Subgroups — C₃×C₅×#2231 #14275265

Order 1920.

240 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	C₃×C₅×#499	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×#2231	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×#125	—	2	1	no	—
10	C₃×C₅×M₃₂	C₂²	1	2	yes	—
11	C₃×C₅×#108	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	—	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×#2231	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#499	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	C₃×C₅×Q₁₆	Q₈	1	1	yes	characteristic
24	C₃×C₅×SD₁₆	—	2	1	no	—
25	C₃×C₅×D₈	Q₈	1	1	yes	characteristic
26	C₃×C₅×M₁₆	—	2	1	no	—
27	C₃×C₅×M₁₆	—	4	1	no	—
28	C₂×C₃×C₅×C₈	—	4	1	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	4	2	no	—
33	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₃×C₅×C₁₆	—	2	2	no	—
35	C₃×#499	C₂×C₅	1	1	yes	characteristic, complemented
36	C₃×#443	C₂×C₅	1	1	yes	characteristic
37	C₃×#365	C₂×C₅	1	2	yes	complemented
38	C₃×#364	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#360	C₂×C₅	1	2	yes	—
40	C₅×#135	C₂²×C₃	1	1	yes	characteristic
41	C₅×#131	—	2	1	no	—
42	C₅×#125	—	2	1	no	—
43	C₅×M₃₂	C₂²×C₃	1	2	yes	—
44	C₅×#108	C₂²×C₃	1	1	yes	characteristic
45	C₅×#107	—	2	1	no	—
46	C₅×#106	C₂²×C₃	1	2	yes	—
47	C₅×#104	—	2	1	no	—
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	#2231	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₃×C₅×Q₈	—	2	1	no	—
51	C₃×C₅×D₄	—	2	1	no	—
52	C₃×C₅×D₄	—	4	1	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	4	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₃×C₅×C₈	—	4	1	no	—
58	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
59	C₃×C₅×C₈	SD₁₆	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₃×#135	C₂²×C₅	1	1	yes	characteristic
63	C₃×#131	—	2	1	no	—
64	C₃×#125	—	2	1	no	—
65	C₃×M₃₂	C₂²×C₅	1	2	yes	—
66	C₃×#108	C₂²×C₅	1	1	yes	characteristic
67	C₃×#107	—	2	1	no	—
68	C₃×#106	C₂²×C₅	1	2	yes	—
69	C₃×#104	—	2	1	no	—
70	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₅×#41	—	2	1	no	—
72	C₅×Q₁₆	C₃×Q₈	1	1	yes	characteristic
73	C₅×SD₁₆	—	2	1	no	—
74	C₅×D₈	C₃×Q₈	1	1	yes	characteristic
75	C₅×M₁₆	—	2	1	no	—
76	C₅×M₁₆	—	4	1	no	—
77	C₂×C₅×C₈	—	4	1	no	—
78	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
79	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
83	C₅×C₁₆	—	2	2	no	—
84	#499	C₂×C₃×C₅	1	1	yes	characteristic, complemented
85	#443	C₂×C₃×C₅	1	1	yes	characteristic
86	#365	C₂×C₃×C₅	1	2	yes	complemented
87	#364	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	#360	C₂×C₃×C₅	1	2	yes	—
89	C₂²×C₃×C₅	—	4	1	no	—
90	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
91	C₃×C₄×C₅	—	4	1	no	—
92	C₃×C₄×C₅	—	4	1	no	—
93	C₃×C₄×C₅	—	8	2	no	—
94	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
95	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
96	C₃×#41	—	2	1	no	—
97	C₃×Q₁₆	C₅×Q₈	1	1	yes	characteristic
98	C₃×SD₁₆	—	2	1	no	—
99	C₃×D₈	C₅×Q₈	1	1	yes	characteristic
100	C₃×M₁₆	—	2	1	no	—
101	C₃×M₁₆	—	4	1	no	—
102	C₂×C₃×C₈	—	4	1	no	—
103	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄⋊C₄	—	4	2	no	—
107	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
108	C₃×C₁₆	—	2	2	no	—
109	C₅×Q₈	—	2	1	no	—
110	C₅×D₄	—	2	1	no	—
111	C₅×D₄	—	4	1	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	4	1	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
116	C₅×C₈	—	4	1	no	—
117	C₅×C₈	C₃×SD₁₆	1	1	yes	characteristic
118	C₅×C₈	C₃×SD₁₆	1	1	yes	characteristic
