Subgroups — C₃×C₅×#3261 #14276295

Order 1920.

644 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#541	C₂	1	2	yes	maximal
2	C₃×C₅×#479	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#470	C₂	1	2	yes	maximal
4	C₃×C₅×#467	C₂	1	2	yes	maximal
5	C₃×C₅×#462	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic
7	C₅×#3261	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×#126	—	2	4	no	—
14	C₃×C₅×#125	—	2	4	no	—
15	C₃×C₅×#125	C₂²	1	2	yes	—
16	C₃×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
17	C₃×C₄×C₅×Q₈	C₂²	1	4	yes	—
18	C₃×C₅×#117	C₂²	1	1	yes	characteristic
19	C₃×C₅×#107	C₂²	1	2	yes	—
20	C₃×C₅×#106	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
22	C₃×C₅×#103	C₂²	1	4	yes	—
23	C₃×C₅×#103	C₂²	1	4	yes	—
24	C₃×#3261	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#541	C₂×C₃	1	2	yes	—
26	C₅×#479	C₂×C₃	1	4	yes	complemented
27	C₅×#470	C₂×C₃	1	2	yes	—
28	C₅×#467	C₂×C₃	1	2	yes	—
29	C₅×#462	C₂×C₃	1	4	yes	complemented
30	C₅×#424	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₃×C₅×Q₁₆	—	4	2	no	—
34	C₃×C₅×Q₁₆	—	4	2	no	—
35	C₃×C₅×M₁₆	—	4	1	no	—
36	C₂×C₃×C₅×C₈	—	4	1	no	—
37	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	4	no	—
40	C₃×C₅×C₄⋊C₄	—	2	4	no	—
41	C₃×C₅×C₄⋊C₄	—	2	4	no	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
44	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
45	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
46	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
47	C₃×C₄²×C₅	—	2	4	no	—
48	C₃×C₄²×C₅	C₂³	1	2	yes	—
49	C₃×#541	C₂×C₅	1	2	yes	—
50	C₃×#479	C₂×C₅	1	4	yes	complemented
51	C₃×#470	C₂×C₅	1	2	yes	—
52	C₃×#467	C₂×C₅	1	2	yes	—
53	C₃×#462	C₂×C₅	1	4	yes	complemented
54	C₃×#424	C₂×C₅	1	1	yes	characteristic
55	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
56	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
57	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
60	C₅×#126	—	2	4	no	—
61	C₅×#125	—	2	4	no	—
62	C₅×#125	C₂²×C₃	1	2	yes	—
63	C₅×C₂²⋊Q₈	C₂²×C₃	1	4	yes	—
64	C₄×C₅×Q₈	C₂²×C₃	1	4	yes	—
65	C₅×#117	C₂²×C₃	1	1	yes	characteristic
66	C₅×#107	C₂²×C₃	1	2	yes	—
67	C₅×#106	C₂²×C₃	1	2	yes	—
68	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
69	C₅×#103	C₂²×C₃	1	4	yes	—
70	C₅×#103	C₂²×C₃	1	4	yes	—
71	#3261	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	2	4	no	—
74	C₃×C₅×Q₈	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	4	no	—
78	C₂×C₃×C₄×C₅	—	2	4	no	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₃×C₅×C₈	—	2	2	no	—
83	C₃×C₅×C₈	—	2	2	no	—
84	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
85	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
86	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
89	C₃×#126	—	2	4	no	—
90	C₃×#125	—	2	4	no	—
91	C₃×#125	C₂²×C₅	1	2	yes	—
92	C₃×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
93	C₃×C₄×Q₈	C₂²×C₅	1	4	yes	—
94	C₃×#117	C₂²×C₅	1	1	yes	characteristic
95	C₃×#107	C₂²×C₅	1	2	yes	—
96	C₃×#106	C₂²×C₅	1	2	yes	—
97	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
98	C₃×#103	C₂²×C₅	1	4	yes	—
99	C₃×#103	C₂²×C₅	1	4	yes	—
100	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
101	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₅×Q₁₆	—	4	2	no	—
103	C₅×Q₁₆	—	4	2	no	—
104	C₅×M₁₆	—	4	1	no	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
107	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
108	C₅×C₄⋊C₄	—	2	4	no	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₅×C₄⋊C₄	—	2	4	no	—
111	C₅×C₄⋊C₄	—	2	4	no	—
112	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
113	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
114	C₅×C₂²⋊C₄	—	2	4	no	—
115	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
116	C₄²×C₅	—	2	4	no	—
117	C₄²×C₅	C₂³×C₃	1	2	yes	—
118	#541	C₂×C₃×C₅	1	2	yes	—
119	#479	C₂×C₃×C₅	1	4	yes	complemented
120	#470	C₂×C₃×C₅	1	2	yes	—
121	#467	C₂×C₃×C₅	1	2	yes	—
122	#462	C₂×C₃×C₅	1	4	yes	complemented
123	#424	C₂×C₃×C₅	1	1	yes	characteristic
124	C₂²×C₃×C₅	—	2	1	no	—
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₃×C₄×C₅	—	2	1	no	—
128	C₃×C₄×C₅	—	2	4	no	—
129	C₃×C₄×C₅	—	4	4	no	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
132	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
