Subgroups — C₃×C₅×#2216 #14275250

Order 1920.

308 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#467	C₂	1	2	yes	maximal
4	C₃×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
6	C₅×#2216	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
10	C₃×C₅×Q₃₂	—	2	2	no	—
11	C₂×C₃×C₅×C₁₆	—	2	2	no	—
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	2	yes	—
15	C₃×C₅×#103	—	2	2	no	—
16	C₃×#2216	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic
19	C₅×#467	C₂×C₃	1	2	yes	—
20	C₅×#366	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#358	C₂×C₃	1	2	yes	complemented
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₃×C₅×Q₁₆	—	2	2	no	—
25	C₃×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	1	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	4	2	no	—
32	C₃×C₅×C₂²⋊C₄	—	4	2	no	—
33	C₃×C₅×C₁₆	—	2	1	no	—
34	C₃×C₅×C₁₆	D₄	1	2	yes	—
35	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic
37	C₃×#467	C₂×C₅	1	2	yes	—
38	C₃×#366	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#358	C₂×C₅	1	2	yes	complemented
40	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×C₂²⋊Q₈	—	2	2	no	—
43	C₅×Q₃₂	—	2	2	no	—
44	C₂×C₅×C₁₆	—	2	2	no	—
45	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
46	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
47	C₅×#107	C₂²×C₃	1	2	yes	—
48	C₅×#103	—	2	2	no	—
49	#2216	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
51	C₃×C₅×Q₈	—	4	2	no	—
52	C₃×C₅×Q₈	—	4	2	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	2	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
63	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₃×C₂²⋊Q₈	—	2	2	no	—
65	C₃×Q₃₂	—	2	2	no	—
66	C₂×C₃×C₁₆	—	2	2	no	—
67	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
69	C₃×#107	C₂²×C₅	1	2	yes	—
70	C₃×#103	—	2	2	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	2	no	—
74	C₅×Q₁₆	—	2	2	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
77	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	4	2	no	—
81	C₅×C₂²⋊C₄	—	4	2	no	—
82	C₅×C₁₆	—	2	1	no	—
83	C₅×C₁₆	C₃×D₄	1	2	yes	—
84	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
85	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
86	#467	C₂×C₃×C₅	1	2	yes	—
87	#366	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	#358	C₂×C₃×C₅	1	2	yes	complemented
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	Q₃₂	1	2	yes	complemented
92	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
93	C₃×C₄×C₅	—	2	1	no	—
94	C₃×C₄×C₅	—	8	2	no	—
95	C₃×C₄×C₅	—	8	2	no	—
96	C₃×C₄×C₅	#121	1	1	yes	characteristic
97	C₃×C₄×C₅	#135	1	1	yes	characteristic
98	C₂×C₃×Q₈	—	2	2	no	—
99	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
100	C₃×Q₁₆	—	2	2	no	—
101	C₃×Q₁₆	—	2	2	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	—	2	1	no	—
104	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
105	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
106	C₃×C₄⋊C₄	—	2	2	no	—
107	C₃×C₄⋊C₄	—	4	2	no	—
108	C₃×C₂²⋊C₄	—	4	2	no	—
109	C₃×C₁₆	—	2	1	no	—
110	C₃×C₁₆	C₅×D₄	1	2	yes	—
111	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
112	C₅×Q₈	—	4	2	no	—
113	C₅×Q₈	—	4	2	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
122	C₅×C₈	C₃×#41	1	1	yes	characteristic
123	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
124	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
125	C₂²⋊Q₈	—	2	2	no	—
126	Q₃₂	—	2	2	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
130	#107	C₂²×C₃×C₅	1	2	yes	—
131	#103	—	2	2	no	—
132	C₂×C₃×C₅	—	2	2	no	—
133	C₂×C₃×C₅	#466	1	1	yes	characteristic
134	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
135	C₂×C₃×C₅	#508	1	1	yes	characteristic
