Finite Group Databasebeta

← #14460162

Subgroups — #14460162

Order 1920.

796 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#65889C211yesmaximal, characteristic, complemented
2#11752715C211yesmaximal, characteristic, complemented
3C4×#6715C211yesmaximal, characteristic, complemented
4C5×#5008C211yesmaximal, characteristic, complemented
5#11749026C212yesmaximal, complemented
6#11748961C212yesmaximal, complemented
7#11747443C212yesmaximal, complemented
8#11747205C212yesmaximal, complemented
9#11747085C211yesmaximal, characteristic, complemented
10#11746996C211yesmaximal, characteristic, complemented
11#11746924C211yesmaximal, characteristic, complemented
12#10612807—31nomaximal, Hall
13C2×C3×#711C2212yes—
14C3×C4×C5×D4C2211yescharacteristic
15C2×#6715C2211yescharacteristic
16C4×D60C2211yescharacteristic
17C5×#869C2211yescharacteristic
18C3×#4076—21no—
19C2×C3×C4⋊Dic5—21no—
20C3×#4043C2212yes—
21C3×#4032—22no—
22C3×C4×D20C2211yescharacteristic
23C3×#4015C2211yescharacteristic
24#91465—21no—
25#91455—22no—
26#91450C2212yes—
27#91358C2212yes—
28#91347—21no—
29C2×#6658C2212yes—
30C2×D5×C3⋊C8C2212yes—
31#91116C2211yescharacteristic
32#91093C2211yescharacteristic
33#91091C2211yescharacteristic
34C4×C3⋊#168C2211yescharacteristic
35C5×#832C2212yes—
36C5×#808C2212yes—
37C5×C4⋊(C3⋊C8)C2211yescharacteristic
38#90971C2212yes—
39#90970C2212yes—
40#90970C2212yes—
41#90955C2212yes—
42#83511—51nomaximal, Hall
43#66068—31no—
44#65889D311yescharacteristic, complemented
45C5×#476—31no—
46#65382—31no—
47C4×#686—31no—
48#65156—32no—
49#65153—32no—
50#65130—31no—
51#65123—31no—
52#65070—32no—
53#65061—32no—
54C2×C3×C5×D4C2311yescharacteristic
55C22×C3×C4×C5—21no—
56C2×D60C2311yescharacteristic
57C2×C4×D15—21no—
58C2×C5×Dic6—21no—
59C2×C3×C22⋊D5—22no—
60C22×C3×Dic5—22no—
61C3×#711—24no—
62C3×#711—42no—
63C2×C3×D20D411yescharacteristic
64C2×C3×C4×D5C2312yes—
65C2×C3×Dic10C2312yes—
66C2×#1202—22no—
67C3×C5×C4⋊C4C2311yescharacteristic
68C3×C5×C22⋊C4—21no—
69C3×C42×C5C2311yescharacteristic
70#6715—22no—
71#6714—21no—
72C4⋊Dic15C2311yescharacteristic
73C2×C3⋊#168C2311yescharacteristic
74C2×C3⋊#168C2311yescharacteristic
75C5×C4⋊Dic3—21no—
76C5×C4⋊Dic3C2312yes—
77C2×C5×C3⋊C8C2312yes—
78C3×#685—22no—
79C3×#685D412yes—
80C3×C4⋊Dic5—21no—
81C3×C4⋊Dic5—21no—
82C3×C4⋊Dic5—22no—
83C3×C4⋊Dic5D411yescharacteristic
84#6667—22no—
85#6665—22no—
86Dic3×Dic5—22no—
87#6658—42no—
88D5×C3⋊C8—42no—
89C3×#547—51no—
90#5563—51no—
91#5238—52no—
92#5233—52no—
93#5083—51no—
94#5079—51no—
95#5048—52no—
96#5045—52no—
97C4×#213—51no—
98D4×C3⋊C8—51no—
99#5008D511yescharacteristic, complemented
100C2×#711C2×D312yes—
101C5×C4⋊Q8—31no—
102C4×C5×D4C2×D311yescharacteristic
103#4093—62no—
104C2×#686—31no—
105#4076—21no—
106C2×C4⋊Dic5—21no—
107C2×#677—32no—
108C2×C8×D5—32no—
109#4043C2×D312yes—
110#4041—32no—
111#4041—61no—
112#4038—32no—
113#4038—61no—
