Finite Group Databasebeta

← C109×#12755 #42130749

Subgroups — C109×#12755 #42130749

Order 27904.

864 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3537C211yesmaximal, characteristic
2C109×#2134C211yesmaximal, characteristic, complemented
3C109×#2122C211yesmaximal, characteristic, complemented
4C109×#2120C211yesmaximal, characteristic, complemented
5C109×#1914C211yesmaximal, characteristic, complemented
6C109×#1910C211yesmaximal, characteristic, complemented
7C109×#1891C211yesmaximal, characteristic, complemented
8#568×C109C2211yescharacteristic, complemented
9#560×C109—21no—
10#560×C109C412yescomplemented
11#553×C109—21no—
12#553×C109—21no—
13D42×C109—21no—
14D42×C109—21no—
15D42×C109—21no—
16D42×C109C412yescomplemented
17#540×C109—21no—
18#538×C109—22no—
19#535×C109C2211yescharacteristic
20#535×C109C2211yescharacteristic
21#534×C109—22no—
22#525×C109C2211yescharacteristic
23C2×#121×C109—21no—
24#418×C109—21no—
25#413×C109C2211yescharacteristic, complemented
26#413×C109C2211yescharacteristic, complemented
27C2×#100×C109—21no—
28#410×C109C2211yescharacteristic, complemented
29#407×C109—21no—
30#355×C109—22no—
31#352×C109—21no—
32AΣL(1,𝔽16)×C109—21no—
33AΣL(1,𝔽16)×C109—22no—
34#344×C109—21no—
35#344×C109—21no—
36H(3,ℤ/4)×C109—21no—
37C2×#41×C109C2311yescharacteristic
38C22×D4×C109—21no—
39C22×D4×C109—22no—
40C22×D4×C109—22no—
41C22×D4×C109—22no—
42C22×D4×C109D411yescharacteristic
43C22×D4×C109D411yescharacteristic
44C23×C4×C109D411yescharacteristic
45#131×C109—42no—
46C2×M16×C109—21no—
47C2×M16×C109—21no—
48C2×M16×C109—21no—
49C2×M16×C109—21no—
50C2×M16×C109—21no—
51#127×C109—21no—
52#127×C109—21no—
53#127×C109C2×C412yes—
54#126×C109—22no—
55#126×C109—22no—
56#125×C109—21no—
57#124×C109—21no—
58#124×C109D412yes—
59#123×C109—22no—
60#123×C109—22no—
61#123×C109—22no—
62#123×C109—22no—
63#123×C109C2×C411yescharacteristic
64#123×C109D411yescharacteristic
65C22⋊Q8×C109—21no—
66#121×C109—22no—
67#121×C109—21no—
68#121×C109—21no—
69#121×C109—22no—
70#121×C109—22no—
71#121×C109—22no—
72#121×C109—21no—
73#121×C109—22no—
74#121×C109—21no—
75#121×C109—21no—
76#121×C109—21no—
77#121×C109—21no—
78#121×C109—22no—
79#121×C109—22no—
80#121×C109—22no—
81#120×C109—22no—
82#120×C109—22no—
83#120×C109—22no—
84#120×C109—22no—
85#120×C109—21no—
86#120×C109—21no—
87#120×C109C2×C411yescharacteristic
88#120×C109C2×C412yes—
89#120×C109D411yescharacteristic
90#120×C109D412yes—
91C4×D4×C109—22no—
92C4×D4×C109—21no—
93C4×D4×C109—21no—
94C4×D4×C109—21no—
95C4×D4×C109—21no—
96C4×D4×C109—22no—
97#117×C109—21no—
98#117×C109D411yescharacteristic
99#117×C109D411yescharacteristic
100#117×C109D411yescharacteristic
101C2×C4⋊C4×C109—21no—
102C2×C22⋊C4×C109—22no—
103#102×C109—42no—
104#101×C109—21no—
105#101×C109—21no—
106#100×C109—21no—
107#100×C109—21no—
108#100×C109—42no—
109#99×C109—21no—
110#99×C109—21no—
111C22⋊C8×C109—42no—
112C24×C109—22no—
113C24×C109—21no—
114C24×C109C22⋊C411yescharacteristic
115C24×C109C22⋊C412yes—
116#41×C109—41no—
117#41×C109—41no—
118C2×Q8×C109C2×D411yescharacteristic
