Subgroups — C₃×C₅×#1692 #14274726

Order 1920.

292 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#498	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#477	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#462	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#446	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#333	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic
8	C₅×#1692	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	—	2	1	no	—
14	C₃×C₅×#106	—	2	1	no	—
15	C₃×C₅×#106	—	2	1	no	—
16	C₃×C₅×C₄⋊C₈	—	2	1	no	—
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	—	2	1	no	—
21	C₃×C₅×#103	—	2	1	no	—
22	C₃×C₅×#97	—	2	1	no	—
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₃×#1692	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#498	C₂×C₃	1	1	yes	characteristic
26	C₅×#477	C₂×C₃	1	1	yes	characteristic
27	C₅×#462	C₂×C₃	1	1	yes	characteristic
28	C₅×#446	C₂×C₃	1	1	yes	characteristic
29	C₅×#335	C₂×C₃	1	1	yes	characteristic
30	C₅×#333	C₂×C₃	1	1	yes	characteristic
31	C₅×#332	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×Q₈	—	2	1	no	—
34	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
35	C₃×C₅×Q₁₆	—	4	2	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	—	2	1	no	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×#498	C₂×C₅	1	1	yes	characteristic
50	C₃×#477	C₂×C₅	1	1	yes	characteristic
51	C₃×#462	C₂×C₅	1	1	yes	characteristic
52	C₃×#446	C₂×C₅	1	1	yes	characteristic
53	C₃×#335	C₂×C₅	1	1	yes	characteristic
54	C₃×#333	C₂×C₅	1	1	yes	characteristic
55	C₃×#332	C₂×C₅	1	1	yes	characteristic
56	C₂×C₅×Q₁₆	—	2	1	no	—
57	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
58	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
59	C₄×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
60	C₅×#107	—	2	1	no	—
61	C₅×#106	—	2	1	no	—
62	C₅×#106	—	2	1	no	—
63	C₅×C₄⋊C₈	—	2	1	no	—
64	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
66	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
67	C₅×#103	—	2	1	no	—
68	C₅×#103	—	2	1	no	—
69	C₅×#97	—	2	1	no	—
70	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
71	#1692	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₃×C₅×Q₈	—	2	1	no	—
73	C₃×C₅×Q₈	—	2	1	no	—
74	C₃×C₅×Q₈	—	4	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	8	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
84	C₃×C₅×C₈	—	2	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×C₅×C₈	Q₁₆	1	2	yes	complemented
89	C₂×C₃×Q₁₆	—	2	1	no	—
90	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
91	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
92	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
93	C₃×#107	—	2	1	no	—
94	C₃×#106	—	2	1	no	—
95	C₃×#106	—	2	1	no	—
96	C₃×C₄⋊C₈	—	2	1	no	—
97	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
100	C₃×#103	—	2	1	no	—
101	C₃×#103	—	2	1	no	—
102	C₃×#97	—	2	1	no	—
103	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₅×Q₈	—	2	1	no	—
105	C₂×C₅×Q₈	—	2	1	no	—
106	C₂×C₅×Q₈	C₃×D₄	1	1	yes	characteristic
107	C₅×Q₁₆	—	4	2	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	—	2	1	no	—
111	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	1	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
119	C₄²×C₅	—	2	1	no	—
120	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
121	#498	C₂×C₃×C₅	1	1	yes	characteristic
122	#477	C₂×C₃×C₅	1	1	yes	characteristic
123	#462	C₂×C₃×C₅	1	1	yes	characteristic
124	#446	C₂×C₃×C₅	1	1	yes	characteristic
125	#335	C₂×C₃×C₅	1	1	yes	characteristic
126	#333	C₂×C₃×C₅	1	1	yes	characteristic
127	#332	C₂×C₃×C₅	1	1	yes	characteristic
128	C₂²×C₃×C₅	#121	1	1	yes	characteristic
129	C₃×C₄×C₅	—	2	1	no	—
130	C₃×C₄×C₅	—	4	1	no	—
131	C₃×C₄×C₅	—	4	1	no	—
132	C₃×C₄×C₅	—	8	1	no	—
133	C₃×C₄×C₅	—	8	1	no	—
134	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
135	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₃×C₄×C₅	#137	1	1	yes	characteristic
137	C₃×C₄×C₅	#137	1	1	yes	characteristic
138	C₂×C₃×Q₈	—	2	1	no	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
141	C₃×Q₁₆	—	4	2	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	1	no	—
145	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
153	C₃×C₄²	—	2	1	no	—
154	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
155	C₅×Q₈	—	2	1	no	—
