Finite Group Databasebeta

← C109×#18299 #42136293

Subgroups — C109×#18299 #42136293

Order 27904.

974 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3460C211yesmaximal, characteristic
2C109×#3190C212yesmaximal, complemented
3C109×#2638C211yesmaximal, characteristic, complemented
4C109×#1612C212yesmaximal, complemented
5C109×#1611C211yesmaximal, characteristic, complemented
6C109×#1610C211yesmaximal, characteristic, complemented
7C109×#1606C212yesmaximal, complemented
8C109×#1605C211yesmaximal, characteristic, complemented
9C109×#1604C211yesmaximal, characteristic, complemented
10C109×#1516C212yesmaximal, complemented
11C109×#1459C211yesmaximal, characteristic, complemented
12#544×C109C2211yescharacteristic
13#542×C109—22no—
14#540×C109—22no—
15#538×C109—21no—
16#538×C109C2212yes—
17#537×C109C2211yescharacteristic
18#535×C109—21no—
19#532×C109—22no—
20#529×C109—22no—
21#525×C109—21no—
22#525×C109C2212yes—
23C2×C22⋊Q8×C109C2211yescharacteristic
24C4×#41×C109—21no—
25C2×C4×D4×C109C2211yescharacteristic
26#485×C109—22no—
27#483×C109C2212yes—
28#480×C109C2212yes—
29#467×C109—22no—
30#465×C109—22no—
31#452×C109—22no—
32#451×C109C2212yes—
33#448×C109C2212yes—
34C2×C22⋊C8×C109C2212yes—
35#398×C109C2211yescharacteristic
36#397×C109—21no—
37#397×C109C2211yescharacteristic
38#395×C109C2211yescharacteristic
39#394×C109—21no—
40#393×C109C2211yescharacteristic
41#387×C109—21no—
42#385×C109—21no—
43#381×C109C2211yescharacteristic
44#328×C109C2212yes—
45#328×C109C2212yes—
46#327×C109C2212yes—
47#327×C109C2212yes—
48#324×C109C2211yescharacteristic
49#324×C109C2211yescharacteristic
50#323×C109C2212yes—
51C2×#41×C109—21no—
52C2×#41×C109—21no—
53C2×#41×C109D412yes—
54C22×Q8×C109C2311yescharacteristic
55C22×D4×C109D411yescharacteristic
56C23×C4×C109D411yescharacteristic
57C23×C4×C109C2311yescharacteristic
58C2×Q16×C109—22no—
59C2×SD16×C109—22no—
60C22×C8×C109—22no—
61C4⋊Q8×C109—22no—
62#127×C109—22no—
63#126×C109—22no—
64#126×C109—22no—
65#125×C109—22no—
66#124×C109—22no—
67#124×C109—22no—
68#124×C109—22no—
69#123×C109—22no—
70#123×C109—22no—
71#123×C109—22no—
72#123×C109—22no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109D412yes—
79C22⋊Q8×C109C2312yes—
80#121×C109—21no—
81#121×C109—22no—
82#121×C109—22no—
83#121×C109—22no—
84#121×C109—22no—
85#121×C109—22no—
86#121×C109—21no—
87#121×C109D412yes—
88#121×C109C2312yes—
89#120×C109—22no—
90#120×C109—22no—
91C4×Q8×C109—21no—
92C4×Q8×C109—21no—
93C4×D4×C109—21no—
94C4×D4×C109—21no—
95C4×D4×C109—22no—
96C4×D4×C109—22no—
97C4×D4×C109—22no—
98C4×D4×C109—22no—
99C4×D4×C109—22no—
100C4×D4×C109—22no—
101#117×C109—22no—
102#117×C109—21no—
103#117×C109—21no—
104#117×C109D412yes—
105C2×C4⋊C4×C109—21no—
106C2×C4⋊C4×C109—21no—
107C2×C4⋊C4×C109—21no—
108C2×C4⋊C4×C109C2311yescharacteristic
109C2×C4⋊C4×C109C2311yescharacteristic
110C2×C22⋊C4×C109—21no—
111C2×C22⋊C4×C109—21no—
112C2×C22⋊C4×C109—21no—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109D411yescharacteristic
115C2×C22⋊C4×C109C2311yescharacteristic
116C2×C42×C109—22no—
117C2×C42×C109D411yescharacteristic
118#107×C109—22no—
119#106×C109—22no—
120#103×C109—22no—
121#103×C109—22no—
122#103×C109—22no—
123#102×C109—22no—
124C22⋊C8×C109—22no—
