Finite Group Databasebeta

← C109×#19372 #42137366

Subgroups — C109×#19372 #42137366

Order 27904.

588 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3255C211yesmaximal, characteristic, complemented
2C109×#3216C211yesmaximal, characteristic
3C109×#3182C211yesmaximal, characteristic
4C109×#2962C211yesmaximal, characteristic, complemented
5C109×#1702C211yesmaximal, characteristic, complemented
6C109×#1700C211yesmaximal, characteristic, complemented
7C109×#1694C211yesmaximal, characteristic, complemented
8C109×#1686C211yesmaximal, characteristic, complemented
9C109×#1682C211yesmaximal, characteristic, complemented
10C109×#1680C211yesmaximal, characteristic, complemented
11C109×#1674C211yesmaximal, characteristic, complemented
12C109×#1666C211yesmaximal, characteristic, complemented
13C109×#1569C211yesmaximal, characteristic, complemented
14C109×#1552C211yesmaximal, characteristic, complemented
15C109×#1547C211yesmaximal, characteristic
16#533×C109C2211yescharacteristic
17#533×C109C2211yescharacteristic
18#532×C109C2211yescharacteristic
19#500×C109C2211yescharacteristic
20#497×C109—21no—
21#492×C109C2211yescharacteristic
22#489×C109—21no—
23#487×C109C2211yescharacteristic
24#486×C109C2211yescharacteristic
25#475×C109—21no—
26#475×C109C2211yescharacteristic
27#475×C109C2211yescharacteristic
28#470×C109—21no—
29#470×C109—21no—
30#468×C109—21no—
31#468×C109—21no—
32#462×C109—21no—
33#462×C109C2211yescharacteristic
34#462×C109C2211yescharacteristic
35#462×C109C2211yescharacteristic
36#462×C109C2211yescharacteristic
37#460×C109C2211yescharacteristic
38#460×C109C2211yescharacteristic
39#452×C109—21no—
40#452×C109—21no—
41#446×C109—21no—
42#446×C109—21no—
43#446×C109C2211yescharacteristic
44#446×C109C2211yescharacteristic
45C8×Q8×C109C2211yescharacteristic
46C8×Q8×C109C2211yescharacteristic
47#440×C109—21no—
48#440×C109—21no—
49#432×C109C2211yescharacteristic
50#424×C109—21no—
51#423×C109C2211yescharacteristic
52C2×C4⋊C8×C109C2211yescharacteristic
53#405×C109—21no—
54C4×M16×C109C2211yescharacteristic
55#335×C109C2211yescharacteristic
56#335×C109C2211yescharacteristic
57#334×C109C2211yescharacteristic
58#334×C109C2211yescharacteristic
59#332×C109C2211yescharacteristic
60#332×C109C2211yescharacteristic
61#332×C109C2211yescharacteristic
62#332×C109C2211yescharacteristic
63#332×C109C2211yescharacteristic
64#332×C109C2211yescharacteristic
65#331×C109C2211yescharacteristic
66#331×C109C2211yescharacteristic
67C2×#41×C109—21no—
68C2×Q16×C109—21no—
69C2×Q16×C109—21no—
70C2×Q16×C109—21no—
71C2×Q16×C109—21no—
72C2×SD16×C109—21no—
73C2×SD16×C109—21no—
74C2×SD16×C109—21no—
75C2×SD16×C109—21no—
76C2×M16×C109—21no—
77C2×M16×C109D411yescharacteristic
78C2×M16×C109D411yescharacteristic
79C22×C8×C109—21no—
80C4⋊Q8×C109D411yescharacteristic
81C4⋊Q8×C109D411yescharacteristic
82C4⋊Q8×C109C2311yescharacteristic
83C4⋊Q8×C109C2311yescharacteristic
84C4⋊Q8×C109C2311yescharacteristic
