Finite Group Databasebeta

← #14439471

Subgroups — #14439471

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91831C211yesmaximal, characteristic, complemented
2C22×C5×#215C211yesmaximal, characteristic, complemented
3C3×#66150C211yesmaximal, characteristic, complemented
4#11754606C211yesmaximal, characteristic, complemented
5C2×#91311C211yesmaximal, characteristic, complemented
6C2×#91299C211yesmaximal, characteristic, complemented
7#11749340C212yesmaximal, complemented
8#11749339C212yesmaximal, complemented
9#11749313C212yesmaximal, complemented
10#11749312C212yesmaximal, complemented
11#11748318C211yesmaximal, characteristic, complemented
12#10615585—31nomaximal, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C2×#6817C2211yescharacteristic
15C22×C5×D12C2211yescharacteristic
16C2×C3×#709C2211yescharacteristic
17C2×#6718C2212yescomplemented
18#91831—24no—
19C2×#6717C2212yescomplemented
20C2×#6710C2211yescharacteristic
21C2×C5×#215—24no—
22C2×C5×#215C2212yes—
23C2×C5×#215C2212yes—
24C22×C5×C3⋊C8C2211yescharacteristic
25C3×#4093C2212yescomplemented
26C3×#4092—22no—
27C3×#4090C2212yescomplemented
28C3×#4075—22no—
29C3×#4072C2211yescharacteristic
30#91570—22no—
31#91466—22no—
32#91451C2212yes—
33#91373C2212yes—
34C2×#6662C2211yescharacteristic, complemented
35C2×#6658C2211yescharacteristic, complemented
36#91311—24no—
37C2×#6657C2211yescharacteristic, complemented
38C2×#6656C2212yes—
39C2×#6653C2211yescharacteristic, complemented
40#91299—24no—
41C2×#6652C2212yes—
42#90952C2212yescomplemented
43Dic5×C3⋊C8C2212yescomplemented
44#83402—51nomaximal, Hall
45C22×C5×SD16—31no—
46#66150D311yescharacteristic, complemented
47C2×#4088—31no—
48#66139—31no—
49C2×#4078—31no—
50C2×#4050—31no—
51#65465—32no—
52#65464—32no—
53#65457—32no—
54#65456—32no—
55#65390—31no—
56C23×C5×D3—21no—
57C2×C3×C5×Q8—21no—
58C2×C3×C5×Q8—22no—
59C2×C3×C5×Q8—22no—
60C2×C3×C5×Q8C2312yes—
61C22×C3×C4×C5—21no—
62C22×C3×C4×C5C2311yescharacteristic
63C2×#1220—21no—
64#6817—22no—
65#6817—22no—
66C2×D60C2311yescharacteristic
67C2×C4×D15—21no—
68C2×Dic30C2311yescharacteristic
69C2×C5×D12—22no—
70C2×C5×D12—22no—
71C2×C5×D12—21no—
72C2×C5×D12C2312yes—
73C2×C3×C22⋊D5—21no—
74C3×#709—22no—
75C3×#709—22no—
76C2×C3×D20C2311yescharacteristic
77C2×C3×C4×D5—21no—
78C2×C3×Dic10C2311yescharacteristic
79C2×#1203—24no—
80#6718—24no—
81#6717—24no—
82#6710D414yescomplemented
83C2×C3⋊#168C2312yes—
84C5×#215—24no—
85C5×#215—24no—
86C2×C5×C3⋊C8D414yescomplemented
87C2×C5×C3⋊C8C2311yescharacteristic
88C2×C5×C3⋊C8C2311yescharacteristic
89C3×C22⋊Dic5D412yes—
90C3×#685—24no—
91C3×C4⋊Dic5D412yes—
92C3×#683—24no—
