Subgroups — C₃×C₅×#2215 #14275249

Order 1920.

348 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#466	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₅×#2215	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#121	—	2	2	no	—
10	C₃×C₅×D₁₆	—	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×#102	—	2	2	no	—
16	C₃×#2215	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂²×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
19	C₅×#466	C₂×C₃	1	2	yes	complemented
20	C₅×#366	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#357	C₂×C₃	1	2	yes	complemented
22	C₂×C₃×C₅×D₄	—	2	2	no	—
23	C₂×C₃×C₅×D₄	—	4	2	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₃×C₅×D₈	—	2	2	no	—
26	C₃×C₅×D₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	2	1	no	—
29	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	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	complemented
35	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂²×C₃×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×#466	C₂×C₅	1	2	yes	complemented
38	C₃×#366	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#357	C₂×C₅	1	2	yes	complemented
40	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×#121	—	2	2	no	—
43	C₅×D₁₆	—	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×#102	—	2	2	no	—
49	#2215	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	4	2	no	—
51	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
52	C₃×C₅×D₄	—	4	2	no	—
53	C₃×C₅×D₄	—	4	2	no	—
54	C₃×C₅×D₄	—	8	4	no	—
55	C₂×C₃×C₄×C₅	—	2	1	no	—
56	C₂×C₃×C₄×C₅	—	2	1	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
62	C₃×C₅×C₈	#41	1	1	yes	characteristic
63	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
64	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₃×#121	—	2	2	no	—
66	C₃×D₁₆	—	2	2	no	—
67	C₂×C₃×C₁₆	—	2	2	no	—
68	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
69	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
70	C₃×#107	C₂²×C₅	1	2	yes	—
71	C₃×#102	—	2	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	4	2	no	—
74	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
75	C₅×D₈	—	2	2	no	—
76	C₅×D₈	—	2	2	no	—
77	C₂×C₅×C₈	—	2	1	no	—
78	C₂×C₅×C₈	—	2	1	no	—
79	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
80	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
81	C₅×C₄⋊C₄	—	2	2	no	—
82	C₅×C₂²⋊C₄	—	4	2	no	—
83	C₅×C₁₆	—	2	1	no	—
84	C₅×C₁₆	C₃×D₄	1	2	yes	complemented
85	C₂×D₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₂²×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	#466	C₂×C₃×C₅	1	2	yes	complemented
88	#366	C₂×C₃×C₅	1	1	yes	characteristic, complemented
89	#357	C₂×C₃×C₅	1	2	yes	complemented
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	—	2	1	no	—
92	C₂²×C₃×C₅	—	8	2	no	—
93	C₂²×C₃×C₅	—	8	2	no	—
94	C₂²×C₃×C₅	—	8	2	no	—
95	C₂²×C₃×C₅	D₁₆	1	2	yes	complemented
96	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
97	C₃×C₄×C₅	—	2	1	no	—
98	C₃×C₄×C₅	—	8	2	no	—
99	C₃×C₄×C₅	#121	1	1	yes	characteristic
100	C₃×C₄×C₅	#135	1	1	yes	characteristic
101	C₂×C₃×D₄	—	2	2	no	—
102	C₂×C₃×D₄	—	4	2	no	—
103	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
104	C₃×D₈	—	2	2	no	—
105	C₃×D₈	—	2	2	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	—	2	1	no	—
108	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
109	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
110	C₃×C₄⋊C₄	—	2	2	no	—
111	C₃×C₂²⋊C₄	—	4	2	no	—
112	C₃×C₁₆	—	2	1	no	—
113	C₃×C₁₆	C₅×D₄	1	2	yes	complemented
114	C₂³×C₅	—	4	2	no	—
115	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
116	C₅×D₄	—	4	2	no	—
117	C₅×D₄	—	4	2	no	—
118	C₅×D₄	—	8	4	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	4	2	no	—
122	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
123	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
124	C₅×C₈	—	2	1	no	—
125	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
126	C₅×C₈	C₃×#41	1	1	yes	characteristic
127	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
128	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
129	#121	—	2	2	no	—
130	D₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
133	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
134	#107	C₂²×C₃×C₅	1	2	yes	—
135	#102	—	2	2	no	—
