Finite Group Databasebeta

← C109×#18160 #42136154

Subgroups — C109×#18160 #42136154

Order 27904.

976 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3461C211yesmaximal, characteristic, complemented
2C109×#3114C212yesmaximal, complemented
3C109×#2536C211yesmaximal, characteristic, complemented
4C109×#1611C211yesmaximal, characteristic, complemented
5C109×#1610C211yesmaximal, characteristic, complemented
6C109×#1609C212yesmaximal, complemented
7C109×#1599C211yesmaximal, characteristic, complemented
8C109×#1598C211yesmaximal, characteristic
9C109×#1597C212yesmaximal, complemented
10C109×#1512C212yesmaximal, complemented
11C109×#1467C211yesmaximal, characteristic, complemented
12#544×C109C2211yescharacteristic
13#542×C109—22no—
14#541×C109—22no—
15#539×C109—21no—
16#539×C109C2212yes—
17#537×C109C2211yescharacteristic
18#536×C109—21no—
19#533×C109—22no—
20#529×C109—22no—
21#529×C109—21no—
22#529×C109C2212yes—
23C2×C22⋊Q8×C109—21no—
24C4×#41×C109—21no—
25C2×C4×Q8×C109C2211yescharacteristic
26C2×C4×D4×C109C2211yescharacteristic
27#483×C109C2212yes—
28#480×C109C2212yes—
29#478×C109—22no—
30#463×C109—22no—
31#450×C109C2212yes—
32#447×C109C2212yes—
33#442×C109—22no—
34#440×C109—22no—
35#432×C109C2212yes—
36#398×C109C2211yescharacteristic
37#398×C109C2211yescharacteristic
38#394×C109C2211yescharacteristic
39#394×C109C2211yescharacteristic
40#388×C109—21no—
41#388×C109—21no—
42#382×C109C2211yescharacteristic
43#380×C109—21no—
44#328×C109C2212yes—
45#328×C109C2212yes—
46#327×C109C2212yes—
47#327×C109C2212yes—
48#324×C109C2212yes—
49#324×C109C2211yescharacteristic
50#324×C109C2211yescharacteristic
51C2×#41×C109—21no—
52C22×Q8×C109D411yescharacteristic
53C22×D4×C109C2311yescharacteristic
54C23×C4×C109—21no—
55C2×SD16×C109—22no—
56C2×D8×C109—22no—
57C4⋊Q8×C109—22no—
58C4⋊Q8×C109—22no—
59#126×C109—22no—
60#126×C109—22no—
61#126×C109—22no—
62#126×C109—22no—
63#125×C109—21no—
64#125×C109—22no—
65#125×C109—22no—
66#125×C109—22no—
67#125×C109D412yes—
68#124×C109—22no—
69#124×C109—21no—
70#124×C109D412yes—
71#123×C109—22no—
72#123×C109—22no—
73#123×C109—22no—
74#123×C109—22no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109—22no—
79C22⋊Q8×C109—21no—
80C22⋊Q8×C109—22no—
81C22⋊Q8×C109—44no—
82C22⋊Q8×C109C2312yes—
83#121×C109—22no—
84#121×C109—21no—
85#121×C109C2312yes—
86C4×Q8×C109—21no—
87C4×Q8×C109—21no—
88C4×Q8×C109—21no—
89C4×Q8×C109—22no—
90C4×Q8×C109—22no—
91C4×Q8×C109D412yes—
92C4×D4×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×C109D412yes—
100#117×C109—22no—
101#117×C109—22no—
102#117×C109—21no—
103#117×C109—22no—
104C2×C4⋊C4×C109—21no—
105C2×C4⋊C4×C109—21no—
106C2×C4⋊C4×C109—21no—
107C2×C4⋊C4×C109D411yescharacteristic
108C2×C4⋊C4×C109C2311yescharacteristic
109C2×C4⋊C4×C109C2311yescharacteristic
110C2×C4⋊C4×C109C2311yescharacteristic
111C2×C22⋊C4×C109—21no—
112C2×C22⋊C4×C109—21no—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109—21no—
115C2×C22⋊C4×C109—21no—
116C2×C42×C109—22no—
117C2×C42×C109D411yescharacteristic
118C2×C42×C109D411yescharacteristic
119C2×C42×C109C2311yescharacteristic
120#107×C109—22no—
121#106×C109—22no—
122C4⋊C8×C109—22no—
123#103×C109—22no—
