Finite Group Databasebeta

← C109×#18159 #42136153

Subgroups — C109×#18159 #42136153

Order 27904.

998 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3459C211yesmaximal, characteristic
2C109×#3117C212yesmaximal, complemented
3C109×#2580C211yesmaximal, characteristic, complemented
4C109×#1618C212yesmaximal, complemented
5C109×#1614C211yesmaximal, characteristic, complemented
6C109×#1613C211yesmaximal, characteristic, complemented
7C109×#1606C212yesmaximal, complemented
8C109×#1602C211yesmaximal, characteristic, complemented
9C109×#1601C211yesmaximal, characteristic, complemented
10C109×#1512C212yesmaximal, complemented
11C109×#1466C211yesmaximal, characteristic, complemented
12#543×C109C2211yescharacteristic
13#539×C109—22no—
14#539×C109—21no—
15#538×C109—22no—
16#538×C109C2212yes—
17#534×C109—21no—
18#534×C109C2211yescharacteristic
19#529×C109—22no—
20#529×C109C2212yes—
21#525×C109—21no—
22#525×C109—22no—
23C2×C4×Q8×C109C2211yescharacteristic, complemented
24C2×C4×D4×C109—21no—
25C2×C4×D4×C109C2211yescharacteristic
26#484×C109C2212yescomplemented
27#481×C109C2212yes—
28#479×C109—22no—
29#464×C109—22no—
30#451×C109C2212yescomplemented
31#448×C109C2212yes—
32#441×C109—22no—
33#440×C109—22no—
34#432×C109C2212yescomplemented
35#399×C109C2211yescharacteristic
36#399×C109C2211yescharacteristic, complemented
37#393×C109C2211yescharacteristic
38#393×C109C2211yescharacteristic, complemented
39#390×C109—21no—
40#389×C109—21no—
41#385×C109—21no—
42#385×C109—21no—
43#382×C109C2211yescharacteristic, complemented
44#328×C109C2212yescomplemented
45#328×C109C2212yescomplemented
46#327×C109C2212yes—
47#327×C109C2212yes—
48#323×C109C2211yescharacteristic
49#323×C109C2211yescharacteristic
50#323×C109C2212yescomplemented
51C2×#41×C109—21no—
52C22×Q8×C109C2311yescharacteristic
53C22×D4×C109—21no—
54C22×D4×C109D411yescharacteristic
55C23×C4×C109—21no—
56C23×C4×C109D411yescharacteristic
57C23×C4×C109D411yescharacteristic
58C2×Q16×C109—22no—
59C2×SD16×C109—22no—
60#126×C109—22no—
61#126×C109—22no—
62#126×C109—22no—
63#125×C109—22no—
64#125×C109D412yes—
65#124×C109—22no—
66#124×C109D412yes—
67#123×C109—22no—
68#123×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—22no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109—21no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109C2312yes—
79#121×C109—22no—
80#121×C109—21no—
81#121×C109—22no—
82#121×C109—22no—
83#121×C109—21no—
84#121×C109C2312yes—
85#120×C109—22no—
86#120×C109—22no—
87#120×C109—22no—
88C4×Q8×C109—22no—
89C4×Q8×C109—22no—
90C4×Q8×C109D412yes—
91C4×D4×C109—22no—
92C4×D4×C109—22no—
93C4×D4×C109—22no—
94C4×D4×C109—22no—
95C4×D4×C109—22no—
96C4×D4×C109—22no—
97C4×D4×C109—22no—
98C4×D4×C109D412yes—
99#117×C109—21no—
100#117×C109—22no—
101C2×C4⋊C4×C109—21no—
102C2×C4⋊C4×C109—21no—
103C2×C4⋊C4×C109—21no—
104C2×C4⋊C4×C109—21no—
105C2×C4⋊C4×C109—21no—
106C2×C4⋊C4×C109D411yescharacteristic
107C2×C4⋊C4×C109C2311yescharacteristic
108C2×C22⋊C4×C109—21no—
109C2×C22⋊C4×C109—21no—
110C2×C22⋊C4×C109—21no—
111C2×C22⋊C4×C109—21no—
112C2×C22⋊C4×C109—21no—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109C2311yescharacteristic
115C2×C22⋊C4×C109C2311yescharacteristic
116C2×C42×C109—21no—
117C2×C42×C109—21no—
118C2×C42×C109C2311yescharacteristic
119#107×C109—22no—
120#106×C109—22no—