125C22⋊C8×C109C2312yes—
126C22⋊C8×C109C2312yes—
127#95×C109—21no—
128#95×C109—21no—
129#95×C109—21no—
130#95×C109—21no—
131#95×C109—21no—
132#95×C109C2311yescharacteristic
133#95×C109C2311yescharacteristic
134C24×C109C2×D411yescharacteristic
135C24×C109C2×D411yescharacteristic
136#41×C109—42no—
137#41×C109—42no—
138#41×C109—82no—
139#41×C109—82no—
140C2×Q8×C109—21no—
141C2×Q8×C109—21no—
142C2×Q8×C109—22no—
143C2×Q8×C109—81no—
144C2×Q8×C109C2×D412yes—
145C2×D4×C109—21no—
146C2×D4×C109—22no—
147C2×D4×C109—22no—
148C2×D4×C109—22no—
149C2×D4×C109—22no—
150C2×D4×C109—21no—
151C2×D4×C109—22no—
152C2×D4×C109—22no—
153C2×D4×C109—22no—
154C2×D4×C109—41no—
155C2×D4×C109C2×D412yes—
156C22×C4×C109—22no—
157C22×C4×C109—22no—
158C22×C4×C109—22no—
159C22×C4×C109—21no—
160C22×C4×C109—21no—
161C22×C4×C109—21no—
162C22×C4×C109—21no—
163C22×C4×C109—21no—
164C22×C4×C109—21no—
165C22×C4×C109—81no—
166C22×C4×C109—81no—
167C22×C4×C109C2×D411yescharacteristic
168C22×C4×C109C2×D411yescharacteristic
169C22×C4×C109C2×D411yescharacteristic
170C22×C4×C109C2×D411yescharacteristic
171C22×C4×C109C2×D412yes—
172C22×C4×C109C2×D412yes—
173C22×C4×C109C2411yescharacteristic
174Q16×C109—82no—
175SD16×C109—82no—
176C2×C8×C109—22no—
177C2×C8×C109—22no—
178C2×C8×C109—82no—
179C4⋊C4×C109—21no—
180C4⋊C4×C109—21no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—21no—
183C4⋊C4×C109—21no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—22no—
189C4⋊C4×C109—41no—
190C4⋊C4×C109—41no—
191C4⋊C4×C109—42no—
192C4⋊C4×C109—81no—
193C4⋊C4×C109C2×D412yes—
194C4⋊C4×C109C2×D412yes—
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—22no—
203C22⋊C4×C109—22no—
204C22⋊C4×C109—22no—
205C22⋊C4×C109—22no—
206C22⋊C4×C109—42no—
207C22⋊C4×C109—42no—
208C22⋊C4×C109—81no—
209C22⋊C4×C109—81no—
210C42×C109—22no—
211C42×C109—22no—
212C42×C109—21no—
213C42×C109—22no—
214C42×C109—21no—
215C23×C109—21no—
216C23×C109—21no—
217C23×C109—21no—
218C23×C109—22no—
219C23×C109—21no—
220C23×C109—21no—
221C23×C109—41no—
222C23×C109—41no—
223C23×C109—41no—
224C23×C109—41no—
225C23×C109#12012yes—
226C23×C109C22×D411yescharacteristic
227Q8×C109—42no—
228Q8×C109—42no—
229Q8×C109—42no—
230Q8×C109—42no—
231Q8×C109—81no—
232D4×C109—44no—
233D4×C109—42no—
234D4×C109—42no—
235D4×C109—42no—
236D4×C109—42no—
237D4×C109—82no—
238D4×C109—82no—
239D4×C109—82no—
240D4×C109—81no—
241D4×C109—82no—
242C2×C4×C109—21no—
243C2×C4×C109—21no—
244C2×C4×C109—21no—
245C2×C4×C109—22no—
246C2×C4×C109—22no—
247C2×C4×C109—22no—
248C2×C4×C109—21no—
249C2×C4×C109—42no—
250C2×C4×C109—41no—
251C2×C4×C109—42no—
252C2×C4×C109—41no—
253C2×C4×C109—42no—
254C2×C4×C109—44no—
255C2×C4×C109—44no—
256C2×C4×C109—41no—
257C2×C4×C109—41no—
258C2×C4×C109—41no—
259C2×C4×C109—41no—
260C2×C4×C109—41no—
261C2×C4×C109—41no—
262C2×C4×C109—41no—
263C2×C4×C109—41no—
264C2×C4×C109—41no—
265C2×C4×C109—41no—
266C2×C4×C109—41no—
267C2×C4×C109—41no—
268C2×C4×C109—41no—
269C2×C4×C109—41no—
270C2×C4×C109—81no—
271C2×C4×C109—82no—
272C2×C4×C109#12011yescharacteristic
273C2×C4×C109#12011yescharacteristic
274C2×C4×C109#13512yes—
275C2×C4×C109C22×D412yes—
276C8×C109—42no—
277C8×C109—42no—
278C22×C109—21no—
279C22×C109—21no—
280C22×C109—22no—
281C22×C109—22no—
282C22×C109—22no—
283C22×C109—42no—
