Finite Group Databasebeta

← #14424431

Subgroups — #14424431

Order 1920.

1464 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×2−1+4C211yesmaximal, characteristic, complemented
2#11756589C211yesmaximal, characteristic, complemented
3C2×#91838C211yesmaximal, characteristic, complemented
4C22×#6717C211yesmaximal, characteristic, complemented
5#11751339C214yesmaximal, complemented
6#11751335C211yesmaximal, characteristic, complemented
7#11751334C211yesmaximal, characteristic, complemented
8#11751328C214yesmaximal, complemented
9#11751326C211yesmaximal, characteristic, complemented
10#10615844—31nomaximal, Hall
11C3×C5×2−1+4—24no—
12C3×C5×2−1+4—24no—
13C2×C3×C5×#41—22no—
14C2×C3×C5×#41—21no—
15C2×C3×C5×#41C2211yescharacteristic
16C2×C3×C5×#41C2211yescharacteristic
17C2×C3×C5×#41C2212yescomplemented
18C22×C3×C5×Q8—21no—
19C22×C3×C5×Q8C2211yescharacteristic
20C22×C3×C5×Q8C2212yescomplemented
21C22×D60C2211yescharacteristic
22#91838—24no—
23#91836—21no—
24C2×#6717—24no—
25C2×#6717C2214yes—
26C2×#6717C2212yes—
27#91829—21no—
28#91826—22no—
29C2×#6715C2212yes—
30#91819C2214yes—
31#91815C2211yescharacteristic
32C2×#6710C2211yescharacteristic
33C22×C3⋊#168C2211yescharacteristic
34#91119C2214yescomplemented
35#91119C2212yescomplemented
36#91116C2212yescomplemented
37#91114C2214yescomplemented
38#84113—51nomaximal, Hall
39C2×C5×2−1+4D311yescharacteristic, complemented
40#66164—31no—
41C2×#4095—31no—
42C22×#688—31no—
43#65527—34no—
44#65523—31no—
45#65522—31no—
46#65516—34no—
47#65514—31no—
48C23×D15—21no—
49C3×C5×#41—216no—
50C3×C5×#41—28no—
51C3×C5×#41—24no—
52C3×C5×#41—24no—
53C3×C5×#41—22no—
54C3×C5×#41—22no—
55C3×C5×#41—22no—
56C3×C5×#41D414yescomplemented
57C2×C3×C5×Q8—28no—
58C2×C3×C5×Q8—24no—
59C2×C3×C5×Q8—24no—
60C2×C3×C5×Q8—21no—
61C2×C3×C5×Q8—22no—
62C2×C3×C5×Q8—22no—
63C2×C3×C5×Q8D411yescharacteristic
64C2×C3×C5×Q8D414yescomplemented
65C2×C3×C5×Q8C2312yes—
66C2×C3×C5×Q8C2311yescharacteristic
67C2×C3×C5×D4—21no—
68C2×C3×C5×D4—22no—
69C2×C3×C5×D4D412yes—
70C2×C3×C5×D4D411yescharacteristic
71C2×C3×C5×D4C2311yescharacteristic
72C22×C3×C4×C5—21no—
73C22×C3×C4×C5—22no—
74C22×C3×C4×C5—21no—
75C22×C3×C4×C5—22no—
76C22×C3×C4×C5C2312yes—
77C22×C3×C4×C5C2311yescharacteristic
78C2×#1220—24no—
79C2×D60—22no—
80C2×D60—22no—
81C2×D60—21no—
82C2×D60C2312yes—
83#6719—21no—
84#6717—28no—
85#6717—24no—
86#6715—24no—
87#6714—24no—
88C4⋊Dic15C2312yes—
89C4×Dic15—21no—
90#6710—22no—
91C2×C3⋊#168—22no—
92C2×C3⋊#168C2312yes—
93C2×C3⋊#168C2312yes—
94C2×C3×2−1+4D511yescharacteristic, complemented
95#5830—51no—
96C2×#949—51no—
97C22×#215—51no—
98#5238—54no—
99#5234—51no—
100#5233—51no—
101#5227—54no—
102#5225—51no—
