Finite Group Databasebeta

← C109×#18260 #42136254

Subgroups — C109×#18260 #42136254

Order 27904.

1336 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3536C211yesmaximal, characteristic, complemented
2C109×#3003C211yesmaximal, characteristic, complemented
3C2×#447×C109C211yesmaximal, characteristic, complemented
4C109×#2435C211yesmaximal, characteristic, complemented
5C109×#1598C214yesmaximal, complemented
6C109×#1597C214yesmaximal, complemented
7C109×#1507C211yesmaximal, characteristic, complemented
8C109×#1501C211yesmaximal, characteristic, complemented
9C109×#1472C211yesmaximal, characteristic, complemented
10C23×D4×C109C2211yescharacteristic
11C2×AGL(1,ℤ/8)×C109—21no—
12C22×D8×C109—21no—
13#562×C109—22no—
14#554×C109—21no—
15#548×C109—21no—
16#547×C109—21no—
17#546×C109—21no—
18#546×C109—22no—
19#543×C109—21no—
20#542×C109—21no—
21#539×C109C2212yescomplemented
22#538×C109—21no—
23#537×C109—21no—
24#537×C109C2212yescomplemented
25C2×#124×C109—21no—
26#525×C109C2211yescharacteristic
27C2×C22⋊Q8×C109C2211yescharacteristic
28C2×#121×C109C2211yescharacteristic
29#450×C109—21no—
30#450×C109C2212yescomplemented
31#447×C109—22no—
32#447×C109—21no—
33#447×C109C2214yes—
34#447×C109C2212yescomplemented
35#418×C109—21no—
36C2×#102×C109—21no—
37#407×C109C2211yescharacteristic
38C2×C22⋊C8×C109C2211yescharacteristic
39#394×C109C2214yescomplemented
40#390×C109—21no—
41#379×C109—22no—
42C4×C22⋊C4×C109C2211yescharacteristic
43#328×C109C2212yescomplemented
44#327×C109C2212yescomplemented
45#327×C109C2214yescomplemented
46#324×C109C2214yescomplemented
47C25×C109—21no—
48C2×#41×C109—21no—
49C2×#41×C109—22no—
50C2×#41×C109D411yescharacteristic
51C22×Q8×C109D411yescharacteristic
52C22×D4×C109—21no—
53C22×D4×C109—21no—
54C22×D4×C109—21no—
55C22×D4×C109—21no—
56C22×D4×C109—21no—
57C22×D4×C109—21no—
58C22×D4×C109—44no—
59C22×D4×C109D411yescharacteristic
60C22×D4×C109C2312yes—
61C23×C4×C109C2311yescharacteristic
62AGL(1,ℤ/8)×C109—44no—
63C2×SD16×C109—22no—
64C2×D8×C109—24no—
65C2×D8×C109—22no—
66C2×D8×C109—44no—
67C2×M16×C109—21no—
68C22×C8×C109—21no—
69C4⋊Q8×C109—21no—
70#126×C109—22no—
71#125×C109—22no—
72#124×C109—22no—
73#124×C109—21no—
74#124×C109—22no—
75#124×C109—44no—
76#123×C109—22no—
77#123×C109—22no—
78#123×C109—22no—
79#123×C109—21no—
80#123×C109—22no—
81C22⋊Q8×C109—21no—
82C22⋊Q8×C109—22no—
83C22⋊Q8×C109—22no—
84C22⋊Q8×C109—22no—
85C22⋊Q8×C109—22no—
86C22⋊Q8×C109—21no—
87C22⋊Q8×C109—21no—
88C22⋊Q8×C109—21no—
89C22⋊Q8×C109—21no—
90C22⋊Q8×C109—22no—
91C22⋊Q8×C109—22no—
92C22⋊Q8×C109D411yescharacteristic
93C22⋊Q8×C109C2311yescharacteristic
94#121×C109—21no—
95#121×C109—22no—
96#121×C109—22no—
97#121×C109—22no—
98#121×C109—22no—
99#121×C109—21no—
100#121×C109—21no—
101#121×C109—22no—
102#121×C109D412yes—
103#121×C109D411yescharacteristic
104#121×C109C2312yes—
105#121×C109C2311yescharacteristic
106#120×C109—22no—
107#120×C109—21no—
108C4×Q8×C109—21no—
109C4×D4×C109—22no—
110C4×D4×C109—21no—
111C4×D4×C109—21no—
112C4×D4×C109—21no—
113C4×D4×C109—22no—
114#117×C109D411yescharacteristic
115C2×C4⋊C4×C109—21no—
116C2×C4⋊C4×C109D411yescharacteristic
117C2×C4⋊C4×C109D411yescharacteristic