85#127×C109C2311yescharacteristic
86#125×C109—21no—
87#125×C109—21no—
88#124×C109—21no—
89C22⋊Q8×C109—21no—
90C22⋊Q8×C109—21no—
91C22⋊Q8×C109—21no—
92C22⋊Q8×C109D411yescharacteristic
93C22⋊Q8×C109D411yescharacteristic
94#121×C109—21no—
95#121×C109—21no—
96C4×Q8×C109—21no—
97C4×Q8×C109—21no—
98C4×Q8×C109—21no—
99C4×Q8×C109C2311yescharacteristic
100C4×Q8×C109C2311yescharacteristic
101C4×D4×C109—21no—
102C4×D4×C109—21no—
103#117×C109—21no—
104#117×C109—21no—
105C2×C42×C109C2311yescharacteristic
106#107×C109—21no—
107#107×C109—21no—
108#106×C109—21no—
109#106×C109—21no—
110C4⋊C8×C109—21no—
111C4⋊C8×C109—21no—
112C4⋊C8×C109—21no—
113C4⋊C8×C109—21no—
114C4⋊C8×C109—21no—
115C4⋊C8×C109—21no—
116C4⋊C8×C109C2311yescharacteristic
117C4⋊C8×C109C2311yescharacteristic
118C4⋊C8×C109C2311yescharacteristic
119C4⋊C8×C109C2311yescharacteristic
120C4⋊C8×C109C2311yescharacteristic
121C4⋊C8×C109C2311yescharacteristic
122#103×C109—21no—
123#103×C109—21no—
124#103×C109—21no—
125#103×C109—21no—
126#103×C109—21no—
127#103×C109—21no—
128#103×C109—21no—
129#103×C109—21no—
130#102×C109—21no—
131#102×C109—21no—
132#102×C109—21no—
133#102×C109—21no—
134C22⋊C8×C109—21no—
135#97×C109—21no—
136#97×C109—21no—
137#97×C109—21no—
138#97×C109D411yescharacteristic
139#97×C109D411yescharacteristic
140C4×C8×C109—21no—
141C4×C8×C109—21no—
142C4×C8×C109C2311yescharacteristic
143C4×C8×C109C2311yescharacteristic
144#41×C109—81no—
145#41×C109—81no—
146C2×Q8×C109—21no—
147C2×Q8×C109—21no—
148C2×Q8×C109—21no—
149C2×Q8×C109—21no—
150C2×Q8×C109C2×D411yescharacteristic
151C2×Q8×C109C2×D411yescharacteristic
152C2×D4×C109—21no—
153C2×D4×C109—21no—
154C2×D4×C109—21no—
155C2×D4×C109—41no—
156C22×C4×C109—41no—
157C22×C4×C109C2×D411yescharacteristic
158C22×C4×C109C2×D411yescharacteristic
159C22×C4×C109#4111yescharacteristic
160Q16×C109—81no—
161Q16×C109—81no—
162Q16×C109—81no—
163Q16×C109—81no—
164SD16×C109—42no—
165SD16×C109—42no—
166SD16×C109—81no—
167SD16×C109—81no—
168M16×C109—22no—
169M16×C109—41no—
170M16×C109—81no—
171C2×C8×C109—21no—
172C2×C8×C109—21no—
173C2×C8×C109—21no—
174C2×C8×C109—21no—
175C2×C8×C109—21no—
176C2×C8×C109—21no—
177C2×C8×C109—42no—
178C2×C8×C109C2×D411yescharacteristic
179C2×C8×C109C2×D411yescharacteristic
180C2×C8×C109C2×D411yescharacteristic
181C2×C8×C109C2×D411yescharacteristic
182C4⋊C4×C109—21no—
183C4⋊C4×C109—21no—
184C4⋊C4×C109—21no—
185C4⋊C4×C109—21no—
186C4⋊C4×C109—21no—
187C4⋊C4×C109—21no—
188C4⋊C4×C109—21no—
189C4⋊C4×C109—21no—
190C4⋊C4×C109—21no—
191C4⋊C4×C109—21no—
192C4⋊C4×C109—21no—
193C4⋊C4×C109—21no—
194C4⋊C4×C109—21no—
195C4⋊C4×C109—41no—
196C4⋊C4×C109—41no—
197C4⋊C4×C109C2×D411yescharacteristic
198C4⋊C4×C109C2×D411yescharacteristic
199C22⋊C4×C109—21no—
200C22⋊C4×C109—21no—
201C22⋊C4×C109—41no—
202C22⋊C4×C109—41no—
203C22⋊C4×C109—41no—
204C42×C109—21no—
205C42×C109—21no—
206C42×C109—41no—
207C42×C109C2×D411yescharacteristic
208C42×C109C2×D411yescharacteristic
