Subgroups — C₃×C₅×#1868 #14274902

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#424	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#408	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#344	C₂	1	2	yes	maximal, complemented
6	C₅×#1868	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₅×#131	—	2	4	no	—
9	C₂×C₃×C₅×M₁₆	—	2	2	no	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#117	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	—	2	4	no	—
15	C₃×C₅×#99	C₄	1	4	yes	—
16	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
17	C₃×#1868	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₅×#568	C₂×C₃	1	1	yes	characteristic
19	C₅×#424	C₂×C₃	1	2	yes	complemented
20	C₅×#410	C₂×C₃	1	1	yes	characteristic
21	C₅×#408	C₂×C₃	1	1	yes	characteristic
22	C₅×#344	C₂×C₃	1	2	yes	complemented
23	C₃×C₅×#41	—	2	2	no	—
24	C₃×C₅×#41	—	2	2	no	—
25	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
26	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×D₄	Q₈	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×M₁₆	—	2	4	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₃×C₅×M₁₆	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	4	no	—
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×#568	C₂×C₅	1	1	yes	characteristic
43	C₃×#424	C₂×C₅	1	2	yes	complemented
44	C₃×#410	C₂×C₅	1	1	yes	characteristic
45	C₃×#408	C₂×C₅	1	1	yes	characteristic
46	C₃×#344	C₂×C₅	1	2	yes	complemented
47	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
48	C₅×#131	—	2	4	no	—
49	C₂×C₅×M₁₆	—	2	2	no	—
50	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
52	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×#117	C₂²×C₃	1	2	yes	—
54	C₅×C₄⋊C₈	—	2	4	no	—
55	C₅×#99	C₃×C₄	1	4	yes	—
56	C₅×C₂²⋊C₈	—	2	2	no	—
57	#1868	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
59	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₃×C₅×Q₈	—	2	2	no	—
61	C₃×C₅×D₄	—	2	4	no	—
62	C₃×C₅×D₄	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₃×C₅×C₈	—	2	4	no	—
72	C₃×C₅×C₈	—	4	1	no	—
73	C₃×C₅×C₈	—	4	1	no	—
74	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
75	C₃×#131	—	2	4	no	—
76	C₂×C₃×M₁₆	—	2	2	no	—
77	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
79	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×#117	C₂²×C₅	1	2	yes	—
81	C₃×C₄⋊C₈	—	2	4	no	—
82	C₃×#99	C₄×C₅	1	4	yes	—
83	C₃×C₂²⋊C₈	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₅×#41	—	2	2	no	—
86	C₂×C₅×Q₈	C₃×Q₈	1	1	yes	characteristic
87	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
88	C₂×C₅×D₄	C₃×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	2	2	no	—
93	C₅×M₁₆	—	2	2	no	—
94	C₅×M₁₆	—	2	2	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	1	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	1	no	—
99	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₂²⋊C₄	C₂×C₃×C₄	1	4	yes	—
102	C₄²×C₅	—	2	2	no	—
103	#568	C₂×C₃×C₅	1	1	yes	characteristic
104	#424	C₂×C₃×C₅	1	2	yes	complemented
105	#410	C₂×C₃×C₅	1	1	yes	characteristic
106	#408	C₂×C₃×C₅	1	1	yes	characteristic
107	#344	C₂×C₃×C₅	1	2	yes	complemented
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	2	no	—
111	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
113	C₃×C₄×C₅	—	2	2	no	—
114	C₃×C₄×C₅	—	4	4	no	—
115	C₃×C₄×C₅	#120	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
117	C₃×C₄×C₅	#123	1	1	yes	characteristic
118	C₃×#41	—	2	2	no	—
119	C₃×#41	—	2	2	no	—
120	C₂×C₃×Q₈	C₅×Q₈	1	1	yes	characteristic
121	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
122	C₂×C₃×D₄	C₅×Q₈	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
124	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
125	C₃×M₁₆	—	2	4	no	—
126	C₃×M₁₆	—	2	2	no	—
127	C₃×M₁₆	—	2	2	no	—
128	C₃×M₁₆	—	2	2	no	—
129	C₂×C₃×C₈	—	2	4	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	2	no	—
133	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
136	C₃×C₄²	—	2	2	no	—
137	C₂³×C₅	C₃×C₄⋊C₄	1	2	yes	—
138	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