118C2×C4⋊C4×C109C2312yes—
119C2×C22⋊C4×C109—22no—
120C2×C22⋊C4×C109—21no—
121C2×C22⋊C4×C109—21no—
122C2×C22⋊C4×C109—24no—
123C2×C22⋊C4×C109—24no—
124C2×C22⋊C4×C109D412yes—
125C2×C42×C109—21no—
126#102×C109—24no—
127#102×C109—22no—
128#102×C109—22no—
129C22⋊C8×C109—21no—
130C22⋊C8×C109—21no—
131C22⋊C8×C109C2312yes—
132C22⋊C8×C109C2312yes—
133#95×C109C2312yes—
134C24×C109—22no—
135C24×C109—21no—
136C24×C109—44no—
137C24×C109—44no—
138C24×C109—41no—
139C24×C109—41no—
140C24×C109—82no—
141C24×C109C2×D411yescharacteristic
142#41×C109—42no—
143#41×C109—84no—
144#41×C109—82no—
145C2×Q8×C109—22no—
146C2×Q8×C109—21no—
147C2×Q8×C109—21no—
148C2×Q8×C109—42no—
149C2×Q8×C109C2×D411yescharacteristic
150C2×D4×C109—24no—
151C2×D4×C109—24no—
152C2×D4×C109—24no—
153C2×D4×C109—21no—
154C2×D4×C109—21no—
155C2×D4×C109—22no—
156C2×D4×C109—22no—
157C2×D4×C109—22no—
158C2×D4×C109—21no—
159C2×D4×C109—22no—
160C2×D4×C109—21no—
161C2×D4×C109—22no—
162C2×D4×C109—22no—
163C2×D4×C109—42no—
164C2×D4×C109—42no—
165C2×D4×C109—42no—
166C2×D4×C109—42no—
167C2×D4×C109—44no—
168C2×D4×C109—44no—
169C2×D4×C109—44no—
170C2×D4×C109—41no—
171C2×D4×C109—41no—
172C2×D4×C109—82no—
173C2×D4×C109C2×D411yescharacteristic
174C2×D4×C109C2×D412yes—
175C22×C4×C109—22no—
176C22×C4×C109—22no—
177C22×C4×C109—22no—
178C22×C4×C109—21no—
179C22×C4×C109—21no—
180C22×C4×C109—22no—
181C22×C4×C109—42no—
182C22×C4×C109C2×D412yes—
183C22×C4×C109C2×D411yescharacteristic
184C22×C4×C109C2×D411yescharacteristic
185C22×C4×C109C2×D411yescharacteristic
186C22×C4×C109C2×D412yes—
187C22×C4×C109C2×D411yescharacteristic
188C22×C4×C109C2×D411yescharacteristic
189C22×C4×C109C2411yescharacteristic
190SD16×C109—44no—
191D8×C109—48no—
192D8×C109—44no—
193M16×C109—42no—
194C2×C8×C109—22no—
195C2×C8×C109—22no—
196C2×C8×C109—42no—
197C4⋊C4×C109—22no—
198C4⋊C4×C109—22no—
199C4⋊C4×C109—22no—
200C4⋊C4×C109—22no—
201C4⋊C4×C109—21no—
202C4⋊C4×C109—21no—
203C4⋊C4×C109—21no—
204C4⋊C4×C109—21no—
205C4⋊C4×C109—22no—
206C4⋊C4×C109—22no—
207C4⋊C4×C109—21no—
208C4⋊C4×C109C2×D412yes—
209C4⋊C4×C109C2×D411yescharacteristic
210C4⋊C4×C109C2×D411yescharacteristic
211C22⋊C4×C109—22no—
212C22⋊C4×C109—22no—
213C22⋊C4×C109—22no—
214C22⋊C4×C109—22no—
215C22⋊C4×C109—22no—
216C22⋊C4×C109—21no—
217C22⋊C4×C109—22no—
218C22⋊C4×C109—22no—
219C22⋊C4×C109—21no—
220C22⋊C4×C109—21no—
221C22⋊C4×C109—22no—
222C22⋊C4×C109—21no—
223C22⋊C4×C109—22no—
224C22⋊C4×C109—21no—
225C22⋊C4×C109—42no—
226C22⋊C4×C109—48no—
227C22⋊C4×C109—84no—
228C42×C109—22no—
229C42×C109—21no—
230C42×C109—42no—
231C23×C109—22no—
232C23×C109—22no—
233C23×C109—24no—
234C23×C109—22no—
235C23×C109—21no—
236C23×C109—41no—
237C23×C109—44no—
238C23×C109—44no—
239C23×C109—42no—
240C23×C109—44no—
241C23×C109—42no—
242C23×C109—44no—
243C23×C109—42no—
244C23×C109—81no—
245C23×C109—81no—
246C23×C109—82no—
247C23×C109—82no—
248C23×C109—82no—
249C23×C109—816no—
250C23×C109—82no—
251C23×C109#12011yescharacteristic
252C23×C109#12011yescharacteristic
253C23×C109C22×D411yescharacteristic
254Q8×C109—41no—
255Q8×C109—42no—
256Q8×C109—81no—
257Q8×C109—82no—
258D4×C109—24no—
259D4×C109—24no—
260D4×C109—42no—
261D4×C109—41no—
262D4×C109—42no—
