Subgroups — C₃×C₅×#2221 #14275255

Order 1920.

296 conjugacy classes of subgroups.

The subgroup lattice has 220 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	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#472	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#361	C₂	1	2	yes	maximal, complemented
6	C₅×#2221	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₅×AGL(1,ℤ/8)	—	2	2	no	—
8	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
9	C₃×C₅×#131	C₂²	1	1	yes	characteristic
10	C₃×C₅×D₁₆	—	2	1	no	—
11	C₃×C₅×D₁₆	—	2	1	no	—
12	C₃×C₅×M₃₂	—	2	1	no	—
13	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₁₆	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₃×C₅×#108	C₂²	1	2	yes	—
17	C₃×C₅×#100	—	2	2	no	—
18	C₃×#2221	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#504	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#472	C₂×C₃	1	2	yes	complemented
22	C₅×#368	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#361	C₂×C₃	1	2	yes	complemented
24	C₃×C₅×#41	D₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₃×C₅×SD₁₆	—	4	2	no	—
27	C₃×C₅×D₈	—	2	2	no	—
28	C₃×C₅×D₈	—	2	2	no	—
29	C₃×C₅×D₈	—	4	2	no	—
30	C₃×C₅×M₁₆	—	2	1	no	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	D₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₃×C₅×C₁₆	—	2	1	no	—
36	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
37	C₃×C₅×C₁₆	D₄	1	1	yes	characteristic
38	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#504	C₂×C₅	1	1	yes	characteristic, complemented
40	C₃×#472	C₂×C₅	1	2	yes	complemented
41	C₃×#368	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃×#361	C₂×C₅	1	2	yes	complemented
43	C₅×AGL(1,ℤ/8)	—	2	2	no	—
44	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
45	C₅×#131	C₂²×C₃	1	1	yes	characteristic
46	C₅×D₁₆	—	2	1	no	—
47	C₅×D₁₆	—	2	1	no	—
48	C₅×M₃₂	—	2	1	no	—
49	C₅×M₃₂	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×C₁₆	—	2	1	no	—
51	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
52	C₅×#108	C₂²×C₃	1	2	yes	—
53	C₅×#100	—	2	2	no	—
54	#2221	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₃×C₅	—	4	2	no	—
56	C₃×C₅×Q₈	D₈	1	1	yes	characteristic
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	—	4	2	no	—
59	C₃×C₅×D₄	—	4	2	no	—
60	C₃×C₅×D₄	D₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
66	C₃×C₅×C₈	#41	1	1	yes	characteristic
67	C₃×AGL(1,ℤ/8)	—	2	2	no	—
68	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
69	C₃×#131	C₂²×C₅	1	1	yes	characteristic
70	C₃×D₁₆	—	2	1	no	—
71	C₃×D₁₆	—	2	1	no	—
72	C₃×M₃₂	—	2	1	no	—
73	C₃×M₃₂	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×C₁₆	—	2	1	no	—
75	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
76	C₃×#108	C₂²×C₅	1	2	yes	—
77	C₃×#100	—	2	2	no	—
78	C₅×#41	C₃×D₄	1	1	yes	characteristic
79	C₂×C₅×D₄	—	2	2	no	—
80	C₅×SD₁₆	—	4	2	no	—
81	C₅×D₈	—	2	2	no	—
82	C₅×D₈	—	2	2	no	—
83	C₅×D₈	—	4	2	no	—
84	C₅×M₁₆	—	2	1	no	—
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	C₃×D₄	1	1	yes	characteristic
87	C₂×C₅×C₈	—	2	1	no	—
88	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
89	C₅×C₁₆	—	2	1	no	—
90	C₅×C₁₆	C₃×D₄	1	1	yes	characteristic
91	C₅×C₁₆	C₃×D₄	1	1	yes	characteristic
92	C₂×D₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
93	#504	C₂×C₃×C₅	1	1	yes	characteristic, complemented
94	#472	C₂×C₃×C₅	1	2	yes	complemented
95	#368	C₂×C₃×C₅	1	1	yes	characteristic, complemented
96	#361	C₂×C₃×C₅	1	2	yes	complemented
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	8	2	no	—
99	C₂²×C₃×C₅	—	16	2	no	—
100	C₂²×C₃×C₅	#135	1	1	yes	characteristic
101	C₃×C₄×C₅	—	2	1	no	—
102	C₃×C₄×C₅	#121	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
104	C₃×#41	C₅×D₄	1	1	yes	characteristic
105	C₂×C₃×D₄	—	2	2	no	—
106	C₃×SD₁₆	—	4	2	no	—
107	C₃×D₈	—	2	2	no	—
108	C₃×D₈	—	2	2	no	—
109	C₃×D₈	—	4	2	no	—
110	C₃×M₁₆	—	2	1	no	—
111	C₃×M₁₆	—	2	2	no	—
112	C₃×M₁₆	C₅×D₄	1	1	yes	characteristic
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
115	C₃×C₁₆	—	2	1	no	—
116	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
117	C₃×C₁₆	C₅×D₄	1	1	yes	characteristic
118	C₂³×C₅	—	4	2	no	—
119	C₅×Q₈	C₃×D₈	1	1	yes	characteristic
