Subgroups — C₃×C₅×#3292 #14276326

Order 1920.

952 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×#576	C₂	1	4	yes	maximal, complemented
2	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#472	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#453	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#443	C₂	1	2	yes	maximal, complemented
6	C₅×#3292	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×H(5,𝔽₂)	C₂²	1	4	yes	complemented
8	C₃×C₅×AGL(1,ℤ/8)	—	2	8	no	—
9	C₃×C₅×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
10	C₃×C₅×#135	—	2	4	no	—
11	C₃×C₅×#135	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₈	—	2	4	no	—
13	C₂×C₃×C₅×D₈	C₂²	1	4	yes	complemented
14	C₃×C₅×#131	C₂²	1	4	yes	complemented
15	C₃×C₅×#127	C₂²	1	2	yes	complemented
16	C₃×C₅×#108	C₂²	1	2	yes	complemented
17	C₃×C₅×#104	C₂²	1	4	yes	complemented
18	C₃×C₅×#100	C₂²	1	4	yes	complemented
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
20	C₃×#3292	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#576	C₂×C₃	1	4	yes	complemented
22	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#472	C₂×C₃	1	4	yes	complemented
24	C₅×#453	C₂×C₃	1	4	yes	complemented
25	C₅×#443	C₂×C₃	1	2	yes	complemented
26	C₃×C₅×#41	—	2	8	no	—
27	C₃×C₅×#41	C₂³	1	4	yes	—
28	C₂×C₃×C₅×D₄	—	2	8	no	—
29	C₂×C₃×C₅×D₄	—	4	2	no	—
30	C₂×C₃×C₅×D₄	—	4	4	no	—
31	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
32	C₃×C₅×Q₁₆	D₄	1	2	yes	complemented
33	C₃×C₅×SD₁₆	—	2	8	no	—
34	C₃×C₅×SD₁₆	D₄	1	4	yes	complemented
35	C₃×C₅×D₈	—	2	8	no	—
36	C₃×C₅×D₈	—	2	4	no	—
37	C₃×C₅×D₈	—	2	4	no	—
38	C₃×C₅×D₈	D₄	1	2	yes	complemented
39	C₃×C₅×M₁₆	—	2	4	no	—
40	C₃×C₅×M₁₆	C₂³	1	4	yes	—
41	C₂×C₃×C₅×C₈	—	2	4	no	—
42	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₃×#576	C₂×C₅	1	4	yes	complemented
45	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
46	C₃×#472	C₂×C₅	1	4	yes	complemented
47	C₃×#453	C₂×C₅	1	4	yes	complemented
48	C₃×#443	C₂×C₅	1	2	yes	complemented
49	C₅×H(5,𝔽₂)	C₂²×C₃	1	4	yes	complemented
50	C₅×AGL(1,ℤ/8)	—	2	8	no	—
51	C₅×AGL(1,ℤ/8)	C₂²×C₃	1	8	yes	complemented
52	C₅×#135	—	2	4	no	—
53	C₅×#135	C₂²×C₃	1	2	yes	—
54	C₂×C₅×D₈	—	2	4	no	—
55	C₂×C₅×D₈	C₂²×C₃	1	4	yes	complemented
56	C₅×#131	C₂²×C₃	1	4	yes	complemented
57	C₅×#127	C₂²×C₃	1	2	yes	complemented
58	C₅×#108	C₂²×C₃	1	2	yes	complemented
59	C₅×#104	C₂²×C₃	1	4	yes	complemented
60	C₅×#100	C₂²×C₃	1	4	yes	complemented
61	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic, complemented
62	#3292	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₂³×C₃×C₅	—	2	4	no	—
64	C₂³×C₃×C₅	—	4	4	no	—
65	C₃×C₅×Q₈	C₂×D₄	1	4	yes	—
66	C₃×C₅×D₄	—	2	4	no	—
67	C₃×C₅×D₄	—	2	8	no	—
68	C₃×C₅×D₄	—	4	8	no	—
69	C₃×C₅×D₄	—	8	4	no	—
70	C₃×C₅×D₄	C₂×D₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	—	2	1	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	4	4	no	—
