Subgroups — C₃×C₅×#1833 #14274867

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 264 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×#405	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#344	C₂	1	2	yes	maximal
5	C₃×C₅×H(3,ℤ/4)	C₂	1	2	yes	maximal
6	C₅×#1833	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₂²	1	2	yes	—
11	C₃×C₅×#117	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	—	2	1	no	—
13	C₃×C₅×#108	—	2	1	no	—
14	C₃×C₅×#107	—	2	1	no	—
15	C₃×C₅×#106	—	2	1	no	—
16	C₃×C₅×#97	C₄	1	2	yes	complemented
17	C₃×C₄×C₅×C₈	C₄	1	2	yes	complemented
18	C₃×#1833	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#108	C₂×C₃	1	1	yes	characteristic
20	C₅×#427	C₂×C₃	1	1	yes	characteristic
21	C₅×#405	C₂×C₃	1	1	yes	characteristic
22	C₅×#344	C₂×C₃	1	2	yes	—
23	C₅×H(3,ℤ/4)	C₂×C₃	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₃×C₅×M₁₆	—	2	1	no	—
26	C₃×C₅×M₁₆	—	2	1	no	—
27	C₃×C₅×M₁₆	—	2	2	no	—
28	C₃×C₅×M₁₆	—	2	2	no	—
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	Q₈	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	—	2	2	no	—
39	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
43	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
44	C₃×#427	C₂×C₅	1	1	yes	characteristic
45	C₃×#405	C₂×C₅	1	1	yes	characteristic
46	C₃×#344	C₂×C₅	1	2	yes	—
47	C₃×H(3,ℤ/4)	C₂×C₅	1	2	yes	—
48	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
49	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×#117	C₂²×C₃	1	2	yes	—
52	C₅×#117	C₂²×C₃	1	1	yes	characteristic
53	C₅×#108	—	2	1	no	—
54	C₅×#108	—	2	1	no	—
55	C₅×#107	—	2	1	no	—
56	C₅×#106	—	2	1	no	—
57	C₅×#97	C₃×C₄	1	2	yes	complemented
58	C₄×C₅×C₈	C₃×C₄	1	2	yes	complemented
59	#1833	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
60	C₂³×C₃×C₅	#41	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
69	C₃×C₅×C₈	—	2	2	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
72	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
76	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
77	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₃×#117	C₂²×C₅	1	2	yes	—
79	C₃×#117	C₂²×C₅	1	1	yes	characteristic
80	C₃×#108	—	2	1	no	—
81	C₃×#108	—	2	1	no	—
82	C₃×#107	—	2	1	no	—
83	C₃×#106	—	2	1	no	—
84	C₃×#97	C₄×C₅	1	2	yes	complemented
85	C₃×C₄×C₈	C₄×C₅	1	2	yes	complemented
86	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₅×M₁₆	—	2	1	no	—
88	C₅×M₁₆	—	2	1	no	—
89	C₅×M₁₆	—	2	2	no	—
90	C₅×M₁₆	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
93	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
98	C₂×C₅×C₈	C₃×Q₈	1	1	yes	characteristic
99	C₅×C₄⋊C₄	—	2	1	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₂²⋊C₄	—	2	1	no	—
102	C₅×C₂²⋊C₄	—	2	2	no	—
103	C₄²×C₅	—	2	2	no	—
104	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
105	C₂×#108	C₂×C₃×C₅	1	1	yes	characteristic
106	#427	C₂×C₃×C₅	1	1	yes	characteristic
107	#405	C₂×C₃×C₅	1	1	yes	characteristic
108	#344	C₂×C₃×C₅	1	2	yes	—
109	H(3,ℤ/4)	C₂×C₃×C₅	1	2	yes	—
110	C₂²×C₃×C₅	—	4	1	no	—
111	C₂²×C₃×C₅	#117	1	1	yes	characteristic
112	C₂²×C₃×C₅	#124	1	1	yes	characteristic
113	C₂²×C₃×C₅	#125	1	1	yes	characteristic
114	C₃×C₄×C₅	—	4	2	no	—
115	C₃×C₄×C₅	—	8	2	no	—
116	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃×C₄×C₅	#117	1	1	yes	characteristic
118	C₃×C₄×C₅	#124	1	1	yes	characteristic
119	C₃×C₄×C₅	#125	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
121	C₃×M₁₆	—	2	1	no	—
122	C₃×M₁₆	—	2	1	no	—
123	C₃×M₁₆	—	2	2	no	—
124	C₃×M₁₆	—	2	2	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
127	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
128	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
130	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
132	C₂×C₃×C₈	C₅×Q₈	1	1	yes	characteristic
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₂²⋊C₄	—	2	1	no	—
136	C₃×C₂²⋊C₄	—	2	2	no	—
137	C₃×C₄²	—	2	2	no	—
138	C₃×C₄²	C₂×C₄×C₅	1	2	yes	—
139	C₂³×C₅	C₃×#41	1	1	yes	characteristic
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
146	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
147	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
148	C₅×C₈	—	2	2	no	—
149	C₅×C₈	—	4	2	no	—
150	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
151	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
152	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
153	C₅×C₈	C₃×C₄⋊C₄	1	1	yes	characteristic
154	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
155	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
