Finite Group Databasebeta

← C109×#19871 #42137865

Subgroups — C109×#19871 #42137865

Order 27904.

632 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3399C211yesmaximal, characteristic, complemented
2C109×#3396C211yesmaximal, characteristic, complemented
3C109×#3394C211yesmaximal, characteristic, complemented
4C109×#3304C211yesmaximal, characteristic, complemented
5C109×#1721C211yesmaximal, characteristic, complemented
6C109×#1686C211yesmaximal, characteristic, complemented
7C109×#1682C211yesmaximal, characteristic, complemented
8C109×#1668C211yesmaximal, characteristic, complemented
9C109×#1667C211yesmaximal, characteristic, complemented
10C109×#1640C211yesmaximal, characteristic, complemented
11C109×#1628C211yesmaximal, characteristic, complemented
12C109×#1621C211yesmaximal, characteristic, complemented
13C109×#1620C211yesmaximal, characteristic, complemented
14C109×#1584C211yesmaximal, characteristic, complemented
15C109×SD16⋊C8C211yesmaximal, characteristic, complemented
16#559×C109C2211yescharacteristic
17#559×C109C2211yescharacteristic
18#556×C109C2211yescharacteristic
19#550×C109C2211yescharacteristic
20#496×C109—21no—
21#495×C109—21no—
22#495×C109—21no—
23#493×C109C2211yescharacteristic
24#493×C109C2211yescharacteristic
25#493×C109C2211yescharacteristic
26#492×C109C2211yescharacteristic
27#486×C109C2211yescharacteristic
28#486×C109C2211yescharacteristic
29#482×C109—21no—
30#475×C109C2211yescharacteristic
31#465×C109—21no—
32#464×C109—21no—
33#463×C109—21no—
34#463×C109—21no—
35#463×C109—21no—
36#463×C109—21no—
37#462×C109C2211yescharacteristic
38#460×C109C2211yescharacteristic
39#460×C109C2211yescharacteristic
40#460×C109C2211yescharacteristic
41#460×C109C2211yescharacteristic
42#459×C109—21no—
43#459×C109C2211yescharacteristic
44#459×C109C2211yescharacteristic
45#452×C109—21no—
46#446×C109C2211yescharacteristic
47C8×Q8×C109C2211yescharacteristic
48C8×Q8×C109C2211yescharacteristic
49#442×C109—21no—
50C4×Q16×C109C2211yescharacteristic
51C4×SD16×C109—21no—
52C4×SD16×C109C2211yescharacteristic
53C4×D8×C109—21no—
54C8×D4×C109C2211yescharacteristic
55#416×C109—21no—
56#335×C109C2211yescharacteristic
57#335×C109C2211yescharacteristic
58#332×C109C2211yescharacteristic
59#331×C109C2211yescharacteristic
60#331×C109C2211yescharacteristic
61#331×C109C2211yescharacteristic
62Q8⋊C8×C109C2211yescharacteristic
63Q8⋊C8×C109C2211yescharacteristic
64Q8⋊C8×C109C2211yescharacteristic
65D4⋊C8×C109C2211yescharacteristic
66#322×C109C2211yescharacteristic
67C2×#41×C109—21no—
68C2×Q16×C109—21no—
69C2×Q16×C109D411yescharacteristic
70C2×SD16×C109—21no—
71C2×SD16×C109—21no—
72C2×SD16×C109—21no—
73C2×SD16×C109—21no—
74C2×SD16×C109—21no—
75C2×SD16×C109—21no—
76C2×SD16×C109D411yescharacteristic
77C2×D8×C109—21no—
78C2×D8×C109—21no—
79C2×D8×C109—21no—
80C2×D8×C109—21no—
81C2×D8×C109—21no—
82C2×D8×C109—21no—
83C2×D8×C109—41no—
84C22×C8×C109—21no—
85C4⋊Q8×C109C2311yescharacteristic
86#127×C109—21no—
87#127×C109C2311yescharacteristic
88#127×C109C2311yescharacteristic
89#127×C109C2311yescharacteristic