93C3×C4×Dic5C2312yes—
94(C2×D3)⋊Dic5—24no—
95#6662—22no—
96#6658—22no—
97#6657—22no—
98#6656—24no—
99#6653—22no—
100#6652—24no—
101C3×#536—51no—
102C2×#950—51no—
103C2×#949—51no—
104C2×#942—51no—
105C22×#215D511yescharacteristic, complemented
106#5525—51no—
107#5086—52no—
108#5085—52no—
109#5059—52no—
110#5058—52no—
111#4997—51no—
112C22×C5×Q8C2×D311yescharacteristic
113C22×C5×D4—31no—
114C2×#709—31no—
115C2×#709C2×D311yescharacteristic
116C2×C5×SD16—32no—
117C2×C5×SD16—32no—
118C2×C5×SD16—64no—
119C22×C5×C8—31no—
120#4099—62no—
121#4093C2×D312yescomplemented
122#4092—22no—
123#4090C2×D312yescomplemented
124C2×#689—32no—
125#4088—64no—
126C2×#688—32no—
127#4086—32no—
128#4084—62no—
129#4082—32no—
130C2×#687—32no—
131#4078—64no—
132C2×#686—32no—
133#4075—22no—
134#4072C2×D311yescharacteristic
135C2×#681—31no—
136C2×Dic20—31no—
137#4050—64no—
138C2×D40—31no—
139C2×#677—31no—
140C2×#677—31no—
141#3947—32no—
142C8×Dic5—32no—
143#3154—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×D15—21no—
146C22×C5×D3—22no—
147C22×C5×D3—42no—
148C22×C5×D3—82no—
149C22×C3×D5—21no—
150C3×C5×Q8—22no—
151C3×C5×Q8—22no—
152C3×C5×Q8—42no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—42no—
155C2×C3×C4×C5C2×D411yescharacteristic
156C2×C3×C4×C5C2×D412yes—
157C2×C3×C4×C5C2×D412yes—
158C2×C3×C4×C5C2411yescharacteristic
159#1220—42no—
160C2×Dic15—21no—
161D60—21no—
162D60—21no—
163C4×D15—42no—
164Dic30—21no—
165Dic30—21no—
166C5×D12—22no—
167C5×D12—22no—
168C5×D12—42no—
169C3×C22⋊D5—42no—
170C2×C3×Dic5—21no—
171C2×C3×Dic5C2×D414yes—
172C3×D20—21no—
173C3×D20—21no—
174C3×C4×D5—42no—
175C3×Dic10—21no—
176C3×Dic10—21no—
177#1203—84no—
178C3⋊#168C2×D414yes—
179C5×C3⋊C8C2×D412yes—
180C5×C3⋊C8C2×D412yes—
181C2×C3×#41—51no—
182C22×C3×Q8C2×D511yescharacteristic
183C2×#235—51no—
184C22×D12C2×D511yescharacteristic
185C3×C4⋊Q8—52no—
186C3×#124—52no—
187C3×#123—102no—
188C3×C22⋊Q8—102no—
189C3×#117—51no—
190#950—104no—
191#949—104no—
192C2×#216—52no—
193C2×#216—51no—
194#942—104no—
195C2×#215—24no—
196C2×#215—52no—
197C2×#215—52no—
198C2×#215—51no—
199C2×#215C2×D512yes—
200C2×#215C2×D512yes—
201C2×#214—51no—
202C2×#213—52no—
203C2×#213—51no—
204C2×#208—51no—
205C22×C3⋊C8C2×D511yescharacteristic
206#895—102no—
207#882—102no—
208#875—52no—
209#874—52no—
210#803—52no—
211C4×C3⋊C8—52no—
212C24×C5—61no—
213C2×C5×Q8—21no—
214C2×C5×Q8—22no—
215C2×C5×Q8—22no—
216C2×C5×Q8C22×D312yes—
217C2×C5×D4—32no—
218C2×C5×D4—62no—
219C2×C5×D4—62no—
220C2×C5×D4—61no—
221C22×C4×C5—21no—
222C22×C4×C5C22×D311yescharacteristic
223C2×C22⋊D5—21no—
224C2×C22⋊D5—61no—
225C2×C22⋊D5—64no—
226#709—22no—
227#709—22no—
228#709—62no—