120	C₅×D₄	—	2	1	no	—
121	C₅×D₄	—	4	2	no	—
122	C₅×D₄	—	4	2	no	—
123	C₅×D₄	C₃×D₈	1	1	yes	characteristic
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₅×C₈	—	2	1	no	—
127	C₅×C₈	—	4	2	no	—
128	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
129	C₅×C₈	C₃×#41	1	1	yes	characteristic
130	AGL(1,ℤ/8)	—	2	2	no	—
131	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
132	#131	C₂²×C₃×C₅	1	1	yes	characteristic
133	D₁₆	—	2	1	no	—
134	D₁₆	—	2	1	no	—
135	M₃₂	—	2	1	no	—
136	M₃₂	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
139	#108	C₂²×C₃×C₅	1	2	yes	—
140	#100	—	2	2	no	—
141	C₂×C₃×C₅	—	2	1	no	—
142	C₂×C₃×C₅	—	4	1	no	—
143	C₂×C₃×C₅	—	16	2	no	—
144	C₂×C₃×C₅	#466	1	1	yes	characteristic
145	C₂³×C₃	—	4	2	no	—
146	C₃×Q₈	C₅×D₈	1	1	yes	characteristic
147	C₃×D₄	—	2	1	no	—
148	C₃×D₄	—	4	2	no	—
149	C₃×D₄	—	4	2	no	—
150	C₃×D₄	C₅×D₈	1	1	yes	characteristic
151	C₂×C₃×C₄	—	2	1	no	—
152	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
153	C₃×C₈	—	2	1	no	—
154	C₃×C₈	—	4	2	no	—
155	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
156	C₃×C₈	C₅×#41	1	1	yes	characteristic
157	C₂²×C₅	—	2	1	no	—
158	C₂²×C₅	—	8	2	no	—
159	C₂²×C₅	—	16	2	no	—
160	C₂²×C₅	C₃×#135	1	1	yes	characteristic
161	C₄×C₅	—	2	1	no	—
162	C₄×C₅	C₃×#121	1	1	yes	characteristic
163	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
164	#41	C₃×C₅×D₄	1	1	yes	characteristic
165	C₂×D₄	—	2	2	no	—
166	SD₁₆	—	4	2	no	—
167	D₈	—	2	2	no	—
168	D₈	—	2	2	no	—
169	D₈	—	4	2	no	—
170	M₁₆	—	2	1	no	—
171	M₁₆	—	2	2	no	—
172	M₁₆	C₃×C₅×D₄	1	1	yes	characteristic
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
175	C₁₆	—	2	1	no	—
176	C₁₆	C₃×C₅×D₄	1	1	yes	characteristic
177	C₁₆	C₃×C₅×D₄	1	1	yes	characteristic
178	C₃×C₅	#2221	1	1	yes	characteristic, direct product factor, Hall
179	C₂²×C₃	—	2	1	no	—
180	C₂²×C₃	—	8	2	no	—
181	C₂²×C₃	—	16	2	no	—
182	C₂²×C₃	C₅×#135	1	1	yes	characteristic
183	C₃×C₄	—	2	1	no	—
184	C₃×C₄	C₅×#121	1	1	yes	characteristic
185	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
186	C₂×C₅	—	2	1	no	—
187	C₂×C₅	—	4	1	no	—
188	C₂×C₅	—	16	2	no	—
189	C₂×C₅	C₃×#466	1	1	yes	characteristic
190	C₂³	—	4	2	no	—
191	Q₈	C₃×C₅×D₈	1	1	yes	characteristic
192	D₄	—	2	1	no	—
193	D₄	—	4	2	no	—
194	D₄	—	4	2	no	—
195	D₄	C₃×C₅×D₈	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
198	C₈	—	2	1	no	—
199	C₈	—	4	2	no	—
200	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
201	C₈	C₃×C₅×#41	1	1	yes	characteristic
202	C₂×C₃	—	2	1	no	—
203	C₂×C₃	—	4	1	no	—
204	C₂×C₃	—	16	2	no	—
205	C₂×C₃	C₅×#466	1	1	yes	characteristic
206	C₅	C₃×#2221	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	2	1	no	—
208	C₂²	—	8	2	no	—
209	C₂²	—	16	2	no	—
210	C₂²	C₃×C₅×#135	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	C₃×C₅×#121	1	1	yes	characteristic
213	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
214	C₃	C₅×#2221	1	1	yes	characteristic, direct product factor, Sylow
215	C₂	—	2	1	no	—
216	C₂	—	4	1	no	—
217	C₂	—	16	2	no	—
218	C₂	C₃×C₅×#466	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₅	#466
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₅×#131 ◁ 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. C₃×C₅×#504	C₂	semidirect
3. C₃×C₅×#472	C₂	semidirect
4. C₃×C₅×#368	C₂	semidirect
5. C₃×C₅×#361	C₂	semidirect
6. C₅×#2221	C₃	direct (N × H)
18. C₃×#2221	C₅	direct (N × H)
19. C₂×C₅×D₁₆	C₂×C₃	semidirect
20. C₅×#504	C₂×C₃	semidirect
21. C₅×#472	C₂×C₃	semidirect
22. C₅×#368	C₂×C₃	semidirect
23. C₅×#361	C₂×C₃	semidirect
38. C₂×C₃×D₁₆	C₂×C₅	semidirect
39. C₃×#504	C₂×C₅	semidirect
40. C₃×#472	C₂×C₅	semidirect
41. C₃×#368	C₂×C₅	semidirect
42. C₃×#361	C₂×C₅	semidirect
54. #2221	C₃×C₅	direct (N × H)
92. C₂×D₁₆	C₂×C₃×C₅	semidirect
93. #504	C₂×C₃×C₅	semidirect
94. #472	C₂×C₃×C₅	semidirect
95. #368	C₂×C₃×C₅	semidirect
96. #361	C₂×C₃×C₅	semidirect
178. C₃×C₅	#2221	direct (N × H)
206. C₅	C₃×#2221	direct (N × H)
214. C₃	C₅×#2221	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
C₃×C₅×#504	2
C₃×C₅×#472	2
C₃×C₅×#368	2
C₃×C₅×#361	2
C₅×#2221	3
C₃×#2221	5

Subgroups — C3×C5×#2221 #14275255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2221 #14275255