90#126×C109—41no—
91#124×C109—21no—
92#124×C109—21no—
93#124×C109—21no—
94#124×C109—21no—
95#124×C109—21no—
96#123×C109—41no—
97#123×C109—41no—
98C22⋊Q8×C109—21no—
99#121×C109—21no—
100#121×C109—41no—
101C4×Q8×C109—21no—
102C4×Q8×C109C2311yescharacteristic
103C4×Q8×C109C2311yescharacteristic
104C4×Q8×C109C2311yescharacteristic
105C4×D4×C109—21no—
106C4×D4×C109—21no—
107C4×D4×C109—21no—
108C4×D4×C109—21no—
109C4×D4×C109—21no—
110C4×D4×C109C2311yescharacteristic
111#117×C109—21no—
112#107×C109D411yescharacteristic
113#106×C109—21no—
114#106×C109D411yescharacteristic
115C4⋊C8×C109—21no—
116C4⋊C8×C109—21no—
117C4⋊C8×C109—21no—
118C4⋊C8×C109C2311yescharacteristic
119C4⋊C8×C109C2311yescharacteristic
120C4⋊C8×C109C2311yescharacteristic
121C4⋊C8×C109C2311yescharacteristic
122#103×C109—21no—
123#103×C109—21no—
124#103×C109D411yescharacteristic
125#103×C109D411yescharacteristic
126#103×C109D411yescharacteristic
127#102×C109—21no—
128#102×C109—21no—
129#102×C109—21no—
130#102×C109—21no—
131#102×C109—21no—
132#102×C109—21no—
133#102×C109—21no—
134#102×C109—21no—
135#102×C109—41no—
136#102×C109D411yescharacteristic
137C22⋊C8×C109—21no—
138#97×C109—21no—
139C4×C8×C109—21no—
140C4×C8×C109—21no—
141C4×C8×C109C2311yescharacteristic
142C4×C8×C109C2311yescharacteristic
143C4×C8×C109C2311yescharacteristic
144#41×C109—42no—
145#41×C109—81no—
146C2×Q8×C109—21no—
147C2×Q8×C109C2×D411yescharacteristic
148C2×Q8×C109C2×D411yescharacteristic
149C2×Q8×C109C2×D411yescharacteristic
150C2×D4×C109—21no—
151C2×D4×C109—21no—
152C2×D4×C109—21no—
153C2×D4×C109—21no—
154C2×D4×C109—21no—
155C2×D4×C109—21no—
156C2×D4×C109—21no—
157C2×D4×C109—21no—
158C2×D4×C109—21no—
159C2×D4×C109—41no—
160C2×D4×C109—41no—
161C2×D4×C109C2×D411yescharacteristic
162C22×C4×C109—21no—
163C22×C4×C109—41no—
164C22×C4×C109—41no—
165C22×C4×C109—41no—
166Q16×C109—41no—
167Q16×C109—81no—
168SD16×C109—22no—
169SD16×C109—42no—
170SD16×C109—81no—
171SD16×C109—81no—
172SD16×C109—81no—
173SD16×C109—81no—
174SD16×C109—81no—
175D8×C109—42no—
176D8×C109—42no—
177D8×C109—81no—
178D8×C109—81no—
179D8×C109—81no—
180D8×C109—81no—
181D8×C109—82no—
182C2×C8×C109—21no—
183C2×C8×C109—21no—
184C2×C8×C109—21no—
185C2×C8×C109—21no—
186C2×C8×C109—21no—
187C2×C8×C109—81no—
188C2×C8×C109C2×D411yescharacteristic
189C2×C8×C109C2×D411yescharacteristic
190C2×C8×C109C2×D411yescharacteristic
191C2×C8×C109C2×D411yescharacteristic
192C4⋊C4×C109—21no—
193C4⋊C4×C109—21no—
194C4⋊C4×C109—21no—
195C4⋊C4×C109—21no—
196C4⋊C4×C109—21no—
197C4⋊C4×C109—41no—
198C4⋊C4×C109C2×D411yescharacteristic
199C4⋊C4×C109C2×D411yescharacteristic
200C4⋊C4×C109C2×D411yescharacteristic
201C4⋊C4×C109C2×D411yescharacteristic
202C22⋊C4×C109—21no—
203C22⋊C4×C109—41no—
204C22⋊C4×C109—41no—
205C22⋊C4×C109—41no—
206C22⋊C4×C109—41no—
207C22⋊C4×C109—41no—
208C22⋊C4×C109—41no—
209C22⋊C4×C109—41no—
210C22⋊C4×C109—41no—
211C22⋊C4×C109—41no—
212C42×C109—21no—
213C42×C109—21no—
214C42×C109—21no—
215C42×C109—41no—
216C42×C109C2411yescharacteristic
217C23×C109—21no—
218C23×C109—41no—
219C23×C109—41no—
