Subgroups — C₃×C₅×#2228 #14275262

Order 1920.

272 conjugacy classes of subgroups.

The subgroup lattice has 252 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×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₅×#357	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#357	C₂	1	1	yes	maximal, characteristic
8	C₅×#2228	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#125	—	2	1	no	—
11	C₃×C₄×C₅×D₄	—	2	1	no	—
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	1	yes	characteristic
15	C₃×C₅×#107	C₂²	1	1	yes	characteristic
16	C₃×C₅×#107	C₂²	1	1	yes	characteristic
17	C₃×C₅×#106	—	2	1	no	—
18	C₃×C₅×#102	—	2	1	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×#2228	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#499	C₂×C₃	1	1	yes	characteristic, complemented
22	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic
23	C₅×#367	C₂×C₃	1	1	yes	characteristic, complemented
24	C₅×#366	C₂×C₃	1	1	yes	characteristic, complemented
25	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
26	C₅×#357	C₂×C₃	1	1	yes	characteristic
27	C₅×#357	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×C₅×D₄	—	2	1	no	—
29	C₂²×C₃×C₄×C₅	—	4	1	no	—
30	C₃×C₅×D₈	—	2	1	no	—
31	C₃×C₅×D₈	Q₈	1	2	yes	—
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	—	2	1	no	—
37	C₃×C₅×C₄⋊C₄	—	4	1	no	—
38	C₃×C₅×C₄⋊C₄	—	4	1	no	—
39	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₅×C₁₆	—	2	1	no	—
42	C₃×C₅×C₁₆	—	2	1	no	—
43	C₃×#499	C₂×C₅	1	1	yes	characteristic, complemented
44	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
45	C₃×#367	C₂×C₅	1	1	yes	characteristic, complemented
46	C₃×#366	C₂×C₅	1	1	yes	characteristic, complemented
47	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×#357	C₂×C₅	1	1	yes	characteristic
49	C₃×#357	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×#125	—	2	1	no	—
52	C₄×C₅×D₄	—	2	1	no	—
53	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
55	C₅×#107	C₂²×C₃	1	1	yes	characteristic
56	C₅×#107	C₂²×C₃	1	1	yes	characteristic
57	C₅×#107	C₂²×C₃	1	1	yes	characteristic
58	C₅×#106	—	2	1	no	—
59	C₅×#102	—	2	1	no	—
60	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
61	#2228	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₂³×C₃×C₅	—	4	1	no	—
63	C₃×C₅×D₄	—	2	2	no	—
64	C₃×C₅×D₄	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	4	1	no	—
66	C₂×C₃×C₄×C₅	—	4	1	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	—	8	2	no	—
69	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₃×C₅×C₈	—	2	1	no	—
73	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₃×C₅×C₈	#41	1	1	yes	characteristic
75	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
76	C₃×#125	—	2	1	no	—
77	C₃×C₄×D₄	—	2	1	no	—
78	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
80	C₃×#107	C₂²×C₅	1	1	yes	characteristic
81	C₃×#107	C₂²×C₅	1	1	yes	characteristic
82	C₃×#107	C₂²×C₅	1	1	yes	characteristic
83	C₃×#106	—	2	1	no	—
84	C₃×#102	—	2	1	no	—
85	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₅×D₄	—	2	1	no	—
87	C₂²×C₄×C₅	—	4	1	no	—
88	C₅×D₈	—	2	1	no	—
89	C₅×D₈	C₃×Q₈	1	2	yes	—
90	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₄⋊C₄	—	2	1	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₅×C₄⋊C₄	—	4	1	no	—
97	C₅×C₂²⋊C₄	—	4	1	no	—
98	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
99	C₅×C₁₆	—	2	1	no	—
100	C₅×C₁₆	—	2	1	no	—
101	#499	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic
103	#367	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#366	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	#357	C₂×C₃×C₅	1	1	yes	characteristic
107	#357	C₂×C₃×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₅	—	4	2	no	—
109	C₂²×C₃×C₅	—	8	1	no	—
110	C₂²×C₃×C₅	—	8	1	no	—
111	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	8	1	no	—
115	C₃×C₄×C₅	—	8	1	no	—
116	C₃×C₄×C₅	—	8	1	no	—
117	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
118	C₃×C₄×C₅	#137	1	1	yes	characteristic
119	C₂×C₃×D₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	4	1	no	—
121	C₃×D₈	—	2	1	no	—
122	C₃×D₈	C₅×Q₈	1	2	yes	—
123	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
125	C₃×C₄⋊C₄	—	2	1	no	—
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₄⋊C₄	—	4	1	no	—
129	C₃×C₄⋊C₄	—	4	1	no	—
130	C₃×C₂²⋊C₄	—	4	1	no	—
131	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
132	C₃×C₁₆	—	2	1	no	—
133	C₃×C₁₆	—	2	1	no	—
134	C₂³×C₅	—	4	1	no	—
135	C₅×D₄	—	2	2	no	—
