Subgroups — C₂×C₅×#4491 #14313779

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4491	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×#357	—	3	1	no	maximal, Hall
6	C₂²×C₅×D₁₂	—	2	1	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
9	C₂×C₅×D₂₄	—	2	2	no	—
10	C₂×C₅×D₂₄	—	2	1	no	—
11	C₂×C₅×D₂₄	C₄	1	4	yes	complemented
12	C₂×C₅×D₂₄	C₂²	1	2	yes	—
13	C₃×C₅×#107	—	2	1	no	—
14	C₃×C₅×#107	C₂²	1	2	yes	complemented
15	C₂×C₅×C₃⋊C₁₆	—	2	2	no	—
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	complemented
17	C₂×#4491	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂²×C₅×D₈	—	3	1	no	—
19	C₂²×C₅×C₁₆	—	3	1	no	—
20	C₂×C₅×#107	D₃	1	1	yes	characteristic, complemented
21	C₅×#357	—	3	4	no	—
22	C₂³×C₅×D₃	—	4	1	no	—
23	C₂²×C₃×C₄×C₅	—	4	1	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×D₁₂	—	2	4	no	—
26	C₂×C₅×D₁₂	—	2	2	no	—
27	C₂×C₅×D₁₂	—	4	1	no	—
28	C₂×C₅×D₁₂	—	4	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	4	1	no	—
35	C₅×D₂₄	—	2	4	no	—
36	C₅×D₂₄	—	2	2	no	—
37	C₅×D₂₄	C₂×C₄	1	4	yes	complemented
38	C₅×C₃⋊C₁₆	—	2	2	no	—
39	C₂²×D₂₄	C₂×C₅	1	1	yes	characteristic
40	C₂×C₃×#107	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂²×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
42	#4491	C₂×C₅	1	4	yes	direct product factor
43	C₂²×C₅×D₄	—	6	1	no	—
44	C₂×C₅×D₈	—	3	4	no	—
45	C₂×C₅×D₈	—	3	2	no	—
46	C₂×C₅×D₈	—	6	2	no	—
47	C₂×C₅×D₈	—	6	1	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	—	2	1	no	—
50	C₂×C₅×C₁₆	—	3	2	no	—
51	C₂×C₅×C₁₆	—	6	2	no	—
52	C₅×#107	—	2	1	no	—
53	C₅×#107	C₂×D₃	1	2	yes	complemented
54	C₂×#357	—	3	1	no	Sylow
55	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
56	C₂²×C₅×D₃	—	4	4	no	—
57	C₂²×C₅×D₃	—	4	2	no	—
58	C₂²×C₅×D₃	—	8	2	no	—
59	C₂²×C₅×D₃	—	8	2	no	—
60	C₂×C₃×C₄×C₅	—	4	2	no	—
61	C₂×C₃×C₄×C₅	—	8	2	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₅×D₁₂	—	2	4	no	—
67	C₅×D₁₂	—	4	4	no	—
68	C₅×D₁₂	—	4	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂²×D₁₂	—	2	1	no	—
73	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₂×C₃×C₄⋊C₄	—	2	1	no	—
75	C₂×D₂₄	—	2	2	no	—
76	C₂×D₂₄	—	2	1	no	—
77	C₂×D₂₄	C₄×C₅	1	4	yes	complemented
78	C₂×D₂₄	C₂²×C₅	1	2	yes	—
79	C₃×#107	—	2	1	no	—
80	C₃×#107	C₂²×C₅	1	2	yes	complemented
81	C₂×C₃⋊C₁₆	—	2	2	no	—
82	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	complemented
83	C₂⁴×C₅	—	12	1	no	—
84	C₂×C₅×D₄	—	6	4	no	—
85	C₂×C₅×D₄	—	6	2	no	—
86	C₂×C₅×D₄	—	12	1	no	—
87	C₂×C₅×D₄	—	12	2	no	—
88	C₂²×C₄×C₅	—	4	1	no	—
89	C₂²×C₄×C₅	#67	1	1	yes	characteristic
90	C₅×D₈	—	3	4	no	—
91	C₅×D₈	—	6	4	no	—
92	C₅×D₈	—	6	2	no	—
93	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
94	C₂×C₅×C₈	D₁₂	1	2	yes	—
95	C₂×C₅×C₈	#67	1	1	yes	characteristic
96	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	4	1	no	—
99	C₅×C₁₆	—	6	2	no	—
100	C₂²×D₈	—	3	1	no	—
101	C₂²×C₁₆	—	3	1	no	—
102	C₂×#107	C₅×D₃	1	1	yes	characteristic, complemented
103	#357	—	3	4	no	—
104	C₂²×C₃×C₅	#102	1	2	yes	—
105	C₂²×C₃×C₅	D₁₆	1	2	yes	—
106	C₂²×C₃×C₅	SD₃₂	1	2	yes	—
107	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
108	C₂×C₅×D₃	—	4	4	no	—
109	C₂×C₅×D₃	—	8	8	no	—
110	C₂×C₅×D₃	—	8	2	no	—
111	C₂×C₅×D₃	—	8	2	no	—
112	C₃×C₄×C₅	—	8	2	no	—
113	C₃×C₄×C₅	#102	1	2	yes	—
