Finite Group Databasebeta

← #14297257

Subgroups — #14297257

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#92004C211yesmaximal, characteristic
2C2×#91089C211yesmaximal, characteristic, complemented
3C2×C5×#805C211yesmaximal, characteristic, complemented
4C3×#65218C211yesmaximal, characteristic, complemented
5#11748413C211yesmaximal, characteristic, complemented
6#11748384C211yesmaximal, characteristic, complemented
7C2×#90961C211yesmaximal, characteristic, complemented
8#10601180—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C22×C3×Dic10C2211yescharacteristic
12C2×#6782—22no—
13C2×D15⋊Q8—21no—
14C2×#6769—21no—
15#92004—22no—
16#92004—22no—
17#92004—22no—
18#92004C414yescomplemented
19C2×#6768—21no—
20C2×D3×Dic10—21no—
21C2×C3×C42×C5C2211yescharacteristic
22C2×#6710C2211yescharacteristic
23C2×C5×#208C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25#91089—22no—
26C5×#805—22no—
27C3×#3963—22no—
28#90974—21no—
29#90972—21no—
30#90968—21no—
31#90966—21no—
32#90961—22no—
33#69692—51nomaximal, Hall
34C2×#4150—31no—
35C2×C5×#104—31no—
36#65521—31no—
37C2×#3956—31no—
38#65422—31no—
39#65218D311yescharacteristic, complemented
40C2×#3937—31no—
41C22×C3×C4×C5—21no—
42C22×C3×C4×C5C2311yescharacteristic
43C2×#1220—21no—
44#6817—21no—
45#6817—21no—
46#6817C2×C412yes—
47#6817D412yes—
48C2×D60D411yescharacteristic
49C2×C4×D15—21no—
50C2×Dic30D411yescharacteristic
51C2×C5×#67—21no—
52C5×#235—21no—
53C5×#235—21no—
54C5×#235C2×C412yes—
55C5×#235D412yes—
56C2×C5×D12D411yescharacteristic
57C2×C4×C5×D3—21no—
58C2×C5×Dic6D411yescharacteristic
59C22×C3×Dic5—21no—
60C22×C3×Dic5D412yes—
61C2×C3×Dic10—21no—
62C2×C3×Dic10—21no—
63C2×C3×Dic10—21no—
64C2×C3×Dic10—21no—
65C2×C3×Dic10C2×C412yes—
66C2×C3×Dic10D412yes—
67C2×#1204—22no—
68C2×#1203—22no—
69C2×#1200—22no—
70#6782—28no—
71#6782—28no—
72C2×D3×Dic5—22no—
73D15⋊Q8—24no—
74D15⋊Q8—24no—
75#6769—24no—
76#6769—24no—
77#6768—24no—
78#6768—24no—
79D3×Dic10—24no—
80D3×Dic10—24no—
81C3×C42×C5—21no—
82C3×C42×C5—21no—
83#6710—21no—
84#6710—21no—
85C2×C3⋊#168—21no—
86C5×#208—21no—
87C5×#208—21no—
88C2×C5×C3⋊C8—21no—
89C3×C4⋊Dic5—21no—
90C3×C4⋊Dic5—21no—
91C2×#1007—51no—
92C3×#391—51no—
93C2×#836—51no—
94#4966—51no—
95#4966—51no—
96C2×#805—51no—
97C2×#805D511yescharacteristic, complemented
98C2×C5×#41—31no—
99#4150—34no—
100#4150—62no—
101#4150—62no—
102#4150—62no—
103C2×Q8×D5—61no—
104C2×#711—61no—
105C2×#711—62no—
106C2×#709—31no—
107C2×#709—61no—
108C22×Dic10—61no—
109C22×Dic10C2×D311yescharacteristic
110C2×C5×M16—31no—
111C2×C42×C5C2×D311yescharacteristic
112C2×C4⋊Dic5C2×D311yescharacteristic
