Subgroups — C₃×C₅×#1688 #14274722

Order 1920.

368 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	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#460	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	2	yes	maximal, complemented
6	C₅×#1688	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×SD₁₆	—	2	2	no	—
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₃×C₅×#127	C₂²	1	2	yes	—
11	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	—	2	1	no	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#102	—	2	2	no	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1688	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#493	C₂×C₃	1	1	yes	characteristic
20	C₅×#460	C₂×C₃	1	2	yes	complemented
21	C₅×#446	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#331	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₃×C₅×SD₁₆	—	4	4	no	—
29	C₃×C₅×D₈	—	4	2	no	—
30	C₃×C₅×D₈	—	4	2	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₃×#493	C₂×C₅	1	1	yes	characteristic
40	C₃×#460	C₂×C₅	1	2	yes	complemented
41	C₃×#446	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#331	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×SD₁₆	—	2	2	no	—
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	—	2	1	no	—
47	C₅×#127	C₂²×C₃	1	2	yes	—
48	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
52	C₅×#102	—	2	2	no	—
53	C₅×#97	—	2	1	no	—
54	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
55	#1688	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	4	2	no	—
57	C₃×C₅×Q₈	—	2	2	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₃×C₅×D₄	—	4	2	no	—
60	C₃×C₅×D₄	—	4	2	no	—
61	C₃×C₅×D₄	—	4	2	no	—
62	C₃×C₅×D₄	—	8	2	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	1	no	—
69	C₃×C₅×C₈	—	4	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	SD₁₆	1	2	yes	complemented
72	C₂×C₃×SD₁₆	—	2	2	no	—
73	C₂×C₃×D₈	—	2	1	no	—
74	C₂×C₃×D₈	—	2	1	no	—
75	C₃×#127	C₂²×C₅	1	2	yes	—
76	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
77	C₃×C₄⋊C₈	—	2	1	no	—
78	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
79	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
80	C₃×#102	—	2	2	no	—
81	C₃×#97	—	2	1	no	—
82	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
83	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
84	C₂×C₅×D₄	—	2	2	no	—
85	C₂×C₅×D₄	—	2	2	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₅×SD₁₆	—	4	4	no	—
88	C₅×D₈	—	4	2	no	—
89	C₅×D₈	—	4	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
93	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	1	no	—
95	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
96	C₄²×C₅	—	2	1	no	—
97	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
98	#493	C₂×C₃×C₅	1	1	yes	characteristic
99	#460	C₂×C₃×C₅	1	2	yes	complemented
100	#446	C₂×C₃×C₅	1	1	yes	characteristic, complemented
101	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
102	#331	C₂×C₃×C₅	1	2	yes	complemented
103	C₂²×C₃×C₅	—	8	2	no	—
104	C₂²×C₃×C₅	—	8	2	no	—
105	C₂²×C₃×C₅	—	8	2	no	—
106	C₂²×C₃×C₅	#121	1	1	yes	characteristic
107	C₃×C₄×C₅	—	2	1	no	—
108	C₃×C₄×C₅	—	4	1	no	—
109	C₃×C₄×C₅	—	4	1	no	—
110	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
111	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
113	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
114	C₂×C₃×D₄	—	2	2	no	—
115	C₂×C₃×D₄	—	2	2	no	—
116	C₂×C₃×D₄	—	2	2	no	—
117	C₃×SD₁₆	—	4	4	no	—
118	C₃×D₈	—	4	2	no	—
119	C₃×D₈	—	4	2	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	2	no	—
122	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
123	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
124	C₃×C₄⋊C₄	—	2	1	no	—
125	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
126	C₃×C₄²	—	2	1	no	—
127	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
128	C₂³×C₅	—	4	2	no	—
129	C₅×Q₈	—	2	2	no	—
130	C₅×D₄	—	4	2	no	—
131	C₅×D₄	—	4	2	no	—
132	C₅×D₄	—	4	2	no	—
133	C₅×D₄	—	4	2	no	—
134	C₅×D₄	—	8	2	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	—	4	1	no	—
142	C₅×C₈	—	4	2	no	—
143	C₅×C₈	C₃×SD₁₆	1	2	yes	complemented
144	C₂×SD₁₆	—	2	2	no	—
145	C₂×D₈	—	2	1	no	—
