Subgroups — C₃×C₅×#1985 #14275019

Order 1920.

560 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#336	C₂	1	4	yes	maximal
5	C₅×#1985	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
9	C₃×C₄×C₅×Q₈	—	2	2	no	—
10	C₃×C₄×C₅×Q₈	—	2	2	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	C₃×C₅×#97	—	2	2	no	—
17	C₃×#1985	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×#97	C₂×C₃	1	1	yes	characteristic
21	C₅×#336	C₂×C₃	1	4	yes	—
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×Q₈	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	—	2	2	no	—
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	4	no	—
33	C₂×C₃×C₅×C₈	Q₈	1	4	yes	complemented
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₄²×C₅	—	2	2	no	—
37	C₃×C₄²×C₅	—	2	2	no	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	D₄	1	2	yes	—
40	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
42	C₂×C₃×#97	C₂×C₅	1	1	yes	characteristic
43	C₃×#336	C₂×C₅	1	4	yes	—
44	C₂²×C₅×Q₈	C₃×C₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
46	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
47	C₄×C₅×Q₈	—	2	2	no	—
48	C₄×C₅×Q₈	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₃×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
51	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
52	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₄⋊C₈	—	2	2	no	—
54	C₅×#97	—	2	2	no	—
55	#1985	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	2	8	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₃×C₅×C₈	—	2	4	no	—
72	C₃×C₅×C₈	—	4	2	no	—
73	C₂²×C₃×Q₈	C₄×C₅	1	1	yes	characteristic
74	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
75	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
76	C₃×C₄×Q₈	—	2	2	no	—
77	C₃×C₄×Q₈	—	2	2	no	—
78	C₂×C₃×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
79	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
81	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄⋊C₈	—	2	2	no	—
83	C₃×#97	—	2	2	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂×C₅×Q₈	—	2	4	no	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	C₃×Q₈	1	4	yes	complemented
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₄²×C₅	—	2	2	no	—
99	C₄²×C₅	—	2	2	no	—
100	C₄²×C₅	C₃×D₄	1	2	yes	—
101	C₄²×C₅	C₃×D₄	1	2	yes	—
102	C₂×C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
104	C₂×#97	C₂×C₃×C₅	1	1	yes	characteristic
105	#336	C₂×C₃×C₅	1	4	yes	—
106	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×Q₈	1	2	yes	—
108	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
109	C₂²×C₃×C₅	#131	1	2	yes	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	2	4	no	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	4	2	no	—
114	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
115	C₃×C₄×C₅	#124	1	1	yes	characteristic
116	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	—	2	4	no	—
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
123	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	—	2	2	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	C₅×Q₈	1	4	yes	complemented
129	C₃×C₄⋊C₄	—	2	4	no	—
130	C₃×C₄⋊C₄	—	2	2	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	—	2	2	no	—
133	C₃×C₄²	C₅×D₄	1	2	yes	—
134	C₃×C₄²	C₅×D₄	1	2	yes	—
135	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
136	C₅×Q₈	—	2	8	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
145	C₂×C₄×C₅	C₃×M₁₆	1	4	yes	—
146	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₃×#41	1	2	yes	—
150	C₅×C₈	—	2	4	no	—
151	C₅×C₈	—	4	2	no	—
152	C₂²×Q₈	C₃×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
154	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
155	C₄×Q₈	—	2	2	no	—
156	C₄×Q₈	—	2	2	no	—
157	C₂×C₄⋊C₄	C₃×C₄×C₅	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	—
159	C₂×C₄²	C₂²×C₃×C₅	1	2	yes	—
