Finite Group Databasebeta

← #14459957

Subgroups — #14459957

Order 1920.

772 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756343C211yesmaximal, characteristic, complemented
2#11756314C211yesmaximal, characteristic
3C3×C4×C5×SD16C211yesmaximal, characteristic, complemented
4#11751274C212yesmaximal, complemented
5#11751132C211yesmaximal, characteristic, complemented
6#11751108C211yesmaximal, characteristic, complemented
7#11750969C212yesmaximal, complemented
8#11750914C212yesmaximal, complemented
9#11750873C212yesmaximal, complemented
10C8×D60C211yesmaximal, characteristic, complemented
11#11750782C211yesmaximal, characteristic, complemented
12#10611411—31nomaximal, Hall
13C2×#6819C2212yes—
14C2×C3×C5×SD16C2211yescharacteristic
15C3×C4×C5×Q8C2211yescharacteristic
16C3×C4×C5×D4C2211yescharacteristic
17#91835C2212yes—
18C2×#6718C2212yes—
19#91819—21no—
20C2×C4⋊Dic15—21no—
21C2×C8×D15C2212yes—
22#91787—22no—
23#91786C2212yes—
24#91781C2212yes—
25#91775—22no—
26#91764—21no—
27C4×D60C2211yescharacteristic
28#91758C2211yescharacteristic
29#91758C2211yescharacteristic
30C4×Dic30—21no—
31C3×C5×#106C2211yescharacteristic
32C3×C5×#103C2211yescharacteristic
33C3×C5×#102C2211yescharacteristic
34C3×C4×C5×C8C2211yescharacteristic
35#91116C2212yes—
36#91104C2212yes—
37#91100C2212yes—
38#91100C2212yes—
39#91094C2212yes—
40#91091C2212yes—
41C4⋊(C3⋊#168)C2211yescharacteristic
42#79680—51nomaximal, Hall
43#65918—31no—
44#65889—31no—
45C4×C5×SD16D311yescharacteristic, complemented
46#65462—32no—
47#65320—31no—
48#65296—31no—
49#65156—32no—
50#65102—32no—
51#65061—32no—
52C8×D20—31no—
53#64970—31no—
54C2×C3×C5×Q8C2311yescharacteristic
55C2×C3×C5×D4C2311yescharacteristic
56C22×C3×C4×C5—21no—
57C2×#1220—22no—
58C22×Dic15—22no—
59#6819—24no—
60#6819—42no—
61C2×D60D411yescharacteristic
62C2×C4×D15C2312yes—
63C2×Dic30—21no—
64C2×Dic30C2312yes—
65C3×C5×SD16—22no—
66C2×C3×C5×C8C2311yescharacteristic
67C2×C3×C5×C8C2311yescharacteristic
68C3×C5×C4⋊C4—21no—
69C3×C5×C4⋊C4C2311yescharacteristic
70C3×C5×C4⋊C4C2311yescharacteristic
71C3×C5×C22⋊C4—21no—
72C3×C42×C5—21no—
73C3×C42×C5C2311yescharacteristic
74#6718—42no—
75#6714—22no—
76#6714—22no—
77#6714—22no—
78#6714D412yes—
79C4⋊Dic15—21no—
80C4⋊Dic15—21no—
81C4⋊Dic15—21no—
82C4⋊Dic15—22no—
83C4⋊Dic15D411yescharacteristic
84C4⋊Dic15C2312yes—
85#6712—22no—
86C4×Dic15—22no—
87C2×C3⋊#168C2312yes—
88C8×D15—42no—
89#5584—51no—
90#5555—51no—
91C3×C4×SD16D511yescharacteristic, complemented
92#5173—52no—
93#5031—51no—
94#5007—51no—
95#4868—52no—
96#4813—52no—
97#4772—52no—
98C8×D12—51no—
99#4681—51no—
100C2×#711—32no—
101C2×C5×SD16C2×D311yescharacteristic
102C4×C5×Q8C2×D311yescharacteristic
103C4×C5×D4C2×D311yescharacteristic
104#4092—32no—
105C2×#689—32no—
106#4076—61no—
107C2×C4⋊Dic5—61no—
108C2×C8×D5—32no—
109#4044—62no—
110#4043—32no—
111#4038—32no—
112#4032—62no—
113#4021—61no—
114C4×D20—31no—
115#4015—31no—
