Subgroups — C₃×C₅×#2226 #14275260

Order 1920.

284 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	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#367	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#357	C₂	1	2	yes	maximal
6	C₅×#2226	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₄×C₅×D₄	—	2	1	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₅×#107	—	2	1	no	—
12	C₃×C₅×#107	C₂²	1	2	yes	—
13	C₃×C₅×#107	C₂²	1	1	yes	characteristic
14	C₃×C₅×#102	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×#2226	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
18	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic
19	C₅×#367	C₂×C₃	1	2	yes	complemented
20	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#357	C₂×C₃	1	2	yes	—
22	C₂×C₃×C₅×Q₈	—	4	1	no	—
23	C₂×C₃×C₅×D₄	—	2	1	no	—
24	C₂²×C₃×C₄×C₅	—	4	1	no	—
25	C₃×C₅×D₈	—	2	1	no	—
26	C₃×C₅×D₈	Q₈	1	2	yes	complemented
27	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	2	1	no	—
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	4	1	no	—
32	C₃×C₅×C₂²⋊C₄	—	4	1	no	—
33	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
34	C₃×C₅×C₁₆	—	2	2	no	—
35	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
36	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
37	C₃×#367	C₂×C₅	1	2	yes	complemented
38	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#357	C₂×C₅	1	2	yes	—
40	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
41	C₅×C₄⋊Q₈	—	2	1	no	—
42	C₄×C₅×D₄	—	2	1	no	—
43	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
44	C₅×#107	—	2	1	no	—
45	C₅×#107	C₂²×C₃	1	2	yes	—
46	C₅×#107	C₂²×C₃	1	1	yes	characteristic
47	C₅×#102	—	2	1	no	—
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	#2226	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	4	1	no	—
51	C₃×C₅×Q₈	—	8	2	no	—
52	C₃×C₅×D₄	—	2	2	no	—
53	C₃×C₅×D₄	—	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	D₈	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
62	C₃×C₅×C₈	#41	1	1	yes	characteristic
63	C₂×C₃×D₈	C₂²×C₅	1	1	yes	characteristic
64	C₃×C₄⋊Q₈	—	2	1	no	—
65	C₃×C₄×D₄	—	2	1	no	—
66	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
67	C₃×#107	—	2	1	no	—
68	C₃×#107	C₂²×C₅	1	1	yes	characteristic
69	C₃×#107	C₂²×C₅	1	2	yes	—
70	C₃×#102	—	2	1	no	—
71	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₂×C₅×Q₈	—	4	1	no	—
73	C₂×C₅×D₄	—	2	1	no	—
74	C₂²×C₄×C₅	—	4	1	no	—
75	C₅×D₈	—	2	1	no	—
76	C₅×D₈	C₃×Q₈	1	2	yes	complemented
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	1	no	—
80	C₅×C₄⋊C₄	—	2	2	no	—
81	C₅×C₄⋊C₄	—	4	1	no	—
82	C₅×C₂²⋊C₄	—	4	1	no	—
83	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
84	C₅×C₁₆	—	2	2	no	—
85	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
86	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic
87	#367	C₂×C₃×C₅	1	2	yes	complemented
88	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
89	#357	C₂×C₃×C₅	1	2	yes	—
90	C₂²×C₃×C₅	—	4	2	no	—
91	C₂²×C₃×C₅	—	8	1	no	—
92	C₂²×C₃×C₅	—	8	1	no	—
93	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	—	8	1	no	—
96	C₃×C₄×C₅	—	8	2	no	—
97	C₃×C₄×C₅	SD₃₂	1	2	yes	—
98	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
99	C₃×C₄×C₅	#137	1	1	yes	characteristic
100	C₂×C₃×Q₈	—	4	1	no	—
101	C₂×C₃×D₄	—	2	1	no	—
102	C₂²×C₃×C₄	—	4	1	no	—
103	C₃×D₈	—	2	1	no	—
104	C₃×D₈	C₅×Q₈	1	2	yes	complemented
105	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
106	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
107	C₃×C₄⋊C₄	—	2	1	no	—
108	C₃×C₄⋊C₄	—	2	2	no	—
109	C₃×C₄⋊C₄	—	4	1	no	—
110	C₃×C₂²⋊C₄	—	4	1	no	—
111	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
112	C₃×C₁₆	—	2	2	no	—
113	C₂³×C₅	—	4	1	no	—
114	C₅×Q₈	—	8	2	no	—
115	C₅×D₄	—	2	2	no	—
116	C₅×D₄	—	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×D₈	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
125	C₅×C₈	C₃×#41	1	1	yes	characteristic
126	C₂×D₈	C₂²×C₃×C₅	1	1	yes	characteristic
127	C₄⋊Q₈	—	2	1	no	—