220C23×C109—41no—
221Q8×C109—21no—
222Q8×C109—21no—
223Q8×C109—21no—
224Q8×C109—21no—
225Q8×C109—21no—
226Q8×C109—21no—
227Q8×C109—41no—
228Q8×C109—41no—
229D4×C109—21no—
230D4×C109—21no—
231D4×C109—41no—
232D4×C109—41no—
233D4×C109—41no—
234D4×C109—41no—
235D4×C109—41no—
236D4×C109—41no—
237D4×C109—41no—
238D4×C109—41no—
239D4×C109—41no—
240D4×C109—41no—
241D4×C109—41no—
242D4×C109—41no—
243D4×C109—81no—
244D4×C109—81no—
245D4×C109—81no—
246D4×C109—82no—
247D4×C109—161no—
248C2×C4×C109—21no—
249C2×C4×C109—21no—
250C2×C4×C109—21no—
251C2×C4×C109—21no—
252C2×C4×C109—21no—
253C2×C4×C109—21no—
254C2×C4×C109—21no—
255C2×C4×C109—41no—
256C2×C4×C109—41no—
257C2×C4×C109—41no—
258C2×C4×C109—82no—
259C2×C4×C109—161no—
260C2×C4×C109—161no—
261C2×C4×C109C22×D411yescharacteristic
262C2×C4×C109C22×D411yescharacteristic
263C2×C4×C109H(5,𝔽2)11yescharacteristic
264C8×C109—21no—
265C8×C109—21no—
266C8×C109—21no—
267C8×C109—41no—
268C8×C109—41no—
269C8×C109—41no—
270C8×C109—41no—
271C8×C109—41no—
272C8×C109#13512yes—
273C22×C109—41no—
274C22×C109—41no—
275C22×C109—41no—
276C22×C109—81no—
277C22×C109—81no—
278C22×C109—81no—
279C22×C109—81no—
280C22×C109—81no—
281C22×C109—81no—
282C22×C109—81no—
283C22×C109—81no—
284C22×C109—81no—
285C22×C109D4211yescharacteristic
286C4×C109—21no—
287C4×C109—41no—
288C4×C109—41no—
289C4×C109—41no—
290C4×C109—41no—
291C4×C109—41no—
292C4×C109—41no—
293C4×C109—41no—
294C4×C109—81no—
295C4×C109C2×#13511yescharacteristic
296C4×C109#57511yescharacteristic
297C4×C109#57611yescharacteristic
298C4×C109#57611yescharacteristic
299#19871C10911yesmaximal, characteristic, direct product factor, Sylow
300C2×C109—81no—
301C2×C109—161no—
302C2×C109—161no—
303C2×C109—161no—
304C2×C109#326811yescharacteristic
305C2×C109#328711yescharacteristic
306C2×C109#328811yescharacteristic
307#3399C2×C10911yescharacteristic, complemented
308#3396C2×C10911yescharacteristic, complemented
309#3394C2×C10911yescharacteristic, complemented
310#3304C2×C10911yescharacteristic, complemented
311#1721C2×C10911yescharacteristic, complemented
312#1686C2×C10911yescharacteristic, complemented
313#1682C2×C10911yescharacteristic, complemented
314#1668C2×C10911yescharacteristic, complemented
315#1667C2×C10911yescharacteristic, complemented
316#1640C2×C10911yescharacteristic, complemented
317#1628C2×C10911yescharacteristic, complemented
318#1621C2×C10911yescharacteristic, complemented
319#1620C2×C10911yescharacteristic, complemented
320#1584C2×C10911yescharacteristic, complemented
321SD16⋊C8C2×C10911yescharacteristic, complemented
322C109#1987111yescharacteristic, direct product factor, Sylow
323#559C22×C10911yescharacteristic
324#559C22×C10911yescharacteristic
325#556C22×C10911yescharacteristic
326#550C22×C10911yescharacteristic
327#496—21no—
328#495—21no—
329#495—21no—
330#493C22×C10911yescharacteristic
331#493C22×C10911yescharacteristic
332#493C22×C10911yescharacteristic
333#492C22×C10911yescharacteristic
334#486C22×C10911yescharacteristic
335#486C22×C10911yescharacteristic
336#482—21no—
