Subgroups — C₃×C₅×#1693 #14274727

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 244 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, complemented
2	C₃×C₅×#461	C₂	1	2	yes	maximal
3	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#332	C₂	1	2	yes	maximal, complemented
6	C₅×#1693	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
8	C₂×C₃×C₅×M₁₆	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₃×C₅×#107	—	2	1	no	—
12	C₃×C₅×#107	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1693	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#461	C₂×C₃	1	2	yes	—
21	C₅×#436	C₂×C₃	1	1	yes	characteristic
22	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#332	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	—	2	1	no	—
27	C₃×C₅×SD₁₆	—	4	4	no	—
28	C₃×C₅×M₁₆	—	4	2	no	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
36	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
37	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#461	C₂×C₅	1	2	yes	—
40	C₃×#436	C₂×C₅	1	1	yes	characteristic
41	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#332	C₂×C₅	1	2	yes	complemented
43	C₂×C₅×SD₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	—	2	1	no	—
45	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
46	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
47	C₅×#107	—	2	1	no	—
48	C₅×#107	—	2	1	no	—
49	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
50	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
51	C₅×#103	—	2	2	no	—
52	C₅×C₂²⋊C₈	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#1693	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	2	1	no	—
56	C₃×C₅×Q₈	—	4	2	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×D₄	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
66	C₃×C₅×C₈	—	2	1	no	—
67	C₃×C₅×C₈	—	4	1	no	—
68	C₃×C₅×C₈	—	4	2	no	—
69	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
70	C₂×C₃×SD₁₆	—	2	2	no	—
71	C₂×C₃×M₁₆	—	2	1	no	—
72	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
73	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
74	C₃×#107	—	2	1	no	—
75	C₃×#107	—	2	1	no	—
76	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
77	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₃×#103	—	2	2	no	—
79	C₃×C₂²⋊C₈	—	2	1	no	—
80	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	—	2	1	no	—
84	C₅×SD₁₆	—	4	4	no	—
85	C₅×M₁₆	—	4	2	no	—
86	C₂×C₅×C₈	—	2	1	no	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
93	C₅×C₂²⋊C₄	—	2	1	no	—
94	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
95	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
96	#461	C₂×C₃×C₅	1	2	yes	—
97	#436	C₂×C₃×C₅	1	1	yes	characteristic
98	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	#332	C₂×C₃×C₅	1	2	yes	complemented
100	C₂²×C₃×C₅	—	4	1	no	—
101	C₂²×C₃×C₅	—	4	1	no	—
102	C₂²×C₃×C₅	—	4	1	no	—
103	C₂²×C₃×C₅	#121	1	1	yes	characteristic
104	C₃×C₄×C₅	—	2	1	no	—
105	C₃×C₄×C₅	—	4	1	no	—
106	C₃×C₄×C₅	—	8	2	no	—
107	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
108	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
109	C₃×C₄×C₅	#137	1	1	yes	characteristic
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₃×SD₁₆	—	4	4	no	—
114	C₃×M₁₆	—	4	2	no	—
115	C₂×C₃×C₈	—	2	1	no	—
116	C₂×C₃×C₈	—	2	2	no	—
117	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
122	C₃×C₂²⋊C₄	—	2	1	no	—
123	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
124	C₂³×C₅	—	2	1	no	—
125	C₅×Q₈	—	4	2	no	—
126	C₅×Q₈	—	4	2	no	—
127	C₅×D₄	—	2	2	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
135	C₅×C₈	—	2	1	no	—
136	C₅×C₈	—	4	1	no	—
137	C₅×C₈	—	4	2	no	—
138	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
139	C₂×SD₁₆	—	2	2	no	—
140	C₂×M₁₆	—	2	1	no	—
141	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
