Subgroups — C₃×C₅×#1965 #14274999

Order 1920.

792 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×#100	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1965	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×D₄	C₄	1	4	yes	—
7	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
10	C₃×C₅×#127	—	2	1	no	—
11	C₃×C₅×#127	—	2	1	no	—
12	C₃×C₅×#127	—	2	2	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	C₃×C₅×#100	—	2	8	no	—
16	C₃×C₅×#97	—	2	1	no	—
17	C₃×C₄×C₅×C₈	—	2	1	no	—
18	C₃×#1965	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#127	C₂×C₃	1	1	yes	characteristic
20	C₅×#423	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₅×#100	C₂×C₃	1	4	yes	complemented
22	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂⁴×C₃×C₅	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₅×D₄	—	2	4	no	—
25	C₂×C₃×C₅×D₄	—	2	4	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	—	2	2	no	—
28	C₂×C₃×C₅×D₄	—	2	4	no	—
29	C₂×C₃×C₅×D₄	—	2	4	no	—
30	C₂×C₃×C₅×D₄	—	4	8	no	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₃×C₅×M₁₆	—	2	2	no	—
34	C₃×C₅×M₁₆	—	2	4	no	—
35	C₃×C₅×M₁₆	D₄	1	4	yes	complemented
36	C₂×C₃×C₅×C₈	—	2	2	no	—
37	C₂×C₃×C₅×C₈	—	2	2	no	—
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	2	yes	—
41	C₂×C₃×#127	C₂×C₅	1	1	yes	characteristic
42	C₃×#423	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₃×#100	C₂×C₅	1	4	yes	complemented
44	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic, complemented
45	C₂²×C₅×D₄	C₃×C₄	1	4	yes	—
46	C₂²×C₅×D₄	C₂²×C₃	1	2	yes	—
47	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
48	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
49	C₅×#127	—	2	1	no	—
50	C₅×#127	—	2	1	no	—
51	C₅×#127	—	2	2	no	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×#100	—	2	8	no	—
55	C₅×#97	—	2	1	no	—
56	C₄×C₅×C₈	—	2	1	no	—
57	#1965	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	—	2	4	no	—
59	C₂³×C₃×C₅	—	2	8	no	—
60	C₂³×C₃×C₅	—	4	8	no	—
61	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₃×C₅×D₄	—	4	8	no	—
63	C₃×C₅×D₄	—	4	8	no	—
64	C₃×C₅×D₄	—	4	4	no	—
65	C₃×C₅×D₄	—	4	2	no	—
66	C₃×C₅×D₄	—	4	2	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	2	yes	—
74	C₃×C₅×C₈	—	2	4	no	—
75	C₃×C₅×C₈	—	4	2	no	—
76	C₂²×C₃×D₄	C₄×C₅	1	4	yes	—
77	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
78	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
79	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
80	C₃×#127	—	2	1	no	—
81	C₃×#127	—	2	1	no	—
82	C₃×#127	—	2	2	no	—
83	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
84	C₃×C₄⋊C₈	—	2	2	no	—
85	C₃×#100	—	2	8	no	—
86	C₃×#97	—	2	1	no	—
87	C₃×C₄×C₈	—	2	1	no	—
88	C₂⁴×C₅	C₂×C₃×C₄	1	4	yes	—
89	C₂×C₅×D₄	—	2	4	no	—
90	C₂×C₅×D₄	—	2	4	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	2	2	no	—
93	C₂×C₅×D₄	—	2	4	no	—
94	C₂×C₅×D₄	—	2	4	no	—
95	C₂×C₅×D₄	—	4	8	no	—
96	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
97	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
98	C₅×M₁₆	—	2	2	no	—
99	C₅×M₁₆	—	2	4	no	—
100	C₅×M₁₆	C₃×D₄	1	4	yes	complemented
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
