Subgroups — C₃×C₅×#2229 #14275263

Order 1920.

248 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#499	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#367	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#366	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₃×C₅×#358	C₂	1	1	yes	maximal, characteristic
7	C₃×C₅×#358	C₂	1	1	yes	maximal, characteristic
8	C₅×#2229	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₃×C₅×#125	—	2	1	no	—
11	C₃×C₄×C₅×Q₈	—	2	1	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	1	yes	characteristic
15	C₃×C₅×#107	C₂²	1	1	yes	characteristic
16	C₃×C₅×#107	C₂²	1	1	yes	characteristic
17	C₃×C₅×#106	—	2	1	no	—
18	C₃×C₅×#103	—	2	1	no	—
19	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₃×#2229	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₅×#499	C₂×C₃	1	1	yes	characteristic
22	C₄×C₅×Q₁₆	C₂×C₃	1	1	yes	characteristic
23	C₅×#367	C₂×C₃	1	1	yes	characteristic
24	C₅×#366	C₂×C₃	1	1	yes	characteristic
25	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic
26	C₅×#358	C₂×C₃	1	1	yes	characteristic
27	C₅×#358	C₂×C₃	1	1	yes	characteristic
28	C₂×C₃×C₅×Q₈	—	2	1	no	—
29	C₃×C₅×Q₁₆	—	2	1	no	—
30	C₃×C₅×Q₁₆	Q₈	1	2	yes	—
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	—	2	1	no	—
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	—	4	1	no	—
37	C₃×C₅×C₄⋊C₄	—	4	1	no	—
38	C₃×C₅×C₄⋊C₄	—	4	1	no	—
39	C₃×C₄²×C₅	—	4	1	no	—
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₅×C₁₆	—	2	1	no	—
42	C₃×C₅×C₁₆	—	2	1	no	—
43	C₃×#499	C₂×C₅	1	1	yes	characteristic
44	C₃×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
45	C₃×#367	C₂×C₅	1	1	yes	characteristic
46	C₃×#366	C₂×C₅	1	1	yes	characteristic
47	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
48	C₃×#358	C₂×C₅	1	1	yes	characteristic
49	C₃×#358	C₂×C₅	1	1	yes	characteristic
50	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
51	C₅×#125	—	2	1	no	—
52	C₄×C₅×Q₈	—	2	1	no	—
53	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
55	C₅×#107	C₂²×C₃	1	1	yes	characteristic
56	C₅×#107	C₂²×C₃	1	1	yes	characteristic
57	C₅×#107	C₂²×C₃	1	1	yes	characteristic
58	C₅×#106	—	2	1	no	—
59	C₅×#103	—	2	1	no	—
60	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
61	#2229	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
62	C₃×C₅×Q₈	—	2	2	no	—
63	C₃×C₅×Q₈	—	4	1	no	—
64	C₂×C₃×C₄×C₅	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	4	1	no	—
66	C₂×C₃×C₄×C₅	—	4	1	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×C₈	—	2	1	no	—
72	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₃×C₅×C₈	#41	1	1	yes	characteristic
74	C₂×C₃×Q₁₆	C₂²×C₅	1	1	yes	characteristic
75	C₃×#125	—	2	1	no	—
76	C₃×C₄×Q₈	—	2	1	no	—
77	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
79	C₃×#107	C₂²×C₅	1	1	yes	characteristic
80	C₃×#107	C₂²×C₅	1	1	yes	characteristic
81	C₃×#107	C₂²×C₅	1	1	yes	characteristic
82	C₃×#106	—	2	1	no	—
83	C₃×#103	—	2	1	no	—
84	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
85	C₂×C₅×Q₈	—	2	1	no	—
86	C₅×Q₁₆	—	2	1	no	—
87	C₅×Q₁₆	C₃×Q₈	1	2	yes	—
88	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
89	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	1	no	—
91	C₅×C₄⋊C₄	—	2	1	no	—
92	C₅×C₄⋊C₄	—	2	1	no	—
93	C₅×C₄⋊C₄	—	4	1	no	—
94	C₅×C₄⋊C₄	—	4	1	no	—
95	C₅×C₄⋊C₄	—	4	1	no	—
96	C₄²×C₅	—	4	1	no	—
97	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
98	C₅×C₁₆	—	2	1	no	—
99	C₅×C₁₆	—	2	1	no	—
100	#499	C₂×C₃×C₅	1	1	yes	characteristic
101	C₄×Q₁₆	C₂×C₃×C₅	1	1	yes	characteristic
102	#367	C₂×C₃×C₅	1	1	yes	characteristic
103	#366	C₂×C₃×C₅	1	1	yes	characteristic
104	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic
105	#358	C₂×C₃×C₅	1	1	yes	characteristic
106	#358	C₂×C₃×C₅	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	4	2	no	—
111	C₃×C₄×C₅	—	8	1	no	—
112	C₃×C₄×C₅	—	8	1	no	—
113	C₃×C₄×C₅	—	8	1	no	—
114	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
115	C₃×C₄×C₅	#137	1	1	yes	characteristic
116	C₂×C₃×Q₈	—	2	1	no	—
117	C₃×Q₁₆	—	2	1	no	—
118	C₃×Q₁₆	C₅×Q₈	1	2	yes	—
119	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
120	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