113C2×#681—31no—
114C5×#104—62no—
115#3967—61no—
116#3964—61no—
117#3963—22no—
118#3956—62no—
119#3953—61no—
120#3948—61no—
121#3937—62no—
122#1793—151noSylow
123C23×C3×C5C2×D411yescharacteristic
124C22×D15—21no—
125C22×C5×D3—21no—
126C2×C3×C4×C5—22no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5—41no—
129C2×C3×C4×C5C22×C411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5C2×D411yescharacteristic
135#1220—22no—
136#1220—22no—
137C2×Dic15—21no—
138D60—21no—
139D60C22⋊C412yes—
140C4×D15—22no—
141C4×D15—22no—
142Dic30—21no—
143Dic30C22⋊C412yes—
144C5×#67—22no—
145C5×#67—22no—
146C2×C5×Dic3—21no—
147C5×D12—21no—
148C5×D12C22⋊C412yes—
149C4×C5×D3—22no—
150C4×C5×D3—22no—
151C5×Dic6—21no—
152C5×Dic6C22⋊C412yes—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—42no—
156C2×C3×Dic5—41no—
157C2×C3×Dic5C22⋊C414yes—
158C3×Dic10—22no—
159C3×Dic10—22no—
160C3×Dic10—22no—
161C3×Dic10—22no—
162#1204—28no—
163#1203—28no—
164#1200—28no—
165D3×Dic5—28no—
166C3⋊#168—41no—
167C5×C3⋊C8—41no—
168C22×C3×Q8—51no—
169#1007—54no—
170#1007—102no—
171#1007—102no—
172#1007—102no—
173C2×#239—101no—
174C2×#239—101no—
175C2×D3×Q8—101no—
176C2×D3×Q8—101no—
177C2×#235—51no—
178C2×#235—102no—
179C2×#235C2×D511yescharacteristic
180C2×C3×C4⋊C4—51no—
181C2×C3×C42C2×D511yescharacteristic
182C2×#208—51no—
183C2×#208C2×D511yescharacteristic
184C3×#95—102no—
185#836—102no—
186#810—101no—
187#810—101no—
188#809—101no—
189#809—101no—
190#805—22no—
191#805—102no—
192C5×#41—32no—
193C5×#41—32no—
194C5×#41—61no—
195C5×#41—61no—
196C2×C5×Q8—31no—
197C2×C5×D4—31no—
198C2×C5×D4—61no—
199C22×C4×C5—21no—
200C22×C4×C5—61no—
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—62no—
203C2×C22⋊D5—61no—
204C22×Dic5—21no—
205C22×Dic5—62no—
206C22×Dic5#6712yes—
207Q8×D5—64no—
208Q8×D5—64no—
209#711—64no—
210#711—64no—
211#711—68no—
212#711—68no—
213#709—32no—
214#709—32no—
215#709—64no—
216#709—64no—
217#709—61no—
218#709—61no—
219C2×D20—31no—
220C2×C4×D5—62no—
221C2×C4×D5—61no—
222C2×Dic10—21no—
223C2×Dic10—21no—
224C2×Dic10—21no—
225C2×Dic10—21no—
226C2×Dic10—31no—
227C2×Dic10—64no—
228C2×Dic10—64no—
229C2×Dic10—62no—
230C2×Dic10C4×D312yes—
231C2×Dic10D1212yes—
232C5×M16—61no—
233C5×M16—61no—
234C2×C5×C8—61no—
235C42×C5—21no—
236C42×C5—21no—
237C4⋊Dic5—21no—
238C4⋊Dic5—21no—
239#681—61no—
240#681—61no—
241C2×#168—61no—
242C2×2−1+4—151no—
243C2×#104—31no—
244C2×#104—151no—
245#417—151no—
246#417—151no—
247C2×#101—151no—
248#391—51no—
249C22×C3×C5—21no—
250C22×C3×C5—21no—
251C22×C3×C5C2×C22⋊C411yescharacteristic
252C22×C3×C5#12011yescharacteristic
253C22×C3×C5#12011yescharacteristic
