Subgroups — C₃×C₅×#2251 #14275285

Order 1920.

356 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×SD₃₂	C₂	1	2	yes	maximal
3	C₃×C₅×#495	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2251	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
11	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
13	C₃×C₅×#124	—	2	1	no	—
14	C₃×C₅×Q₃₂	—	2	4	no	—
15	C₃×C₅×SD₃₂	—	2	4	no	—
16	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#2251	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×Q₃₂	C₂×C₃	1	2	yes	complemented
20	C₂×C₅×SD₃₂	C₂×C₃	1	2	yes	—
21	C₅×#495	C₂×C₃	1	1	yes	characteristic
22	C₅×#494	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#346	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂×C₃×C₅×Q₈	—	2	1	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	1	no	—
27	C₃×C₅×Q₁₆	—	2	2	no	—
28	C₃×C₅×Q₁₆	—	2	2	no	—
29	C₃×C₅×Q₁₆	—	2	2	no	—
30	C₃×C₅×Q₁₆	—	4	2	no	—
31	C₃×C₅×SD₁₆	—	4	2	no	—
32	C₃×C₅×D₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	4	2	no	—
36	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₁₆	D₄	1	4	yes	—
39	C₂×C₃×Q₃₂	C₂×C₅	1	2	yes	complemented
40	C₂×C₃×SD₃₂	C₂×C₅	1	2	yes	—
41	C₃×#495	C₂×C₅	1	1	yes	characteristic
42	C₃×#494	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#346	C₂×C₅	1	1	yes	characteristic, complemented
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
46	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₅×SD₁₆	—	2	1	no	—
48	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×#124	—	2	1	no	—
51	C₅×Q₃₂	—	2	4	no	—
52	C₅×SD₃₂	—	2	4	no	—
53	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	#2251	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	4	1	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×Q₈	—	4	1	no	—
59	C₃×C₅×Q₈	—	4	1	no	—
60	C₃×C₅×Q₈	—	4	2	no	—
61	C₃×C₅×D₄	—	4	1	no	—
62	C₃×C₅×D₄	—	4	1	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×C₈	D₈	1	2	yes	—
68	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₃×Q₁₆	—	2	1	no	—
70	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
71	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₃×SD₁₆	—	2	1	no	—
73	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
74	C₃×C₄⋊Q₈	—	2	1	no	—
75	C₃×#124	—	2	1	no	—
76	C₃×Q₃₂	—	2	4	no	—
77	C₃×SD₃₂	—	2	4	no	—
78	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
79	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₅×Q₈	—	2	1	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×D₄	—	2	1	no	—
83	C₅×Q₁₆	—	2	2	no	—
84	C₅×Q₁₆	—	2	2	no	—
85	C₅×Q₁₆	—	2	2	no	—
86	C₅×Q₁₆	—	4	2	no	—
87	C₅×SD₁₆	—	4	2	no	—
88	C₅×D₈	—	2	2	no	—
89	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
90	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
91	C₅×C₄⋊C₄	—	4	2	no	—
92	C₅×C₂²⋊C₄	—	4	2	no	—
93	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
94	C₅×C₁₆	C₃×D₄	1	4	yes	—
95	C₂×Q₃₂	C₂×C₃×C₅	1	2	yes	complemented
96	C₂×SD₃₂	C₂×C₃×C₅	1	2	yes	—
97	#495	C₂×C₃×C₅	1	1	yes	characteristic
98	#494	C₂×C₃×C₅	1	1	yes	characteristic, complemented
99	#346	C₂×C₃×C₅	1	1	yes	characteristic, complemented
100	C₂²×C₃×C₅	—	8	1	no	—
101	C₂²×C₃×C₅	—	8	2	no	—
102	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	8	1	no	—
105	C₃×C₄×C₅	—	8	2	no	—
106	C₃×C₄×C₅	#127	1	1	yes	characteristic
107	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
108	C₂×C₃×Q₈	—	2	1	no	—
109	C₂×C₃×Q₈	—	2	2	no	—
110	C₂×C₃×D₄	—	2	1	no	—
111	C₃×Q₁₆	—	2	2	no	—
112	C₃×Q₁₆	—	2	2	no	—
113	C₃×Q₁₆	—	2	2	no	—
114	C₃×Q₁₆	—	4	2	no	—
115	C₃×SD₁₆	—	4	2	no	—
116	C₃×D₈	—	2	2	no	—
117	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	4	2	no	—
120	C₃×C₂²⋊C₄	—	4	2	no	—
121	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
122	C₃×C₁₆	C₅×D₄	1	4	yes	—
123	C₂³×C₅	—	4	1	no	—
124	C₅×Q₈	—	4	2	no	—
125	C₅×Q₈	—	4	1	no	—
