Subgroups — C₃×C₅×#2313 #14275347

Order 1920.

1024 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#387	C₂	1	8	yes	maximal, complemented
3	C₃×C₅×#381	C₂	1	2	yes	maximal
4	C₃×C₅×#380	C₂	1	2	yes	maximal
5	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	2	yes	maximal
6	C₅×#2313	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×C₄⋊C₄	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	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	4	yes	—
16	C₃×C₅×#95	C₂²	1	4	yes	—
17	C₃×C₅×#95	C₂²	1	8	yes	—
18	C₃×#2313	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂²×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
20	C₅×#387	C₂×C₃	1	8	yes	complemented
21	C₅×#381	C₂×C₃	1	2	yes	—
22	C₅×#380	C₂×C₃	1	2	yes	—
23	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	2	yes	—
24	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	4	no	—
26	C₂²×C₃×C₄×C₅	—	4	2	no	—
27	C₂²×C₃×C₄×C₅	—	4	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	8	yes	—
32	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
35	C₃×C₅×C₄⋊C₄	—	4	4	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
37	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
38	C₃×C₄²×C₅	—	4	4	no	—
39	C₂²×C₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
40	C₃×#387	C₂×C₅	1	8	yes	complemented
41	C₃×#381	C₂×C₅	1	2	yes	—
42	C₃×#380	C₂×C₅	1	2	yes	—
43	C₃×C₄×C₂²⋊C₄	C₂×C₅	1	2	yes	—
44	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
46	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₅×C₄⋊C₄	C₃×C₄	1	8	yes	complemented
48	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
49	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
51	C₂×C₄²×C₅	C₂²×C₃	1	4	yes	—
52	C₅×#95	C₂²×C₃	1	4	yes	—
53	C₅×#95	C₂²×C₃	1	4	yes	—
54	C₅×#95	C₂²×C₃	1	8	yes	—
55	#2313	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
56	C₂³×C₃×C₅	—	2	2	no	—
57	C₂³×C₃×C₅	—	2	2	no	—
58	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
59	C₂³×C₃×C₅	#41	1	4	yes	—
60	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	8	no	—
62	C₂×C₃×C₄×C₅	—	2	4	no	—
63	C₂×C₃×C₄×C₅	—	2	4	no	—
64	C₂×C₃×C₄×C₅	—	2	4	no	—
65	C₂×C₃×C₄×C₅	—	2	8	no	—
66	C₂×C₃×C₄×C₅	—	2	8	no	—
67	C₂×C₃×C₄×C₅	—	2	8	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	—	4	4	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
73	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
74	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
75	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₃×C₄⋊C₄	C₄×C₅	1	8	yes	complemented
77	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
79	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
80	C₂×C₃×C₄²	C₂²×C₅	1	4	yes	—
81	C₃×#95	C₂²×C₅	1	4	yes	—
82	C₃×#95	C₂²×C₅	1	4	yes	—
83	C₃×#95	C₂²×C₅	1	8	yes	—
84	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
85	C₂²×C₄×C₅	—	2	4	no	—
86	C₂²×C₄×C₅	—	4	2	no	—
87	C₂²×C₄×C₅	—	4	2	no	—
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
89	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
90	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
91	C₂²×C₄×C₅	C₂³×C₃	1	8	yes	—
92	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
93	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
94	C₂²×C₄×C₅	C₂³×C₃	1	2	yes	—
95	C₅×C₄⋊C₄	—	4	4	no	—
96	C₅×C₄⋊C₄	C₂×C₃×C₄	1	16	yes	—
97	C₅×C₂²⋊C₄	—	2	8	no	—
98	C₄²×C₅	—	4	4	no	—
99	C₂²×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic
100	#387	C₂×C₃×C₅	1	8	yes	complemented
101	#381	C₂×C₃×C₅	1	2	yes	—
102	#380	C₂×C₃×C₅	1	2	yes	—
103	C₄×C₂²⋊C₄	C₂×C₃×C₅	1	2	yes	—
104	C₂²×C₃×C₅	—	2	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	8	no	—
107	C₂²×C₃×C₅	#117	1	4	yes	—
108	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
110	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
111	C₂²×C₃×C₅	2₋¹⁺⁴	1	2	yes	—
112	C₃×C₄×C₅	—	2	4	no	—
113	C₃×C₄×C₅	—	2	4	no	—
114	C₃×C₄×C₅	—	2	4	no	—
115	C₃×C₄×C₅	—	4	8	no	—
116	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
117	C₂²×C₃×C₄	—	2	4	no	—
118	C₂²×C₃×C₄	—	4	2	no	—
119	C₂²×C₃×C₄	—	4	2	no	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
121	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
122	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
123	C₂²×C₃×C₄	C₂³×C₅	1	8	yes	—
124	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
125	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
126	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
127	C₃×C₄⋊C₄	—	4	4	no	—
128	C₃×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
129	C₃×C₂²⋊C₄	—	2	8	no	—
130	C₃×C₄²	—	4	4	no	—
131	C₂³×C₅	—	2	2	no	—
132	C₂³×C₅	—	2	2	no	—
133	C₂³×C₅	C₂²×C₃×C₄	1	2	yes	—
134	C₂³×C₅	C₃×#41	1	4	yes	—
135	C₂³×C₅	C₂⁴×C₃	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	8	no	—
137	C₂×C₄×C₅	—	2	4	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	8	no	—
