Finite Group Databasebeta

← #14301975

Subgroups — #14301975

Order 1920.

884 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
2#11754526C211yesmaximal, characteristic
3#11752415C211yesmaximal, characteristic, complemented
4C2×#91156C212yesmaximal, complemented
5#11746526C212yesmaximal, complemented
6#10604489—31nomaximal, Hall
7C23×C3×AGL(1,𝔽5)—21no—
8C22×C3×C4×D5C2211yescharacteristic
9C2×#6772C2211yescharacteristic
10C2×#6758C2212yes—
11C2×C3×C4×AGL(1,𝔽5)—21no—
12C2×C3×C4×AGL(1,𝔽5)—24no—
13C2×C3×C4×AGL(1,𝔽5)C2212yes—
14#91813C411yescharacteristic, complemented
15C5×#930C411yescharacteristic, complemented
16#91563—21no—
17Dic5×#67—21no—
18#91449C411yescharacteristic, complemented
19#91447—21no—
20#91440—21no—
21#91360C411yescharacteristic, complemented
22#91358C2211yescharacteristic
23#91241—21no—
24#91232—21no—
25#91156—24no—
26#71532—51nomaximal, Hall
27C22×C4×AGL(1,𝔽5)D311yescharacteristic, complemented
28#65940—31no—
29C2×#4008—32no—
30#65756—31no—
31#64875—32no—
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—24no—
34C22×C3×AGL(1,𝔽5)—24no—
35C22×C3×AGL(1,𝔽5)—24no—
36C22×C3×AGL(1,𝔽5)—22no—
37C22×C3×C4×C5C2×C411yescharacteristic
38C2×Dic30C2×C411yescharacteristic, complemented
39C2×C5×#67—21no—
40C2×C5×D12C2×C411yescharacteristic, complemented
41C22×C3×Dic5D411yescharacteristic
42C2×C3×C4×D5—22no—
43C2×C3×C4×D5—22no—
44C2×C3×C4×D5C2×C411yescharacteristic
45C2×C3×C4×D5D411yescharacteristic
46C2×C3×C4×D5D411yescharacteristic
47C2×C3×C4×D5C2311yescharacteristic
48C2×#1201—21no—
49C2×#1201—21no—
50C2×D3×Dic5—21no—
51#6772—22no—
52#6772—22no—
53C2×C3⋊#170—22no—
54#6758—22no—
55#6758—22no—
56C3×C4×AGL(1,𝔽5)—24no—
57C3×C4×AGL(1,𝔽5)—22no—
58C3×C4×AGL(1,𝔽5)D414yescomplemented
59#6719—21no—
60C4⋊Dic15C2×C411yescharacteristic, complemented
61#6712—21no—
62C5×(C2×D3)⋊C4—21no—
63C5×C4⋊Dic3C2×C411yescharacteristic, complemented
64Dic3⋊Dic5—21no—
65(C2×D3)⋊Dic5—21no—
66#6665—21no—
67#6665—21no—
68Dic3×Dic5—21no—
69C22×C3×C42—51no—
70#5791—51no—
71#5207—51no—
72C2×#805—52no—
73#4523—52no—
74C23×AGL(1,𝔽5)—21no—
75C2×#711—31no—
76C22×C4×D5C2×D311yescharacteristic
77C2×C4×AGL(1,𝔽5)—21no—
78C2×C4×AGL(1,𝔽5)—24no—
79C2×C4×AGL(1,𝔽5)C2×D312yes—
80C2×#700—32no—
81C5×#121—31no—
82#4084—31no—
83#4083—61no—
84D4×Dic5—61no—
85#4070—31no—
86#4043—31no—
87#4038—31no—
88#4029—61no—
89#4027—61no—
90C22⋊#170—61no—
91#4008—64no—
92#4003—61no—
93#1896—151noSylow
94C23×C3×C5C22⋊C411yescharacteristic
95C22×C5×D3—21no—
96C22×C3×D5—22no—
97C22×C3×D5—22no—
98C22×C3×D5—21no—
99C22×C3×D5—21no—
100C22×C3×D5C22⋊C411yescharacteristic
101C22×C3×D5C2×D411yescharacteristic
102C2×C3×AGL(1,𝔽5)—216no—
103C2×C3×AGL(1,𝔽5)—24no—
104C2×C3×AGL(1,𝔽5)—24no—
105C2×C3×AGL(1,𝔽5)—24no—
106C2×C3×C4×C5—22no—
107C2×C3×C4×C5C22⋊C411yescharacteristic
108C2×C3×C4×C5C22×C411yescharacteristic
109C2×Dic15—21no—
110C2×Dic15—21no—
111Dic30—22no—
112C5×#67—42no—
113C2×C5×Dic3—21no—
114C5×D12—22no—
115C2×C3×Dic5—22no—
116C2×C3×Dic5C22⋊C411yescharacteristic
117C2×C3×Dic5C2×D411yescharacteristic
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—22no—
