Finite Group Databasebeta

← #72229937

Subgroups — #72229937

Order 2880.

1716 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
2C2×D5×#638C211yesmaximal, characteristic
3C2×#10624098C211yesmaximal, characteristic, complemented
4C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)C212yesmaximal, complemented
5(C2×AGL(1,𝔽5))⋊(C3⋊Dic3)C212yesmaximal, complemented
6#11752022—34nomaximal
7C23×C32×D5C2211yescharacteristic
8C22×D5×C3⋊D3C2211yescharacteristic
9C22×C3⋊ΓL(1,𝔽16)C2212yes—
10C22×C32×AGL(1,𝔽5)—22no—
11C22×C32×AGL(1,𝔽5)—24no—
12C22×C32×AGL(1,𝔽5)C2212yes—
13C2×#67184C411yescharacteristic, complemented
14C2×C5×#638C411yescharacteristic, complemented
15D5×#638—22no—
16D5×#638—22no—
17C2×#67139C411yescharacteristic, complemented
18C2×#67138C411yescharacteristic, complemented
19C2×D5×C3⋊Dic3C2211yescharacteristic
20#10624098—22no—
21(C2×AGL(1,𝔽5))⋊(C3⋊D3)—24no—
22#10592912—51nomaximal, Hall
23C23×C3×AGL(1,𝔽5)D314yescomplemented
24C2×D5×#67—34no—
25C2×#6763—34no—
26C2×(C2×D3)⋊AGL(1,𝔽5)—38no—
27(C2×AGL(1,𝔽5))⋊Dic3—38no—
28C23×C32×C5C2×C411yescharacteristic
29C22×C3⋊D15C2×C411yescharacteristic, complemented
30C22×C5×C3⋊D3C2×C411yescharacteristic, complemented
31C22×C32×D5—22no—
32C22×C32×D5—22no—
33C22×C32×D5C2×C411yescharacteristic
34C22×C32×D5D412yes—
35C22×C32×D5D412yes—
36C22×C32×D5C2311yescharacteristic
37C2×D5×C3⋊D3—21no—
38C2×D5×C3⋊D3—21no—
39C2×D5×C3⋊D3—22no—
40C2×D5×C3⋊D3—22no—
41C2×C3⋊ΓL(1,𝔽16)—22no—
42C2×C3⋊ΓL(1,𝔽16)—22no—
43C2×C3⋊ΓL(1,𝔽16)—22no—
44C2×C32×AGL(1,𝔽5)—28no—
45C2×C32×AGL(1,𝔽5)—22no—
46C2×C32×AGL(1,𝔽5)—22no—
47C2×C32×AGL(1,𝔽5)D414yescomplemented
48#67184—22no—
49C2×C3⋊Dic15C2×C411yescharacteristic, complemented
50C5×#638—22no—
51C2×C5×C3⋊Dic3C2×C411yescharacteristic, complemented
52#67139—22no—
53#67138—22no—
54D5×C3⋊Dic3—21no—
55D5×C3⋊Dic3—21no—
56#65779—91noHall
57C23×C32×C4—51no—
58C22×#638—51no—
59C2×#3757—51no—
60C2×#3752—52no—
61#63658—52no—
62C23×C3×D5C2×D314yes—
63C22×D3×D5—34no—
64C22×ΓL(1,𝔽16)—38no—
65C22×C3×AGL(1,𝔽5)—28no—
66C22×C3×AGL(1,𝔽5)—216no—
67C22×C3×AGL(1,𝔽5)C2×D318yes—
68C2×#1220—34no—
69C2×C5×#67—34no—
70D5×#67—68no—
71D5×#67—68no—
72C2×#1202—34no—
73C2×#1201—34no—
74C2×D5×Dic3—34no—
75#6763—68no—
76(C2×D3)⋊AGL(1,𝔽5)—616no—
77#5213—154no—
78C22×C32×C5—22no—
79C22×C32×C5C22⋊C412yescomplemented
80C22×C32×C5C22×C411yescharacteristic
81C2×C3⋊D15—21no—
82C2×C3⋊D15—22no—
83C2×C5×C3⋊D3—21no—
84C2×C5×C3⋊D3—22no—
85C2×C32×D5—22no—
