Subgroups — C₃×C₅×#1668 #14274702

Order 1920.

376 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#460	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×C₈×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	2	yes	maximal, complemented
6	C₅×#1668	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₃×C₅×#127	C₂²	1	2	yes	—
11	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	—	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×#102	—	2	2	no	—
16	C₃×C₄×C₅×C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1668	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#493	C₂×C₃	1	1	yes	characteristic
20	C₅×#460	C₂×C₃	1	2	yes	complemented
21	C₅×C₈×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#335	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#331	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₃×C₅×SD₁₆	—	4	4	no	—
29	C₃×C₅×D₈	—	4	2	no	—
30	C₃×C₅×D₈	—	4	2	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₃×#493	C₂×C₅	1	1	yes	characteristic
40	C₃×#460	C₂×C₅	1	2	yes	complemented
41	C₃×C₈×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#331	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×SD₁₆	—	2	2	no	—
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	—	2	1	no	—
47	C₅×#127	C₂²×C₃	1	2	yes	—
48	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	1	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	C₄×C₅×C₈	—	2	1	no	—
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	#1668	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	4	2	no	—
57	C₃×C₅×Q₈	—	2	1	no	—
58	C₃×C₅×Q₈	D₈	1	2	yes	complemented
59	C₃×C₅×D₄	—	4	2	no	—
60	C₃×C₅×D₄	—	4	2	no	—
61	C₃×C₅×D₄	—	4	2	no	—
62	C₃×C₅×D₄	—	4	2	no	—
63	C₃×C₅×D₄	—	8	2	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
69	C₃×C₅×C₈	—	2	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	—	4	1	no	—
72	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
73	C₂×C₃×SD₁₆	—	2	2	no	—
74	C₂×C₃×D₈	—	2	1	no	—
75	C₂×C₃×D₈	—	2	1	no	—
76	C₃×#127	C₂²×C₅	1	2	yes	—
77	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
78	C₃×C₄⋊C₈	—	2	1	no	—
79	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
81	C₃×#102	—	2	2	no	—
82	C₃×C₄×C₈	—	2	1	no	—
83	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₅×SD₁₆	—	4	4	no	—
89	C₅×D₈	—	4	2	no	—
90	C₅×D₈	—	4	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
95	C₅×C₄⋊C₄	—	2	1	no	—
96	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
97	C₄²×C₅	—	2	1	no	—
98	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
99	#493	C₂×C₃×C₅	1	1	yes	characteristic
100	#460	C₂×C₃×C₅	1	2	yes	complemented
101	C₈×Q₈	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	#335	C₂×C₃×C₅	1	1	yes	characteristic, complemented
103	#331	C₂×C₃×C₅	1	2	yes	complemented
104	C₂²×C₃×C₅	—	8	2	no	—
105	C₂²×C₃×C₅	—	8	2	no	—
106	C₂²×C₃×C₅	—	8	2	no	—
107	C₂²×C₃×C₅	#121	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	4	1	no	—
111	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
112	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
113	C₃×C₄×C₅	#135	1	1	yes	characteristic
114	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
115	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
116	C₂×C₃×D₄	—	2	2	no	—
117	C₂×C₃×D₄	—	2	2	no	—
118	C₂×C₃×D₄	—	2	2	no	—
119	C₃×SD₁₆	—	4	4	no	—
120	C₃×D₈	—	4	2	no	—
121	C₃×D₈	—	4	2	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
125	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
126	C₃×C₄⋊C₄	—	2	1	no	—
127	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
128	C₃×C₄²	—	2	1	no	—
129	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
130	C₂³×C₅	—	4	2	no	—
131	C₅×Q₈	—	2	1	no	—
132	C₅×Q₈	C₃×D₈	1	2	yes	complemented
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	4	2	no	—
135	C₅×D₄	—	4	2	no	—
136	C₅×D₄	—	4	2	no	—
137	C₅×D₄	—	8	2	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
143	C₅×C₈	—	2	1	no	—
144	C₅×C₈	—	4	2	no	—
145	C₅×C₈	—	4	1	no	—
146	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
147	C₂×SD₁₆	—	2	2	no	—
148	C₂×D₈	—	2	1	no	—
149	C₂×D₈	—	2	1	no	—
150	#127	C₂²×C₃×C₅	1	2	yes	—
151	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₄⋊C₈	—	2	1	no	—