136	C₅×D₄	—	4	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	1	no	—
140	C₂×C₄×C₅	—	8	2	no	—
141	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
144	C₅×C₈	—	2	1	no	—
145	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₅×C₈	C₃×#41	1	1	yes	characteristic
147	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
148	#125	—	2	1	no	—
149	C₄×D₄	—	2	1	no	—
150	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
152	#107	C₂²×C₃×C₅	1	1	yes	characteristic
153	#107	C₂²×C₃×C₅	1	1	yes	characteristic
154	#107	C₂²×C₃×C₅	1	1	yes	characteristic
155	#106	—	2	1	no	—
156	#102	—	2	1	no	—
157	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₂×C₃×C₅	—	8	2	no	—
159	C₂×C₃×C₅	#474	1	1	yes	characteristic
160	C₂×C₃×C₅	#508	1	1	yes	characteristic
161	C₂×C₃×C₅	#509	1	1	yes	characteristic
162	C₂³×C₃	—	4	1	no	—
163	C₃×D₄	—	2	2	no	—
164	C₃×D₄	—	4	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	—	8	2	no	—
169	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
170	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
172	C₃×C₈	—	2	1	no	—
173	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
174	C₃×C₈	C₅×#41	1	1	yes	characteristic
175	C₂²×C₅	—	4	2	no	—
176	C₂²×C₅	—	8	1	no	—
177	C₂²×C₅	—	8	1	no	—
178	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	8	1	no	—
182	C₄×C₅	—	8	1	no	—
183	C₄×C₅	—	8	1	no	—
184	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
185	C₄×C₅	C₃×#137	1	1	yes	characteristic
186	C₂×D₄	—	2	1	no	—
187	C₂²×C₄	—	4	1	no	—
188	D₈	—	2	1	no	—
189	D₈	C₃×C₅×Q₈	1	2	yes	—
190	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
191	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	4	1	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₂²⋊C₄	—	4	1	no	—
198	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
199	C₁₆	—	2	1	no	—
200	C₁₆	—	2	1	no	—
201	C₃×C₅	#2228	1	1	yes	characteristic, direct product factor, Hall
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	—	8	1	no	—
204	C₂²×C₃	—	8	1	no	—
205	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	8	1	no	—
209	C₃×C₄	—	8	1	no	—
210	C₃×C₄	—	8	1	no	—
211	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
212	C₃×C₄	C₅×#137	1	1	yes	characteristic
213	C₂×C₅	—	8	2	no	—
214	C₂×C₅	C₃×#474	1	1	yes	characteristic
215	C₂×C₅	C₃×#508	1	1	yes	characteristic
216	C₂×C₅	C₃×#509	1	1	yes	characteristic
217	C₂³	—	4	1	no	—
218	D₄	—	2	2	no	—
219	D₄	—	4	1	no	—
220	C₂×C₄	—	4	1	no	—
221	C₂×C₄	—	4	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
225	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
226	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
229	C₈	C₃×C₅×#41	1	1	yes	characteristic
230	C₂×C₃	—	8	2	no	—
231	C₂×C₃	C₅×#474	1	1	yes	characteristic
232	C₂×C₃	C₅×#508	1	1	yes	characteristic
233	C₂×C₃	C₅×#509	1	1	yes	characteristic
234	C₅	C₃×#2228	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	4	2	no	—
236	C₂²	—	8	1	no	—
237	C₂²	—	8	1	no	—
238	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	8	1	no	—
242	C₄	—	8	1	no	—
243	C₄	—	8	1	no	—
244	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
245	C₄	C₃×C₅×#137	1	1	yes	characteristic
246	C₃	C₅×#2228	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	8	2	no	—
248	C₂	C₃×C₅×#474	1	1	yes	characteristic
249	C₂	C₃×C₅×#508	1	1	yes	characteristic
250	C₂	C₃×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₈	C₂²
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. C₃×C₅×#499	C₂	semidirect
3. C₃×C₅×#367	C₂	semidirect
4. C₃×C₅×#366	C₂	semidirect
5. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
8. C₅×#2228	C₃	direct (N × H)
20. C₃×#2228	C₅	direct (N × H)
21. C₅×#499	C₂×C₃	semidirect
23. C₅×#367	C₂×C₃	semidirect
24. C₅×#366	C₂×C₃	semidirect
25. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
43. C₃×#499	C₂×C₅	semidirect
45. C₃×#367	C₂×C₅	semidirect
46. C₃×#366	C₂×C₅	semidirect
47. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
61. #2228	C₃×C₅	direct (N × H)
101. #499	C₂×C₃×C₅	semidirect
103. #367	C₂×C₃×C₅	semidirect
104. #366	C₂×C₃×C₅	semidirect
105. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
201. C₃×C₅	#2228	direct (N × H)
234. C₅	C₃×#2228	direct (N × H)
246. C₃	C₅×#2228	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×C₄×C₅×D₈	2
C₃×C₅×#367	2
C₃×C₅×#366	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#357	2
C₅×#2228	3
C₃×#2228	5

Subgroups — C3×C5×#2228 #14275262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2228 #14275262