Finite Group Databasebeta

← C109×#10322 #42128316

Subgroups — C109×#10322 #42128316

Order 27904.

988 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3187C211yesmaximal, characteristic, complemented
2C109×#2848C211yesmaximal, characteristic, complemented
3C109×#2847C211yesmaximal, characteristic
4C109×#1779C211yesmaximal, characteristic, complemented
5C109×#1520C211yesmaximal, characteristic, complemented
6C109×#1519C211yesmaximal, characteristic, complemented
7C109×#1289C211yesmaximal, characteristic, complemented
8C24×C4×C109C2211yescharacteristic
9C2×C22⋊Q8×C109—21no—
10C2×C22⋊Q8×C109C2211yescharacteristic
11C2×#121×C109—21no—
12C2×#121×C109C2211yescharacteristic
13C2×#120×C109—21no—
14#483×C109—21no—
15#481×C109—21no—
16#465×C109—22no—
17#450×C109—21no—
18#448×C109—21no—
19C2×C22⋊C8×C109—22no—
20C2×C22⋊C8×C109C2211yescharacteristic
21C2×C22⋊C8×C109C2211yescharacteristic
22#386×C109—21no—
23#386×C109—22no—
24#385×C109—21no—
25#385×C109—22no—
26#380×C109—21no—
27#380×C109—21no—
28#380×C109C2211yescharacteristic
29#380×C109C2211yescharacteristic
30#379×C109—21no—
31#328×C109—22no—
32#327×C109—22no—
33C25×C109D411yescharacteristic
34C22×Q8×C109D411yescharacteristic, complemented
35C22×D4×C109—22no—
36C22×D4×C109—21no—
37C22×D4×C109D411yescharacteristic
38C23×C4×C109—21no—
39C23×C4×C109—21no—
40C23×C4×C109—21no—
41C23×C4×C109—21no—
42C23×C4×C109—21no—
43C23×C4×C109—21no—
44C23×C4×C109—44no—
45C23×C4×C109D411yescharacteristic
46C23×C4×C109C2311yescharacteristic
47C2×SD16×C109—22no—
48C22×C8×C109—21no—
49C22×C8×C109D412yescomplemented
50C22⋊Q8×C109—21no—
51C22⋊Q8×C109—21no—
52C22⋊Q8×C109—21no—
53C22⋊Q8×C109—21no—
54C22⋊Q8×C109—44no—
55#121×C109—21no—
56#121×C109—21no—
57#121×C109—21no—
58#121×C109—21no—
59#121×C109—44no—
60#120×C109—42no—
61#120×C109—42no—
62C2×C4⋊C4×C109—22no—
63C2×C4⋊C4×C109—21no—
64C2×C4⋊C4×C109—21no—
65C2×C4⋊C4×C109—21no—
66C2×C4⋊C4×C109D411yescharacteristic
67C2×C4⋊C4×C109D411yescharacteristic, complemented
68C2×C22⋊C4×C109—22no—
69C2×C22⋊C4×C109—22no—
70C2×C22⋊C4×C109—22no—
71C2×C22⋊C4×C109—22no—
72C2×C22⋊C4×C109—21no—
73C2×C22⋊C4×C109—21no—
74C2×C22⋊C4×C109—21no—
75C2×C22⋊C4×C109—21no—
76#106×C109—22no—
77#103×C109—22no—
78#102×C109—22no—
79C22⋊C8×C109—21no—
80C22⋊C8×C109—21no—
81C22⋊C8×C109—21no—
82C22⋊C8×C109—21no—
83C22⋊C8×C109—44no—
84#95×C109—22no—
85#95×C109—22no—
86#95×C109D412yes—
87#95×C109D412yescomplemented
88C24×C109—21no—
89C24×C109—21no—
90C24×C109—21no—
91C24×C109—21no—
92C24×C109—21no—
93C24×C109—21no—
94C24×C109—21no—
95C24×C109—42no—
96C24×C109—42no—
97C24×C109SD1612yescomplemented
98C24×C109C2×D411yescharacteristic
99C2×Q8×C109—21no—
100C2×Q8×C109—21no—
101C2×Q8×C109—42no—
102C2×D4×C109—21no—
103C2×D4×C109—21no—
104C2×D4×C109—42no—
105C2×D4×C109—44no—
106C2×D4×C109—44no—
107C2×D4×C109—44no—
108C2×D4×C109—41no—
109C2×D4×C109—41no—
110C2×D4×C109—41no—
111C2×D4×C109—41no—
112C2×D4×C109—41no—
113C2×D4×C109—41no—
114C22×C4×C109—24no—
115C22×C4×C109—24no—
116C22×C4×C109—24no—
117C22×C4×C109—21no—
118C22×C4×C109—21no—
