Finite Group Databasebeta

← #14459175

Subgroups — #14459175

Order 1920.

840 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×D20C211yesmaximal, characteristic, complemented
2#11752533C211yesmaximal, characteristic
3C4×#6717C211yesmaximal, characteristic, complemented
4C5×#5020C211yesmaximal, characteristic, complemented
5#11749268C212yesmaximal, complemented
6D5×#835C212yesmaximal, complemented
7#11747443C212yesmaximal, complemented
8#11747193C212yesmaximal, complemented
9#11747085C211yesmaximal, characteristic, complemented
10#11746996C211yesmaximal, characteristic, complemented
11#11746924C211yesmaximal, characteristic, complemented
12#10612753—31nomaximal, Hall
13C2×C3×Q8×D5C2212yes—
14C2×#6766—21no—
15C3×C4×C5×Q8C2211yescharacteristic
16C2×#6717C2211yescharacteristic
17C4×D60C2211yescharacteristic
18C5×#869C2211yescharacteristic
19C3×#4043—22no—
20C3×#4043C2212yes—
21C3×C4×D20—21no—
22C3×C4×D20C2211yescharacteristic
23C3×#4015—21no—
24C3×#4015C2211yescharacteristic
25#91450C2212yes—
26#91413—22no—
27D5×C4⋊Dic3C2212yes—
28#91383—21no—
29C2×#6658C2212yes—
30C2×D5×C3⋊C8C2212yes—
31#91119C2211yescharacteristic
32#91093C2211yescharacteristic
33#91092C2211yescharacteristic
34C4×C3⋊#168C2211yescharacteristic
35C5×#835C2212yes—
36C5×#808C2212yes—
37C5×C4⋊(C3⋊C8)C2211yescharacteristic
38#90971C2212yes—
39#90970C2212yes—
40#90970C2212yes—
41#90955C2212yes—
42#83550—51nomaximal, Hall
43#66060—31no—
44Q8×D20D311yescharacteristic, complemented
45C5×#475—31no—
46#65382—31no—
47C4×#688—31no—
48#65153—32no—
49#65151—32no—
50#65130—31no—
51#65123—31no—
52#65095—32no—
53D5×#103—32no—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×D60C2311yescharacteristic
56C2×C4×D15—21no—
57C2×C5×Dic6—21no—
58C3×Q8×D5—24no—
59C3×Q8×D5—42no—
60C2×C3×D20D411yescharacteristic
61C2×C3×C4×D5—22no—
62C2×C3×C4×D5C2312yes—
63C2×C3×Dic10—22no—
64C2×C3×Dic10C2312yes—
65C2×#1202—22no—
66C2×D5×Dic3—22no—
67#6766—44no—
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4C2311yescharacteristic
70C3×C42×C5—21no—
71C3×C42×C5C2311yescharacteristic
72#6717—22no—
73#6714—21no—
74C4⋊Dic15C2311yescharacteristic
75C2×C3⋊#168C2311yescharacteristic
76C2×C3⋊#168C2311yescharacteristic
77C5×C4⋊Dic3—21no—
78C5×C4⋊Dic3C2312yes—
79C2×C5×C3⋊C8C2312yes—
80C3×#685—22no—
81C3×#685D412yes—
82C3×C4⋊Dic5—21no—
83C3×C4⋊Dic5—21no—
84C3×C4⋊Dic5—22no—
85C3×C4⋊Dic5D411yescharacteristic
86#6668—22no—
87#6667—22no—
88#6658—42no—
89D5×C3⋊C8—42no—
90C3×D4×Q8—51no—
91#5555—51no—
92#5227—52no—
93C2×#835—52no—
94#5083—51no—
95#5079—51no—
96#5048—52no—
97#5043—52no—
98C4×#215—51no—
99#5020D511yescharacteristic, complemented
100D4×C3⋊C8—51no—
101C2×#713—61no—
102C2×Q8×D5C2×D312yes—
103C5×C4⋊Q8—31no—
104C4×C5×Q8C2×D311yescharacteristic
105#4092—61no—
106C2×#688—31no—
107C2×#677—32no—
108C2×C8×D5—32no—
109#4043—22no—
110#4043C2×D312yes—
111#4041—32no—
112#4040—62no—