284C22×C109—42no—
285C22×C109—41no—
286C22×C109—42no—
287C22×C109—41no—
288C22×C109—41no—
289C22×C109—42no—
290C22×C109—81no—
291C22×C109—81no—
292C22×C109—81no—
293C22×C109C2×#12011yescharacteristic
294C22×C109C2×#13511yescharacteristic
295C22×C109#57511yescharacteristic
296C4×C109—22no—
297C4×C109—22no—
298C4×C109—21no—
299C4×C109—21no—
300C4×C109—42no—
301C4×C109—41no—
302C4×C109—42no—
303C4×C109—41no—
304C4×C109—42no—
305C4×C109—81no—
306C4×C109—81no—
307C4×C109—81no—
308C4×C109—81no—
309#18299C10911yesmaximal, characteristic, direct product factor, Sylow
310C2×C109—21no—
311C2×C109—21no—
312C2×C109—42no—
313C2×C109—82no—
314C2×C109—81no—
315C2×C109#300211yescharacteristic
316C2×C109#302211yescharacteristic
317C2×C109#302611yescharacteristic
318#3460C2×C10911yescharacteristic
319#3190C2×C10912yescomplemented
320#2638C2×C10911yescharacteristic, complemented
321#1612C2×C10912yescomplemented
322#1611C2×C10911yescharacteristic, complemented
323#1610C2×C10911yescharacteristic, complemented
324#1606C2×C10912yescomplemented
325#1605C2×C10911yescharacteristic, complemented
326#1604C2×C10911yescharacteristic, complemented
327#1516C2×C10912yescomplemented
328#1459C2×C10911yescharacteristic, complemented
329C109#1829911yescharacteristic, direct product factor, Sylow
330#544C22×C10911yescharacteristic
331#542—22no—
332#540—22no—
333#538—21no—
334#538C22×C10912yes—
335#537C22×C10911yescharacteristic
336#535—21no—
337#532—22no—
338#529—22no—
339#525—21no—
340#525C22×C10912yes—
341C2×C22⋊Q8C22×C10911yescharacteristic
342C4×#41—21no—
343C2×C4×D4C22×C10911yescharacteristic
344#485—22no—
345#483C22×C10912yes—
346#480C22×C10912yes—
347#467—22no—
348#465—22no—
349#452—22no—
350#451C22×C10912yes—
351#448C22×C10912yes—
352C2×C22⋊C8C22×C10912yes—
353#398C22×C10911yescharacteristic
354#397—21no—
355#397C22×C10911yescharacteristic
356#395C22×C10911yescharacteristic
357#394—21no—
358#393C22×C10911yescharacteristic
359#387—21no—
360#385—21no—
361#381C22×C10911yescharacteristic
362#328C22×C10912yes—
363#328C22×C10912yes—
364#327C22×C10912yes—
365#327C22×C10912yes—
366#324C22×C10911yescharacteristic
367#324C22×C10911yescharacteristic
368#323C22×C10912yes—
369C2×#41—21no—
370C2×#41—21no—
371C2×#41D4×C10912yes—
372C22×Q8C23×C10911yescharacteristic
373C22×D4D4×C10911yescharacteristic
374C23×C4D4×C10911yescharacteristic
375C23×C4C23×C10911yescharacteristic
376C2×Q16—22no—
377C2×SD16—22no—
378C22×C8—22no—
379C4⋊Q8—22no—
380#127—22no—
381#126—22no—
382#126—22no—
383#125—22no—
384#124—22no—
385#124—22no—
386#124—22no—
387#123—22no—
388#123—22no—
389#123—22no—
390#123—22no—
391C22⋊Q8—21no—
392C22⋊Q8—21no—
393C22⋊Q8—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8D4×C10912yes—
397C22⋊Q8C23×C10912yes—
398#121—21no—
399#121—22no—
400#121—22no—
401#121—22no—
402#121—22no—
403#121—22no—
404#121—21no—
405#121D4×C10912yes—
406#121C23×C10912yes—
407#120—22no—
408#120—22no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419#117—22no—
420#117—21no—
421#117—21no—
422#117D4×C10912yes—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4C23×C10911yescharacteristic
427C2×C4⋊C4C23×C10911yescharacteristic
428C2×C22⋊C4—21no—
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4D4×C10911yescharacteristic