119C22×C4×C109—21no—
120C22×C4×C109—21no—
121C22×C4×C109—21no—
122C22×C4×C109—22no—
123C22×C4×C109—22no—
124C22×C4×C109—21no—
125C22×C4×C109—42no—
126C22×C4×C109—42no—
127C22×C4×C109—42no—
128C22×C4×C109—42no—
129C22×C4×C109—42no—
130C22×C4×C109—42no—
131C22×C4×C109—42no—
132C22×C4×C109—42no—
133C22×C4×C109—42no—
134C22×C4×C109—42no—
135C22×C4×C109—42no—
136C22×C4×C109—41no—
137C22×C4×C109—41no—
138C22×C4×C109—41no—
139C22×C4×C109—41no—
140C22×C4×C109C2×D411yescharacteristic
141C22×C4×C109C2×D411yescharacteristic
142C22×C4×C109C2×D411yescharacteristic
143C22×C4×C109C2×D411yescharacteristic
144C22×C4×C109C2×D411yescharacteristic
145C22×C4×C109#4111yescharacteristic
146SD16×C109—82no—
147C2×C8×C109—22no—
148C2×C8×C109—42no—
149C2×C8×C109—42no—
150C2×C8×C109—41no—
151C4⋊C4×C109—21no—
152C4⋊C4×C109—21no—
153C4⋊C4×C109—21no—
154C4⋊C4×C109—21no—
155C4⋊C4×C109—42no—
156C4⋊C4×C109—42no—
157C4⋊C4×C109—44no—
158C4⋊C4×C109—41no—
159C4⋊C4×C109—41no—
160C22⋊C4×C109—44no—
161C22⋊C4×C109—44no—
162C22⋊C4×C109—44no—
163C22⋊C4×C109—44no—
164C22⋊C4×C109—41no—
165C22⋊C4×C109—41no—
166C22⋊C4×C109—41no—
167C22⋊C4×C109—41no—
168C22⋊C4×C109—41no—
169C22⋊C4×C109—41no—
170C22⋊C4×C109—41no—
171C22⋊C4×C109—41no—
172C23×C109—22no—
173C23×C109—22no—
174C23×C109—22no—
175C23×C109—22no—
176C23×C109—22no—
177C23×C109—22no—
178C23×C109—22no—
179C23×C109—22no—
180C23×C109—22no—
181C23×C109—41no—
182C23×C109—44no—
183C23×C109—41no—
184C23×C109—41no—
185C23×C109—41no—
186C23×C109—41no—
187C23×C109—41no—
188C23×C109—41no—
189C23×C109—41no—
190C23×C109—44no—
191C23×C109—44no—
192C23×C109—44no—
193C23×C109—42no—
194C23×C109—42no—
195C23×C109—41no—
196C23×C109—41no—
197C23×C109—41no—
198C23×C109—41no—
199C23×C109—41no—
200C23×C109—41no—
201C23×C109—41no—
202C23×C109#12011yescharacteristic
203C23×C109#12011yescharacteristic
204C23×C109#12111yescharacteristic
205C23×C109C2×SD1611yescharacteristic
206C23×C109#13511yescharacteristic
207C23×C109AGL(1,ℤ/8)11yescharacteristic
208C23×C109#13711yescharacteristic
209Q8×C109—42no—
210Q8×C109—81no—
211D4×C109—42no—
212D4×C109—81no—
213D4×C109—88no—
214D4×C109—81no—
215D4×C109—81no—
216D4×C109—81no—
217D4×C109—81no—
218C2×C4×C109—22no—
219C2×C4×C109—22no—
220C2×C4×C109—22no—
221C2×C4×C109—22no—
222C2×C4×C109—22no—
223C2×C4×C109—22no—
224C2×C4×C109—41no—
225C2×C4×C109—41no—
226C2×C4×C109—41no—
227C2×C4×C109—44no—
228C2×C4×C109—44no—
229C2×C4×C109—44no—
230C2×C4×C109—41no—
231C2×C4×C109—41no—
232C2×C4×C109—41no—
233C2×C4×C109—41no—
234C2×C4×C109—44no—
235C2×C4×C109—42no—
236C2×C4×C109—42no—
237C2×C4×C109—42no—
238C2×C4×C109—42no—
239C2×C4×C109—42no—
240C2×C4×C109—42no—
241C2×C4×C109—41no—
242C2×C4×C109—41no—
243C2×C4×C109—41no—
244C2×C4×C109—42no—
245C2×C4×C109—42no—
246C2×C4×C109—41no—
247C2×C4×C109—41no—
248C2×C4×C109#12011yescharacteristic
249C2×C4×C109#12111yescharacteristic
250C2×C4×C109#12111yescharacteristic
251C2×C4×C109#12711yescharacteristic
252C8×C109—42no—
253C8×C109—81no—
254C22×C109—21no—
255C22×C109—21no—
256C22×C109—21no—
257C22×C109—21no—