141	C₂×C₄×C₅	—	2	8	no	—
142	C₂×C₄×C₅	—	2	8	no	—
143	C₂×C₄×C₅	—	4	4	no	—
144	C₂×C₄×C₅	—	4	4	no	—
145	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
146	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
147	C₂×C₄×C₅	C₂²×C₃×C₄	1	4	yes	—
148	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₃×C₄×C₅	1	8	yes	complemented
152	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
153	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
154	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
155	C₂×C₄²	C₂²×C₃×C₅	1	4	yes	—
156	#95	C₂²×C₃×C₅	1	4	yes	—
157	#95	C₂²×C₃×C₅	1	4	yes	—
158	#95	C₂²×C₃×C₅	1	8	yes	—
159	C₂×C₃×C₅	—	2	4	no	—
160	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
161	C₂×C₃×C₅	#518	1	1	yes	characteristic
162	C₂×C₃×C₅	#519	1	1	yes	characteristic
163	C₂×C₃×C₅	#539	1	4	yes	—
164	C₂³×C₃	—	2	2	no	—
165	C₂³×C₃	—	2	2	no	—
166	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
167	C₂³×C₃	C₅×#41	1	4	yes	—
168	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	—	2	8	no	—
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	4	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	—	2	8	no	—
174	C₂×C₃×C₄	—	2	8	no	—
175	C₂×C₃×C₄	—	2	8	no	—
176	C₂×C₃×C₄	—	4	4	no	—
177	C₂×C₃×C₄	—	4	4	no	—
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
179	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
180	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	—	2	2	no	—
183	C₂²×C₅	—	2	8	no	—
184	C₂²×C₅	C₃×#117	1	4	yes	—
185	C₂²×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
186	C₂²×C₅	C₂×C₃×#41	1	2	yes	—
187	C₂²×C₅	C₃×H(5,𝔽₂)	1	2	yes	—
188	C₂²×C₅	C₃×2₋¹⁺⁴	1	2	yes	—
189	C₄×C₅	—	2	4	no	—
190	C₄×C₅	—	2	4	no	—
191	C₄×C₅	—	2	4	no	—
192	C₄×C₅	—	4	8	no	—
193	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
198	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
199	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
200	C₂²×C₄	C₂³×C₃×C₅	1	8	yes	—
201	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
202	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
203	C₂²×C₄	C₂³×C₃×C₅	1	2	yes	—
204	C₄⋊C₄	—	4	4	no	—
205	C₄⋊C₄	C₂×C₃×C₄×C₅	1	16	yes	—
206	C₂²⋊C₄	—	2	8	no	—
207	C₄²	—	4	4	no	—
208	C₃×C₅	#2313	1	1	yes	characteristic, direct product factor, Hall
209	C₂²×C₃	—	2	2	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	8	no	—
212	C₂²×C₃	C₅×#117	1	4	yes	—
213	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
214	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
215	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
216	C₂²×C₃	C₅×2₋¹⁺⁴	1	2	yes	—
217	C₃×C₄	—	2	4	no	—
218	C₃×C₄	—	2	4	no	—
219	C₃×C₄	—	2	4	no	—
220	C₃×C₄	—	4	8	no	—
221	C₂×C₅	—	2	4	no	—
222	C₂×C₅	C₂×C₃×#117	1	1	yes	characteristic
223	C₂×C₅	C₃×#518	1	1	yes	characteristic
224	C₂×C₅	C₃×#519	1	1	yes	characteristic
225	C₂×C₅	C₃×#539	1	4	yes	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	C₂²×C₃×C₄×C₅	1	2	yes	—
229	C₂³	C₃×C₅×#41	1	4	yes	—
230	C₂³	C₂⁴×C₃×C₅	1	1	yes	characteristic
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	8	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
241	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
242	C₂×C₄	C₂²×C₃×C₄×C₅	1	4	yes	—
243	C₂×C₃	—	2	4	no	—
244	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
245	C₂×C₃	C₅×#518	1	1	yes	characteristic
246	C₂×C₃	C₅×#519	1	1	yes	characteristic
247	C₂×C₃	C₅×#539	1	4	yes	—
248	C₅	C₃×#2313	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	8	no	—
252	C₂²	C₃×C₅×#117	1	4	yes	—
253	C₂²	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
254	C₂²	C₂×C₃×C₅×#41	1	2	yes	—
255	C₂²	C₃×C₅×H(5,𝔽₂)	1	2	yes	—
256	C₂²	C₃×C₅×2₋¹⁺⁴	1	2	yes	—
257	C₄	—	2	4	no	—
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	8	no	—
261	C₃	C₅×#2313	1	1	yes	characteristic, direct product factor, Sylow
262	C₂	—	2	4	no	—
263	C₂	C₂×C₃×C₅×#117	1	1	yes	characteristic
264	C₂	C₃×C₅×#518	1	1	yes	characteristic
265	C₂	C₃×C₅×#519	1	1	yes	characteristic
266	C₂	C₃×C₅×#539	1	4	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
2. C₃×C₅×#387	C₂	semidirect
6. C₅×#2313	C₃	direct (N × H)
10. C₂×C₃×C₅×C₄⋊C₄	C₄	semidirect
18. C₃×#2313	C₅	direct (N × H)
20. C₅×#387	C₂×C₃	semidirect
40. C₃×#387	C₂×C₅	semidirect
47. C₂×C₅×C₄⋊C₄	C₃×C₄	semidirect
55. #2313	C₃×C₅	direct (N × H)
76. C₂×C₃×C₄⋊C₄	C₄×C₅	semidirect
100. #387	C₂×C₃×C₅	semidirect
151. C₂×C₄⋊C₄	C₃×C₄×C₅	semidirect
208. C₃×C₅	#2313	direct (N × H)
248. C₅	C₃×#2313	direct (N × H)
261. C₃	C₅×#2313	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃×C₅×C₄⋊C₄	2
C₃×C₅×#387	2
C₃×C₅×#381	2
C₃×C₅×#380	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₅×#2313	3
C₃×#2313	5

Subgroups — C3×C5×#2313 #14275347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2313 #14275347