Subgroups — C₃×C₅×#2078 #14275112

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#417	C₂	1	2	yes	maximal
3	C₃×C₅×#408	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#343	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2078	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₅×#107	—	2	2	no	—
13	C₃×C₅×#103	—	2	2	no	—
14	C₃×C₅×#102	—	2	2	no	—
15	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
16	C₃×#2078	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic, complemented
18	C₅×#417	C₂×C₃	1	2	yes	—
19	C₅×#408	C₂×C₃	1	1	yes	characteristic
20	C₅×#343	C₂×C₃	1	2	yes	complemented
21	C₅×#341	C₂×C₃	1	1	yes	characteristic, complemented
22	C₃×C₅×#41	—	2	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	—	2	1	no	—
26	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×M₁₆	—	2	2	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	2	1	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#417	C₂×C₅	1	2	yes	—
41	C₃×#408	C₂×C₅	1	1	yes	characteristic
42	C₃×#343	C₂×C₅	1	2	yes	complemented
43	C₃×#341	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₅×#41	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₅×#107	—	2	2	no	—
50	C₅×#103	—	2	2	no	—
51	C₅×#102	—	2	2	no	—
52	C₅×C₂²⋊C₈	—	2	2	no	—
53	#2078	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	—	2	1	no	—
55	C₂³×C₃×C₅	#41	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	2	no	—
57	C₃×C₅×D₄	—	2	2	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	—	8	2	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	2	yes	—
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₈	—	4	2	no	—
70	C₃×C₅×C₈	—	4	1	no	—
71	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
74	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
76	C₃×#107	—	2	2	no	—
77	C₃×#103	—	2	2	no	—
78	C₃×#102	—	2	2	no	—
79	C₃×C₂²⋊C₈	—	2	2	no	—
80	C₅×#41	—	2	2	no	—
81	C₅×#41	—	2	2	no	—
82	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
83	C₂×C₅×D₄	—	2	1	no	—
84	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	1	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₅×M₁₆	—	2	2	no	—
89	C₅×M₁₆	—	2	2	no	—
90	C₅×M₁₆	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₅×C₄⋊C₄	—	2	2	no	—
97	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic, complemented
98	#417	C₂×C₃×C₅	1	2	yes	—
99	#408	C₂×C₃×C₅	1	1	yes	characteristic
100	#343	C₂×C₃×C₅	1	2	yes	complemented
101	#341	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	C₂²×C₃×C₅	—	2	1	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	—	4	1	no	—
105	C₂²×C₃×C₅	—	4	1	no	—
106	C₂²×C₃×C₅	—	4	1	no	—
107	C₂²×C₃×C₅	#121	1	1	yes	characteristic
108	C₂²×C₃×C₅	#126	1	2	yes	—
109	C₃×C₄×C₅	—	4	1	no	—
110	C₃×C₄×C₅	—	8	2	no	—
111	C₃×C₄×C₅	#123	1	1	yes	characteristic
112	C₃×C₄×C₅	#123	1	2	yes	—
113	C₃×C₄×C₅	#124	1	1	yes	characteristic
114	C₃×#41	—	2	2	no	—
115	C₃×#41	—	2	2	no	—
116	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
117	C₂×C₃×D₄	—	2	1	no	—
118	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	2	1	no	—
120	C₂²×C₃×C₄	—	2	2	no	—
121	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
122	C₃×M₁₆	—	2	2	no	—
123	C₃×M₁₆	—	2	2	no	—
124	C₃×M₁₆	—	2	2	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₂³×C₅	—	2	1	no	—
132	C₂³×C₅	C₃×#41	1	1	yes	characteristic
133	C₅×Q₈	—	2	2	no	—
134	C₅×D₄	—	2	2	no	—
135	C₅×D₄	—	4	2	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	1	no	—
139	C₂×C₄×C₅	—	4	2	no	—
140	C₂×C₄×C₅	—	8	2	no	—
141	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₃×#41	1	2	yes	—
144	C₂×C₄×C₅	C₃×#41	1	2	yes	—
145	C₅×C₈	—	2	2	no	—