116#4015—31no—
117C4×Dic10—61no—
118C5×#106C2×D311yescharacteristic
119C5×#103C2×D311yescharacteristic
120C5×#102C2×D311yescharacteristic
121C4×C5×C8C2×D311yescharacteristic
122#3964—32no—
123#3952—32no—
124#3948—32no—
125#3948—32no—
126#3942—32no—
127#3939—32no—
128C4⋊#168—31no—
129#3302—151noSylow
130C23×C3×C5—21no—
131C22×D15C2×D412yes—
132C3×C5×Q8—21no—
133C3×C5×Q8—21no—
134C3×C5×D4—21no—
135C3×C5×D4#4112yes—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—42no—
140C2×C3×C4×C5C2×D411yescharacteristic
141C2×C3×C4×C5C2×D411yescharacteristic
142C2×C3×C4×C5C2411yescharacteristic
143#1220—44no—
144C2×Dic15—22no—
145C2×Dic15—22no—
146C2×Dic15—44no—
147C2×Dic15C2×D412yes—
148D60—22no—
149C4×D15—22no—
150C4×D15—22no—
151Dic30—22no—
152Dic30—22no—
153Dic30—42no—
154C3×C5×C8—21no—
155C3×C5×C8—21no—
156C3⋊#168—22no—
157C2×#237—52no—
158C2×C3×SD16C2×D511yescharacteristic
159C3×C4×Q8C2×D511yescharacteristic
160C3×C4×D4C2×D511yescharacteristic
161#946—52no—
162C2×#216—52no—
163#930—101no—
164C2×C4⋊Dic3—101no—
165C2×D3×C8—52no—
166#898—102no—
167#897—52no—
168#892—52no—
169#886—102no—
170#875—101no—
171C4×D12—51no—
172#869—51no—
173#869—51no—
174C4×Dic6—101no—
175C3×#106C2×D511yescharacteristic
176C3×#103C2×D511yescharacteristic
177C3×#102C2×D511yescharacteristic
178C3×C4×C8C2×D511yescharacteristic
179#832—52no—
180(C2×D3)⋊C8—52no—
181#816—52no—
182#816—52no—
183#810—52no—
184#807—52no—
185C4⋊(C3⋊C8)—51no—
186C2×C5×Q8C22×D311yescharacteristic
187C2×C5×D4C22×D311yescharacteristic
188C22×C4×C5—21no—
189C2×C22⋊D5—62no—
190C22×Dic5—62no—
191#711—64no—
192#711—122no—
193C2×D20—31no—
194C2×C4×D5—32no—
195C2×Dic10—32no—
196C2×Dic10—61no—
197C5×SD16—22no—
198C2×C5×C8C22×D311yescharacteristic
199C2×C5×C8C22×D311yescharacteristic
200C5×C4⋊C4—21no—
201C5×C4⋊C4C22×D311yescharacteristic
202C5×C4⋊C4C22×D311yescharacteristic
203C5×C22⋊C4—21no—
204C42×C5—21no—
205C42×C5C22×D311yescharacteristic
206#689—122no—
207#685—32no—
208#685—62no—
209#685—62no—
210#685—62no—
211C4⋊Dic5—31no—
212C4⋊Dic5—32no—
213C4⋊Dic5—62no—
214C4⋊Dic5—61no—
215C4⋊Dic5—61no—
216C4⋊Dic5—61no—
217#683—62no—
218C4×Dic5—62no—
219C2×#168—32no—
220C8×D5—122no—
221#556—151no—
222#547—151no—
223#487—151no—
224#485—152no—
225#474—151no—
226#467—152no—
227#462—151no—
228#452—152no—
229C4×SD16D1511yescharacteristic, complemented
230C8×D4—151no—
231#416—152no—
232C22×C3×C5—22no—
233C22×C3×C5—41no—
234C22×C3×C5—41no—
235C22×C3×C5C22×D411yescharacteristic
236C2×D15—22no—
237C2×D15—22no—
238C2×D15—22no—
239C3×C4×C5—21no—
240C3×C4×C5—41no—
241C3×C4×C5—41no—
242C3×C4×C5—41no—
243C3×C4×C5#13512yes—
244C3×C4×C5C2×#4111yescharacteristic
245C3×C4×C5H(5,𝔽2)11yescharacteristic
246Dic15—22no—
247Dic15—42no—
248Dic15—42no—
249C2×C3×Q8C22×D511yescharacteristic
250C2×C3×D4C22×D511yescharacteristic
251C22×C3×C4—21no—
252C2×#67—102no—
253C22×Dic3—102no—
254#237—104no—
255#237—202no—
256C2×D12—51no—
