Subgroups — C₃×C₅×#1669 #14274703

Order 1920.

344 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×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#474	C₂	1	2	yes	maximal
3	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#332	C₂	1	2	yes	maximal, complemented
6	C₅×#1669	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	C₂²×C₃×C₅×C₈	—	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×#1669	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×C₈×D₄	C₂×C₃	1	1	yes	characteristic
22	C₅×#335	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	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	4	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×C₈×D₄	C₂×C₅	1	1	yes	characteristic
42	C₃×#335	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	C₂²×C₅×C₈	—	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	#1669	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	1	no	—
60	C₃×C₅×D₄	Q₁₆	1	2	yes	complemented
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	—	4	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	D₈	1	2	yes	—
72	C₂×C₃×Q₁₆	—	2	1	no	—
73	C₂×C₃×Q₁₆	—	2	1	no	—
74	C₂²×C₃×C₈	—	2	1	no	—
75	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
76	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
77	C₃×#107	—	2	2	no	—
78	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
79	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
80	C₃×#102	—	2	2	no	—
81	C₃×C₂²⋊C₈	—	2	1	no	—
82	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₅×Q₁₆	—	4	2	no	—
87	C₅×Q₁₆	—	4	2	no	—
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	2	no	—
90	C₂×C₅×C₈	—	4	2	no	—
91	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
92	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	2	2	no	—
95	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
96	C₅×C₂²⋊C₄	—	2	1	no	—
97	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
98	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	#474	C₂×C₃×C₅	1	2	yes	—
100	C₈×D₄	C₂×C₃×C₅	1	1	yes	characteristic
101	#335	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	#332	C₂×C₃×C₅	1	2	yes	complemented
103	C₂²×C₃×C₅	—	2	2	no	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	—	4	1	no	—
106	C₂²×C₃×C₅	#121	1	1	yes	characteristic
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	4	1	no	—
109	C₃×C₄×C₅	—	8	2	no	—
110	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
111	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
112	C₃×C₄×C₅	#135	1	1	yes	characteristic
113	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₃×Q₁₆	—	4	2	no	—
118	C₃×Q₁₆	—	4	2	no	—
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	4	2	no	—
122	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	2	2	no	—
125	C₃×C₄⋊C₄	—	2	2	no	—
126	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
127	C₃×C₂²⋊C₄	—	2	1	no	—
128	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
129	C₂³×C₅	—	2	1	no	—
130	C₅×Q₈	—	4	2	no	—
131	C₅×Q₈	—	4	2	no	—
132	C₅×D₄	—	2	1	no	—
133	C₅×D₄	C₃×Q₁₆	1	2	yes	complemented
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
141	C₅×C₈	—	2	1	no	—
142	C₅×C₈	—	4	1	no	—
143	C₅×C₈	—	4	2	no	—
144	C₅×C₈	C₃×D₈	1	2	yes	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
149	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
150	#107	—	2	2	no	—
151	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
153	#102	—	2	2	no	—
154	C₂²⋊C₈	—	2	1	no	—
155	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
156	C₂×C₃×C₅	—	4	2	no	—
157	C₂×C₃×C₅	#459	1	1	yes	characteristic
158	C₂×C₃×C₅	#467	1	1	yes	characteristic
159	C₂×C₃×C₅	#473	1	1	yes	characteristic
160	C₂³×C₃	—	2	1	no	—
161	C₃×Q₈	—	4	2	no	—
162	C₃×Q₈	—	4	2	no	—
163	C₃×D₄	—	2	1	no	—
164	C₃×D₄	C₅×Q₁₆	1	2	yes	complemented
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	—	4	2	no	—
169	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
172	C₃×C₈	—	2	1	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	—	4	2	no	—
175	C₃×C₈	C₅×D₈	1	2	yes	—
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	4	1	no	—
178	C₂²×C₅	—	4	1	no	—
179	C₂²×C₅	C₃×#121	1	1	yes	characteristic
180	C₄×C₅	—	2	1	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	—	8	2	no	—
183	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
184	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
185	C₄×C₅	C₃×#135	1	1	yes	characteristic
186	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	Q₁₆	—	4	2	no	—
191	Q₁₆	—	4	2	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
196	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
200	C₂²⋊C₄	—	2	1	no	—
201	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
202	C₃×C₅	#1669	1	1	yes	characteristic, direct product factor, Hall
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	4	1	no	—
205	C₂²×C₃	—	4	1	no	—
206	C₂²×C₃	C₅×#121	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	4	1	no	—
209	C₃×C₄	—	8	2	no	—
210	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
211	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
212	C₃×C₄	C₅×#135	1	1	yes	characteristic
213	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
214	C₂×C₅	—	4	2	no	—
215	C₂×C₅	C₃×#459	1	1	yes	characteristic
216	C₂×C₅	C₃×#467	1	1	yes	characteristic
217	C₂×C₅	C₃×#473	1	1	yes	characteristic
218	C₂³	—	2	1	no	—
219	Q₈	—	4	2	no	—
220	Q₈	—	4	2	no	—
221	D₄	—	2	1	no	—
222	D₄	C₃×C₅×Q₁₆	1	2	yes	complemented
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
229	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	4	1	no	—
232	C₈	—	4	2	no	—
233	C₈	C₃×C₅×D₈	1	2	yes	—
234	C₂×C₃	—	4	2	no	—
235	C₂×C₃	C₅×#459	1	1	yes	characteristic
236	C₂×C₃	C₅×#467	1	1	yes	characteristic
237	C₂×C₃	C₅×#473	1	1	yes	characteristic
238	C₅	C₃×#1669	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	C₃×C₅×#121	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	4	1	no	—
245	C₄	—	8	2	no	—
246	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
247	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
248	C₄	C₃×C₅×#135	1	1	yes	characteristic
249	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
250	C₃	C₅×#1669	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	2	no	—
252	C₂	C₃×C₅×#459	1	1	yes	characteristic
253	C₂	C₃×C₅×#467	1	1	yes	characteristic
254	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×#335	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#1669	C₃	direct (N × H)
18. C₃×#1669	C₅	direct (N × H)
19. C₅×#494	C₂×C₃	semidirect
22. C₅×#335	C₂×C₃	semidirect
23. C₅×#332	C₂×C₃	semidirect
39. C₃×#494	C₂×C₅	semidirect
42. C₃×#335	C₂×C₅	semidirect
43. C₃×#332	C₂×C₅	semidirect
55. #1669	C₃×C₅	direct (N × H)
60. C₃×C₅×D₄	Q₁₆	semidirect
98. #494	C₂×C₃×C₅	semidirect
101. #335	C₂×C₃×C₅	semidirect
102. #332	C₂×C₃×C₅	semidirect
133. C₅×D₄	C₃×Q₁₆	semidirect
164. C₃×D₄	C₅×Q₁₆	semidirect
202. C₃×C₅	#1669	direct (N × H)
222. D₄	C₃×C₅×Q₁₆	semidirect
238. C₅	C₃×#1669	direct (N × H)
250. C₃	C₅×#1669	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×C₈×D₄	2
C₃×C₅×#335	2
C₃×C₅×#332	2
C₅×#1669	3
C₃×#1669	5

Subgroups — C3×C5×#1669 #14274703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1669 #14274703