113D5×C4⋊C4—32no—
114C4×D20—21no—
115C4×D20—31no—
116C4×D20C2×D311yescharacteristic
117#4015—21no—
118#4015C2×D311yescharacteristic
119C5×#106—32no—
120C5×C4⋊C8—31no—
121C5×#103—32no—
122#3967—31no—
123#3952—32no—
124#3948—32no—
125#3942—32no—
126#3942—32no—
127#3941—31no—
128#3940—31no—
129C4×#168—31no—
130#3298—151noSylow
131C22×D15—21no—
132C22×C3×D5C2×D412yes—
133C3×C5×Q8—21no—
134C3×C5×Q8#4112yescomplemented
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
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—42no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5C2×D412yes—
148C3×D20SD1614yescomplemented
149C3×C4×D5—22no—
150C3×C4×D5—22no—
151C3×C4×D5—44no—
152C3×Dic10—22no—
153C3×Dic10—22no—
154C3×Dic10—44no—
155#1202—44no—
156D5×Dic3—44no—
157C3⋊#168—21no—
158C3⋊#168—21no—
159C5×C3⋊C8—22no—
160C22×C3×Q8—52no—
161C2×#237—101no—
162C3×C4⋊Q8—51no—
163C3×C4⋊Q8—101no—
164C3×C22⋊Q8—52no—
165C3×C22⋊Q8—102no—
166C3×C4×Q8C2×D511yescharacteristic
167C3×C4×D4—51no—
168C3×C4×D4—101no—
169C2×#215—51no—
170C2×#215—52no—
171#930—52no—
172C2×C4⋊Dic3—52no—
173C22×C3⋊C8—52no—
174#897—101no—
175#886—102no—
176C4×D12—51no—
177#869C2×D511yescharacteristic
178#835—51no—
179#835—52no—
180#835—52no—
181#835—52no—
182#835C2×D512yes—
183#830—52no—
184#809—51no—
185#808—51no—
186#808C2×D512yes—
187C4⋊(C3⋊C8)C2×D511yescharacteristic
188C4×C3⋊C8—51no—
189C2×C5×Q8—61no—
190C2×C5×Q8C22×D311yescharacteristic
191#713—124no—
192Q8×D5—24no—
193Q8×D5—42no—
194C2×D20—31no—
195C2×D20—62no—
196C2×D20#6711yescharacteristic
197C2×C4×D5—22no—
198C2×C4×D5—62no—
199C2×C4×D5—61no—
200C2×C4×D5C22×D312yes—
201C2×Dic10—22no—
202C2×Dic10C22×D312yes—
203C2×C5×C8—32no—
204C5×C4⋊C4—21no—
205C5×C4⋊C4—32no—
206C5×C4⋊C4—61no—
207C5×C4⋊C4C22×D311yescharacteristic
208C42×C5—21no—
209C42×C5C22×D311yescharacteristic
210#688—62no—
211#685—22no—
212#685—61no—
213#685—62no—
214#685#6712yes—
215C4⋊Dic5—21no—
216C4⋊Dic5—21no—
217C4⋊Dic5—22no—
218C4⋊Dic5—31no—
219C4⋊Dic5#6711yescharacteristic
220#683—62no—
221C2×#168—31no—
222C2×#168—31no—
223#677—122no—
224C8×D5—122no—
225D4×Q8—51no—
226#547—151no—
227#487—151no—
228#483—152no—
229#475—31no—
230#465—152no—
231#461—151no—
232#448—152no—
233C4×SD16—151no—
234C8×D4—151no—
235C2×#103—152no—
236C22×C3×C5C22×D411yescharacteristic
237C2×D15—41no—
238C2×D15—41no—
239C2×D15—41no—
240C2×C3×D5—22no—
241C2×C3×D5—22no—
242C2×C3×D5C2×SD1614yes—
243C3×C4×C5—22no—
244C3×C4×C5—21no—
245C3×C4×C5—41no—
246C3×C4×C5C2×SD1612yes—
247C3×C4×C5C2×#4111yescharacteristic
248C3×C4×C5H(5,𝔽2)11yescharacteristic
249Dic15—41no—
250C3×Dic5—22no—
251C3×Dic5—42no—
252C5×Dic3—42no—
253C2×C3×Q8—52no—
254C2×C3×Q8—102no—
255C2×C3×Q8—104no—
256C2×C3×Q8—202no—
257C2×C3×Q8C22×D511yescharacteristic
258C2×C3×D4—51no—
259C22×C3×C4—52no—
260C22×C3×C4—102no—
261C2×#67—102no—
262C22×Dic3—102no—
263#237—204no—
264C2×D12—51no—
265C2×C4×D3—101no—