257C2×C4×D3—52no—
258C2×Dic6—52no—
259C2×Dic6—101no—
260C3×SD16—22no—
261C2×C3×C8C22×D511yescharacteristic
262C2×C3×C8C22×D511yescharacteristic
263C3×C4⋊C4—21no—
264C3×C4⋊C4C22×D511yescharacteristic
265C3×C4⋊C4C22×D511yescharacteristic
266C3×C22⋊C4—21no—
267C3×C42—21no—
268C3×C42C22×D511yescharacteristic
269#216—202no—
270(C2×D3)⋊C4—52no—
271(C2×D3)⋊C4—102no—
272(C2×D3)⋊C4—102no—
273(C2×D3)⋊C4—102no—
274C4⋊Dic3—51no—
275C4⋊Dic3—52no—
276C4⋊Dic3—102no—
277C4⋊Dic3—101no—
278C4⋊Dic3—101no—
279C4⋊Dic3—101no—
280Dic3⋊C4—102no—
281C4×Dic3—102no—
282C2×C3⋊C8—52no—
283D3×C8—202no—
284C23×C5—21no—
285C22×D5—32no—
286C5×Q8—21no—
287C5×Q8—21no—
288C5×D4—21no—
289C5×D4#23512yes—
290C2×C4×C5—21no—
291C2×C4×C5—21no—
292C2×C4×C5—21no—
293C2×C4×C5—42no—
294C2×C4×C5D3×D411yescharacteristic
295C2×C4×C5D3×D411yescharacteristic
296C2×C4×C5C23×D311yescharacteristic
297C22⋊D5—124no—
298C2×Dic5—32no—
299C2×Dic5—62no—
300C2×Dic5—62no—
301C2×Dic5—124no—
302D20—62no—
303C4×D5—62no—
304C4×D5—62no—
305Dic10—62no—
306Dic10—62no—
307Dic10—122no—
308C5×C8—21no—
309C5×C8—21no—
310#168—62no—
311C2×#41—152no—
312C2×Q16—152no—
313C2×SD16C2×D1511yescharacteristic
314C22×C8—152no—
315C4⋊Q8—151no—
316C4⋊Q8—151no—
317#126—302no—
318#124—301no—
319#123—302no—
320C22⋊Q8—152no—
321C22⋊Q8—152no—
322#121—301no—
323C4×Q8—301no—
324C4×Q8C2×D1511yescharacteristic
325C4×D4—151no—
326C4×D4C2×D1511yescharacteristic
327#117—152no—
328C2×C4⋊C4—301no—
329#107—152no—
330#106C2×D1511yescharacteristic
331C4⋊C8—151no—
332#103—152no—
333#103—152no—
334#103—152no—
335#103C2×D1511yescharacteristic
336#102—152no—
337#102C2×D1511yescharacteristic
338C22⋊C8—152no—
339C4×C8C2×D1511yescharacteristic
340C2×C3×C5—42no—
341C2×C3×C5#54611yescharacteristic
342C2×C3×C5C2×#13511yescharacteristic
343C2×C3×C5#57811yescharacteristic
344D15—42no—
345C23×C3—21no—
346C22×D3—52no—
347C3×Q8—21no—
348C3×Q8—21no—
349C3×D4—21no—
350C3×D4#70912yes—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—42no—
355C2×C3×C4D4×D511yescharacteristic
356C2×C3×C4D4×D511yescharacteristic
357C2×C3×C4C23×D511yescharacteristic
358#67—204no—
359C2×Dic3—52no—
360C2×Dic3—102no—
361C2×Dic3—102no—
362C2×Dic3—204no—
363D12—102no—
364C4×D3—102no—
365C4×D3—102no—
366Dic6—102no—
367Dic6—102no—
368Dic6—202no—
369C3×C8—21no—
370C3×C8—21no—
371C3⋊C8—102no—
372C22×C5—22no—
373C22×C5—41no—
374C22×C5—41no—
375C22×C5C2×D3×D411yescharacteristic
376C2×D5—62no—
377C2×D5—62no—
378C2×D5—62no—
379C4×C5—21no—
380C4×C5—41no—
381C4×C5—41no—
382C4×C5—41no—
383C4×C5#91612yes—
384C4×C5C2×#23511yescharacteristic
385C4×C5#100911yescharacteristic
386Dic5—62no—
387Dic5—122no—
388Dic5—122no—
389#41—304no—
390#41—602no—
391C2×Q8—152no—
392C2×Q8—301no—
393C2×Q8C22×D1511yescharacteristic
394C2×D4—151no—
395C2×D4—302no—
396C2×D4C22×D1511yescharacteristic
397C22×C4—21no—
398C22×C4—152no—
399C22×C4—302no—
400Q16—602no—
401SD16—22no—
402C2×C8—152no—
403C2×C8—602no—