121C3×C4×D5—22no—
122C3×C4×D5C2×D412yes—
123C3×C4×D5#4112yes—
124#1201—42no—
125#1201—41no—
126#1201—41no—
127D3×Dic5—42no—
128C3⋊#170—42no—
129C23×C3×C4—51no—
130C23×C3×C4—101no—
131C2×#235—51no—
132C2×C3×C42—52no—
133C2×C3×C42—104no—
134C2×C3×C42—101no—
135#953—101no—
136#930—51no—
137#930AGL(1,𝔽5)11yescharacteristic, complemented
138#929—101no—
139C4×#67—101no—
140C4×#67—101no—
141#926—51no—
142#924—51no—
143#924—51no—
144C2×#208—52no—
145#830—101no—
146#830—101no—
147#805—104no—
148C23×D5#6711yescharacteristic
149C22×AGL(1,𝔽5)—24no—
150C22×AGL(1,𝔽5)—24no—
151C22×AGL(1,𝔽5)—24no—
152C22×AGL(1,𝔽5)—22no—
153C2×C5×D4—31no—
154C2×C5×D4—61no—
155C22×C4×C5C4×D311yescharacteristic
156C2×C22⋊D5—61no—
157C2×C22⋊D5—61no—
158C22×Dic5—61no—
159C22×Dic5D1211yescharacteristic
160#711—62no—
161#711—62no—
162C2×C4×D5—22no—
163C2×C4×D5—22no—
164C2×C4×D5C4×D311yescharacteristic
165C2×C4×D5D1211yescharacteristic
166C2×C4×D5#6711yescharacteristic
167C2×C4×D5C22×D311yescharacteristic
168C2×Dic10—31no—
169C2×#170—62no—
170C4×AGL(1,𝔽5)—24no—
171C4×AGL(1,𝔽5)—22no—
172C4×AGL(1,𝔽5)#6714yescomplemented
173#700—62no—
174#700—62no—
175C5×C4⋊C4—31no—
176C5×C22⋊C4—61no—
177C22⋊Dic5—61no—
178C22⋊Dic5—61no—
179#685—61no—
180#685—61no—
181C4⋊Dic5—31no—
182C4⋊Dic5—61no—
183#683—61no—
184C4×Dic5—61no—
185#525—151no—
186C22×C42—51no—
187C2×#104—152no—
188#407—151no—
189#339—152no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5#10412yescomplemented
193C22×C3×C5C2×C22⋊C411yescharacteristic
194C2×C5×D3—41no—
195C2×C5×D3—41no—
196C2×C5×D3—41no—
197C2×C3×D5—24no—
198C2×C3×D5—24no—
199C2×C3×D5—21no—
200C2×C3×D5—21no—
201C2×C3×D5—21no—
202C2×C3×D5—21no—
203C2×C3×D5#10412yescomplemented
204C2×C3×D5#12011yescharacteristic
205C2×C3×D5#12311yescharacteristic
206C3×AGL(1,𝔽5)—28no—
207C3×C4×C5—21no—
208C3×C4×C5C4×D412yes—
209Dic15—41no—
210Dic15—41no—
211C3×Dic5—21no—
212C3×Dic5#12112yes—
213C5×Dic3—41no—
214C24×C3—51no—
215C22×C3×C4—51no—
216C22×C3×C4—51no—
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—102no—
221C22×C3×C4—102no—
222C22×C3×C4—102no—
223C22×C3×C4—104no—
224C22×C3×C4—104no—
225C22×C3×C4—104no—
226C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
227C2×#67—21no—
228C2×#67—101no—
229C2×#67—101no—
230#235—102no—
231#235—102no—
232C2×D12C2×AGL(1,𝔽5)11yescharacteristic, complemented
233C2×C4×D3—101no—
234C2×Dic6—51no—
235C3×C42—54no—
236C3×C42—102no—
237C3×C42—104no—
238#217—101no—
239#217—101no—
240#217—101no—
241(C2×D3)⋊C4—21no—
242(C2×D3)⋊C4—101no—
243C4⋊Dic3—51no—
244C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic, complemented
245Dic3⋊C4—101no—
246Dic3⋊C4—101no—
247C4×Dic3—101no—
248#208—102no—
249#208—102no—
250C2×C3⋊C8—102no—
251C23×C5—61no—
252C23×C5(C2×D3)⋊C411yescharacteristic
253C22×D5—22no—
254C22×D5—22no—
255C22×D5—21no—
256C22×D5—21no—
257C22×D5(C2×D3)⋊C411yescharacteristic
258C22×D5C2×#6711yescharacteristic
259C2×AGL(1,𝔽5)—216no—
260C2×AGL(1,𝔽5)—24no—
261C2×AGL(1,𝔽5)—24no—
262C2×AGL(1,𝔽5)—24no—
263C5×D4—62no—
264C5×D4—122no—
265C2×C4×C5—22no—
