Subgroups — C₃×C₅×#1946 #14274980

Order 1920.

400 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#424	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#339	C₂	1	2	yes	maximal
6	C₅×#1946	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	2	no	—
13	C₃×C₅×#107	C₄	1	2	yes	complemented
14	C₃×C₅×#106	C₄	1	2	yes	complemented
15	C₃×C₅×C₄⋊C₈	—	2	4	no	—
16	C₃×C₅×#97	—	2	2	no	—
17	C₃×#1946	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
19	C₅×#427	C₂×C₃	1	1	yes	characteristic
20	C₅×#424	C₂×C₃	1	2	yes	—
21	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
22	C₅×#339	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	—	2	1	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₃×C₅×M₁₆	—	2	2	no	—
26	C₃×C₅×M₁₆	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
35	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
36	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	—	2	2	no	—
39	C₃×C₄²×C₅	Q₈	1	2	yes	—
40	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
41	C₃×#427	C₂×C₅	1	1	yes	characteristic
42	C₃×#424	C₂×C₅	1	2	yes	—
43	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
44	C₃×#339	C₂×C₅	1	2	yes	—
45	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
46	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₅×#117	C₂²×C₃	1	2	yes	—
49	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₅×#108	—	2	2	no	—
51	C₅×#107	C₃×C₄	1	2	yes	complemented
52	C₅×#106	C₃×C₄	1	2	yes	complemented
53	C₅×C₄⋊C₈	—	2	4	no	—
54	C₅×#97	—	2	2	no	—
55	#1946	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	#41	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	—	2	4	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	4	2	no	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
66	C₃×C₅×C₈	—	2	2	no	—
67	C₃×C₅×C₈	—	4	2	no	—
68	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
69	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
71	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×#117	C₂²×C₅	1	2	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₃×#108	—	2	2	no	—
76	C₃×#107	C₄×C₅	1	2	yes	complemented
77	C₃×#106	C₄×C₅	1	2	yes	complemented
78	C₃×C₄⋊C₈	—	2	4	no	—
79	C₃×#97	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	1	no	—
81	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
82	C₅×M₁₆	—	2	2	no	—
83	C₅×M₁₆	—	2	2	no	—
84	C₂×C₅×C₈	—	2	1	no	—
85	C₂×C₅×C₈	—	2	1	no	—
86	C₂×C₅×C₈	—	2	2	no	—
87	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
90	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
91	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
92	C₅×C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
93	C₅×C₂²⋊C₄	—	2	2	no	—
94	C₄²×C₅	—	2	1	no	—
95	C₄²×C₅	—	2	2	no	—
96	C₄²×C₅	C₃×Q₈	1	2	yes	—
97	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
98	#427	C₂×C₃×C₅	1	1	yes	characteristic
99	#424	C₂×C₃×C₅	1	2	yes	—
100	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
101	#339	C₂×C₃×C₅	1	2	yes	—
102	C₂²×C₃×C₅	—	2	2	no	—
103	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₃×C₅	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂²×C₃×C₅	#125	1	1	yes	characteristic
106	C₃×C₄×C₅	—	4	2	no	—
107	C₃×C₄×C₅	—	4	4	no	—
108	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
110	C₃×C₄×C₅	#121	1	1	yes	characteristic
111	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
114	C₃×M₁₆	—	2	2	no	—
115	C₃×M₁₆	—	2	2	no	—
116	C₂×C₃×C₈	—	2	1	no	—
117	C₂×C₃×C₈	—	2	1	no	—
118	C₂×C₃×C₈	—	2	2	no	—
119	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
122	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
123	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
124	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
125	C₃×C₂²⋊C₄	—	2	2	no	—
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	C₅×Q₈	1	2	yes	—
129	C₂³×C₅	C₃×#41	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
139	C₅×C₈	—	2	2	no	—
140	C₅×C₈	—	4	2	no	—
141	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
142	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
143	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
144	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
