Subgroups — C₃×C₅×#2316 #14275350

Order 1920.

1200 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	2	yes	maximal
2	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#389	C₂	1	8	yes	maximal, complemented
4	C₃×C₅×#380	C₂	1	4	yes	maximal
5	C₅×#2316	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	2	yes	—
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	—	2	8	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₄	1	8	yes	complemented
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	8	yes	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	4	yes	—
15	C₃×C₅×#95	C₂²	1	8	yes	—
16	C₃×#2316	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#117	C₂×C₃	1	2	yes	—
18	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₅×#389	C₂×C₃	1	8	yes	complemented
20	C₅×#380	C₂×C₃	1	4	yes	—
21	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	—	4	4	no	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	8	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
26	C₂²×C₃×C₄×C₅	Q₈	1	4	yes	—
27	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
28	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	8	no	—
31	C₃×C₅×C₄⋊C₄	—	4	8	no	—
32	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
33	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
34	C₃×C₄²×C₅	—	2	8	no	—
35	C₂×C₃×#117	C₂×C₅	1	2	yes	—
36	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
37	C₃×#389	C₂×C₅	1	8	yes	complemented
38	C₃×#380	C₂×C₅	1	4	yes	—
39	C₂³×C₄×C₅	C₂²×C₃	1	2	yes	—
40	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
41	C₅×#117	—	2	8	no	—
42	C₂×C₅×C₄⋊C₄	C₃×C₄	1	8	yes	complemented
43	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
44	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	8	yes	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
48	C₅×#95	C₂²×C₃	1	8	yes	—
49	#2316	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	—	2	2	no	—
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
53	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
54	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
55	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	—	2	8	no	—
57	C₂×C₃×C₄×C₅	—	2	8	no	—
58	C₂×C₃×C₄×C₅	—	2	8	no	—
59	C₂×C₃×C₄×C₅	—	2	8	no	—
60	C₂×C₃×C₄×C₅	—	4	8	no	—
61	C₂×C₃×C₄×C₅	C₄⋊C₄	1	16	yes	complemented
62	C₂×C₃×C₄×C₅	C₂²×C₄	1	8	yes	—
63	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
66	C₂³×C₃×C₄	C₂²×C₅	1	2	yes	—
67	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
68	C₃×#117	—	2	8	no	—
69	C₂×C₃×C₄⋊C₄	C₄×C₅	1	8	yes	complemented
70	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
71	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	8	yes	—
72	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
73	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
74	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
75	C₃×#95	C₂²×C₅	1	8	yes	—
76	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	4	4	no	—
78	C₂²×C₄×C₅	C₂×C₃×C₄	1	8	yes	—
79	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
80	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
81	C₂²×C₄×C₅	C₃×Q₈	1	4	yes	—
82	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
83	C₂²×C₄×C₅	C₂³×C₃	1	4	yes	—
84	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
85	C₅×C₄⋊C₄	—	2	8	no	—
86	C₅×C₄⋊C₄	—	4	8	no	—
87	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
88	C₅×C₂²⋊C₄	—	2	8	no	—
89	C₄²×C₅	—	2	8	no	—
90	C₂×#117	C₂×C₃×C₅	1	2	yes	—
91	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
92	#389	C₂×C₃×C₅	1	8	yes	complemented
93	#380	C₂×C₃×C₅	1	4	yes	—
94	C₂²×C₃×C₅	—	2	2	no	—
95	C₂²×C₃×C₅	—	2	2	no	—
96	C₂²×C₃×C₅	—	2	4	no	—
97	C₂²×C₃×C₅	—	2	4	no	—
98	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	4	yes	—
99	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
102	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
103	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
104	C₃×C₄×C₅	—	2	8	no	—
105	C₃×C₄×C₅	—	4	8	no	—
106	C₃×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
107	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
108	C₂²×C₃×C₄	—	4	4	no	—
109	C₂²×C₃×C₄	C₂×C₄×C₅	1	8	yes	—
110	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
111	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
112	C₂²×C₃×C₄	C₅×Q₈	1	4	yes	—
113	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
114	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
115	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
116	C₃×C₄⋊C₄	—	2	8	no	—
117	C₃×C₄⋊C₄	—	4	8	no	—
118	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
119	C₃×C₂²⋊C₄	—	2	8	no	—
120	C₃×C₄²	—	2	8	no	—
121	C₂³×C₅	—	2	2	no	—
122	C₂³×C₅	—	2	2	no	—
123	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
124	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
125	C₂³×C₅	C₂×C₃×Q₈	1	2	yes	—
126	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
127	C₂×C₄×C₅	—	2	8	no	—
128	C₂×C₄×C₅	—	2	8	no	—
129	C₂×C₄×C₅	—	2	8	no	—
130	C₂×C₄×C₅	—	2	8	no	—
131	C₂×C₄×C₅	—	4	8	no	—
132	C₂×C₄×C₅	C₃×C₄⋊C₄	1	16	yes	complemented
133	C₂×C₄×C₅	C₂²×C₃×C₄	1	8	yes	—
134	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
135	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
136	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
137	C₂³×C₄	C₂²×C₃×C₅	1	2	yes	—
138	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
139	#117	—	2	8	no	—
140	C₂×C₄⋊C₄	C₃×C₄×C₅	1	8	yes	complemented
141	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
