Subgroups — C₃×C₅×#2213 #14275247

Order 1920.

340 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#509	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#364	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#361	C₂	1	2	yes	maximal, complemented
6	C₅×#2213	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×#135	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
10	C₃×C₅×#131	—	2	1	no	—
11	C₃×C₅×#127	—	2	1	no	—
12	C₃×C₅×SD₃₂	—	2	2	no	—
13	C₃×C₅×D₁₆	—	2	2	no	—
14	C₃×C₅×M₃₂	C₂²	1	2	yes	—
15	C₃×C₅×#108	C₂²	1	1	yes	characteristic
16	C₃×C₅×#104	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#2213	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#509	C₂×C₃	1	2	yes	complemented
20	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#443	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#364	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#361	C₂×C₃	1	2	yes	complemented
24	C₃×C₅×#41	—	2	1	no	—
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	4	2	no	—
27	C₃×C₅×Q₁₆	D₄	1	1	yes	characteristic, complemented
28	C₃×C₅×SD₁₆	—	2	1	no	—
29	C₃×C₅×D₈	—	2	2	no	—
30	C₃×C₅×D₈	—	2	2	no	—
31	C₃×C₅×D₈	—	4	1	no	—
32	C₃×C₅×D₈	—	4	1	no	—
33	C₃×C₅×D₈	D₄	1	1	yes	characteristic, complemented
34	C₃×C₅×M₁₆	—	2	1	no	—
35	C₃×C₅×M₁₆	—	4	1	no	—
36	C₂×C₃×C₅×C₈	—	4	1	no	—
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₅×C₁₆	—	2	2	no	—
41	C₃×#509	C₂×C₅	1	2	yes	complemented
42	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#443	C₂×C₅	1	1	yes	characteristic, complemented
44	C₃×#364	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×#361	C₂×C₅	1	2	yes	complemented
46	C₅×#135	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₅×D₈	—	2	1	no	—
48	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
49	C₅×#131	—	2	1	no	—
50	C₅×#127	—	2	1	no	—
51	C₅×SD₃₂	—	2	2	no	—
52	C₅×D₁₆	—	2	2	no	—
53	C₅×M₃₂	C₂²×C₃	1	2	yes	—
54	C₅×#108	C₂²×C₃	1	1	yes	characteristic
55	C₅×#104	—	2	1	no	—
56	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	#2213	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	—	4	2	no	—
59	C₃×C₅×Q₈	—	2	1	no	—
60	C₃×C₅×D₄	—	2	1	no	—
61	C₃×C₅×D₄	—	4	1	no	—
62	C₃×C₅×D₄	—	4	2	no	—
63	C₃×C₅×D₄	—	4	2	no	—
64	C₃×C₅×D₄	—	8	4	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	4	1	no	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×C₈	—	4	1	no	—
69	C₃×C₅×C₈	D₈	1	1	yes	characteristic
70	C₃×C₅×C₈	D₈	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×C₈	#41	1	1	yes	characteristic
73	C₃×#135	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×D₈	—	2	1	no	—
75	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
76	C₃×#131	—	2	1	no	—
77	C₃×#127	—	2	1	no	—
78	C₃×SD₃₂	—	2	2	no	—
79	C₃×D₁₆	—	2	2	no	—
80	C₃×M₃₂	C₂²×C₅	1	2	yes	—
81	C₃×#108	C₂²×C₅	1	1	yes	characteristic
82	C₃×#104	—	2	1	no	—
83	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₅×#41	—	2	1	no	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	4	2	no	—
87	C₅×Q₁₆	C₃×D₄	1	1	yes	characteristic, complemented
88	C₅×SD₁₆	—	2	1	no	—
