Finite Group Databasebeta

← C109×#18267 #42136261

Subgroups — C109×#18267 #42136261

Order 27904.

1056 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3536C211yesmaximal, characteristic
2C109×#3092C211yesmaximal, characteristic, complemented
3C2×#484×C109C211yesmaximal, characteristic, complemented
4C109×#2501C211yesmaximal, characteristic, complemented
5C109×#1618C214yesmaximal, complemented
6C109×#1614C214yesmaximal, complemented
7C109×#1508C211yesmaximal, characteristic, complemented
8C109×#1501C211yesmaximal, characteristic, complemented
9C109×#1471C211yesmaximal, characteristic, complemented
10#562×C109—22no—
11#554×C109—21no—
12#548×C109—21no—
13#547×C109—21no—
14#546×C109—22no—
15#546×C109—21no—
16#543×C109C2212yes—
17#542×C109—21no—
18#539×C109—21no—
19#538×C109C2212yes—
20#537×C109—21no—
21#537×C109—21no—
22#525×C109C2211yescharacteristic
23C2×#123×C109—22no—
24C2×C22⋊Q8×C109C2211yescharacteristic
25C2×#121×C109C2211yescharacteristic
26C22×C4⋊C4×C109C2211yescharacteristic
27#484×C109—22no—
28#484×C109C2214yes—
29#484×C109C2212yes—
30#483×C109—21no—
31#481×C109C2212yes—
32#480×C109—21no—
33#428×C109—21no—
34C2×#107×C109—21no—
35#417×C109—21no—
36C2×#103×C109—21no—
37#407×C109C2211yescharacteristic
38C2×C22⋊C8×C109C2211yescharacteristic
39#399×C109C2214yes—
40#390×C109—21no—
41#384×C109—21no—
42C4×C22⋊C4×C109C2211yescharacteristic
43#328×C109C2212yes—
44#328×C109C2214yes—
45#327×C109C2212yes—
46#323×C109C2214yes—
47C2×#41×C109—22no—
48C2×#41×C109—21no—
49C2×#41×C109D411yescharacteristic
50C22×Q8×C109D411yescharacteristic
51C22×D4×C109D412yes—
52C22×D4×C109D411yescharacteristic
53C23×C4×C109—21no—
54C23×C4×C109C2311yescharacteristic
55C2×M16×C109—21no—
56C22×C8×C109—21no—
57C4⋊Q8×C109—21no—
58#126×C109—22no—
59#125×C109—22no—
60#124×C109—21no—
61#124×C109—22no—
62#124×C109—22no—
63#123×C109—22no—
64#123×C109—22no—
65#123×C109—22no—
66#123×C109—21no—
67#123×C109—22no—
68#123×C109—48no—
69C22⋊Q8×C109—21no—
70C22⋊Q8×C109—22no—
71C22⋊Q8×C109—22no—
72C22⋊Q8×C109—22no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109D412yes—
79C22⋊Q8×C109D411yescharacteristic
80C22⋊Q8×C109C2312yes—
81C22⋊Q8×C109C2311yescharacteristic
82#121×C109—21no—
83#121×C109—22no—
84#121×C109—22no—
85#121×C109—22no—
86#121×C109—22no—
87#121×C109—22no—
88#121×C109—21no—
89#121×C109—21no—
90#121×C109—21no—
91#121×C109—21no—
92#121×C109D411yescharacteristic
93#121×C109C2311yescharacteristic
94#120×C109—21no—
95#120×C109—22no—
96C4×Q8×C109—21no—
97C4×D4×C109—22no—
98C4×D4×C109—21no—
99C4×D4×C109—21no—
100C4×D4×C109—21no—
101C4×D4×C109—22no—
102#117×C109D411yescharacteristic
103C2×C4⋊C4×C109—21no—
104C2×C4⋊C4×C109—21no—
105C2×C4⋊C4×C109—21no—
106C2×C4⋊C4×C109—21no—
107C2×C4⋊C4×C109—21no—
108C2×C4⋊C4×C109—21no—
109C2×C4⋊C4×C109—21no—
110C2×C4⋊C4×C109—24no—
111C2×C4⋊C4×C109D411yescharacteristic
112C2×C4⋊C4×C109D411yescharacteristic
113C2×C4⋊C4×C109C2312yes—
114C2×C22⋊C4×C109—21no—
115C2×C22⋊C4×C109—21no—
116C2×C22⋊C4×C109—22no—
117C2×C22⋊C4×C109—24no—
118C2×C22⋊C4×C109C2312yes—
119C2×C42×C109—21no—
120#107×C109—24no—
121#107×C109—22no—
122#106×C109—22no—
123#103×C109—24no—
124#103×C109—22no—
125#102×C109—22no—
126C22⋊C8×C109—21no—
127C22⋊C8×C109—21no—
128C22⋊C8×C109C2312yes—
129C22⋊C8×C109C2312yes—
130#95×C109C2312yes—
