Subgroups — C₃×C₅×#2138 #14275172

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₅×#2138	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	1	no	—
8	C₃×C₅×#135	—	2	2	no	—
9	C₃×C₅×#135	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
12	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×#117	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	—	2	2	no	—
16	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
17	C₃×C₅×#107	C₂²	1	2	yes	—
18	C₃×C₅×#106	—	2	1	no	—
19	C₃×#2138	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₂×C₅×#135	C₂×C₃	1	1	yes	characteristic
21	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#427	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#358	C₂×C₃	1	2	yes	complemented
24	C₅×#357	C₂×C₃	1	2	yes	complemented
25	C₃×C₅×#41	—	2	4	no	—
26	C₃×C₅×#41	—	4	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₂×C₃×C₅×D₄	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂²×C₃×C₄×C₅	—	4	1	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×Q₁₆	—	2	1	no	—
33	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	—
34	C₃×C₅×SD₁₆	—	2	4	no	—
35	C₃×C₅×D₈	—	2	1	no	—
36	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
43	C₃×C₄²×C₅	—	4	1	no	—
44	C₃×C₅×C₁₆	—	2	2	no	—
45	C₂×C₃×#135	C₂×C₅	1	1	yes	characteristic
46	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
47	C₃×#427	C₂×C₅	1	1	yes	characteristic, complemented
48	C₃×#358	C₂×C₅	1	2	yes	complemented
49	C₃×#357	C₂×C₅	1	2	yes	complemented
50	C₂×C₅×#41	—	2	1	no	—
51	C₅×#135	—	2	2	no	—
52	C₅×#135	C₃×C₄	1	4	yes	complemented
53	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×SD₁₆	—	2	1	no	—
55	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	C₅×#117	—	2	1	no	—
58	C₂×C₅×C₁₆	—	2	2	no	—
59	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
60	C₅×#107	C₂²×C₃	1	2	yes	—
61	C₅×#106	—	2	1	no	—
62	#2138	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
63	C₂³×C₃×C₅	—	4	1	no	—
64	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	2	2	no	—
66	C₃×C₅×Q₈	—	4	1	no	—
67	C₃×C₅×D₄	—	2	2	no	—
68	C₃×C₅×D₄	—	4	4	no	—
69	C₃×C₅×D₄	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	—	4	4	no	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×#41	—	2	1	no	—
81	C₃×#135	—	2	2	no	—
82	C₃×#135	C₄×C₅	1	4	yes	complemented
83	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
84	C₂×C₃×SD₁₆	—	2	1	no	—
85	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
86	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₃×#117	—	2	1	no	—
88	C₂×C₃×C₁₆	—	2	2	no	—
89	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
90	C₃×#107	C₂²×C₅	1	2	yes	—
91	C₃×#106	—	2	1	no	—
92	C₅×#41	—	2	4	no	—
93	C₅×#41	—	4	2	no	—
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	4	1	no	—
97	C₂²×C₄×C₅	—	4	1	no	—
98	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
99	C₅×Q₁₆	—	2	1	no	—
100	C₅×Q₁₆	C₂×C₃×C₄	1	2	yes	—
101	C₅×SD₁₆	—	2	4	no	—
102	C₅×D₈	—	2	1	no	—
103	C₅×D₈	C₂×C₃×C₄	1	2	yes	—
104	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
105	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
107	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₂²⋊C₄	—	4	1	no	—
110	C₄²×C₅	—	4	1	no	—
111	C₅×C₁₆	—	2	2	no	—
112	C₂×#135	C₂×C₃×C₅	1	1	yes	characteristic
113	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
114	#427	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	#358	C₂×C₃×C₅	1	2	yes	complemented
116	#357	C₂×C₃×C₅	1	2	yes	complemented
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	—	4	2	no	—
119	C₂²×C₃×C₅	—	8	2	no	—
120	C₂²×C₃×C₅	#102	1	2	yes	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₃×C₄×C₅	—	4	2	no	—
123	C₃×C₄×C₅	—	8	2	no	—
124	C₃×C₄×C₅	#102	1	2	yes	—
125	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
127	C₃×#41	—	2	4	no	—
128	C₃×#41	—	4	2	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	4	1	no	—
132	C₂²×C₃×C₄	—	4	1	no	—
133	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
134	C₃×Q₁₆	—	2	1	no	—
135	C₃×Q₁₆	C₂×C₄×C₅	1	2	yes	—
136	C₃×SD₁₆	—	2	4	no	—
137	C₃×D₈	—	2	1	no	—
138	C₃×D₈	C₂×C₄×C₅	1	2	yes	—
139	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
140	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
141	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
142	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₂²⋊C₄	—	4	1	no	—
145	C₃×C₄²	—	4	1	no	—
146	C₃×C₁₆	—	2	2	no	—
147	C₂³×C₅	—	4	1	no	—
148	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
149	C₅×Q₈	—	2	2	no	—
150	C₅×Q₈	—	4	1	no	—
151	C₅×D₄	—	2	2	no	—
152	C₅×D₄	—	4	4	no	—
153	C₅×D₄	—	4	1	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	4	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
158	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