263D4×C109—42no—
264D4×C109—44no—
265D4×C109—44no—
266D4×C109—42no—
267D4×C109—42no—
268D4×C109—44no—
269D4×C109—42no—
270D4×C109—82no—
271D4×C109—81no—
272D4×C109—82no—
273D4×C109—82no—
274C2×C4×C109—22no—
275C2×C4×C109—22no—
276C2×C4×C109—22no—
277C2×C4×C109—21no—
278C2×C4×C109—22no—
279C2×C4×C109—21no—
280C2×C4×C109—22no—
281C2×C4×C109—21no—
282C2×C4×C109—22no—
283C2×C4×C109—24no—
284C2×C4×C109—41no—
285C2×C4×C109—41no—
286C2×C4×C109—41no—
287C2×C4×C109—42no—
288C2×C4×C109—42no—
289C2×C4×C109—42no—
290C2×C4×C109—42no—
291C2×C4×C109—42no—
292C2×C4×C109—81no—
293C2×C4×C109—82no—
294C2×C4×C109—82no—
295C2×C4×C109—82no—
296C2×C4×C109#12011yescharacteristic
297C2×C4×C109#12011yescharacteristic
298C2×C4×C109C22×D411yescharacteristic
299C2×C4×C109C22×D411yescharacteristic
300C8×C109—42no—
301C8×C109—42no—
302C22×C109—21no—
303C22×C109—21no—
304C22×C109—21no—
305C22×C109—21no—
306C22×C109—21no—
307C22×C109—21no—
308C22×C109—21no—
309C22×C109—44no—
310C22×C109—48no—
311C22×C109—41no—
312C22×C109—42no—
313C22×C109—41no—
314C22×C109—42no—
315C22×C109—41no—
316C22×C109—42no—
317C22×C109—44no—
318C22×C109—84no—
319C22×C109—88no—
320C22×C109—88no—
321C22×C109—82no—
322C22×C109—82no—
323C22×C109—162no—
324C22×C109#45412yes—
325C22×C109C2×#12011yescharacteristic
326C22×C109#57612yes—
327C4×C109—21no—
328C4×C109—21no—
329C4×C109—21no—
330C4×C109—42no—
331C4×C109—42no—
332C4×C109—41no—
333C4×C109—41no—
334C4×C109—41no—
335C4×C109—42no—
336C4×C109—82no—
337C4×C109B(4,𝔽2)12yes—
338#18260C10911yesmaximal, characteristic, direct product factor, Sylow
339C2×C109—22no—
340C2×C109—22no—
341C2×C109—41no—
342C2×C109—81no—
343C2×C109—84no—
344C2×C109—82no—
345C2×C109#300811yescharacteristic
346C2×C109C2×#45411yescharacteristic
347C2×C109C2×B(4,𝔽2)11yescharacteristic
348#3536C2×C10911yescharacteristic, complemented
349#3003C2×C10911yescharacteristic, complemented
350C2×#447C2×C10911yescharacteristic, complemented
351#2435C2×C10911yescharacteristic, complemented
352#1598C2×C10914yescomplemented
353#1597C2×C10914yescomplemented
354#1507C2×C10911yescharacteristic, complemented
355#1501C2×C10911yescharacteristic, complemented
356#1472C2×C10911yescharacteristic, complemented
357C109#1826011yescharacteristic, direct product factor, Sylow
358C23×D4C22×C10911yescharacteristic
359C2×AGL(1,ℤ/8)—21no—
360C22×D8—21no—
361#562—22no—
362#554—21no—
363#548—21no—
364#547—21no—
365#546—21no—
366#546—22no—
367#543—21no—
368#542—21no—
369#539C22×C10912yescomplemented
370#538—21no—
371#537—21no—
372#537C22×C10912yescomplemented
373C2×#124—21no—
374#525C22×C10911yescharacteristic
375C2×C22⋊Q8C22×C10911yescharacteristic
376C2×#121C22×C10911yescharacteristic
377#450—21no—
378#450C22×C10912yescomplemented
379#447—22no—
380#447—21no—
381#447C22×C10914yes—
382#447C22×C10912yescomplemented
383#418—21no—
384C2×#102—21no—
385#407C22×C10911yescharacteristic
386C2×C22⋊C8C22×C10911yescharacteristic
387#394C22×C10914yescomplemented
388#390—21no—
389#379—22no—
390C4×C22⋊C4C22×C10911yescharacteristic
391#328C22×C10912yescomplemented
392#327C22×C10914yescomplemented
393#327C22×C10912yescomplemented
394#324C22×C10914yescomplemented