119C2×D4×C109—22no—
120C2×D4×C109—22no—
121C2×D4×C109—21no—
122C2×D4×C109—21no—
123C2×D4×C109—22no—
124C2×D4×C109—22no—
125C2×D4×C109—22no—
126C2×D4×C109—22no—
127C2×D4×C109—22no—
128C2×D4×C109—22no—
129C2×D4×C109—22no—
130C2×D4×C109—21no—
131C2×D4×C109—21no—
132C2×D4×C109—21no—
133C2×D4×C109—41no—
134C2×D4×C109—41no—
135C2×D4×C109—41no—
136C2×D4×C109—41no—
137C2×D4×C109—81no—
138C2×D4×C109—81no—
139C2×D4×C109—84no—
140C2×D4×C109—84no—
141C2×D4×C109—84no—
142C2×D4×C109C22⋊C412yes—
143C2×D4×C109C22×C411yescharacteristic
144C2×D4×C109C2×D411yescharacteristic
145C2×D4×C109C2×D411yescharacteristic
146C22×C4×C109—22no—
147C22×C4×C109—22no—
148C22×C4×C109—22no—
149C22×C4×C109—21no—
150C22×C4×C109—21no—
151C22×C4×C109—41no—
152C22×C4×C109—41no—
153C22×C4×C109C22⋊C411yescharacteristic
154C22×C4×C109C2×D411yescharacteristic
155C22×C4×C109C2×D411yescharacteristic
156C22×C4×C109C2×D411yescharacteristic
157M16×C109—22no—
158M16×C109—41no—
159M16×C109—41no—
160M16×C109—41no—
161M16×C109—41no—
162M16×C109—41no—
163M16×C109—41no—
164M16×C109—41no—
165C2×C8×C109—22no—
166C2×C8×C109—22no—
167C2×C8×C109—41no—
168C2×C8×C109—41no—
169C2×C8×C109—41no—
170C4⋊C4×C109—22no—
171C4⋊C4×C109—22no—
172C4⋊C4×C109—22no—
173C4⋊C4×C109—21no—
174C4⋊C4×C109—21no—
175C4⋊C4×C109—21no—
176C4⋊C4×C109—21no—
177C4⋊C4×C109—21no—
178C4⋊C4×C109—41no—
179C22⋊C4×C109—22no—
180C22⋊C4×C109—22no—
181C22⋊C4×C109—22no—
182C22⋊C4×C109—22no—
183C22⋊C4×C109—21no—
184C22⋊C4×C109—21no—
185C22⋊C4×C109—21no—
186C22⋊C4×C109—21no—
187C22⋊C4×C109—22no—
188C22⋊C4×C109—22no—
189C22⋊C4×C109—22no—
190C22⋊C4×C109—21no—
191C22⋊C4×C109—22no—
192C22⋊C4×C109—41no—
193C22⋊C4×C109C22⋊C412yes—
194C22⋊C4×C109C22⋊C411yescharacteristic
195C22⋊C4×C109C22⋊C411yescharacteristic
196C42×C109—21no—
197C42×C109—21no—
198C42×C109—41no—
199C42×C109C22⋊C412yes—
200C23×C109—21no—
201C23×C109—22no—
202C23×C109—22no—
203C23×C109—41no—
204C23×C109—42no—
205C23×C109—44no—
206C23×C109—42no—
207C23×C109—81no—
208C23×C109—82no—
209C23×C109—82no—
210C23×C109—82no—
211C23×C109—82no—
212C23×C109C2×C22⋊C411yescharacteristic
213C23×C109C2×C22⋊C411yescharacteristic
214C23×C109#12011yescharacteristic
215Q8×C109—41no—
216D4×C109—41no—
217D4×C109—41no—
218D4×C109—41no—
219D4×C109—42no—
220D4×C109—42no—
221D4×C109—42no—
222D4×C109—44no—
223D4×C109—44no—
224D4×C109—41no—
225D4×C109—41no—
226D4×C109—41no—
227D4×C109—41no—
228D4×C109—81no—
229D4×C109—82no—
230D4×C109—82no—
231D4×C109—82no—
232D4×C109—82no—
233D4×C109—82no—
234D4×C109—82no—
235D4×C109—82no—
236C2×C4×C109—21no—
237C2×C4×C109—21no—
238C2×C4×C109—21no—
239C2×C4×C109—21no—
240C2×C4×C109—21no—
241C2×C4×C109—21no—
242C2×C4×C109—22no—
243C2×C4×C109—21no—
244C2×C4×C109—21no—
245C2×C4×C109—22no—
246C2×C4×C109—21no—
247C2×C4×C109—41no—
248C2×C4×C109—41no—
249C2×C4×C109—41no—
250C2×C4×C109—81no—
251C2×C4×C109—82no—
252C2×C4×C109—82no—
253C2×C4×C109—82no—
254C2×C4×C109—81no—
255C2×C4×C109C2×C22⋊C411yescharacteristic
256C2×C4×C109#12011yescharacteristic
257C2×C4×C109#12011yescharacteristic
258C2×C4×C109#12011yescharacteristic