86C2×C32×D5—22no—
87C2×C32×D5—24no—
88C2×C32×D5—22no—
89C2×C32×D5C22⋊C412yescomplemented
90C2×C32×D5C2×D412yes—
91C2×C32×D5C2×D411yescharacteristic
92C2×C32×D5C2×D411yescharacteristic
93C2×C32×D5#4112yes—
94D5×C3⋊D3—41no—
95D5×C3⋊D3—42no—
96D5×C3⋊D3—41no—
97C3⋊ΓL(1,𝔽16)—42no—
98C32×AGL(1,𝔽5)—24no—
99C3⋊Dic15—21no—
100C5×C3⋊Dic3—21no—
101C23×AGL(1,𝔽5)C3⋊D311yescharacteristic, complemented
102C2×D4×D5—91no—
103C2×C22⋊AGL(1,𝔽5)—91no—
104C2×C22⋊AGL(1,𝔽5)—92no—
105(C2×AGL(1,𝔽5))⋊C4—92no—
106C24×C32—51no—
107C23×C3⋊D3—51no—
108C22×C32×C4—52no—
109C22×C32×C4—104no—
110C22×C32×C4—102no—
111C2×#638—51no—
112C2×#638—51no—
113C2×#638—51no—
114C2×#638—102no—
115C2×#638—102no—
116C2×#638AGL(1,𝔽5)11yescharacteristic, complemented
117C22×C3⋊Dic3—51no—
118C22×C3⋊Dic3—52no—
119#3757—102no—
120#3752—104no—
121C23×C3×C5C4×D314yes—
122C22×D15—34no—
123C22×C5×D3—34no—
124C22×C3×D5—28no—
125C22×C3×D5—28no—
126C22×C3×D5C4×D314yes—
127C22×C3×D5D1218yes—
128C22×C3×D5#6718yes—
129C22×C3×D5C22×D314yes—
130C2×D3×D5—64no—
131C2×D3×D5—68no—
132C2×D3×D5—68no—
133C2×D3×D5—64no—
134C2×ΓL(1,𝔽16)—68no—
135C2×ΓL(1,𝔽16)—68no—
136C2×ΓL(1,𝔽16)—68no—
137C2×C3×AGL(1,𝔽5)—232no—
138C2×C3×AGL(1,𝔽5)—28no—
139C2×C3×AGL(1,𝔽5)—28no—
140C2×C3×AGL(1,𝔽5)#67116yescomplemented
141#1220—68no—
142C2×Dic15—34no—
143C5×#67—68no—
144C2×C5×Dic3—34no—
145#1202—68no—
146#1201—68no—
147D5×Dic3—64no—
148D5×Dic3—64no—
149C23×C3×C4—54no—
150C22×#67—154no—
151C2×#217—154no—
152C2×(C2×D3)⋊C4—158no—
153C3⋊#95—158no—
154C2×C32×C5—22no—
155C2×C32×C5C2×C22⋊C411yescharacteristic
156C2×C32×C5C4×D412yes—
157C3⋊D15—41no—
158C5×C3⋊D3—41no—
159C32×D5—22no—
160C32×D5#12011yescharacteristic
161C32×D5#12112yes—
162C32×D5#12311yescharacteristic
163C23×D5—91no—
164C23×D5C2×C3⋊D311yescharacteristic
165C22×AGL(1,𝔽5)—22no—
166C22×AGL(1,𝔽5)—24no—
167C22×AGL(1,𝔽5)—92no—
168C22×AGL(1,𝔽5)C2×C3⋊D312yes—
169C2×C5×D4—91no—
170C2×C22⋊D5—91no—
171C2×C22⋊D5—91no—
172D4×D5—182no—
173D4×D5—182no—
174C2×D20—91no—
175C2×C4×D5—91no—
176C22⋊AGL(1,𝔽5)—182no—
177C22⋊AGL(1,𝔽5)—184no—
178C23×C32—51no—
179C23×C32—52no—
180C23×C32—52no—
181C23×C32—51no—
182C23×C32—102no—
183C23×C32—102no—
184C23×C32C2×AGL(1,𝔽5)11yescharacteristic
185C22×C3⋊D3—51no—
186C22×C3⋊D3—102no—
187C22×C3⋊D3—102no—
188C22×C3⋊D3—101no—
189C22×C3⋊D3—101no—
190C22×C3⋊D3C2×AGL(1,𝔽5)11yescharacteristic, complemented
191C2×C32×C4—54no—
192C2×C32×C4—102no—
193C2×C32×C4—102no—
194C2×C32×C4—108no—
195#638—22no—
196#638—102no—