337#475C22×C10911yescharacteristic
338#465—21no—
339#464—21no—
340#463—21no—
341#463—21no—
342#463—21no—
343#463—21no—
344#462C22×C10911yescharacteristic
345#460C22×C10911yescharacteristic
346#460C22×C10911yescharacteristic
347#460C22×C10911yescharacteristic
348#460C22×C10911yescharacteristic
349#459—21no—
350#459C22×C10911yescharacteristic
351#459C22×C10911yescharacteristic
352#452—21no—
353#446C22×C10911yescharacteristic
354C8×Q8C22×C10911yescharacteristic
355C8×Q8C22×C10911yescharacteristic
356#442—21no—
357C4×Q16C22×C10911yescharacteristic
358C4×SD16—21no—
359C4×SD16C22×C10911yescharacteristic
360C4×D8—21no—
361C8×D4C22×C10911yescharacteristic
362#416—21no—
363#335C22×C10911yescharacteristic
364#335C22×C10911yescharacteristic
365#332C22×C10911yescharacteristic
366#331C22×C10911yescharacteristic
367#331C22×C10911yescharacteristic
368#331C22×C10911yescharacteristic
369Q8⋊C8C22×C10911yescharacteristic
370Q8⋊C8C22×C10911yescharacteristic
371Q8⋊C8C22×C10911yescharacteristic
372D4⋊C8C22×C10911yescharacteristic
373#322C22×C10911yescharacteristic
374C2×#41—21no—
375C2×Q16—21no—
376C2×Q16D4×C10911yescharacteristic
377C2×SD16—21no—
378C2×SD16—21no—
379C2×SD16—21no—
380C2×SD16—21no—
381C2×SD16—21no—
382C2×SD16—21no—
383C2×SD16D4×C10911yescharacteristic
384C2×D8—21no—
385C2×D8—21no—
386C2×D8—21no—
387C2×D8—21no—
388C2×D8—21no—
389C2×D8—21no—
390C2×D8—41no—
391C22×C8—21no—
392C4⋊Q8C23×C10911yescharacteristic
393#127—21no—
394#127C23×C10911yescharacteristic
395#127C23×C10911yescharacteristic
396#127C23×C10911yescharacteristic
397#126—41no—
398#124—21no—
399#124—21no—
400#124—21no—
401#124—21no—
402#124—21no—
403#123—41no—
404#123—41no—
405C22⋊Q8—21no—
406#121—21no—
407#121—41no—
408C4×Q8—21no—
409C4×Q8C23×C10911yescharacteristic
410C4×Q8C23×C10911yescharacteristic
411C4×Q8C23×C10911yescharacteristic
412C4×D4—21no—
413C4×D4—21no—
414C4×D4—21no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4C23×C10911yescharacteristic
418#117—21no—
419#107D4×C10911yescharacteristic
420#106—21no—
421#106D4×C10911yescharacteristic
422C4⋊C8—21no—
423C4⋊C8—21no—
424C4⋊C8—21no—
425C4⋊C8C23×C10911yescharacteristic
426C4⋊C8C23×C10911yescharacteristic
427C4⋊C8C23×C10911yescharacteristic
428C4⋊C8C23×C10911yescharacteristic
429#103—21no—
430#103—21no—
431#103D4×C10911yescharacteristic
432#103D4×C10911yescharacteristic
433#103D4×C10911yescharacteristic
434#102—21no—
435#102—21no—
436#102—21no—
437#102—21no—
438#102—21no—
439#102—21no—
440#102—21no—
441#102—21no—
442#102—41no—
443#102D4×C10911yescharacteristic
444C22⋊C8—21no—
445#97—21no—
446C4×C8—21no—
447C4×C8—21no—
448C4×C8C23×C10911yescharacteristic
449C4×C8C23×C10911yescharacteristic
450C4×C8C23×C10911yescharacteristic
451#41—42no—
452#41—81no—
453C2×Q8—21no—
454C2×Q8C2×D4×C10911yescharacteristic
455C2×Q8C2×D4×C10911yescharacteristic
456C2×Q8C2×D4×C10911yescharacteristic
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4C2×D4×C10911yescharacteristic
469C22×C4—21no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473Q16—41no—
474Q16—81no—
475SD16—22no—
476SD16—42no—
