Subgroups — C₃×C₅×#1723 #14274757

Order 1920.

352 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#487	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic
8	C₅×#1723	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	—	2	2	no	—
15	C₃×C₅×#107	—	2	2	no	—
16	C₃×C₅×#106	—	2	2	no	—
17	C₃×C₅×#103	—	2	2	no	—
18	C₃×C₅×#103	—	2	2	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₃×#1723	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#500	C₂×C₃	1	1	yes	characteristic
24	C₅×#500	C₂×C₃	1	1	yes	characteristic
25	C₅×#498	C₂×C₃	1	1	yes	characteristic
26	C₅×#494	C₂×C₃	1	1	yes	characteristic
27	C₅×#487	C₂×C₃	1	1	yes	characteristic
28	C₅×#487	C₂×C₃	1	1	yes	characteristic
29	C₅×#322	C₂×C₃	1	1	yes	characteristic
30	C₂×C₃×C₅×Q₈	—	2	1	no	—
31	C₂×C₃×C₅×Q₈	—	2	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₃×C₅×Q₁₆	—	4	4	no	—
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×#500	C₂×C₅	1	1	yes	characteristic
48	C₃×#500	C₂×C₅	1	1	yes	characteristic
49	C₃×#498	C₂×C₅	1	1	yes	characteristic
50	C₃×#494	C₂×C₅	1	1	yes	characteristic
51	C₃×#487	C₂×C₅	1	1	yes	characteristic
52	C₃×#487	C₂×C₅	1	1	yes	characteristic
53	C₃×#322	C₂×C₅	1	1	yes	characteristic
54	C₂×C₅×Q₁₆	—	2	2	no	—
55	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
56	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×#107	—	2	2	no	—
60	C₅×#107	—	2	2	no	—
61	C₅×#106	—	2	2	no	—
62	C₅×#103	—	2	2	no	—
63	C₅×#103	—	2	2	no	—
64	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
66	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
67	#1723	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
68	C₃×C₅×Q₈	—	4	2	no	—
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	4	1	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₃×C₅×C₈	—	2	2	no	—
80	C₃×C₅×C₈	—	2	2	no	—
81	C₃×C₅×C₈	Q₁₆	1	4	yes	complemented
82	C₂×C₃×Q₁₆	—	2	2	no	—
83	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
85	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
86	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
87	C₃×#107	—	2	2	no	—
88	C₃×#107	—	2	2	no	—
89	C₃×#106	—	2	2	no	—
90	C₃×#103	—	2	2	no	—
91	C₃×#103	—	2	2	no	—
92	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	2	1	no	—
97	C₂×C₅×Q₈	—	2	1	no	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₅×Q₁₆	—	4	4	no	—
100	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
101	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
102	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
103	C₅×C₄⋊C₄	—	2	1	no	—
104	C₅×C₄⋊C₄	—	2	1	no	—
105	C₅×C₄⋊C₄	—	2	1	no	—
106	C₅×C₄⋊C₄	—	2	1	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	1	no	—
110	C₅×C₄⋊C₄	—	2	1	no	—
111	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
112	#500	C₂×C₃×C₅	1	1	yes	characteristic
113	#500	C₂×C₃×C₅	1	1	yes	characteristic
114	#498	C₂×C₃×C₅	1	1	yes	characteristic
115	#494	C₂×C₃×C₅	1	1	yes	characteristic
116	#487	C₂×C₃×C₅	1	1	yes	characteristic
117	#487	C₂×C₃×C₅	1	1	yes	characteristic
118	#322	C₂×C₃×C₅	1	1	yes	characteristic
119	C₂²×C₃×C₅	#127	1	1	yes	characteristic
120	C₃×C₄×C₅	—	8	1	no	—
121	C₃×C₄×C₅	—	8	1	no	—
122	C₃×C₄×C₅	—	8	1	no	—
123	C₃×C₄×C₅	—	8	1	no	—
124	C₃×C₄×C₅	C₂×Q₁₆	1	2	yes	—
125	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
126	C₃×C₄×C₅	#137	1	2	yes	—
127	C₂×C₃×Q₈	—	2	1	no	—
128	C₂×C₃×Q₈	—	2	1	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×Q₈	—	2	1	no	—
131	C₃×Q₁₆	—	4	4	no	—
132	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
133	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
134	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
135	C₃×C₄⋊C₄	—	2	1	no	—
136	C₃×C₄⋊C₄	—	2	1	no	—
137	C₃×C₄⋊C₄	—	2	1	no	—