209C42×C109#4111yescharacteristic
210C42×C109C2411yescharacteristic
211C23×C109—41no—
212C23×C109#12111yescharacteristic
213Q8×C109—21no—
214Q8×C109—21no—
215Q8×C109—21no—
216Q8×C109—21no—
217Q8×C109—41no—
218Q8×C109—41no—
219Q8×C109—41no—
220Q8×C109—41no—
221Q8×C109—41no—
222Q8×C109—41no—
223Q8×C109—81no—
224D4×C109—41no—
225D4×C109—41no—
226D4×C109—41no—
227D4×C109—41no—
228D4×C109—81no—
229D4×C109—82no—
230C2×C4×C109—21no—
231C2×C4×C109—21no—
232C2×C4×C109—21no—
233C2×C4×C109—21no—
234C2×C4×C109—21no—
235C2×C4×C109—21no—
236C2×C4×C109—21no—
237C2×C4×C109—21no—
238C2×C4×C109—41no—
239C2×C4×C109—41no—
240C2×C4×C109—41no—
241C2×C4×C109—41no—
242C2×C4×C109—82no—
243C2×C4×C109#12111yescharacteristic
244C2×C4×C109#12111yescharacteristic
245C2×C4×C109#12111yescharacteristic
246C2×C4×C109C22×D411yescharacteristic
247C2×C4×C109C22×D411yescharacteristic
248C2×C4×C109C2×#4111yescharacteristic
249C8×C109—21no—
250C8×C109—21no—
251C8×C109—21no—
252C8×C109—41no—
253C8×C109—41no—
254C8×C109—41no—
255C8×C109—41no—
256C8×C109—41no—
257C8×C109—41no—
258C8×C109#13712yes—
259C22×C109—21no—
260C22×C109—21no—
261C22×C109—21no—
262C22×C109—81no—
263C22×C109—81no—
264C22×C109—81no—
265C22×C109C2×#12111yescharacteristic
266C4×C109—21no—
267C4×C109—21no—
268C4×C109—21no—
269C4×C109—21no—
270C4×C109—41no—
271C4×C109—41no—
272C4×C109—41no—
273C4×C109—41no—
274C4×C109—81no—
275C4×C109—81no—
276C4×C109—81no—
277C4×C109C2×#13711yescharacteristic
278C4×C109#57511yescharacteristic
279C4×C109#57611yescharacteristic
280C4×C109#57711yescharacteristic
281#19372C10911yesmaximal, characteristic, direct product factor, Sylow
282C2×C109—41no—
283C2×C109—161no—
284C2×C109#303711yescharacteristic
285C2×C109#306211yescharacteristic
286C2×C109#306911yescharacteristic
287#3255C2×C10911yescharacteristic, complemented
288#3216C2×C10911yescharacteristic
289#3182C2×C10911yescharacteristic
290#2962C2×C10911yescharacteristic, complemented
291#1702C2×C10911yescharacteristic, complemented
292#1700C2×C10911yescharacteristic, complemented
293#1694C2×C10911yescharacteristic, complemented
294#1686C2×C10911yescharacteristic, complemented
295#1682C2×C10911yescharacteristic, complemented
296#1680C2×C10911yescharacteristic, complemented
297#1674C2×C10911yescharacteristic, complemented
298#1666C2×C10911yescharacteristic, complemented
299#1569C2×C10911yescharacteristic, complemented
300#1552C2×C10911yescharacteristic, complemented
301#1547C2×C10911yescharacteristic
302C109#1937211yescharacteristic, direct product factor, Sylow
303#533C22×C10911yescharacteristic
304#533C22×C10911yescharacteristic
305#532C22×C10911yescharacteristic
306#500C22×C10911yescharacteristic
307#497—21no—
308#492C22×C10911yescharacteristic
309#489—21no—
310#487C22×C10911yescharacteristic
311#486C22×C10911yescharacteristic
312#475—21no—
313#475C22×C10911yescharacteristic
314#475C22×C10911yescharacteristic
315#470—21no—
316#470—21no—