103C5×2−1+4—24no—
104C5×2−1+4—24no—
105C2×C5×#41—22no—
106C2×C5×#41—21no—
107C2×C5×#41C2×D311yescharacteristic
108C2×C5×#41C2×D311yescharacteristic
109C2×C5×#41C2×D312yescomplemented
110C22×C5×Q8—21no—
111C22×C5×Q8C2×D311yescharacteristic
112C22×C5×Q8C2×D312yescomplemented
113C22×D20—31no—
114#4095—64no—
115#4093—61no—
116C2×#688—34no—
117C2×#688—32no—
118C2×#688—64no—
119#4086—61no—
120#4083—62no—
121C2×#686—32no—
122#4076—34no—
123#4072—31no—
124C2×#681—31no—
125C22×#168—31no—
126#3967—34no—
127#3967—32no—
128#3964—32no—
129C22⋊#168—34no—
130#3010—151noSylow
131C23×C3×C5—21no—
132C23×C3×C5C2×D412yes—
133C23×C3×C5C2×D411yescharacteristic
134C22×D15—22no—
135C22×D15—42no—
136C22×D15—82no—
137C3×C5×Q8—28no—
138C3×C5×Q8—22no—
139C3×C5×Q8—24no—
140C3×C5×Q8—21no—
141C3×C5×Q8—22no—
142C3×C5×Q8C2×D412yes—
143C3×C5×Q8C2×D414yes—
144C3×C5×D4—28no—
145C3×C5×D4—24no—
146C3×C5×D4—24no—
147C3×C5×D4—22no—
148C3×C5×D4—21no—
149C3×C5×D4C2×D412yes—
150C2×C3×C4×C5—28no—
151C2×C3×C4×C5—28no—
152C2×C3×C4×C5—24no—
153C2×C3×C4×C5—22no—
154C2×C3×C4×C5—21no—
155C2×C3×C4×C5—24no—
156C2×C3×C4×C5—22no—
157C2×C3×C4×C5—22no—
158C2×C3×C4×C5C2×D412yes—
159C2×C3×C4×C5C2×D411yescharacteristic
160C2×C3×C4×C5C2×D412yes—
161C2×C3×C4×C5C2×D412yes—
162C2×C3×C4×C5C2411yescharacteristic
163#1220—84no—
164C2×Dic15—22no—
165D60—22no—
166D60—22no—
167D60—42no—
168C3⋊#168—22no—
169C3⋊#168—22no—
170C3×2−1+4—24no—
171C3×2−1+4—24no—
172C2×C3×#41—22no—
173C2×C3×#41—21no—
174C2×C3×#41C2×D511yescharacteristic
175C2×C3×#41C2×D511yescharacteristic
176C2×C3×#41C2×D512yescomplemented
177C22×C3×Q8—21no—
178C22×C3×Q8C2×D511yescharacteristic
179C22×C3×Q8C2×D512yescomplemented
180C22×D12—51no—
181#949—104no—
182#947—101no—
183C2×#215—52no—
184C2×#215—54no—
185C2×#215—104no—
186#940—101no—
187#937—102no—
188C2×#213—52no—
189#930—54no—
190#926—51no—
191C2×#208—51no—
192C22×C3⋊C8—51no—
193#835—54no—
194#835—52no—
195#832—52no—
196#830—54no—
197C23×D5—61no—
198C5×#41—216no—
199C5×#41—28no—
200C5×#41—24no—
201C5×#41—24no—
202C5×#41—22no—
203C5×#41—22no—
204C5×#41—22no—
205C5×#41#6714yescomplemented
206C2×C5×Q8—28no—
207C2×C5×Q8—24no—
208C2×C5×Q8—24no—
209C2×C5×Q8—21no—
210C2×C5×Q8—22no—
211C2×C5×Q8—22no—
212C2×C5×Q8#6711yescharacteristic
213C2×C5×Q8#6714yescomplemented
214C2×C5×Q8C22×D312yes—
215C2×C5×Q8C22×D311yescharacteristic
216C2×C5×D4—21no—
217C2×C5×D4—22no—
218C2×C5×D4#6712yes—
219C2×C5×D4#6711yescharacteristic
220C2×C5×D4C22×D311yescharacteristic
221C22×C4×C5—21no—
222C22×C4×C5—22no—
223C22×C4×C5—21no—
224C22×C4×C5—22no—
225C22×C4×C5C22×D312yes—
226C22×C4×C5C22×D311yescharacteristic
227C2×C22⋊D5—64no—
228C2×D20—32no—
229C2×D20—62no—
230C2×D20—62no—
231C2×D20—61no—
