Subgroups — C₃×C₅×#1847 #14274881

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 252 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	2	yes	maximal
2	C₂×C₃×C₅×#106	C₂	1	2	yes	maximal
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic
6	C₅×#1847	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	—	2	4	no	—
14	C₃×C₅×#106	—	2	4	no	—
15	C₃×C₅×#97	C₄	1	4	yes	complemented
16	C₃×C₅×#95	—	2	2	no	—
17	C₃×#1847	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#107	C₂×C₃	1	2	yes	—
19	C₂×C₅×#106	C₂×C₃	1	2	yes	—
20	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
21	C₅×#384	C₂×C₃	1	1	yes	characteristic
22	C₅×#382	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	2	no	—
24	C₂²×C₃×C₄×C₅	—	2	1	no	—
25	C₂²×C₃×C₄×C₅	—	2	1	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
30	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	4	4	no	—
36	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
37	C₃×C₄²×C₅	Q₈	1	2	yes	—
38	C₂×C₃×#107	C₂×C₅	1	2	yes	—
39	C₂×C₃×#106	C₂×C₅	1	2	yes	—
40	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
41	C₃×#384	C₂×C₅	1	1	yes	characteristic
42	C₃×#382	C₂×C₅	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	—	2	2	no	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×#107	—	2	4	no	—
50	C₅×#106	—	2	4	no	—
51	C₅×#97	C₃×C₄	1	4	yes	complemented
52	C₅×#95	—	2	2	no	—
53	#1847	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	—	4	1	no	—
59	C₂×C₃×C₄×C₅	—	4	1	no	—
60	C₂×C₃×C₄×C₅	—	4	4	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₃×C₅×C₈	C₄⋊C₄	1	8	yes	complemented
69	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
70	C₂×C₃×C₄⋊C₄	—	2	2	no	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
73	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
75	C₃×#107	—	2	4	no	—
76	C₃×#106	—	2	4	no	—
77	C₃×#97	C₄×C₅	1	4	yes	complemented
78	C₃×#95	—	2	2	no	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	1	no	—
81	C₂²×C₄×C₅	—	2	1	no	—
82	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×C₃×C₄	1	4	yes	—
85	C₂×C₅×C₈	C₃×D₄	1	4	yes	—
86	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
87	C₅×C₄⋊C₄	—	2	2	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	2	no	—
91	C₅×C₄⋊C₄	—	4	4	no	—
92	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
93	C₄²×C₅	C₃×Q₈	1	2	yes	—
94	C₂×#107	C₂×C₃×C₅	1	2	yes	—
95	C₂×#106	C₂×C₃×C₅	1	2	yes	—
96	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
97	#384	C₂×C₃×C₅	1	1	yes	characteristic
98	#382	C₂×C₃×C₅	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	#127	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
102	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
103	C₂²×C₃×C₅	#137	1	2	yes	—
104	C₃×C₄×C₅	—	2	4	no	—
105	C₃×C₄×C₅	—	8	2	no	—
106	C₃×C₄×C₅	—	8	1	no	—
107	C₃×C₄×C₅	—	8	1	no	—
108	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
109	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
110	C₂²×C₃×C₄	—	2	1	no	—
111	C₂²×C₃×C₄	—	2	1	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
114	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂×C₄×C₅	1	4	yes	—
116	C₂×C₃×C₈	C₅×D₄	1	4	yes	—
117	C₂×C₃×C₈	C₅×Q₈	1	4	yes	—
118	C₃×C₄⋊C₄	—	2	2	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₄⋊C₄	—	2	2	no	—
122	C₃×C₄⋊C₄	—	4	4	no	—
123	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
124	C₃×C₄²	C₅×Q₈	1	2	yes	—
125	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	2	no	—
133	C₂×C₄×C₅	—	4	2	no	—
134	C₂×C₄×C₅	C₃×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
137	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
139	C₅×C₈	C₃×C₄⋊C₄	1	8	yes	complemented
140	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	1	yes	characteristic
144	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
145	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
146	#107	—	2	4	no	—
147	#106	—	2	4	no	—
148	#97	C₃×C₄×C₅	1	4	yes	complemented
149	#95	—	2	2	no	—
150	C₂×C₃×C₅	#384	1	1	yes	characteristic