104	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
105	C₄²×C₅	C₃×D₄	1	2	yes	—
106	C₂×#127	C₂×C₃×C₅	1	1	yes	characteristic
107	#423	C₂×C₃×C₅	1	1	yes	characteristic, complemented
108	C₂×#100	C₂×C₃×C₅	1	4	yes	complemented
109	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	C₂²×C₃×C₅	—	2	2	no	—
111	C₂²×C₃×C₅	—	4	4	no	—
112	C₂²×C₃×C₅	—	4	8	no	—
113	C₂²×C₃×C₅	—	8	8	no	—
114	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	#100	1	4	yes	complemented
118	C₃×C₄×C₅	#121	1	2	yes	—
119	C₃×C₄×C₅	#124	1	1	yes	characteristic
120	C₃×C₄×C₅	#127	1	1	yes	characteristic
121	C₂⁴×C₃	C₂×C₄×C₅	1	4	yes	—
122	C₂×C₃×D₄	—	2	4	no	—
123	C₂×C₃×D₄	—	2	4	no	—
124	C₂×C₃×D₄	—	2	4	no	—
125	C₂×C₃×D₄	—	2	2	no	—
126	C₂×C₃×D₄	—	2	2	no	—
127	C₂×C₃×D₄	—	2	4	no	—
128	C₂×C₃×D₄	—	4	8	no	—
129	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
130	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
131	C₃×M₁₆	—	2	2	no	—
132	C₃×M₁₆	—	2	4	no	—
133	C₃×M₁₆	C₅×D₄	1	4	yes	complemented
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
137	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
138	C₃×C₄²	C₅×D₄	1	2	yes	—
139	C₂³×C₅	—	2	4	no	—
140	C₂³×C₅	—	2	8	no	—
141	C₂³×C₅	—	4	8	no	—
142	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
143	C₅×D₄	—	4	8	no	—
144	C₅×D₄	—	4	8	no	—
145	C₅×D₄	—	4	4	no	—
146	C₅×D₄	—	4	2	no	—
147	C₅×D₄	—	4	2	no	—
148	C₂×C₄×C₅	—	2	4	no	—
149	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
153	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₃×#41	1	2	yes	—
155	C₅×C₈	—	2	4	no	—
156	C₅×C₈	—	4	2	no	—
157	C₂²×D₄	C₃×C₄×C₅	1	4	yes	—
158	C₂²×D₄	C₂²×C₃×C₅	1	2	yes	—
159	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
160	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
161	#127	—	2	1	no	—
162	#127	—	2	1	no	—
163	#127	—	2	2	no	—
164	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	#100	—	2	8	no	—
167	#97	—	2	1	no	—
168	C₄×C₈	—	2	1	no	—
169	C₂×C₃×C₅	—	2	2	no	—
170	C₂×C₃×C₅	—	8	4	no	—
171	C₂×C₃×C₅	#390	1	1	yes	characteristic
172	C₂×C₃×C₅	C₂×#100	1	2	yes	—
173	C₂³×C₃	—	2	4	no	—
174	C₂³×C₃	—	2	8	no	—
175	C₂³×C₃	—	4	8	no	—
176	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₃×D₄	—	4	8	no	—
178	C₃×D₄	—	4	4	no	—
179	C₃×D₄	—	4	2	no	—
180	C₃×D₄	—	4	2	no	—
181	C₃×D₄	—	4	8	no	—
182	C₂×C₃×C₄	—	2	4	no	—
183	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
184	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
185	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
188	C₂×C₃×C₄	C₅×#41	1	2	yes	—
189	C₃×C₈	—	2	4	no	—
190	C₃×C₈	—	4	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	4	4	no	—
193	C₂²×C₅	—	4	8	no	—
194	C₂²×C₅	—	8	8	no	—
195	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
196	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
197	C₄×C₅	—	2	2	no	—
198	C₄×C₅	C₃×#100	1	4	yes	complemented
199	C₄×C₅	C₃×#121	1	2	yes	—
200	C₄×C₅	C₃×#124	1	1	yes	characteristic
201	C₄×C₅	C₃×#127	1	1	yes	characteristic
202	C₂⁴	C₂×C₃×C₄×C₅	1	4	yes	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	4	8	no	—
210	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
