Subgroups — C₃×C₅×#3166 #14276200

Order 1920.

696 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	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#125	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#501	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×#499	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#428	C₂	1	4	yes	maximal
6	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
7	C₅×#3166	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃×C₅×#125	—	2	2	no	—
11	C₃×C₅×#125	C₂²	1	2	yes	—
12	C₃×C₅×#125	C₂²	1	2	yes	—
13	C₃×C₅×#125	C₂²	1	2	yes	—
14	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
15	C₃×C₄×C₅×Q₈	—	2	2	no	—
16	C₃×C₅×#117	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
20	C₃×C₅×#107	C₂²	1	8	yes	—
21	C₃×C₅×#106	C₂²	1	8	yes	—
22	C₃×C₅×#97	C₂²	1	2	yes	—
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	C₃×#3166	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#533	C₂×C₃	1	1	yes	characteristic
26	C₂×C₅×#125	C₂×C₃	1	1	yes	characteristic
27	C₅×#501	C₂×C₃	1	4	yes	complemented
28	C₅×#499	C₂×C₃	1	4	yes	complemented
29	C₅×#428	C₂×C₃	1	4	yes	—
30	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂²×C₃×C₄×C₅	—	2	2	no	—
33	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₅×M₁₆	Q₈	1	8	yes	complemented
36	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
37	C₃×C₅×C₄⋊C₄	—	2	4	no	—
38	C₃×C₅×C₄⋊C₄	—	2	4	no	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	—	2	4	no	—
41	C₃×C₅×C₄⋊C₄	—	2	2	no	—
42	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
44	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
45	C₃×C₄²×C₅	—	2	2	no	—
46	C₃×C₄²×C₅	D₄	1	2	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×#533	C₂×C₅	1	1	yes	characteristic
50	C₂×C₃×#125	C₂×C₅	1	1	yes	characteristic
51	C₃×#501	C₂×C₅	1	4	yes	complemented
52	C₃×#499	C₂×C₅	1	4	yes	complemented
53	C₃×#428	C₂×C₅	1	4	yes	—
54	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
55	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
56	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
57	C₅×#125	—	2	2	no	—
58	C₅×#125	C₂²×C₃	1	2	yes	—
59	C₅×#125	C₂²×C₃	1	2	yes	—
60	C₅×#125	C₂²×C₃	1	2	yes	—
61	C₅×C₂²⋊Q₈	—	2	2	no	—
62	C₄×C₅×Q₈	—	2	2	no	—
63	C₅×#117	C₂²×C₃	1	2	yes	—
64	C₂×C₅×C₄⋊C₄	—	2	2	no	—
65	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	2	yes	—
66	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
67	C₅×#107	C₂²×C₃	1	8	yes	—
68	C₅×#106	C₂²×C₃	1	8	yes	—
69	C₅×#97	C₂²×C₃	1	2	yes	—
70	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
71	#3166	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₅×Q₈	—	4	4	no	—
74	C₂×C₃×C₄×C₅	—	2	4	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	—	4	2	no	—
77	C₂×C₃×C₄×C₅	—	4	4	no	—
78	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
83	C₃×C₅×C₈	C₂×Q₈	1	8	yes	—
84	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
85	C₃×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
86	C₃×#125	—	2	2	no	—
87	C₃×#125	C₂²×C₅	1	2	yes	—
88	C₃×#125	C₂²×C₅	1	2	yes	—
89	C₃×#125	C₂²×C₅	1	2	yes	—
90	C₃×C₂²⋊Q₈	—	2	2	no	—
91	C₃×C₄×Q₈	—	2	2	no	—
92	C₃×#117	C₂²×C₅	1	2	yes	—
93	C₂×C₃×C₄⋊C₄	—	2	2	no	—
94	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
95	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
96	C₃×#107	C₂²×C₅	1	8	yes	—
97	C₃×#106	C₂²×C₅	1	8	yes	—
98	C₃×#97	C₂²×C₅	1	2	yes	—
99	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂²×C₄×C₅	—	2	2	no	—
102	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
103	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
104	C₅×M₁₆	C₃×Q₈	1	8	yes	complemented
105	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
106	C₅×C₄⋊C₄	—	2	4	no	—
107	C₅×C₄⋊C₄	—	2	4	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
112	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	C₃×D₄	1	2	yes	—
116	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
117	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
118	#533	C₂×C₃×C₅	1	1	yes	characteristic
119	C₂×#125	C₂×C₃×C₅	1	1	yes	characteristic
120	#501	C₂×C₃×C₅	1	4	yes	complemented
121	#499	C₂×C₃×C₅	1	4	yes	complemented
122	#428	C₂×C₃×C₅	1	4	yes	—
123	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
126	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
127	C₃×C₄×C₅	—	2	2	no	—
128	C₃×C₄×C₅	—	2	2	no	—
129	C₃×C₄×C₅	—	4	4	no	—
130	C₃×C₄×C₅	—	4	4	no	—
131	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
132	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
133	C₂×C₃×Q₈	—	2	2	no	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
136	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
137	C₃×M₁₆	C₅×Q₈	1	8	yes	complemented
138	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
139	C₃×C₄⋊C₄	—	2	4	no	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	4	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