74	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
75	C₃×C₅×C₈	—	2	4	no	—
76	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
77	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
78	C₃×H(5,𝔽₂)	C₂²×C₅	1	4	yes	complemented
79	C₃×AGL(1,ℤ/8)	—	2	8	no	—
80	C₃×AGL(1,ℤ/8)	C₂²×C₅	1	8	yes	complemented
81	C₃×#135	—	2	4	no	—
82	C₃×#135	C₂²×C₅	1	2	yes	—
83	C₂×C₃×D₈	—	2	4	no	—
84	C₂×C₃×D₈	C₂²×C₅	1	4	yes	complemented
85	C₃×#131	C₂²×C₅	1	4	yes	complemented
86	C₃×#127	C₂²×C₅	1	2	yes	complemented
87	C₃×#108	C₂²×C₅	1	2	yes	complemented
88	C₃×#104	C₂²×C₅	1	4	yes	complemented
89	C₃×#100	C₂²×C₅	1	4	yes	complemented
90	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic, complemented
91	C₅×#41	—	2	8	no	—
92	C₅×#41	C₂³×C₃	1	4	yes	—
93	C₂×C₅×D₄	—	2	8	no	—
94	C₂×C₅×D₄	—	4	2	no	—
95	C₂×C₅×D₄	—	4	4	no	—
96	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
97	C₅×Q₁₆	C₃×D₄	1	2	yes	complemented
98	C₅×SD₁₆	—	2	8	no	—
99	C₅×SD₁₆	C₃×D₄	1	4	yes	complemented
100	C₅×D₈	—	2	8	no	—
101	C₅×D₈	—	2	4	no	—
102	C₅×D₈	—	2	4	no	—
103	C₅×D₈	C₃×D₄	1	2	yes	complemented
104	C₅×M₁₆	—	2	4	no	—
105	C₅×M₁₆	C₂³×C₃	1	4	yes	—
106	C₂×C₅×C₈	—	2	4	no	—
107	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
108	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
109	#576	C₂×C₃×C₅	1	4	yes	complemented
110	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	#472	C₂×C₃×C₅	1	4	yes	complemented
112	#453	C₂×C₃×C₅	1	4	yes	complemented
113	#443	C₂×C₃×C₅	1	2	yes	complemented
114	C₂²×C₃×C₅	—	2	4	no	—
115	C₂²×C₃×C₅	—	4	4	no	—
116	C₂²×C₃×C₅	—	8	4	no	—
117	C₂²×C₃×C₅	—	16	4	no	—
118	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	—	2	4	no	—
121	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
122	C₃×#41	—	2	8	no	—
123	C₃×#41	C₂³×C₅	1	4	yes	—
124	C₂×C₃×D₄	—	2	8	no	—
125	C₂×C₃×D₄	—	4	2	no	—
126	C₂×C₃×D₄	—	4	4	no	—
127	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
128	C₃×Q₁₆	C₅×D₄	1	2	yes	complemented
129	C₃×SD₁₆	—	2	8	no	—
130	C₃×SD₁₆	C₅×D₄	1	4	yes	complemented
131	C₃×D₈	—	2	8	no	—
132	C₃×D₈	—	2	4	no	—
133	C₃×D₈	—	2	4	no	—
134	C₃×D₈	C₅×D₄	1	2	yes	complemented
135	C₃×M₁₆	—	2	4	no	—
136	C₃×M₁₆	C₂³×C₅	1	4	yes	—
137	C₂×C₃×C₈	—	2	4	no	—
138	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
139	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
140	C₂³×C₅	—	2	4	no	—
141	C₂³×C₅	—	4	4	no	—
142	C₅×Q₈	C₂×C₃×D₄	1	4	yes	—
143	C₅×D₄	—	2	4	no	—
144	C₅×D₄	—	2	8	no	—
145	C₅×D₄	—	4	8	no	—
146	C₅×D₄	—	8	4	no	—
147	C₅×D₄	C₂×C₃×D₄	1	4	yes	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	4	no	—
150	C₂×C₄×C₅	—	4	4	no	—
151	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
152	C₅×C₈	—	2	4	no	—
153	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
154	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