146	C₂×D₈	—	2	1	no	—
147	#127	C₂²×C₃×C₅	1	2	yes	—
148	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₄⋊C₈	—	2	1	no	—
150	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
152	#102	—	2	2	no	—
153	#97	—	2	1	no	—
154	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
155	C₂×C₃×C₅	—	16	2	no	—
156	C₂×C₃×C₅	#460	1	1	yes	characteristic
157	C₂×C₃×C₅	#469	1	1	yes	characteristic
158	C₂×C₃×C₅	#472	1	1	yes	characteristic
159	C₂³×C₃	—	4	2	no	—
160	C₃×Q₈	—	2	2	no	—
161	C₃×D₄	—	4	2	no	—
162	C₃×D₄	—	4	2	no	—
163	C₃×D₄	—	4	2	no	—
164	C₃×D₄	—	4	2	no	—
165	C₃×D₄	—	8	2	no	—
166	C₂×C₃×C₄	—	2	1	no	—
167	C₂×C₃×C₄	—	2	1	no	—
168	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
170	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
171	C₃×C₈	—	2	1	no	—
172	C₃×C₈	—	4	1	no	—
173	C₃×C₈	—	4	2	no	—
174	C₃×C₈	C₅×SD₁₆	1	2	yes	complemented
175	C₂²×C₅	—	8	2	no	—
176	C₂²×C₅	—	8	2	no	—
177	C₂²×C₅	—	8	2	no	—
178	C₂²×C₅	C₃×#121	1	1	yes	characteristic
179	C₄×C₅	—	2	1	no	—
180	C₄×C₅	—	4	1	no	—
181	C₄×C₅	—	4	1	no	—
182	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
183	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
185	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	2	no	—
189	SD₁₆	—	4	4	no	—
190	D₈	—	4	2	no	—
191	D₈	—	4	2	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
195	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
198	C₄²	—	2	1	no	—
199	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
200	C₃×C₅	#1688	1	1	yes	characteristic, direct product factor, Hall
201	C₂²×C₃	—	8	2	no	—
202	C₂²×C₃	—	8	2	no	—
203	C₂²×C₃	—	8	2	no	—
204	C₂²×C₃	C₅×#121	1	1	yes	characteristic
205	C₃×C₄	—	2	1	no	—
206	C₃×C₄	—	4	1	no	—
207	C₃×C₄	—	4	1	no	—
208	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
209	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
211	C₂×C₅	—	16	2	no	—
212	C₂×C₅	C₃×#460	1	1	yes	characteristic
213	C₂×C₅	C₃×#469	1	1	yes	characteristic
214	C₂×C₅	C₃×#472	1	1	yes	characteristic
215	C₂³	—	4	2	no	—
216	Q₈	—	2	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	4	2	no	—
221	D₄	—	8	2	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
225	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
226	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
227	C₈	—	2	1	no	—
228	C₈	—	4	2	no	—
229	C₈	—	4	1	no	—
230	C₈	C₃×C₅×SD₁₆	1	2	yes	complemented
231	C₂×C₃	—	16	2	no	—
232	C₂×C₃	C₅×#460	1	1	yes	characteristic
233	C₂×C₃	C₅×#469	1	1	yes	characteristic
234	C₂×C₃	C₅×#472	1	1	yes	characteristic
235	C₅	C₃×#1688	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	8	2	no	—
237	C₂²	—	8	2	no	—
238	C₂²	—	8	2	no	—
239	C₂²	C₃×C₅×#121	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	4	1	no	—
242	C₄	—	4	1	no	—
243	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
244	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
245	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
246	C₃	C₅×#1688	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	16	2	no	—
248	C₂	C₃×C₅×#460	1	1	yes	characteristic
249	C₂	C₃×C₅×#469	1	1	yes	characteristic
250	C₂	C₃×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#460	C₂	semidirect
3. C₃×C₅×#446	C₂	semidirect
4. C₃×C₅×#334	C₂	semidirect
5. C₃×C₅×#331	C₂	semidirect
6. C₅×#1688	C₃	direct (N × H)
18. C₃×#1688	C₅	direct (N × H)
20. C₅×#460	C₂×C₃	semidirect
21. C₅×#446	C₂×C₃	semidirect
22. C₅×#334	C₂×C₃	semidirect
23. C₅×#331	C₂×C₃	semidirect
40. C₃×#460	C₂×C₅	semidirect
41. C₃×#446	C₂×C₅	semidirect
42. C₃×#334	C₂×C₅	semidirect
43. C₃×#331	C₂×C₅	semidirect
55. #1688	C₃×C₅	direct (N × H)
71. C₃×C₅×C₈	SD₁₆	semidirect
99. #460	C₂×C₃×C₅	semidirect
100. #446	C₂×C₃×C₅	semidirect
101. #334	C₂×C₃×C₅	semidirect
102. #331	C₂×C₃×C₅	semidirect
143. C₅×C₈	C₃×SD₁₆	semidirect
174. C₃×C₈	C₅×SD₁₆	semidirect
200. C₃×C₅	#1688	direct (N × H)
230. C₈	C₃×C₅×SD₁₆	semidirect
235. C₅	C₃×#1688	direct (N × H)
246. C₃	C₅×#1688	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#493	2
C₃×C₅×#460	2
C₃×C₅×#446	2
C₃×C₅×#334	2
C₃×C₅×#331	2
C₅×#1688	3
C₃×#1688	5

Subgroups — C3×C5×#1688 #14274722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1688 #14274722