Subgroups — C₃×C₅×#2209 #14275243

Order 1920.

308 conjugacy classes of subgroups.

The subgroup lattice has 228 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	2	yes	maximal
2	C₃×C₅×#494	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#358	C₂	1	2	yes	maximal
6	C₅×#2209	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×Q₁₆	—	2	1	no	—
8	C₂×C₃×C₅×Q₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
11	C₃×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₅×Q₃₂	—	2	4	no	—
13	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
14	C₃×C₅×#107	C₂²	1	1	yes	characteristic
15	C₃×C₅×#103	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#2209	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×Q₃₂	C₂×C₃	1	2	yes	—
19	C₅×#494	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
21	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
22	C₅×#358	C₂×C₃	1	2	yes	—
23	C₂×C₃×C₅×Q₈	—	2	1	no	—
24	C₂×C₃×C₅×Q₈	—	2	2	no	—
25	C₃×C₅×Q₁₆	—	2	2	no	—
26	C₃×C₅×Q₁₆	—	2	1	no	—
27	C₃×C₅×Q₁₆	—	2	2	no	—
28	C₃×C₅×Q₁₆	—	4	2	no	—
29	C₃×C₅×Q₁₆	D₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	1	no	—
33	C₃×C₅×C₄⋊C₄	—	4	1	no	—
34	C₃×C₅×C₄⋊C₄	—	4	1	no	—
35	C₃×C₅×C₄⋊C₄	—	4	1	no	—
36	C₃×C₄²×C₅	—	4	1	no	—
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₅×C₁₆	—	2	2	no	—
39	C₂×C₃×Q₃₂	C₂×C₅	1	2	yes	—
40	C₃×#494	C₂×C₅	1	1	yes	characteristic
41	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
42	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
43	C₃×#358	C₂×C₅	1	2	yes	—
44	C₂×C₅×Q₁₆	—	2	1	no	—
45	C₂×C₅×Q₁₆	C₂²×C₃	1	2	yes	—
46	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
47	C₅×C₄⋊Q₈	—	2	1	no	—
48	C₄×C₅×Q₈	—	2	1	no	—
49	C₅×Q₃₂	—	2	4	no	—
50	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
51	C₅×#107	C₂²×C₃	1	1	yes	characteristic
52	C₅×#103	—	2	1	no	—
53	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
54	#2209	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
55	C₃×C₅×Q₈	—	2	2	no	—
56	C₃×C₅×Q₈	—	4	1	no	—
57	C₃×C₅×Q₈	—	4	2	no	—
58	C₃×C₅×Q₈	—	4	2	no	—
59	C₂×C₃×C₄×C₅	—	4	1	no	—
60	C₂×C₃×C₄×C₅	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₃×C₅×C₈	—	2	1	no	—
66	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×C₈	#41	1	1	yes	characteristic
68	C₂×C₃×Q₁₆	—	2	1	no	—
69	C₂×C₃×Q₁₆	C₂²×C₅	1	2	yes	—
70	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
71	C₃×C₄⋊Q₈	—	2	1	no	—
72	C₃×C₄×Q₈	—	2	1	no	—
73	C₃×Q₃₂	—	2	4	no	—
74	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
75	C₃×#107	C₂²×C₅	1	1	yes	characteristic
76	C₃×#103	—	2	1	no	—
77	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₅×Q₈	—	2	1	no	—
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₅×Q₁₆	—	2	2	no	—
81	C₅×Q₁₆	—	2	1	no	—
82	C₅×Q₁₆	—	2	2	no	—
83	C₅×Q₁₆	—	4	2	no	—
84	C₅×Q₁₆	C₃×D₄	1	2	yes	—
85	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
87	C₅×C₄⋊C₄	—	2	1	no	—
88	C₅×C₄⋊C₄	—	4	1	no	—
89	C₅×C₄⋊C₄	—	4	1	no	—
90	C₅×C₄⋊C₄	—	4	1	no	—
91	C₄²×C₅	—	4	1	no	—
92	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
93	C₅×C₁₆	—	2	2	no	—
94	C₂×Q₃₂	C₂×C₃×C₅	1	2	yes	—
95	#494	C₂×C₃×C₅	1	1	yes	characteristic
96	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
97	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
98	#358	C₂×C₃×C₅	1	2	yes	—
99	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
100	C₃×C₄×C₅	—	2	1	no	—
101	C₃×C₄×C₅	—	4	2	no	—
102	C₃×C₄×C₅	—	8	2	no	—
103	C₃×C₄×C₅	—	8	1	no	—
104	C₃×C₄×C₅	Q₃₂	1	2	yes	complemented
105	C₃×C₄×C₅	#121	1	1	yes	characteristic
106	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₃×Q₈	—	2	1	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₃×Q₁₆	—	2	2	no	—
110	C₃×Q₁₆	—	2	1	no	—
111	C₃×Q₁₆	—	2	2	no	—
112	C₃×Q₁₆	—	4	2	no	—
113	C₃×Q₁₆	C₅×D₄	1	2	yes	—
114	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
115	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
116	C₃×C₄⋊C₄	—	2	1	no	—
117	C₃×C₄⋊C₄	—	4	1	no	—
118	C₃×C₄⋊C₄	—	4	1	no	—
119	C₃×C₄⋊C₄	—	4	1	no	—
120	C₃×C₄²	—	4	1	no	—
121	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
122	C₃×C₁₆	—	2	2	no	—
123	C₅×Q₈	—	2	2	no	—