114#4032—22no—
115C4×D20—31no—
116C4×D20C2×D311yescharacteristic
117#4015C2×D311yescharacteristic
118C5×#106—32no—
119C5×C4⋊C8—31no—
120C5×#102—32no—
121#3964—31no—
122#3952—32no—
123#3948—32no—
124#3942—32no—
125#3942—32no—
126#3941—31no—
127#3939—31no—
128C4×#168—31no—
129#3303—151noSylow
130C23×C3×C5—21no—
131C22×D15—21no—
132C22×C3×D5C2×D412yes—
133C3×C5×D4—21no—
134C3×C5×D4#4112yes—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—42no—
137C2×C3×C4×C5C2×D411yescharacteristic
138C2×C3×C4×C5C2×D411yescharacteristic
139C2×C3×C4×C5C2411yescharacteristic
140C2×Dic15—21no—
141D60—21no—
142D60—21no—
143C4×D15—42no—
144C2×C5×Dic3—22no—
145C5×Dic6—81no—
146C3×C22⋊D5—44no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—44no—
149C2×C3×Dic5C2×D412yes—
150C3×D20—22no—
151C3×C4×D5—22no—
152C3×C4×D5—22no—
153C3×Dic10—22no—
154C3×Dic10—22no—
155#1202—82no—
156C3⋊#168—21no—
157C3⋊#168—21no—
158C5×C3⋊C8—22no—
159C2×C3×#41—52no—
160C3×C4⋊Q8—51no—
161C3×#123—102no—
162C3×C22⋊Q8—52no—
163C3×#121—101no—
164C3×C4×D4—51no—
165C3×C4×D4C2×D511yescharacteristic
166C2×C3×C4⋊C4—101no—
167C2×#215—52no—
168#938—101no—
169C2×#213—51no—
170#930—52no—
171#926—52no—
172C22×C3⋊C8—52no—
173#892—101no—
174#885—102no—
175C4×D12—51no—
176#869C2×D511yescharacteristic
177#835—52no—
178#835—52no—
179#835—52no—
180#832—51no—
181#832C2×D512yes—
182#830—52no—
183#809—51no—
184#808C2×D512yes—
185#807—51no—
186C4⋊(C3⋊C8)C2×D511yescharacteristic
187C4×C3⋊C8—51no—
188C2×C5×Q8—61no—
189C2×C5×D4C22×D311yescharacteristic
190C22×C4×C5—21no—
191C2×C22⋊D5—22no—
192C22×Dic5—22no—
193#711—24no—
194#711—42no—
195C2×D20—31no—
196C2×D20—62no—
197C2×D20#6711yescharacteristic
198C2×C4×D5—61no—
199C2×C4×D5C22×D312yes—
200C2×Dic10C22×D312yes—
201C2×C5×C8—32no—
202C5×C4⋊C4—32no—
203C5×C4⋊C4—61no—
204C5×C4⋊C4C22×D311yescharacteristic
205C5×C22⋊C4—21no—
206C42×C5C22×D311yescharacteristic
207#686—62no—
208#685—22no—
209#685—62no—
210#685—61no—
211#685—62no—
212#685#6712yes—
213C4⋊Dic5—21no—
214C4⋊Dic5—21no—
215C4⋊Dic5—22no—
216C4⋊Dic5—31no—
217C4⋊Dic5#6711yescharacteristic
218C4×Dic5—62no—
219C2×#168—31no—
220C2×#168—31no—
221#677—122no—
222C8×D5—122no—
223#555—151no—
224#547—51no—
225#487—151no—
226#485—152no—
227#476—31no—
228#465—152no—
229#461—151no—
230#449—152no—
231C4×D8—151no—
232C8×D4—151no—
233#416—152no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5C22×D411yescharacteristic
238C2×D15—41no—
239C2×D15—41no—
240C2×D15—41no—
241C2×C3×D5—22no—
242C2×C3×D5—22no—
243C2×C3×D5—22no—
244C3×C4×C5—21no—
245C3×C4×C5—41no—
246C3×C4×C5#13512yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—41no—
250C3×Dic5—22no—
251C3×Dic5—42no—
252C5×Dic3—42no—
253C3×#41—104no—
254C3×#41—202no—
255C2×C3×Q8—52no—
256C2×C3×D4—51no—
257C2×C3×D4—102no—
258C2×C3×D4C22×D511yescharacteristic
259C22×C3×C4—21no—
260C22×C3×C4—52no—
261C22×C3×C4—102no—
262C2×#67—102no—
263C2×D12—51no—
