Finite Group Databasebeta

← C109×#18265 #42136259

Subgroups — C109×#18265 #42136259

Order 27904.

1124 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3537C211yesmaximal, characteristic
2C109×#3092C211yesmaximal, characteristic, complemented
3C2×#481×C109C211yesmaximal, characteristic, complemented
4C109×#2501C211yesmaximal, characteristic, complemented
5C109×#1618C214yesmaximal, complemented
6C109×#1613C214yesmaximal, complemented
7C109×#1507C211yesmaximal, characteristic, complemented
8C109×#1501C211yesmaximal, characteristic, complemented
9C109×#1471C211yesmaximal, characteristic, complemented
10#560×C109—22no—
11#553×C109—21no—
12#553×C109—21no—
13D42×C109—22no—
14D42×C109—21no—
15D42×C109—21no—
16#540×C109—21no—
17#538×C109—21no—
18#538×C109C2212yes—
19#535×C109—21no—
20#534×C109—21no—
21#534×C109C2212yes—
22#525×C109C2211yescharacteristic
23C2×#123×C109—22no—
24C2×#121×C109C2211yescharacteristic
25C2×#121×C109C2211yescharacteristic
26C22×C4⋊C4×C109C2211yescharacteristic, complemented
27#484×C109C2212yescomplemented
28#483×C109—21no—
29#481×C109—22no—
30#481×C109C2214yescomplemented
31#481×C109C2212yescomplemented
32#480×C109—21no—
33#428×C109—21no—
34C2×#106×C109—21no—
35#417×C109—21no—
36C2×#102×C109—21no—
37#407×C109C2211yescharacteristic, complemented
38C2×C22⋊C8×C109C2211yescharacteristic, complemented
39#399×C109C2214yescomplemented
40#390×C109—21no—
41#384×C109—21no—
42C4×C22⋊C4×C109C2211yescharacteristic, complemented
43#328×C109C2212yescomplemented
44#327×C109C2212yescomplemented
45#327×C109C2214yescomplemented
46#323×C109C2214yescomplemented
47C2×#41×C109D411yescharacteristic
48C22×D4×C109—22no—
49C22×D4×C109—21no—
50C22×D4×C109—21no—
51C22×D4×C109—22no—
52C22×D4×C109D412yes—
53C22×D4×C109D411yescharacteristic
54C22×D4×C109D411yescharacteristic
55C23×C4×C109—21no—
56C23×C4×C109C2311yescharacteristic
57C2×M16×C109—21no—
58C22×C8×C109—21no—
59#127×C109—22no—
60#127×C109—21no—
61#126×C109—22no—
62#126×C109—22no—
63#125×C109—21no—
64#124×C109—22no—
65#123×C109—22no—
66#123×C109—21no—
67#123×C109—22no—
68#123×C109—22no—
69#123×C109—22no—
70#123×C109—48no—
71C22⋊Q8×C109D411yescharacteristic
72C22⋊Q8×C109C2311yescharacteristic
73#121×C109—22no—
74#121×C109—21no—
75#121×C109—21no—
76#121×C109—22no—
77#121×C109—22no—
78#121×C109—22no—
79#121×C109—22no—
80#121×C109—21no—
81#121×C109—21no—
82#121×C109—21no—
83#121×C109—21no—
84#121×C109—21no—
85#121×C109—21no—
86#121×C109—22no—
87#121×C109D412yes—
88#121×C109D411yescharacteristic
89#121×C109C2312yes—
90#121×C109C2311yescharacteristic
91#120×C109—22no—
92#120×C109—22no—
93#120×C109—22no—
94#120×C109—21no—
95#120×C109—22no—
96#120×C109—22no—
97#120×C109—22no—
98C4×D4×C109—22no—
99C4×D4×C109—21no—
100C4×D4×C109—21no—
101C4×D4×C109—21no—
102C4×D4×C109—21no—
103C4×D4×C109—22no—
104#117×C109—21no—
105#117×C109D411yescharacteristic
106C2×C4⋊C4×C109—21no—