121C4⋊C8×C109—22no—
122#103×C109—22no—
123#103×C109—22no—
124#103×C109—22no—
125#102×C109—22no—
126C22⋊C8×C109C2312yes—
127C22⋊C8×C109C2312yes—
128#97×C109—22no—
129#95×C109—21no—
130#95×C109—21no—
131#95×C109—21no—
132#95×C109C2311yescharacteristic
133#95×C109C2311yescharacteristic
134C24×C109—21no—
135C24×C109C2×D411yescharacteristic
136C24×C109C2×D411yescharacteristic
137#41×C109—82no—
138C2×Q8×C109—21no—
139C2×Q8×C109—22no—
140C2×Q8×C109—42no—
141C2×Q8×C109C2×D412yes—
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—41no—
150C2×D4×C109—42no—
151C2×D4×C109—42no—
152C2×D4×C109C2×D412yes—
153C22×C4×C109—21no—
154C22×C4×C109—21no—
155C22×C4×C109—21no—
156C22×C4×C109—21no—
157C22×C4×C109—22no—
158C22×C4×C109—21no—
159C22×C4×C109—21no—
160C22×C4×C109—22no—
161C22×C4×C109—21no—
162C22×C4×C109—21no—
163C22×C4×C109—21no—
164C22×C4×C109—21no—
165C22×C4×C109—21no—
166C22×C4×C109—81no—
167C22×C4×C109—81no—
168C22×C4×C109—82no—
169C22×C4×C109C2×D411yescharacteristic
170C22×C4×C109C2×D411yescharacteristic
171C22×C4×C109C2×D411yescharacteristic
172C22×C4×C109C2×D411yescharacteristic
173C22×C4×C109C2411yescharacteristic
174Q16×C109—44no—
175SD16×C109—44no—
176C2×C8×C109—22no—
177C2×C8×C109—22no—
178C4⋊C4×C109—22no—
179C4⋊C4×C109—21no—
180C4⋊C4×C109—21no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—42no—
189C4⋊C4×C109—41no—
190C4⋊C4×C109—41no—
191C4⋊C4×C109—81no—
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—22no—
207C22⋊C4×C109—42no—
208C22⋊C4×C109—42no—
209C22⋊C4×C109—81no—
210C22⋊C4×C109—81no—
211C42×C109—22no—
212C42×C109—22no—
213C42×C109—42no—
214C42×C109C2×D412yes—
215C42×C109C2×D412yes—
216C23×C109—21no—
217C23×C109—21no—
218C23×C109—21no—
219C23×C109—22no—
220C23×C109—21no—
221C23×C109—21no—
222C23×C109—41no—
223C23×C109—41no—
224C23×C109—41no—
225C23×C109—41no—
226C23×C109—42no—
227C23×C109—81no—
228C23×C109—81no—
229C23×C109#13712yes—
230C23×C109C22×D411yescharacteristic
231Q8×C109—24no—
232Q8×C109—42no—
233Q8×C109—42no—
234Q8×C109—81no—
235D4×C109—44no—
236D4×C109—42no—
237D4×C109—44no—
238D4×C109—42no—
239D4×C109—42no—
240D4×C109—82no—
241D4×C109—81no—
242D4×C109—82no—
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—21no—
250C2×C4×C109—22no—
251C2×C4×C109—41no—
252C2×C4×C109—41no—
253C2×C4×C109—42no—
254C2×C4×C109—42no—
255C2×C4×C109—44no—
256C2×C4×C109—42no—
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—81no—
272C2×C4×C109—81no—
273C2×C4×C109—82no—
274C2×C4×C109—81no—
275C2×C4×C109#12011yescharacteristic
276C2×C4×C109#12012yes—
277C2×C4×C109#12011yescharacteristic
278C2×C4×C109C22×D412yes—
279C8×C109—42no—
280C8×C109—42no—
281C22×C109—21no—
282C22×C109—21no—
283C22×C109—22no—
284C22×C109—42no—
285C22×C109—42no—
286C22×C109—41no—
287C22×C109—41no—
288C22×C109—41no—
289C22×C109—42no—
290C22×C109—41no—
291C22×C109—42no—
292C22×C109—42no—
293C22×C109—41no—
294C22×C109—81no—
295C22×C109—81no—
296C22×C109—84no—
297C22×C109C2×#12011yescharacteristic
298C22×C109C2×#13711yescharacteristic
299C22×C109#57511yescharacteristic
300C4×C109—22no—
301C4×C109—21no—
302C4×C109—21no—
303C4×C109—22no—
304C4×C109—42no—
305C4×C109—41no—
306C4×C109—41no—
307C4×C109—42no—
308C4×C109—42no—
309C4×C109—42no—
310C4×C109—81no—
311C4×C109—81no—
312C4×C109—81no—
313#18159C10911yesmaximal, characteristic, direct product factor, Sylow