211	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	C₃×C₅×D₄	1	4	yes	complemented
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
218	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
219	C₄²	C₃×C₅×D₄	1	2	yes	—
220	C₃×C₅	#1965	1	1	yes	characteristic, direct product factor, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	4	4	no	—
223	C₂²×C₃	—	4	8	no	—
224	C₂²×C₃	—	8	8	no	—
225	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
226	C₂²×C₃	C₄×C₅×D₄	1	2	yes	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	C₅×#100	1	4	yes	complemented
229	C₃×C₄	C₅×#121	1	2	yes	—
230	C₃×C₄	C₅×#124	1	1	yes	characteristic
231	C₃×C₄	C₅×#127	1	1	yes	characteristic
232	C₂×C₅	—	2	2	no	—
233	C₂×C₅	—	8	4	no	—
234	C₂×C₅	C₃×#390	1	1	yes	characteristic
235	C₂×C₅	C₂×C₃×#100	1	2	yes	—
236	C₂³	—	2	4	no	—
237	C₂³	—	2	8	no	—
238	C₂³	—	4	8	no	—
239	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
240	D₄	—	4	8	no	—
241	D₄	—	4	4	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
247	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
248	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
249	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
250	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
251	C₂×C₄	C₃×C₅×#41	1	2	yes	—
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₂×C₃	—	2	2	no	—
255	C₂×C₃	—	8	4	no	—
256	C₂×C₃	C₅×#390	1	1	yes	characteristic
257	C₂×C₃	C₂×C₅×#100	1	2	yes	—
258	C₅	C₃×#1965	1	1	yes	characteristic, direct product factor, Sylow
259	C₂²	—	2	2	no	—
260	C₂²	—	4	4	no	—
261	C₂²	—	4	8	no	—
262	C₂²	—	8	8	no	—
263	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
264	C₂²	C₃×C₄×C₅×D₄	1	2	yes	—
265	C₄	—	2	2	no	—
266	C₄	C₃×C₅×#100	1	4	yes	complemented
267	C₄	C₃×C₅×#121	1	2	yes	—
268	C₄	C₃×C₅×#124	1	1	yes	characteristic
269	C₄	C₃×C₅×#127	1	1	yes	characteristic
270	C₃	C₅×#1965	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	2	2	no	—
272	C₂	—	8	4	no	—
273	C₂	C₃×C₅×#390	1	1	yes	characteristic
274	C₂	C₂×C₃×C₅×#100	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×#127	C₂
LCS	G ▷ C₂³ ▷ 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
2. C₃×C₅×#423	C₂	semidirect
3. C₂×C₃×C₅×#100	C₂	semidirect
4. C₃×C₄×C₅×M₁₆	C₂	semidirect
5. C₅×#1965	C₃	direct (N × H)
18. C₃×#1965	C₅	direct (N × H)
20. C₅×#423	C₂×C₃	semidirect
21. C₂×C₅×#100	C₂×C₃	semidirect
22. C₄×C₅×M₁₆	C₂×C₃	semidirect
35. C₃×C₅×M₁₆	D₄	semidirect
42. C₃×#423	C₂×C₅	semidirect
43. C₂×C₃×#100	C₂×C₅	semidirect
44. C₃×C₄×M₁₆	C₂×C₅	semidirect
57. #1965	C₃×C₅	direct (N × H)
100. C₅×M₁₆	C₃×D₄	semidirect
107. #423	C₂×C₃×C₅	semidirect
108. C₂×#100	C₂×C₃×C₅	semidirect
109. C₄×M₁₆	C₂×C₃×C₅	semidirect
117. C₃×C₄×C₅	#100	semidirect
133. C₃×M₁₆	C₅×D₄	semidirect
198. C₄×C₅	C₃×#100	semidirect
214. M₁₆	C₃×C₅×D₄	semidirect
220. C₃×C₅	#1965	direct (N × H)
228. C₃×C₄	C₅×#100	semidirect
258. C₅	C₃×#1965	direct (N × H)
266. C₄	C₃×C₅×#100	semidirect
270. C₃	C₅×#1965	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×#127	2
C₃×C₅×#423	2
C₂×C₃×C₅×#100	2
C₃×C₄×C₅×M₁₆	2
C₅×#1965	3
C₃×#1965	5

Subgroups — C3×C5×#1965 #14274999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1965 #14274999