258C22×C109—21no—
259C22×C109—21no—
260C22×C109—24no—
261C22×C109—24no—
262C22×C109—24no—
263C22×C109—41no—
264C22×C109—41no—
265C22×C109—41no—
266C22×C109—41no—
267C22×C109—42no—
268C22×C109—42no—
269C22×C109—42no—
270C22×C109—42no—
271C22×C109—42no—
272C22×C109—42no—
273C22×C109—42no—
274C22×C109—42no—
275C22×C109—42no—
276C22×C109—42no—
277C22×C109—42no—
278C22×C109—42no—
279C22×C109—82no—
280C22×C109—82no—
281C22×C109—81no—
282C22×C109—81no—
283C22×C109—81no—
284C22×C109#39211yescharacteristic
285C22×C109#44811yescharacteristic
286C22×C109#44911yescharacteristic
287C22×C109#45011yescharacteristic
288C22×C109#45211yescharacteristic
289C22×C109#46511yescharacteristic
290C22×C109#46811yescharacteristic
291C4×C109—21no—
292C4×C109—21no—
293C4×C109—21no—
294C4×C109—42no—
295C4×C109—82no—
296C4×C109—82no—
297C4×C109—81no—
298C4×C109—81no—
299C4×C109—81no—
300C4×C109B(4,𝔽2)12yes—
301#10322C10911yesmaximal, characteristic, direct product factor, Sylow
302C2×C109—21no—
303C2×C109—21no—
304C2×C109—21no—
305C2×C109—21no—
306C2×C109—21no—
307C2×C109—21no—
308C2×C109—44no—
309C2×C109—161no—
310C2×C109#200011yescharacteristic
311C2×C109#201011yescharacteristic
312C2×C109#202111yescharacteristic
313#3187C2×C10911yescharacteristic, complemented
314#2848C2×C10911yescharacteristic, complemented
315#2847C2×C10911yescharacteristic
316#1779C2×C10911yescharacteristic, complemented
317#1520C2×C10911yescharacteristic, complemented
318#1519C2×C10911yescharacteristic, complemented
319#1289C2×C10911yescharacteristic, complemented
320C109#1032211yescharacteristic, direct product factor, Sylow
321C24×C4C22×C10911yescharacteristic
322C2×C22⋊Q8—21no—
323C2×C22⋊Q8C22×C10911yescharacteristic
324C2×#121—21no—
325C2×#121C22×C10911yescharacteristic
326C2×#120—21no—
327#483—21no—
328#481—21no—
329#465—22no—
330#450—21no—
331#448—21no—
332C2×C22⋊C8—22no—
333C2×C22⋊C8C22×C10911yescharacteristic
334C2×C22⋊C8C22×C10911yescharacteristic
335#386—21no—
336#386—22no—
337#385—21no—
338#385—22no—
339#380—21no—
340#380—21no—
341#380C22×C10911yescharacteristic
342#380C22×C10911yescharacteristic
343#379—21no—
344#328—22no—
345#327—22no—
346C25D4×C10911yescharacteristic
347C22×Q8D4×C10911yescharacteristic, complemented
348C22×D4—22no—
349C22×D4—21no—
350C22×D4D4×C10911yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—21no—
355C23×C4—21no—
356C23×C4—21no—
357C23×C4—44no—
358C23×C4D4×C10911yescharacteristic
359C23×C4C23×C10911yescharacteristic
360C2×SD16—22no—
361C22×C8—21no—
362C22×C8D4×C10912yescomplemented
363C22⋊Q8—21no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—21no—
367C22⋊Q8—44no—
368#121—21no—
369#121—21no—
370#121—21no—
371#121—21no—
372#121—44no—
373#120—42no—
374#120—42no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4D4×C10911yescharacteristic
380C2×C4⋊C4D4×C10911yescharacteristic, complemented
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389#106—22no—
390#103—22no—
391#102—22no—
392C22⋊C8—21no—
393C22⋊C8—21no—
394C22⋊C8—21no—
395C22⋊C8—21no—
396C22⋊C8—44no—
397#95—22no—