197#638—102no—
198#638—102no—
199C2×C3⋊Dic3—51no—
200C2×C3⋊Dic3—101no—
201C2×C3⋊Dic3—101no—
202C2×C3⋊Dic3—102no—
203C2×C3⋊Dic3—102no—
204C2×C3⋊Dic3—102no—
205C2×C3⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
206#386—451noSylow
207C22×C3×C5—28no—
208C22×C3×C5(C2×D3)⋊C418yescomplemented
209C22×C3×C5C2×C4×D314yes—
210C2×D15—64no—
211C2×D15—68no—
212C2×C5×D3—64no—
213C2×C5×D3—68no—
214C2×C3×D5—28no—
215C2×C3×D5—28no—
216C2×C3×D5—216no—
217C2×C3×D5—28no—
218C2×C3×D5(C2×D3)⋊C418yescomplemented
219C2×C3×D5C2×D1214yes—
220C2×C3×D5#23518yes—
221C2×C3×D5C2×#6714yes—
222C2×C3×D5C2×#6718yes—
223D3×D5—124no—
224D3×D5—128no—
225D3×D5—124no—
226ΓL(1,𝔽16)—128no—
227C3×AGL(1,𝔽5)—216no—
228Dic15—64no—
229C5×Dic3—64no—
230C24×C3—54no—
231C23×D3—154no—
232C22×C3×C4—58no—
233C22×C3×C4—1016no—
234C22×C3×C4—108no—
235C2×#67—34no—
236C2×#67—154no—
237C2×#67—154no—
238C2×#67—154no—
239C2×#67—308no—
240C2×#67—308no—
241C22×Dic3—154no—
242C22×Dic3—158no—
243#217—308no—
244(C2×D3)⋊C4—3016no—
245C32×C5#38611yescharacteristic, complemented, Hall
246C23×C5—91no—
247C23×C5C4×C3⋊D311yescharacteristic
248C22×D5—22no—
249C22×D5—22no—
250C22×D5—91no—
251C22×D5—181no—
252C22×D5—182no—
253C22×D5—182no—
254C22×D5—181no—
255C22×D5C4×C3⋊D311yescharacteristic
256C22×D5C4⋊(C3⋊D3)12yes—
257C22×D5#63812yes—
258C22×D5C22×C3⋊D311yescharacteristic
259C2×AGL(1,𝔽5)—28no—
260C2×AGL(1,𝔽5)—22no—
261C2×AGL(1,𝔽5)—22no—
262C2×AGL(1,𝔽5)—182no—
263C2×AGL(1,𝔽5)—182no—
264C2×AGL(1,𝔽5)—182no—
265C2×AGL(1,𝔽5)#63814yescomplemented
266C5×D4—182no—
267C2×C4×C5—91no—
268C22⋊D5—182no—
269C22⋊D5—182no—
270C2×Dic5—91no—
271D20—182no—
272C4×D5—181no—
273C4×D5—181no—
274C22×C32—22no—
275C22×C32—52no—
276C22×C32—52no—
277C22×C32—51no—
278C22×C32—51no—
279C22×C32—52no—
280C22×C32—102no—
281C22×C32—104no—
282C22×C32—102no—
283C22×C32—102no—
284C22×C32C22⋊AGL(1,𝔽5)12yescomplemented
285C22×C32C22×AGL(1,𝔽5)11yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—21no—
288C2×C3⋊D3—102no—
289C2×C3⋊D3—101no—
290C2×C3⋊D3—201no—
291C2×C3⋊D3—202no—
292C2×C3⋊D3—201no—
293C32×C4—104no—
294C3⋊Dic3—21no—
295C3⋊Dic3—101no—
296C3⋊Dic3—202no—
297C22×D4—451no—
298C23×C4—51no—
299C2×C22⋊C4—452no—
300C2×C22⋊C4—451no—
301#95—452no—
302C2×C3×C5—28no—
303C2×C3×C5C2×(C2×D3)⋊C414yes—
304C2×C3×C5C4×#6718yes—
305D15—124no—
306C3×D5—28no—
307C3×D5#92914yes—
308C3×D5#93018yes—
309C3×D5#95314yes—
310C5×D3—124no—
311C23×C3—54no—
312C23×C3—58no—
313C23×C3—58no—
314C23×C3—54no—
