Subgroups — C₃×C₅×#2113 #14275147

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₅×#2113	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
7	C₃×C₅×#131	—	2	2	no	—
8	C₃×C₅×#131	—	2	2	no	—
9	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₅×M₃₂	—	2	2	no	—
14	C₂×C₃×C₅×C₁₆	—	2	2	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
16	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
17	C₃×#2113	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#131	C₂×C₃	1	1	yes	characteristic
19	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×C₂²⋊C₁₆	C₂×C₃	1	4	yes	complemented
22	C₃×C₅×#41	—	2	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	C₈	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	C₈	1	2	yes	—
26	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₃×C₅×M₁₆	—	2	4	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₃×C₅×C₁₆	—	2	2	no	—
40	C₂×C₃×#131	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×C₂²⋊C₁₆	C₂×C₅	1	4	yes	complemented
44	C₂×C₅×#41	C₃×C₄	1	1	yes	characteristic
45	C₅×#131	—	2	2	no	—
46	C₅×#131	—	2	2	no	—
47	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
48	C₂×C₅×M₁₆	C₃×C₄	1	2	yes	—
49	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
50	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×M₃₂	—	2	2	no	—
52	C₂×C₅×C₁₆	—	2	2	no	—
53	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
54	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
55	#2113	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
57	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
58	C₃×C₅×Q₈	—	2	2	no	—
59	C₃×C₅×D₄	—	2	4	no	—
60	C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×#41	C₄×C₅	1	1	yes	characteristic
73	C₃×#131	—	2	2	no	—
74	C₃×#131	—	2	2	no	—
75	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₄×C₅	1	2	yes	—
77	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
78	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
79	C₃×M₃₂	—	2	2	no	—
80	C₂×C₃×C₁₆	—	2	2	no	—
81	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
82	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
83	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₂×C₅×Q₈	C₃×C₈	1	1	yes	characteristic
86	C₂×C₅×D₄	C₃×C₈	1	2	yes	—
87	C₂×C₅×D₄	C₃×C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
90	C₅×M₁₆	—	2	2	no	—
91	C₅×M₁₆	—	2	4	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
97	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
98	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
99	C₅×C₁₆	—	2	2	no	—
100	C₅×C₁₆	—	2	2	no	—
101	C₂×#131	C₂×C₃×C₅	1	1	yes	characteristic
102	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
103	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	C₂²⋊C₁₆	C₂×C₃×C₅	1	4	yes	complemented
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	2	no	—
107	C₂²×C₃×C₅	—	2	4	no	—
108	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
112	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₃×#41	—	2	2	no	—
115	C₃×#41	—	2	2	no	—
116	C₂×C₃×Q₈	C₅×C₈	1	1	yes	characteristic
117	C₂×C₃×D₄	C₅×C₈	1	2	yes	—
118	C₂×C₃×D₄	C₅×C₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	2	4	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
128	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
129	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
130	C₃×C₁₆	—	2	2	no	—
131	C₃×C₁₆	—	2	2	no	—
132	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
133	C₂³×C₅	C₂×C₃×C₈	1	1	yes	characteristic
134	C₅×Q₈	—	2	2	no	—
135	C₅×D₄	—	2	4	no	—
136	C₅×D₄	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
141	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
142	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
146	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
147	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×#41	C₃×C₄×C₅	1	1	yes	characteristic
149	#131	—	2	2	no	—
150	#131	—	2	2	no	—
151	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
152	C₂×M₁₆	C₃×C₄×C₅	1	2	yes	—
153	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
154	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
155	M₃₂	—	2	2	no	—
156	C₂×C₁₆	—	2	2	no	—
157	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
158	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	—	4	2	no	—
161	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₃×C₅	#504	1	2	yes	—
163	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
164	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
165	C₃×Q₈	—	2	2	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
177	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
178	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	C₃×C₂²⋊C₈	1	2	yes	—
183	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	C₃×C₂²⋊C₈	1	2	yes	—
186	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	C₃×C₅×C₈	1	1	yes	characteristic
191	C₂×D₄	C₃×C₅×C₈	1	2	yes	—
192	C₂×D₄	C₃×C₅×C₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
194	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
200	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
201	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
202	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
203	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
204	C₁₆	—	2	2	no	—
205	C₁₆	—	2	2	no	—
206	C₃×C₅	#2113	1	1	yes	characteristic, direct product factor, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
211	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
214	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	4	2	no	—
218	C₂×C₅	C₂×C₃×C₂²⋊C₈	1	1	yes	characteristic
219	C₂×C₅	C₃×#504	1	2	yes	—
220	C₂³	C₂×C₃×C₅×C₈	1	2	yes	—
221	C₂³	C₂×C₃×C₅×C₈	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
228	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
229	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
230	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
231	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₃×C₅×C₂²⋊C₄	1	2	yes	—
234	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
235	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	4	2	no	—
238	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₃	C₅×#504	1	2	yes	—
240	C₅	C₃×#2113	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	2	no	—
244	C₂²	C₃×C₅×C₂²⋊C₈	1	2	yes	—
245	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	C₃×C₅×C₂²⋊C₈	1	2	yes	—
248	C₄	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
249	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
250	C₃	C₅×#2113	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₂×C₃×C₅×C₂²⋊C₈	1	1	yes	characteristic
254	C₂	C₃×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₅×C₈	C₂³
commutator	C₂²	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×C₅×#41	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂²×C₃×C₅×C₁₆	C₂	semidirect
4. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
5. C₅×#2113	C₃	direct (N × H)
17. C₃×#2113	C₅	direct (N × H)
19. C₂×C₅×M₃₂	C₂×C₃	semidirect
20. C₂²×C₅×C₁₆	C₂×C₃	semidirect
21. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
41. C₂×C₃×M₃₂	C₂×C₅	semidirect
42. C₂²×C₃×C₁₆	C₂×C₅	semidirect
43. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
55. #2113	C₃×C₅	direct (N × H)
102. C₂×M₃₂	C₂×C₃×C₅	semidirect
103. C₂²×C₁₆	C₂×C₃×C₅	semidirect
104. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
206. C₃×C₅	#2113	direct (N × H)
240. C₅	C₃×#2113	direct (N × H)
250. C₃	C₅×#2113	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×#131	2
C₂×C₃×C₅×M₃₂	2
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×C₂²⋊C₁₆	2
C₅×#2113	3
C₃×#2113	5

Subgroups — C3×C5×#2113 #14275147

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2113 #14275147