×C₈	C₅×SD₁₆	1	1	yes	characteristic
148	C₃×C₈	C₅×SD₁₆	1	1	yes	characteristic
149	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
150	C₃×C₈	C₅×#41	1	1	yes	characteristic
151	C₂²×C₅	—	4	1	no	—
152	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	—	4	1	no	—
155	C₄×C₅	—	8	2	no	—
156	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
157	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
158	#41	—	2	1	no	—
159	C₂×Q₈	—	4	1	no	—
160	Q₁₆	C₃×C₅×Q₈	1	1	yes	characteristic, complemented
161	SD₁₆	—	2	1	no	—
162	D₈	C₃×C₅×Q₈	1	1	yes	characteristic, complemented
163	M₁₆	—	2	1	no	—
164	M₁₆	—	4	1	no	—
165	C₂×C₈	—	4	1	no	—
166	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
167	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	1	no	—
170	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₃×C₅	#2230	1	1	yes	characteristic, direct product factor, Hall
173	C₂²×C₃	—	4	1	no	—
174	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	4	1	no	—
177	C₃×C₄	—	8	2	no	—
178	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
179	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	—	8	1	no	—
182	C₂×C₅	C₃×#476	1	1	yes	characteristic
183	Q₈	—	2	1	no	—
184	Q₈	—	8	2	no	—
185	D₄	—	2	1	no	—
186	D₄	—	4	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	4	1	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
191	C₈	—	4	1	no	—
192	C₈	C₃×C₅×SD₁₆	1	1	yes	characteristic
193	C₈	C₃×C₅×SD₁₆	1	1	yes	characteristic
194	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
195	C₈	C₃×C₅×#41	1	1	yes	characteristic
196	C₂×C₃	—	2	1	no	—
197	C₂×C₃	—	8	1	no	—
198	C₂×C₃	C₅×#476	1	1	yes	characteristic
199	C₅	C₃×#2230	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	4	1	no	—
201	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
202	C₄	—	2	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	8	2	no	—
205	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
206	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
207	C₃	C₅×#2230	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	8	1	no	—
210	C₂	C₃×C₅×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#476
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#498	C₂	semidirect
3. C₃×C₅×#365	C₂	semidirect
4. C₃×C₅×#364	C₂	semidirect
6. C₅×#2230	C₃	direct (N × H)
16. C₃×#2230	C₅	direct (N × H)
17. C₅×#498	C₂×C₃	semidirect
19. C₅×#365	C₂×C₃	semidirect
20. C₅×#364	C₂×C₃	semidirect
24. C₃×C₅×Q₁₆	Q₈	semidirect
26. C₃×C₅×D₈	Q₈	semidirect
36. C₃×#498	C₂×C₅	semidirect
38. C₃×#365	C₂×C₅	semidirect
39. C₃×#364	C₂×C₅	semidirect
50. #2230	C₃×C₅	direct (N × H)
75. C₅×Q₁₆	C₃×Q₈	semidirect
77. C₅×D₈	C₃×Q₈	semidirect
87. #498	C₂×C₃×C₅	semidirect
89. #365	C₂×C₃×C₅	semidirect
90. #364	C₂×C₃×C₅	semidirect
101. C₃×Q₁₆	C₅×Q₈	semidirect
103. C₃×D₈	C₅×Q₈	semidirect
160. Q₁₆	C₃×C₅×Q₈	semidirect
162. D₈	C₃×C₅×Q₈	semidirect
172. C₃×C₅	#2230	direct (N × H)
199. C₅	C₃×#2230	direct (N × H)
207. C₃	C₅×#2230	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×#498	2
C₃×C₅×#443	2
C₃×C₅×#365	2
C₃×C₅×#364	2
C₃×C₅×#360	2
C₅×#2230	3
C₃×#2230	5

Subgroups — C3×C5×#2230 #14275264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2230 #14275264