Subgroups — C₃×C₅×#2351 #14275385

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#389	C₂	1	6	yes	maximal
4	C₃×C₅×#382	C₂	1	3	yes	maximal
5	C₃×C₄×C₅×C₄⋊C₄	C₂	1	3	yes	maximal
6	C₅×#2351	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	—	2	8	no	—
9	C₃×C₄×C₅×Q₈	C₄	1	8	yes	complemented
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	6	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	3	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	6	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	—
15	C₃×C₅×#95	C₂²	1	1	yes	characteristic
16	C₃×C₅×#95	C₂²	1	3	yes	—
17	C₃×C₅×#95	C₂²	1	6	yes	—
18	C₃×#2351	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₄×C₅×Q₈	C₂×C₃	1	2	yes	—
20	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
21	C₅×#389	C₂×C₃	1	6	yes	—
22	C₅×#382	C₂×C₃	1	3	yes	—
23	C₄×C₅×C₄⋊C₄	C₂×C₃	1	3	yes	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₃×C₅×Q₈	D₄	1	4	yes	—
26	C₂×C₃×C₅×Q₈	Q₈	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	6	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
32	C₃×C₅×C₄⋊C₄	—	2	12	no	—
33	C₃×C₅×C₄⋊C₄	—	2	12	no	—
34	C₃×C₅×C₄⋊C₄	C₂×C₄	1	12	yes	—
35	C₃×C₄²×C₅	—	2	12	no	—
36	C₃×C₄²×C₅	C₂×C₄	1	12	yes	—
37	C₂×C₃×C₄×Q₈	C₂×C₅	1	2	yes	—
38	C₂×C₃×C₄×Q₈	C₂×C₅	1	1	yes	characteristic
39	C₃×#389	C₂×C₅	1	6	yes	—
40	C₃×#382	C₂×C₅	1	3	yes	—
41	C₃×C₄×C₄⋊C₄	C₂×C₅	1	3	yes	—
42	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₄×C₅×Q₈	—	2	8	no	—
44	C₄×C₅×Q₈	C₃×C₄	1	8	yes	complemented
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
46	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	6	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	3	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	6	yes	—
49	C₂×C₄²×C₅	C₂²×C₃	1	3	yes	—
50	C₅×#95	C₂²×C₃	1	1	yes	characteristic
51	C₅×#95	C₂²×C₃	1	3	yes	—
52	C₅×#95	C₂²×C₃	1	6	yes	—
53	#2351	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
54	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
55	C₃×C₅×Q₈	C₄⋊C₄	1	16	yes	complemented
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	—	2	6	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	6	no	—
61	C₂×C₃×C₄×C₅	—	2	6	no	—
62	C₂×C₃×C₄×C₅	—	2	6	no	—
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	6	yes	—
68	C₂×C₃×C₄×C₅	#41	1	4	yes	—
69	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
70	C₃×C₄×Q₈	—	2	8	no	—
71	C₃×C₄×Q₈	C₄×C₅	1	8	yes	complemented
72	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
73	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	6	yes	—
74	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	3	yes	—
75	C₂×C₃×C₄²	C₂²×C₅	1	6	yes	—
76	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	—
77	C₃×#95	C₂²×C₅	1	1	yes	characteristic
78	C₃×#95	C₂²×C₅	1	3	yes	—
79	C₃×#95	C₂²×C₅	1	6	yes	—
80	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
81	C₂×C₅×Q₈	C₃×D₄	1	4	yes	—
82	C₂×C₅×Q₈	C₃×Q₈	1	4	yes	—
83	C₂²×C₄×C₅	C₂³×C₃	1	6	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
86	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
87	C₂²×C₄×C₅	C₂³×C₃	1	3	yes	—
88	C₅×C₄⋊C₄	—	2	12	no	—
89	C₅×C₄⋊C₄	—	2	12	no	—
90	C₅×C₄⋊C₄	C₂×C₃×C₄	1	12	yes	—