264C2×C4×D3—101no—
265C2×Dic6—21no—
266C3×C4⋊C4—51no—
267C3×C4⋊C4—102no—
268C3×C4⋊C4—101no—
269C3×C4⋊C4—101no—
270C3×C4⋊C4C22×D511yescharacteristic
271C3×C22⋊C4—21no—
272C3×C22⋊C4—52no—
273C3×C22⋊C4—102no—
274C3×C42C22×D511yescharacteristic
275#217—102no—
276#215—202no—
277#213—102no—
278(C2×D3)⋊C4—102no—
279(C2×D3)⋊C4—101no—
280C4⋊Dic3—21no—
281C4⋊Dic3—51no—
282C4⋊Dic3C22×D512yes—
283C4×Dic3—102no—
284C2×C3⋊C8—51no—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—202no—
287C2×C3⋊C8C22×D512yes—
288C23×C5—21no—
289C22×D5—61no—
290C22×D5C2×#6712yes—
291C5×Q8—241no—
292C5×D4—21no—
293C5×D4#23712yes—
294C2×C4×C5—21no—
295C2×C4×C5—42no—
296C2×C4×C5—62no—
297C2×C4×C5C2×#6711yescharacteristic
298C2×C4×C5C2×#6711yescharacteristic
299C2×C4×C5C23×D311yescharacteristic
300C22⋊D5—44no—
301C2×Dic5—22no—
302C2×Dic5—44no—
303C2×Dic5—61no—
304C2×Dic5C2×#6712yes—
305D20—22no—
306D20—61no—
307D20—61no—
308D20—242no—
309C4×D5—22no—
310C4×D5—22no—
311C4×D5—122no—
312Dic10—22no—
313Dic10—22no—
314C5×C8—62no—
315#168—61no—
316#168—61no—
317C2×#41—52no—
318C2×SD16—152no—
319C2×D8—151no—
320C22×C8—152no—
321C4⋊Q8—31no—
322C4⋊Q8—51no—
323#124—302no—
324#123—102no—
325C22⋊Q8—52no—
326#121—101no—
327#121—152no—
328#121—301no—
329C4×D4—51no—
330C4×D4—151no—
331C4×D4D3×D511yescharacteristic
332#117—152no—
333#117—301no—
334C2×C4⋊C4—101no—
335#107—151no—
336#106—32no—
337C4⋊C8—31no—
338#103—152no—
339#103—152no—
340#103—152no—
341#102—32no—
342#102—151no—
343#102—151no—
344C22⋊C8—152no—
345C4×C8—151no—
346C2×C3×C5—42no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C2×#13511yescharacteristic
349C2×C3×C5#57711yescharacteristic
350D15—81no—
351C3×D5—42no—
352C23×C3—21no—
353C23×C3—52no—
354C22×D3—101no—
355C3×Q8—102no—
356C3×Q8—102no—
357C3×D4—21no—
358C3×D4—102no—
359C3×D4—204no—
360C3×D4#71312yes—
361C2×C3×C4—21no—
362C2×C3×C4—42no—
363C2×C3×C4—52no—
364C2×C3×C4—102no—
365C2×C3×C4—102no—
366C2×C3×C4—102no—
367C2×C3×C4—204no—
368C2×C3×C4D4×D511yescharacteristic
369C2×C3×C4D4×D511yescharacteristic
370C2×C3×C4C23×D511yescharacteristic
371#67—402no—
372C2×Dic3—22no—
373C2×Dic3—101no—
374D12—101no—
375D12—101no—
376C4×D3—202no—
377Dic6—81no—
378C3⋊C8—22no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5—22no—
382C22×C5—41no—
383C22×C5—41no—
384C22×C5C22×#6711yescharacteristic
385C2×D5—22no—
386C2×D5—22no—
387C2×D5—22no—
388C2×D5—121no—
389C2×D5—121no—
390C2×D5—121no—
391C4×C5—21no—
392C4×C5—41no—
393C4×C5—122no—
394C4×C5#95012yes—
395C4×C5C2×#23711yescharacteristic
396C4×C5#100411yescharacteristic
397Dic5—22no—
398Dic5—42no—
399Dic5—121no—
400#41—104no—
401#41—202no—
402C2×Q8—52no—
403C2×Q8—61no—
404C2×D4—51no—
405C2×D4—102no—
406C2×D4—151no—
407C2×D4—302no—
408C2×D4C2×D3×D511yescharacteristic
409C22×C4—21no—
410C22×C4—52no—
411C22×C4—102no—
412C22×C4—301no—
413SD16—602no—
414D8—302no—
415C2×C8—32no—
416C2×C8—151no—