151	C₂×C₃×C₅	#428	1	2	yes	—
152	C₂×C₃×C₅	#496	1	1	yes	characteristic
153	C₂×C₃×C₅	#497	1	1	yes	characteristic
154	C₂×C₃×C₅	#501	1	2	yes	—
155	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	—	4	4	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	C₅×C₄⋊C₄	1	4	yes	—
165	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
167	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
168	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
169	C₃×C₈	C₅×C₄⋊C₄	1	8	yes	complemented
170	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
171	C₂²×C₅	C₃×#127	1	1	yes	characteristic
172	C₂²×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
173	C₂²×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
174	C₂²×C₅	C₃×#137	1	2	yes	—
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	8	2	no	—
177	C₄×C₅	—	8	1	no	—
178	C₄×C₅	—	8	1	no	—
179	C₄×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
180	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
185	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
186	C₂×C₈	C₂×C₃×C₄×C₅	1	4	yes	—
187	C₂×C₈	C₃×C₅×D₄	1	4	yes	—
188	C₂×C₈	C₃×C₅×Q₈	1	4	yes	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
195	C₄²	C₃×C₅×Q₈	1	2	yes	—
196	C₃×C₅	#1847	1	1	yes	characteristic, direct product factor, Hall
197	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
198	C₂²×C₃	C₅×#127	1	1	yes	characteristic
199	C₂²×C₃	C₅×C₄⋊Q₈	1	1	yes	characteristic
200	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
201	C₂²×C₃	C₅×#137	1	2	yes	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	8	2	no	—
204	C₃×C₄	—	8	1	no	—
205	C₃×C₄	—	8	1	no	—
206	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
207	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
208	C₂×C₅	C₃×#384	1	1	yes	characteristic
209	C₂×C₅	C₃×#428	1	2	yes	—
210	C₂×C₅	C₃×#496	1	1	yes	characteristic
211	C₂×C₅	C₃×#497	1	1	yes	characteristic
212	C₂×C₅	C₃×#501	1	2	yes	—
213	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	C₃×C₅×C₄⋊C₄	1	4	yes	—
223	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
224	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
225	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
226	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
227	C₈	C₃×C₅×C₄⋊C₄	1	8	yes	complemented
228	C₂×C₃	C₅×#384	1	1	yes	characteristic
229	C₂×C₃	C₅×#428	1	2	yes	—
230	C₂×C₃	C₅×#496	1	1	yes	characteristic
231	C₂×C₃	C₅×#497	1	1	yes	characteristic
232	C₂×C₃	C₅×#501	1	2	yes	—
233	C₅	C₃×#1847	1	1	yes	characteristic, direct product factor, Sylow
234	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
235	C₂²	C₃×C₅×#127	1	1	yes	characteristic
236	C₂²	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
237	C₂²	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
238	C₂²	C₃×C₅×#137	1	2	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	8	2	no	—
241	C₄	—	8	1	no	—
242	C₄	—	8	1	no	—
243	C₄	C₂×C₃×C₅×C₄⋊C₄	1	2	yes	—
244	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
245	C₃	C₅×#1847	1	1	yes	characteristic, direct product factor, Sylow
246	C₂	C₃×C₅×#384	1	1	yes	characteristic
247	C₂	C₃×C₅×#428	1	2	yes	—
248	C₂	C₃×C₅×#496	1	1	yes	characteristic
249	C₂	C₃×C₅×#497	1	1	yes	characteristic
250	C₂	C₃×C₅×#501	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1847	C₃	direct (N × H)
15. C₃×C₅×#97	C₄	semidirect
17. C₃×#1847	C₅	direct (N × H)
51. C₅×#97	C₃×C₄	semidirect
53. #1847	C₃×C₅	direct (N × H)
68. C₃×C₅×C₈	C₄⋊C₄	semidirect
77. C₃×#97	C₄×C₅	semidirect
139. C₅×C₈	C₃×C₄⋊C₄	semidirect
148. #97	C₃×C₄×C₅	semidirect
169. C₃×C₈	C₅×C₄⋊C₄	semidirect
196. C₃×C₅	#1847	direct (N × H)
227. C₈	C₃×C₅×C₄⋊C₄	semidirect
233. C₅	C₃×#1847	direct (N × H)
245. C₃	C₅×#1847	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
C₂×C₃×C₅×#106	2
C₂×C₃×C₅×#97	2
C₃×C₅×#384	2
C₃×C₅×#382	2
C₅×#1847	3
C₃×#1847	5

Subgroups — C3×C5×#1847 #14274881

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1847 #14274881