Subgroups — C₃×C₅×#2134 #14275168

Order 1920.

524 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#535	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#413	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#355	C₂	1	4	yes	maximal, complemented
4	C₅×#2134	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
6	C₂²×C₃×C₅×D₄	C₄	1	1	yes	characteristic
7	C₂×C₃×C₅×M₁₆	—	2	2	no	—
8	C₃×C₅×#127	C₄	1	2	yes	—
9	C₃×C₅×#124	C₂²	1	2	yes	—
10	C₃×C₅×#121	—	2	2	no	—
11	C₃×C₅×#120	—	2	2	no	—
12	C₃×C₅×#117	C₄	1	1	yes	characteristic
13	C₃×C₅×#101	C₂²	1	4	yes	complemented
14	C₃×C₅×#100	—	2	2	no	—
15	C₃×#2134	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₅×#535	C₂×C₃	1	1	yes	characteristic
17	C₅×#413	C₂×C₃	1	2	yes	complemented
18	C₅×#355	C₂×C₃	1	4	yes	complemented
19	C₂⁴×C₃×C₅	—	2	1	no	—
20	C₃×C₅×#41	—	4	2	no	—
21	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
22	C₂×C₃×C₅×D₄	—	2	4	no	—
23	C₂×C₃×C₅×D₄	—	2	1	no	—
24	C₂×C₃×C₅×D₄	—	4	2	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×D₄	D₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₃×C₅×M₁₆	—	2	4	no	—
31	C₃×C₅×M₁₆	—	4	2	no	—
32	C₂×C₃×C₅×C₈	—	4	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	1	no	—
34	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
35	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
36	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₃×#535	C₂×C₅	1	1	yes	characteristic
38	C₃×#413	C₂×C₅	1	2	yes	complemented
39	C₃×#355	C₂×C₅	1	4	yes	complemented
40	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×D₄	C₃×C₄	1	1	yes	characteristic
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₅×#127	C₃×C₄	1	2	yes	—
44	C₅×#124	C₂²×C₃	1	2	yes	—
45	C₅×#121	—	2	2	no	—
46	C₅×#120	—	2	2	no	—
47	C₅×#117	C₃×C₄	1	1	yes	characteristic
48	C₅×#101	C₂²×C₃	1	4	yes	complemented
49	C₅×#100	—	2	2	no	—
50	#2134	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	—	4	2	no	—
53	C₂³×C₃×C₅	—	8	2	no	—
54	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
55	C₂³×C₃×C₅	C₂²⋊C₄	1	2	yes	—
56	C₃×C₅×Q₈	—	4	1	no	—
57	C₃×C₅×D₄	—	4	1	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₃×C₅×D₄	—	4	2	no	—
60	C₃×C₅×D₄	—	4	2	no	—
61	C₃×C₅×D₄	—	8	4	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₃×C₅×C₈	—	4	4	no	—
70	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
71	C₂²×C₃×D₄	C₄×C₅	1	1	yes	characteristic
72	C₂×C₃×M₁₆	—	2	2	no	—
73	C₃×#127	C₄×C₅	1	2	yes	—
74	C₃×#124	C₂²×C₅	1	2	yes	—
75	C₃×#121	—	2	2	no	—
76	C₃×#120	—	2	2	no	—
77	C₃×#117	C₄×C₅	1	1	yes	characteristic
78	C₃×#101	C₂²×C₅	1	4	yes	complemented
79	C₃×#100	—	2	2	no	—
80	C₂⁴×C₅	—	2	1	no	—
81	C₅×#41	—	4	2	no	—
82	C₂×C₅×Q₈	C₂³×C₃	1	1	yes	characteristic
83	C₂×C₅×D₄	—	2	4	no	—
84	C₂×C₅×D₄	—	2	1	no	—
85	C₂×C₅×D₄	—	4	2	no	—
86	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
87	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×C₅×D₄	C₃×D₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
91	C₅×M₁₆	—	2	4	no	—
92	C₅×M₁₆	—	4	2	no	—
93	C₂×C₅×C₈	—	4	2	no	—
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₂²⋊C₄	—	2	1	no	—
96	C₅×C₂²⋊C₄	—	2	4	no	—
97	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
98	#535	C₂×C₃×C₅	1	1	yes	characteristic
99	#413	C₂×C₃×C₅	1	2	yes	complemented
100	#355	C₂×C₃×C₅	1	4	yes	complemented
101	C₂²×C₃×C₅	—	2	1	no	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	—	4	2	no	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	—	4	4	no	—
106	C₂²×C₃×C₅	—	4	2	no	—
107	C₂²×C₃×C₅	—	8	4	no	—
108	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	—	4	2	no	—
112	C₃×C₄×C₅	#99	1	2	yes	—
113	C₂⁴×C₃	—	2	1	no	—
114	C₃×#41	—	4	2	no	—
115	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
116	C₂×C₃×D₄	—	2	4	no	—
117	C₂×C₃×D₄	—	2	1	no	—
118	C₂×C₃×D₄	—	4	2	no	—
119	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
120	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₃×D₄	C₅×D₄	1	2	yes	—
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
124	C₃×M₁₆	—	2	4	no	—
125	C₃×M₁₆	—	4	2	no	—
126	C₂×C₃×C₈	—	4	2	no	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₂²⋊C₄	—	2	1	no	—
129	C₃×C₂²⋊C₄	—	2	4	no	—
130	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	4	2	no	—
133	C₂³×C₅	—	8	2	no	—
134	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