232C22⋊Dic5—61no—
233#688—68no—
234#688—64no—
235#686—64no—
236#685—64no—
237C4⋊Dic5—32no—
238C4×Dic5—61no—
239#681—62no—
240C2×#168—32no—
241C2×#168—32no—
242C2×#168—62no—
243C2×2−1+4D1511yescharacteristic, complemented
244C2×AGL(1,ℤ/8)—151no—
245C22×SD16—151no—
246#535—151no—
247#449—154no—
248#448—154no—
249#419—151no—
250#416—151no—
251#408—151no—
252C22×C3×C5—22no—
253C22×C3×C5—22no—
254C22×C3×C5—22no—
255C22×C3×C5—24no—
256C22×C3×C5—42no—
257C22×C3×C5#12012yes—
258C22×C3×C5C22×D411yescharacteristic
259C2×D15—42no—
260C2×D15—44no—
261C2×D15—84no—
262C3×C4×C5—24no—
263C3×C4×C5—22no—
264C3×C4×C5—22no—
265C3×C4×C5#12012yes—
266C3×C4×C5C22×D411yescharacteristic
267C3×C4×C5C22×D411yescharacteristic
268Dic15—42no—
269C23×D3—101no—
270C3×#41—216no—
271C3×#41—28no—
272C3×#41—24no—
273C3×#41—24no—
274C3×#41—22no—
275C3×#41—22no—
276C3×#41—22no—
277C3×#41C22⋊D514yescomplemented
278C2×C3×Q8—28no—
279C2×C3×Q8—24no—
280C2×C3×Q8—24no—
281C2×C3×Q8—21no—
282C2×C3×Q8—22no—
283C2×C3×Q8—22no—
284C2×C3×Q8C22⋊D511yescharacteristic
285C2×C3×Q8C22⋊D514yescomplemented
286C2×C3×Q8C22×D512yes—
287C2×C3×Q8C22×D511yescharacteristic
288C2×C3×D4—21no—
289C2×C3×D4—22no—
290C2×C3×D4C22⋊D512yes—
291C2×C3×D4C22⋊D511yescharacteristic
292C2×C3×D4C22×D511yescharacteristic
293C22×C3×C4—21no—
294C22×C3×C4—22no—
295C22×C3×C4—21no—
296C22×C3×C4—22no—
297C22×C3×C4C22×D512yes—
298C22×C3×C4C22×D511yescharacteristic
299C2×#67—104no—
300C2×D12—52no—
301C2×D12—102no—
302C2×D12—102no—
303C2×D12—101no—
304#217—101no—
305#215—108no—
306#215—104no—
307#213—104no—
308(C2×D3)⋊C4—104no—
309C4⋊Dic3—52no—
310C4×Dic3—101no—
311#208—102no—
312C2×C3⋊C8—52no—
313C2×C3⋊C8—52no—
314C2×C3⋊C8—102no—
315C23×C5—21no—
316C23×C5C2×#6712yes—
317C23×C5C2×#6711yescharacteristic
318C22×D5—62no—
319C22×D5—122no—
320C22×D5—242no—
321C5×Q8—28no—
322C5×Q8—22no—
323C5×Q8—24no—
324C5×Q8—21no—
325C5×Q8—22no—
326C5×Q8C2×#6712yes—
327C5×Q8C2×#6714yes—
328C5×D4—28no—
329C5×D4—24no—
330C5×D4—24no—
331C5×D4—22no—
332C5×D4—21no—
333C5×D4C2×#6712yes—
334C2×C4×C5—28no—
335C2×C4×C5—28no—
336C2×C4×C5—24no—
337C2×C4×C5—22no—
338C2×C4×C5—21no—
339C2×C4×C5—24no—
340C2×C4×C5—22no—
341C2×C4×C5—22no—
342C2×C4×C5C2×#6712yes—
343C2×C4×C5C2×#6711yescharacteristic
344C2×C4×C5C2×#6712yes—
345C2×C4×C5C2×#6712yes—
346C2×C4×C5C23×D311yescharacteristic
347C22⋊D5—244no—
348C2×Dic5—62no—
349D20—62no—
350D20—62no—
351D20—122no—
352#168—62no—
353#168—62no—
3542−1+4—24no—
3552−1+4—24no—
356C2×#41—22no—
357C2×#41—21no—
358C2×#41C2×D1511yescharacteristic
359C2×#41C2×D1511yescharacteristic
360C2×#41C2×D1512yescomplemented
361C22×Q8—21no—
362C22×Q8C2×D1511yescharacteristic
363C22×Q8C2×D1512yescomplemented