254C2×D15—22no—
255C2×D15—41no—
256C2×D15—41no—
257C2×C5×D3—22no—
258C2×C5×D3—41no—
259C2×C5×D3—41no—
260C3×C4×C5—42no—
261C3×C4×C5C2×C22⋊C411yescharacteristic
262C3×C4×C5C2×C22⋊C411yescharacteristic
263C3×C4×C5#12011yescharacteristic
264C3×C4×C5#12011yescharacteristic
265Dic15—22no—
266C3×Dic5—24no—
267C3×Dic5—81no—
268C5×Dic3—22no—
269C2×C3×Q8—52no—
270C2×C3×Q8—52no—
271C2×C3×Q8—101no—
272C2×C3×Q8—101no—
273C2×C3×Q8—101no—
274C2×C3×Q8—101no—
275C22×C3×C4—21no—
276C22×C3×C4—52no—
277C22×C3×C4—101no—
278C22×C3×C4C22×D511yescharacteristic
279C2×#67—21no—
280C2×#67—101no—
281#239—104no—
282#239—104no—
283#239—104no—
284#239—104no—
285D3×Q8—104no—
286D3×Q8—104no—
287D3×Q8—104no—
288D3×Q8—104no—
289#235—21no—
290#235—21no—
291#235—52no—
292#235—52no—
293#235—108no—
294#235—108no—
295#235—101no—
296#235—101no—
297#235C4×D512yes—
298#235C22⋊D512yes—
299C2×D12—51no—
300C2×D12—102no—
301C2×D12D2011yescharacteristic
302C2×C4×D3—21no—
303C2×C4×D3—102no—
304C2×C4×D3—102no—
305C2×C4×D3—101no—
306C2×Dic6—51no—
307C2×Dic6—102no—
308C2×Dic6D2011yescharacteristic
309C3×C4⋊C4—101no—
310C3×C4⋊C4—101no—
311C3×C42—21no—
312C3×C42—21no—
313#208—21no—
314#208—21no—
315#208—101no—
316#208—101no—
317C2×C3⋊C8—21no—
318C2×C3⋊C8—101no—
319C23×C5—61no—
320C23×C5C2×#6711yescharacteristic
321C22×D5—61no—
322C5×Q8—32no—
323C5×Q8—61no—
324C5×D4—32no—
325C5×D4—62no—
326C5×D4—62no—
327C5×D4—61no—
328C2×C4×C5—22no—
329C2×C4×C5—41no—
330C2×C4×C5—41no—
331C2×C4×C5—61no—
332C2×C4×C5—62no—
333C2×C4×C5—62no—
334C2×C4×C5C2×C4×D311yescharacteristic
335C2×C4×C5C2×D1211yescharacteristic
336C2×C4×C5D3×D411yescharacteristic
337C2×C4×C5D3×D411yescharacteristic
338C2×C4×C5D3×D411yescharacteristic
339C2×C4×C5D3×D411yescharacteristic
340C22⋊D5—68no—
341C22⋊D5—62no—
342C22⋊D5—62no—
343C2×Dic5—22no—
344C2×Dic5—22no—
345C2×Dic5—42no—
346C2×Dic5—41no—
347C2×Dic5—68no—
348C2×Dic5—61no—
349C2×Dic5(C2×D3)⋊C414yes—
350D20—32no—
351D20—61no—
352C4×D5—68no—
353C4×D5—62no—
354C4×D5—62no—
355Dic10—22no—
356Dic10—22no—
357Dic10—22no—
358Dic10—22no—
359Dic10—32no—
360Dic10—68no—
361Dic10—61no—
362C5×C8—121no—
363#168—121no—
3642−1+4—154no—
3652−1+4—302no—
3662−1+4—302no—
3672−1+4—302no—
368C2×#41—31no—
369C2×#41—151no—
370C2×#41—302no—
371C2×#41—301no—
372C2×#41—301no—
373C22×Q8—51no—
374C22×Q8—301no—
375C22×Q8—301no—
376C2×M16—31no—
377C2×M16—151no—
378C2×C4⋊C4—51no—
379C2×C42D3×D511yescharacteristic
380#104—62no—
381#104—302no—
382#103—301no—
383#103—301no—
384#102—301no—
385#102—301no—
386#101—302no—
387#95—102no—
388C2×C3×C5—21no—
389C2×C3×C5—21no—
390C2×C3×C5#37911yescharacteristic
391C2×C3×C5#45411yescharacteristic