135	C₂³×C₅	C₃×C₂²⋊C₄	1	2	yes	—
136	C₅×Q₈	—	4	1	no	—
137	C₅×D₄	—	4	1	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	4	2	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	8	4	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
149	C₅×C₈	—	4	4	no	—
150	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂²×D₄	C₃×C₄×C₅	1	1	yes	characteristic
152	C₂×M₁₆	—	2	2	no	—
153	#127	C₃×C₄×C₅	1	2	yes	—
154	#124	C₂²×C₃×C₅	1	2	yes	—
155	#121	—	2	2	no	—
156	#120	—	2	2	no	—
157	#117	C₃×C₄×C₅	1	1	yes	characteristic
158	#101	C₂²×C₃×C₅	1	4	yes	complemented
159	#100	—	2	2	no	—
160	C₂×C₃×C₅	—	2	1	no	—
161	C₂×C₃×C₅	—	4	2	no	—
162	C₂×C₃×C₅	—	4	1	no	—
163	C₂×C₃×C₅	—	8	2	no	—
164	C₂×C₃×C₅	C₂×#99	1	1	yes	characteristic
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	—	4	2	no	—
167	C₂³×C₃	—	8	2	no	—
168	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂³×C₃	C₅×C₂²⋊C₄	1	2	yes	—
170	C₃×Q₈	—	4	1	no	—
171	C₃×D₄	—	4	1	no	—
172	C₃×D₄	—	4	2	no	—
173	C₃×D₄	—	4	2	no	—
174	C₃×D₄	—	4	2	no	—
175	C₃×D₄	—	8	4	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
183	C₃×C₈	—	4	4	no	—
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	—	2	2	no	—
186	C₂²×C₅	—	4	2	no	—
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	—	4	4	no	—
189	C₂²×C₅	—	4	2	no	—
190	C₂²×C₅	—	8	4	no	—
191	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
192	C₄×C₅	—	2	2	no	—
193	C₄×C₅	—	2	1	no	—
194	C₄×C₅	—	4	2	no	—
195	C₄×C₅	C₃×#99	1	2	yes	—
196	C₂⁴	—	2	1	no	—
197	#41	—	4	2	no	—
198	C₂×Q₈	C₂³×C₃×C₅	1	1	yes	characteristic
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
203	C₂×D₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
204	C₂×D₄	C₃×C₅×D₄	1	2	yes	—
205	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
206	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	4	2	no	—
209	C₂×C₈	—	4	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
214	C₃×C₅	#2134	1	1	yes	characteristic, direct product factor, Hall
215	C₂²×C₃	—	2	1	no	—
216	C₂²×C₃	—	2	2	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	4	4	no	—
220	C₂²×C₃	—	4	2	no	—
221	C₂²×C₃	—	8	4	no	—
222	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
223	C₃×C₄	—	2	2	no	—
224	C₃×C₄	—	2	1	no	—
225	C₃×C₄	—	4	2	no	—
226	C₃×C₄	C₅×#99	1	2	yes	—
227	C₂×C₅	—	2	1	no	—
228	C₂×C₅	—	4	1	no	—
229	C₂×C₅	—	4	2	no	—
230	C₂×C₅	—	8	2	no	—
231	C₂×C₅	C₂×C₃×#99	1	1	yes	characteristic
232	C₂³	—	2	2	no	—
233	C₂³	—	4	2	no	—
234	C₂³	—	8	2	no	—
235	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₂³	C₃×C₅×C₂²⋊C₄	1	2	yes	—
237	Q₈	—	4	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	8	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
250	C₈	—	4	4	no	—
251	C₂×C₃	—	2	1	no	—
252	C₂×C₃	—	4	1	no	—
253	C₂×C₃	—	4	2	no	—
254	C₂×C₃	—	8	2	no	—
255	C₂×C₃	C₂×C₅×#99	1	1	yes	characteristic
256	C₅	C₃×#2134	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	2	1	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	4	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	4	no	—
264	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	4	2	no	—
268	C₄	C₃×C₅×#99	1	2	yes	—
269	C₃	C₅×#2134	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	—	2	1	no	—
271	C₂	—	4	1	no	—
272	C₂	—	4	2	no	—
273	C₂	—	8	2	no	—
274	C₂	C₂×C₃×C₅×#99	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	C₂×#99
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×Q₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#535	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₅×#41 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#413	C₂	semidirect
3. C₃×C₅×#355	C₂	semidirect
4. C₅×#2134	C₃	direct (N × H)
13. C₃×C₅×#101	C₂²	semidirect
15. C₃×#2134	C₅	direct (N × H)
17. C₅×#413	C₂×C₃	semidirect
18. C₅×#355	C₂×C₃	semidirect
38. C₃×#413	C₂×C₅	semidirect
39. C₃×#355	C₂×C₅	semidirect
48. C₅×#101	C₂²×C₃	semidirect
50. #2134	C₃×C₅	direct (N × H)
78. C₃×#101	C₂²×C₅	semidirect
99. #413	C₂×C₃×C₅	semidirect
100. #355	C₂×C₃×C₅	semidirect
158. #101	C₂²×C₃×C₅	semidirect
214. C₃×C₅	#2134	direct (N × H)
256. C₅	C₃×#2134	direct (N × H)
269. C₃	C₅×#2134	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×#535	2
C₃×C₅×#413	2
C₃×C₅×#355	2
C₅×#2134	3
C₃×#2134	5

Subgroups — C3×C5×#2134 #14275168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2134 #14275168