128	C₄×D₄	—	2	1	no	—
129	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
130	#107	—	2	1	no	—
131	#107	C₂²×C₃×C₅	1	2	yes	—
132	#107	C₂²×C₃×C₅	1	1	yes	characteristic
133	#102	—	2	1	no	—
134	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₂×C₃×C₅	—	8	2	no	—
136	C₂×C₃×C₅	#474	1	1	yes	characteristic
137	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
138	C₂×C₃×C₅	#509	1	1	yes	characteristic
139	C₂³×C₃	—	4	1	no	—
140	C₃×Q₈	—	8	2	no	—
141	C₃×D₄	—	2	2	no	—
142	C₃×D₄	—	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×D₈	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
151	C₃×C₈	C₅×#41	1	1	yes	characteristic
152	C₂²×C₅	—	4	2	no	—
153	C₂²×C₅	—	8	1	no	—
154	C₂²×C₅	—	8	1	no	—
155	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
156	C₄×C₅	—	2	1	no	—
157	C₄×C₅	—	8	1	no	—
158	C₄×C₅	—	8	2	no	—
159	C₄×C₅	C₃×SD₃₂	1	2	yes	—
160	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₅	C₃×#137	1	1	yes	characteristic
162	C₂×Q₈	—	4	1	no	—
163	C₂×D₄	—	2	1	no	—
164	C₂²×C₄	—	4	1	no	—
165	D₈	—	2	1	no	—
166	D₈	C₃×C₅×Q₈	1	2	yes	complemented
167	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
168	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₄⋊C₄	—	4	1	no	—
172	C₂²⋊C₄	—	4	1	no	—
173	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
174	C₁₆	—	2	2	no	—
175	C₃×C₅	#2226	1	1	yes	characteristic, direct product factor, Hall
176	C₂²×C₃	—	4	2	no	—
177	C₂²×C₃	—	8	1	no	—
178	C₂²×C₃	—	8	1	no	—
179	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
180	C₃×C₄	—	2	1	no	—
181	C₃×C₄	—	8	1	no	—
182	C₃×C₄	—	8	2	no	—
183	C₃×C₄	C₅×SD₃₂	1	2	yes	—
184	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
185	C₃×C₄	C₅×#137	1	1	yes	characteristic
186	C₂×C₅	—	8	2	no	—
187	C₂×C₅	C₃×#474	1	1	yes	characteristic
188	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
189	C₂×C₅	C₃×#509	1	1	yes	characteristic
190	C₂³	—	4	1	no	—
191	Q₈	—	8	2	no	—
192	D₄	—	2	2	no	—
193	D₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
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×Q₈	1	1	yes	characteristic
202	C₈	C₃×C₅×#41	1	1	yes	characteristic
203	C₂×C₃	—	8	2	no	—
204	C₂×C₃	C₅×#474	1	1	yes	characteristic
205	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
206	C₂×C₃	C₅×#509	1	1	yes	characteristic
207	C₅	C₃×#2226	1	1	yes	characteristic, direct product factor, Sylow
208	C₂²	—	4	2	no	—
209	C₂²	—	8	1	no	—
210	C₂²	—	8	1	no	—
211	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	8	1	no	—
214	C₄	—	8	2	no	—
215	C₄	C₃×C₅×SD₃₂	1	2	yes	—
216	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
217	C₄	C₃×C₅×#137	1	1	yes	characteristic
218	C₃	C₅×#2226	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	8	2	no	—
220	C₂	C₃×C₅×#474	1	1	yes	characteristic
221	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic
222	C₂	C₃×C₅×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂×C₃×C₅×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#500	C₂	semidirect
3. C₃×C₅×#367	C₂	semidirect
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₅×#2226	C₃	direct (N × H)
16. C₃×#2226	C₅	direct (N × H)
17. C₅×#500	C₂×C₃	semidirect
19. C₅×#367	C₂×C₃	semidirect
20. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
26. C₃×C₅×D₈	Q₈	semidirect
35. C₃×#500	C₂×C₅	semidirect
37. C₃×#367	C₂×C₅	semidirect
38. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
49. #2226	C₃×C₅	direct (N × H)
76. C₅×D₈	C₃×Q₈	semidirect
85. #500	C₂×C₃×C₅	semidirect
87. #367	C₂×C₃×C₅	semidirect
88. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
104. C₃×D₈	C₅×Q₈	semidirect
166. D₈	C₃×C₅×Q₈	semidirect
175. C₃×C₅	#2226	direct (N × H)
207. C₅	C₃×#2226	direct (N × H)
218. C₃	C₅×#2226	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×C₅×#500	2
C₃×C₄×C₅×D₈	2
C₃×C₅×#367	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#357	2
C₅×#2226	3
C₃×#2226	5

Subgroups — C3×C5×#2226 #14275260

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2226 #14275260