392C2×C3×C5#45611yescharacteristic
393D15—42no—
394C5×D3—42no—
395C23×C3D4×D511yescharacteristic
396C22×D3—21no—
397C22×D3—101no—
398C3×Q8—102no—
399C3×Q8—102no—
400C3×Q8—102no—
401C3×Q8—102no—
402C2×C3×C4—22no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—54no—
406C2×C3×C4—102no—
407C2×C3×C4—102no—
408C2×C3×C4—202no—
409C2×C3×C4—201no—
410C2×C3×C4C2×C4×D511yescharacteristic
411C2×C3×C4C2×D2011yescharacteristic
412C2×C3×C4D4×D511yescharacteristic
413C2×C3×C4D4×D511yescharacteristic
414C2×C3×C4D4×D511yescharacteristic
415C2×C3×C4C2×C22⋊D511yescharacteristic
416#67—22no—
417#67—22no—
418#67—102no—
419#67—102no—
420C2×Dic3—21no—
421C2×Dic3—101no—
422D12—21no—
423D12—52no—
424D12—108no—
425D12—101no—
426D12#68512yes—
427C4×D3—22no—
428C4×D3—22no—
429C4×D3—108no—
430C4×D3—108no—
431C4×D3—102no—
432C4×D3—102no—
433Dic6—21no—
434Dic6—52no—
435Dic6—108no—
436Dic6—101no—
437Dic6#68512yes—
438C3⋊C8—41no—
439C3⋊C8—201no—
440C22×C5—21no—
441C22×C5—21no—
442C22×C5—62no—
443C22×C5—121no—
444C22×C5—121no—
445C22×C5C2×(C2×D3)⋊C411yescharacteristic
446C22×C5#93711yescharacteristic
447C22×C5#93711yescharacteristic
448C2×D5—62no—
449C2×D5—121no—
450C2×D5—121no—
451C4×C5—42no—
452C4×C5—62no—
453C4×C5D3×C22⋊C411yescharacteristic
454C4×C5D3×C22⋊C411yescharacteristic
455C4×C5#88211yescharacteristic
456C4×C5#88211yescharacteristic
457Dic5—24no—
458Dic5—62no—
459Dic5—81no—
460#41—32no—
461#41—32no—
462#41—61no—
463#41—61no—
464#41—152no—
465#41—152no—
466#41—308no—
467#41—304no—
468#41—304no—
469#41—308no—
470#41—304no—
471#41—304no—
472#41—301no—
473#41—301no—
474C2×Q8—31no—
475C2×Q8—52no—
476C2×Q8—52no—
477C2×Q8—101no—
478C2×Q8—101no—
479C2×Q8—101no—
480C2×Q8—101no—
481C2×Q8—151no—
482C2×Q8—302no—
483C2×Q8—304no—
484C2×Q8—304no—
485C2×Q8—304no—
486C2×Q8—304no—
487C2×D4—31no—
488C2×D4—61no—
489C2×D4—151no—
490C2×D4—302no—
491C2×D4—301no—
492C22×C4—21no—
493C22×C4—52no—
494C22×C4—61no—
495C22×C4—101no—
496C22×C4—302no—
497C22×C4—302no—
498C22×C4—301no—
499C22×C4C2×D3×D511yescharacteristic
500M16—61no—
501M16—61no—
502M16—301no—
503M16—301no—
504C2×C8—61no—
505C2×C8—301no—
506C4⋊C4—101no—
507C4⋊C4—101no—
508C42—21no—
509C42—21no—
510C3×C5#179311yescharacteristic, complemented, Hall
511C22×C3—21no—
512C22×C3—21no—
513C22×C3D5×C22⋊C411yescharacteristic
514C22×C3#402811yescharacteristic
515C22×C3#408311yescharacteristic
516C2×D3—22no—
517C2×D3—41no—
518C2×D3—41no—
519C2×D3—102no—
520C2×D3—201no—
521C2×D3—201no—
522C3×C4—42no—
523C3×C4—104no—
524C3×C4—401no—
525C3×C4D5×C22⋊C411yescharacteristic
526C3×C4#402811yescharacteristic
527C3×C4C2×#68511yescharacteristic