153	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
155	#102	—	2	2	no	—
156	C₄×C₈	—	2	1	no	—
157	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
158	C₂×C₃×C₅	—	16	2	no	—
159	C₂×C₃×C₅	#460	1	1	yes	characteristic
160	C₂×C₃×C₅	#466	1	1	yes	characteristic
161	C₂×C₃×C₅	#472	1	1	yes	characteristic
162	C₂³×C₃	—	4	2	no	—
163	C₃×Q₈	—	2	1	no	—
164	C₃×Q₈	C₅×D₈	1	2	yes	complemented
165	C₃×D₄	—	4	2	no	—
166	C₃×D₄	—	4	2	no	—
167	C₃×D₄	—	4	2	no	—
168	C₃×D₄	—	4	2	no	—
169	C₃×D₄	—	8	2	no	—
170	C₂×C₃×C₄	—	2	1	no	—
171	C₂×C₃×C₄	—	2	1	no	—
172	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
175	C₃×C₈	—	2	1	no	—
176	C₃×C₈	—	4	2	no	—
177	C₃×C₈	—	4	1	no	—
178	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
179	C₂²×C₅	—	8	2	no	—
180	C₂²×C₅	—	8	2	no	—
181	C₂²×C₅	—	8	2	no	—
182	C₂²×C₅	C₃×#121	1	1	yes	characteristic
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	—	4	1	no	—
186	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
187	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
188	C₄×C₅	C₃×#135	1	1	yes	characteristic
189	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	SD₁₆	—	4	4	no	—
195	D₈	—	4	2	no	—
196	D₈	—	4	2	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
200	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
203	C₄²	—	2	1	no	—
204	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
205	C₃×C₅	#1668	1	1	yes	characteristic, direct product factor, Hall
206	C₂²×C₃	—	8	2	no	—
207	C₂²×C₃	—	8	2	no	—
208	C₂²×C₃	—	8	2	no	—
209	C₂²×C₃	C₅×#121	1	1	yes	characteristic
210	C₃×C₄	—	2	2	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	4	1	no	—
213	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
214	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
215	C₃×C₄	C₅×#135	1	1	yes	characteristic
216	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
217	C₂×C₅	—	16	2	no	—
218	C₂×C₅	C₃×#460	1	1	yes	characteristic
219	C₂×C₅	C₃×#466	1	1	yes	characteristic
220	C₂×C₅	C₃×#472	1	1	yes	characteristic
221	C₂³	—	4	2	no	—
222	Q₈	—	2	1	no	—
223	Q₈	C₃×C₅×D₈	1	2	yes	complemented
224	D₄	—	4	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
232	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
233	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	—	4	1	no	—
237	C₈	C₃×C₅×SD₁₆	1	2	yes	complemented
238	C₂×C₃	—	16	2	no	—
239	C₂×C₃	C₅×#460	1	1	yes	characteristic
240	C₂×C₃	C₅×#466	1	1	yes	characteristic
241	C₂×C₃	C₅×#472	1	1	yes	characteristic
242	C₅	C₃×#1668	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²	—	8	2	no	—
244	C₂²	—	8	2	no	—
245	C₂²	—	8	2	no	—
246	C₂²	C₃×C₅×#121	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
251	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
252	C₄	C₃×C₅×#135	1	1	yes	characteristic
253	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₃	C₅×#1668	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	16	2	no	—
256	C₂	C₃×C₅×#460	1	1	yes	characteristic
257	C₂	C₃×C₅×#466	1	1	yes	characteristic
258	C₂	C₃×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#493	C₂
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
2. C₃×C₅×#460	C₂	semidirect
3. C₃×C₅×C₈×Q₈	C₂	semidirect
4. C₃×C₅×#335	C₂	semidirect
5. C₃×C₅×#331	C₂	semidirect
6. C₅×#1668	C₃	direct (N × H)
18. C₃×#1668	C₅	direct (N × H)
20. C₅×#460	C₂×C₃	semidirect
21. C₅×C₈×Q₈	C₂×C₃	semidirect
22. C₅×#335	C₂×C₃	semidirect
23. C₅×#331	C₂×C₃	semidirect
40. C₃×#460	C₂×C₅	semidirect
41. C₃×C₈×Q₈	C₂×C₅	semidirect
42. C₃×#335	C₂×C₅	semidirect
43. C₃×#331	C₂×C₅	semidirect
55. #1668	C₃×C₅	direct (N × H)
58. C₃×C₅×Q₈	D₈	semidirect
72. C₃×C₅×C₈	SD₁₆	semidirect
100. #460	C₂×C₃×C₅	semidirect
101. C₈×Q₈	C₂×C₃×C₅	semidirect
102. #335	C₂×C₃×C₅	semidirect
103. #331	C₂×C₃×C₅	semidirect
132. C₅×Q₈	C₃×D₈	semidirect
146. C₅×C₈	C₃×SD₁₆	semidirect
164. C₃×Q₈	C₅×D₈	semidirect
178. C₃×C₈	C₅×SD₁₆	semidirect
205. C₃×C₅	#1668	direct (N × H)
223. Q₈	C₃×C₅×D₈	semidirect
237. C₈	C₃×C₅×SD₁₆	semidirect
242. C₅	C₃×#1668	direct (N × H)
254. C₃	C₅×#1668	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×#493	2
C₃×C₅×#460	2
C₃×C₅×C₈×Q₈	2
C₃×C₅×#335	2
C₃×C₅×#331	2
C₅×#1668	3
C₃×#1668	5

Subgroups — C3×C5×#1668 #14274702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1668 #14274702