314C2×C109—21no—
315C2×C109—21no—
316C2×C109—42no—
317C2×C109—81no—
318C2×C109—82no—
319C2×C109#300711yescharacteristic
320C2×C109#301911yescharacteristic
321C2×C109#302511yescharacteristic
322#3459C2×C10911yescharacteristic
323#3117C2×C10912yescomplemented
324#2580C2×C10911yescharacteristic, complemented
325#1618C2×C10912yescomplemented
326#1614C2×C10911yescharacteristic, complemented
327#1613C2×C10911yescharacteristic, complemented
328#1606C2×C10912yescomplemented
329#1602C2×C10911yescharacteristic, complemented
330#1601C2×C10911yescharacteristic, complemented
331#1512C2×C10912yescomplemented
332#1466C2×C10911yescharacteristic, complemented
333C109#1815911yescharacteristic, direct product factor, Sylow
334#543C22×C10911yescharacteristic
335#539—22no—
336#539—21no—
337#538—22no—
338#538C22×C10912yes—
339#534—21no—
340#534C22×C10911yescharacteristic
341#529—22no—
342#529C22×C10912yes—
343#525—21no—
344#525—22no—
345C2×C4×Q8C22×C10911yescharacteristic, complemented
346C2×C4×D4—21no—
347C2×C4×D4C22×C10911yescharacteristic
348#484C22×C10912yescomplemented
349#481C22×C10912yes—
350#479—22no—
351#464—22no—
352#451C22×C10912yescomplemented
353#448C22×C10912yes—
354#441—22no—
355#440—22no—
356#432C22×C10912yescomplemented
357#399C22×C10911yescharacteristic
358#399C22×C10911yescharacteristic, complemented
359#393C22×C10911yescharacteristic
360#393C22×C10911yescharacteristic, complemented
361#390—21no—
362#389—21no—
363#385—21no—
364#385—21no—
365#382C22×C10911yescharacteristic, complemented
366#328C22×C10912yescomplemented
367#328C22×C10912yescomplemented
368#327C22×C10912yes—
369#327C22×C10912yes—
370#323C22×C10911yescharacteristic
371#323C22×C10911yescharacteristic
372#323C22×C10912yescomplemented
373C2×#41—21no—
374C22×Q8C23×C10911yescharacteristic
375C22×D4—21no—
376C22×D4D4×C10911yescharacteristic
377C23×C4—21no—
378C23×C4D4×C10911yescharacteristic
379C23×C4D4×C10911yescharacteristic
380C2×Q16—22no—
381C2×SD16—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#125—22no—
386#125D4×C10912yes—
387#124—22no—
388#124D4×C10912yes—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—22no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8C23×C10912yes—
401#121—22no—
402#121—21no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121C23×C10912yes—
407#120—22no—
408#120—22no—
409#120—22no—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8D4×C10912yes—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4D4×C10912yes—
421#117—21no—
422#117—22no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4D4×C10911yescharacteristic
429C2×C4⋊C4C23×C10911yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4C23×C10911yescharacteristic
437C2×C22⋊C4C23×C10911yescharacteristic
438C2×C42—21no—
439C2×C42—21no—
440C2×C42C23×C10911yescharacteristic
441#107—22no—
442#106—22no—
443C4⋊C8—22no—
444#103—22no—
445#103—22no—
446#103—22no—
447#102—22no—
448C22⋊C8C23×C10912yes—
449C22⋊C8C23×C10912yes—
450#97—22no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95C23×C10911yescharacteristic
455#95C23×C10911yescharacteristic
456C24—21no—
457C24C2×D4×C10911yescharacteristic
458C24C2×D4×C10911yescharacteristic
459#41—82no—
460C2×Q8—21no—
461C2×Q8—22no—
462C2×Q8—42no—