160	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	#97	—	2	2	no	—
163	C₂×C₃×C₅	#391	1	1	yes	characteristic
164	C₂×C₃×C₅	#407	1	1	yes	characteristic
165	C₂×C₃×C₅	#408	1	1	yes	characteristic
166	C₂×C₃×C₅	#446	1	4	yes	—
167	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
168	C₃×Q₈	—	2	8	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	—	2	2	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	2	2	no	—
174	C₂×C₃×C₄	—	2	4	no	—
175	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
176	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
177	C₂×C₃×C₄	C₅×M₁₆	1	4	yes	—
178	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
181	C₂×C₃×C₄	C₅×#41	1	2	yes	—
182	C₃×C₈	—	2	4	no	—
183	C₃×C₈	—	4	2	no	—
184	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
185	C₂²×C₅	C₃×C₄×Q₈	1	2	yes	—
186	C₂²×C₅	C₂×C₃×M₁₆	1	2	yes	—
187	C₂²×C₅	C₃×#131	1	2	yes	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	2	2	no	—
191	C₄×C₅	—	4	2	no	—
192	C₄×C₅	C₃×C₂²⋊Q₈	1	2	yes	—
193	C₄×C₅	C₃×#124	1	1	yes	characteristic
194	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×Q₈	—	2	2	no	—
196	C₂×Q₈	—	2	4	no	—
197	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
198	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
201	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₃×C₅×Q₈	1	4	yes	complemented
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	2	no	—
211	C₄²	C₃×C₅×D₄	1	2	yes	—
212	C₄²	C₃×C₅×D₄	1	2	yes	—
213	C₃×C₅	#1985	1	1	yes	characteristic, direct product factor, Hall
214	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₃	C₄×C₅×Q₈	1	2	yes	—
216	C₂²×C₃	C₂×C₅×M₁₆	1	2	yes	—
217	C₂²×C₃	C₅×#131	1	2	yes	—
218	C₃×C₄	—	2	2	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	—	4	2	no	—
222	C₃×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
223	C₃×C₄	C₅×#124	1	1	yes	characteristic
224	C₃×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
225	C₂×C₅	C₃×#391	1	1	yes	characteristic
226	C₂×C₅	C₃×#407	1	1	yes	characteristic
227	C₂×C₅	C₃×#408	1	1	yes	characteristic
228	C₂×C₅	C₃×#446	1	4	yes	—
229	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
230	Q₈	—	2	8	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
238	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
239	C₂×C₄	C₃×C₅×M₁₆	1	4	yes	—
240	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
242	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
243	C₂×C₄	C₃×C₅×#41	1	2	yes	—
244	C₈	—	2	4	no	—
245	C₈	—	4	2	no	—
246	C₂×C₃	C₅×#391	1	1	yes	characteristic
247	C₂×C₃	C₅×#407	1	1	yes	characteristic
248	C₂×C₃	C₅×#408	1	1	yes	characteristic
249	C₂×C₃	C₅×#446	1	4	yes	—
250	C₅	C₃×#1985	1	1	yes	characteristic, direct product factor, Sylow
251	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
252	C₂²	C₃×C₄×C₅×Q₈	1	2	yes	—
253	C₂²	C₂×C₃×C₅×M₁₆	1	2	yes	—
254	C₂²	C₃×C₅×#131	1	2	yes	—
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	2	no	—
259	C₄	C₃×C₅×C₂²⋊Q₈	1	2	yes	—
260	C₄	C₃×C₅×#124	1	1	yes	characteristic
261	C₄	C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
262	C₃	C₅×#1985	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	C₃×C₅×#391	1	1	yes	characteristic
264	C₂	C₃×C₅×#407	1	1	yes	characteristic
265	C₂	C₃×C₅×#408	1	1	yes	characteristic
266	C₂	C₃×C₅×#446	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
5. C₅×#1985	C₃	direct (N × H)
17. C₃×#1985	C₅	direct (N × H)
33. C₂×C₃×C₅×C₈	Q₈	semidirect
55. #1985	C₃×C₅	direct (N × H)
95. C₂×C₅×C₈	C₃×Q₈	semidirect
128. C₂×C₃×C₈	C₅×Q₈	semidirect
206. C₂×C₈	C₃×C₅×Q₈	semidirect
213. C₃×C₅	#1985	direct (N × H)
250. C₅	C₃×#1985	direct (N × H)
262. C₃	C₅×#1985	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×C₄×C₅×Q₈	2
C₂×C₃×C₅×C₄⋊C₈	2
C₂×C₃×C₅×#97	2
C₃×C₅×#336	2
C₅×#1985	3
C₃×#1985	5

Subgroups — C3×C5×#1985 #14275019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1985 #14275019