Subgroups — C₃×C₅×#2159 #14275193

Order 1920.

312 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#367	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#366	C₂	1	2	yes	maximal, complemented
6	C₅×#2159	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	—	2	1	no	—
9	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
10	C₃×C₅×M₃₂	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
12	C₃×C₅×#107	—	2	1	no	—
13	C₃×C₅×#107	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	2	yes	—
15	C₃×C₅×#107	C₂²	1	1	yes	characteristic
16	C₃×C₅×#106	—	2	1	no	—
17	C₃×#2159	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
19	C₅×#427	C₂×C₃	1	1	yes	characteristic
20	C₂×C₅×#107	C₂×C₃	1	1	yes	characteristic
21	C₅×#367	C₂×C₃	1	2	yes	complemented
22	C₅×#366	C₂×C₃	1	2	yes	complemented
23	C₂²×C₃×C₄×C₅	—	4	1	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	2	no	—
30	C₃×C₅×C₄⋊C₄	—	2	1	no	—
31	C₃×C₅×C₄⋊C₄	—	2	1	no	—
32	C₃×C₅×C₄⋊C₄	—	4	1	no	—
33	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
34	C₃×C₄²×C₅	—	4	1	no	—
35	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
36	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
37	C₃×#427	C₂×C₅	1	1	yes	characteristic
38	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic
39	C₃×#367	C₂×C₅	1	2	yes	complemented
40	C₃×#366	C₂×C₅	1	2	yes	complemented
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×#117	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₅×M₃₂	C₃×C₄	1	4	yes	complemented
45	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
46	C₅×#107	—	2	1	no	—
47	C₅×#107	C₂²×C₃	1	2	yes	—
48	C₅×#107	C₂²×C₃	1	1	yes	characteristic
49	C₅×#107	C₂²×C₃	1	1	yes	characteristic
50	C₅×#106	—	2	1	no	—
51	#2159	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	—	4	1	no	—
54	C₂×C₃×C₄×C₅	—	4	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	8	2	no	—
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
58	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	—
62	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
63	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₃×#117	—	2	1	no	—
66	C₂×C₃×C₄⋊C₄	—	2	1	no	—
67	C₃×M₃₂	C₄×C₅	1	4	yes	complemented
68	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
69	C₃×#107	—	2	1	no	—
70	C₃×#107	C₂²×C₅	1	1	yes	characteristic
71	C₃×#107	C₂²×C₅	1	2	yes	—
72	C₃×#107	C₂²×C₅	1	1	yes	characteristic
73	C₃×#106	—	2	1	no	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
78	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
79	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	2	1	no	—
82	C₅×C₄⋊C₄	—	2	1	no	—
83	C₅×C₄⋊C₄	—	4	1	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₄²×C₅	—	4	1	no	—
86	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	complemented
87	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic
88	#427	C₂×C₃×C₅	1	1	yes	characteristic
89	C₂×#107	C₂×C₃×C₅	1	1	yes	characteristic
90	#367	C₂×C₃×C₅	1	2	yes	complemented
91	#366	C₂×C₃×C₅	1	2	yes	complemented
92	C₂²×C₃×C₅	—	2	1	no	—
93	C₂²×C₃×C₅	—	2	1	no	—
94	C₂²×C₃×C₅	#107	1	2	yes	—
95	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
96	C₃×C₄×C₅	—	8	1	no	—
97	C₃×C₄×C₅	—	8	2	no	—
98	C₃×C₄×C₅	—	8	1	no	—
99	C₃×C₄×C₅	#107	1	2	yes	—
100	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
102	C₂²×C₃×C₄	—	4	1	no	—
103	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₅×Q₈	1	2	yes	—
107	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
108	C₃×C₄⋊C₄	—	2	1	no	—
109	C₃×C₄⋊C₄	—	2	1	no	—
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₄⋊C₄	—	4	1	no	—
112	C₃×C₂²⋊C₄	—	4	1	no	—
113	C₃×C₄²	—	4	1	no	—
114	C₃×C₁₆	C₂×C₄×C₅	1	4	yes	complemented
115	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
116	C₂×C₄×C₅	—	4	1	no	—
117	C₂×C₄×C₅	—	4	1	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	—	8	2	no	—
120	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₃×Q₁₆	1	2	yes	—
123	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
124	C₅×C₈	C₃×C₄⋊C₄	1	2	yes	—
125	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
126	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
127	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
128	#117	—	2	1	no	—
129	C₂×C₄⋊C₄	—	2	1	no	—