145	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
146	#117	C₂²×C₃×C₅	1	2	yes	—
147	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
148	#108	—	2	2	no	—
149	#107	C₃×C₄×C₅	1	2	yes	complemented
150	#106	C₃×C₄×C₅	1	2	yes	complemented
151	C₄⋊C₈	—	2	4	no	—
152	#97	—	2	2	no	—
153	C₂×C₃×C₅	—	2	1	no	—
154	C₂×C₃×C₅	—	2	1	no	—
155	C₂×C₃×C₅	#389	1	1	yes	characteristic
156	C₂×C₃×C₅	#444	1	2	yes	—
157	C₂³×C₃	C₅×#41	1	1	yes	characteristic
158	C₂×C₃×C₄	—	2	4	no	—
159	C₂×C₃×C₄	—	2	2	no	—
160	C₂×C₃×C₄	—	4	2	no	—
161	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
166	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
167	C₃×C₈	—	2	2	no	—
168	C₃×C₈	—	4	2	no	—
169	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
170	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
171	C₂²×C₅	—	2	2	no	—
172	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
173	C₂²×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
174	C₂²×C₅	C₃×#125	1	1	yes	characteristic
175	C₄×C₅	—	4	2	no	—
176	C₄×C₅	—	4	4	no	—
177	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
178	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₅	C₃×#121	1	1	yes	characteristic
180	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
183	M₁₆	—	2	2	no	—
184	M₁₆	—	2	2	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
189	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
190	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
191	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
192	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
193	C₄⋊C₄	C₂×C₃×C₄×C₅	1	4	yes	—
194	C₂²⋊C₄	—	2	2	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	2	no	—
197	C₄²	C₃×C₅×Q₈	1	2	yes	—
198	C₃×C₅	#1946	1	1	yes	characteristic, direct product factor, Hall
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
201	C₂²×C₃	C₅×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂²×C₃	C₅×#125	1	1	yes	characteristic
203	C₃×C₄	—	4	2	no	—
204	C₃×C₄	—	4	4	no	—
205	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
206	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
207	C₃×C₄	C₅×#121	1	1	yes	characteristic
208	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
209	C₂×C₅	—	2	1	no	—
210	C₂×C₅	—	2	1	no	—
211	C₂×C₅	C₃×#389	1	1	yes	characteristic
212	C₂×C₅	C₃×#444	1	2	yes	—
213	C₂³	C₃×C₅×#41	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
218	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
219	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
221	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
222	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	2	no	—
225	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
226	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	C₅×#389	1	1	yes	characteristic
230	C₂×C₃	C₅×#444	1	2	yes	—
231	C₅	C₃×#1946	1	1	yes	characteristic, direct product factor, Sylow
232	C₂²	—	2	2	no	—
233	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
234	C₂²	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂²	C₃×C₅×#125	1	1	yes	characteristic
236	C₄	—	4	2	no	—
237	C₄	—	4	4	no	—
238	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
239	C₄	C₃×C₄×C₅×Q₈	1	1	yes	characteristic
240	C₄	C₃×C₅×#121	1	1	yes	characteristic
241	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
242	C₃	C₅×#1946	1	1	yes	characteristic, direct product factor, Sylow
243	C₂	—	2	1	no	—
244	C₂	—	2	1	no	—
245	C₂	C₃×C₅×#389	1	1	yes	characteristic
246	C₂	C₃×C₅×#444	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1946	C₃	direct (N × H)
13. C₃×C₅×#107	C₄	semidirect
14. C₃×C₅×#106	C₄	semidirect
17. C₃×#1946	C₅	direct (N × H)
51. C₅×#107	C₃×C₄	semidirect
52. C₅×#106	C₃×C₄	semidirect
55. #1946	C₃×C₅	direct (N × H)
76. C₃×#107	C₄×C₅	semidirect
77. C₃×#106	C₄×C₅	semidirect
149. #107	C₃×C₄×C₅	semidirect
150. #106	C₃×C₄×C₅	semidirect
198. C₃×C₅	#1946	direct (N × H)
231. C₅	C₃×#1946	direct (N × H)
242. C₃	C₅×#1946	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×#108	2
C₃×C₅×#427	2
C₃×C₅×#424	2
C₂×C₃×C₅×#97	2
C₃×C₅×#339	2
C₅×#1946	3
C₃×#1946	5

Subgroups — C3×C5×#1946 #14274980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1946 #14274980