155	H(5,𝔽₂)	C₂²×C₃×C₅	1	4	yes	complemented
156	AGL(1,ℤ/8)	—	2	8	no	—
157	AGL(1,ℤ/8)	C₂²×C₃×C₅	1	8	yes	complemented
158	#135	—	2	4	no	—
159	#135	C₂²×C₃×C₅	1	2	yes	—
160	C₂×D₈	—	2	4	no	—
161	C₂×D₈	C₂²×C₃×C₅	1	4	yes	complemented
162	#131	C₂²×C₃×C₅	1	4	yes	complemented
163	#127	C₂²×C₃×C₅	1	2	yes	complemented
164	#108	C₂²×C₃×C₅	1	2	yes	complemented
165	#104	C₂²×C₃×C₅	1	4	yes	complemented
166	#100	C₂²×C₃×C₅	1	4	yes	complemented
167	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
168	C₂×C₃×C₅	—	2	1	no	—
169	C₂×C₃×C₅	—	4	4	no	—
170	C₂×C₃×C₅	—	8	4	no	—
171	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
172	C₂³×C₃	—	2	4	no	—
173	C₂³×C₃	—	4	4	no	—
174	C₃×Q₈	C₂×C₅×D₄	1	4	yes	—
175	C₃×D₄	—	2	4	no	—
176	C₃×D₄	—	2	8	no	—
177	C₃×D₄	—	4	8	no	—
178	C₃×D₄	—	8	4	no	—
179	C₃×D₄	C₂×C₅×D₄	1	4	yes	—
180	C₂×C₃×C₄	—	2	1	no	—
181	C₂×C₃×C₄	—	2	4	no	—
182	C₂×C₃×C₄	—	4	4	no	—
183	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
184	C₃×C₈	—	2	4	no	—
185	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
186	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
187	C₂²×C₅	—	2	4	no	—
188	C₂²×C₅	—	4	4	no	—
189	C₂²×C₅	—	8	4	no	—
190	C₂²×C₅	—	16	4	no	—
191	C₂²×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	4	no	—
194	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
195	#41	—	2	8	no	—
196	#41	C₂³×C₃×C₅	1	4	yes	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	4	2	no	—
199	C₂×D₄	—	4	4	no	—
200	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
201	Q₁₆	C₃×C₅×D₄	1	2	yes	complemented
202	SD₁₆	—	2	8	no	—
203	SD₁₆	C₃×C₅×D₄	1	4	yes	complemented
204	D₈	—	2	8	no	—
205	D₈	—	2	4	no	—
206	D₈	—	2	4	no	—
207	D₈	C₃×C₅×D₄	1	2	yes	complemented
208	M₁₆	—	2	4	no	—
209	M₁₆	C₂³×C₃×C₅	1	4	yes	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
212	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
213	C₃×C₅	#3292	1	1	yes	characteristic, direct product factor, Hall
214	C₂²×C₃	—	2	4	no	—
215	C₂²×C₃	—	4	4	no	—
216	C₂²×C₃	—	8	4	no	—
217	C₂²×C₃	—	16	4	no	—
218	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
219	C₃×C₄	—	2	2	no	—
220	C₃×C₄	—	2	4	no	—
221	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
222	C₂×C₅	—	2	1	no	—
223	C₂×C₅	—	4	4	no	—
224	C₂×C₅	—	8	4	no	—
225	C₂×C₅	C₃×D₄²	1	1	yes	characteristic
226	C₂³	—	2	4	no	—
227	C₂³	—	4	4	no	—
228	Q₈	C₂×C₃×C₅×D₄	1	4	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	8	4	no	—
233	D₄	C₂×C₃×C₅×D₄	1	4	yes	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
240	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	—	4	4	no	—
243	C₂×C₃	—	8	4	no	—
244	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
245	C₅	C₃×#3292	1	1	yes	characteristic, direct product factor, Sylow
246	C₂²	—	2	4	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	16	4	no	—