139	C₅×Q₈	—	2	2	no	—
140	C₅×D₄	—	2	4	no	—
141	C₅×D₄	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
146	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₃×#41	1	2	yes	—
150	C₅×C₈	—	2	4	no	—
151	C₅×C₈	—	4	1	no	—
152	C₅×C₈	—	4	1	no	—
153	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
154	#131	—	2	4	no	—
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
158	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
159	#117	C₂²×C₃×C₅	1	2	yes	—
160	C₄⋊C₈	—	2	4	no	—
161	#99	C₃×C₄×C₅	1	4	yes	—
162	C₂²⋊C₈	—	2	2	no	—
163	C₂×C₃×C₅	—	2	2	no	—
164	C₂×C₃×C₅	—	2	1	no	—
165	C₂×C₃×C₅	—	4	2	no	—
166	C₂×C₃×C₅	#385	1	1	yes	characteristic
167	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
168	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
169	C₃×Q₈	—	2	2	no	—
170	C₃×D₄	—	2	4	no	—
171	C₃×D₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
179	C₂×C₃×C₄	C₅×#41	1	2	yes	—
180	C₃×C₈	—	2	4	no	—
181	C₃×C₈	—	4	1	no	—
182	C₃×C₈	—	4	1	no	—
183	C₂²×C₅	—	2	2	no	—
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	—	4	2	no	—
186	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
187	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	4	4	no	—
190	C₄×C₅	C₃×#120	1	1	yes	characteristic
191	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
192	C₄×C₅	C₃×#123	1	1	yes	characteristic
193	#41	—	2	2	no	—
194	#41	—	2	2	no	—
195	C₂×Q₈	C₃×C₅×Q₈	1	1	yes	characteristic
196	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
197	C₂×D₄	C₃×C₅×Q₈	1	1	yes	characteristic
198	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
199	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	2	no	—
202	M₁₆	—	2	2	no	—
203	M₁₆	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
211	C₄²	—	2	2	no	—
212	C₃×C₅	#1868	1	1	yes	characteristic, direct product factor, Hall
213	C₂²×C₃	—	2	2	no	—
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	—	4	2	no	—
216	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
217	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	4	4	no	—
220	C₃×C₄	C₅×#120	1	1	yes	characteristic
221	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
222	C₃×C₄	C₅×#123	1	1	yes	characteristic
223	C₂×C₅	—	2	2	no	—
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	4	2	no	—
226	C₂×C₅	C₃×#385	1	1	yes	characteristic
227	C₂³	C₃×C₅×C₄⋊C₄	1	2	yes	—
228	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
229	Q₈	—	2	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
237	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
239	C₂×C₄	C₃×C₅×#41	1	2	yes	—
240	C₈	—	2	4	no	—
241	C₈	—	4	1	no	—
242	C₈	—	4	1	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	4	2	no	—
246	C₂×C₃	C₅×#385	1	1	yes	characteristic
247	C₅	C₃×#1868	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	2	no	—
251	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
252	C₂²	C₃×C₄×C₅×D₄	1	2	yes	—
253	C₄	—	2	2	no	—
254	C₄	—	4	4	no	—
255	C₄	C₃×C₅×#120	1	1	yes	characteristic
256	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
257	C₄	C₃×C₅×#123	1	1	yes	characteristic
258	C₃	C₅×#1868	1	1	yes	characteristic, direct product factor, Sylow
259	C₂	—	2	2	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	C₃×C₅×#385	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#120
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#424	C₂	semidirect
5. C₃×C₅×#344	C₂	semidirect
6. C₅×#1868	C₃	direct (N × H)
17. C₃×#1868	C₅	direct (N × H)
19. C₅×#424	C₂×C₃	semidirect
22. C₅×#344	C₂×C₃	semidirect
43. C₃×#424	C₂×C₅	semidirect
46. C₃×#344	C₂×C₅	semidirect
57. #1868	C₃×C₅	direct (N × H)
104. #424	C₂×C₃×C₅	semidirect
107. #344	C₂×C₃×C₅	semidirect
212. C₃×C₅	#1868	direct (N × H)
247. C₅	C₃×#1868	direct (N × H)
258. C₃	C₅×#1868	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×#568	2
C₃×C₅×#424	2
C₃×C₅×#410	2
C₃×C₅×#408	2
C₃×C₅×#344	2
C₅×#1868	3
C₃×#1868	5

Subgroups — C3×C5×#1868 #14274902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1868 #14274902