Finite Group Databasebeta

← C109×#18245 #42136239

Subgroups — C109×#18245 #42136239

Order 27904.

908 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3559C211yesmaximal, characteristic
2C109×#3117C211yesmaximal, characteristic, complemented
3C109×#3103C211yesmaximal, characteristic, complemented
4C109×#2634C211yesmaximal, characteristic, complemented
5C109×#1616C212yesmaximal, complemented
6C109×#1615C212yesmaximal, complemented
7C109×#1604C212yesmaximal, complemented
8C109×#1603C212yesmaximal, complemented
9C109×#1521C211yesmaximal, characteristic, complemented
10C109×#1514C211yesmaximal, characteristic, complemented
11C109×#1470C211yesmaximal, characteristic, complemented
12#563×C109—21no—
13#562×C109—21no—
14#554×C109—21no—
15#553×C109—21no—
16#553×C109—21no—
17#547×C109—22no—
18#547×C109—21no—
19#544×C109—21no—
20#540×C109—21no—
21#539×C109—22no—
22#539×C109C2212yes—
23#537×C109C2212yes—
24#532×C109C2211yescharacteristic
25#529×C109C2211yescharacteristic
26C2×#125×C109C2211yescharacteristic
27C2×C4×Q8×C109C2211yescharacteristic
28#484×C109C2211yescharacteristic
29#481×C109C2212yes—
30#481×C109C2211yescharacteristic
31#479×C109—21no—
32#475×C109—21no—
33#464×C109—21no—
34#460×C109—21no—
35#451×C109C2211yescharacteristic
36#448×C109C2212yes—
37#448×C109C2211yescharacteristic
38#441×C109—21no—
39#440×C109—21no—
40C4×SD16×C109—22no—
41#432×C109C2211yescharacteristic
42#431×C109C2211yescharacteristic
43#400×C109—21no—
44#400×C109C2212yes—
45#398×C109—21no—
46#395×C109C2212yes—
47#387×C109—22no—
48#383×C109C2211yescharacteristic
49#328×C109C2212yes—
50#327×C109C2212yes—
51#327×C109C2212yes—
52#327×C109C2212yes—
53#324×C109C2212yes—
54#324×C109C2212yes—
55C2×#41×C109—21no—
56C2×#41×C109—22no—
57C22×Q8×C109C2311yescharacteristic
58C2×Q16×C109—21no—
59C2×SD16×C109—22no—
60C2×SD16×C109—21no—
61C4⋊Q8×C109—21no—
62C4⋊Q8×C109—22no—
63C4⋊Q8×C109D411yescharacteristic
64#127×C109D411yescharacteristic
65#126×C109—22no—
66#126×C109—22no—
67#126×C109—21no—
68#125×C109—21no—
69#125×C109—22no—
70#125×C109—21no—
71#125×C109—21no—
72#125×C109—21no—
73#125×C109—21no—
74#125×C109D411yescharacteristic
75#124×C109—21no—
76#124×C109D411yescharacteristic
77#123×C109—22no—
78#123×C109—22no—
79#123×C109—22no—
80#123×C109—22no—
81#123×C109—22no—
82#123×C109—21no—
83#123×C109—22no—
84C22⋊Q8×C109—22no—
85C22⋊Q8×C109—22no—
86C22⋊Q8×C109—22no—
87C22⋊Q8×C109—22no—
88C22⋊Q8×C109—22no—
89C22⋊Q8×C109—22no—
90C22⋊Q8×C109C2311yescharacteristic
91#121×C109—21no—
92#121×C109—22no—
93#121×C109—21no—
94#121×C109—22no—
95#121×C109—22no—
96#121×C109—21no—
97#121×C109—22no—
98#121×C109C2311yescharacteristic
99#121×C109C2312yes—
100C4×Q8×C109—21no—
101C4×Q8×C109—21no—
102C4×Q8×C109—21no—
103C4×Q8×C109—21no—
104C4×Q8×C109D411yescharacteristic
105C4×D4×C109—22no—
106C4×D4×C109—21no—
107C4×D4×C109—21no—
108C4×D4×C109—21no—
109C4×D4×C109—22no—
110C4×D4×C109—21no—
111C4×D4×C109D411yescharacteristic
112C4×D4×C109D412yes—
113#117×C109—21no—
114#117×C109—21no—
115C2×C4⋊C4×C109—22no—
116C2×C4⋊C4×C109—21no—
117C2×C4⋊C4×C109—21no—
118C2×C4⋊C4×C109—22no—
119C2×C4⋊C4×C109D412yes—
120C2×C4⋊C4×C109D411yescharacteristic
121C2×C4⋊C4×C109D411yescharacteristic
122C2×C4⋊C4×C109C2311yescharacteristic
123C2×C4⋊C4×C109C2312yes—
124C2×C42×C109—21no—
125C2×C42×C109C2311yescharacteristic
126#107×C109—21no—
127#106×C109—22no—