107C2×C4⋊C4×C109—21no—
108C2×C4⋊C4×C109—21no—
109C2×C4⋊C4×C109—21no—
110C2×C4⋊C4×C109—21no—
111C2×C4⋊C4×C109—24no—
112C2×C4⋊C4×C109D411yescharacteristic
113C2×C4⋊C4×C109D411yescharacteristic
114C2×C4⋊C4×C109C2312yes—
115C2×C22⋊C4×C109—21no—
116C2×C22⋊C4×C109—21no—
117C2×C22⋊C4×C109—24no—
118C2×C22⋊C4×C109C2312yes—
119C2×C42×C109—21no—
120#107×C109—22no—
121#106×C109—24no—
122#106×C109—22no—
123#103×C109—22no—
124#102×C109—24no—
125#102×C109—22no—
126C22⋊C8×C109—21no—
127C22⋊C8×C109—21no—
128C22⋊C8×C109C2312yes—
129C22⋊C8×C109C2312yes—
130#95×C109C2312yes—
131C24×C109—22no—
132C24×C109—21no—
133C24×C109C2×D411yescharacteristic
134C24×C109C2×D412yes—
135#41×C109—42no—
136C2×Q8×C109C2×D411yescharacteristic
137C2×D4×C109—22no—
138C2×D4×C109—21no—
139C2×D4×C109—21no—
140C2×D4×C109—22no—
141C2×D4×C109—22no—
142C2×D4×C109—22no—
143C2×D4×C109—22no—
144C2×D4×C109—22no—
145C2×D4×C109—22no—
146C2×D4×C109—21no—
147C2×D4×C109—22no—
148C2×D4×C109—22no—
149C2×D4×C109—21no—
150C2×D4×C109—21no—
151C2×D4×C109—21no—
152C2×D4×C109—42no—
153C2×D4×C109—42no—
154C2×D4×C109—44no—
155C2×D4×C109—82no—
156C2×D4×C109—82no—
157C2×D4×C109—84no—
158C2×D4×C109—84no—
159C2×D4×C109C2×D411yescharacteristic
160C2×D4×C109C2×D412yes—
161C22×C4×C109—22no—
162C22×C4×C109—22no—
163C22×C4×C109—22no—
164C22×C4×C109—21no—
165C22×C4×C109—21no—
166C22×C4×C109—22no—
167C22×C4×C109—42no—
168C22×C4×C109—44no—
169C22×C4×C109—41no—
170C22×C4×C109—41no—
171C22×C4×C109C2×D411yescharacteristic
172C22×C4×C109C2×D411yescharacteristic
173C22×C4×C109C2×D411yescharacteristic
174C22×C4×C109C2×D412yes—
175C22×C4×C109C2×D411yescharacteristic
176C22×C4×C109C2×D411yescharacteristic
177C22×C4×C109C2411yescharacteristic
178M16×C109—42no—
179C2×C8×C109—22no—
180C2×C8×C109—22no—
181C2×C8×C109—42no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—21no—
185C4⋊C4×C109—21no—
186C4⋊C4×C109—21no—
187C4⋊C4×C109—21no—
188C4⋊C4×C109—21no—
189C4⋊C4×C109—22no—
190C4⋊C4×C109—22no—
191C4⋊C4×C109—48no—
192C4⋊C4×C109—41no—
193C4⋊C4×C109—41no—
194C4⋊C4×C109C2×D412yes—
195C4⋊C4×C109C2×D411yescharacteristic
196C4⋊C4×C109C2×D411yescharacteristic
197C22⋊C4×C109—24no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—22no—
203C22⋊C4×C109—21no—
204C22⋊C4×C109—21no—
205C22⋊C4×C109—21no—
206C22⋊C4×C109—22no—
207C22⋊C4×C109—21no—
208C22⋊C4×C109—22no—
209C22⋊C4×C109—22no—
210C22⋊C4×C109—22no—
211C22⋊C4×C109—21no—
212C22⋊C4×C109—48no—
213C42×C109—22no—
214C42×C109—21no—
215C42×C109—42no—
216C23×C109—22no—
217C23×C109—22no—
218C23×C109—22no—
219C23×C109—22no—
220C23×C109—21no—
221C23×C109—24no—
222C23×C109—41no—
223C23×C109—42no—
224C23×C109—42no—
225C23×C109—42no—
226C23×C109—81no—
227C23×C109—81no—
228C23×C109—82no—
229C23×C109—82no—
230C23×C109—82no—
231C23×C109#12011yescharacteristic
232C23×C109#12011yescharacteristic
233C23×C109C22×D411yescharacteristic
234Q8×C109—41no—
235D4×C109—42no—
236D4×C109—41no—