317#468—21no—
318#468—21no—
319#462—21no—
320#462C22×C10911yescharacteristic
321#462C22×C10911yescharacteristic
322#462C22×C10911yescharacteristic
323#462C22×C10911yescharacteristic
324#460C22×C10911yescharacteristic
325#460C22×C10911yescharacteristic
326#452—21no—
327#452—21no—
328#446—21no—
329#446—21no—
330#446C22×C10911yescharacteristic
331#446C22×C10911yescharacteristic
332C8×Q8C22×C10911yescharacteristic
333C8×Q8C22×C10911yescharacteristic
334#440—21no—
335#440—21no—
336#432C22×C10911yescharacteristic
337#424—21no—
338#423C22×C10911yescharacteristic
339C2×C4⋊C8C22×C10911yescharacteristic
340#405—21no—
341C4×M16C22×C10911yescharacteristic
342#335C22×C10911yescharacteristic
343#335C22×C10911yescharacteristic
344#334C22×C10911yescharacteristic
345#334C22×C10911yescharacteristic
346#332C22×C10911yescharacteristic
347#332C22×C10911yescharacteristic
348#332C22×C10911yescharacteristic
349#332C22×C10911yescharacteristic
350#332C22×C10911yescharacteristic
351#332C22×C10911yescharacteristic
352#331C22×C10911yescharacteristic
353#331C22×C10911yescharacteristic
354C2×#41—21no—
355C2×Q16—21no—
356C2×Q16—21no—
357C2×Q16—21no—
358C2×Q16—21no—
359C2×SD16—21no—
360C2×SD16—21no—
361C2×SD16—21no—
362C2×SD16—21no—
363C2×M16—21no—
364C2×M16D4×C10911yescharacteristic
365C2×M16D4×C10911yescharacteristic
366C22×C8—21no—
367C4⋊Q8D4×C10911yescharacteristic
368C4⋊Q8D4×C10911yescharacteristic
369C4⋊Q8C23×C10911yescharacteristic
370C4⋊Q8C23×C10911yescharacteristic
371C4⋊Q8C23×C10911yescharacteristic
372#127C23×C10911yescharacteristic
373#125—21no—
374#125—21no—
375#124—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8D4×C10911yescharacteristic
380C22⋊Q8D4×C10911yescharacteristic
381#121—21no—
382#121—21no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—21no—
386C4×Q8C23×C10911yescharacteristic
387C4×Q8C23×C10911yescharacteristic
388C4×D4—21no—
389C4×D4—21no—
390#117—21no—
391#117—21no—
392C2×C42C23×C10911yescharacteristic
393#107—21no—
394#107—21no—
395#106—21no—
396#106—21no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8C23×C10911yescharacteristic
404C4⋊C8C23×C10911yescharacteristic
405C4⋊C8C23×C10911yescharacteristic
406C4⋊C8C23×C10911yescharacteristic
407C4⋊C8C23×C10911yescharacteristic
408C4⋊C8C23×C10911yescharacteristic
409#103—21no—
410#103—21no—
411#103—21no—
412#103—21no—
413#103—21no—
414#103—21no—
415#103—21no—
416#103—21no—
417#102—21no—
418#102—21no—
419#102—21no—
420#102—21no—
421C22⋊C8—21no—
422#97—21no—
423#97—21no—
424#97—21no—
425#97D4×C10911yescharacteristic
426#97D4×C10911yescharacteristic
427C4×C8—21no—
428C4×C8—21no—
429C4×C8C23×C10911yescharacteristic
430C4×C8C23×C10911yescharacteristic
431#41—81no—
432#41—81no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8C2×D4×C10911yescharacteristic
438C2×Q8C2×D4×C10911yescharacteristic
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C22×C4—41no—
444C22×C4C2×D4×C10911yescharacteristic