404C2×C8C22×D1511yescharacteristic
405C2×C8C22×D1511yescharacteristic
406C4⋊C4—21no—
407C4⋊C4—151no—
408C4⋊C4—152no—
409C4⋊C4—302no—
410C4⋊C4—302no—
411C4⋊C4—301no—
412C4⋊C4—301no—
413C4⋊C4—301no—
414C4⋊C4C22×D1511yescharacteristic
415C4⋊C4C22×D1511yescharacteristic
416C22⋊C4—21no—
417C22⋊C4—152no—
418C22⋊C4—302no—
419C22⋊C4—302no—
420C22⋊C4—302no—
421C42—21no—
422C42—302no—
423C42C22×D1511yescharacteristic
424C3×C5#330211yescharacteristic, complemented, Hall
425C22×C3—22no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3C2×D4×D511yescharacteristic
429C2×D3—102no—
430C2×D3—102no—
431C2×D3—102no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4#406212yes—
437C3×C4C2×#70911yescharacteristic
438C3×C4#414911yescharacteristic
439Dic3—102no—
440Dic3—202no—
441Dic3—202no—
442C2×C5—42no—
443C2×C5#555011yescharacteristic
444C2×C5C2×#91611yescharacteristic
445C2×C5#577111yescharacteristic
446D5—122no—
447C23—21no—
448C23—152no—
449Q8—21no—
450Q8—21no—
451Q8—302no—
452Q8—302no—
453Q8—602no—
454D4—21no—
455D4—302no—
456D4—604no—
457D4#681712yes—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—42no—
462C2×C4—152no—
463C2×C4—302no—
464C2×C4—302no—
465C2×C4—302no—
466C2×C4—302no—
467C2×C4—604no—
468C2×C4D4×D1511yescharacteristic
469C2×C4D4×D1511yescharacteristic
470C2×C4C23×D1511yescharacteristic
471C8—21no—
472C8—21no—
473C8—302no—
474C2×C3—42no—
475C2×C3#6588411yescharacteristic
476C2×C3C2×#406211yescharacteristic
477C2×C3#6610511yescharacteristic
478D3—202no—
479C5#7968011yescharacteristic, complemented, Sylow
480C22—22no—
481C22—41no—
482C22—41no—
483C22—302no—
484C22—302no—
485C22—302no—
486C22C2×D4×D1511yescharacteristic
487C4—21no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—302no—
492C4—602no—
493C4—602no—
494C4#9180512yes—
495C4C2×#681711yescharacteristic
496C4#9210011yescharacteristic
497C3#1061141111yescharacteristic, complemented, Sylow
498C2—42no—
499C2—602no—
500C2#1175630911yescharacteristic
501C2C2×#9180511yescharacteristic
502C2#1175653011yescharacteristic
subgroupG/H
centerC22C2×D4×D15
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C4×C5×SD16C2
socleC22×C3×C5C22×D4
prime radical#11756314C2
hypercenterC4×SD16D15
hypocenterC3×C5#3302
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #11756343C2semidirect
3. C3×C4×C5×SD16C2semidirect
4. #11751274C2semidirect
5. #11751132C2semidirect
6. #11751108C2semidirect
7. #11750969C2semidirect
8. #11750914C2semidirect
9. #11750873C2semidirect
10. C8×D60C2semidirect
11. #11750782C2semidirect
45. C4×C5×SD16D3semidirect
91. C3×C4×SD16D5semidirect
229. C4×SD16D15semidirect
424. C3×C5#3302semidirect
479. C5#79680semidirect
497. C3#10611411semidirect

Maximal subgroups

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

subgroupindex
#117563432
#117563142
C3×C4×C5×SD162
#117512742
#117511322
#117511082
#117509692
#117509142
#117508732
C8×D602
#117507822
#106114113
#796805