237D4×C109—44no—
238D4×C109—42no—
239D4×C109—44no—
240D4×C109—42no—
241D4×C109—42no—
242D4×C109—42no—
243D4×C109—81no—
244D4×C109—82no—
245D4×C109—82no—
246D4×C109—81no—
247D4×C109—82no—
248D4×C109—82no—
249D4×C109—82no—
250D4×C109—82no—
251D4×C109—82no—
252C2×C4×C109—22no—
253C2×C4×C109—22no—
254C2×C4×C109—22no—
255C2×C4×C109—24no—
256C2×C4×C109—21no—
257C2×C4×C109—22no—
258C2×C4×C109—22no—
259C2×C4×C109—21no—
260C2×C4×C109—21no—
261C2×C4×C109—41no—
262C2×C4×C109—41no—
263C2×C4×C109—41no—
264C2×C4×C109—42no—
265C2×C4×C109—42no—
266C2×C4×C109—42no—
267C2×C4×C109—42no—
268C2×C4×C109—44no—
269C2×C4×C109—42no—
270C2×C4×C109—81no—
271C2×C4×C109—82no—
272C2×C4×C109—82no—
273C2×C4×C109—82no—
274C2×C4×C109—82no—
275C2×C4×C109#12011yescharacteristic
276C2×C4×C109#12011yescharacteristic
277C2×C4×C109C22×D411yescharacteristic
278C2×C4×C109C22×D411yescharacteristic
279C8×C109—42no—
280C8×C109—42no—
281C22×C109—21no—
282C22×C109—21no—
283C22×C109—21no—
284C22×C109—21no—
285C22×C109—21no—
286C22×C109—21no—
287C22×C109—21no—
288C22×C109—44no—
289C22×C109—42no—
290C22×C109—42no—
291C22×C109—41no—
292C22×C109—42no—
293C22×C109—42no—
294C22×C109—41no—
295C22×C109—41no—
296C22×C109—42no—
297C22×C109—42no—
298C22×C109—84no—
299C22×C109—162no—
300C22×C109—164no—
301C22×C109#45312yes—
302C22×C109C2×#12011yescharacteristic
303C22×C109#57712yes—
304C4×C109—21no—
305C4×C109—21no—
306C4×C109—21no—
307C4×C109—44no—
308C4×C109—42no—
309C4×C109—41no—
310C4×C109—42no—
311C4×C109—41no—
312C4×C109—41no—
313C4×C109—82no—
314C4×C109#45812yes—
315#18265C10911yesmaximal, characteristic, direct product factor, Sylow
316C2×C109—22no—
317C2×C109—22no—
318C2×C109—41no—
319C2×C109—82no—
320C2×C109—82no—
321C2×C109—81no—
322C2×C109#300911yescharacteristic
323C2×C109C2×#45311yescharacteristic
324C2×C109C2×#45811yescharacteristic
325#3537C2×C10911yescharacteristic
326#3092C2×C10911yescharacteristic, complemented
327C2×#481C2×C10911yescharacteristic, complemented
328#2501C2×C10911yescharacteristic, complemented
329#1618C2×C10914yescomplemented
330#1613C2×C10914yescomplemented
331#1507C2×C10911yescharacteristic, complemented
332#1501C2×C10911yescharacteristic, complemented
333#1471C2×C10911yescharacteristic, complemented
334C109#1826511yescharacteristic, direct product factor, Sylow
335#560—22no—
336#553—21no—
337#553—21no—
338D42—22no—
339D42—21no—
340D42—21no—
341#540—21no—
342#538—21no—
343#538C22×C10912yes—
344#535—21no—
345#534—21no—
346#534C22×C10912yes—
347#525C22×C10911yescharacteristic
348C2×#123—22no—
349C2×#121C22×C10911yescharacteristic
350C2×#121C22×C10911yescharacteristic
351C22×C4⋊C4C22×C10911yescharacteristic, complemented
352#484C22×C10912yescomplemented
353#483—21no—
354#481—22no—
355#481C22×C10914yescomplemented
356#481C22×C10912yescomplemented
357#480—21no—
358#428—21no—
359C2×#106—21no—
360#417—21no—
361C2×#102—21no—