156	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
157	#117	C₂²×C₃×C₅	1	2	yes	—
158	#117	C₂²×C₃×C₅	1	1	yes	characteristic
159	#108	—	2	1	no	—
160	#108	—	2	1	no	—
161	#107	—	2	1	no	—
162	#106	—	2	1	no	—
163	#97	C₃×C₄×C₅	1	2	yes	complemented
164	C₄×C₈	C₃×C₄×C₅	1	2	yes	complemented
165	C₂×C₃×C₅	—	2	1	no	—
166	C₂×C₃×C₅	—	2	1	no	—
167	C₂×C₃×C₅	—	2	1	no	—
168	C₂×C₃×C₅	#382	1	1	yes	characteristic
169	C₂³×C₃	C₅×#41	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
175	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
176	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
177	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
178	C₃×C₈	—	2	2	no	—
179	C₃×C₈	—	4	2	no	—
180	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
181	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
182	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
183	C₃×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂²×C₅	—	4	1	no	—
185	C₂²×C₅	C₃×#117	1	1	yes	characteristic
186	C₂²×C₅	C₃×#124	1	1	yes	characteristic
187	C₂²×C₅	C₃×#125	1	1	yes	characteristic
188	C₄×C₅	—	4	2	no	—
189	C₄×C₅	—	8	2	no	—
190	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
191	C₄×C₅	C₃×#117	1	1	yes	characteristic
192	C₄×C₅	C₃×#124	1	1	yes	characteristic
193	C₄×C₅	C₃×#125	1	1	yes	characteristic
194	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
195	M₁₆	—	2	1	no	—
196	M₁₆	—	2	1	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
201	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
202	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
203	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
204	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
205	C₂×C₈	C₃×C₅×Q₈	1	1	yes	characteristic
206	C₂×C₈	C₃×C₅×Q₈	1	1	yes	characteristic
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₄²	—	2	2	no	—
212	C₄²	C₂×C₃×C₄×C₅	1	2	yes	—
213	C₃×C₅	#1833	1	1	yes	characteristic, direct product factor, Hall
214	C₂²×C₃	—	4	1	no	—
215	C₂²×C₃	C₅×#117	1	1	yes	characteristic
216	C₂²×C₃	C₅×#124	1	1	yes	characteristic
217	C₂²×C₃	C₅×#125	1	1	yes	characteristic
218	C₃×C₄	—	4	2	no	—
219	C₃×C₄	—	8	2	no	—
220	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
221	C₃×C₄	C₅×#117	1	1	yes	characteristic
222	C₃×C₄	C₅×#124	1	1	yes	characteristic
223	C₃×C₄	C₅×#125	1	1	yes	characteristic
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	1	no	—
226	C₂×C₅	—	2	1	no	—
227	C₂×C₅	C₃×#382	1	1	yes	characteristic
228	C₂³	C₃×C₅×#41	1	1	yes	characteristic
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
232	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
233	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
234	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
235	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
236	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	4	2	no	—
239	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
240	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
241	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
242	C₈	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	1	no	—
246	C₂×C₃	C₅×#382	1	1	yes	characteristic
247	C₅	C₃×#1833	1	1	yes	characteristic, direct product factor, Sylow
248	C₂²	—	4	1	no	—
249	C₂²	C₃×C₅×#117	1	1	yes	characteristic
250	C₂²	C₃×C₅×#124	1	1	yes	characteristic
251	C₂²	C₃×C₅×#125	1	1	yes	characteristic
252	C₄	—	4	2	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
255	C₄	C₃×C₅×#117	1	1	yes	characteristic
256	C₄	C₃×C₅×#124	1	1	yes	characteristic
257	C₄	C₃×C₅×#125	1	1	yes	characteristic
258	C₃	C₅×#1833	1	1	yes	characteristic, direct product factor, Sylow
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	C₃×C₅×#382	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#382
prime radical	C₂³×C₃×C₅	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₃×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1833	C₃	direct (N × H)
16. C₃×C₅×#97	C₄	semidirect
17. C₃×C₄×C₅×C₈	C₄	semidirect
18. C₃×#1833	C₅	direct (N × H)
57. C₅×#97	C₃×C₄	semidirect
58. C₄×C₅×C₈	C₃×C₄	semidirect
59. #1833	C₃×C₅	direct (N × H)
84. C₃×#97	C₄×C₅	semidirect
85. C₃×C₄×C₈	C₄×C₅	semidirect
163. #97	C₃×C₄×C₅	semidirect
164. C₄×C₈	C₃×C₄×C₅	semidirect
213. C₃×C₅	#1833	direct (N × H)
247. C₅	C₃×#1833	direct (N × H)
258. C₃	C₅×#1833	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×#405	2
C₃×C₅×#344	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#1833	3
C₃×#1833	5

Subgroups — C3×C5×#1833 #14274867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1833 #14274867