138	C₃×C₄⋊C₄	—	2	1	no	—
139	C₃×C₄⋊C₄	—	2	1	no	—
140	C₃×C₄⋊C₄	—	2	1	no	—
141	C₃×C₄⋊C₄	—	2	1	no	—
142	C₃×C₄⋊C₄	—	2	1	no	—
143	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	4	2	no	—
147	C₅×Q₈	—	4	2	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	—	4	1	no	—
150	C₂×C₄×C₅	—	4	1	no	—
151	C₂×C₄×C₅	—	4	1	no	—
152	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
155	C₅×C₈	—	2	2	no	—
156	C₅×C₈	—	2	2	no	—
157	C₅×C₈	C₃×Q₁₆	1	4	yes	complemented
158	C₂×Q₁₆	—	2	2	no	—
159	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
160	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
163	#107	—	2	2	no	—
164	#107	—	2	2	no	—
165	#106	—	2	2	no	—
166	#103	—	2	2	no	—
167	#103	—	2	2	no	—
168	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
169	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
170	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
171	C₂×C₃×C₅	#494	1	1	yes	characteristic
172	C₂×C₃×C₅	#496	1	1	yes	characteristic
173	C₂×C₃×C₅	#497	1	1	yes	characteristic
174	C₃×Q₈	—	4	2	no	—
175	C₃×Q₈	—	4	2	no	—
176	C₃×Q₈	—	4	2	no	—
177	C₃×Q₈	—	4	2	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	—	4	1	no	—
182	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	2	2	no	—
187	C₃×C₈	C₅×Q₁₆	1	4	yes	complemented
188	C₂²×C₅	C₃×#127	1	1	yes	characteristic
189	C₄×C₅	—	8	1	no	—
190	C₄×C₅	—	8	1	no	—
191	C₄×C₅	—	8	1	no	—
192	C₄×C₅	—	8	1	no	—
193	C₄×C₅	C₂×C₃×Q₁₆	1	2	yes	—
194	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
195	C₄×C₅	C₃×#137	1	2	yes	—
196	C₂×Q₈	—	2	1	no	—
197	C₂×Q₈	—	2	1	no	—
198	C₂×Q₈	—	2	1	no	—
199	C₂×Q₈	—	2	1	no	—
200	Q₁₆	—	4	4	no	—
201	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
202	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
203	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
213	C₃×C₅	#1723	1	1	yes	characteristic, direct product factor, Hall
214	C₂²×C₃	C₅×#127	1	1	yes	characteristic
215	C₃×C₄	—	8	1	no	—
216	C₃×C₄	—	8	1	no	—
217	C₃×C₄	—	8	1	no	—
218	C₃×C₄	—	8	1	no	—
219	C₃×C₄	C₂×C₅×Q₁₆	1	2	yes	—
220	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
221	C₃×C₄	C₅×#137	1	2	yes	—
222	C₂×C₅	C₃×#494	1	1	yes	characteristic
223	C₂×C₅	C₃×#496	1	1	yes	characteristic
224	C₂×C₅	C₃×#497	1	1	yes	characteristic
225	Q₈	—	4	2	no	—
226	Q₈	—	4	2	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	4	2	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
234	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	C₃×C₅×Q₁₆	1	4	yes	complemented
239	C₂×C₃	C₅×#494	1	1	yes	characteristic
240	C₂×C₃	C₅×#496	1	1	yes	characteristic
241	C₂×C₃	C₅×#497	1	1	yes	characteristic
242	C₅	C₃×#1723	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	C₃×C₅×#127	1	1	yes	characteristic
244	C₄	—	8	1	no	—
245	C₄	—	8	1	no	—
246	C₄	—	8	1	no	—
247	C₄	—	8	1	no	—
248	C₄	C₂×C₃×C₅×Q₁₆	1	2	yes	—
249	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
250	C₄	C₃×C₅×#137	1	2	yes	—
251	C₃	C₅×#1723	1	1	yes	characteristic, direct product factor, Sylow
252	C₂	C₃×C₅×#494	1	1	yes	characteristic
253	C₂	C₃×C₅×#496	1	1	yes	characteristic
254	C₂	C₃×C₅×#497	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#1723	C₃	direct (N × H)
22. C₃×#1723	C₅	direct (N × H)
67. #1723	C₃×C₅	direct (N × H)
81. C₃×C₅×C₈	Q₁₆	semidirect
157. C₅×C₈	C₃×Q₁₆	semidirect
187. C₃×C₈	C₅×Q₁₆	semidirect
213. C₃×C₅	#1723	direct (N × H)
238. C₈	C₃×C₅×Q₁₆	semidirect
242. C₅	C₃×#1723	direct (N × H)
251. C₃	C₅×#1723	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×#500	2
C₃×C₅×#498	2
C₃×C₅×#494	2
C₃×C₅×#487	2
C₃×C₅×#322	2
C₅×#1723	3
C₃×#1723	5

Subgroups — C3×C5×#1723 #14274757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1723 #14274757