Finite Group Databasebeta

← C109×#19375 #42137369

Subgroups — C109×#19375 #42137369

Order 27904.

554 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3255C211yesmaximal, characteristic
2C109×#3237C211yesmaximal, characteristic
3C109×#3185C211yesmaximal, characteristic
4C109×#2962C211yesmaximal, characteristic
5C109×#1704C211yesmaximal, characteristic, complemented
6C109×#1700C211yesmaximal, characteristic, complemented
7C109×#1692C211yesmaximal, characteristic, complemented
8C109×#1690C211yesmaximal, characteristic, complemented
9C109×#1684C211yesmaximal, characteristic, complemented
10C109×#1680C211yesmaximal, characteristic, complemented
11C109×#1672C211yesmaximal, characteristic, complemented
12C109×#1670C211yesmaximal, characteristic, complemented
13C109×#1569C211yesmaximal, characteristic
14C109×#1552C211yesmaximal, characteristic
15C109×#1550C211yesmaximal, characteristic
16#533×C109C2211yescharacteristic
17#533×C109C2211yescharacteristic
18#533×C109C2211yescharacteristic
19#501×C109—21no—
20#498×C109C2211yescharacteristic
21#494×C109C2211yescharacteristic
22#491×C109—21no—
23#487×C109C2211yescharacteristic
24#487×C109C2211yescharacteristic
25#485×C109—21no—
26#485×C109—21no—
27#477×C109C2211yescharacteristic
28#477×C109C2211yescharacteristic
29#475×C109—21no—
30#475×C109C2211yescharacteristic
31#475×C109C2211yescharacteristic
32#475×C109C2211yescharacteristic
33#475×C109C2211yescharacteristic
34#470×C109—21no—
35#470×C109—21no—
36#470×C109—21no—
37#470×C109—21no—
38#462×C109—21no—
39#462×C109C2211yescharacteristic
40#462×C109C2211yescharacteristic
41#446×C109—21no—
42#446×C109—21no—
43#446×C109C2211yescharacteristic
44#446×C109C2211yescharacteristic
45C8×Q8×C109C2211yescharacteristic
46C8×Q8×C109C2211yescharacteristic
47#441×C109—21no—
48#441×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#333×C109C2211yescharacteristic
60#333×C109C2211yescharacteristic
61#332×C109C2211yescharacteristic
62#332×C109C2211yescharacteristic
63#332×C109C2211yescharacteristic
64#332×C109C2211yescharacteristic
65#332×C109C2211yescharacteristic
66#332×C109C2211yescharacteristic
67C2×Q16×C109—21no—
68C2×Q16×C109—21no—
69C2×Q16×C109—21no—
70C2×Q16×C109—21no—
71C2×M16×C109—21no—
72C2×M16×C109D411yescharacteristic
73C2×M16×C109D411yescharacteristic
74C22×C8×C109—21no—
75C4⋊Q8×C109D411yescharacteristic
76C4⋊Q8×C109D411yescharacteristic
77C4⋊Q8×C109C2311yescharacteristic
78C4⋊Q8×C109C2311yescharacteristic
79C4⋊Q8×C109C2311yescharacteristic
80C4⋊Q8×C109C2311yescharacteristic
81#125×C109—21no—
82#125×C109—21no—
83#125×C109—21no—
84C22⋊Q8×C109—21no—
85C22⋊Q8×C109—21no—
86C22⋊Q8×C109—21no—
87C22⋊Q8×C109—21no—
88C22⋊Q8×C109—21no—
89C22⋊Q8×C109D411yescharacteristic
90C22⋊Q8×C109D411yescharacteristic
91C4×Q8×C109—21no—
92C4×Q8×C109—21no—
93C4×Q8×C109—21no—
94C4×Q8×C109—21no—
95C4×Q8×C109—21no—
96C4×Q8×C109C2311yescharacteristic
97C4×Q8×C109C2311yescharacteristic
98#117×C109—21no—
99#117×C109—21no—
100#117×C109—21no—
101C2×C42×C109C2311yescharacteristic
102#107×C109—21no—
103#107×C109—21no—
104#106×C109—21no—
105#106×C109—21no—
106#106×C109—21no—
107#106×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#103×C109—21no—