145	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
146	C₃×C₂²⋊C₄	—	2	2	no	—
147	C₃×C₄²	—	2	2	no	—
148	C₃×C₄²	C₅×D₄	1	2	yes	—
149	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
150	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
151	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₅×Q₈	—	4	4	no	—
153	C₂×C₄×C₅	—	2	4	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	4	no	—
157	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
158	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
161	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
162	C₅×C₈	C₂×C₃×Q₈	1	8	yes	—
163	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
164	C₄⋊Q₈	C₂²×C₃×C₅	1	2	yes	—
165	#125	—	2	2	no	—
166	#125	C₂²×C₃×C₅	1	2	yes	—
167	#125	C₂²×C₃×C₅	1	2	yes	—
168	#125	C₂²×C₃×C₅	1	2	yes	—
169	C₂²⋊Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	#117	C₂²×C₃×C₅	1	2	yes	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	2	yes	—
174	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
175	#107	C₂²×C₃×C₅	1	8	yes	—
176	#106	C₂²×C₃×C₅	1	8	yes	—
177	#97	C₂²×C₃×C₅	1	2	yes	—
178	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
179	C₂×C₃×C₅	—	2	2	no	—
180	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₂×C₃×C₅	#575	1	2	yes	—
182	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
183	C₃×Q₈	—	4	4	no	—
184	C₂×C₃×C₄	—	2	4	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	4	2	no	—
187	C₂×C₃×C₄	—	4	4	no	—
188	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
190	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
192	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
193	C₃×C₈	C₂×C₅×Q₈	1	8	yes	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
196	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	—	2	2	no	—
199	C₄×C₅	—	4	4	no	—
200	C₄×C₅	—	4	4	no	—
201	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
202	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
203	C₂×Q₈	—	2	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
206	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
207	M₁₆	C₃×C₅×Q₈	1	8	yes	complemented
208	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
209	C₄⋊C₄	—	2	4	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
215	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	—	2	2	no	—
218	C₄²	C₃×C₅×D₄	1	2	yes	—
219	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
220	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
221	C₃×C₅	#3166	1	1	yes	characteristic, direct product factor, Hall
222	C₂²×C₃	—	2	2	no	—
223	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
224	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	4	4	no	—
228	C₃×C₄	—	4	4	no	—
229	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
230	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
231	C₂×C₅	—	2	2	no	—
232	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
233	C₂×C₅	C₃×#575	1	2	yes	—
234	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
235	Q₈	—	4	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
241	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
242	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
244	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
245	C₈	C₂×C₃×C₅×Q₈	1	8	yes	—
246	C₂×C₃	—	2	2	no	—
247	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
248	C₂×C₃	C₅×#575	1	2	yes	—
249	C₅	C₃×#3166	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	2	no	—
251	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
252	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	4	4	no	—
256	C₄	—	4	4	no	—
257	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
258	C₄	C₂²×C₃×C₅×Q₈	1	2	yes	—
259	C₃	C₅×#3166	1	1	yes	characteristic, direct product factor, Sylow
260	C₂	—	2	2	no	—
261	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
262	C₂	C₃×C₅×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
prime radical	C₂³×C₃×C₅	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#501	C₂	semidirect
4. C₃×C₅×#499	C₂	semidirect
7. C₅×#3166	C₃	direct (N × H)
24. C₃×#3166	C₅	direct (N × H)
27. C₅×#501	C₂×C₃	semidirect
28. C₅×#499	C₂×C₃	semidirect
35. C₃×C₅×M₁₆	Q₈	semidirect
51. C₃×#501	C₂×C₅	semidirect
52. C₃×#499	C₂×C₅	semidirect
71. #3166	C₃×C₅	direct (N × H)
104. C₅×M₁₆	C₃×Q₈	semidirect
120. #501	C₂×C₃×C₅	semidirect
121. #499	C₂×C₃×C₅	semidirect
137. C₃×M₁₆	C₅×Q₈	semidirect
207. M₁₆	C₃×C₅×Q₈	semidirect
221. C₃×C₅	#3166	direct (N × H)
249. C₅	C₃×#3166	direct (N × H)
259. C₃	C₅×#3166	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×#533	2
C₂×C₃×C₅×#125	2
C₃×C₅×#501	2
C₃×C₅×#499	2
C₃×C₅×#428	2
C₃×C₄×C₅×M₁₆	2
C₅×#3166	3
C₃×#3166	5

Subgroups — C3×C5×#3166 #14276200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3166 #14276200