91	C₄²×C₅	—	2	12	no	—
92	C₄²×C₅	C₂×C₃×C₄	1	12	yes	—
93	C₂×C₄×Q₈	C₂×C₃×C₅	1	2	yes	—
94	C₂×C₄×Q₈	C₂×C₃×C₅	1	1	yes	characteristic
95	#389	C₂×C₃×C₅	1	6	yes	—
96	#382	C₂×C₃×C₅	1	3	yes	—
97	C₄×C₄⋊C₄	C₂×C₃×C₅	1	3	yes	—
98	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
99	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
102	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
103	C₃×C₄×C₅	—	2	6	no	—
104	C₃×C₄×C₅	—	2	2	no	—
105	C₃×C₄×C₅	—	2	4	no	—
106	C₃×C₄×C₅	—	4	6	no	—
107	C₃×C₄×C₅	C₂×C₄⋊C₄	1	12	yes	—
108	C₂×C₃×Q₈	C₂×C₄×C₅	1	4	yes	—
109	C₂×C₃×Q₈	C₅×D₄	1	4	yes	—
110	C₂×C₃×Q₈	C₅×Q₈	1	4	yes	—
111	C₂²×C₃×C₄	C₂³×C₅	1	6	yes	—
112	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
114	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
115	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
116	C₃×C₄⋊C₄	—	2	12	no	—
117	C₃×C₄⋊C₄	—	2	12	no	—
118	C₃×C₄⋊C₄	C₂×C₄×C₅	1	12	yes	—
119	C₃×C₄²	—	2	12	no	—
120	C₃×C₄²	C₂×C₄×C₅	1	12	yes	—
121	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
122	C₅×Q₈	C₃×C₄⋊C₄	1	16	yes	complemented
123	C₂×C₄×C₅	—	2	2	no	—
124	C₂×C₄×C₅	—	2	6	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	6	no	—
128	C₂×C₄×C₅	—	2	6	no	—
129	C₂×C₄×C₅	—	2	6	no	—
130	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
131	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
132	C₂×C₄×C₅	C₂²×C₃×C₄	1	6	yes	—
133	C₂×C₄×C₅	C₂×C₃×D₄	1	6	yes	—
134	C₂×C₄×C₅	C₂×C₃×Q₈	1	6	yes	—
135	C₂×C₄×C₅	C₃×#41	1	4	yes	—
136	C₂²×Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₄×Q₈	—	2	8	no	—
138	C₄×Q₈	C₃×C₄×C₅	1	8	yes	complemented
139	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
140	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	6	yes	—
141	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	3	yes	—
142	C₂×C₄²	C₂²×C₃×C₅	1	6	yes	—
143	C₂×C₄²	C₂²×C₃×C₅	1	3	yes	—
144	#95	C₂²×C₃×C₅	1	1	yes	characteristic
145	#95	C₂²×C₃×C₅	1	3	yes	—
146	#95	C₂²×C₃×C₅	1	6	yes	—
147	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
149	C₂×C₃×C₅	#519	1	1	yes	characteristic
150	C₂×C₃×C₅	#548	1	2	yes	—
151	C₂×C₃×C₅	#557	1	2	yes	—
152	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
153	C₃×Q₈	C₅×C₄⋊C₄	1	16	yes	complemented
154	C₂×C₃×C₄	—	2	2	no	—
155	C₂×C₃×C₄	—	2	6	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	2	6	no	—
159	C₂×C₃×C₄	—	2	6	no	—
160	C₂×C₃×C₄	—	2	6	no	—
161	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
162	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
163	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
164	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
165	C₂×C₃×C₄	C₂×C₅×Q₈	1	6	yes	—
166	C₂×C₃×C₄	C₅×#41	1	4	yes	—
167	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
168	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
169	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
170	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
171	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
172	C₄×C₅	—	2	6	no	—
173	C₄×C₅	—	2	2	no	—
174	C₄×C₅	—	2	4	no	—
175	C₄×C₅	—	4	6	no	—
176	C₄×C₅	C₂×C₃×C₄⋊C₄	1	12	yes	—
177	C₂×Q₈	C₂×C₃×C₄×C₅	1	4	yes	—
178	C₂×Q₈	C₃×C₅×D₄	1	4	yes	—
179	C₂×Q₈	C₃×C₅×Q₈	1	4	yes	—