131#103×C109—21no—
132#103×C109—21no—
133#103×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
144C2×Q8×C109—21no—
145C2×Q8×C109—21no—
146C2×Q8×C109—21no—
147C2×Q8×C109—21no—
148C2×Q8×C109—21no—
149C2×Q8×C109C2×D411yescharacteristic
150C2×Q8×C109C2×D411yescharacteristic
151C22×C4×C109C2×D411yescharacteristic
152C22×C4×C109C2×D411yescharacteristic
153C22×C4×C109#4111yescharacteristic
154Q16×C109—42no—
155Q16×C109—42no—
156Q16×C109—81no—
157Q16×C109—81no—
158M16×C109—22no—
159M16×C109—41no—
160M16×C109—81no—
161C2×C8×C109—21no—
162C2×C8×C109—21no—
163C2×C8×C109—21no—
164C2×C8×C109—21no—
165C2×C8×C109—21no—
166C2×C8×C109—21no—
167C2×C8×C109—42no—
168C2×C8×C109C2×D411yescharacteristic
169C2×C8×C109C2×D411yescharacteristic
170C2×C8×C109C2×D411yescharacteristic
171C2×C8×C109C2×D411yescharacteristic
172C4⋊C4×C109—21no—
173C4⋊C4×C109—21no—
174C4⋊C4×C109—21no—
175C4⋊C4×C109—21no—
176C4⋊C4×C109—21no—
177C4⋊C4×C109—21no—
178C4⋊C4×C109—21no—
179C4⋊C4×C109—21no—
180C4⋊C4×C109—21no—
181C4⋊C4×C109—21no—
182C4⋊C4×C109—21no—
183C4⋊C4×C109—21no—
184C4⋊C4×C109—21no—
185C4⋊C4×C109—21no—
186C4⋊C4×C109—21no—
187C4⋊C4×C109—41no—
188C4⋊C4×C109—41no—
189C4⋊C4×C109—41no—
190C4⋊C4×C109—41no—
191C4⋊C4×C109C2×D411yescharacteristic
192C4⋊C4×C109C2×D411yescharacteristic
193C22⋊C4×C109—21no—
194C22⋊C4×C109—21no—
195C22⋊C4×C109—41no—
196C22⋊C4×C109—41no—
197C42×C109—21no—
198C42×C109—21no—
199C42×C109—41no—
200C42×C109—41no—
201C42×C109C2×D411yescharacteristic
202C42×C109C2×D411yescharacteristic
203C42×C109#4111yescharacteristic
204C42×C109C2411yescharacteristic
205C23×C109#12111yescharacteristic
206Q8×C109—21no—
207Q8×C109—21no—
208Q8×C109—21no—
209Q8×C109—21no—
210Q8×C109—41no—
211Q8×C109—41no—
212Q8×C109—41no—
213Q8×C109—41no—
214Q8×C109—41no—
215Q8×C109—41no—
216Q8×C109—81no—
217Q8×C109—81no—
218C2×C4×C109—21no—
219C2×C4×C109—21no—
220C2×C4×C109—21no—
221C2×C4×C109—21no—
222C2×C4×C109—21no—
223C2×C4×C109—21no—
224C2×C4×C109—21no—
225C2×C4×C109—21no—
226C2×C4×C109—41no—
227C2×C4×C109—41no—
228C2×C4×C109—41no—
229C2×C4×C109—41no—
230C2×C4×C109—41no—
231C2×C4×C109#12111yescharacteristic
232C2×C4×C109#12111yescharacteristic
233C2×C4×C109#12111yescharacteristic
234C2×C4×C109C22×D411yescharacteristic
235C2×C4×C109C22×D411yescharacteristic
236C2×C4×C109C2×#4111yescharacteristic
237C8×C109—21no—
238C8×C109—21no—
239C8×C109—21no—
240C8×C109—41no—
241C8×C109—41no—
242C8×C109—41no—
243C8×C109—41no—
244C8×C109—41no—
245C8×C109—41no—
246C8×C109#13712yes—
247C22×C109—21no—
248C22×C109—21no—
249C22×C109—21no—
250C22×C109C2×#12111yescharacteristic
251C4×C109—21no—
252C4×C109—21no—
253C4×C109—21no—
254C4×C109—21no—
255C4×C109—41no—
256C4×C109—41no—
257C4×C109—41no—
258C4×C109—41no—
259C4×C109—81no—
260C4×C109—81no—
261C4×C109—81no—
262C4×C109—81no—
263C4×C109C2×#13711yescharacteristic
264C4×C109#57511yescharacteristic
265C4×C109#57711yescharacteristic
266C4×C109#57811yescharacteristic
267#19375C10911yesmaximal, characteristic, direct product factor, Sylow
268C2×C109—41no—
269C2×C109#303711yescharacteristic
270C2×C109#306311yescharacteristic
271C2×C109#306911yescharacteristic
272#3255C2×C10911yescharacteristic
273#3237C2×C10911yescharacteristic