477SD16—81no—
478SD16—81no—
479SD16—81no—
480SD16—81no—
481SD16—81no—
482D8—42no—
483D8—42no—
484D8—81no—
485D8—81no—
486D8—81no—
487D8—81no—
488D8—82no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—81no—
495C2×C8C2×D4×C10911yescharacteristic
496C2×C8C2×D4×C10911yescharacteristic
497C2×C8C2×D4×C10911yescharacteristic
498C2×C8C2×D4×C10911yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—41no—
505C4⋊C4C2×D4×C10911yescharacteristic
506C4⋊C4C2×D4×C10911yescharacteristic
507C4⋊C4C2×D4×C10911yescharacteristic
508C4⋊C4C2×D4×C10911yescharacteristic
509C22⋊C4—21no—
510C22⋊C4—41no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42C24×C10911yescharacteristic
524C23—21no—
525C23—41no—
526C23—41no—
527C23—41no—
528Q8—21no—
529Q8—21no—
530Q8—21no—
531Q8—21no—
532Q8—21no—
533Q8—21no—
534Q8—41no—
535Q8—41no—
536D4—21no—
537D4—21no—
538D4—41no—
539D4—41no—
540D4—41no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—41no—
546D4—41no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—82no—
554D4—161no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—82no—
566C2×C4—161no—
567C2×C4—161no—
568C2×C4C22×D4×C10911yescharacteristic
569C2×C4C22×D4×C10911yescharacteristic
570C2×C4H(5,𝔽2)×C10911yescharacteristic
571C8—21no—
572C8—21no—
573C8—21no—
574C8—41no—
575C8—41no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8#135×C10912yes—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22D42×C10911yescharacteristic
593C4—21no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4C2×#135×C10911yescharacteristic
603C4#575×C10911yescharacteristic
604C4#576×C10911yescharacteristic
605C4#576×C10911yescharacteristic
606C2—81no—
607C2—161no—
608C2—161no—
609C2—161no—
610C2C109×#326811yescharacteristic
611C2C109×#328711yescharacteristic
612C2C109×#328811yescharacteristic
subgroupG/H
center, socleC22×C109D42
commutator, Frattini, cosocleC42C24×C109
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#3399C2semidirect
2. C109×#3396C2semidirect
3. C109×#3394C2semidirect
4. C109×#3304C2semidirect
5. C109×#1721C2semidirect
6. C109×#1686C2semidirect
7. C109×#1682C2semidirect
8. C109×#1668C2semidirect
9. C109×#1667C2semidirect
10. C109×#1640C2semidirect
11. C109×#1628C2semidirect
12. C109×#1621C2semidirect
13. C109×#1620C2semidirect
14. C109×#1584C2semidirect
15. C109×SD16⋊C8C2semidirect
299. #19871C109direct (N × H)
307. #3399C2×C109semidirect
308. #3396C2×C109semidirect
309. #3394C2×C109semidirect
310. #3304C2×C109semidirect
311. #1721C2×C109semidirect
312. #1686C2×C109semidirect
313. #1682C2×C109semidirect
314. #1668C2×C109semidirect
315. #1667C2×C109semidirect
316. #1640C2×C109semidirect
317. #1628C2×C109semidirect
318. #1621C2×C109semidirect
319. #1620C2×C109semidirect
320. #1584C2×C109semidirect
321. SD16⋊C8C2×C109semidirect
322. C109#19871direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#33992
C109×#33962
C109×#33942
C109×#33042
C109×#17212
C109×#16862
C109×#16822
C109×#16682
C109×#16672
C109×#16402
C109×#16282
C109×#16212
C109×#16202
C109×#15842
C109×SD16⋊C82
#19871109