124#102×C109—22no—
125#102×C109—22no—
126#102×C109—22no—
127C22⋊C8×C109C2312yes—
128C22⋊C8×C109C2312yes—
129#97×C109—22no—
130#95×C109—21no—
131#95×C109—21no—
132#95×C109—21no—
133#95×C109C2311yescharacteristic
134#95×C109C2311yescharacteristic
135C24×C109—21no—
136#41×C109—42no—
137#41×C109—42no—
138C2×Q8×C109—21no—
139C2×Q8×C109—21no—
140C2×Q8×C109—22no—
141C2×Q8×C109—22no—
142C2×Q8×C109—41no—
143C2×Q8×C109C2×D412yes—
144C2×D4×C109—21no—
145C2×D4×C109—22no—
146C2×D4×C109—21no—
147C2×D4×C109—22no—
148C2×D4×C109—42no—
149C2×D4×C109C2×D412yes—
150C22×C4×C109—22no—
151C22×C4×C109—22no—
152C22×C4×C109—22no—
153C22×C4×C109—21no—
154C22×C4×C109—21no—
155C22×C4×C109—21no—
156C22×C4×C109—42no—
157C22×C4×C109—42no—
158C22×C4×C109—42no—
159C22×C4×C109C2×D411yescharacteristic
160C22×C4×C109C2×D411yescharacteristic
161C22×C4×C109C2×D411yescharacteristic
162C22×C4×C109C2×D411yescharacteristic
163C22×C4×C109C2×D411yescharacteristic
164C22×C4×C109C2×D411yescharacteristic
165C22×C4×C109C2411yescharacteristic
166SD16×C109—44no—
167D8×C109—44no—
168C2×C8×C109—22no—
169C2×C8×C109—22no—
170C4⋊C4×C109—21no—
171C4⋊C4×C109—22no—
172C4⋊C4×C109—21no—
173C4⋊C4×C109—22no—
174C4⋊C4×C109—22no—
175C4⋊C4×C109—21no—
176C4⋊C4×C109—21no—
177C4⋊C4×C109—22no—
178C4⋊C4×C109—22no—
179C4⋊C4×C109—22no—
180C4⋊C4×C109—22no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—42no—
187C4⋊C4×C109—42no—
188C4⋊C4×C109—42no—
189C4⋊C4×C109C2×D412yes—
190C4⋊C4×C109C2×D412yes—
191C22⋊C4×C109—22no—
192C22⋊C4×C109—22no—
193C22⋊C4×C109—22no—
194C22⋊C4×C109—22no—
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—42no—
201C22⋊C4×C109—42no—
202C22⋊C4×C109—42no—
203C22⋊C4×C109—41no—
204C22⋊C4×C109—41no—
205C22⋊C4×C109—42no—
206C42×C109—22no—
207C42×C109—22no—
208C42×C109—21no—
209C42×C109—22no—
210C42×C109—21no—
211C42×C109—21no—
212C42×C109—21no—
213C42×C109C2×D412yes—
214C42×C109C2×D412yes—
215C23×C109—22no—
216C23×C109—22no—
217C23×C109—41no—
218C23×C109—41no—
219C23×C109—41no—
220C23×C109—41no—
221C23×C109—41no—
222C23×C109—41no—
223C23×C109C22×D411yescharacteristic
224Q8×C109—44no—
225Q8×C109—42no—
226Q8×C109—42no—
227Q8×C109—81no—
228D4×C109—24no—
229D4×C109—42no—
230D4×C109—42no—
231D4×C109—42no—
232D4×C109—42no—
233D4×C109—42no—
234D4×C109—81no—
235C2×C4×C109—21no—
236C2×C4×C109—21no—
237C2×C4×C109—21no—
238C2×C4×C109—21no—
239C2×C4×C109—21no—
240C2×C4×C109—22no—
241C2×C4×C109—22no—
242C2×C4×C109—21no—
243C2×C4×C109—21no—
244C2×C4×C109—22no—
245C2×C4×C109—21no—
246C2×C4×C109—21no—
247C2×C4×C109—41no—
248C2×C4×C109—41no—
249C2×C4×C109—44no—
250C2×C4×C109—44no—
251C2×C4×C109—41no—
252C2×C4×C109—41no—
253C2×C4×C109—41no—
254C2×C4×C109—41no—
255C2×C4×C109—41no—
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—84no—
263C2×C4×C109—82no—
264C2×C4×C109#12011yescharacteristic
265C2×C4×C109#12011yescharacteristic
266C2×C4×C109#12012yes—
267C2×C4×C109AGL(1,ℤ/8)12yes—
268C2×C4×C109C22×D412yes—
269C8×C109—42no—
270C8×C109—42no—
271C22×C109—21no—
272C22×C109—21no—
273C22×C109—42no—
274C22×C109—42no—
275C22×C109—42no—
276C22×C109—42no—
277C22×C109—42no—
278C22×C109—42no—
279C22×C109—81no—
280C22×C109—81no—
281C22×C109—81no—