315C23×C3—108no—
316C23×C3—108no—
317C23×C3D3×AGL(1,𝔽5)14yes—
318C22×D3—34no—
319C22×D3—154no—
320C22×D3—308no—
321C22×D3—304no—
322C22×D3—304no—
323C22×D3—308no—
324C2×C3×C4—516no—
325C2×C3×C4—108no—
326C2×C3×C4—108no—
327C2×C3×C4—1032no—
328#67—68no—
329#67—308no—
330#67—308no—
331#67—308no—
332C2×Dic3—34no—
333C2×Dic3—154no—
334C2×Dic3—304no—
335C2×Dic3—304no—
336C2×Dic3—308no—
337C2×Dic3—308no—
338C2×Dic3—308no—
339C22×C5—22no—
340C22×C5—181no—
341C22×C5—182no—
342C22×C5#375212yescomplemented
343C22×C5C2×C4×C3⋊D311yescharacteristic
344C2×D5—22no—
345C2×D5—22no—
346C2×D5—24no—
347C2×D5—22no—
348C2×D5—181no—
349C2×D5—182no—
350C2×D5—361no—
351C2×D5—362no—
352C2×D5—361no—
353C2×D5#375212yescomplemented
354C2×D5C2×C4⋊(C3⋊D3)11yescharacteristic
355C2×D5#382812yes—
356C2×D5C2×#63811yescharacteristic
357C2×D5C2×#63812yes—
358AGL(1,𝔽5)—24no—
359AGL(1,𝔽5)—362no—
360C4×C5—181no—
361Dic5—181no—
362C2×C32—22no—
363C2×C32—51no—
364C2×C32—52no—
365C2×C32—51no—
366C2×C32—102no—
367C2×C32D4×AGL(1,𝔽5)12yes—
368C2×C32C2×C22⋊AGL(1,𝔽5)11yescharacteristic
369C3⋊D3—41no—
370C3⋊D3—201no—
371C24—51no—
372C24—451no—
373C2×D4—91no—
374C2×D4—451no—
375C2×D4—451no—
376C2×D4—451no—
377C2×D4—902no—
378C2×D4—902no—
379C22×C4—52no—
380C22×C4—104no—
381C22×C4—102no—
382C22×C4—451no—
383C22×C4—452no—
384C22⋊C4—904no—
385C22⋊C4—902no—
386C3×C5#521314yescomplemented
387C22×C3—28no—
388C22×C3—54no—
389C22×C3—54no—
390C22×C3—58no—
391C22×C3—58no—
392C22×C3—58no—
393C22×C3—108no—
394C22×C3—1016no—
395C22×C3—108no—
396C22×C3—108no—
397C22×C3(C2×D3)⋊AGL(1,𝔽5)18yescomplemented
398C22×C3C2×D3×AGL(1,𝔽5)14yes—
399C2×D3—64no—
400C2×D3—68no—
401C2×D3—304no—
402C2×D3—308no—
403C2×D3—608no—
404C2×D3—604no—
405C2×D3—604no—
406C3×C4—1016no—
407Dic3—64no—
408Dic3—304no—
409Dic3—608no—
410C2×C5—22no—
411C2×C5—361no—
412C2×C5C2×#375211yescharacteristic
413C2×C5C4×#63812yes—
414D5—22no—
415D5—361no—
416D5#6413811yescharacteristic
417D5#6413912yes—
418D5#6416211yescharacteristic
419C32#6577911yescharacteristic, complemented, Sylow
420C23—51no—
421C23—51no—
422C23—52no—
423C23—52no—
424C23—91no—
425C23—102no—
426C23—102no—
427C23—451no—
428C23—902no—
429C23—901no—
430C23—901no—
431C23—902no—
432C23C3⋊D3×AGL(1,𝔽5)11yescharacteristic
433D4—182no—
434D4—902no—
435D4—902no—
436D4—902no—
437C2×C4—54no—
438C2×C4—91no—
439C2×C4—102no—
440C2×C4—102no—
441C2×C4—108no—
442C2×C4—451no—
443C2×C4—901no—
444C2×C4—901no—
445C2×C4—902no—