136	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
137	C₃×Q₈	—	4	2	no	—
138	C₃×Q₈	—	4	2	no	—
139	C₂×C₃×C₄	—	2	1	no	—
140	C₂×C₃×C₄	—	2	1	no	—
141	C₂×C₃×C₄	—	4	2	no	—
142	C₂×C₃×C₄	—	4	2	no	—
143	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
144	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
147	C₃×C₈	C₅×#41	1	1	yes	characteristic
148	C₂²×C₅	—	2	1	no	—
149	C₂²×C₅	—	2	1	no	—
150	C₂²×C₅	C₃×Q₃₂	1	2	yes	complemented
151	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	8	2	no	—
154	C₄×C₅	—	8	2	no	—
155	C₄×C₅	C₃×#121	1	1	yes	characteristic
156	C₄×C₅	C₃×#135	1	1	yes	characteristic
157	C₂×Q₈	—	2	2	no	—
158	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
159	Q₁₆	—	2	2	no	—
160	Q₁₆	—	2	2	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
164	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
165	C₄⋊C₄	—	2	2	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₂²⋊C₄	—	4	2	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	C₃×C₅×D₄	1	2	yes	—
170	C₃×C₅	#2216	1	1	yes	characteristic, direct product factor, Hall
171	C₂²×C₃	—	2	1	no	—
172	C₂²×C₃	—	2	1	no	—
173	C₂²×C₃	C₅×Q₃₂	1	2	yes	complemented
174	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
175	C₃×C₄	—	2	1	no	—
176	C₃×C₄	—	8	2	no	—
177	C₃×C₄	—	8	2	no	—
178	C₃×C₄	C₅×#121	1	1	yes	characteristic
179	C₃×C₄	C₅×#135	1	1	yes	characteristic
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	C₃×#466	1	1	yes	characteristic
182	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
183	C₂×C₅	C₃×#508	1	1	yes	characteristic
184	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
185	Q₈	—	4	2	no	—
186	Q₈	—	4	2	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	4	2	no	—
190	C₂×C₄	—	4	2	no	—
191	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
192	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
193	C₈	—	2	1	no	—
194	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
195	C₈	C₃×C₅×#41	1	1	yes	characteristic
196	C₂×C₃	—	2	2	no	—
197	C₂×C₃	C₅×#466	1	1	yes	characteristic
198	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
199	C₂×C₃	C₅×#508	1	1	yes	characteristic
200	C₅	C₃×#2216	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₃×C₅×Q₃₂	1	2	yes	complemented
204	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	8	2	no	—
207	C₄	—	8	2	no	—
208	C₄	C₃×C₅×#121	1	1	yes	characteristic
209	C₄	C₃×C₅×#135	1	1	yes	characteristic
210	C₃	C₅×#2216	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	2	no	—
212	C₂	C₃×C₅×#466	1	1	yes	characteristic
213	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic
214	C₂	C₃×C₅×#508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂²×C₃×C₄×C₅ ◁ C₂²×C₃×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×Q₃₂	C₂	semidirect
4. C₃×C₅×#366	C₂	semidirect
5. C₃×C₅×#358	C₂	semidirect
6. C₅×#2216	C₃	direct (N × H)
16. C₃×#2216	C₅	direct (N × H)
17. C₂×C₅×Q₃₂	C₂×C₃	semidirect
20. C₅×#366	C₂×C₃	semidirect
21. C₅×#358	C₂×C₃	semidirect
35. C₂×C₃×Q₃₂	C₂×C₅	semidirect
38. C₃×#366	C₂×C₅	semidirect
39. C₃×#358	C₂×C₅	semidirect
49. #2216	C₃×C₅	direct (N × H)
84. C₂×Q₃₂	C₂×C₃×C₅	semidirect
87. #366	C₂×C₃×C₅	semidirect
88. #358	C₂×C₃×C₅	semidirect
91. C₂²×C₃×C₅	Q₃₂	semidirect
150. C₂²×C₅	C₃×Q₃₂	semidirect
170. C₃×C₅	#2216	direct (N × H)
173. C₂²×C₃	C₅×Q₃₂	semidirect
200. C₅	C₃×#2216	direct (N × H)
203. C₂²	C₃×C₅×Q₃₂	semidirect
210. C₃	C₅×#2216	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×Q₃₂	2
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×#467	2
C₃×C₅×#366	2
C₃×C₅×#358	2
C₅×#2216	3
C₃×#2216	5

Subgroups — C3×C5×#2216 #14275250

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2216 #14275250