229#709—62no—
230C2×D20—31no—
231C2×D20C22×D311yescharacteristic
232C2×C4×D5—21no—
233C2×C4×D5—61no—
234C2×Dic10—31no—
235C2×Dic10C22×D311yescharacteristic
236C5×SD16—64no—
237C5×SD16—64no—
238C2×C5×C8—34no—
239C2×C5×C8—31no—
240C2×C5×C8—31no—
241C22⋊Dic5—64no—
242C22⋊Dic5#6712yes—
243#689—64no—
244#688—64no—
245#687—64no—
246#686—64no—
247#685—24no—
248C4⋊Dic5#6712yes—
249#683—24no—
250C4×Dic5C22×D312yes—
251#681—34no—
252C2×#168—32no—
253Dic20—62no—
254D40—62no—
255#677—62no—
256#677—62no—
257C2×#137—151no—
258C2×AGL(1,ℤ/8)—151no—
259C2×#135—151no—
260C22×SD16—31no—
261#536—51no—
262#535—151no—
263#497—152no—
264#496—152no—
265#495—152no—
266#492—152no—
267#405—151no—
268C22×C3×C5—22no—
269C22×C3×C5#12712yes—
270C22×C3×C5C22×D411yescharacteristic
271C2×D15—41no—
272C2×D15—42no—
273C2×C5×D3—42no—
274C2×C5×D3—44no—
275C2×C5×D3—84no—
276C2×C3×D5—41no—
277C2×C3×D5—42no—
278C3×C4×C5—42no—
279C3×C4×C5#12712yes—
280C3×C4×C5C22×D411yescharacteristic
281C3×C4×C5C22×D411yescharacteristic
282Dic15—41no—
283C3×Dic5—24no—
284C3×Dic5—41no—
285C23×D3—21no—
286C3×#41—102no—
287C3×#41—102no—
288C2×C3×Q8—21no—
289C2×C3×Q8—22no—
290C2×C3×Q8—22no—
291C2×C3×Q8—51no—
292C2×C3×Q8C22×D512yes—
293C2×C3×D4—51no—
294C2×C3×D4—101no—
295C22×C3×C4—21no—
296C22×C3×C4—101no—
297C22×C3×C4C22×D511yescharacteristic
298C2×#67—101no—
299#235—102no—
300#235—102no—
301C2×D12—22no—
302C2×D12—22no—
303C2×D12—21no—
304C2×D12—51no—
305C2×D12—104no—
306C2×D12C22×D512yes—
307C2×C4×D3—101no—
308C2×Dic6—51no—
309C3×C4⋊C4—52no—
310C3×C4⋊C4—104no—
311C3×C22⋊C4—52no—
312C3×C22⋊C4—104no—
313C3×C42—52no—
314#216—104no—
315#216—102no—
316#215—24no—
317#215—24no—
318#215—104no—
319#215—104no—
320#215—102no—
321#214—102no—
322#213—104no—
323#213—102no—
324(C2×D3)⋊C4—104no—
325#208—54no—
326C2×C3⋊C8—52no—
327C2×C3⋊C8C22⋊D514yescomplemented
328C2×C3⋊C8C22×D511yescharacteristic
329C2×C3⋊C8C22×D511yescharacteristic
330C23×C5—62no—
331C23×C5—122no—
332C23×C5—242no—
333C23×C5C2×#6711yescharacteristic
334C22×D5—21no—
335C22×D5—61no—
336C5×Q8—22no—
337C5×Q8—22no—
338C5×Q8—42no—
339C5×D4—62no—
340C5×D4—62no—
341C5×D4—122no—
342C2×C4×C5—22no—
343C2×C4×C5—42no—
344C2×C4×C5D3×D412yes—
345C2×C4×C5D3×D412yes—
346C2×C4×C5C2×#6711yescharacteristic
347C2×C4×C5C23×D311yescharacteristic
348C22⋊D5—42no—
349C22⋊D5—122no—
350C22⋊D5—244no—
351C2×Dic5—21no—
352C2×Dic5—61no—
353C2×Dic5C2×#6714yes—
354D20—21no—
355D20—21no—
356D20—61no—
357D20—61no—
358C4×D5—42no—
359C4×D5—122no—
360Dic10—21no—
361Dic10—21no—
362Dic10—61no—
363Dic10—61no—
364C5×C8—32no—
365C5×C8—32no—
366#168—34no—
367C2×#41—51no—
368C2×#41—151no—
369C22×Q8D3×D511yescharacteristic