133	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
134	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
135	C₃×Q₁₆	—	4	2	no	—
136	C₃×Q₁₆	—	4	2	no	—
137	C₃×M₁₆	—	4	1	no	—
138	C₂×C₃×C₈	—	4	1	no	—
139	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
140	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
141	C₃×C₄⋊C₄	—	2	4	no	—
142	C₃×C₄⋊C₄	—	2	4	no	—
143	C₃×C₄⋊C₄	—	2	4	no	—
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
146	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
147	C₃×C₂²⋊C₄	—	2	4	no	—
148	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
149	C₃×C₄²	—	2	4	no	—
150	C₃×C₄²	C₂³×C₅	1	2	yes	—
151	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₅×Q₈	—	2	4	no	—
153	C₅×Q₈	—	2	4	no	—
154	C₂×C₄×C₅	—	2	1	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
161	C₅×C₈	—	2	2	no	—
162	C₅×C₈	—	2	2	no	—
163	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
164	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
165	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
168	#126	—	2	4	no	—
169	#125	—	2	4	no	—
170	#125	C₂²×C₃×C₅	1	2	yes	—
171	C₂²⋊Q₈	C₂²×C₃×C₅	1	4	yes	—
172	C₄×Q₈	C₂²×C₃×C₅	1	4	yes	—
173	#117	C₂²×C₃×C₅	1	1	yes	characteristic
174	#107	C₂²×C₃×C₅	1	2	yes	—
175	#106	C₂²×C₃×C₅	1	2	yes	—
176	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
177	#103	C₂²×C₃×C₅	1	4	yes	—
178	#103	C₂²×C₃×C₅	1	4	yes	—
179	C₂×C₃×C₅	—	4	1	no	—
180	C₂×C₃×C₅	#537	1	1	yes	characteristic
181	C₂×C₃×C₅	#578	1	2	yes	—
182	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
183	C₃×Q₈	—	2	4	no	—
184	C₃×Q₈	—	2	4	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
190	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	2	2	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₄×C₅	—	2	4	no	—
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	4	no	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
202	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
203	C₂×Q₈	C₂³×C₃×C₅	1	4	yes	—
204	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
205	Q₁₆	—	4	2	no	—
206	Q₁₆	—	4	2	no	—
207	M₁₆	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
210	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
216	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
219	C₄²	—	2	4	no	—
220	C₄²	C₂³×C₃×C₅	1	2	yes	—
221	C₃×C₅	#3261	1	1	yes	characteristic, direct product factor, Hall
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	2	no	—
224	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	4	no	—
227	C₃×C₄	—	4	4	no	—
228	C₃×C₄	—	4	4	no	—
229	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
230	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂×C₅	—	4	1	no	—
232	C₂×C₅	C₃×#537	1	1	yes	characteristic
233	C₂×C₅	C₃×#578	1	2	yes	—
234	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
235	Q₈	—	2	4	no	—
236	Q₈	—	2	4	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
242	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₂×C₃	—	4	1	no	—
247	C₂×C₃	C₅×#537	1	1	yes	characteristic
248	C₂×C₃	C₅×#578	1	2	yes	—
249	C₅	C₃×#3261	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	2	no	—
252	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
253	C₄	—	2	1	no	—
254	C₄	—	2	4	no	—
255	C₄	—	4	4	no	—
256	C₄	—	4	4	no	—
257	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
258	C₄	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
259	C₃	C₅×#3261	1	1	yes	characteristic, direct product factor, Sylow
260	C₂	—	4	1	no	—
261	C₂	C₃×C₅×#537	1	1	yes	characteristic
262	C₂	C₃×C₅×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#479	C₂	semidirect
5. C₃×C₅×#462	C₂	semidirect
7. C₅×#3261	C₃	direct (N × H)
24. C₃×#3261	C₅	direct (N × H)
26. C₅×#479	C₂×C₃	semidirect
29. C₅×#462	C₂×C₃	semidirect
50. C₃×#479	C₂×C₅	semidirect
53. C₃×#462	C₂×C₅	semidirect
71. #3261	C₃×C₅	direct (N × H)
119. #479	C₂×C₃×C₅	semidirect
122. #462	C₂×C₃×C₅	semidirect
221. C₃×C₅	#3261	direct (N × H)
249. C₅	C₃×#3261	direct (N × H)
259. C₃	C₅×#3261	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×#541	2
C₃×C₅×#479	2
C₃×C₅×#470	2
C₃×C₅×#467	2
C₃×C₅×#462	2
C₃×C₅×#424	2
C₅×#3261	3
C₃×#3261	5

Subgroups — C3×C5×#3261 #14276295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3261 #14276295