259C8×C109—42no—
260C8×C109—81no—
261C8×C109—81no—
262C22×C109—22no—
263C22×C109—21no—
264C22×C109—21no—
265C22×C109—21no—
266C22×C109—41no—
267C22×C109—41no—
268C22×C109—41no—
269C22×C109—41no—
270C22×C109—42no—
271C22×C109—42no—
272C22×C109—42no—
273C22×C109—42no—
274C22×C109—42no—
275C22×C109—41no—
276C22×C109—42no—
277C22×C109—42no—
278C22×C109—81no—
279C22×C109—162no—
280C22×C109—164no—
281C22×C109—162no—
282C22×C109#37911yescharacteristic
283C4×C109—21no—
284C4×C109—21no—
285C4×C109—21no—
286C4×C109—42no—
287C4×C109—41no—
288C4×C109—41no—
289C4×C109—42no—
290C4×C109—41no—
291C4×C109—161no—
292C4×C109B(4,𝔽2)11yescharacteristic
293C4×C109#45811yescharacteristic
294#12755C10911yesmaximal, characteristic, direct product factor, Sylow
295C2×C109—21no—
296C2×C109—41no—
297C2×C109—41no—
298C2×C109—42no—
299C2×C109—41no—
300C2×C109—82no—
301C2×C109—82no—
302C2×C109#179011yescharacteristic
303#3537C2×C10911yescharacteristic
304#2134C2×C10911yescharacteristic, complemented
305#2122C2×C10911yescharacteristic, complemented
306#2120C2×C10911yescharacteristic, complemented
307#1914C2×C10911yescharacteristic, complemented
308#1910C2×C10911yescharacteristic, complemented
309#1891C2×C10911yescharacteristic, complemented
310C109#1275511yescharacteristic, direct product factor, Sylow
311#568C22×C10911yescharacteristic, complemented
312#560—21no—
313#560C4×C10912yescomplemented
314#553—21no—
315#553—21no—
316D42—21no—
317D42—21no—
318D42—21no—
319D42C4×C10912yescomplemented
320#540—21no—
321#538—22no—
322#535C22×C10911yescharacteristic
323#535C22×C10911yescharacteristic
324#534—22no—
325#525C22×C10911yescharacteristic
326C2×#121—21no—
327#418—21no—
328#413C22×C10911yescharacteristic, complemented
329#413C22×C10911yescharacteristic, complemented
330C2×#100—21no—
331#410C22×C10911yescharacteristic, complemented
332#407—21no—
333#355—22no—
334#352—21no—
335AΣL(1,𝔽16)—21no—
336AΣL(1,𝔽16)—22no—
337#344—21no—
338#344—21no—
339H(3,ℤ/4)—21no—
340C2×#41C23×C10911yescharacteristic
341C22×D4—21no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4—22no—
345C22×D4D4×C10911yescharacteristic
346C22×D4D4×C10911yescharacteristic
347C23×C4D4×C10911yescharacteristic
348#131—42no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354#127—21no—
355#127—21no—
356#127C2×C4×C10912yes—
357#126—22no—
358#126—22no—
359#125—21no—
360#124—21no—
361#124D4×C10912yes—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—22no—
366#123C2×C4×C10911yescharacteristic
367#123D4×C10911yescharacteristic
368C22⋊Q8—21no—
369#121—22no—
370#121—21no—
371#121—21no—
372#121—22no—
373#121—22no—
374#121—22no—
375#121—21no—
376#121—22no—
377#121—21no—
378#121—21no—
379#121—21no—
380#121—21no—
381#121—22no—
382#121—22no—
383#121—22no—
384#120—22no—
385#120—22no—
386#120—22no—
387#120—22no—
388#120—21no—
389#120—21no—
390#120C2×C4×C10911yescharacteristic
391#120C2×C4×C10912yes—
392#120D4×C10911yescharacteristic
393#120D4×C10912yes—
394C4×D4—22no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—22no—
400#117—21no—
401#117D4×C10911yescharacteristic
