Subgroups — C₃×C₅×#1689 #14274723

Order 1920.

332 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×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#474	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×#1689	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
9	C₂×C₃×C₅×M₁₆	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	—	2	2	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#102	—	2	2	no	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1689	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#474	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×Q₁₆	—	4	2	no	—
28	C₃×C₅×Q₁₆	—	4	2	no	—
29	C₃×C₅×M₁₆	—	4	2	no	—
30	C₂×C₃×C₅×C₈	—	2	1	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#474	C₂×C₅	1	2	yes	—
41	C₃×#436	C₂×C₅	1	1	yes	characteristic
42	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#332	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	—	2	1	no	—
46	C₂×C₅×M₁₆	—	2	1	no	—
47	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
48	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
49	C₅×#107	—	2	2	no	—
50	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
52	C₅×#102	—	2	2	no	—
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	#1689	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	2	1	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×Q₈	—	4	2	no	—
59	C₃×C₅×D₄	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₃×C₅×C₈	—	2	1	no	—
68	C₃×C₅×C₈	—	4	1	no	—
69	C₃×C₅×C₈	—	4	2	no	—
70	C₃×C₅×C₈	D₈	1	2	yes	—
71	C₂×C₃×Q₁₆	—	2	1	no	—
72	C₂×C₃×Q₁₆	—	2	1	no	—
73	C₂×C₃×M₁₆	—	2	1	no	—
74	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
75	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
76	C₃×#107	—	2	2	no	—
77	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
78	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
79	C₃×#102	—	2	2	no	—
80	C₃×C₂²⋊C₈	—	2	1	no	—
81	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	—	2	1	no	—
85	C₅×Q₁₆	—	4	2	no	—
86	C₅×Q₁₆	—	4	2	no	—
87	C₅×M₁₆	—	4	2	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
91	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
95	C₅×C₂²⋊C₄	—	2	1	no	—
96	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
97	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
98	#474	C₂×C₃×C₅	1	2	yes	—
99	#436	C₂×C₃×C₅	1	1	yes	characteristic
100	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	#332	C₂×C₃×C₅	1	2	yes	complemented
102	C₂²×C₃×C₅	—	4	1	no	—
103	C₂²×C₃×C₅	—	4	1	no	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	#121	1	1	yes	characteristic
106	C₃×C₄×C₅	—	2	1	no	—
107	C₃×C₄×C₅	—	4	1	no	—
108	C₃×C₄×C₅	—	8	2	no	—
109	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
110	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₃×C₄×C₅	#137	1	2	yes	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₃×Q₁₆	—	4	2	no	—
116	C₃×Q₁₆	—	4	2	no	—
117	C₃×M₁₆	—	4	2	no	—
118	C₂×C₃×C₈	—	2	1	no	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
122	C₃×C₄⋊C₄	—	2	2	no	—
123	C₃×C₄⋊C₄	—	2	2	no	—
124	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
125	C₃×C₂²⋊C₄	—	2	1	no	—
126	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
127	C₂³×C₅	—	2	1	no	—
128	C₅×Q₈	—	4	2	no	—
129	C₅×Q₈	—	4	2	no	—
130	C₅×D₄	—	2	2	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
138	C₅×C₈	—	2	1	no	—
139	C₅×C₈	—	4	1	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	C₃×D₈	1	2	yes	—
142	C₂×Q₁₆	—	2	1	no	—
143	C₂×Q₁₆	—	2	1	no	—
144	C₂×M₁₆	—	2	1	no	—
145	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
146	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