121	C₃×C₄⋊C₄	—	2	1	no	—
122	C₃×C₄⋊C₄	—	2	1	no	—
123	C₃×C₄⋊C₄	—	2	1	no	—
124	C₃×C₄⋊C₄	—	4	1	no	—
125	C₃×C₄⋊C₄	—	4	1	no	—
126	C₃×C₄⋊C₄	—	4	1	no	—
127	C₃×C₄²	—	4	1	no	—
128	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
129	C₃×C₁₆	—	2	1	no	—
130	C₃×C₁₆	—	2	1	no	—
131	C₅×Q₈	—	2	2	no	—
132	C₅×Q₈	—	4	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	—	4	1	no	—
135	C₂×C₄×C₅	—	4	1	no	—
136	C₂×C₄×C₅	—	4	1	no	—
137	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
138	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
140	C₅×C₈	—	2	1	no	—
141	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
142	C₅×C₈	C₃×#41	1	1	yes	characteristic
143	C₂×Q₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
144	#125	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
147	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
148	#107	C₂²×C₃×C₅	1	1	yes	characteristic
149	#107	C₂²×C₃×C₅	1	1	yes	characteristic
150	#107	C₂²×C₃×C₅	1	1	yes	characteristic
151	#106	—	2	1	no	—
152	#103	—	2	1	no	—
153	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
154	C₂×C₃×C₅	#474	1	1	yes	characteristic
155	C₂×C₃×C₅	#508	1	1	yes	characteristic
156	C₂×C₃×C₅	#510	1	1	yes	characteristic
157	C₃×Q₈	—	2	2	no	—
158	C₃×Q₈	—	4	1	no	—
159	C₂×C₃×C₄	—	4	1	no	—
160	C₂×C₃×C₄	—	4	1	no	—
161	C₂×C₃×C₄	—	4	1	no	—
162	C₂×C₃×C₄	—	4	1	no	—
163	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
164	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₃×C₈	—	2	1	no	—
167	C₃×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
168	C₃×C₈	C₅×#41	1	1	yes	characteristic
169	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
170	C₄×C₅	—	2	1	no	—
171	C₄×C₅	—	2	1	no	—
172	C₄×C₅	—	4	2	no	—
173	C₄×C₅	—	8	1	no	—
174	C₄×C₅	—	8	1	no	—
175	C₄×C₅	—	8	1	no	—
176	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
177	C₄×C₅	C₃×#137	1	1	yes	characteristic
178	C₂×Q₈	—	2	1	no	—
179	Q₁₆	—	2	1	no	—
180	Q₁₆	C₃×C₅×Q₈	1	2	yes	—
181	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
183	C₄⋊C₄	—	2	1	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	4	1	no	—
187	C₄⋊C₄	—	4	1	no	—
188	C₄⋊C₄	—	4	1	no	—
189	C₄²	—	4	1	no	—
190	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
191	C₁₆	—	2	1	no	—
192	C₁₆	—	2	1	no	—
193	C₃×C₅	#2229	1	1	yes	characteristic, direct product factor, Hall
194	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
195	C₃×C₄	—	2	1	no	—
196	C₃×C₄	—	2	1	no	—
197	C₃×C₄	—	4	2	no	—
198	C₃×C₄	—	8	1	no	—
199	C₃×C₄	—	8	1	no	—
200	C₃×C₄	—	8	1	no	—
201	C₃×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
202	C₃×C₄	C₅×#137	1	1	yes	characteristic
203	C₂×C₅	C₃×#474	1	1	yes	characteristic
204	C₂×C₅	C₃×#508	1	1	yes	characteristic
205	C₂×C₅	C₃×#510	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	1	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
213	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
214	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
217	C₈	C₃×C₅×#41	1	1	yes	characteristic
218	C₂×C₃	C₅×#474	1	1	yes	characteristic
219	C₂×C₃	C₅×#508	1	1	yes	characteristic
220	C₂×C₃	C₅×#510	1	1	yes	characteristic
221	C₅	C₃×#2229	1	1	yes	characteristic, direct product factor, Sylow
222	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
223	C₄	—	2	1	no	—
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	—	8	1	no	—
227	C₄	—	8	1	no	—
228	C₄	—	8	1	no	—
229	C₄	C₃×C₅×C₂²⋊Q₈	1	1	yes	characteristic
230	C₄	C₃×C₅×#137	1	1	yes	characteristic
231	C₃	C₅×#2229	1	1	yes	characteristic, direct product factor, Sylow
232	C₂	C₃×C₅×#474	1	1	yes	characteristic
233	C₂	C₃×C₅×#508	1	1	yes	characteristic
234	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
8. C₅×#2229	C₃	direct (N × H)
20. C₃×#2229	C₅	direct (N × H)
61. #2229	C₃×C₅	direct (N × H)
193. C₃×C₅	#2229	direct (N × H)
221. C₅	C₃×#2229	direct (N × H)
231. C₃	C₅×#2229	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×#499	2
C₃×C₄×C₅×Q₁₆	2
C₃×C₅×#367	2
C₃×C₅×#366	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×#358	2
C₅×#2229	3
C₃×#2229	5

Subgroups — C3×C5×#2229 #14275263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2229 #14275263