266C2×Dic6—21no—
267C3×C4⋊C4—21no—
268C3×C4⋊C4—51no—
269C3×C4⋊C4—102no—
270C3×C4⋊C4—101no—
271C3×C4⋊C4—101no—
272C3×C4⋊C4C22×D511yescharacteristic
273C3×C22⋊C4—52no—
274C3×C22⋊C4—102no—
275C3×C42—21no—
276C3×C42C22×D511yescharacteristic
277#215—102no—
278#215—202no—
279(C2×D3)⋊C4—102no—
280(C2×D3)⋊C4—101no—
281C4⋊Dic3—21no—
282C4⋊Dic3—51no—
283C4⋊Dic3—102no—
284C4⋊Dic3C22×D512yes—
285C2×C3⋊C8—51no—
286C2×C3⋊C8—51no—
287C2×C3⋊C8—202no—
288C2×C3⋊C8C22×D512yes—
289C22×D5—61no—
290C22×D5C2×#6712yes—
291C5×Q8—21no—
292C5×Q8—122no—
293C5×Q8#23712yescomplemented
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—62no—
297C2×C4×C5C2×#6711yescharacteristic
298C2×C4×C5C2×#6711yescharacteristic
299C2×C4×C5C23×D311yescharacteristic
300C2×Dic5—22no—
301C2×Dic5—61no—
302C2×Dic5C2×#6712yes—
303D20—61no—
304D20—61no—
305D20—124no—
306D20#21514yescomplemented
307C4×D5—22no—
308C4×D5—22no—
309C4×D5—44no—
310C4×D5—124no—
311C4×D5—122no—
312Dic10—22no—
313Dic10—22no—
314Dic10—44no—
315C5×C8—62no—
316#168—61no—
317#168—61no—
318C2×#41—301no—
319C22×Q8—52no—
320C2×SD16—152no—
321C2×SD16—151no—
322C22×C8—152no—
323C4⋊Q8—31no—
324C4⋊Q8—51no—
325C4⋊Q8—101no—
326#123—302no—
327C22⋊Q8—52no—
328C22⋊Q8—102no—
329C22⋊Q8—301no—
330#121—152no—
331C4×Q8D3×D511yescharacteristic
332C4×D4—51no—
333C4×D4—101no—
334C4×D4—151no—
335C2×C4⋊C4—152no—
336#106—32no—
337#106—151no—
338C4⋊C8—31no—
339#103—32no—
340#103—152no—
341#103—152no—
342#103—152no—
343#103—151no—
344#102—151no—
345C22⋊C8—152no—
346C4×C8—151no—
347C2×C3×C5#54611yescharacteristic
348C2×C3×C5C22×SD1611yescharacteristic
349C2×C3×C5#57811yescharacteristic
350D15—81no—
351C3×D5—24no—
352C23×C3—52no—
353C22×D3—101no—
354C3×Q8—21no—
355C3×Q8—102no—
356C3×Q8—102no—
357C3×Q8—204no—
358C3×Q8#71312yescomplemented
359C3×D4—54no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—52no—
363C2×C3×C4—102no—
364C2×C3×C4—102no—
365C2×C3×C4—102no—
366C2×C3×C4—204no—
367C2×C3×C4D4×D511yescharacteristic
368C2×C3×C4D4×D511yescharacteristic
369C2×C3×C4C23×D511yescharacteristic
370#67—204no—
371C2×Dic3—22no—
372C2×Dic3—101no—
373C2×Dic3—204no—
374D12—101no—
375D12—101no—
376C4×D3—202no—
377Dic6—42no—
378C3⋊C8—22no—
379C3⋊C8—101no—
380C3⋊C8—101no—
381C22×C5C22×#6711yescharacteristic
382C2×D5—22no—
383C2×D5—22no—
384C2×D5—121no—
385C2×D5—121no—
386C2×D5—121no—
387C2×D5C2×#21514yes—
388C4×C5—22no—
389C4×C5—21no—
390C4×C5—41no—
391C4×C5—122no—
392C4×C5C2×#21512yes—
393C4×C5C2×#23711yescharacteristic
394C4×C5#100411yescharacteristic
395Dic5—22no—
396Dic5—42no—
397Dic5—121no—
398#41—604no—
399C2×Q8—52no—
400C2×Q8—61no—
401C2×Q8—102no—
402C2×Q8—104no—
403C2×Q8—202no—
404C2×Q8C2×D3×D511yescharacteristic
405C2×D4—51no—
406C2×D4—151no—
407C2×D4—302no—
408C22×C4—52no—
409C22×C4—102no—
410C22×C4—302no—
411C22×C4—301no—
412SD16—302no—
413SD16—602no—
414C2×C8—32no—
415C2×C8—151no—
416C2×C8—151no—