124	C₅×Q₈	—	4	1	no	—
125	C₅×Q₈	—	4	2	no	—
126	C₅×Q₈	—	4	2	no	—
127	C₂×C₄×C₅	—	4	1	no	—
128	C₂×C₄×C₅	—	4	1	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
131	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
132	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
133	C₅×C₈	—	2	1	no	—
134	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
135	C₅×C₈	C₃×#41	1	1	yes	characteristic
136	C₂×Q₁₆	—	2	1	no	—
137	C₂×Q₁₆	C₂²×C₃×C₅	1	2	yes	—
138	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
139	C₄⋊Q₈	—	2	1	no	—
140	C₄×Q₈	—	2	1	no	—
141	Q₃₂	—	2	4	no	—
142	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
143	#107	C₂²×C₃×C₅	1	1	yes	characteristic
144	#103	—	2	1	no	—
145	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
146	C₂×C₃×C₅	#459	1	1	yes	characteristic
147	C₂×C₃×C₅	C₂×Q₃₂	1	1	yes	characteristic
148	C₂×C₃×C₅	#510	1	1	yes	characteristic
149	C₃×Q₈	—	2	2	no	—
150	C₃×Q₈	—	4	1	no	—
151	C₃×Q₈	—	4	2	no	—
152	C₃×Q₈	—	4	2	no	—
153	C₂×C₃×C₄	—	4	1	no	—
154	C₂×C₃×C₄	—	4	1	no	—
155	C₂×C₃×C₄	—	4	2	no	—
156	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
157	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
158	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
159	C₃×C₈	—	2	1	no	—
160	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
161	C₃×C₈	C₅×#41	1	1	yes	characteristic
162	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
163	C₄×C₅	—	2	1	no	—
164	C₄×C₅	—	4	2	no	—
165	C₄×C₅	—	8	1	no	—
166	C₄×C₅	—	8	2	no	—
167	C₄×C₅	C₃×Q₃₂	1	2	yes	complemented
168	C₄×C₅	C₃×#121	1	1	yes	characteristic
169	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	2	no	—
172	Q₁₆	—	2	2	no	—
173	Q₁₆	—	2	1	no	—
174	Q₁₆	—	2	2	no	—
175	Q₁₆	—	4	2	no	—
176	Q₁₆	C₃×C₅×D₄	1	2	yes	—
177	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	4	1	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₄²	—	4	1	no	—
184	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
185	C₁₆	—	2	2	no	—
186	C₃×C₅	#2209	1	1	yes	characteristic, direct product factor, Hall
187	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
188	C₃×C₄	—	2	1	no	—
189	C₃×C₄	—	4	2	no	—
190	C₃×C₄	—	8	2	no	—
191	C₃×C₄	—	8	1	no	—
192	C₃×C₄	C₅×Q₃₂	1	2	yes	complemented
193	C₃×C₄	C₅×#121	1	1	yes	characteristic
194	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂×C₅	C₃×#459	1	1	yes	characteristic
196	C₂×C₅	C₂×C₃×Q₃₂	1	1	yes	characteristic
197	C₂×C₅	C₃×#510	1	1	yes	characteristic
198	Q₈	—	2	2	no	—
199	Q₈	—	4	1	no	—
200	Q₈	—	4	2	no	—
201	Q₈	—	4	2	no	—
202	C₂×C₄	—	4	1	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
206	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
207	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
208	C₈	—	2	1	no	—
209	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
210	C₈	C₃×C₅×#41	1	1	yes	characteristic
211	C₂×C₃	C₅×#459	1	1	yes	characteristic
212	C₂×C₃	C₂×C₅×Q₃₂	1	1	yes	characteristic
213	C₂×C₃	C₅×#510	1	1	yes	characteristic
214	C₅	C₃×#2209	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	4	2	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	1	no	—
220	C₄	C₃×C₅×Q₃₂	1	2	yes	complemented
221	C₄	C₃×C₅×#121	1	1	yes	characteristic
222	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₃	C₅×#2209	1	1	yes	characteristic, direct product factor, Sylow
224	C₂	C₃×C₅×#459	1	1	yes	characteristic
225	C₂	C₂×C₃×C₅×Q₃₂	1	1	yes	characteristic
226	C₂	C₃×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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 ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2209	C₃	direct (N × H)
17. C₃×#2209	C₅	direct (N × H)
54. #2209	C₃×C₅	direct (N × H)
104. C₃×C₄×C₅	Q₃₂	semidirect
167. C₄×C₅	C₃×Q₃₂	semidirect
186. C₃×C₅	#2209	direct (N × H)
192. C₃×C₄	C₅×Q₃₂	semidirect
214. C₅	C₃×#2209	direct (N × H)
220. C₄	C₃×C₅×Q₃₂	semidirect
223. C₃	C₅×#2209	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
C₃×C₅×#494	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#358	2
C₅×#2209	3
C₃×#2209	5

Subgroups — C3×C5×#2209 #14275243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2209 #14275243