370C22×D4—31no—
371#137—304no—
372AGL(1,ℤ/8)—304no—
373#135—304no—
374C2×Q16—151no—
375C2×Q16—152no—
376C2×SD16—32no—
377C2×SD16—32no—
378C2×SD16—64no—
379C2×SD16—151no—
380C2×SD16—151no—
381C2×SD16—152no—
382C2×SD16—152no—
383C2×D8—151no—
384C2×D8—152no—
385C2×M16—151no—
386C22×C8—31no—
387C4⋊Q8—52no—
388#127—152no—
389#124—52no—
390#124—152no—
391#123—102no—
392C22⋊Q8—102no—
393#121—302no—
394#120—302no—
395#117—51no—
396#97—152no—
397C4×C8—152no—
398C2×C3×C5—22no—
399C2×C3×C5C2×#12711yescharacteristic
400C2×C3×C5#57712yes—
401D15—81no—
402C3×D5—81no—
403C5×D3—82no—
404C23×C3—101no—
405C23×C3D4×D511yescharacteristic
406C22×D3—22no—
407C22×D3—42no—
408C22×D3—82no—
409C22×D3—101no—
410C3×Q8—22no—
411C3×Q8—22no—
412C3×Q8—42no—
413C3×Q8—101no—
414C3×Q8—101no—
415C3×D4—101no—
416C3×D4—101no—
417C3×D4—202no—
418C2×C3×C4—22no—
419C2×C3×C4—42no—
420C2×C3×C4—54no—
421C2×C3×C4—101no—
422C2×C3×C4—202no—
423C2×C3×C4D4×D511yescharacteristic
424C2×C3×C4D4×D512yes—
425C2×C3×C4C2×C22⋊D512yes—
426C2×C3×C4C23×D511yescharacteristic
427#67—202no—
428C2×Dic3—101no—
429D12—22no—
430D12—22no—
431D12—42no—
432D12—101no—
433D12—101no—
434D12—404no—
435C4×D3—202no—
436Dic6—101no—
437Dic6—101no—
438C3⋊C8—54no—
439C3⋊C8C2×C22⋊D512yes—
440C3⋊C8C2×C22⋊D512yes—
441C22×C5—22no—
442C22×C5—122no—
443C22×C5—124no—
444C22×C5—244no—
445C22×C5#94012yes—
446C22×C5C22×#6711yescharacteristic
447C2×D5—41no—
448C2×D5—42no—
449C2×D5—121no—
450C2×D5—122no—
451C4×C5—42no—
452C4×C5#94012yes—
453C4×C5C2×D3×D411yescharacteristic
454C4×C5C2×D3×D411yescharacteristic
455Dic5—24no—
456Dic5—41no—
457Dic5—121no—
458C24—61no—
459#41—102no—
460#41—102no—
461#41—302no—
462#41—302no—
463C2×Q8—21no—
464C2×Q8—22no—
465C2×Q8—22no—
466C2×Q8—51no—
467C2×Q8—151no—
468C2×Q8C2×D3×D512yes—
469C2×D4—32no—
470C2×D4—51no—
471C2×D4—61no—
472C2×D4—62no—
473C2×D4—62no—
474C2×D4—101no—
475C2×D4—151no—
476C2×D4—304no—
477C2×D4—301no—
478C22×C4—21no—
479C22×C4—101no—
480C22×C4—301no—
481C22×C4C2×D3×D511yescharacteristic
482Q16—302no—
483Q16—304no—
484SD16—64no—
485SD16—64no—
486SD16—302no—
487SD16—302no—
488SD16—304no—
489SD16—304no—
490D8—302no—
491D8—304no—
492M16—154no—
493C2×C8—34no—
494C2×C8—31no—
495C2×C8—31no—
496C2×C8—152no—
497C4⋊C4—52no—
498C4⋊C4—104no—
499C22⋊C4—52no—
500C22⋊C4—104no—
501C22⋊C4—304no—
502C42—52no—
503C3×C5#315411yescharacteristic, complemented, Hall
504C22×C3—22no—
505C22×C3—202no—
506C22×C3—201no—
507C22×C3#408612yes—
508C22×C3C2×D4×D511yescharacteristic
509C2×D3—44no—
510C2×D3—42no—
511C2×D3—84no—
512C2×D3—202no—
513C2×D3—201no—
514C3×C4—42no—
515C3×C4—104no—
516C3×C4—201no—
517C3×C4#408612yes—