180	C₂²×C₄	C₂³×C₃×C₅	1	6	yes	—
181	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
182	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
183	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
184	C₂²×C₄	C₂³×C₃×C₅	1	3	yes	—
185	C₄⋊C₄	—	2	12	no	—
186	C₄⋊C₄	—	2	12	no	—
187	C₄⋊C₄	C₂×C₃×C₄×C₅	1	12	yes	—
188	C₄²	—	2	12	no	—
189	C₄²	C₂×C₃×C₄×C₅	1	12	yes	—
190	C₃×C₅	#2351	1	1	yes	characteristic, direct product factor, Hall
191	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
192	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
193	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
194	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
195	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
196	C₃×C₄	—	2	6	no	—
197	C₃×C₄	—	2	2	no	—
198	C₃×C₄	—	2	4	no	—
199	C₃×C₄	—	4	6	no	—
200	C₃×C₄	C₂×C₅×C₄⋊C₄	1	12	yes	—
201	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₅	C₃×C₄×#41	1	1	yes	characteristic
203	C₂×C₅	C₃×#519	1	1	yes	characteristic
204	C₂×C₅	C₃×#548	1	2	yes	—
205	C₂×C₅	C₃×#557	1	2	yes	—
206	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
207	Q₈	C₃×C₅×C₄⋊C₄	1	16	yes	complemented
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	6	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	6	no	—
213	C₂×C₄	—	2	6	no	—
214	C₂×C₄	—	2	6	no	—
215	C₂×C₄	C₂²×C₃×C₄×C₅	1	2	yes	—
216	C₂×C₄	C₂²×C₃×C₄×C₅	1	6	yes	—
217	C₂×C₄	C₂²×C₃×C₄×C₅	1	6	yes	—
218	C₂×C₄	C₂×C₃×C₅×D₄	1	6	yes	—
219	C₂×C₄	C₂×C₃×C₅×Q₈	1	6	yes	—
220	C₂×C₄	C₃×C₅×#41	1	4	yes	—
221	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
222	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
223	C₂×C₃	C₅×#519	1	1	yes	characteristic
224	C₂×C₃	C₅×#548	1	2	yes	—
225	C₂×C₃	C₅×#557	1	2	yes	—
226	C₅	C₃×#2351	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
228	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
229	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
230	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
231	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
232	C₄	—	2	6	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	6	no	—
236	C₄	C₂×C₃×C₅×C₄⋊C₄	1	12	yes	—
237	C₃	C₅×#2351	1	1	yes	characteristic, direct product factor, Sylow
238	C₂	C₂²×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
239	C₂	C₃×C₄×C₅×#41	1	1	yes	characteristic
240	C₂	C₃×C₅×#519	1	1	yes	characteristic
241	C₂	C₃×C₅×#548	1	2	yes	—
242	C₂	C₃×C₅×#557	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#2351	C₃	direct (N × H)
9. C₃×C₄×C₅×Q₈	C₄	semidirect
18. C₃×#2351	C₅	direct (N × H)
44. C₄×C₅×Q₈	C₃×C₄	semidirect
53. #2351	C₃×C₅	direct (N × H)
55. C₃×C₅×Q₈	C₄⋊C₄	semidirect
71. C₃×C₄×Q₈	C₄×C₅	semidirect
122. C₅×Q₈	C₃×C₄⋊C₄	semidirect
138. C₄×Q₈	C₃×C₄×C₅	semidirect
153. C₃×Q₈	C₅×C₄⋊C₄	semidirect
190. C₃×C₅	#2351	direct (N × H)
207. Q₈	C₃×C₅×C₄⋊C₄	semidirect
226. C₅	C₃×#2351	direct (N × H)
237. C₃	C₅×#2351	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×C₄×C₅×Q₈	2
C₃×C₅×#389	2
C₃×C₅×#382	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#2351	3
C₃×#2351	5

Subgroups — C3×C5×#2351 #14275385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2351 #14275385