159	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
160	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
161	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
162	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
163	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×#41	—	2	1	no	—
165	#135	—	2	2	no	—
166	#135	C₃×C₄×C₅	1	4	yes	complemented
167	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
171	#117	—	2	1	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
174	#107	C₂²×C₃×C₅	1	2	yes	—
175	#106	—	2	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	—	8	2	no	—
178	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
179	C₂×C₃×C₅	#508	1	2	yes	—
180	C₂³×C₃	—	4	1	no	—
181	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
182	C₃×Q₈	—	2	2	no	—
183	C₃×Q₈	—	4	1	no	—
184	C₃×D₄	—	2	2	no	—
185	C₃×D₄	—	4	4	no	—
186	C₃×D₄	—	4	1	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	4	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
191	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
192	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
193	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
194	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
195	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
196	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	4	2	no	—
199	C₂²×C₅	—	8	2	no	—
200	C₂²×C₅	C₃×#102	1	2	yes	—
201	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
202	C₄×C₅	—	4	2	no	—
203	C₄×C₅	—	8	2	no	—
204	C₄×C₅	C₃×#102	1	2	yes	—
205	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
207	#41	—	2	4	no	—
208	#41	—	4	2	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
214	Q₁₆	—	2	1	no	—
215	Q₁₆	C₂×C₃×C₄×C₅	1	2	yes	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	1	no	—
218	D₈	C₂×C₃×C₄×C₅	1	2	yes	—
219	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
220	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
221	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
222	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	4	1	no	—
225	C₄²	—	4	1	no	—
226	C₁₆	—	2	2	no	—
227	C₃×C₅	#2138	1	1	yes	characteristic, direct product factor, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	4	2	no	—
230	C₂²×C₃	—	8	2	no	—
231	C₂²×C₃	C₅×#102	1	2	yes	—
232	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	—	8	2	no	—
235	C₃×C₄	C₅×#102	1	2	yes	—
236	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
237	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	8	2	no	—
240	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
241	C₂×C₅	C₃×#508	1	2	yes	—
242	C₂³	—	4	1	no	—
243	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	4	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
254	C₂×C₄	C₃×C₅×SD₁₆	1	2	yes	—
255	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
256	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
257	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
258	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	8	2	no	—
261	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
262	C₂×C₃	C₅×#508	1	2	yes	—
263	C₅	C₃×#2138	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	C₃×C₅×#102	1	2	yes	—
268	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	8	2	no	—
271	C₄	C₃×C₅×#102	1	2	yes	—
272	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
273	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
274	C₃	C₅×#2138	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	8	2	no	—
277	C₂	C₂×C₃×C₅×#102	1	1	yes	characteristic
278	C₂	C₃×C₅×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×C₅×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₃×C₅×#427	C₂	semidirect
4. C₃×C₅×#358	C₂	semidirect
5. C₃×C₅×#357	C₂	semidirect
6. C₅×#2138	C₃	direct (N × H)
9. C₃×C₅×#135	C₄	semidirect
19. C₃×#2138	C₅	direct (N × H)
21. C₂²×C₅×C₁₆	C₂×C₃	semidirect
22. C₅×#427	C₂×C₃	semidirect
23. C₅×#358	C₂×C₃	semidirect
24. C₅×#357	C₂×C₃	semidirect
46. C₂²×C₃×C₁₆	C₂×C₅	semidirect
47. C₃×#427	C₂×C₅	semidirect
48. C₃×#358	C₂×C₅	semidirect
49. C₃×#357	C₂×C₅	semidirect
52. C₅×#135	C₃×C₄	semidirect
62. #2138	C₃×C₅	direct (N × H)
82. C₃×#135	C₄×C₅	semidirect
113. C₂²×C₁₆	C₂×C₃×C₅	semidirect
114. #427	C₂×C₃×C₅	semidirect
115. #358	C₂×C₃×C₅	semidirect
116. #357	C₂×C₃×C₅	semidirect
166. #135	C₃×C₄×C₅	semidirect
227. C₃×C₅	#2138	direct (N × H)
263. C₅	C₃×#2138	direct (N × H)
274. C₃	C₅×#2138	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×#135	2
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×#427	2
C₃×C₅×#358	2
C₃×C₅×#357	2
C₅×#2138	3
C₃×#2138	5

Subgroups — C3×C5×#2138 #14275172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2138 #14275172