126	C₅×Q₈	—	4	1	no	—
127	C₅×Q₈	—	4	2	no	—
128	C₅×D₄	—	4	1	no	—
129	C₅×D₄	—	4	1	no	—
130	C₂×C₄×C₅	—	4	1	no	—
131	C₂×C₄×C₅	—	4	2	no	—
132	C₂×C₄×C₅	C₃×D₈	1	2	yes	—
133	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
134	C₅×C₈	C₃×D₈	1	2	yes	—
135	C₅×C₈	C₂×C₃×D₄	1	2	yes	—
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
138	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₂×SD₁₆	—	2	1	no	—
140	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	1	no	—
142	#124	—	2	1	no	—
143	Q₃₂	—	2	4	no	—
144	SD₃₂	—	2	4	no	—
145	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
146	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂×C₃×C₅	—	16	1	no	—
148	C₂×C₃×C₅	#493	1	1	yes	characteristic
149	C₂×C₃×C₅	#510	1	2	yes	—
150	C₂³×C₃	—	4	1	no	—
151	C₃×Q₈	—	4	2	no	—
152	C₃×Q₈	—	4	1	no	—
153	C₃×Q₈	—	4	1	no	—
154	C₃×Q₈	—	4	2	no	—
155	C₃×D₄	—	4	1	no	—
156	C₃×D₄	—	4	1	no	—
157	C₂×C₃×C₄	—	4	1	no	—
158	C₂×C₃×C₄	—	4	2	no	—
159	C₂×C₃×C₄	C₅×D₈	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
161	C₃×C₈	C₅×D₈	1	2	yes	—
162	C₃×C₈	C₂×C₅×D₄	1	2	yes	—
163	C₂²×C₅	—	8	2	no	—
164	C₂²×C₅	—	8	1	no	—
165	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
166	C₄×C₅	—	2	2	no	—
167	C₄×C₅	—	8	1	no	—
168	C₄×C₅	—	8	2	no	—
169	C₄×C₅	C₃×#127	1	1	yes	characteristic
170	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×D₄	—	2	1	no	—
174	Q₁₆	—	2	2	no	—
175	Q₁₆	—	2	2	no	—
176	Q₁₆	—	2	2	no	—
177	Q₁₆	—	4	2	no	—
178	SD₁₆	—	4	2	no	—
179	D₈	—	2	2	no	—
180	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
181	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
182	C₄⋊C₄	—	4	2	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
185	C₁₆	C₃×C₅×D₄	1	4	yes	—
186	C₃×C₅	#2251	1	1	yes	characteristic, direct product factor, Hall
187	C₂²×C₃	—	8	1	no	—
188	C₂²×C₃	—	8	2	no	—
189	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
190	C₃×C₄	—	2	2	no	—
191	C₃×C₄	—	8	1	no	—
192	C₃×C₄	—	8	2	no	—
193	C₃×C₄	C₅×#127	1	1	yes	characteristic
194	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
195	C₂×C₅	—	16	1	no	—
196	C₂×C₅	C₃×#493	1	1	yes	characteristic
197	C₂×C₅	C₃×#510	1	2	yes	—
198	C₂³	—	4	1	no	—
199	Q₈	—	4	2	no	—
200	Q₈	—	4	1	no	—
201	Q₈	—	4	1	no	—
202	Q₈	—	4	2	no	—
203	D₄	—	4	1	no	—
204	D₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₃×C₅×D₈	1	2	yes	—
208	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
209	C₈	C₃×C₅×D₈	1	2	yes	—
210	C₈	C₂×C₃×C₅×D₄	1	2	yes	—
211	C₂×C₃	—	16	1	no	—
212	C₂×C₃	C₅×#493	1	1	yes	characteristic
213	C₂×C₃	C₅×#510	1	2	yes	—
214	C₅	C₃×#2251	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	—	8	1	no	—
216	C₂²	—	8	2	no	—
217	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	1	no	—
220	C₄	—	8	2	no	—
221	C₄	C₃×C₅×#127	1	1	yes	characteristic
222	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
223	C₃	C₅×#2251	1	1	yes	characteristic, direct product factor, Sylow
224	C₂	—	16	1	no	—
225	C₂	C₃×C₅×#493	1	1	yes	characteristic
226	C₂	C₃×C₅×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×Q₃₂	C₂	semidirect
4. C₃×C₅×#494	C₂	semidirect
5. C₃×C₅×#346	C₂	semidirect
6. C₅×#2251	C₃	direct (N × H)
18. C₃×#2251	C₅	direct (N × H)
19. C₂×C₅×Q₃₂	C₂×C₃	semidirect
22. C₅×#494	C₂×C₃	semidirect
23. C₅×#346	C₂×C₃	semidirect
39. C₂×C₃×Q₃₂	C₂×C₅	semidirect
42. C₃×#494	C₂×C₅	semidirect
43. C₃×#346	C₂×C₅	semidirect
55. #2251	C₃×C₅	direct (N × H)
95. C₂×Q₃₂	C₂×C₃×C₅	semidirect
98. #494	C₂×C₃×C₅	semidirect
99. #346	C₂×C₃×C₅	semidirect
186. C₃×C₅	#2251	direct (N × H)
214. C₅	C₃×#2251	direct (N × H)
223. C₃	C₅×#2251	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×Q₃₂	2
C₂×C₃×C₅×SD₃₂	2
C₃×C₅×#495	2
C₃×C₅×#494	2
C₃×C₅×#346	2
C₅×#2251	3
C₃×#2251	5

Subgroups — C3×C5×#2251 #14275285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2251 #14275285