364C22×D4—151no—
365AGL(1,ℤ/8)—304no—
366C2×SD16—154no—
367C2×SD16—152no—
368C2×SD16—304no—
369C2×D8—152no—
370C2×M16—151no—
371C22×C8—151no—
372#127—301no—
373#124—301no—
374#121—154no—
375#120—302no—
376#117—151no—
377#103—152no—
378#103—154no—
379#102—152no—
380C22⋊C8—154no—
381C2×C3×C5—22no—
382C2×C3×C5—42no—
383C2×C3×C5—42no—
384C2×C3×C5C2×#12011yescharacteristic
385C2×C3×C5#57712yes—
386D15—82no—
387C23×C3—21no—
388C23×C3C2×C22⋊D512yes—
389C23×C3C2×C22⋊D511yescharacteristic
390C22×D3—102no—
391C22×D3—202no—
392C22×D3—402no—
393C3×Q8—28no—
394C3×Q8—22no—
395C3×Q8—24no—
396C3×Q8—21no—
397C3×Q8—22no—
398C3×Q8C2×C22⋊D512yes—
399C3×Q8C2×C22⋊D514yes—
400C3×D4—28no—
401C3×D4—24no—
402C3×D4—24no—
403C3×D4—22no—
404C3×D4—21no—
405C3×D4C2×C22⋊D512yes—
406C2×C3×C4—28no—
407C2×C3×C4—28no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—21no—
411C2×C3×C4—24no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4C2×C22⋊D512yes—
415C2×C3×C4C2×C22⋊D511yescharacteristic
416C2×C3×C4C2×C22⋊D512yes—
417C2×C3×C4C2×C22⋊D512yes—
418C2×C3×C4C23×D511yescharacteristic
419#67—404no—
420C2×Dic3—102no—
421D12—102no—
422D12—102no—
423D12—202no—
424C3⋊C8—102no—
425C3⋊C8—102no—
426C22×C5—22no—
427C22×C5—24no—
428C22×C5—22no—
429C22×C5—22no—
430C22×C5—42no—
431C22×C5#95312yes—
432C22×C5C22×#6711yescharacteristic
433C2×D5—124no—
434C2×D5—122no—
435C2×D5—244no—
436C4×C5—24no—
437C4×C5—22no—
438C4×C5—22no—
439C4×C5#95312yes—
440C4×C5C22×#6711yescharacteristic
441C4×C5C22×#6711yescharacteristic
442Dic5—122no—
443C24—301no—
444#41—216no—
445#41—28no—
446#41—24no—
447#41—24no—
448#41—22no—
449#41—22no—
450#41—22no—
451#41#122014yescomplemented
452C2×Q8—28no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—21no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8#122011yescharacteristic
459C2×Q8#122014yescomplemented
460C2×Q8C22×D1512yes—
461C2×Q8C22×D1511yescharacteristic
462C2×D4—21no—
463C2×D4—22no—
464C2×D4—152no—
465C2×D4—304no—
466C2×D4—302no—
467C2×D4—302no—
468C2×D4—301no—
469C2×D4#122012yes—
470C2×D4#122011yescharacteristic
471C2×D4C22×D1511yescharacteristic
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4C22×D1512yes—
477C22×C4C22×D1511yescharacteristic
478SD16—308no—
479SD16—304no—
480D8—304no—
481M16—302no—
482C2×C8—152no—
483C2×C8—152no—
484C2×C8—302no—
485C4⋊C4—152no—
486C22⋊C4—301no—
487C22⋊C4—304no—
488C42—301no—
489C3×C5#301011yescharacteristic, complemented, Hall
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—42no—
495C22×C3#409912yes—
496C22×C3C22×C22⋊D511yescharacteristic
497C2×D3—204no—
498C2×D3—202no—
499C2×D3—404no—
500C3×C4—24no—
501C3×C4—22no—
502C3×C4—22no—
503C3×C4#409912yes—