131C24×C109C2×D411yescharacteristic
132C24×C109C2×D412yes—
133#41×C109—42no—
134#41×C109—84no—
135#41×C109—82no—
136C2×Q8×C109—22no—
137C2×Q8×C109—21no—
138C2×Q8×C109—42no—
139C2×Q8×C109C2×D411yescharacteristic
140C2×Q8×C109C2×D412yes—
141C2×D4×C109—21no—
142C2×D4×C109—22no—
143C2×D4×C109—21no—
144C2×D4×C109—22no—
145C2×D4×C109—22no—
146C2×D4×C109—21no—
147C2×D4×C109—22no—
148C2×D4×C109—21no—
149C2×D4×C109—21no—
150C2×D4×C109—42no—
151C2×D4×C109—44no—
152C2×D4×C109C2×D411yescharacteristic
153C22×C4×C109—21no—
154C22×C4×C109—22no—
155C22×C4×C109—22no—
156C22×C4×C109—22no—
157C22×C4×C109—22no—
158C22×C4×C109—21no—
159C22×C4×C109—21no—
160C22×C4×C109—22no—
161C22×C4×C109—42no—
162C22×C4×C109—44no—
163C22×C4×C109—41no—
164C22×C4×C109—41no—
165C22×C4×C109C2×D411yescharacteristic
166C22×C4×C109C2×D411yescharacteristic
167C22×C4×C109C2×D411yescharacteristic
168C22×C4×C109C2×D412yes—
169C22×C4×C109C2×D411yescharacteristic
170C22×C4×C109C2×D411yescharacteristic
171C22×C4×C109C2411yescharacteristic
172M16×C109—42no—
173C2×C8×C109—22no—
174C2×C8×C109—22no—
175C2×C8×C109—42no—
176C4⋊C4×C109—22no—
177C4⋊C4×C109—22no—
178C4⋊C4×C109—22no—
179C4⋊C4×C109—22no—
180C4⋊C4×C109—21no—
181C4⋊C4×C109—21no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—21no—
184C4⋊C4×C109—21no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—21no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—22no—
189C4⋊C4×C109—48no—
190C4⋊C4×C109—41no—
191C4⋊C4×C109—41no—
192C4⋊C4×C109C2×D411yescharacteristic
193C4⋊C4×C109C2×D412yes—
194C4⋊C4×C109C2×D411yescharacteristic
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—24no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—21no—
203C22⋊C4×C109—21no—
204C22⋊C4×C109—22no—
205C22⋊C4×C109—21no—
206C22⋊C4×C109—21no—
207C22⋊C4×C109—22no—
208C22⋊C4×C109—22no—
209C22⋊C4×C109—21no—
210C22⋊C4×C109—48no—
211C42×C109—22no—
212C42×C109—21no—
213C42×C109—42no—
214C23×C109—22no—
215C23×C109—22no—
216C23×C109—22no—
217C23×C109—21no—
218C23×C109—24no—
219C23×C109—41no—
220C23×C109—42no—
221C23×C109—42no—
222C23×C109#12011yescharacteristic
223C23×C109#12011yescharacteristic
224C23×C109C22×D411yescharacteristic
225Q8×C109—42no—
226Q8×C109—41no—
227Q8×C109—42no—
228Q8×C109—82no—
229D4×C109—41no—
230D4×C109—42no—
231D4×C109—44no—
232D4×C109—42no—
233D4×C109—42no—
234D4×C109—81no—
235D4×C109—82no—
236D4×C109—81no—
237D4×C109—82no—
238D4×C109—82no—
239C2×C4×C109—22no—
240C2×C4×C109—22no—
241C2×C4×C109—22no—
242C2×C4×C109—24no—
243C2×C4×C109—21no—
244C2×C4×C109—22no—
245C2×C4×C109—21no—
246C2×C4×C109—22no—
247C2×C4×C109—21no—
248C2×C4×C109—22no—
249C2×C4×C109—41no—
250C2×C4×C109—41no—
251C2×C4×C109—41no—
252C2×C4×C109—42no—
253C2×C4×C109—42no—
254C2×C4×C109—42no—
255C2×C4×C109—42no—
256C2×C4×C109—42no—
257C2×C4×C109—44no—
258C2×C4×C109—42no—
259C2×C4×C109—81no—
260C2×C4×C109—82no—
261C2×C4×C109—82no—
262C2×C4×C109—82no—
263C2×C4×C109—82no—
264C2×C4×C109—82no—
265C2×C4×C109#12011yescharacteristic
266C2×C4×C109#12011yescharacteristic
267C2×C4×C109C22×D411yescharacteristic
268C2×C4×C109C22×D411yescharacteristic
269C8×C109—42no—
270C8×C109—42no—
271C22×C109—21no—
272C22×C109—21no—
273C22×C109—21no—
274C22×C109—21no—
275C22×C109—21no—
276C22×C109—21no—
277C22×C109—21no—
278C22×C109—44no—
279C22×C109—41no—
