Finite Group Databasebeta

← C5×C3⋊D3×C22⋊D5 #72231464

Subgroups — C5×C3⋊D3×C22⋊D5 #72231464

Order 3600.

840 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C52×C3⋊D3C211yesmaximal, characteristic, complemented
2C2×C5×D5×C3⋊D3C211yesmaximal, characteristic, complemented
3C5×#67184C211yesmaximal, characteristic, complemented
4C32×C5×C22⋊D5C211yesmaximal, characteristic, complemented
5C5×#67140C211yesmaximal, characteristic, complemented
6C5×#67138C211yesmaximal, characteristic, complemented
7C5×C3⋊D3×Dic5C211yesmaximal, characteristic, complemented
8C5×D3×C22⋊D5—34nomaximal
9C22×C32×C52C2211yescharacteristic, complemented
10C2×C5×C3⋊D15C2211yescharacteristic, complemented
11C2×C52×C3⋊D3—22no—
12C2×C52×C3⋊D3C2211yescharacteristic
13C2×C52×C3⋊D3C2211yescharacteristic, complemented
14C2×C32×C5×D5C2211yescharacteristic, complemented
15C5×D5×C3⋊D3—22no—
16C5×C3⋊Dic15C2211yescharacteristic, complemented
17C32×C5×Dic5C2211yescharacteristic, complemented
18C5×D4×C3⋊D3—51nomaximal
19C3⋊D3×C22⋊D5C511yesmaximal, characteristic, direct product factor
20C22×C52×D3—34no—
21C2×C5×D3×D5—34no—
22C5×#1220—34no—
23C3×C5×C22⋊D5D314yescomplemented
24C5×#1203—34no—
25C5×#1201—34no—
26C5×D3×Dic5—34no—
27C2×C32×C52—21no—
28C2×C32×C52C2311yescharacteristic
29C5×C3⋊D15—21no—
30C52×C3⋊D3—21no—
31C52×C3⋊D3D412yescomplemented
32C32×C5×D5—21no—
33C2×C5×C22⋊D5—91noHall
34C22×C5×C3⋊D3—22no—
35C22×C5×C3⋊D3—51no—
36C22×C5×C3⋊D3D511yescharacteristic, complemented
37C22×C5×C3⋊D3C2×C511yescharacteristic, complemented
38C2×D5×C3⋊D3C2×C511yescharacteristic, complemented
39C32×C5×D4—51no—
40#67184C2×C511yescharacteristic, complemented
41C5×#638—51no—
42C5×#638—51no—
43C5×C4⋊(C3⋊D3)—51no—
44C4×C5×C3⋊D3—51no—
45C32×C22⋊D5C2×C511yescharacteristic, complemented
46#67140C2×C511yescharacteristic, complemented
47#67138C2×C511yescharacteristic, complemented
48C3⋊D3×Dic5C2×C511yescharacteristic, complemented
49C22×C3×C52C2×D314yescomplemented
50C2×C5×D15—34no—
51C2×C52×D3—34no—
52C2×C52×D3—34no—
53C2×C52×D3—68no—
54C2×C3×C5×D5C2×D314yescomplemented
55C5×D3×D5—68no—
56C5×Dic15—34no—
57C3×C5×Dic5C2×D314yescomplemented
58C5×D3×D4—154no—
59D3×C22⋊D5—34no—
60C32×C52C2×D411yescharacteristic, complemented, Hall
61C23×C52—91no—
62C22×C5×D5—91no—
63C5×C22⋊D5—91no—
64C5×C22⋊D5—91no—
65C5×C22⋊D5—91no—
66C5×C22⋊D5C3⋊D311yescharacteristic, direct product factor
67C2×C5×Dic5—91no—
68C22×C32×C5—22no—
69C22×C32×C5—51no—
70C22×C32×C5C2×D511yescharacteristic, complemented
71C22×C32×C5C22×C511yescharacteristic, complemented
72C2×C3⋊D15C22×C511yescharacteristic, complemented
73C2×C5×C3⋊D3—24no—
74C2×C5×C3⋊D3—24no—
75C2×C5×C3⋊D3—22no—
76C2×C5×C3⋊D3—22no—
77C2×C5×C3⋊D3—22no—