528C3×C4#408311yescharacteristic
529Dic3—22no—
530Dic3—102no—
531C2×C5—21no—
532C2×C5—21no—
533C2×C5—122no—
534C2×C5#495511yescharacteristic
535C2×C5#506211yescharacteristic
536C2×C5#509511yescharacteristic
537D5—122no—
538C23—61no—
539C23—301no—
540C23D5×#6711yescharacteristic
541Q8—32no—
542Q8—61no—
543Q8—102no—
544Q8—102no—
545Q8—102no—
546Q8—102no—
547Q8—152no—
548Q8—308no—
549Q8—301no—
550D4—32no—
551D4—62no—
552D4—62no—
553D4—61no—
554D4—152no—
555D4—308no—
556D4—302no—
557D4—302no—
558D4—301no—
559C2×C4—22no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—54no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—102no—
567C2×C4—102no—
568C2×C4—202no—
569C2×C4—201no—
570C2×C4—308no—
571C2×C4—308no—
572C2×C4—301no—
573C2×C4—302no—
574C2×C4—302no—
575C2×C4C4×D3×D511yescharacteristic
576C2×C4D5×D1211yescharacteristic
577C2×C4D3×D2011yescharacteristic
578C2×C4#677711yescharacteristic
579C2×C4D3×C22⋊D511yescharacteristic
580C2×C4#679011yescharacteristic
581C8—121no—
582C8—601no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3#6511511yescharacteristic
586C2×C3#6511611yescharacteristic
587C2×C3#6524411yescharacteristic
588D3—42no—
589D3—202no—
590C5#6969211yescharacteristic, complemented, Sylow
591C22—21no—
592C22—21no—
593C22—62no—
594C22—121no—
595C22—121no—
596C22—302no—
597C22—601no—
598C22—601no—
599C22D5×(C2×D3)⋊C411yescharacteristic
600C22#9147411yescharacteristic
601C22#9157711yescharacteristic
602C4—42no—
603C4—62no—
604C4—104no—
605C4—302no—
606C4—401no—
607C4D3×#68511yescharacteristic
608C4#9147311yescharacteristic
609C4#9147511yescharacteristic
610C4#9147611yescharacteristic
611C3#1060118011yescharacteristic, complemented, Sylow
612C2—21no—
613C2—21no—
614C2—122no—
615C2—602no—
616C2#1174712011yescharacteristic
617C2#1174712111yescharacteristic
618C2#1174894011yescharacteristic
subgroupG/H
centerC22#91474
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C42×C5C22
socleC22×C3×C5#120
prime radicalC2×#92004C2
cosocleC22×C3×C4×C5C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#1793
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×#91089C2semidirect
3. C2×C5×#805C2semidirect
4. C3×#65218C2semidirect
5. #11748413C2semidirect
6. #11748384C2semidirect
7. C2×#90961C2semidirect
18. #92004C4semidirect
39. #65218D3semidirect
97. C2×#805D5semidirect
510. C3×C5#1793semidirect
590. C5#69692semidirect
611. C3#10601180semidirect

Maximal subgroups

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

subgroupindex
C2×#920042
C2×#910892
C2×C5×#8052
C3×#652182
#117484132
#117483842
C2×#909612
#106011803
#696925