128#106×C109—21no—
129C4⋊C8×C109—21no—
130C4⋊C8×C109—21no—
131#103×C109—21no—
132#103×C109—22no—
133#103×C109—21no—
134#103×C109—21no—
135#102×C109—22no—
136#102×C109—21no—
137C22⋊C8×C109C2311yescharacteristic
138C22⋊C8×C109C2311yescharacteristic
139C22⋊C8×C109C2312yes—
140#97×C109—21no—
141C4×C8×C109—21no—
142#95×C109—21no—
143#95×C109—21no—
144#95×C109—22no—
145#95×C109—22no—
146#95×C109—22no—
147#95×C109C2312yes—
148#41×C109—82no—
149#41×C109—84no—
150C2×Q8×C109—22no—
151C2×Q8×C109—21no—
152C2×Q8×C109—21no—
153C2×Q8×C109—21no—
154C2×Q8×C109—42no—
155C2×Q8×C109C2×D411yescharacteristic
156C2×D4×C109—21no—
157C2×D4×C109—22no—
158C2×D4×C109—22no—
159C2×D4×C109—22no—
160C2×D4×C109—21no—
161C2×D4×C109—21no—
162C2×D4×C109C2×D411yescharacteristic
163C2×D4×C109C2×D412yes—
164C22×C4×C109—21no—
165C22×C4×C109—22no—
166C22×C4×C109—22no—
167C22×C4×C109—22no—
168C22×C4×C109—21no—
169C22×C4×C109—22no—
170C22×C4×C109—22no—
171C22×C4×C109—21no—
172C22×C4×C109C2×D412yes—
173C22×C4×C109C2×D411yescharacteristic
174C22×C4×C109C2×D412yes—
175C22×C4×C109C2×D411yescharacteristic
176C22×C4×C109C2411yescharacteristic
177Q16×C109—42no—
178SD16×C109—44no—
179SD16×C109—42no—
180C2×C8×C109—22no—
181C2×C8×C109—21no—
182C2×C8×C109—21no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—21no—
188C4⋊C4×C109—21no—
189C4⋊C4×C109—22no—
190C4⋊C4×C109—21no—
191C4⋊C4×C109—21no—
192C4⋊C4×C109—22no—
193C4⋊C4×C109—21no—
194C4⋊C4×C109—21no—
195C4⋊C4×C109—21no—
196C4⋊C4×C109—21no—
197C4⋊C4×C109—22no—
198C4⋊C4×C109—22no—
199C4⋊C4×C109—21no—
200C4⋊C4×C109—21no—
201C4⋊C4×C109—21no—
202C4⋊C4×C109—44no—
203C4⋊C4×C109—41no—
204C4⋊C4×C109—41no—
205C4⋊C4×C109C2×D412yes—
206C4⋊C4×C109C2×D411yescharacteristic
207C4⋊C4×C109C2×D411yescharacteristic
208C22⋊C4×C109—22no—
209C22⋊C4×C109—22no—
210C22⋊C4×C109—22no—
211C22⋊C4×C109—22no—
212C22⋊C4×C109—22no—
213C22⋊C4×C109—21no—
214C22⋊C4×C109—21no—
215C22⋊C4×C109—21no—
216C22⋊C4×C109—21no—
217C22⋊C4×C109—22no—
218C22⋊C4×C109—21no—
219C22⋊C4×C109—22no—
220C42×C109—22no—
221C42×C109—21no—
222C42×C109—41no—
223C42×C109—41no—
224C42×C109C2×D411yescharacteristic
225C42×C109C2×D411yescharacteristic
226C42×C109C2×D411yescharacteristic
227C42×C109C2×D411yescharacteristic
228C23×C109—21no—
229C23×C109—22no—
230C23×C109C22×D411yescharacteristic
231Q8×C109—22no—
232Q8×C109—22no—
233Q8×C109—41no—
234Q8×C109—41no—
235Q8×C109—41no—
236Q8×C109—81no—
237Q8×C109—82no—
238D4×C109—42no—
239D4×C109—41no—
240D4×C109—42no—
241D4×C109—44no—
242D4×C109—42no—
243D4×C109—81no—
244D4×C109—82no—
245D4×C109—82no—
246C2×C4×C109—22no—
247C2×C4×C109—22no—
248C2×C4×C109—21no—
249C2×C4×C109—21no—
250C2×C4×C109—22no—
251C2×C4×C109—22no—
252C2×C4×C109—21no—
253C2×C4×C109—22no—
254C2×C4×C109—24no—
255C2×C4×C109—41no—
256C2×C4×C109—41no—
257C2×C4×C109—42no—
258C2×C4×C109—41no—
259C2×C4×C109—41no—
260C2×C4×C109—42no—
261C2×C4×C109—42no—
262C2×C4×C109—42no—
263C2×C4×C109—42no—
264C2×C4×C109—41no—
265C2×C4×C109—42no—
266C2×C4×C109—82no—
267C2×C4×C109—82no—
268C2×C4×C109—82no—
269C2×C4×C109—82no—
270C2×C4×C109#12011yescharacteristic
271C2×C4×C109#12011yescharacteristic
272C2×C4×C109#12011yescharacteristic
273C2×C4×C109#12011yescharacteristic
274C2×C4×C109C22×D411yescharacteristic
275C2×C4×C109C22×D411yescharacteristic
276C8×C109—42no—
277C8×C109—41no—
278C8×C109—41no—
279C22×C109—21no—
280C22×C109—21no—
281C22×C109—42no—