130	M₃₂	C₃×C₄×C₅	1	4	yes	complemented
131	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
132	#107	—	2	1	no	—
133	#107	C₂²×C₃×C₅	1	1	yes	characteristic
134	#107	C₂²×C₃×C₅	1	2	yes	—
135	#107	C₂²×C₃×C₅	1	1	yes	characteristic
136	#106	—	2	1	no	—
137	C₂×C₃×C₅	—	2	2	no	—
138	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
139	C₂×C₃×C₅	#509	1	1	yes	characteristic
140	C₂×C₃×C₅	#510	1	1	yes	characteristic
141	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
142	C₂×C₃×C₄	—	4	1	no	—
143	C₂×C₃×C₄	—	4	1	no	—
144	C₂×C₃×C₄	—	4	2	no	—
145	C₂×C₃×C₄	—	8	2	no	—
146	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
147	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
148	C₂×C₃×C₄	C₅×Q₁₆	1	2	yes	—
149	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
150	C₃×C₈	C₅×C₄⋊C₄	1	2	yes	—
151	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
152	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	C₃×#107	1	2	yes	—
156	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
157	C₄×C₅	—	8	1	no	—
158	C₄×C₅	—	8	2	no	—
159	C₄×C₅	—	8	1	no	—
160	C₄×C₅	C₃×#107	1	2	yes	—
161	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₅	C₂×C₃×Q₁₆	1	1	yes	characteristic
163	C₂²×C₄	—	4	1	no	—
164	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
165	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
166	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
167	C₂×C₈	C₃×C₅×Q₈	1	2	yes	—
168	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	4	1	no	—
173	C₂²⋊C₄	—	4	1	no	—
174	C₄²	—	4	1	no	—
175	C₁₆	C₂×C₃×C₄×C₅	1	4	yes	complemented
176	C₃×C₅	#2159	1	1	yes	characteristic, direct product factor, Hall
177	C₂²×C₃	—	2	1	no	—
178	C₂²×C₃	—	2	1	no	—
179	C₂²×C₃	C₅×#107	1	2	yes	—
180	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
181	C₃×C₄	—	8	1	no	—
182	C₃×C₄	—	8	2	no	—
183	C₃×C₄	—	8	1	no	—
184	C₃×C₄	C₅×#107	1	2	yes	—
185	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
186	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
187	C₂×C₅	—	2	2	no	—
188	C₂×C₅	C₂×C₃×#107	1	1	yes	characteristic
189	C₂×C₅	C₃×#509	1	1	yes	characteristic
190	C₂×C₅	C₃×#510	1	1	yes	characteristic
191	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	C₃×C₅×C₄⋊C₄	1	2	yes	—
197	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
198	C₂×C₄	C₃×C₅×Q₁₆	1	2	yes	—
199	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
200	C₈	C₃×C₅×C₄⋊C₄	1	2	yes	—
201	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
202	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
203	C₂×C₃	—	2	2	no	—
204	C₂×C₃	C₂×C₅×#107	1	1	yes	characteristic
205	C₂×C₃	C₅×#509	1	1	yes	characteristic
206	C₂×C₃	C₅×#510	1	1	yes	characteristic
207	C₅	C₃×#2159	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	C₃×C₅×#107	1	2	yes	—
211	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
212	C₄	—	8	2	no	—
213	C₄	—	8	1	no	—
214	C₄	—	8	1	no	—
215	C₄	C₃×C₅×#107	1	2	yes	—
216	C₄	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
217	C₄	C₂×C₃×C₅×Q₁₆	1	1	yes	characteristic
218	C₃	C₅×#2159	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	2	2	no	—
220	C₂	C₂×C₃×C₅×#107	1	1	yes	characteristic
221	C₂	C₃×C₅×#509	1	1	yes	characteristic
222	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	D₈
LCS	G ▷ 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
4. C₃×C₅×#367	C₂	semidirect
5. C₃×C₅×#366	C₂	semidirect
6. C₅×#2159	C₃	direct (N × H)
10. C₃×C₅×M₃₂	C₄	semidirect
17. C₃×#2159	C₅	direct (N × H)
21. C₅×#367	C₂×C₃	semidirect
22. C₅×#366	C₂×C₃	semidirect
35. C₃×C₅×C₁₆	C₂×C₄	semidirect
39. C₃×#367	C₂×C₅	semidirect
40. C₃×#366	C₂×C₅	semidirect
44. C₅×M₃₂	C₃×C₄	semidirect
51. #2159	C₃×C₅	direct (N × H)
67. C₃×M₃₂	C₄×C₅	semidirect
86. C₅×C₁₆	C₂×C₃×C₄	semidirect
90. #367	C₂×C₃×C₅	semidirect
91. #366	C₂×C₃×C₅	semidirect
114. C₃×C₁₆	C₂×C₄×C₅	semidirect
130. M₃₂	C₃×C₄×C₅	semidirect
175. C₁₆	C₂×C₃×C₄×C₅	semidirect
176. C₃×C₅	#2159	direct (N × H)
207. C₅	C₃×#2159	direct (N × H)
218. C₃	C₅×#2159	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×M₃₂	2
C₃×C₅×#427	2
C₂×C₃×C₅×#107	2
C₃×C₅×#367	2
C₃×C₅×#366	2
C₅×#2159	3
C₃×#2159	5

Subgroups — C3×C5×#2159 #14275193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2159 #14275193