119	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
120	C₅×C₈	C₃×#41	1	1	yes	characteristic
121	#135	C₂²×C₃×C₅	1	1	yes	characteristic
122	#131	—	2	1	no	—
123	#125	—	2	1	no	—
124	M₃₂	C₂²×C₃×C₅	1	2	yes	—
125	#108	C₂²×C₃×C₅	1	1	yes	characteristic
126	#107	—	2	1	no	—
127	#106	C₂²×C₃×C₅	1	2	yes	—
128	#104	—	2	1	no	—
129	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	8	1	no	—
132	C₂×C₃×C₅	#476	1	1	yes	characteristic
133	C₃×Q₈	—	2	1	no	—
134	C₃×D₄	—	2	1	no	—
135	C₃×D₄	—	4	1	no	—
136	C₂×C₃×C₄	—	2	1	no	—
137	C₂×C₃×C₄	—	4	1	no	—
138	C₂×C₃×C₄	—	4	2	no	—
139	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
140	C₃×C₈	—	4	1	no	—
141	C₃×C₈	C₅×SD₁₆	1	1	yes	characteristic
142	C₃×C₈	C₅×SD₁₆	1	1	yes	characteristic
143	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
144	C₃×C₈	C₅×#41	1	1	yes	characteristic
145	C₂²×C₅	—	4	1	no	—
146	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₄×C₅	—	4	1	no	—
148	C₄×C₅	—	4	1	no	—
149	C₄×C₅	—	8	2	no	—
150	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
151	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
152	#41	—	2	1	no	—
153	Q₁₆	C₃×C₅×Q₈	1	1	yes	characteristic
154	SD₁₆	—	2	1	no	—
155	D₈	C₃×C₅×Q₈	1	1	yes	characteristic
156	M₁₆	—	2	1	no	—
157	M₁₆	—	4	1	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
160	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
164	C₁₆	—	2	2	no	—
165	C₃×C₅	#2231	1	1	yes	characteristic, direct product factor, Hall
166	C₂²×C₃	—	4	1	no	—
167	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₃×C₄	—	4	1	no	—
169	C₃×C₄	—	4	1	no	—
170	C₃×C₄	—	8	2	no	—
171	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
172	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
173	C₂×C₅	—	2	1	no	—
174	C₂×C₅	—	8	1	no	—
175	C₂×C₅	C₃×#476	1	1	yes	characteristic
176	Q₈	—	2	1	no	—
177	D₄	—	2	1	no	—
178	D₄	—	4	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	4	1	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
183	C₈	—	4	1	no	—
184	C₈	C₃×C₅×SD₁₆	1	1	yes	characteristic
185	C₈	C₃×C₅×SD₁₆	1	1	yes	characteristic
186	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
187	C₈	C₃×C₅×#41	1	1	yes	characteristic
188	C₂×C₃	—	2	1	no	—
189	C₂×C₃	—	8	1	no	—
190	C₂×C₃	C₅×#476	1	1	yes	characteristic
191	C₅	C₃×#2231	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	—	4	1	no	—
193	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
194	C₄	—	4	1	no	—
195	C₄	—	4	1	no	—
196	C₄	—	8	2	no	—
197	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
198	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
199	C₃	C₅×#2231	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	—	2	1	no	—
201	C₂	—	8	1	no	—
202	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×#499	C₂	semidirect
3. C₃×C₅×#365	C₂	semidirect
4. C₃×C₅×#364	C₂	semidirect
6. C₅×#2231	C₃	direct (N × H)
16. C₃×#2231	C₅	direct (N × H)
17. C₅×#499	C₂×C₃	semidirect
19. C₅×#365	C₂×C₃	semidirect
20. C₅×#364	C₂×C₃	semidirect
35. C₃×#499	C₂×C₅	semidirect
37. C₃×#365	C₂×C₅	semidirect
38. C₃×#364	C₂×C₅	semidirect
49. #2231	C₃×C₅	direct (N × H)
84. #499	C₂×C₃×C₅	semidirect
86. #365	C₂×C₃×C₅	semidirect
87. #364	C₂×C₃×C₅	semidirect
165. C₃×C₅	#2231	direct (N × H)
191. C₅	C₃×#2231	direct (N × H)
199. C₃	C₅×#2231	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×#499	2
C₃×C₅×#443	2
C₃×C₅×#365	2
C₃×C₅×#364	2
C₃×C₅×#360	2
C₅×#2231	3
C₃×#2231	5

Subgroups — C3×C5×#2231 #14275265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2231 #14275265