114	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
116	C₂³×D₃	—	4	1	no	—
117	C₂²×C₃×C₄	—	4	1	no	—
118	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
119	C₂×D₁₂	—	2	4	no	—
120	C₂×D₁₂	—	2	2	no	—
121	C₂×D₁₂	—	4	1	no	—
122	C₂×D₁₂	—	4	2	no	—
123	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
125	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
126	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
127	C₃×C₄⋊C₄	—	2	2	no	—
128	C₃×C₄⋊C₄	—	4	1	no	—
129	D₂₄	—	2	4	no	—
130	D₂₄	—	2	2	no	—
131	D₂₄	C₂×C₄×C₅	1	4	yes	complemented
132	C₃⋊C₁₆	—	2	2	no	—
133	C₂³×C₅	—	12	4	no	—
134	C₂³×C₅	—	12	2	no	—
135	C₂³×C₅	—	24	2	no	—
136	C₂³×C₅	—	24	2	no	—
137	C₂³×C₅	#213	1	1	yes	characteristic
138	C₅×D₄	—	6	4	no	—
139	C₅×D₄	—	12	4	no	—
140	C₅×D₄	—	12	2	no	—
141	C₂×C₄×C₅	—	4	2	no	—
142	C₂×C₄×C₅	—	8	2	no	—
143	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
144	C₂×C₄×C₅	#213	1	1	yes	characteristic
145	C₂×C₄×C₅	#215	1	2	yes	—
146	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
147	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
148	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
149	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
150	C₂²×D₄	—	6	1	no	—
151	C₂×D₈	—	3	4	no	—
152	C₂×D₈	—	3	2	no	—
153	C₂×D₈	—	6	2	no	—
154	C₂×D₈	—	6	1	no	—
155	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	1	no	—
157	C₂×C₁₆	—	3	2	no	—
158	C₂×C₁₆	—	6	2	no	—
159	#107	—	2	1	no	—
160	#107	C₂×C₅×D₃	1	2	yes	complemented
161	C₂×C₃×C₅	#357	1	4	yes	complemented
162	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
165	C₅×D₃	—	8	4	no	—
166	C₂³×C₃	C₅×D₈	1	1	yes	characteristic
167	C₂²×D₃	—	4	2	no	—
168	C₂²×D₃	—	4	4	no	—
169	C₂²×D₃	—	8	2	no	—
170	C₂²×D₃	—	8	2	no	—
171	C₂×C₃×C₄	—	4	2	no	—
172	C₂×C₃×C₄	—	8	2	no	—
173	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
174	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
175	C₂×C₃×C₄	C₅×SD₁₆	1	2	yes	—
176	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
177	D₁₂	—	2	4	no	—
178	D₁₂	—	4	2	no	—
179	D₁₂	—	4	4	no	—
180	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
181	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
182	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
183	C₂²×C₅	—	12	4	no	—
184	C₂²×C₅	—	24	2	no	—
185	C₂²×C₅	—	24	2	no	—
186	C₂²×C₅	—	24	8	no	—
187	C₂²×C₅	#809	1	2	yes	—
188	C₂²×C₅	#826	1	2	yes	—
189	C₂²×C₅	#828	1	2	yes	—
190	C₂²×C₅	C₂×#213	1	1	yes	characteristic
191	C₄×C₅	—	8	2	no	—
192	C₄×C₅	#809	1	2	yes	—
193	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
194	C₄×C₅	C₂×#215	1	1	yes	characteristic
195	C₂⁴	—	12	1	no	—
196	C₂×D₄	—	6	4	no	—
197	C₂×D₄	—	6	2	no	—
198	C₂×D₄	—	12	1	no	—
199	C₂×D₄	—	12	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	C₅×#67	1	1	yes	characteristic
202	D₈	—	3	4	no	—
203	D₈	—	6	4	no	—
204	D₈	—	6	2	no	—
205	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
206	C₂×C₈	C₅×D₁₂	1	2	yes	—
207	C₂×C₈	C₅×#67	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₁₆	—	6	2	no	—
212	C₃×C₅	C₂×#357	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	C₅×#102	1	2	yes	—
214	C₂²×C₃	C₅×D₁₆	1	2	yes	—
215	C₂²×C₃	C₅×SD₃₂	1	2	yes	—
216	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