518C3×C4C2×D4×D511yescharacteristic
519C3×C4C22×C22⋊D511yescharacteristic
520Dic3—201no—
521C2×C5—22no—
522C2×C5—242no—
523C2×C5C2×#94011yescharacteristic
524C2×C5#583712yes—
525D5—81no—
526D5—241no—
527C23—62no—
528C23—101no—
529C23—122no—
530C23—242no—
531C23—301no—
532C23D5×#6711yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—42no—
536Q8—101no—
537Q8—101no—
538Q8—301no—
539Q8—301no—
540D4—62no—
541D4—62no—
542D4—101no—
543D4—101no—
544D4—122no—
545D4—202no—
546D4—301no—
547D4—301no—
548D4—602no—
549D4—1204no—
550C2×C4—22no—
551C2×C4—42no—
552C2×C4—54no—
553C2×C4—101no—
554C2×C4—202no—
555C2×C4—301no—
556C2×C4—602no—
557C2×C4D5×#6711yescharacteristic
558C2×C4D3×C22⋊D512yes—
559C2×C4#679012yes—
560C2×C4C22×D3×D511yescharacteristic
561C8—32no—
562C8—32no—
563C8—154no—
564C2×C3—22no—
565C2×C3—401no—
566C2×C3#6609612yes—
567C2×C3C2×#408611yescharacteristic
568D3—82no—
569D3—401no—
570C5#8340211yescharacteristic, complemented, Sylow
571C22—22no—
572C22—124no—
573C22—122no—
574C22—202no—
575C22—201no—
576C22—244no—
577C22—602no—
578C22—601no—
579C22#9156712yes—
580C22C2×D5×#6711yescharacteristic
581C4—42no—
582C4—104no—
583C4—201no—
584C4—601no—
585C4#9156712yes—
586C4C2×D3×C22⋊D511yescharacteristic
587C4C2×#679011yescharacteristic
588C3#1061558511yescharacteristic, complemented, Sylow
589C2—22no—
590C2—242no—
591C2—401no—
592C2—1201no—
593C2#1175508512yes—
594C2C2×#9156711yescharacteristic
subgroupG/H
centerC22C2×D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC22×C3×C5C22×D4
hypercenterC22×Q8D3×D5
hypocenterC3×C5#3154
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#91831C2semidirect
2. C22×C5×#215C2semidirect
3. C3×#66150C2semidirect
4. #11754606C2semidirect
5. C2×#91311C2semidirect
6. C2×#91299C2semidirect
7. #11749340C2semidirect
8. #11749339C2semidirect
9. #11749313C2semidirect
10. #11749312C2semidirect
11. #11748318C2semidirect
17. C2×#6718C22semidirect
19. C2×#6717C22semidirect
25. C3×#4093C22semidirect
27. C3×#4090C22semidirect
34. C2×#6662C22semidirect
35. C2×#6658C22semidirect
37. C2×#6657C22semidirect
39. C2×#6653C22semidirect
42. #90952C22semidirect
43. Dic5×C3⋊C8C22semidirect
46. #66150D3semidirect
82. #6710D4semidirect
86. C2×C5×C3⋊C8D4semidirect
105. C22×#215D5semidirect
121. #4093C2×D3semidirect
123. #4090C2×D3semidirect
327. C2×C3⋊C8C22⋊D5semidirect
503. C3×C5#3154semidirect
570. C5#83402semidirect
588. C3#10615585semidirect

Maximal subgroups

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

subgroupindex
C2×#918312
C22×C5×#2152
C3×#661502
#117546062
C2×#913112
C2×#912992
#117493402
#117493392
#117493132
#117493122
#117483182
#106155853
#834025