282C22×C109—41no—
283C22×C109—41no—
284C22×C109—42no—
285C22×C109—42no—
286C22×C109—41no—
287C22×C109C2×#12011yescharacteristic
288C22×C109#57712yes—
289C4×C109—21no—
290C4×C109—21no—
291C4×C109—21no—
292C4×C109—21no—
293C4×C109—21no—
294C4×C109—21no—
295C4×C109—42no—
296C4×C109—42no—
297C4×C109—41no—
298C4×C109—41no—
299C4×C109—42no—
300C4×C109—42no—
301C4×C109—41no—
302C4×C109—82no—
303C4×C109—81no—
304#18245C10911yesmaximal, characteristic, direct product factor, Sylow
305C2×C109—22no—
306C2×C109—81no—
307C2×C109—82no—
308C2×C109#300911yescharacteristic
309C2×C109#302011yescharacteristic
310C2×C109#302511yescharacteristic
311#3559C2×C10911yescharacteristic
312#3117C2×C10911yescharacteristic, complemented
313#3103C2×C10911yescharacteristic, complemented
314#2634C2×C10911yescharacteristic, complemented
315#1616C2×C10912yescomplemented
316#1615C2×C10912yescomplemented
317#1604C2×C10912yescomplemented
318#1603C2×C10912yescomplemented
319#1521C2×C10911yescharacteristic, complemented
320#1514C2×C10911yescharacteristic, complemented
321#1470C2×C10911yescharacteristic, complemented
322C109#1824511yescharacteristic, direct product factor, Sylow
323#563—21no—
324#562—21no—
325#554—21no—
326#553—21no—
327#553—21no—
328#547—22no—
329#547—21no—
330#544—21no—
331#540—21no—
332#539—22no—
333#539C22×C10912yes—
334#537C22×C10912yes—
335#532C22×C10911yescharacteristic
336#529C22×C10911yescharacteristic
337C2×#125C22×C10911yescharacteristic
338C2×C4×Q8C22×C10911yescharacteristic
339#484C22×C10911yescharacteristic
340#481C22×C10912yes—
341#481C22×C10911yescharacteristic
342#479—21no—
343#475—21no—
344#464—21no—
345#460—21no—
346#451C22×C10911yescharacteristic
347#448C22×C10912yes—
348#448C22×C10911yescharacteristic
349#441—21no—
350#440—21no—
351C4×SD16—22no—
352#432C22×C10911yescharacteristic
353#431C22×C10911yescharacteristic
354#400—21no—
355#400C22×C10912yes—
356#398—21no—
357#395C22×C10912yes—
358#387—22no—
359#383C22×C10911yescharacteristic
360#328C22×C10912yes—
361#327C22×C10912yes—
362#327C22×C10912yes—
363#327C22×C10912yes—
364#324C22×C10912yes—
365#324C22×C10912yes—
366C2×#41—21no—
367C2×#41—22no—
368C22×Q8C23×C10911yescharacteristic
369C2×Q16—21no—
370C2×SD16—22no—
371C2×SD16—21no—
372C4⋊Q8—21no—
373C4⋊Q8—22no—
374C4⋊Q8D4×C10911yescharacteristic
375#127D4×C10911yescharacteristic
376#126—22no—
377#126—22no—
378#126—21no—
379#125—21no—
380#125—22no—
381#125—21no—
382#125—21no—
383#125—21no—
384#125—21no—
385#125D4×C10911yescharacteristic
386#124—21no—
387#124D4×C10911yescharacteristic
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—21no—
394#123—22no—
395C22⋊Q8—22no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—22no—
399C22⋊Q8—22no—
400C22⋊Q8—22no—
401C22⋊Q8C23×C10911yescharacteristic
402#121—21no—
403#121—22no—
404#121—21no—
405#121—22no—
406#121—22no—
407#121—21no—
408#121—22no—
409#121C23×C10911yescharacteristic
410#121C23×C10912yes—
411C4×Q8—21no—
412C4×Q8—21no—
413C4×Q8—21no—
414C4×Q8—21no—
415C4×Q8D4×C10911yescharacteristic
416C4×D4—22no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—22no—
421C4×D4—21no—
422C4×D4D4×C10911yescharacteristic
423C4×D4D4×C10912yes—
424#117—21no—
425#117—21no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—22no—
430C2×C4⋊C4D4×C10912yes—
431C2×C4⋊C4D4×C10911yescharacteristic
432C2×C4⋊C4D4×C10911yescharacteristic