402#117D4×C10911yescharacteristic
403#117D4×C10911yescharacteristic
404C2×C4⋊C4—21no—
405C2×C22⋊C4—22no—
406#102—42no—
407#101—21no—
408#101—21no—
409#100—21no—
410#100—21no—
411#100—42no—
412#99—21no—
413#99—21no—
414C22⋊C8—42no—
415C24—22no—
416C24—21no—
417C24C22⋊C4×C10911yescharacteristic
418C24C22⋊C4×C10912yes—
419#41—41no—
420#41—41no—
421C2×Q8C2×D4×C10911yescharacteristic
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4C22⋊C4×C10912yes—
446C2×D4C22×C4×C10911yescharacteristic
447C2×D4C2×D4×C10911yescharacteristic
448C2×D4C2×D4×C10911yescharacteristic
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4C22⋊C4×C10911yescharacteristic
457C22×C4C2×D4×C10911yescharacteristic
458C22×C4C2×D4×C10911yescharacteristic
459C22×C4C2×D4×C10911yescharacteristic
460M16—22no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4C22⋊C4×C10912yes—
497C22⋊C4C22⋊C4×C10911yescharacteristic
498C22⋊C4C22⋊C4×C10911yescharacteristic
499C42—21no—
500C42—21no—
501C42—41no—
502C42C22⋊C4×C10912yes—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—81no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23C2×C22⋊C4×C10911yescharacteristic
516C23C2×C22⋊C4×C10911yescharacteristic
517C23#120×C10911yescharacteristic
518Q8—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—44no—
526D4—44no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—81no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4C2×C22⋊C4×C10911yescharacteristic
559C2×C4#120×C10911yescharacteristic
560C2×C4#120×C10911yescharacteristic
561C2×C4#120×C10911yescharacteristic
562C8—42no—
563C8—81no—
564C8—81no—
565C22—22no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—162no—
583C22—164no—
584C22—162no—
585C22#379×C10911yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—161no—
595C4B(4,𝔽2)×C10911yescharacteristic
596C4#458×C10911yescharacteristic
597C2—21no—
598C2—41no—
599C2—42no—
600C2—41no—
601C2—41no—
602C2—82no—
603C2—82no—
604C2C109×#179011yescharacteristic
subgroupG/H
center, socleC2×C109#1790
commutatorC2×D4C22×C4×C109
Frattini, cosocleC2×#41C23×C109
prime radicalC109×#3537C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×#41×C109 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C109×#2134C2semidirect
3. C109×#2122C2semidirect
4. C109×#2120C2semidirect
5. C109×#1914C2semidirect
6. C109×#1910C2semidirect
7. C109×#1891C2semidirect
8. #568×C109C22semidirect
10. #560×C109C4semidirect
16. D42×C109C4semidirect
25. #413×C109C22semidirect
26. #413×C109C22semidirect
28. #410×C109C22semidirect
294. #12755C109direct (N × H)
304. #2134C2×C109semidirect
305. #2122C2×C109semidirect
306. #2120C2×C109semidirect
307. #1914C2×C109semidirect
308. #1910C2×C109semidirect
309. #1891C2×C109semidirect
310. C109#12755direct (N × H)
311. #568C22×C109semidirect
313. #560C4×C109semidirect
319. D42C4×C109semidirect
328. #413C22×C109semidirect
329. #413C22×C109semidirect
331. #410C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35372
C109×#21342
C109×#21222
C109×#21202
C109×#19142
C109×#19102
C109×#18912
#12755109