433C2×C22⋊C4C23×C10911yescharacteristic
434C2×C42—22no—
435C2×C42D4×C10911yescharacteristic
436#107—22no—
437#106—22no—
438#103—22no—
439#103—22no—
440#103—22no—
441#102—22no—
442C22⋊C8—22no—
443C22⋊C8C23×C10912yes—
444C22⋊C8C23×C10912yes—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—21no—
449#95—21no—
450#95C23×C10911yescharacteristic
451#95C23×C10911yescharacteristic
452C24C2×D4×C10911yescharacteristic
453C24C2×D4×C10911yescharacteristic
454#41—42no—
455#41—42no—
456#41—82no—
457#41—82no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—22no—
461C2×Q8—81no—
462C2×Q8C2×D4×C10912yes—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—41no—
473C2×D4C2×D4×C10912yes—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—81no—
484C22×C4—81no—
485C22×C4C2×D4×C10911yescharacteristic
486C22×C4C2×D4×C10911yescharacteristic
487C22×C4C2×D4×C10911yescharacteristic
488C22×C4C2×D4×C10911yescharacteristic
489C22×C4C2×D4×C10912yes—
490C22×C4C2×D4×C10912yes—
491C22×C4C24×C10911yescharacteristic
492Q16—82no—
493SD16—82no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—82no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4C2×D4×C10912yes—
512C4⋊C4C2×D4×C10912yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—81no—
527C22⋊C4—81no—
528C42—22no—
529C42—22no—
530C42—21no—
531C42—22no—
532C42—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—21no—
538C23—21no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23#120×C10912yes—
544C23C22×D4×C10911yescharacteristic
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—81no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—82no—
556D4—82no—
557D4—81no—
558D4—82no—
559D4—82no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—81no—
589C2×C4—82no—
590C2×C4#120×C10911yescharacteristic
591C2×C4#120×C10911yescharacteristic
592C2×C4#135×C10912yes—
593C2×C4C22×D4×C10912yes—
594C8—42no—
595C8—42no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22C2×#120×C10911yescharacteristic
612C22C2×#135×C10911yescharacteristic
613C22#575×C10911yescharacteristic
614C4—22no—
615C4—22no—
616C4—21no—
617C4—21no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C2—21no—
628C2—21no—
629C2—42no—
630C2—81no—
631C2—82no—
632C2C109×#300211yescharacteristic
633C2C109×#302211yescharacteristic
634C2C109×#302611yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3460C2
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×#3190C2semidirect
3. C109×#2638C2semidirect
4. C109×#1612C2semidirect
5. C109×#1611C2semidirect
6. C109×#1610C2semidirect
7. C109×#1606C2semidirect
8. C109×#1605C2semidirect
9. C109×#1604C2semidirect
10. C109×#1516C2semidirect
11. C109×#1459C2semidirect
309. #18299C109direct (N × H)
319. #3190C2×C109semidirect
320. #2638C2×C109semidirect
321. #1612C2×C109semidirect
322. #1611C2×C109semidirect
323. #1610C2×C109semidirect
324. #1606C2×C109semidirect
325. #1605C2×C109semidirect
326. #1604C2×C109semidirect
327. #1516C2×C109semidirect
328. #1459C2×C109semidirect
329. C109#18299direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#34602
C109×#31902
C109×#26382
C109×#16122
C109×#16112
C109×#16102
C109×#16062
C109×#16052
C109×#16042
C109×#15162
C109×#14592
#18299109