446C2×C4—902no—
447C2×C4—902no—
448C2×C3—28no—
449C2×C3—58no—
450C2×C3—54no—
451C2×C3—54no—
452C2×C3—108no—
453C2×C3C2×(C2×D3)⋊AGL(1,𝔽5)14yes—
454C2×C3AGL(1,𝔽5)×#6718yes—
455D3—124no—
456D3—604no—
457C5#1059291211yescharacteristic, complemented, Sylow
458C22—22no—
459C22—51no—
460C22—51no—
461C22—52no—
462C22—52no—
463C22—52no—
464C22—102no—
465C22—104no—
466C22—102no—
467C22—102no—
468C22—181no—
469C22—182no—
470C22—901no—
471C22—902no—
472C22—1801no—
473C22—1802no—
474C22—1801no—
475C22(C2×AGL(1,𝔽5))⋊(C3⋊D3)12yescomplemented
476C22C2×C3⋊D3×AGL(1,𝔽5)11yescharacteristic
477C4—104no—
478C4—181no—
479C4—901no—
480C4—1802no—
481C3#1175202214yescomplemented
482C2—22no—
483C2—51no—
484C2—52no—
485C2—51no—
486C2—102no—
487C2—361no—
488C2—1801no—
489C2C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)11yescharacteristic
490C2AGL(1,𝔽5)×#63812yes—
subgroupG/H
center, FrattiniC22C2×C3⋊D3×AGL(1,𝔽5)
commutator, socleC22×C32×C5C22×C4
FittingC23×C32×C5C2×C4
prime radicalC2×D5×#638C2
cosocleC22×C32×D5C23
hypercenterC23C3⋊D3×AGL(1,𝔽5)
hypocenterC32×C5#386
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C32×AGL(1,𝔽5)C2semidirect
3. C2×#10624098C2semidirect
4. C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)C2semidirect
5. (C2×AGL(1,𝔽5))⋊(C3⋊Dic3)C2semidirect
13. C2×#67184C4semidirect
14. C2×C5×#638C4semidirect
17. C2×#67139C4semidirect
18. C2×#67138C4semidirect
23. C23×C3×AGL(1,𝔽5)D3semidirect
29. C22×C3⋊D15C2×C4semidirect
30. C22×C5×C3⋊D3C2×C4semidirect
47. C2×C32×AGL(1,𝔽5)D4semidirect
49. C2×C3⋊Dic15C2×C4semidirect
51. C2×C5×C3⋊Dic3C2×C4semidirect
79. C22×C32×C5C22⋊C4semidirect
89. C2×C32×D5C22⋊C4semidirect
101. C23×AGL(1,𝔽5)C3⋊D3semidirect
116. C2×#638AGL(1,𝔽5)semidirect
140. C2×C3×AGL(1,𝔽5)#67semidirect
190. C22×C3⋊D3C2×AGL(1,𝔽5)semidirect
205. C2×C3⋊Dic3C2×AGL(1,𝔽5)semidirect
208. C22×C3×C5(C2×D3)⋊C4semidirect
218. C2×C3×D5(C2×D3)⋊C4semidirect
245. C32×C5#386semidirect
265. C2×AGL(1,𝔽5)#638semidirect
284. C22×C32C22⋊AGL(1,𝔽5)semidirect
342. C22×C5#3752semidirect
353. C2×D5#3752semidirect
386. C3×C5#5213semidirect
397. C22×C3(C2×D3)⋊AGL(1,𝔽5)semidirect
419. C32#65779semidirect
457. C5#10592912semidirect
475. C22(C2×AGL(1,𝔽5))⋊(C3⋊D3)semidirect
481. C3#11752022semidirect

Maximal subgroups

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

subgroupindex
C23×C32×AGL(1,𝔽5)2
C2×D5×#6382
C2×#106240982
C2×(C2×AGL(1,𝔽5))⋊(C3⋊D3)2
(C2×AGL(1,𝔽5))⋊(C3⋊Dic3)2
#117520223
#105929125