274#3185C2×C10911yescharacteristic
275#2962C2×C10911yescharacteristic
276#1704C2×C10911yescharacteristic, complemented
277#1700C2×C10911yescharacteristic, complemented
278#1692C2×C10911yescharacteristic, complemented
279#1690C2×C10911yescharacteristic, complemented
280#1684C2×C10911yescharacteristic, complemented
281#1680C2×C10911yescharacteristic, complemented
282#1672C2×C10911yescharacteristic, complemented
283#1670C2×C10911yescharacteristic, complemented
284#1569C2×C10911yescharacteristic
285#1552C2×C10911yescharacteristic
286#1550C2×C10911yescharacteristic
287C109#1937511yescharacteristic, direct product factor, Sylow
288#533C22×C10911yescharacteristic
289#533C22×C10911yescharacteristic
290#533C22×C10911yescharacteristic
291#501—21no—
292#498C22×C10911yescharacteristic
293#494C22×C10911yescharacteristic
294#491—21no—
295#487C22×C10911yescharacteristic
296#487C22×C10911yescharacteristic
297#485—21no—
298#485—21no—
299#477C22×C10911yescharacteristic
300#477C22×C10911yescharacteristic
301#475—21no—
302#475C22×C10911yescharacteristic
303#475C22×C10911yescharacteristic
304#475C22×C10911yescharacteristic
305#475C22×C10911yescharacteristic
306#470—21no—
307#470—21no—
308#470—21no—
309#470—21no—
310#462—21no—
311#462C22×C10911yescharacteristic
312#462C22×C10911yescharacteristic
313#446—21no—
314#446—21no—
315#446C22×C10911yescharacteristic
316#446C22×C10911yescharacteristic
317C8×Q8C22×C10911yescharacteristic
318C8×Q8C22×C10911yescharacteristic
319#441—21no—
320#441—21no—
321#432C22×C10911yescharacteristic
322#424—21no—
323#423C22×C10911yescharacteristic
324C2×C4⋊C8C22×C10911yescharacteristic
325#405—21no—
326C4×M16C22×C10911yescharacteristic
327#335C22×C10911yescharacteristic
328#335C22×C10911yescharacteristic
329#334C22×C10911yescharacteristic
330#334C22×C10911yescharacteristic
331#333C22×C10911yescharacteristic
332#333C22×C10911yescharacteristic
333#332C22×C10911yescharacteristic
334#332C22×C10911yescharacteristic
335#332C22×C10911yescharacteristic
336#332C22×C10911yescharacteristic
337#332C22×C10911yescharacteristic
338#332C22×C10911yescharacteristic
339C2×Q16—21no—
340C2×Q16—21no—
341C2×Q16—21no—
342C2×Q16—21no—
343C2×M16—21no—
344C2×M16D4×C10911yescharacteristic
345C2×M16D4×C10911yescharacteristic
346C22×C8—21no—
347C4⋊Q8D4×C10911yescharacteristic
348C4⋊Q8D4×C10911yescharacteristic
349C4⋊Q8C23×C10911yescharacteristic
350C4⋊Q8C23×C10911yescharacteristic
351C4⋊Q8C23×C10911yescharacteristic
352C4⋊Q8C23×C10911yescharacteristic
353#125—21no—
354#125—21no—
355#125—21no—
356C22⋊Q8—21no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361C22⋊Q8D4×C10911yescharacteristic
362C22⋊Q8D4×C10911yescharacteristic
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8C23×C10911yescharacteristic
369C4×Q8C23×C10911yescharacteristic
370#117—21no—
371#117—21no—
372#117—21no—
373C2×C42C23×C10911yescharacteristic
374#107—21no—
375#107—21no—
376#106—21no—
377#106—21no—
378#106—21no—
379#106—21no—
380#106—21no—
381#106—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—21no—
388C4⋊C8C23×C10911yescharacteristic
389C4⋊C8C23×C10911yescharacteristic
390C4⋊C8C23×C10911yescharacteristic
391C4⋊C8C23×C10911yescharacteristic
