Subgroups — C₃×C₅×#2219 #14275253

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#467	C₂	1	2	yes	maximal
4	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#358	C₂	1	2	yes	maximal, complemented
6	C₅×#2219	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₂²⋊Q₈	—	2	2	no	—
10	C₃×C₅×Q₃₂	—	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×#103	—	2	2	no	—
16	C₃×#2219	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×Q₃₂	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₅×M₃₂	C₂×C₃	1	1	yes	characteristic
19	C₅×#467	C₂×C₃	1	2	yes	—
20	C₅×#367	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×#358	C₂×C₃	1	2	yes	complemented
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₃×C₅×Q₁₆	—	2	2	no	—
25	C₃×C₅×Q₁₆	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	1	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
29	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	2	no	—
31	C₃×C₅×C₄⋊C₄	—	4	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	—
35	C₂×C₃×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
36	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic
37	C₃×#467	C₂×C₅	1	2	yes	—
38	C₃×#367	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×#358	C₂×C₅	1	2	yes	complemented
40	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×C₂²⋊Q₈	—	2	2	no	—
43	C₅×Q₃₂	—	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×#103	—	2	2	no	—
49	#2219	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
51	C₃×C₅×Q₈	—	4	2	no	—
52	C₃×C₅×Q₈	—	4	2	no	—
53	C₂×C₃×C₄×C₅	—	2	1	no	—
54	C₂×C₃×C₄×C₅	—	2	1	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
63	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₃×C₂²⋊Q₈	—	2	2	no	—
65	C₃×Q₃₂	—	2	2	no	—
66	C₃×M₃₂	—	2	2	no	—
67	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
69	C₃×#107	C₂²×C₅	1	2	yes	—
70	C₃×#103	—	2	2	no	—
71	C₂×C₅×Q₈	—	2	2	no	—
72	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
73	C₅×Q₁₆	—	2	2	no	—
74	C₅×Q₁₆	—	2	2	no	—
75	C₂×C₅×C₈	—	2	1	no	—
76	C₂×C₅×C₈	—	2	1	no	—
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	2	no	—
80	C₅×C₄⋊C₄	—	4	2	no	—
81	C₅×C₂²⋊C₄	—	4	2	no	—
82	C₅×C₁₆	—	2	1	no	—
83	C₅×C₁₆	C₃×D₄	1	2	yes	—
84	C₂×Q₃₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
85	C₂×M₃₂	C₂×C₃×C₅	1	1	yes	characteristic
86	#467	C₂×C₃×C₅	1	2	yes	—
87	#367	C₂×C₃×C₅	1	1	yes	characteristic, complemented
88	#358	C₂×C₃×C₅	1	2	yes	complemented
89	C₂²×C₃×C₅	—	2	1	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	1	no	—
93	C₃×C₄×C₅	—	8	2	no	—
94	C₃×C₄×C₅	—	8	2	no	—
95	C₃×C₄×C₅	#121	1	1	yes	characteristic
96	C₃×C₄×C₅	#135	1	1	yes	characteristic
97	C₂×C₃×Q₈	—	2	2	no	—
98	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
99	C₃×Q₁₆	—	2	2	no	—
100	C₃×Q₁₆	—	2	2	no	—
101	C₂×C₃×C₈	—	2	1	no	—
102	C₂×C₃×C₈	—	2	1	no	—
103	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
104	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₄⋊C₄	—	4	2	no	—
107	C₃×C₂²⋊C₄	—	4	2	no	—
108	C₃×C₁₆	—	2	1	no	—
109	C₃×C₁₆	C₅×D₄	1	2	yes	—
110	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
111	C₅×Q₈	—	4	2	no	—
112	C₅×Q₈	—	4	2	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	—	4	2	no	—
117	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
119	C₅×C₈	—	2	1	no	—
120	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
121	C₅×C₈	C₃×#41	1	1	yes	characteristic
122	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