445C22×C4C2×D4×C10911yescharacteristic
446C22×C4#41×C10911yescharacteristic
447Q16—81no—
448Q16—81no—
449Q16—81no—
450Q16—81no—
451SD16—42no—
452SD16—42no—
453SD16—81no—
454SD16—81no—
455M16—22no—
456M16—41no—
457M16—81no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—42no—
465C2×C8C2×D4×C10911yescharacteristic
466C2×C8C2×D4×C10911yescharacteristic
467C2×C8C2×D4×C10911yescharacteristic
468C2×C8C2×D4×C10911yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4C2×D4×C10911yescharacteristic
485C4⋊C4C2×D4×C10911yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C42—21no—
492C42—21no—
493C42—41no—
494C42C2×D4×C10911yescharacteristic
495C42C2×D4×C10911yescharacteristic
496C42#41×C10911yescharacteristic
497C42C24×C10911yescharacteristic
498C23—41no—
499C23#121×C10911yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—21no—
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509Q8—41no—
510Q8—81no—
511D4—41no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—81no—
516D4—82no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—82no—
530C2×C4#121×C10911yescharacteristic
531C2×C4#121×C10911yescharacteristic
532C2×C4#121×C10911yescharacteristic
533C2×C4C22×D4×C10911yescharacteristic
534C2×C4C22×D4×C10911yescharacteristic
535C2×C4C2×#41×C10911yescharacteristic
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8#137×C10912yes—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22C2×#121×C10911yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—41no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4C2×#137×C10911yescharacteristic
565C4#575×C10911yescharacteristic
566C4#576×C10911yescharacteristic
567C4#577×C10911yescharacteristic
568C2—41no—
569C2—161no—
570C2C109×#303711yescharacteristic
571C2C109×#306211yescharacteristic
572C2C109×#306911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#121
commutator, Frattini, cosocleC42C24×C109
prime radical#532×C109C22
LCSG ▷ C42 ▷ 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×#3255C2semidirect
4. C109×#2962C2semidirect
5. C109×#1702C2semidirect
6. C109×#1700C2semidirect
7. C109×#1694C2semidirect
8. C109×#1686C2semidirect
9. C109×#1682C2semidirect
10. C109×#1680C2semidirect
11. C109×#1674C2semidirect
12. C109×#1666C2semidirect
13. C109×#1569C2semidirect
14. C109×#1552C2semidirect
281. #19372C109direct (N × H)
287. #3255C2×C109semidirect
290. #2962C2×C109semidirect
291. #1702C2×C109semidirect
292. #1700C2×C109semidirect
293. #1694C2×C109semidirect
294. #1686C2×C109semidirect
295. #1682C2×C109semidirect
296. #1680C2×C109semidirect
297. #1674C2×C109semidirect
298. #1666C2×C109semidirect
299. #1569C2×C109semidirect
300. #1552C2×C109semidirect
302. C109#19372direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32552
C109×#32162
C109×#31822
C109×#29622
C109×#17022
C109×#17002
C109×#16942
C109×#16862
C109×#16822
C109×#16802
C109×#16742
C109×#16662
C109×#15692
C109×#15522
C109×#15472
#19372109