146	C₅×C₈	—	4	2	no	—
147	C₅×C₈	—	4	1	no	—
148	C₂×#41	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
150	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
153	#107	—	2	2	no	—
154	#103	—	2	2	no	—
155	#102	—	2	2	no	—
156	C₂²⋊C₈	—	2	2	no	—
157	C₂×C₃×C₅	—	2	2	no	—
158	C₂×C₃×C₅	—	8	1	no	—
159	C₂×C₃×C₅	#397	1	1	yes	characteristic
160	C₂×C₃×C₅	#472	1	1	yes	characteristic
161	C₂×C₃×C₅	#473	1	1	yes	characteristic
162	C₂³×C₃	—	2	1	no	—
163	C₂³×C₃	C₅×#41	1	1	yes	characteristic
164	C₃×Q₈	—	2	2	no	—
165	C₃×D₄	—	2	2	no	—
166	C₃×D₄	—	4	2	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	—	4	1	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	—	8	2	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	2	yes	—
175	C₂×C₃×C₄	C₅×#41	1	2	yes	—
176	C₃×C₈	—	2	2	no	—
177	C₃×C₈	—	4	2	no	—
178	C₃×C₈	—	4	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	—	2	1	no	—
181	C₂²×C₅	—	4	1	no	—
182	C₂²×C₅	—	4	1	no	—
183	C₂²×C₅	—	4	1	no	—
184	C₂²×C₅	C₃×#121	1	1	yes	characteristic
185	C₂²×C₅	C₃×#126	1	2	yes	—
186	C₄×C₅	—	4	1	no	—
187	C₄×C₅	—	8	2	no	—
188	C₄×C₅	C₃×#123	1	1	yes	characteristic
189	C₄×C₅	C₃×#123	1	2	yes	—
190	C₄×C₅	C₃×#124	1	1	yes	characteristic
191	#41	—	2	2	no	—
192	#41	—	2	2	no	—
193	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₃×C₅	#2078	1	1	yes	characteristic, direct product factor, Hall
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	—	2	1	no	—
211	C₂²×C₃	—	4	1	no	—
212	C₂²×C₃	—	4	1	no	—
213	C₂²×C₃	—	4	1	no	—
214	C₂²×C₃	C₅×#121	1	1	yes	characteristic
215	C₂²×C₃	C₅×#126	1	2	yes	—
216	C₃×C₄	—	4	1	no	—
217	C₃×C₄	—	8	2	no	—
218	C₃×C₄	C₅×#123	1	1	yes	characteristic
219	C₃×C₄	C₅×#123	1	2	yes	—
220	C₃×C₄	C₅×#124	1	1	yes	characteristic
221	C₂×C₅	—	2	2	no	—
222	C₂×C₅	—	8	1	no	—
223	C₂×C₅	C₃×#397	1	1	yes	characteristic
224	C₂×C₅	C₃×#472	1	1	yes	characteristic
225	C₂×C₅	C₃×#473	1	1	yes	characteristic
226	C₂³	—	2	1	no	—
227	C₂³	C₃×C₅×#41	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
238	C₂×C₄	C₃×C₅×#41	1	2	yes	—
239	C₂×C₄	C₃×C₅×#41	1	2	yes	—
240	C₈	—	2	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	1	no	—
243	C₂×C₃	—	2	2	no	—
244	C₂×C₃	—	8	1	no	—
245	C₂×C₃	C₅×#397	1	1	yes	characteristic
246	C₂×C₃	C₅×#472	1	1	yes	characteristic
247	C₂×C₃	C₅×#473	1	1	yes	characteristic
248	C₅	C₃×#2078	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	1	no	—
254	C₂²	C₃×C₅×#121	1	1	yes	characteristic
255	C₂²	C₃×C₅×#126	1	2	yes	—
256	C₄	—	4	1	no	—
257	C₄	—	8	2	no	—
258	C₄	C₃×C₅×#123	1	1	yes	characteristic
259	C₄	C₃×C₅×#123	1	2	yes	—
260	C₄	C₃×C₅×#124	1	1	yes	characteristic
261	C₃	C₅×#2078	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	2	2	no	—
263	C₂	—	8	1	no	—
264	C₂	C₃×C₅×#397	1	1	yes	characteristic
265	C₂	C₃×C₅×#472	1	1	yes	characteristic
266	C₂	C₃×C₅×#473	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#107	C₂	semidirect
4. C₃×C₅×#343	C₂	semidirect
5. C₃×C₅×#341	C₂	semidirect
6. C₅×#2078	C₃	direct (N × H)
16. C₃×#2078	C₅	direct (N × H)
17. C₂×C₅×#107	C₂×C₃	semidirect
20. C₅×#343	C₂×C₃	semidirect
21. C₅×#341	C₂×C₃	semidirect
39. C₂×C₃×#107	C₂×C₅	semidirect
42. C₃×#343	C₂×C₅	semidirect
43. C₃×#341	C₂×C₅	semidirect
53. #2078	C₃×C₅	direct (N × H)
97. C₂×#107	C₂×C₃×C₅	semidirect
100. #343	C₂×C₃×C₅	semidirect
101. #341	C₂×C₃×C₅	semidirect
208. C₃×C₅	#2078	direct (N × H)
248. C₅	C₃×#2078	direct (N × H)
261. C₃	C₅×#2078	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×#107	2
C₃×C₅×#417	2
C₃×C₅×#408	2
C₃×C₅×#343	2
C₃×C₅×#341	2
C₅×#2078	3
C₃×#2078	5

Subgroups — C3×C5×#2078 #14275112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2078 #14275112