156	C₅×Q₈	—	2	1	no	—
157	C₅×Q₈	—	4	1	no	—
158	C₅×Q₈	—	4	1	no	—
159	C₅×Q₈	—	8	1	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	4	1	no	—
163	C₂×C₄×C₅	—	4	1	no	—
164	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
165	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	4	1	no	—
169	C₅×C₈	—	4	1	no	—
170	C₅×C₈	—	4	1	no	—
171	C₅×C₈	C₃×Q₁₆	1	2	yes	complemented
172	C₂×Q₁₆	—	2	1	no	—
173	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
175	C₄×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
176	#107	—	2	1	no	—
177	#106	—	2	1	no	—
178	#106	—	2	1	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
181	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
182	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
183	#103	—	2	1	no	—
184	#103	—	2	1	no	—
185	#97	—	2	1	no	—
186	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
187	C₂×C₃×C₅	#462	1	1	yes	characteristic
188	C₂×C₃×C₅	#468	1	1	yes	characteristic
189	C₂×C₃×C₅	#471	1	1	yes	characteristic
190	C₃×Q₈	—	2	1	no	—
191	C₃×Q₈	—	2	1	no	—
192	C₃×Q₈	—	4	1	no	—
193	C₃×Q₈	—	4	1	no	—
194	C₃×Q₈	—	8	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	4	1	no	—
198	C₂×C₃×C₄	—	4	1	no	—
199	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
202	C₃×C₈	—	2	1	no	—
203	C₃×C₈	—	4	1	no	—
204	C₃×C₈	—	4	1	no	—
205	C₃×C₈	—	4	1	no	—
206	C₃×C₈	C₅×Q₁₆	1	2	yes	complemented
207	C₂²×C₅	C₃×#121	1	1	yes	characteristic
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	4	1	no	—
211	C₄×C₅	—	8	1	no	—
212	C₄×C₅	—	8	1	no	—
213	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
214	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
215	C₄×C₅	C₃×#137	1	1	yes	characteristic
216	C₄×C₅	C₃×#137	1	1	yes	characteristic
217	C₂×Q₈	—	2	1	no	—
218	C₂×Q₈	—	2	1	no	—
219	C₂×Q₈	C₃×C₅×D₄	1	1	yes	characteristic
220	Q₁₆	—	4	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
225	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
232	C₄²	—	2	1	no	—
233	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
234	C₃×C₅	#1692	1	1	yes	characteristic, direct product factor, Hall
235	C₂²×C₃	C₅×#121	1	1	yes	characteristic
236	C₃×C₄	—	2	1	no	—
237	C₃×C₄	—	4	1	no	—
238	C₃×C₄	—	4	1	no	—
239	C₃×C₄	—	8	1	no	—
240	C₃×C₄	—	8	1	no	—
241	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
242	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
243	C₃×C₄	C₅×#137	1	1	yes	characteristic
244	C₃×C₄	C₅×#137	1	1	yes	characteristic
245	C₂×C₅	C₃×#462	1	1	yes	characteristic
246	C₂×C₅	C₃×#468	1	1	yes	characteristic
247	C₂×C₅	C₃×#471	1	1	yes	characteristic
248	Q₈	—	2	1	no	—
249	Q₈	—	2	1	no	—
250	Q₈	—	4	1	no	—
251	Q₈	—	4	1	no	—
252	Q₈	—	8	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
258	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
259	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
260	C₈	—	2	1	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	—	4	1	no	—
264	C₈	C₃×C₅×Q₁₆	1	2	yes	complemented
265	C₂×C₃	C₅×#462	1	1	yes	characteristic
266	C₂×C₃	C₅×#468	1	1	yes	characteristic
267	C₂×C₃	C₅×#471	1	1	yes	characteristic
268	C₅	C₃×#1692	1	1	yes	characteristic, direct product factor, Sylow
269	C₂²	C₃×C₅×#121	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	8	1	no	—
274	C₄	—	8	1	no	—
275	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
276	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
277	C₄	C₃×C₅×#137	1	1	yes	characteristic
278	C₄	C₃×C₅×#137	1	1	yes	characteristic
279	C₃	C₅×#1692	1	1	yes	characteristic, direct product factor, Sylow
280	C₂	C₃×C₅×#462	1	1	yes	characteristic
281	C₂	C₃×C₅×#468	1	1	yes	characteristic
282	C₂	C₃×C₅×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
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
8. C₅×#1692	C₃	direct (N × H)
24. C₃×#1692	C₅	direct (N × H)
71. #1692	C₃×C₅	direct (N × H)
88. C₃×C₅×C₈	Q₁₆	semidirect
171. C₅×C₈	C₃×Q₁₆	semidirect
206. C₃×C₈	C₅×Q₁₆	semidirect
234. C₃×C₅	#1692	direct (N × H)
264. C₈	C₃×C₅×Q₁₆	semidirect
268. C₅	C₃×#1692	direct (N × H)
279. C₃	C₅×#1692	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×#498	2
C₃×C₅×#477	2
C₃×C₅×#462	2
C₃×C₅×#446	2
C₃×C₅×#335	2
C₃×C₅×#333	2
C₃×C₅×#332	2
C₅×#1692	3
C₃×#1692	5

Subgroups — C3×C5×#1692 #14274726

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1692 #14274726