266C2×C4×C5—61no—
267C2×C4×C5(C2×D3)⋊C411yescharacteristic
268C2×C4×C5C2×C4×D311yescharacteristic
269C22⋊D5—122no—
270C22⋊D5—121no—
271C22⋊D5—121no—
272C2×Dic5—22no—
273C2×Dic5—61no—
274C2×Dic5—61no—
275C2×Dic5—122no—
276C2×Dic5(C2×D3)⋊C411yescharacteristic
277C2×Dic5C2×D1211yescharacteristic
278C4×D5—21no—
279C4×D5—21no—
280C4×D5—22no—
281C4×D5—22no—
282C4×D5#23512yes—
283C4×D5C2×#6712yes—
284Dic10—62no—
285#170—122no—
286C2×#41—151no—
287C23×C4—51no—
288C23×C4—101no—
289C2×M16—152no—
290#123—301no—
291C22⋊Q8—151no—
292C22⋊Q8—151no—
293#121—31no—
294#121—151no—
295#120—301no—
296C4×D4—301no—
297C4×D4—301no—
298#117—151no—
299C2×C42—52no—
300C2×C42—104no—
301C2×C42—101no—
302#104—304no—
303C22⋊C8—301no—
304C22⋊C8—301no—
305C2×C3×C5—22no—
306C2×C3×C5#38611yescharacteristic
307C2×C3×C5C2×#10411yescharacteristic
308C2×C3×C5C2×#10411yescharacteristic
309C3×D5—21no—
310C3×D5—22no—
311C3×D5—21no—
312C5×D3—81no—
313C23×C3—51no—
314C23×C3—51no—
315C23×C3—101no—
316C23×C3—102no—
317C23×C3—101no—
318C23×C3—102no—
319C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
320C22×D3—21no—
321C2×C3×C4—22no—
322C2×C3×C4—51no—
323C2×C3×C4—52no—
324C2×C3×C4—52no—
325C2×C3×C4—51no—
326C2×C3×C4—102no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—102no—
330C2×C3×C4—104no—
331C2×C3×C4—1016no—
332C2×C3×C4—104no—
333C2×C3×C4—104no—
334C2×C3×C4—102no—
335C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
336C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
337#67—42no—
338#67—201no—
339#67—201no—
340#67—202no—
341C2×Dic3—21no—
342C2×Dic3—101no—
343C2×Dic3—101no—
344D12—22no—
345C4×D3—202no—
346Dic6—102no—
347C3⋊C8—202no—
348C22×C5—21no—
349C22×C5—21no—
350C22×C5—121no—
351C22×C5—121no—
352C22×C5—121no—
353C22×C5#80512yescomplemented
354C22×C5C2×(C2×D3)⋊C411yescharacteristic
355C2×D5—24no—
356C2×D5—24no—
357C2×D5—21no—
358C2×D5—21no—
359C2×D5—21no—
360C2×D5—21no—
361C2×D5#80512yescomplemented
362C2×D5#92911yescharacteristic
363C2×D5#95311yescharacteristic
364AGL(1,𝔽5)—28no—
365C4×C5—21no—
366C4×C5—121no—
367C4×C5C4×#6712yes—
368Dic5—21no—
369Dic5—121no—
370Dic5—121no—
371Dic5#93012yes—
372C24—51no—
373#41—302no—
374#41—302no—
375C2×Q8—151no—
376C2×D4—31no—
377C2×D4—61no—
378C2×D4—301no—
379C2×D4—301no—
380C22×C4—51no—
381C22×C4—51no—
382C22×C4—51no—
383C22×C4—51no—
384C22×C4—51no—
385C22×C4—102no—
386C22×C4—102no—
387C22×C4—102no—
388C22×C4—104no—
389C22×C4—104no—
390C22×C4—104no—
391C22×C4—301no—
392C22×C4D3×AGL(1,𝔽5)11yescharacteristic
393M16—302no—
394M16—302no—
395C2×C8—302no—
396C4⋊C4—31no—
397C4⋊C4—151no—
398C4⋊C4—301no—
399C4⋊C4—301no—
400C22⋊C4—61no—
401C22⋊C4—301no—
402C22⋊C4—301no—
403C22⋊C4—301no—
404C22⋊C4—301no—
405C42—54no—
406C42—102no—
407C42—104no—
408C42—301no—
409C3×C5#189611yescharacteristic, complemented, Hall
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—52no—
413C22×C3—51no—
414C22×C3—51no—
415C22×C3—104no—
416C22×C3—104no—
417C22×C3—101no—
418C22×C3—101no—
419C22×C3—101no—
420C22×C3—101no—
421C22×C3#400812yescomplemented