280C22×C109—42no—
281C22×C109—41no—
282C22×C109—42no—
283C22×C109—41no—
284C22×C109—42no—
285C22×C109—84no—
286C22×C109#45312yes—
287C22×C109C2×#12011yescharacteristic
288C22×C109#57812yes—
289C4×C109—21no—
290C4×C109—21no—
291C4×C109—21no—
292C4×C109—44no—
293C4×C109—41no—
294C4×C109—42no—
295C4×C109—42no—
296C4×C109—41no—
297C4×C109—41no—
298C4×C109—42no—
299C4×C109—82no—
300C4×C109#45812yes—
301#18267C10911yesmaximal, characteristic, direct product factor, Sylow
302C2×C109—22no—
303C2×C109—22no—
304C2×C109—41no—
305C2×C109—81no—
306C2×C109—82no—
307C2×C109#301111yescharacteristic
308C2×C109C2×#45311yescharacteristic
309C2×C109C2×#45811yescharacteristic
310#3536C2×C10911yescharacteristic
311#3092C2×C10911yescharacteristic, complemented
312C2×#484C2×C10911yescharacteristic, complemented
313#2501C2×C10911yescharacteristic, complemented
314#1618C2×C10914yescomplemented
315#1614C2×C10914yescomplemented
316#1508C2×C10911yescharacteristic, complemented
317#1501C2×C10911yescharacteristic, complemented
318#1471C2×C10911yescharacteristic, complemented
319C109#1826711yescharacteristic, direct product factor, Sylow
320#562—22no—
321#554—21no—
322#548—21no—
323#547—21no—
324#546—22no—
325#546—21no—
326#543C22×C10912yes—
327#542—21no—
328#539—21no—
329#538C22×C10912yes—
330#537—21no—
331#537—21no—
332#525C22×C10911yescharacteristic
333C2×#123—22no—
334C2×C22⋊Q8C22×C10911yescharacteristic
335C2×#121C22×C10911yescharacteristic
336C22×C4⋊C4C22×C10911yescharacteristic
337#484—22no—
338#484C22×C10914yes—
339#484C22×C10912yes—
340#483—21no—
341#481C22×C10912yes—
342#480—21no—
343#428—21no—
344C2×#107—21no—
345#417—21no—
346C2×#103—21no—
347#407C22×C10911yescharacteristic
348C2×C22⋊C8C22×C10911yescharacteristic
349#399C22×C10914yes—
350#390—21no—
351#384—21no—
352C4×C22⋊C4C22×C10911yescharacteristic
353#328C22×C10912yes—
354#328C22×C10914yes—
355#327C22×C10912yes—
356#323C22×C10914yes—
357C2×#41—22no—
358C2×#41—21no—
359C2×#41D4×C10911yescharacteristic
360C22×Q8D4×C10911yescharacteristic
361C22×D4D4×C10912yes—
362C22×D4D4×C10911yescharacteristic
363C23×C4—21no—
364C23×C4C23×C10911yescharacteristic
365C2×M16—21no—
366C22×C8—21no—
367C4⋊Q8—21no—
368#126—22no—
369#125—22no—
370#124—21no—
371#124—22no—
372#124—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—21no—
377#123—22no—
378#123—48no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8—22no—
383C22⋊Q8—21no—
384C22⋊Q8—21no—
385C22⋊Q8—22no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8D4×C10912yes—
389C22⋊Q8D4×C10911yescharacteristic
390C22⋊Q8C23×C10912yes—
391C22⋊Q8C23×C10911yescharacteristic
392#121—21no—
393#121—22no—
394#121—22no—
395#121—22no—
396#121—22no—
397#121—22no—
398#121—21no—
399#121—21no—
400#121—21no—
401#121—21no—
402#121D4×C10911yescharacteristic
403#121C23×C10911yescharacteristic
404#120—21no—
405#120—22no—
406C4×Q8—21no—
407C4×D4—22no—
408C4×D4—21no—
409C4×D4—21no—
410C4×D4—21no—
411C4×D4—22no—
412#117D4×C10911yescharacteristic
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—24no—
421C2×C4⋊C4D4×C10911yescharacteristic
422C2×C4⋊C4D4×C10911yescharacteristic
423C2×C4⋊C4C23×C10912yes—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—24no—
428C2×C22⋊C4C23×C10912yes—
429C2×C42—21no—
430#107—24no—
431#107—22no—
432#106—22no—