89	C₅×D₈	—	2	2	no	—
90	C₅×D₈	—	2	2	no	—
91	C₅×D₈	—	4	1	no	—
92	C₅×D₈	—	4	1	no	—
93	C₅×D₈	C₃×D₄	1	1	yes	characteristic, complemented
94	C₅×M₁₆	—	2	1	no	—
95	C₅×M₁₆	—	4	1	no	—
96	C₂×C₅×C₈	—	4	1	no	—
97	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
99	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
100	C₅×C₁₆	—	2	2	no	—
101	#509	C₂×C₃×C₅	1	2	yes	complemented
102	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
103	#443	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#364	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	#361	C₂×C₃×C₅	1	2	yes	complemented
106	C₂²×C₃×C₅	—	4	1	no	—
107	C₂²×C₃×C₅	—	8	4	no	—
108	C₂²×C₃×C₅	—	8	2	no	—
109	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	4	1	no	—
112	C₃×C₄×C₅	#121	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
114	C₃×#41	—	2	1	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	4	2	no	—
117	C₃×Q₁₆	C₅×D₄	1	1	yes	characteristic, complemented
118	C₃×SD₁₆	—	2	1	no	—
119	C₃×D₈	—	2	2	no	—
120	C₃×D₈	—	2	2	no	—
121	C₃×D₈	—	4	1	no	—
122	C₃×D₈	—	4	1	no	—
123	C₃×D₈	C₅×D₄	1	1	yes	characteristic, complemented
124	C₃×M₁₆	—	2	1	no	—
125	C₃×M₁₆	—	4	1	no	—
126	C₂×C₃×C₈	—	4	1	no	—
127	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
129	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
130	C₃×C₁₆	—	2	2	no	—
131	C₂³×C₅	—	4	2	no	—
132	C₅×Q₈	—	2	1	no	—
133	C₅×D₄	—	2	1	no	—
134	C₅×D₄	—	4	1	no	—
135	C₅×D₄	—	4	2	no	—
136	C₅×D₄	—	4	2	no	—
137	C₅×D₄	—	8	4	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
141	C₅×C₈	—	4	1	no	—
142	C₅×C₈	C₃×D₈	1	1	yes	characteristic
143	C₅×C₈	C₃×D₈	1	1	yes	characteristic
144	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
145	C₅×C₈	C₃×#41	1	1	yes	characteristic
146	#135	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂×D₈	—	2	1	no	—
148	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
149	#131	—	2	1	no	—
150	#127	—	2	1	no	—
151	SD₃₂	—	2	2	no	—
152	D₁₆	—	2	2	no	—
153	M₃₂	C₂²×C₃×C₅	1	2	yes	—
154	#108	C₂²×C₃×C₅	1	1	yes	characteristic
155	#104	—	2	1	no	—
156	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₂×C₃×C₅	—	2	1	no	—
158	C₂×C₃×C₅	—	8	1	no	—
159	C₂×C₃×C₅	—	16	2	no	—
160	C₂×C₃×C₅	#459	1	1	yes	characteristic
161	C₂³×C₃	—	4	2	no	—
162	C₃×Q₈	—	2	1	no	—
163	C₃×D₄	—	2	1	no	—
164	C₃×D₄	—	4	1	no	—
165	C₃×D₄	—	4	2	no	—
166	C₃×D₄	—	4	2	no	—
167	C₃×D₄	—	8	4	no	—
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
171	C₃×C₈	—	4	1	no	—
172	C₃×C₈	C₅×D₈	1	1	yes	characteristic
173	C₃×C₈	C₅×D₈	1	1	yes	characteristic
174	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
175	C₃×C₈	C₅×#41	1	1	yes	characteristic
176	C₂²×C₅	—	4	1	no	—
177	C₂²×C₅	—	8	4	no	—
178	C₂²×C₅	—	8	2	no	—
179	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	C₃×#121	1	1	yes	characteristic
183	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
184	#41	—	2	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	4	2	no	—
187	Q₁₆	C₃×C₅×D₄	1	1	yes	characteristic, complemented
188	SD₁₆	—	2	1	no	—
189	D₈	—	2	2	no	—
190	D₈	—	2	2	no	—