78C2×C5×C3⋊D3—22no—
79C2×C5×C3⋊D3—51no—
80C2×C5×C3⋊D3—102no—
81C2×C5×C3⋊D3C2×D511yescharacteristic
82C2×C5×C3⋊D3C2×D511yescharacteristic, complemented
83C2×C5×C3⋊D3C22×C511yescharacteristic
84C2×C5×C3⋊D3C22×C511yescharacteristic, complemented
85C2×C32×D5C22×C511yescharacteristic, complemented
86D5×C3⋊D3—22no—
87C32×C4×C5—51no—
88C3⋊Dic15C22×C511yescharacteristic, complemented
89C5×C3⋊Dic3—51no—
90C32×Dic5C22×C511yescharacteristic, complemented
91C2×C3×C52—24no—
92C2×C3×C52C22×D314yes—
93C5×D15—64no—
94C52×D3—38no—
95C52×D3—64no—
96C3×C5×D5—24no—
97D4×C3⋊D3—51noHall
98C22×C5×D3—34no—
99C22×C5×D3—34no—
100C22×C5×D3—68no—
101C22×C5×D3—154no—
102C2×D3×D5—34no—
103C3×C5×D4—54no—
104#1220—34no—
105C5×#67—154no—
106C5×#67—154no—
107C5×D12—154no—
108C4×C5×D3—154no—
109C3×C22⋊D5C5×D314yescomplemented
110#1203—34no—
111#1201—34no—
112D3×Dic5—34no—
113C22×C52—91no—
114C22×C52—91no—
115C22×C52—182no—
116C22×C52C2×C3⋊D311yescharacteristic, complemented
117C2×C5×D5—91no—
118C2×C5×D5—182no—
119C2×C5×D5C2×C3⋊D311yescharacteristic, complemented
120C5×Dic5—91no—
121C5×Dic5C2×C3⋊D311yescharacteristic, complemented
122C2×C32×C5—22no—
123C2×C32×C5—22no—
124C2×C32×C5—21no—
125C2×C32×C5—21no—
126C2×C32×C5—22no—
127C2×C32×C5—101no—
128C2×C32×C5C22×D511yescharacteristic
129C2×C32×C5C23×C511yescharacteristic
130C3⋊D15—21no—
131C5×C3⋊D3—22no—
132C5×C3⋊D3—22no—
133C5×C3⋊D3—21no—
134C5×C3⋊D3—21no—
135C5×C3⋊D3—24no—
136C5×C3⋊D3—101no—
137C5×C3⋊D3C22⋊D512yesdirect product factor
138C5×C3⋊D3C5×D412yescomplemented
139C32×D5—21no—
140C2×C5×D4—451no—
141C2×C22⋊D5—91no—
142C3×C52D3×D414yescomplemented
143C22×C3⋊D3—51no—
144C22×C3⋊D3C5×D511yescharacteristic, complemented
145C32×D4—51no—
146#638—51no—
147#638—51no—
148C4⋊(C3⋊D3)—51no—
149C4×C3⋊D3—51no—
150C22×C3×C5—28no—
151C22×C3×C5—54no—
152C22×C3×C5D3×D514yescomplemented
153C22×C3×C5C2×C5×D314yescomplemented
154C2×D15—34no—
155C2×C5×D3—34no—
156C2×C5×D3—34no—
157C2×C5×D3—34no—
158C2×C5×D3—34no—
159C2×C5×D3—68no—
160C2×C5×D3—68no—
161C2×C5×D3—68no—
162C2×C5×D3—68no—
163C2×C5×D3—616no—
164C2×C5×D3—616no—
165C2×C5×D3—154no—
166C2×C5×D3—308no—
167C2×C3×D5C2×C5×D314yescomplemented
168D3×D5—68no—
169C3×C4×C5—54no—
170Dic15—34no—
171C3×Dic5C2×C5×D314yescomplemented
172C5×Dic3—154no—
173C2×C52—21no—
174C2×C52—92no—
175C2×C52—181no—
176C2×C52C22×C3⋊D311yescharacteristic
177C5×D5—21no—
178C5×D5—181no—
179D3×D4—154no—
180C32×C5—22no—
181C32×C5C2×C22⋊D511yescharacteristic, complemented
182C32×C5C2×C5×D411yescharacteristic, complemented
183C23×C5—91no—
184C23×C5—91no—
185C23×C5—182no—
186C23×C5—451no—
187C22×D5—91no—
188C5×D4—51no—
189C5×D4—451no—
190C5×D4—451no—
191C5×D4—451no—
192C2×C4×C5—451no—