398#95—22no—
399#95D4×C10912yes—
400#95D4×C10912yescomplemented
401C24—21no—
402C24—21no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—21no—
407C24—21no—
408C24—42no—
409C24—42no—
410C24SD16×C10912yescomplemented
411C24C2×D4×C10911yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—41no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—41no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4C2×D4×C10911yescharacteristic
454C22×C4C2×D4×C10911yescharacteristic
455C22×C4C2×D4×C10911yescharacteristic
456C22×C4C2×D4×C10911yescharacteristic
457C22×C4C2×D4×C10911yescharacteristic
458C22×C4#41×C10911yescharacteristic
459SD16—82no—
460C2×C8—22no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—41no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—41no—
495C23—44no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—42no—
507C23—42no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23#120×C10911yescharacteristic
516C23#120×C10911yescharacteristic
517C23#121×C10911yescharacteristic
518C23C2×SD16×C10911yescharacteristic
519C23#135×C10911yescharacteristic
520C23AGL(1,ℤ/8)×C10911yescharacteristic
521C23#137×C10911yescharacteristic
522Q8—42no—
523Q8—81no—
524D4—42no—
525D4—81no—
526D4—88no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4#120×C10911yescharacteristic
562C2×C4#121×C10911yescharacteristic
563C2×C4#121×C10911yescharacteristic
564C2×C4#127×C10911yescharacteristic
565C8—42no—
566C8—81no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—81no—
595C22—81no—
596C22—81no—
597C22#392×C10911yescharacteristic
598C22#448×C10911yescharacteristic
599C22#449×C10911yescharacteristic
600C22#450×C10911yescharacteristic
601C22#452×C10911yescharacteristic
602C22#465×C10911yescharacteristic
603C22#468×C10911yescharacteristic
604C4—21no—
605C4—21no—
606C4—21no—
607C4—42no—
608C4—82no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4B(4,𝔽2)×C10912yes—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—21no—
619C2—21no—
620C2—44no—
621C2—161no—
622C2C109×#200011yescharacteristic
623C2C109×#201011yescharacteristic
624C2C109×#202111yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC23×C4C23×C109
prime radicalC109×#2847C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#3187C2semidirect
2. C109×#2848C2semidirect
4. C109×#1779C2semidirect
5. C109×#1520C2semidirect
6. C109×#1519C2semidirect
7. C109×#1289C2semidirect
34. C22×Q8×C109D4semidirect
49. C22×C8×C109D4semidirect
67. C2×C4⋊C4×C109D4semidirect
87. #95×C109D4semidirect
97. C24×C109SD16semidirect
301. #10322C109direct (N × H)
313. #3187C2×C109semidirect
314. #2848C2×C109semidirect
316. #1779C2×C109semidirect
317. #1520C2×C109semidirect
318. #1519C2×C109semidirect
319. #1289C2×C109semidirect
320. C109#10322direct (N × H)
347. C22×Q8D4×C109semidirect
362. C22×C8D4×C109semidirect
380. C2×C4⋊C4D4×C109semidirect
400. #95D4×C109semidirect
410. C24SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#31872
C109×#28482
C109×#28472
C109×#17792
C109×#15202
C109×#15192
C109×#12892
#10322109