362#407C22×C10911yescharacteristic, complemented
363C2×C22⋊C8C22×C10911yescharacteristic, complemented
364#399C22×C10914yescomplemented
365#390—21no—
366#384—21no—
367C4×C22⋊C4C22×C10911yescharacteristic, complemented
368#328C22×C10912yescomplemented
369#327C22×C10912yescomplemented
370#327C22×C10914yescomplemented
371#323C22×C10914yescomplemented
372C2×#41D4×C10911yescharacteristic
373C22×D4—22no—
374C22×D4—21no—
375C22×D4—21no—
376C22×D4—22no—
377C22×D4D4×C10912yes—
378C22×D4D4×C10911yescharacteristic
379C22×D4D4×C10911yescharacteristic
380C23×C4—21no—
381C23×C4C23×C10911yescharacteristic
382C2×M16—21no—
383C22×C8—21no—
384#127—22no—
385#127—21no—
386#126—22no—
387#126—22no—
388#125—21no—
389#124—22no—
390#123—22no—
391#123—21no—
392#123—22no—
393#123—22no—
394#123—22no—
395#123—48no—
396C22⋊Q8D4×C10911yescharacteristic
397C22⋊Q8C23×C10911yescharacteristic
398#121—22no—
399#121—21no—
400#121—21no—
401#121—22no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121—21no—
407#121—21no—
408#121—21no—
409#121—21no—
410#121—21no—
411#121—22no—
412#121D4×C10912yes—
413#121D4×C10911yescharacteristic
414#121C23×C10912yes—
415#121C23×C10911yescharacteristic
416#120—22no—
417#120—22no—
418#120—22no—
419#120—21no—
420#120—22no—
421#120—22no—
422#120—22no—
423C4×D4—22no—
424C4×D4—21no—
425C4×D4—21no—
426C4×D4—21no—
427C4×D4—21no—
428C4×D4—22no—
429#117—21no—
430#117D4×C10911yescharacteristic
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—24no—
437C2×C4⋊C4D4×C10911yescharacteristic
438C2×C4⋊C4D4×C10911yescharacteristic
439C2×C4⋊C4C23×C10912yes—
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—24no—
443C2×C22⋊C4C23×C10912yes—
444C2×C42—21no—
445#107—22no—
446#106—24no—
447#106—22no—
448#103—22no—
449#102—24no—
450#102—22no—
451C22⋊C8—21no—
452C22⋊C8—21no—
453C22⋊C8C23×C10912yes—
454C22⋊C8C23×C10912yes—
455#95C23×C10912yes—
456C24—22no—
457C24—21no—
458C24C2×D4×C10911yescharacteristic
459C24C2×D4×C10912yes—
460#41—42no—
461C2×Q8C2×D4×C10911yescharacteristic
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—22no—
473C2×D4—22no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—44no—
480C2×D4—82no—
481C2×D4—82no—
482C2×D4—84no—
483C2×D4—84no—
484C2×D4C2×D4×C10911yescharacteristic
485C2×D4C2×D4×C10912yes—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—22no—
492C22×C4—42no—
493C22×C4—44no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4C2×D4×C10911yescharacteristic
497C22×C4C2×D4×C10911yescharacteristic
498C22×C4C2×D4×C10911yescharacteristic
499C22×C4C2×D4×C10912yes—
500C22×C4C2×D4×C10911yescharacteristic
501C22×C4C2×D4×C10911yescharacteristic
502C22×C4C24×C10911yescharacteristic
503M16—42no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—42no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—48no—
517C4⋊C4—41no—
518C4⋊C4—41no—
519C4⋊C4C2×D4×C10912yes—
520C4⋊C4C2×D4×C10911yescharacteristic
521C4⋊C4C2×D4×C10911yescharacteristic