136	C₂×C₃×C₅	—	2	2	no	—
137	C₂×C₃×C₅	—	16	2	no	—
138	C₂×C₃×C₅	#466	1	1	yes	characteristic
139	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
140	C₂×C₃×C₅	#508	1	1	yes	characteristic
141	C₂³×C₃	—	4	2	no	—
142	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
143	C₃×D₄	—	4	2	no	—
144	C₃×D₄	—	4	2	no	—
145	C₃×D₄	—	8	4	no	—
146	C₂×C₃×C₄	—	2	1	no	—
147	C₂×C₃×C₄	—	2	1	no	—
148	C₂×C₃×C₄	—	4	2	no	—
149	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
150	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
151	C₃×C₈	—	2	1	no	—
152	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
153	C₃×C₈	C₅×#41	1	1	yes	characteristic
154	C₂²×C₅	—	2	1	no	—
155	C₂²×C₅	—	2	1	no	—
156	C₂²×C₅	—	8	2	no	—
157	C₂²×C₅	—	8	2	no	—
158	C₂²×C₅	—	8	2	no	—
159	C₂²×C₅	C₃×D₁₆	1	2	yes	complemented
160	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	—	8	2	no	—
163	C₄×C₅	C₃×#121	1	1	yes	characteristic
164	C₄×C₅	C₃×#135	1	1	yes	characteristic
165	C₂×D₄	—	2	2	no	—
166	C₂×D₄	—	4	2	no	—
167	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
168	D₈	—	2	2	no	—
169	D₈	—	2	2	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	—	4	2	no	—
176	C₁₆	—	2	1	no	—
177	C₁₆	C₃×C₅×D₄	1	2	yes	complemented
178	C₃×C₅	#2215	1	1	yes	characteristic, direct product factor, Hall
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	2	1	no	—
181	C₂²×C₃	—	8	2	no	—
182	C₂²×C₃	—	8	2	no	—
183	C₂²×C₃	—	8	2	no	—
184	C₂²×C₃	C₅×D₁₆	1	2	yes	complemented
185	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
186	C₃×C₄	—	2	1	no	—
187	C₃×C₄	—	8	2	no	—
188	C₃×C₄	C₅×#121	1	1	yes	characteristic
189	C₃×C₄	C₅×#135	1	1	yes	characteristic
190	C₂×C₅	—	2	2	no	—
191	C₂×C₅	—	16	2	no	—
192	C₂×C₅	C₃×#466	1	1	yes	characteristic
193	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
194	C₂×C₅	C₃×#508	1	1	yes	characteristic
195	C₂³	—	4	2	no	—
196	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
197	D₄	—	4	2	no	—
198	D₄	—	4	2	no	—
199	D₄	—	8	4	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
204	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
207	C₈	C₃×C₅×#41	1	1	yes	characteristic
208	C₂×C₃	—	2	2	no	—
209	C₂×C₃	—	16	2	no	—
210	C₂×C₃	C₅×#466	1	1	yes	characteristic
211	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
212	C₂×C₃	C₅×#508	1	1	yes	characteristic
213	C₅	C₃×#2215	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	—	8	2	no	—
217	C₂²	—	8	2	no	—
218	C₂²	—	8	2	no	—
219	C₂²	C₃×C₅×D₁₆	1	2	yes	complemented
220	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
221	C₄	—	2	1	no	—
222	C₄	—	8	2	no	—
223	C₄	C₃×C₅×#121	1	1	yes	characteristic
224	C₄	C₃×C₅×#135	1	1	yes	characteristic
225	C₃	C₅×#2215	1	1	yes	characteristic, direct product factor, Sylow
226	C₂	—	2	2	no	—
227	C₂	—	16	2	no	—
228	C₂	C₃×C₅×#466	1	1	yes	characteristic
229	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
230	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
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×D₁₆	C₂	semidirect
2. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₃×C₅×#466	C₂	semidirect
4. C₃×C₅×#366	C₂	semidirect
5. C₃×C₅×#357	C₂	semidirect
6. C₅×#2215	C₃	direct (N × H)
16. C₃×#2215	C₅	direct (N × H)
17. C₂×C₅×D₁₆	C₂×C₃	semidirect
18. C₂²×C₅×C₁₆	C₂×C₃	semidirect
19. C₅×#466	C₂×C₃	semidirect
20. C₅×#366	C₂×C₃	semidirect
21. C₅×#357	C₂×C₃	semidirect
34. C₃×C₅×C₁₆	D₄	semidirect
35. C₂×C₃×D₁₆	C₂×C₅	semidirect
36. C₂²×C₃×C₁₆	C₂×C₅	semidirect
37. C₃×#466	C₂×C₅	semidirect
38. C₃×#366	C₂×C₅	semidirect
39. C₃×#357	C₂×C₅	semidirect
49. #2215	C₃×C₅	direct (N × H)
84. C₅×C₁₆	C₃×D₄	semidirect
85. C₂×D₁₆	C₂×C₃×C₅	semidirect
86. C₂²×C₁₆	C₂×C₃×C₅	semidirect
87. #466	C₂×C₃×C₅	semidirect
88. #366	C₂×C₃×C₅	semidirect
89. #357	C₂×C₃×C₅	semidirect
95. C₂²×C₃×C₅	D₁₆	semidirect
113. C₃×C₁₆	C₅×D₄	semidirect
159. C₂²×C₅	C₃×D₁₆	semidirect
177. C₁₆	C₃×C₅×D₄	semidirect
178. C₃×C₅	#2215	direct (N × H)
184. C₂²×C₃	C₅×D₁₆	semidirect
213. C₅	C₃×#2215	direct (N × H)
219. C₂²	C₃×C₅×D₁₆	semidirect
225. C₃	C₅×#2215	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×D₁₆	2
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×#466	2
C₃×C₅×#366	2
C₃×C₅×#357	2
C₅×#2215	3
C₃×#2215	5

Subgroups — C3×C5×#2215 #14275249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2215 #14275249