433C2×C4⋊C4C23×C10911yescharacteristic
434C2×C4⋊C4C23×C10912yes—
435C2×C42—21no—
436C2×C42C23×C10911yescharacteristic
437#107—21no—
438#106—22no—
439#106—21no—
440C4⋊C8—21no—
441C4⋊C8—21no—
442#103—21no—
443#103—22no—
444#103—21no—
445#103—21no—
446#102—22no—
447#102—21no—
448C22⋊C8C23×C10911yescharacteristic
449C22⋊C8C23×C10911yescharacteristic
450C22⋊C8C23×C10912yes—
451#97—21no—
452C4×C8—21no—
453#95—21no—
454#95—21no—
455#95—22no—
456#95—22no—
457#95—22no—
458#95C23×C10912yes—
459#41—82no—
460#41—84no—
461C2×Q8—22no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—42no—
466C2×Q8C2×D4×C10911yescharacteristic
467C2×D4—21no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4C2×D4×C10911yescharacteristic
474C2×D4C2×D4×C10912yes—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—21no—
483C22×C4C2×D4×C10912yes—
484C22×C4C2×D4×C10911yescharacteristic
485C22×C4C2×D4×C10912yes—
486C22×C4C2×D4×C10911yescharacteristic
487C22×C4C24×C10911yescharacteristic
488Q16—42no—
489SD16—44no—
490SD16—42no—
491C2×C8—22no—
492C2×C8—21no—
493C2×C8—21no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—44no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4C2×D4×C10912yes—
517C4⋊C4C2×D4×C10911yescharacteristic
518C4⋊C4C2×D4×C10911yescharacteristic
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—21no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C42—22no—
532C42—21no—
533C42—41no—
534C42—41no—
535C42C2×D4×C10911yescharacteristic
536C42C2×D4×C10911yescharacteristic
537C42C2×D4×C10911yescharacteristic
538C42C2×D4×C10911yescharacteristic
539C23—21no—
540C23—22no—
541C23C22×D4×C10911yescharacteristic
542Q8—22no—
543Q8—22no—
544Q8—41no—
545Q8—41no—
546Q8—41no—
547Q8—81no—
548Q8—82no—
549D4—42no—
550D4—41no—
551D4—42no—
552D4—44no—
553D4—42no—
554D4—81no—
555D4—82no—
556D4—82no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4#120×C10911yescharacteristic
582C2×C4#120×C10911yescharacteristic
583C2×C4#120×C10911yescharacteristic
584C2×C4#120×C10911yescharacteristic
585C2×C4C22×D4×C10911yescharacteristic
586C2×C4C22×D4×C10911yescharacteristic
587C8—42no—
588C8—41no—
589C8—41no—
590C22—21no—
591C22—21no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—41no—
598C22C2×#120×C10911yescharacteristic
599C22#577×C10912yes—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—41no—
613C4—82no—
614C4—81no—
615C2—22no—
616C2—81no—
617C2—82no—
618C2C109×#300911yescharacteristic
619C2C109×#302011yescharacteristic
620C2C109×#302511yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3559C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3117C2semidirect
3. C109×#3103C2semidirect
4. C109×#2634C2semidirect
5. C109×#1616C2semidirect
6. C109×#1615C2semidirect
7. C109×#1604C2semidirect
8. C109×#1603C2semidirect
9. C109×#1521C2semidirect
10. C109×#1514C2semidirect
11. C109×#1470C2semidirect
304. #18245C109direct (N × H)
312. #3117C2×C109semidirect
313. #3103C2×C109semidirect
314. #2634C2×C109semidirect
315. #1616C2×C109semidirect
316. #1615C2×C109semidirect
317. #1604C2×C109semidirect
318. #1603C2×C109semidirect
319. #1521C2×C109semidirect
320. #1514C2×C109semidirect
321. #1470C2×C109semidirect
322. C109#18245direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#35592
C109×#31172
C109×#31032
C109×#26342
C109×#16162
C109×#16152
C109×#16042
C109×#16032
C109×#15212
C109×#15142
C109×#14702
#18245109