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—21no—
533C22⋊C4—22no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—21no—
537C22⋊C4—48no—
538C42—22no—
539C42—21no—
540C42—42no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—21no—
546C23—24no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23—82no—
554C23—82no—
555C23—82no—
556C23#120×C10911yescharacteristic
557C23#120×C10911yescharacteristic
558C23C22×D4×C10911yescharacteristic
559Q8—41no—
560D4—42no—
561D4—41no—
562D4—44no—
563D4—42no—
564D4—44no—
565D4—42no—
566D4—42no—
567D4—42no—
568D4—81no—
569D4—82no—
570D4—82no—
571D4—81no—
572D4—82no—
573D4—82no—
574D4—82no—
575D4—82no—
576D4—82no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#120×C10911yescharacteristic
601C2×C4#120×C10911yescharacteristic
602C2×C4C22×D4×C10911yescharacteristic
603C2×C4C22×D4×C10911yescharacteristic
604C8—42no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—44no—
614C22—42no—
615C22—42no—
616C22—41no—
617C22—42no—
618C22—42no—
619C22—41no—
620C22—41no—
621C22—42no—
622C22—42no—
623C22—84no—
624C22—162no—
625C22—164no—
626C22#453×C10912yes—
627C22C2×#120×C10911yescharacteristic
628C22#577×C10912yes—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—44no—
633C4—42no—
634C4—41no—
635C4—42no—
636C4—41no—
637C4—41no—
638C4—82no—
639C4#458×C10912yes—
640C2—22no—
641C2—22no—
642C2—41no—
643C2—82no—
644C2—82no—
645C2—81no—
646C2C109×#300911yescharacteristic
647C2C2×#453×C10911yescharacteristic
648C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3092C2semidirect
3. C2×#481×C109C2semidirect
4. C109×#2501C2semidirect
5. C109×#1618C2semidirect
6. C109×#1613C2semidirect
7. C109×#1507C2semidirect
8. C109×#1501C2semidirect
9. C109×#1471C2semidirect
26. C22×C4⋊C4×C109C22semidirect
27. #484×C109C22semidirect
30. #481×C109C22semidirect
31. #481×C109C22semidirect
37. #407×C109C22semidirect
38. C2×C22⋊C8×C109C22semidirect
39. #399×C109C22semidirect
42. C4×C22⋊C4×C109C22semidirect
43. #328×C109C22semidirect
44. #327×C109C22semidirect
45. #327×C109C22semidirect
46. #323×C109C22semidirect
315. #18265C109direct (N × H)
326. #3092C2×C109semidirect
327. C2×#481C2×C109semidirect
328. #2501C2×C109semidirect
329. #1618C2×C109semidirect
330. #1613C2×C109semidirect
331. #1507C2×C109semidirect
332. #1501C2×C109semidirect
333. #1471C2×C109semidirect
334. C109#18265direct (N × H)
351. C22×C4⋊C4C22×C109semidirect
352. #484C22×C109semidirect
355. #481C22×C109semidirect
356. #481C22×C109semidirect
362. #407C22×C109semidirect
363. C2×C22⋊C8C22×C109semidirect
364. #399C22×C109semidirect
367. C4×C22⋊C4C22×C109semidirect
368. #328C22×C109semidirect
369. #327C22×C109semidirect
370. #327C22×C109semidirect
371. #323C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35372
C109×#30922
C2×#481×C1092
C109×#25012
C109×#16182
C109×#16132
C109×#15072
C109×#15012
C109×#14712
#18265109