433#103—24no—
434#103—22no—
435#102—22no—
436C22⋊C8—21no—
437C22⋊C8—21no—
438C22⋊C8C23×C10912yes—
439C22⋊C8C23×C10912yes—
440#95C23×C10912yes—
441C24C2×D4×C10911yescharacteristic
442C24C2×D4×C10912yes—
443#41—42no—
444#41—84no—
445#41—82no—
446C2×Q8—22no—
447C2×Q8—21no—
448C2×Q8—42no—
449C2×Q8C2×D4×C10911yescharacteristic
450C2×Q8C2×D4×C10912yes—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—21no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—42no—
461C2×D4—44no—
462C2×D4C2×D4×C10911yescharacteristic
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—22no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4C2×D4×C10911yescharacteristic
476C22×C4C2×D4×C10911yescharacteristic
477C22×C4C2×D4×C10911yescharacteristic
478C22×C4C2×D4×C10912yes—
479C22×C4C2×D4×C10911yescharacteristic
480C22×C4C2×D4×C10911yescharacteristic
481C22×C4C24×C10911yescharacteristic
482M16—42no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—42no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—48no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4C2×D4×C10911yescharacteristic
503C4⋊C4C2×D4×C10912yes—
504C4⋊C4C2×D4×C10911yescharacteristic
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—24no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—22no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—21no—
520C22⋊C4—48no—
521C42—22no—
522C42—21no—
523C42—42no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—21no—
528C23—24no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23#120×C10911yescharacteristic
533C23#120×C10911yescharacteristic
534C23C22×D4×C10911yescharacteristic
535Q8—42no—
536Q8—41no—
537Q8—42no—
538Q8—82no—
539D4—41no—
540D4—42no—
541D4—44no—
542D4—42no—
543D4—42no—
544D4—81no—
545D4—82no—
546D4—81no—
547D4—82no—
548D4—82no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4#120×C10911yescharacteristic
576C2×C4#120×C10911yescharacteristic
577C2×C4C22×D4×C10911yescharacteristic
578C2×C4C22×D4×C10911yescharacteristic
579C8—42no—
580C8—42no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—44no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—42no—
595C22—84no—
596C22#453×C10912yes—
597C22C2×#120×C10911yescharacteristic
598C22#578×C10912yes—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—44no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—42no—
609C4—82no—
610C4#458×C10912yes—
611C2—22no—
612C2—22no—
613C2—41no—
614C2—82no—
615C2—81no—
616C2C109×#301111yescharacteristic
617C2C2×#453×C10911yescharacteristic
618C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3536C2
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
2. C109×#3092C2semidirect
3. C2×#484×C109C2semidirect
4. C109×#2501C2semidirect
5. C109×#1618C2semidirect
6. C109×#1614C2semidirect
7. C109×#1508C2semidirect
8. C109×#1501C2semidirect
9. C109×#1471C2semidirect
301. #18267C109direct (N × H)
311. #3092C2×C109semidirect
312. C2×#484C2×C109semidirect
313. #2501C2×C109semidirect
314. #1618C2×C109semidirect
315. #1614C2×C109semidirect
316. #1508C2×C109semidirect
317. #1501C2×C109semidirect
318. #1471C2×C109semidirect
319. C109#18267direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#35362
C109×#30922
C2×#484×C1092
C109×#25012
C109×#16182
C109×#16142
C109×#15082
C109×#15012
C109×#14712
#18267109