193C22⋊D5—91no—
194C22⋊D5—91no—
195C22⋊D5—91no—
196C22⋊D5C5×C3⋊D311yescharacteristic, direct product factor
197C2×Dic5—91no—
198C22×C32—51no—
199C22×C32C2×C5×D511yescharacteristic, complemented
200C2×C3⋊D3—22no—
201C2×C3⋊D3—51no—
202C2×C3⋊D3—102no—
203C2×C3⋊D3C2×C5×D511yescharacteristic
204C2×C3⋊D3C2×C5×D511yescharacteristic, complemented
205C32×C4—51no—
206C3⋊Dic3—51no—
207C2×C3×C5—28no—
208C2×C3×C5—28no—
209C2×C3×C5—24no—
210C2×C3×C5—24no—
211C2×C3×C5—28no—
212C2×C3×C5—104no—
213C2×C3×C5C2×D3×D514yes—
214C2×C3×C5C22×C5×D314yes—
215D15—64no—
216C3×D5—24no—
217C5×D3—38no—
218C5×D3—38no—
219C5×D3—616no—
220C5×D3—68no—
221C5×D3—68no—
222C5×D3—64no—
223C5×D3—64no—
224C5×D3—304no—
225C52D4×C3⋊D311yescharacteristic, complemented, Sylow
226C22×D3—34no—
227C22×D3—154no—
228C3×D4—54no—
229#67—154no—
230#67—154no—
231D12—154no—
232C4×D3—154no—
233C22×C5—22no—
234C22×C5—51no—
235C22×C5—91no—
236C22×C5—91no—
237C22×C5—91no—
238C22×C5—91no—
239C22×C5—182no—
240C22×C5—182no—
241C22×C5—182no—
242C22×C5—182no—
243C22×C5—184no—
244C22×C5—184no—
245C22×C5—451no—
246C22×C5—902no—
247C22×C5D5×C3⋊D311yescharacteristic, complemented
248C22×C5C2×C5×C3⋊D311yescharacteristic, complemented
249C2×D5—91no—
250C2×D5—182no—
251C2×D5C2×C5×C3⋊D311yescharacteristic, complemented
252C4×C5—51no—
253C4×C5—451no—
254Dic5—91no—
255Dic5C2×C5×C3⋊D311yescharacteristic, complemented
256C2×C32—21no—
257C2×C32—101no—
258C2×C32C22×C5×D511yescharacteristic
259C3⋊D3—21no—
260C3⋊D3—101no—
261C3⋊D3C5×C22⋊D512yesdirect product factor
262C2×D4—451noSylow
263C3×C5—28no—
264C3×C5D3×C22⋊D514yescomplemented
265C3×C5C5×D3×D414yescomplemented
266C22×C3—54no—
267C22×C3C5×D3×D514yescomplemented
268C2×D3—34no—
269C2×D3—34no—
270C2×D3—68no—
271C2×D3—154no—
272C2×D3—308no—
273C3×C4—54no—
274Dic3—154no—
275C2×C5—22no—
276C2×C5—22no—
277C2×C5—21no—
278C2×C5—21no—
279C2×C5—22no—
280C2×C5—92no—
281C2×C5—92no—
282C2×C5—101no—
283C2×C5—184no—
284C2×C5—182no—
285C2×C5—182no—
286C2×C5—181no—
287C2×C5—181no—
288C2×C5—901no—
289C2×C5C2×D5×C3⋊D311yescharacteristic
290C2×C5C22×C5×C3⋊D311yescharacteristic
291D5—21no—
292D5—181no—
293C32C2×C5×C22⋊D511yescharacteristic, complemented, Sylow
294C23—91no—
295C23—451no—
296D4—51no—
297D4—451no—
298D4—451no—
299D4—451no—
300C2×C4—451no—
301C2×C3—24no—
302C2×C3—104no—
303C2×C3C2×C5×D3×D514yes—
304D3—38no—
305D3—64no—
306D3—304no—
307C5—22no—
308C5C3⋊D3×C22⋊D511yescharacteristic, direct product factor
309C5C5×D4×C3⋊D311yescharacteristic, complemented
310C22—51no—
311C22—91no—
312C22—91no—
313C22—182no—
314C22—451no—
315C22—902no—
316C22C5×D5×C3⋊D311yescharacteristic, complemented
317C4—51no—
318C4—451no—
319C3C5×D3×C22⋊D514yescomplemented
320C2—21no—