147	#107	—	2	2	no	—
148	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
150	#102	—	2	2	no	—
151	C₂²⋊C₈	—	2	1	no	—
152	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₂×C₃×C₅	—	8	1	no	—
154	C₂×C₃×C₅	#459	1	1	yes	characteristic
155	C₂×C₃×C₅	#470	1	1	yes	characteristic
156	C₂×C₃×C₅	#473	1	1	yes	characteristic
157	C₂³×C₃	—	2	1	no	—
158	C₃×Q₈	—	4	2	no	—
159	C₃×Q₈	—	4	2	no	—
160	C₃×D₄	—	2	2	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	—	4	1	no	—
164	C₂×C₃×C₄	—	4	2	no	—
165	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
168	C₃×C₈	—	2	1	no	—
169	C₃×C₈	—	4	1	no	—
170	C₃×C₈	—	4	2	no	—
171	C₃×C₈	C₅×D₈	1	2	yes	—
172	C₂²×C₅	—	4	1	no	—
173	C₂²×C₅	—	4	1	no	—
174	C₂²×C₅	—	4	1	no	—
175	C₂²×C₅	C₃×#121	1	1	yes	characteristic
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	4	1	no	—
178	C₄×C₅	—	8	2	no	—
179	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
180	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₄×C₅	C₃×#137	1	2	yes	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	Q₁₆	—	4	2	no	—
186	Q₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
191	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
195	C₂²⋊C₄	—	2	1	no	—
196	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
197	C₃×C₅	#1689	1	1	yes	characteristic, direct product factor, Hall
198	C₂²×C₃	—	4	1	no	—
199	C₂²×C₃	—	4	1	no	—
200	C₂²×C₃	—	4	1	no	—
201	C₂²×C₃	C₅×#121	1	1	yes	characteristic
202	C₃×C₄	—	2	1	no	—
203	C₃×C₄	—	4	1	no	—
204	C₃×C₄	—	8	2	no	—
205	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
206	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₃×C₄	C₅×#137	1	2	yes	—
208	C₂×C₅	—	8	1	no	—
209	C₂×C₅	C₃×#459	1	1	yes	characteristic
210	C₂×C₅	C₃×#470	1	1	yes	characteristic
211	C₂×C₅	C₃×#473	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	Q₈	—	4	2	no	—
214	Q₈	—	4	2	no	—
215	D₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
221	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
222	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
223	C₈	—	2	1	no	—
224	C₈	—	4	1	no	—
225	C₈	—	4	2	no	—
226	C₈	C₃×C₅×D₈	1	2	yes	—
227	C₂×C₃	—	8	1	no	—
228	C₂×C₃	C₅×#459	1	1	yes	characteristic
229	C₂×C₃	C₅×#470	1	1	yes	characteristic
230	C₂×C₃	C₅×#473	1	1	yes	characteristic
231	C₅	C₃×#1689	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	4	1	no	—
233	C₂²	—	4	1	no	—
234	C₂²	—	4	1	no	—
235	C₂²	C₃×C₅×#121	1	1	yes	characteristic
236	C₄	—	2	1	no	—
237	C₄	—	4	1	no	—
238	C₄	—	8	2	no	—
239	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
240	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₄	C₃×C₅×#137	1	2	yes	—
242	C₃	C₅×#1689	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	8	1	no	—
244	C₂	C₃×C₅×#459	1	1	yes	characteristic
245	C₂	C₃×C₅×#470	1	1	yes	characteristic
246	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×#494	C₂	semidirect
4. C₃×C₅×#334	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#1689	C₃	direct (N × H)
18. C₃×#1689	C₅	direct (N × H)
19. C₅×#494	C₂×C₃	semidirect
22. C₅×#334	C₂×C₃	semidirect
23. C₅×#332	C₂×C₃	semidirect
39. C₃×#494	C₂×C₅	semidirect
42. C₃×#334	C₂×C₅	semidirect
43. C₃×#332	C₂×C₅	semidirect
55. #1689	C₃×C₅	direct (N × H)
97. #494	C₂×C₃×C₅	semidirect
100. #334	C₂×C₃×C₅	semidirect
101. #332	C₂×C₃×C₅	semidirect
197. C₃×C₅	#1689	direct (N × H)
231. C₅	C₃×#1689	direct (N × H)
242. C₃	C₅×#1689	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×#494	2
C₃×C₅×#474	2
C₃×C₅×#436	2
C₃×C₅×#334	2
C₃×C₅×#332	2
C₅×#1689	3
C₃×#1689	5

Subgroups — C3×C5×#1689 #14274723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1689 #14274723