395C25—21no—
396C2×#41—21no—
397C2×#41—22no—
398C2×#41D4×C10911yescharacteristic
399C22×Q8D4×C10911yescharacteristic
400C22×D4—21no—
401C22×D4—21no—
402C22×D4—21no—
403C22×D4—21no—
404C22×D4—21no—
405C22×D4—21no—
406C22×D4—44no—
407C22×D4D4×C10911yescharacteristic
408C22×D4C23×C10912yes—
409C23×C4C23×C10911yescharacteristic
410AGL(1,ℤ/8)—44no—
411C2×SD16—22no—
412C2×D8—24no—
413C2×D8—22no—
414C2×D8—44no—
415C2×M16—21no—
416C22×C8—21no—
417C4⋊Q8—21no—
418#126—22no—
419#125—22no—
420#124—22no—
421#124—21no—
422#124—22no—
423#124—44no—
424#123—22no—
425#123—22no—
426#123—22no—
427#123—21no—
428#123—22no—
429C22⋊Q8—21no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8—22no—
433C22⋊Q8—22no—
434C22⋊Q8—21no—
435C22⋊Q8—21no—
436C22⋊Q8—21no—
437C22⋊Q8—21no—
438C22⋊Q8—22no—
439C22⋊Q8—22no—
440C22⋊Q8D4×C10911yescharacteristic
441C22⋊Q8C23×C10911yescharacteristic
442#121—21no—
443#121—22no—
444#121—22no—
445#121—22no—
446#121—22no—
447#121—21no—
448#121—21no—
449#121—22no—
450#121D4×C10912yes—
451#121D4×C10911yescharacteristic
452#121C23×C10912yes—
453#121C23×C10911yescharacteristic
454#120—22no—
455#120—21no—
456C4×Q8—21no—
457C4×D4—22no—
458C4×D4—21no—
459C4×D4—21no—
460C4×D4—21no—
461C4×D4—22no—
462#117D4×C10911yescharacteristic
463C2×C4⋊C4—21no—
464C2×C4⋊C4D4×C10911yescharacteristic
465C2×C4⋊C4D4×C10911yescharacteristic
466C2×C4⋊C4C23×C10912yes—
467C2×C22⋊C4—22no—
468C2×C22⋊C4—21no—
469C2×C22⋊C4—21no—
470C2×C22⋊C4—24no—
471C2×C22⋊C4—24no—
472C2×C22⋊C4D4×C10912yes—
473C2×C42—21no—
474#102—24no—
475#102—22no—
476#102—22no—
477C22⋊C8—21no—
478C22⋊C8—21no—
479C22⋊C8C23×C10912yes—
480C22⋊C8C23×C10912yes—
481#95C23×C10912yes—
482C24—22no—
483C24—21no—
484C24—44no—
485C24—44no—
486C24—41no—
487C24—41no—
488C24—82no—
489C24C2×D4×C10911yescharacteristic
490#41—42no—
491#41—84no—
492#41—82no—
493C2×Q8—22no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—42no—
497C2×Q8C2×D4×C10911yescharacteristic
498C2×D4—24no—
499C2×D4—24no—
500C2×D4—24no—
501C2×D4—21no—
502C2×D4—21no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—21no—
507C2×D4—22no—
508C2×D4—21no—
509C2×D4—22no—
510C2×D4—22no—
511C2×D4—42no—
512C2×D4—42no—
513C2×D4—42no—
514C2×D4—42no—
515C2×D4—44no—
516C2×D4—44no—
517C2×D4—44no—
518C2×D4—41no—
519C2×D4—41no—
520C2×D4—82no—
521C2×D4C2×D4×C10911yescharacteristic
522C2×D4C2×D4×C10912yes—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—42no—
530C22×C4C2×D4×C10912yes—
531C22×C4C2×D4×C10911yescharacteristic
532C22×C4C2×D4×C10911yescharacteristic
533C22×C4C2×D4×C10911yescharacteristic
534C22×C4C2×D4×C10912yes—
535C22×C4C2×D4×C10911yescharacteristic
536C22×C4C2×D4×C10911yescharacteristic
537C22×C4C24×C10911yescharacteristic
538SD16—44no—
539D8—48no—
540D8—44no—
541M16—42no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—42no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—21no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—21no—
556C4⋊C4C2×D4×C10912yes—
557C4⋊C4C2×D4×C10911yescharacteristic
558C4⋊C4C2×D4×C10911yescharacteristic
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—21no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—21no—