217	C₂×D₃	—	4	4	no	—
218	C₂×D₃	—	8	8	no	—
219	C₂×D₃	—	8	2	no	—
220	C₂×D₃	—	8	2	no	—
221	C₃×C₄	—	8	2	no	—
222	C₃×C₄	C₅×#102	1	2	yes	—
223	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
224	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
225	C₂×C₅	—	24	4	no	—
226	C₂×C₅	#4491	1	4	yes	direct product factor
227	C₂×C₅	C₂×#809	1	1	yes	characteristic
228	C₂×C₅	C₂×#826	1	1	yes	characteristic
229	C₂×C₅	C₂×#828	1	1	yes	characteristic
230	C₂³	—	12	2	no	—
231	C₂³	—	12	4	no	—
232	C₂³	—	24	2	no	—
233	C₂³	—	24	2	no	—
234	C₂³	C₅×#213	1	1	yes	characteristic
235	D₄	—	6	4	no	—
236	D₄	—	12	2	no	—
237	D₄	—	12	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
241	C₂×C₄	C₅×#213	1	1	yes	characteristic
242	C₂×C₄	C₅×#215	1	2	yes	—
243	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
244	C₈	C₅×(C₂×D₃)⋊C₄	1	2	yes	—
245	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
246	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
247	C₂×C₃	C₅×#357	1	4	yes	complemented
248	C₂×C₃	C₂×C₅×#102	1	1	yes	characteristic
249	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
250	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
251	D₃	—	8	4	no	—
252	C₅	C₂×#4491	1	1	yes	characteristic, direct product factor, Sylow
253	C₂²	—	12	4	no	—
254	C₂²	—	24	2	no	—
255	C₂²	—	24	8	no	—
256	C₂²	—	24	2	no	—
257	C₂²	C₅×#809	1	2	yes	—
258	C₂²	C₅×#826	1	2	yes	—
259	C₂²	C₅×#828	1	2	yes	—
260	C₂²	C₂×C₅×#213	1	1	yes	characteristic
261	C₄	—	8	2	no	—
262	C₄	C₅×#809	1	2	yes	—
263	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
264	C₄	C₂×C₅×#215	1	1	yes	characteristic
265	C₃	C₂×C₅×#357	1	1	yes	characteristic, complemented, Sylow
266	C₂	—	24	4	no	—
267	C₂	C₅×#4491	1	4	yes	direct product factor
268	C₂	C₂×C₅×#809	1	1	yes	characteristic
269	C₂	C₂×C₅×#826	1	1	yes	characteristic
270	C₂	C₂×C₅×#828	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	#213
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#107	C₂
socle	C₂³×C₃×C₅	D₈
prime radical	C₂²×C₅×D₂₄	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×#107	D₃
hypocenter	C₃	C₂×C₅×#357
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×#107

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#107	C₂	semidirect
3. C₂²×C₅×C₃⋊C₁₆	C₂	semidirect
4. C₅×#4491	C₂	direct (N × H)
11. C₂×C₅×D₂₄	C₄	semidirect
14. C₃×C₅×#107	C₂²	semidirect
16. C₂×C₅×C₃⋊C₁₆	C₂²	semidirect
17. C₂×#4491	C₅	direct (N × H)
20. C₂×C₅×#107	D₃	semidirect
37. C₅×D₂₄	C₂×C₄	semidirect
40. C₂×C₃×#107	C₂×C₅	semidirect
41. C₂²×C₃⋊C₁₆	C₂×C₅	semidirect
42. #4491	C₂×C₅	direct (N × H)
53. C₅×#107	C₂×D₃	semidirect
77. C₂×D₂₄	C₄×C₅	semidirect
80. C₃×#107	C₂²×C₅	semidirect
82. C₂×C₃⋊C₁₆	C₂²×C₅	semidirect
102. C₂×#107	C₅×D₃	semidirect
131. D₂₄	C₂×C₄×C₅	semidirect
160. #107	C₂×C₅×D₃	semidirect
161. C₂×C₃×C₅	#357	semidirect
212. C₃×C₅	C₂×#357	semidirect
226. C₂×C₅	#4491	direct (N × H)
247. C₂×C₃	C₅×#357	semidirect
252. C₅	C₂×#4491	direct (N × H)
265. C₃	C₂×C₅×#357	semidirect
267. C₂	C₅×#4491	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×C₅×D₂₄	2
C₂×C₃×C₅×#107	2
C₂²×C₅×C₃⋊C₁₆	2
C₅×#4491	2
C₂×C₅×#357	3
C₂×#4491	5

Subgroups — C2×C5×#4491 #14313779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4491 #14313779