422C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
423C2×D3—41no—
424C2×D3—41no—
425C2×D3—41no—
426C3×C4—21no—
427C3×C4—52no—
428C3×C4—108no—
429C3×C4—101no—
430C3×C4D4×AGL(1,𝔽5)12yes—
431Dic3—41no—
432Dic3—201no—
433Dic3—201no—
434C2×C5—22no—
435C2×C5—241no—
436C2×C5C2×#80511yescharacteristic
437C2×C5C2×#80511yescharacteristic
438C2×C5#521311yescharacteristic
439D5—21no—
440D5—22no—
441D5—21no—
442C23—51no—
443C23—51no—
444C23—61no—
445C23—101no—
446C23—102no—
447C23—101no—
448C23—102no—
449C23(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
450Q8—302no—
451D4—62no—
452D4—122no—
453D4—601no—
454D4—601no—
455D4—602no—
456C2×C4—22no—
457C2×C4—51no—
458C2×C4—52no—
459C2×C4—52no—
460C2×C4—51no—
461C2×C4—61no—
462C2×C4—102no—
463C2×C4—101no—
464C2×C4—101no—
465C2×C4—102no—
466C2×C4—104no—
467C2×C4—1016no—
468C2×C4—104no—
469C2×C4—104no—
470C2×C4—102no—
471C2×C4—301no—
472C2×C4—301no—
473C2×C4—602no—
474C2×C4(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
475C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
476C8—602no—
477C2×C3—22no—
478C2×C3—101no—
479C2×C3—102no—
480C2×C3—101no—
481C2×C3C2×#400811yescharacteristic
482C2×C3#6577911yescharacteristic
483C2×C3D5⋊#10411yescharacteristic
484D3—81no—
485C5#7153211yescharacteristic, complemented, Sylow
486C22—21no—
487C22—21no—
488C22—52no—
489C22—51no—
490C22—51no—
491C22—104no—
492C22—104no—
493C22—101no—
494C22—101no—
495C22—101no—
496C22—101no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22#9115612yescomplemented
501C22C2×(C2×D3)⋊AGL(1,𝔽5)11yescharacteristic
502C4—21no—
503C4—52no—
504C4—108no—
505C4—101no—
506C4—121no—
507C4—601no—
508C4—601no—
509C4AGL(1,𝔽5)×#6712yes—
510C3#1060448911yescharacteristic, complemented, Sylow
511C2—22no—
512C2—101no—
513C2—102no—
514C2—101no—
515C2—241no—
516C2C2×#9115611yescharacteristic
517C2D5⋊#80511yescharacteristic
518C2#1175202211yescharacteristic
subgroupG/H
centerC22C2×(C2×D3)⋊AGL(1,𝔽5)
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC22×C3×C4×C5C2×C4
socleC22×C3×C5C2×C22⋊C4
prime radical#11754526C2
cosocleC2×C3×C4×D5C23
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#1896
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×AGL(1,𝔽5)C2semidirect
3. #11752415C2semidirect
4. C2×#91156C2semidirect
5. #11746526C2semidirect
14. #91813C4semidirect
15. C5×#930C4semidirect
18. #91449C4semidirect
21. #91360C4semidirect
27. C22×C4×AGL(1,𝔽5)D3semidirect
38. C2×Dic30C2×C4semidirect
40. C2×C5×D12C2×C4semidirect
58. C3×C4×AGL(1,𝔽5)D4semidirect
60. C4⋊Dic15C2×C4semidirect
63. C5×C4⋊Dic3C2×C4semidirect
137. #930AGL(1,𝔽5)semidirect
172. C4×AGL(1,𝔽5)#67semidirect
192. C22×C3×C5#104semidirect
203. C2×C3×D5#104semidirect
232. C2×D12C2×AGL(1,𝔽5)semidirect
244. C4⋊Dic3C2×AGL(1,𝔽5)semidirect
353. C22×C5#805semidirect
361. C2×D5#805semidirect
409. C3×C5#1896semidirect
421. C22×C3#4008semidirect
485. C5#71532semidirect
500. C22#91156semidirect
510. C3#10604489semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×AGL(1,𝔽5)2
#117545262
#117524152
C2×#911562
#117465262
#106044893
#715325