282C22×C109—81no—
283C22×C109C2×#12011yescharacteristic
284C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C109#57511yescharacteristic
286C4×C109—22no—
287C4×C109—22no—
288C4×C109—21no—
289C4×C109—21no—
290C4×C109—22no—
291C4×C109—41no—
292C4×C109—42no—
293C4×C109—42no—
294C4×C109—41no—
295C4×C109—41no—
296C4×C109—42no—
297C4×C109—81no—
298C4×C109—81no—
299C4×C109—81no—
300#18160C10911yesmaximal, characteristic, direct product factor, Sylow
301C2×C109—21no—
302C2×C109—21no—
303C2×C109—82no—
304C2×C109—82no—
305C2×C109#300611yescharacteristic
306C2×C109#302011yescharacteristic
307C2×C109#302511yescharacteristic
308#3461C2×C10911yescharacteristic, complemented
309#3114C2×C10912yescomplemented
310#2536C2×C10911yescharacteristic, complemented
311#1611C2×C10911yescharacteristic, complemented
312#1610C2×C10911yescharacteristic, complemented
313#1609C2×C10912yescomplemented
314#1599C2×C10911yescharacteristic, complemented
315#1598C2×C10911yescharacteristic
316#1597C2×C10912yescomplemented
317#1512C2×C10912yescomplemented
318#1467C2×C10911yescharacteristic, complemented
319C109#1816011yescharacteristic, direct product factor, Sylow
320#544C22×C10911yescharacteristic
321#542—22no—
322#541—22no—
323#539—21no—
324#539C22×C10912yes—
325#537C22×C10911yescharacteristic
326#536—21no—
327#533—22no—
328#529—22no—
329#529—21no—
330#529C22×C10912yes—
331C2×C22⋊Q8—21no—
332C4×#41—21no—
333C2×C4×Q8C22×C10911yescharacteristic
334C2×C4×D4C22×C10911yescharacteristic
335#483C22×C10912yes—
336#480C22×C10912yes—
337#478—22no—
338#463—22no—
339#450C22×C10912yes—
340#447C22×C10912yes—
341#442—22no—
342#440—22no—
343#432C22×C10912yes—
344#398C22×C10911yescharacteristic
345#398C22×C10911yescharacteristic
346#394C22×C10911yescharacteristic
347#394C22×C10911yescharacteristic
348#388—21no—
349#388—21no—
350#382C22×C10911yescharacteristic
351#380—21no—
352#328C22×C10912yes—
353#328C22×C10912yes—
354#327C22×C10912yes—
355#327C22×C10912yes—
356#324C22×C10912yes—
357#324C22×C10911yescharacteristic
358#324C22×C10911yescharacteristic
359C2×#41—21no—
360C22×Q8D4×C10911yescharacteristic
361C22×D4C23×C10911yescharacteristic
362C23×C4—21no—
363C2×SD16—22no—
364C2×D8—22no—
365C4⋊Q8—22no—
366C4⋊Q8—22no—
367#126—22no—
368#126—22no—
369#126—22no—
370#126—22no—
371#125—21no—
372#125—22no—
373#125—22no—
374#125—22no—
375#125D4×C10912yes—
376#124—22no—
377#124—21no—
378#124D4×C10912yes—
379#123—22no—
380#123—22no—
381#123—22no—
382#123—22no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—21no—
388C22⋊Q8—22no—
389C22⋊Q8—44no—
390C22⋊Q8C23×C10912yes—
391#121—22no—
392#121—21no—
393#121C23×C10912yes—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—22no—
398C4×Q8—22no—
399C4×Q8D4×C10912yes—
400C4×D4—21no—
401C4×D4—21no—
402C4×D4—21no—
403C4×D4—22no—
404C4×D4—22no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4D4×C10912yes—
408#117—22no—
409#117—22no—
410#117—21no—
411#117—22no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4D4×C10911yescharacteristic
416C2×C4⋊C4C23×C10911yescharacteristic
417C2×C4⋊C4C23×C10911yescharacteristic
418C2×C4⋊C4C23×C10911yescharacteristic