463C2×Q8C2×D4×C10912yes—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—41no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4C2×D4×C10912yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—82no—
491C22×C4C2×D4×C10911yescharacteristic
492C22×C4C2×D4×C10911yescharacteristic
493C22×C4C2×D4×C10911yescharacteristic
494C22×C4C2×D4×C10911yescharacteristic
495C22×C4C24×C10911yescharacteristic
496Q16—44no—
497SD16—44no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4—81no—
514C4⋊C4—81no—
515C4⋊C4C2×D4×C10912yes—
516C4⋊C4C2×D4×C10912yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—22no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—81no—
532C22⋊C4—81no—
533C42—22no—
534C42—22no—
535C42—42no—
536C42C2×D4×C10912yes—
537C42C2×D4×C10912yes—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—81no—
550C23—81no—
551C23#137×C10912yes—
552C23C22×D4×C10911yescharacteristic
553Q8—24no—
554Q8—42no—
555Q8—42no—
556Q8—81no—
557D4—44no—
558D4—42no—
559D4—44no—
560D4—42no—
561D4—42no—
562D4—81no—
563D4—82no—
564D4—82no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
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—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—82no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4#120×C10911yescharacteristic
598C2×C4#120×C10912yes—
599C2×C4#120×C10911yescharacteristic
600C2×C4C22×D4×C10912yes—
601C8—42no—
602C8—42no—
603C22—21no—
604C22—21no—
605C22—22no—
606C22—42no—
607C22—42no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—81no—
617C22—81no—
618C22—84no—
619C22C2×#120×C10911yescharacteristic
620C22C2×#137×C10911yescharacteristic
621C22#575×C10911yescharacteristic
622C4—22no—
623C4—21no—
624C4—21no—
625C4—22no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—81no—
635C2—21no—
636C2—21no—
637C2—42no—
638C2—81no—
639C2—82no—
640C2C109×#300711yescharacteristic
641C2C109×#301911yescharacteristic
642C2C109×#302511yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3459C2
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×#2580C2semidirect
4. C109×#1618C2semidirect
5. C109×#1614C2semidirect
6. C109×#1613C2semidirect
7. C109×#1606C2semidirect
8. C109×#1602C2semidirect
9. C109×#1601C2semidirect
10. C109×#1512C2semidirect
11. C109×#1466C2semidirect
23. C2×C4×Q8×C109C22semidirect
26. #484×C109C22semidirect
30. #451×C109C22semidirect
34. #432×C109C22semidirect
36. #399×C109C22semidirect
38. #393×C109C22semidirect
43. #382×C109C22semidirect
44. #328×C109C22semidirect
45. #328×C109C22semidirect
50. #323×C109C22semidirect
313. #18159C109direct (N × H)
323. #3117C2×C109semidirect
324. #2580C2×C109semidirect
325. #1618C2×C109semidirect
326. #1614C2×C109semidirect
327. #1613C2×C109semidirect
328. #1606C2×C109semidirect
329. #1602C2×C109semidirect
330. #1601C2×C109semidirect
331. #1512C2×C109semidirect
332. #1466C2×C109semidirect
333. C109#18159direct (N × H)
345. C2×C4×Q8C22×C109semidirect
348. #484C22×C109semidirect
352. #451C22×C109semidirect
356. #432C22×C109semidirect
358. #399C22×C109semidirect
360. #393C22×C109semidirect
365. #382C22×C109semidirect
366. #328C22×C109semidirect
367. #328C22×C109semidirect
372. #323C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34592
C109×#31172
C109×#25802
C109×#16182
C109×#16142
C109×#16132
C109×#16062
C109×#16022
C109×#16012
C109×#15122
C109×#14662
#18159109