417C2×C8—151no—
418C2×C8—602no—
419C4⋊C4—32no—
420C4⋊C4—51no—
421C4⋊C4—61no—
422C4⋊C4—102no—
423C4⋊C4—101no—
424C4⋊C4—101no—
425C4⋊C4—151no—
426C4⋊C4C2×D3×D511yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—52no—
429C22⋊C4—102no—
430C22⋊C4—302no—
431C22⋊C4—302no—
432C22⋊C4—301no—
433C42—302no—
434C42C2×D3×D511yescharacteristic
435C3×C5#330311yescharacteristic, complemented, Hall
436C22×C3—22no—
437C22×C3—41no—
438C22×C3—41no—
439C22×C3—102no—
440C22×C3—102no—
441C22×C3—102no—
442C22×C3C2×D4×D511yescharacteristic
443C2×D3—201no—
444C2×D3—201no—
445C2×D3—201no—
446C3×C4—21no—
447C3×C4—41no—
448C3×C4—102no—
449C3×C4—202no—
450C3×C4#406212yes—
451C3×C4C2×#71311yescharacteristic
452C3×C4#414911yescharacteristic
453Dic3—42no—
454Dic3—201no—
455C2×C5—42no—
456C2×C5#580611yescharacteristic
457C2×C5C2×#95011yescharacteristic
458C2×C5#583711yescharacteristic
459D5—42no—
460D5—241no—
461C23—21no—
462C23—52no—
463C23—301no—
464Q8—102no—
465Q8—102no—
466Q8—241no—
467D4—21no—
468D4—102no—
469D4—204no—
470D4—301no—
471D4—301no—
472D4—1202no—
473D4#676612yes—
474C2×C4—21no—
475C2×C4—42no—
476C2×C4—52no—
477C2×C4—62no—
478C2×C4—102no—
479C2×C4—102no—
480C2×C4—102no—
481C2×C4—204no—
482C2×C4—301no—
483C2×C4—602no—
484C2×C4D5×#6711yescharacteristic
485C2×C4D5×#6711yescharacteristic
486C2×C4C22×D3×D511yescharacteristic
487C8—62no—
488C8—301no—
489C8—301no—
490C2×C3—42no—
491C2×C3—202no—
492C2×C3#6596411yescharacteristic
493C2×C3C2×#406211yescharacteristic
494C2×C3#6610911yescharacteristic
495D3—401no—
496C5#8351111yescharacteristic, complemented, Sylow
497C22—22no—
498C22—41no—
499C22—41no—
500C22—102no—
501C22—102no—
502C22—102no—
503C22—601no—
504C22—601no—
505C22—601no—
506C22C2×D5×#6711yescharacteristic
507C4—21no—
508C4—41no—
509C4—102no—
510C4—122no—
511C4—202no—
512C4—601no—
513C4#9149212yes—
514C4C2×#676611yescharacteristic
515C4#9201711yescharacteristic
516C3#1061280711yescharacteristic, complemented, Sylow
517C2—42no—
518C2—202no—
519C2—1201no—
520C2#1175430811yescharacteristic
521C2C2×#9149211yescharacteristic
522C2#1175531711yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×D4C22
socleC22×C3×C5C22×D4
hypercenterC4×D4D3×D5
hypocenterC3×C5#3303
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×#65889C2semidirect
2. #11752715C2semidirect
3. C4×#6715C2semidirect
4. C5×#5008C2semidirect
5. #11749026C2semidirect
6. #11748961C2semidirect
7. #11747443C2semidirect
8. #11747205C2semidirect
9. #11747085C2semidirect
10. #11746996C2semidirect
11. #11746924C2semidirect
44. #65889D3semidirect
99. #5008D5semidirect
435. C3×C5#3303semidirect
496. C5#83511semidirect
516. C3#10612807semidirect

Maximal subgroups

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

subgroupindex
C3×#658892
#117527152
C4×#67152
C5×#50082
#117490262
#117489612
#117474432
#117472052
#117470852
#117469962
#117469242
#106128073
#835115