321C2—92no—
322C2—101no—
323C2—181no—
324C2—901no—
325C2C2×C5×D5×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C5C2×D5×C3⋊D3
commutator, cosocleC2×C32×C5C23×C5
FrattiniC2C2×C5×D5×C3⋊D3
FittingC22×C32×C52C22
socleC2×C32×C52C23
hypercenterC22×C5D5×C3⋊D3
hypocenterC32×C5C2×C5×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2×C5 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
1. C22×C52×C3⋊D3C2semidirect
2. C2×C5×D5×C3⋊D3C2semidirect
3. C5×#67184C2semidirect
4. C32×C5×C22⋊D5C2semidirect
5. C5×#67140C2semidirect
6. C5×#67138C2semidirect
7. C5×C3⋊D3×Dic5C2semidirect
9. C22×C32×C52C22semidirect
10. C2×C5×C3⋊D15C22semidirect
13. C2×C52×C3⋊D3C22semidirect
14. C2×C32×C5×D5C22semidirect
16. C5×C3⋊Dic15C22semidirect
17. C32×C5×Dic5C22semidirect
19. C3⋊D3×C22⋊D5C5direct (N × H)
23. C3×C5×C22⋊D5D3semidirect
31. C52×C3⋊D3D4semidirect
36. C22×C5×C3⋊D3D5semidirect
37. C22×C5×C3⋊D3C2×C5semidirect
38. C2×D5×C3⋊D3C2×C5semidirect
40. #67184C2×C5semidirect
45. C32×C22⋊D5C2×C5semidirect
46. #67140C2×C5semidirect
47. #67138C2×C5semidirect
48. C3⋊D3×Dic5C2×C5semidirect
49. C22×C3×C52C2×D3semidirect
54. C2×C3×C5×D5C2×D3semidirect
57. C3×C5×Dic5C2×D3semidirect
60. C32×C52C2×D4semidirect
66. C5×C22⋊D5C3⋊D3direct (N × H)
70. C22×C32×C5C2×D5semidirect
71. C22×C32×C5C22×C5semidirect
72. C2×C3⋊D15C22×C5semidirect
82. C2×C5×C3⋊D3C2×D5semidirect
84. C2×C5×C3⋊D3C22×C5semidirect
85. C2×C32×D5C22×C5semidirect
88. C3⋊Dic15C22×C5semidirect
90. C32×Dic5C22×C5semidirect
109. C3×C22⋊D5C5×D3semidirect
116. C22×C52C2×C3⋊D3semidirect
119. C2×C5×D5C2×C3⋊D3semidirect
121. C5×Dic5C2×C3⋊D3semidirect
137. C5×C3⋊D3C22⋊D5direct (N × H)
138. C5×C3⋊D3C5×D4semidirect
142. C3×C52D3×D4semidirect
144. C22×C3⋊D3C5×D5semidirect
152. C22×C3×C5D3×D5semidirect
153. C22×C3×C5C2×C5×D3semidirect
167. C2×C3×D5C2×C5×D3semidirect
171. C3×Dic5C2×C5×D3semidirect
181. C32×C5C2×C22⋊D5semidirect
182. C32×C5C2×C5×D4semidirect
196. C22⋊D5C5×C3⋊D3direct (N × H)
199. C22×C32C2×C5×D5semidirect
204. C2×C3⋊D3C2×C5×D5semidirect
225. C52D4×C3⋊D3semidirect
247. C22×C5D5×C3⋊D3semidirect
248. C22×C5C2×C5×C3⋊D3semidirect
251. C2×D5C2×C5×C3⋊D3semidirect
255. Dic5C2×C5×C3⋊D3semidirect
261. C3⋊D3C5×C22⋊D5direct (N × H)
264. C3×C5D3×C22⋊D5semidirect
265. C3×C5C5×D3×D4semidirect
267. C22×C3C5×D3×D5semidirect
293. C32C2×C5×C22⋊D5semidirect
308. C5C3⋊D3×C22⋊D5direct (N × H)
309. C5C5×D4×C3⋊D3semidirect
316. C22C5×D5×C3⋊D3semidirect
319. C3C5×D3×C22⋊D5semidirect

Maximal subgroups

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

subgroupindex
C22×C52×C3⋊D32
C2×C5×D5×C3⋊D32
C5×#671842
C32×C5×C22⋊D52
C5×#671402
C5×#671382
C5×C3⋊D3×Dic52
C5×D3×C22⋊D53
C5×D4×C3⋊D35
C3⋊D3×C22⋊D55