568C22⋊C4—21no—
569C22⋊C4—22no—
570C22⋊C4—21no—
571C22⋊C4—22no—
572C22⋊C4—21no—
573C22⋊C4—42no—
574C22⋊C4—48no—
575C22⋊C4—84no—
576C42—22no—
577C42—21no—
578C42—42no—
579C23—22no—
580C23—22no—
581C23—24no—
582C23—22no—
583C23—21no—
584C23—41no—
585C23—44no—
586C23—44no—
587C23—42no—
588C23—44no—
589C23—42no—
590C23—44no—
591C23—42no—
592C23—81no—
593C23—81no—
594C23—82no—
595C23—82no—
596C23—82no—
597C23—816no—
598C23—82no—
599C23#120×C10911yescharacteristic
600C23#120×C10911yescharacteristic
601C23C22×D4×C10911yescharacteristic
602Q8—41no—
603Q8—42no—
604Q8—81no—
605Q8—82no—
606D4—24no—
607D4—24no—
608D4—42no—
609D4—41no—
610D4—42no—
611D4—42no—
612D4—44no—
613D4—44no—
614D4—42no—
615D4—42no—
616D4—44no—
617D4—42no—
618D4—82no—
619D4—81no—
620D4—82no—
621D4—82no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—21no—
626C2×C4—22no—
627C2×C4—21no—
628C2×C4—22no—
629C2×C4—21no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—81no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4#120×C10911yescharacteristic
645C2×C4#120×C10911yescharacteristic
646C2×C4C22×D4×C10911yescharacteristic
647C2×C4C22×D4×C10911yescharacteristic
648C8—42no—
649C8—42no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—21no—
657C22—44no—
658C22—48no—
659C22—41no—
660C22—42no—
661C22—41no—
662C22—42no—
663C22—41no—
664C22—42no—
665C22—44no—
666C22—84no—
667C22—88no—
668C22—88no—
669C22—82no—
670C22—82no—
671C22—162no—
672C22#454×C10912yes—
673C22C2×#120×C10911yescharacteristic
674C22#576×C10912yes—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—42no—
679C4—42no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—42no—
684C4—82no—
685C4B(4,𝔽2)×C10912yes—
686C2—22no—
687C2—22no—
688C2—41no—
689C2—84no—
690C2—82no—
691C2—81no—
692C2C109×#300811yescharacteristic
693C2C2×#454×C10911yescharacteristic
694C2C2×B(4,𝔽2)×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
LCSG ▷ C22×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×#3536C2semidirect
2. C109×#3003C2semidirect
3. C2×#447×C109C2semidirect
4. C109×#2435C2semidirect
5. C109×#1598C2semidirect
6. C109×#1597C2semidirect
7. C109×#1507C2semidirect
8. C109×#1501C2semidirect
9. C109×#1472C2semidirect
21. #539×C109C22semidirect
24. #537×C109C22semidirect
30. #450×C109C22semidirect
34. #447×C109C22semidirect
39. #394×C109C22semidirect
43. #328×C109C22semidirect
44. #327×C109C22semidirect
45. #327×C109C22semidirect
46. #324×C109C22semidirect
338. #18260C109direct (N × H)
348. #3536C2×C109semidirect
349. #3003C2×C109semidirect
350. C2×#447C2×C109semidirect
351. #2435C2×C109semidirect
352. #1598C2×C109semidirect
353. #1597C2×C109semidirect
354. #1507C2×C109semidirect
355. #1501C2×C109semidirect
356. #1472C2×C109semidirect
357. C109#18260direct (N × H)
369. #539C22×C109semidirect
372. #537C22×C109semidirect
378. #450C22×C109semidirect
382. #447C22×C109semidirect
387. #394C22×C109semidirect
391. #328C22×C109semidirect
392. #327C22×C109semidirect
393. #327C22×C109semidirect
394. #324C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35362
C109×#30032
C2×#447×C1092
C109×#24352
C109×#15982
C109×#15972
C109×#15072
C109×#15012
C109×#14722
#18260109