Subgroups — C₃×C₅×#2218 #14275252

Order 1920.

340 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×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#466	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
6	C₅×#2218	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	C₃×C₅×M₃₂	—	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×#2218	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic, complemented
19	C₅×#466	C₂×C₃	1	2	yes	complemented
20	C₅×#367	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	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
37	C₃×#466	C₂×C₅	1	2	yes	complemented
38	C₃×#367	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	C₅×M₃₂	—	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	#2218	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	C₃×M₃₂	—	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	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	#466	C₂×C₃×C₅	1	2	yes	complemented
88	#367	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	2	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	C₂×D₈	1	1	yes	characteristic
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	8	2	no	—
98	C₃×C₄×C₅	#121	1	1	yes	characteristic
99	C₃×C₄×C₅	#135	1	1	yes	characteristic
100	C₂×C₃×D₄	—	2	2	no	—
101	C₂×C₃×D₄	—	4	2	no	—
102	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
103	C₃×D₈	—	2	2	no	—
104	C₃×D₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	2	1	no	—
107	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
109	C₃×C₄⋊C₄	—	2	2	no	—
110	C₃×C₂²⋊C₄	—	4	2	no	—
111	C₃×C₁₆	—	2	1	no	—
112	C₃×C₁₆	C₅×D₄	1	2	yes	complemented
113	C₂³×C₅	—	4	2	no	—
114	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
115	C₅×D₄	—	4	2	no	—
116	C₅×D₄	—	4	2	no	—
117	C₅×D₄	—	8	4	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	4	2	no	—
121	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
122	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
123	C₅×C₈	—	2	1	no	—
124	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
125	C₅×C₈	C₃×#41	1	1	yes	characteristic
126	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
127	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
128	#121	—	2	2	no	—
129	D₁₆	—	2	2	no	—
130	M₃₂	—	2	2	no	—
131	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
133	#107	C₂²×C₃×C₅	1	2	yes	—
134	#102	—	2	2	no	—
135	C₂×C₃×C₅	—	4	1	no	—
136	C₂×C₃×C₅	—	16	2	no	—
137	C₂×C₃×C₅	#466	1	1	yes	characteristic
138	C₂×C₃×C₅	#509	1	2	yes	—
139	C₂³×C₃	—	4	2	no	—
140	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
141	C₃×D₄	—	4	2	no	—
142	C₃×D₄	—	4	2	no	—
143	C₃×D₄	—	8	4	no	—
144	C₂×C₃×C₄	—	2	1	no	—
145	C₂×C₃×C₄	—	2	1	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
149	C₃×C₈	—	2	1	no	—
150	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
151	C₃×C₈	C₅×#41	1	1	yes	characteristic
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	2	1	no	—
154	C₂²×C₅	—	8	2	no	—
155	C₂²×C₅	—	8	2	no	—
156	C₂²×C₅	—	8	2	no	—
157	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
158	C₄×C₅	—	2	1	no	—
159	C₄×C₅	—	8	2	no	—
160	C₄×C₅	C₃×#121	1	1	yes	characteristic
161	C₄×C₅	C₃×#135	1	1	yes	characteristic
162	C₂×D₄	—	2	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
165	D₈	—	2	2	no	—
166	D₈	—	2	2	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
170	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
171	C₄⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	4	2	no	—
173	C₁₆	—	2	1	no	—
174	C₁₆	C₃×C₅×D₄	1	2	yes	complemented
175	C₃×C₅	#2218	1	1	yes	characteristic, direct product factor, Hall
176	C₂²×C₃	—	2	1	no	—
177	C₂²×C₃	—	2	2	no	—
178	C₂²×C₃	—	8	2	no	—
179	C₂²×C₃	—	8	2	no	—
180	C₂²×C₃	—	8	2	no	—
181	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
182	C₃×C₄	—	2	1	no	—
183	C₃×C₄	—	8	2	no	—
184	C₃×C₄	C₅×#121	1	1	yes	characteristic
185	C₃×C₄	C₅×#135	1	1	yes	characteristic
186	C₂×C₅	—	4	1	no	—
187	C₂×C₅	—	16	2	no	—
188	C₂×C₅	C₃×#466	1	1	yes	characteristic
189	C₂×C₅	C₃×#509	1	2	yes	—
190	C₂³	—	4	2	no	—
191	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
192	D₄	—	4	2	no	—
193	D₄	—	4	2	no	—
194	D₄	—	8	4	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
199	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
202	C₈	C₃×C₅×#41	1	1	yes	characteristic
203	C₂×C₃	—	4	1	no	—
204	C₂×C₃	—	16	2	no	—
205	C₂×C₃	C₅×#466	1	1	yes	characteristic
206	C₂×C₃	C₅×#509	1	2	yes	—
207	C₅	C₃×#2218	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	8	2	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	8	2	no	—
213	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	8	2	no	—
216	C₄	C₃×C₅×#121	1	1	yes	characteristic
217	C₄	C₃×C₅×#135	1	1	yes	characteristic
218	C₃	C₅×#2218	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	4	1	no	—
220	C₂	—	16	2	no	—
221	C₂	C₃×C₅×#466	1	1	yes	characteristic
222	C₂	C₃×C₅×#509	1	2	yes	—

	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. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₃×C₅×#466	C₂	semidirect
4. C₃×C₅×#367	C₂	semidirect
5. C₃×C₅×#357	C₂	semidirect
6. C₅×#2218	C₃	direct (N × H)
16. C₃×#2218	C₅	direct (N × H)
17. C₂×C₅×D₁₆	C₂×C₃	semidirect
18. C₂×C₅×M₃₂	C₂×C₃	semidirect
19. C₅×#466	C₂×C₃	semidirect
20. C₅×#367	C₂×C₃	semidirect
21. C₅×#357	C₂×C₃	semidirect
34. C₃×C₅×C₁₆	D₄	semidirect
35. C₂×C₃×D₁₆	C₂×C₅	semidirect
36. C₂×C₃×M₃₂	C₂×C₅	semidirect
37. C₃×#466	C₂×C₅	semidirect
38. C₃×#367	C₂×C₅	semidirect
39. C₃×#357	C₂×C₅	semidirect
49. #2218	C₃×C₅	direct (N × H)
84. C₅×C₁₆	C₃×D₄	semidirect
85. C₂×D₁₆	C₂×C₃×C₅	semidirect
86. C₂×M₃₂	C₂×C₃×C₅	semidirect
87. #466	C₂×C₃×C₅	semidirect
88. #367	C₂×C₃×C₅	semidirect
89. #357	C₂×C₃×C₅	semidirect
112. C₃×C₁₆	C₅×D₄	semidirect
174. C₁₆	C₃×C₅×D₄	semidirect
175. C₃×C₅	#2218	direct (N × H)
207. C₅	C₃×#2218	direct (N × H)
218. C₃	C₅×#2218	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
C₂×C₃×C₅×M₃₂	2
C₃×C₅×#466	2
C₃×C₅×#367	2
C₃×C₅×#357	2
C₅×#2218	3
C₃×#2218	5

Subgroups — C3×C5×#2218 #14275252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2218 #14275252