504C3×C4C22×C22⋊D511yescharacteristic
505C3×C4C22×C22⋊D511yescharacteristic
506Dic3—202no—
507C2×C5—22no—
508C2×C5—42no—
509C2×C5—42no—
510C2×C5#583712yes—
511C2×C5C2×#95311yescharacteristic
512D5—242no—
513C23—21no—
514C23—302no—
515C23—602no—
516C23—1202no—
517C23C2×#122012yes—
518C23C2×#122011yescharacteristic
519Q8—28no—
520Q8—22no—
521Q8—24no—
522Q8—21no—
523Q8—22no—
524Q8C2×#122012yes—
525Q8C2×#122014yes—
526D4—28no—
527D4—24no—
528D4—24no—
529D4—22no—
530D4—21no—
531D4—302no—
532D4—302no—
533D4—602no—
534D4—1204no—
535D4C2×#122012yes—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—302no—
545C2×C4C2×#122012yes—
546C2×C4C2×#122011yescharacteristic
547C2×C4C2×#122012yes—
548C2×C4C2×#122012yes—
549C2×C4C23×D1511yescharacteristic
550C8—302no—
551C8—302no—
552C2×C3—22no—
553C2×C3—42no—
554C2×C3—42no—
555C2×C3#6617112yes—
556C2×C3C2×#409911yescharacteristic
557D3—402no—
558C5#8411311yescharacteristic, complemented, Sylow
559C22—22no—
560C22—24no—
561C22—22no—
562C22—22no—
563C22—42no—
564C22—604no—
565C22—602no—
566C22—1204no—
567C22#9184212yes—
568C22C22×#122011yescharacteristic
569C4—24no—
570C4—22no—
571C4—22no—
572C4—602no—
573C4#9184212yes—
574C4C22×#122011yescharacteristic
575C4C22×#122011yescharacteristic
576C3#1061584411yescharacteristic, complemented, Sylow
577C2—22no—
578C2—42no—
579C2—42no—
580C2—1202no—
581C2#1175659612yes—
582C2C2×#9184211yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC2×C3×C5×2−1+4C2
socleC22×C3×C5C22×D4
hypercenterC2×2−1+4D15
hypocenterC3×C5#3010
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×2−1+4C2semidirect
2. #11756589C2semidirect
3. C2×#91838C2semidirect
4. C22×#6717C2semidirect
5. #11751339C2semidirect
6. #11751335C2semidirect
7. #11751334C2semidirect
8. #11751328C2semidirect
9. #11751326C2semidirect
17. C2×C3×C5×#41C22semidirect
20. C22×C3×C5×Q8C22semidirect
34. #91119C22semidirect
35. #91119C22semidirect
36. #91116C22semidirect
37. #91114C22semidirect
39. C2×C5×2−1+4D3semidirect
56. C3×C5×#41D4semidirect
64. C2×C3×C5×Q8D4semidirect
94. C2×C3×2−1+4D5semidirect
109. C2×C5×#41C2×D3semidirect
112. C22×C5×Q8C2×D3semidirect
176. C2×C3×#41C2×D5semidirect
179. C22×C3×Q8C2×D5semidirect
205. C5×#41#67semidirect
213. C2×C5×Q8#67semidirect
243. C2×2−1+4D15semidirect
277. C3×#41C22⋊D5semidirect
285. C2×C3×Q8C22⋊D5semidirect
360. C2×#41C2×D15semidirect
363. C22×Q8C2×D15semidirect
451. #41#1220semidirect
459. C2×Q8#1220semidirect
489. C3×C5#3010semidirect
558. C5#84113semidirect
576. C3#10615844semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C5×2−1+42
#117565892
C2×#918382
C22×#67172
#117513392
#117513352
#117513342
#117513282
#117513262
#106158443
#841135