419C2×C22⋊C4—21no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C42—22no—
425C2×C42D4×C10911yescharacteristic
426C2×C42D4×C10911yescharacteristic
427C2×C42C23×C10911yescharacteristic
428#107—22no—
429#106—22no—
430C4⋊C8—22no—
431#103—22no—
432#102—22no—
433#102—22no—
434#102—22no—
435C22⋊C8C23×C10912yes—
436C22⋊C8C23×C10912yes—
437#97—22no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95C23×C10911yescharacteristic
442#95C23×C10911yescharacteristic
443C24—21no—
444#41—42no—
445#41—42no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—41no—
451C2×Q8C2×D4×C10912yes—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4C2×D4×C10912yes—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4C2×D4×C10911yescharacteristic
468C22×C4C2×D4×C10911yescharacteristic
469C22×C4C2×D4×C10911yescharacteristic
470C22×C4C2×D4×C10911yescharacteristic
471C22×C4C2×D4×C10911yescharacteristic
472C22×C4C2×D4×C10911yescharacteristic
473C22×C4C24×C10911yescharacteristic
474SD16—44no—
475D8—44no—
476C2×C8—22no—
477C2×C8—22no—
478C4⋊C4—21no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C2×D4×C10912yes—
498C4⋊C4C2×D4×C10912yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—41no—
512C22⋊C4—41no—
513C22⋊C4—42no—
514C42—22no—
515C42—22no—
516C42—21no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42C2×D4×C10912yes—
522C42C2×D4×C10912yes—
523C23—22no—
524C23—22no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23C22×D4×C10911yescharacteristic
532Q8—44no—
533Q8—42no—
534Q8—42no—
535Q8—81no—
536D4—24no—
537D4—42no—
538D4—42no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—81no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—82no—
572C2×C4#120×C10911yescharacteristic
573C2×C4#120×C10911yescharacteristic
574C2×C4#120×C10912yes—
575C2×C4AGL(1,ℤ/8)×C10912yes—
576C2×C4C22×D4×C10912yes—
577C8—42no—
578C8—42no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×#120×C10911yescharacteristic
592C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
593C22#575×C10911yescharacteristic
594C4—22no—
595C4—22no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—81no—
606C4—81no—
607C4—81no—
608C2—21no—
609C2—21no—
610C2—82no—
611C2—82no—
612C2C109×#300611yescharacteristic
613C2C109×#302011yescharacteristic
614C2C109×#302511yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#1598C2
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
1. C109×#3461C2semidirect
2. C109×#3114C2semidirect
3. C109×#2536C2semidirect
4. C109×#1611C2semidirect
5. C109×#1610C2semidirect
6. C109×#1609C2semidirect
7. C109×#1599C2semidirect
9. C109×#1597C2semidirect
10. C109×#1512C2semidirect
11. C109×#1467C2semidirect
300. #18160C109direct (N × H)
308. #3461C2×C109semidirect
309. #3114C2×C109semidirect
310. #2536C2×C109semidirect
311. #1611C2×C109semidirect
312. #1610C2×C109semidirect
313. #1609C2×C109semidirect
314. #1599C2×C109semidirect
316. #1597C2×C109semidirect
317. #1512C2×C109semidirect
318. #1467C2×C109semidirect
319. C109#18160direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#34612
C109×#31142
C109×#25362
C109×#16112
C109×#16102
C109×#16092
C109×#15992
C109×#15982
C109×#15972
C109×#15122
C109×#14672
#18160109