142	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
143	#107	—	2	1	no	—
144	#107	—	2	1	no	—
145	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
147	#103	—	2	2	no	—
148	C₂²⋊C₈	—	2	1	no	—
149	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₂×C₃×C₅	—	8	1	no	—
151	C₂×C₃×C₅	#461	1	1	yes	characteristic
152	C₂×C₃×C₅	#469	1	1	yes	characteristic
153	C₂×C₃×C₅	#473	1	1	yes	characteristic
154	C₂³×C₃	—	2	1	no	—
155	C₃×Q₈	—	4	2	no	—
156	C₃×Q₈	—	4	2	no	—
157	C₃×D₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
165	C₃×C₈	—	2	1	no	—
166	C₃×C₈	—	4	1	no	—
167	C₃×C₈	—	4	2	no	—
168	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
169	C₂²×C₅	—	4	1	no	—
170	C₂²×C₅	—	4	1	no	—
171	C₂²×C₅	—	4	1	no	—
172	C₂²×C₅	C₃×#121	1	1	yes	characteristic
173	C₄×C₅	—	2	1	no	—
174	C₄×C₅	—	4	1	no	—
175	C₄×C₅	—	8	2	no	—
176	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
177	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
178	C₄×C₅	C₃×#137	1	1	yes	characteristic
179	C₂×Q₈	—	2	2	no	—
180	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	SD₁₆	—	4	4	no	—
183	M₁₆	—	4	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
187	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
191	C₂²⋊C₄	—	2	1	no	—
192	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
193	C₃×C₅	#1693	1	1	yes	characteristic, direct product factor, Hall
194	C₂²×C₃	—	4	1	no	—
195	C₂²×C₃	—	4	1	no	—
196	C₂²×C₃	—	4	1	no	—
197	C₂²×C₃	C₅×#121	1	1	yes	characteristic
198	C₃×C₄	—	2	1	no	—
199	C₃×C₄	—	4	1	no	—
200	C₃×C₄	—	8	2	no	—
201	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
202	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
203	C₃×C₄	C₅×#137	1	1	yes	characteristic
204	C₂×C₅	—	8	1	no	—
205	C₂×C₅	C₃×#461	1	1	yes	characteristic
206	C₂×C₅	C₃×#469	1	1	yes	characteristic
207	C₂×C₅	C₃×#473	1	1	yes	characteristic
208	C₂³	—	2	1	no	—
209	Q₈	—	4	2	no	—
210	Q₈	—	4	2	no	—
211	D₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
218	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	4	1	no	—
221	C₈	—	4	2	no	—
222	C₈	C₃×C₅×SD₁₆	1	2	yes	complemented
223	C₂×C₃	—	8	1	no	—
224	C₂×C₃	C₅×#461	1	1	yes	characteristic
225	C₂×C₃	C₅×#469	1	1	yes	characteristic
226	C₂×C₃	C₅×#473	1	1	yes	characteristic
227	C₅	C₃×#1693	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²	—	4	1	no	—
229	C₂²	—	4	1	no	—
230	C₂²	—	4	1	no	—
231	C₂²	C₃×C₅×#121	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	8	2	no	—
235	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
236	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
237	C₄	C₃×C₅×#137	1	1	yes	characteristic
238	C₃	C₅×#1693	1	1	yes	characteristic, direct product factor, Sylow
239	C₂	—	8	1	no	—
240	C₂	C₃×C₅×#461	1	1	yes	characteristic
241	C₂	C₃×C₅×#469	1	1	yes	characteristic
242	C₂	C₃×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₄	D₄
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
1. C₃×C₅×#500	C₂	semidirect
4. C₃×C₅×#334	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#1693	C₃	direct (N × H)
18. C₃×#1693	C₅	direct (N × H)
19. C₅×#500	C₂×C₃	semidirect
22. C₅×#334	C₂×C₃	semidirect
23. C₅×#332	C₂×C₃	semidirect
38. C₃×#500	C₂×C₅	semidirect
41. C₃×#334	C₂×C₅	semidirect
42. C₃×#332	C₂×C₅	semidirect
54. #1693	C₃×C₅	direct (N × H)
69. C₃×C₅×C₈	SD₁₆	semidirect
95. #500	C₂×C₃×C₅	semidirect
98. #334	C₂×C₃×C₅	semidirect
99. #332	C₂×C₃×C₅	semidirect
138. C₅×C₈	C₃×SD₁₆	semidirect
168. C₃×C₈	C₅×SD₁₆	semidirect
193. C₃×C₅	#1693	direct (N × H)
222. C₈	C₃×C₅×SD₁₆	semidirect
227. C₅	C₃×#1693	direct (N × H)
238. C₃	C₅×#1693	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×#461	2
C₃×C₅×#436	2
C₃×C₅×#334	2
C₃×C₅×#332	2
C₅×#1693	3
C₃×#1693	5

Subgroups — C3×C5×#1693 #14274727

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1693 #14274727