191	D₈	—	4	1	no	—
192	D₈	—	4	1	no	—
193	D₈	C₃×C₅×D₄	1	1	yes	characteristic, complemented
194	M₁₆	—	2	1	no	—
195	M₁₆	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
198	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
199	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
200	C₁₆	—	2	2	no	—
201	C₃×C₅	#2213	1	1	yes	characteristic, direct product factor, Hall
202	C₂²×C₃	—	4	1	no	—
203	C₂²×C₃	—	8	4	no	—
204	C₂²×C₃	—	8	2	no	—
205	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	C₅×#121	1	1	yes	characteristic
209	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	—	8	1	no	—
212	C₂×C₅	—	16	2	no	—
213	C₂×C₅	C₃×#459	1	1	yes	characteristic
214	C₂³	—	4	2	no	—
215	Q₈	—	2	1	no	—
216	D₄	—	2	1	no	—
217	D₄	—	4	1	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	8	4	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
224	C₈	—	4	1	no	—
225	C₈	C₃×C₅×D₈	1	1	yes	characteristic
226	C₈	C₃×C₅×D₈	1	1	yes	characteristic
227	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
228	C₈	C₃×C₅×#41	1	1	yes	characteristic
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	8	1	no	—
231	C₂×C₃	—	16	2	no	—
232	C₂×C₃	C₅×#459	1	1	yes	characteristic
233	C₅	C₃×#2213	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	—	4	1	no	—
235	C₂²	—	8	4	no	—
236	C₂²	—	8	2	no	—
237	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₄	—	2	2	no	—
239	C₄	—	4	1	no	—
240	C₄	C₃×C₅×#121	1	1	yes	characteristic
241	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
242	C₃	C₅×#2213	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	2	1	no	—
244	C₂	—	8	1	no	—
245	C₂	—	16	2	no	—
246	C₂	C₃×C₅×#459	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#459
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
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×#509	C₂	semidirect
2. C₃×C₅×#493	C₂	semidirect
3. C₃×C₅×#443	C₂	semidirect
4. C₃×C₅×#364	C₂	semidirect
5. C₃×C₅×#361	C₂	semidirect
6. C₅×#2213	C₃	direct (N × H)
18. C₃×#2213	C₅	direct (N × H)
19. C₅×#509	C₂×C₃	semidirect
20. C₅×#493	C₂×C₃	semidirect
21. C₅×#443	C₂×C₃	semidirect
22. C₅×#364	C₂×C₃	semidirect
23. C₅×#361	C₂×C₃	semidirect
27. C₃×C₅×Q₁₆	D₄	semidirect
33. C₃×C₅×D₈	D₄	semidirect
41. C₃×#509	C₂×C₅	semidirect
42. C₃×#493	C₂×C₅	semidirect
43. C₃×#443	C₂×C₅	semidirect
44. C₃×#364	C₂×C₅	semidirect
45. C₃×#361	C₂×C₅	semidirect
57. #2213	C₃×C₅	direct (N × H)
87. C₅×Q₁₆	C₃×D₄	semidirect
93. C₅×D₈	C₃×D₄	semidirect
101. #509	C₂×C₃×C₅	semidirect
102. #493	C₂×C₃×C₅	semidirect
103. #443	C₂×C₃×C₅	semidirect
104. #364	C₂×C₃×C₅	semidirect
105. #361	C₂×C₃×C₅	semidirect
117. C₃×Q₁₆	C₅×D₄	semidirect
123. C₃×D₈	C₅×D₄	semidirect
187. Q₁₆	C₃×C₅×D₄	semidirect
193. D₈	C₃×C₅×D₄	semidirect
201. C₃×C₅	#2213	direct (N × H)
233. C₅	C₃×#2213	direct (N × H)
242. C₃	C₅×#2213	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×#509	2
C₃×C₅×#493	2
C₃×C₅×#443	2
C₃×C₅×#364	2
C₃×C₅×#361	2
C₅×#2213	3
C₃×#2213	5

Subgroups — C3×C5×#2213 #14275247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2213 #14275247