250	C₂²	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	2	4	no	—
253	C₄	C₂²×C₃×C₅×D₄	1	2	yes	—
254	C₃	C₅×#3292	1	1	yes	characteristic, direct product factor, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	4	4	no	—
257	C₂	—	8	4	no	—
258	C₂	C₃×C₅×D₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#576	C₂	semidirect
2. C₃×C₅×#493	C₂	semidirect
3. C₃×C₅×#472	C₂	semidirect
4. C₃×C₅×#453	C₂	semidirect
5. C₃×C₅×#443	C₂	semidirect
6. C₅×#3292	C₃	direct (N × H)
7. C₃×C₅×H(5,𝔽₂)	C₂²	semidirect
9. C₃×C₅×AGL(1,ℤ/8)	C₂²	semidirect
13. C₂×C₃×C₅×D₈	C₂²	semidirect
14. C₃×C₅×#131	C₂²	semidirect
15. C₃×C₅×#127	C₂²	semidirect
16. C₃×C₅×#108	C₂²	semidirect
17. C₃×C₅×#104	C₂²	semidirect
18. C₃×C₅×#100	C₂²	semidirect
19. C₃×C₄×C₅×C₈	C₂²	semidirect
20. C₃×#3292	C₅	direct (N × H)
21. C₅×#576	C₂×C₃	semidirect
22. C₅×#493	C₂×C₃	semidirect
23. C₅×#472	C₂×C₃	semidirect
24. C₅×#453	C₂×C₃	semidirect
25. C₅×#443	C₂×C₃	semidirect
32. C₃×C₅×Q₁₆	D₄	semidirect
34. C₃×C₅×SD₁₆	D₄	semidirect
38. C₃×C₅×D₈	D₄	semidirect
44. C₃×#576	C₂×C₅	semidirect
45. C₃×#493	C₂×C₅	semidirect
46. C₃×#472	C₂×C₅	semidirect
47. C₃×#453	C₂×C₅	semidirect
48. C₃×#443	C₂×C₅	semidirect
49. C₅×H(5,𝔽₂)	C₂²×C₃	semidirect
51. C₅×AGL(1,ℤ/8)	C₂²×C₃	semidirect
55. C₂×C₅×D₈	C₂²×C₃	semidirect
56. C₅×#131	C₂²×C₃	semidirect
57. C₅×#127	C₂²×C₃	semidirect
58. C₅×#108	C₂²×C₃	semidirect
59. C₅×#104	C₂²×C₃	semidirect
60. C₅×#100	C₂²×C₃	semidirect
61. C₄×C₅×C₈	C₂²×C₃	semidirect
62. #3292	C₃×C₅	direct (N × H)
78. C₃×H(5,𝔽₂)	C₂²×C₅	semidirect
80. C₃×AGL(1,ℤ/8)	C₂²×C₅	semidirect
84. C₂×C₃×D₈	C₂²×C₅	semidirect
85. C₃×#131	C₂²×C₅	semidirect
86. C₃×#127	C₂²×C₅	semidirect
87. C₃×#108	C₂²×C₅	semidirect
88. C₃×#104	C₂²×C₅	semidirect
89. C₃×#100	C₂²×C₅	semidirect
90. C₃×C₄×C₈	C₂²×C₅	semidirect
97. C₅×Q₁₆	C₃×D₄	semidirect
99. C₅×SD₁₆	C₃×D₄	semidirect
103. C₅×D₈	C₃×D₄	semidirect
109. #576	C₂×C₃×C₅	semidirect
110. #493	C₂×C₃×C₅	semidirect
111. #472	C₂×C₃×C₅	semidirect
112. #453	C₂×C₃×C₅	semidirect
113. #443	C₂×C₃×C₅	semidirect
128. C₃×Q₁₆	C₅×D₄	semidirect
130. C₃×SD₁₆	C₅×D₄	semidirect
134. C₃×D₈	C₅×D₄	semidirect
155. H(5,𝔽₂)	C₂²×C₃×C₅	semidirect
157. AGL(1,ℤ/8)	C₂²×C₃×C₅	semidirect
161. C₂×D₈	C₂²×C₃×C₅	semidirect
162. #131	C₂²×C₃×C₅	semidirect
163. #127	C₂²×C₃×C₅	semidirect
164. #108	C₂²×C₃×C₅	semidirect
165. #104	C₂²×C₃×C₅	semidirect
166. #100	C₂²×C₃×C₅	semidirect
167. C₄×C₈	C₂²×C₃×C₅	semidirect
201. Q₁₆	C₃×C₅×D₄	semidirect
203. SD₁₆	C₃×C₅×D₄	semidirect
207. D₈	C₃×C₅×D₄	semidirect
213. C₃×C₅	#3292	direct (N × H)
245. C₅	C₃×#3292	direct (N × H)
254. C₃	C₅×#3292	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×#576	2
C₃×C₅×#493	2
C₃×C₅×#472	2
C₃×C₅×#453	2
C₃×C₅×#443	2
C₅×#3292	3
C₃×#3292	5

Subgroups — C3×C5×#3292 #14276326

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3292 #14276326