392C4⋊C8C23×C10911yescharacteristic
393C4⋊C8C23×C10911yescharacteristic
394#103—21no—
395#103—21no—
396#103—21no—
397#103—21no—
398#103—21no—
399#103—21no—
400#103—21no—
401#103—21no—
402#103—21no—
403#103—21no—
404#103—21no—
405#103—21no—
406C22⋊C8—21no—
407#97—21no—
408#97—21no—
409#97—21no—
410#97D4×C10911yescharacteristic
411#97D4×C10911yescharacteristic
412C4×C8—21no—
413C4×C8—21no—
414C4×C8C23×C10911yescharacteristic
415C4×C8C23×C10911yescharacteristic
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×D4×C10911yescharacteristic
422C2×Q8C2×D4×C10911yescharacteristic
423C22×C4C2×D4×C10911yescharacteristic
424C22×C4C2×D4×C10911yescharacteristic
425C22×C4#41×C10911yescharacteristic
426Q16—42no—
427Q16—42no—
428Q16—81no—
429Q16—81no—
430M16—22no—
431M16—41no—
432M16—81no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—42no—
440C2×C8C2×D4×C10911yescharacteristic
441C2×C8C2×D4×C10911yescharacteristic
442C2×C8C2×D4×C10911yescharacteristic
443C2×C8C2×D4×C10911yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4C2×D4×C10911yescharacteristic
464C4⋊C4C2×D4×C10911yescharacteristic
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C42—21no—
470C42—21no—
471C42—41no—
472C42—41no—
473C42C2×D4×C10911yescharacteristic
474C42C2×D4×C10911yescharacteristic
475C42#41×C10911yescharacteristic
476C42C24×C10911yescharacteristic
477C23#121×C10911yescharacteristic
478Q8—21no—
479Q8—21no—
480Q8—21no—
481Q8—21no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—81no—
489Q8—81no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4#121×C10911yescharacteristic
504C2×C4#121×C10911yescharacteristic
505C2×C4#121×C10911yescharacteristic
506C2×C4C22×D4×C10911yescharacteristic
507C2×C4C22×D4×C10911yescharacteristic
508C2×C4C2×#41×C10911yescharacteristic
509C8—21no—
510C8—21no—
511C8—21no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8#137×C10912yes—
519C22—21no—
520C22—21no—
521C22—21no—
522C22C2×#121×C10911yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—41no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4C2×#137×C10911yescharacteristic
536C4#575×C10911yescharacteristic
537C4#577×C10911yescharacteristic
538C4#578×C10911yescharacteristic
539C2—41no—
540C2C109×#303711yescharacteristic
541C2C109×#306311yescharacteristic
542C2C109×#306911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#121
commutator, Frattini, cosocleC42C24×C109
prime radicalC23×C109#121
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
5. C109×#1704C2semidirect
6. C109×#1700C2semidirect
7. C109×#1692C2semidirect
8. C109×#1690C2semidirect
9. C109×#1684C2semidirect
10. C109×#1680C2semidirect
11. C109×#1672C2semidirect
12. C109×#1670C2semidirect
267. #19375C109direct (N × H)
276. #1704C2×C109semidirect
277. #1700C2×C109semidirect
278. #1692C2×C109semidirect
279. #1690C2×C109semidirect
280. #1684C2×C109semidirect
281. #1680C2×C109semidirect
282. #1672C2×C109semidirect
283. #1670C2×C109semidirect
287. C109#19375direct (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×#32372
C109×#31852
C109×#29622
C109×#17042
C109×#17002
C109×#16922
C109×#16902
C109×#16842
C109×#16802
C109×#16722
C109×#16702
C109×#15692
C109×#15522
C109×#15502
#19375109