142	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	8	yes	—
143	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
144	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
145	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
146	#95	C₂²×C₃×C₅	1	8	yes	—
147	C₂×C₃×C₅	—	2	4	no	—
148	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂×C₃×C₅	#518	1	1	yes	characteristic
150	C₂×C₃×C₅	#519	1	1	yes	characteristic
151	C₂×C₃×C₅	#537	1	2	yes	—
152	C₂×C₃×C₅	#544	1	2	yes	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	—	2	2	no	—
155	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
156	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
157	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
158	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
159	C₂×C₃×C₄	—	2	8	no	—
160	C₂×C₃×C₄	—	2	8	no	—
161	C₂×C₃×C₄	—	2	8	no	—
162	C₂×C₃×C₄	—	2	8	no	—
163	C₂×C₃×C₄	—	4	8	no	—
164	C₂×C₃×C₄	C₅×C₄⋊C₄	1	16	yes	complemented
165	C₂×C₃×C₄	C₂²×C₄×C₅	1	8	yes	—
166	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
167	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
168	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	2	4	no	—
172	C₂²×C₅	—	2	4	no	—
173	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	4	yes	—
174	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
175	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
176	C₂²×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
177	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
178	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
179	C₄×C₅	—	2	8	no	—
180	C₄×C₅	—	4	8	no	—
181	C₄×C₅	C₂×C₃×C₄⋊C₄	1	8	yes	—
182	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
183	C₂²×C₄	—	4	4	no	—
184	C₂²×C₄	C₂×C₃×C₄×C₅	1	8	yes	—
185	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
186	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
187	C₂²×C₄	C₃×C₅×Q₈	1	4	yes	—
188	C₂²×C₄	C₂³×C₃×C₅	1	8	yes	—
189	C₂²×C₄	C₂³×C₃×C₅	1	4	yes	—
190	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
194	C₂²⋊C₄	—	2	8	no	—
195	C₄²	—	2	8	no	—
196	C₃×C₅	#2316	1	1	yes	characteristic, direct product factor, Hall
197	C₂²×C₃	—	2	2	no	—
198	C₂²×C₃	—	2	2	no	—
199	C₂²×C₃	—	2	4	no	—
200	C₂²×C₃	—	2	4	no	—
201	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
202	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
203	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
204	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
205	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
206	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
207	C₃×C₄	—	2	8	no	—
208	C₃×C₄	—	4	8	no	—
209	C₃×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
210	C₂×C₅	—	2	4	no	—
211	C₂×C₅	C₂²×C₃×C₄⋊C₄	1	1	yes	characteristic
212	C₂×C₅	C₃×#518	1	1	yes	characteristic
213	C₂×C₅	C₃×#519	1	1	yes	characteristic
214	C₂×C₅	C₃×#537	1	2	yes	—
215	C₂×C₅	C₃×#544	1	2	yes	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
219	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
220	C₂³	C₂×C₃×C₅×Q₈	1	2	yes	—
221	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
222	C₂×C₄	—	2	8	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	2	8	no	—
225	C₂×C₄	—	2	8	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	C₃×C₅×C₄⋊C₄	1	16	yes	complemented
228	C₂×C₄	C₂²×C₃×C₄×C₅	1	8	yes	—
229	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
230	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
231	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
232	C₂×C₃	—	2	4	no	—
233	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₃	C₅×#518	1	1	yes	characteristic
235	C₂×C₃	C₅×#519	1	1	yes	characteristic
236	C₂×C₃	C₅×#537	1	2	yes	—
237	C₂×C₃	C₅×#544	1	2	yes	—
238	C₅	C₃×#2316	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	4	yes	—
244	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
245	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
246	C₂²	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
247	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
248	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
249	C₄	—	2	8	no	—
250	C₄	—	4	8	no	—
251	C₄	C₂×C₃×C₅×C₄⋊C₄	1	8	yes	—
252	C₃	C₅×#2316	1	1	yes	characteristic, direct product factor, Sylow
253	C₂	—	2	4	no	—
254	C₂	C₂²×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
255	C₂	C₃×C₅×#518	1	1	yes	characteristic
256	C₂	C₃×C₅×#519	1	1	yes	characteristic
257	C₂	C₃×C₅×#537	1	2	yes	—
258	C₂	C₃×C₅×#544	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃×C₅	C₂⁴
commutator	C₂²	C₂³×C₃×C₄×C₅
Frattini, cosocle	C₂³	C₂⁴×C₃×C₅
prime radical	C₂⁴×C₃×C₅	C₂³
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
3. C₃×C₅×#389	C₂	semidirect
5. C₅×#2316	C₃	direct (N × H)
9. C₂×C₃×C₅×C₄⋊C₄	C₄	semidirect
16. C₃×#2316	C₅	direct (N × H)
19. C₅×#389	C₂×C₃	semidirect
37. C₃×#389	C₂×C₅	semidirect
42. C₂×C₅×C₄⋊C₄	C₃×C₄	semidirect
49. #2316	C₃×C₅	direct (N × H)
61. C₂×C₃×C₄×C₅	C₄⋊C₄	semidirect
69. C₂×C₃×C₄⋊C₄	C₄×C₅	semidirect
92. #389	C₂×C₃×C₅	semidirect
132. C₂×C₄×C₅	C₃×C₄⋊C₄	semidirect
140. C₂×C₄⋊C₄	C₃×C₄×C₅	semidirect
164. C₂×C₃×C₄	C₅×C₄⋊C₄	semidirect
196. C₃×C₅	#2316	direct (N × H)
227. C₂×C₄	C₃×C₅×C₄⋊C₄	semidirect
238. C₅	C₃×#2316	direct (N × H)
252. C₃	C₅×#2316	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×#117	2
C₂²×C₃×C₅×C₄⋊C₄	2
C₃×C₅×#389	2
C₃×C₅×#380	2
C₅×#2316	3
C₃×#2316	5

Subgroups — C3×C5×#2316 #14275350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2316 #14275350