417C2×C8—602no—
418C4⋊C4—21no—
419C4⋊C4—32no—
420C4⋊C4—51no—
421C4⋊C4—61no—
422C4⋊C4—102no—
423C4⋊C4—101no—
424C4⋊C4—101no—
425C4⋊C4—151no—
426C4⋊C4—302no—
427C4⋊C4C2×D3×D511yescharacteristic
428C22⋊C4—52no—
429C22⋊C4—102no—
430C22⋊C4—302no—
431C22⋊C4—301no—
432C42—21no—
433C42C2×D3×D511yescharacteristic
434C3×C5#329811yescharacteristic, complemented, Hall
435C22×C3—54no—
436C22×C3—102no—
437C22×C3—102no—
438C22×C3C2×D4×D511yescharacteristic
439C2×D3—201no—
440C2×D3—201no—
441C2×D3—201no—
442C3×C4—21no—
443C3×C4—22no—
444C3×C4—41no—
445C3×C4—102no—
446C3×C4—202no—
447C3×C4D5×SD1612yes—
448C3×C4C2×#71311yescharacteristic
449C3×C4#414911yescharacteristic
450Dic3—42no—
451Dic3—201no—
452C2×C5#580611yescharacteristic
453C2×C5C22×#21511yescharacteristic
454C2×C5#583811yescharacteristic
455D5—24no—
456D5—241no—
457C23—52no—
458C23—301no—
459Q8—21no—
460Q8—102no—
461Q8—102no—
462Q8—122no—
463Q8—204no—
464Q8#676612yescomplemented
465D4—54no—
466D4—301no—
467D4—301no—
468D4—604no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—52no—
472C2×C4—62no—
473C2×C4—102no—
474C2×C4—102no—
475C2×C4—102no—
476C2×C4—204no—
477C2×C4—301no—
478C2×C4—604no—
479C2×C4—602no—
480C2×C4D5×#6711yescharacteristic
481C2×C4D5×#6711yescharacteristic
482C2×C4C22×D3×D511yescharacteristic
483C8—62no—
484C8—301no—
485C8—301no—
486C2×C3—104no—
487C2×C3#6596411yescharacteristic
488C2×C3C2×D5×SD1611yescharacteristic
489C2×C3#6611311yescharacteristic
490D3—401no—
491C5#8355011yescharacteristic, complemented, Sylow
492C22—54no—
493C22—102no—
494C22—102no—
495C22—601no—
496C22—601no—
497C22—601no—
498C22C2×D5×#6711yescharacteristic
499C4—22no—
500C4—21no—
501C4—41no—
502C4—102no—
503C4—122no—
504C4—202no—
505C4—601no—
506C4D5×#21512yes—
507C4C2×#676611yescharacteristic
508C4#9201711yescharacteristic
509C3#1061275311yescharacteristic, complemented, Sylow
510C2—104no—
511C2—1201no—
512C2#1175430811yescharacteristic
513C2C2×D5×#21511yescharacteristic
514C2#1175533611yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC3×C4×C5×Q8C22
socleC22×C3×C5C22×D4
prime radical#11752533C2
hypercenterC4×Q8D3×D5
hypocenterC3×C5#3298
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×D20C2semidirect
3. C4×#6717C2semidirect
4. C5×#5020C2semidirect
5. #11749268C2semidirect
6. D5×#835C2semidirect
7. #11747443C2semidirect
8. #11747193C2semidirect
9. #11747085C2semidirect
10. #11746996C2semidirect
11. #11746924C2semidirect
44. Q8×D20D3semidirect
99. #5020D5semidirect
134. C3×C5×Q8#41semidirect
148. C3×D20SD16semidirect
293. C5×Q8#237semidirect
306. D20#215semidirect
358. C3×Q8#713semidirect
434. C3×C5#3298semidirect
464. Q8#6766semidirect
491. C5#83550semidirect
509. C3#10612753semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×D202
#117525332
C4×#67172
C5×#50202
#117492682
D5×#8352
#117474432
#117471932
#117470852
#117469962
#117469242
#106127533
#835505