123	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
124	C₂²⋊Q₈	—	2	2	no	—
125	Q₃₂	—	2	2	no	—
126	M₃₂	—	2	2	no	—
127	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
128	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
129	#107	C₂²×C₃×C₅	1	2	yes	—
130	#103	—	2	2	no	—
131	C₂×C₃×C₅	—	4	1	no	—
132	C₂×C₃×C₅	#466	1	1	yes	characteristic
133	C₂×C₃×C₅	#510	1	2	yes	—
134	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
135	C₃×Q₈	—	4	2	no	—
136	C₃×Q₈	—	4	2	no	—
137	C₂×C₃×C₄	—	2	1	no	—
138	C₂×C₃×C₄	—	2	1	no	—
139	C₂×C₃×C₄	—	4	2	no	—
140	C₂×C₃×C₄	—	4	2	no	—
141	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
142	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
143	C₃×C₈	—	2	1	no	—
144	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
145	C₃×C₈	C₅×#41	1	1	yes	characteristic
146	C₂²×C₅	—	2	2	no	—
147	C₂²×C₅	—	2	1	no	—
148	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	—	8	2	no	—
152	C₄×C₅	C₃×#121	1	1	yes	characteristic
153	C₄×C₅	C₃×#135	1	1	yes	characteristic
154	C₂×Q₈	—	2	2	no	—
155	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
156	Q₁₆	—	2	2	no	—
157	Q₁₆	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
161	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	4	2	no	—
164	C₂²⋊C₄	—	4	2	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	C₃×C₅×D₄	1	2	yes	—
167	C₃×C₅	#2219	1	1	yes	characteristic, direct product factor, Hall
168	C₂²×C₃	—	2	1	no	—
169	C₂²×C₃	—	2	2	no	—
170	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
171	C₃×C₄	—	2	1	no	—
172	C₃×C₄	—	8	2	no	—
173	C₃×C₄	—	8	2	no	—
174	C₃×C₄	C₅×#121	1	1	yes	characteristic
175	C₃×C₄	C₅×#135	1	1	yes	characteristic
176	C₂×C₅	—	4	1	no	—
177	C₂×C₅	C₃×#466	1	1	yes	characteristic
178	C₂×C₅	C₃×#510	1	2	yes	—
179	C₂³	C₃×C₅×D₈	1	1	yes	characteristic
180	Q₈	—	4	2	no	—
181	Q₈	—	4	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
187	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
188	C₈	—	2	1	no	—
189	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
190	C₈	C₃×C₅×#41	1	1	yes	characteristic
191	C₂×C₃	—	4	1	no	—
192	C₂×C₃	C₅×#466	1	1	yes	characteristic
193	C₂×C₃	C₅×#510	1	2	yes	—
194	C₅	C₃×#2219	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²	—	2	1	no	—
196	C₂²	—	2	2	no	—
197	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	8	2	no	—
200	C₄	—	8	2	no	—
201	C₄	C₃×C₅×#121	1	1	yes	characteristic
202	C₄	C₃×C₅×#135	1	1	yes	characteristic
203	C₃	C₅×#2219	1	1	yes	characteristic, direct product factor, Sylow
204	C₂	—	4	1	no	—
205	C₂	C₃×C₅×#466	1	1	yes	characteristic
206	C₂	C₃×C₅×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₂³×C₃×C₅	D₈
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×Q₃₂	C₂	semidirect
4. C₃×C₅×#367	C₂	semidirect
5. C₃×C₅×#358	C₂	semidirect
6. C₅×#2219	C₃	direct (N × H)
16. C₃×#2219	C₅	direct (N × H)
17. C₂×C₅×Q₃₂	C₂×C₃	semidirect
20. C₅×#367	C₂×C₃	semidirect
21. C₅×#358	C₂×C₃	semidirect
35. C₂×C₃×Q₃₂	C₂×C₅	semidirect
38. C₃×#367	C₂×C₅	semidirect
39. C₃×#358	C₂×C₅	semidirect
49. #2219	C₃×C₅	direct (N × H)
84. C₂×Q₃₂	C₂×C₃×C₅	semidirect
87. #367	C₂×C₃×C₅	semidirect
88. #358	C₂×C₃×C₅	semidirect
167. C₃×C₅	#2219	direct (N × H)
194. C₅	C₃×#2219	direct (N × H)
203. C₃	C₅×#2219	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×Q₃₂	2
C₂×C₃×C₅×M₃₂	2
C₃×C₅×#467	2
C₃×C₅×#367	2
C₃×C₅×#358	2
C₅×#2219	3
C₃×#2219	5

Subgroups — C3×C5×#2219 #14275253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2219 #14275253