Finite Group Databasebeta

← C109×#18271 #42136265

Subgroups — C109×#18271 #42136265

Order 27904.

984 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3536C211yesmaximal, characteristic
2C109×#3092C211yesmaximal, characteristic, complemented
3C2×#481×C109C211yesmaximal, characteristic, complemented
4C109×#2503C211yesmaximal, characteristic, complemented
5C109×#1616C214yesmaximal, complemented
6C109×#1615C214yesmaximal, complemented
7C109×#1507C211yesmaximal, characteristic, complemented
8C109×#1501C211yesmaximal, characteristic, complemented
9C109×#1472C211yesmaximal, characteristic, complemented
10#562×C109—22no—
11#554×C109—21no—
12#548×C109—21no—
13#547×C109—21no—
14#546×C109—21no—
15#546×C109—22no—
16#543×C109—21no—
17#542×C109—21no—
18#539×C109C2212yes—
19#538×C109—21no—
20#537×C109—21no—
21#537×C109C2212yes—
22#525×C109C2211yescharacteristic
23C2×C22⋊Q8×C109C2211yescharacteristic
24C2×#121×C109C2211yescharacteristic
25C22×C4⋊C4×C109C2211yescharacteristic
26#484×C109C2212yescomplemented
27#483×C109—21no—
28#481×C109—22no—
29#481×C109C2214yes—
30#481×C109C2212yescomplemented
31#480×C109—21no—
32#428×C109—21no—
33C2×#106×C109—21no—
34#417×C109—21no—
35C2×#102×C109—21no—
36#407×C109C2211yescharacteristic
37C2×C22⋊C8×C109C2211yescharacteristic
38#400×C109C2214yescomplemented
39#391×C109—21no—
40#382×C109—21no—
41#381×C109—22no—
42C4×C22⋊C4×C109C2211yescharacteristic
43#328×C109C2212yescomplemented
44#327×C109C2214yes—
45#327×C109C2212yescomplemented
46#324×C109C2214yescomplemented
47C2×#41×C109—21no—
48C2×#41×C109—22no—
49C2×#41×C109D411yescharacteristic
50C22×Q8×C109D411yescharacteristic
51C22×D4×C109—21no—
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—22no—
61#124×C109—21no—
62#124×C109—22no—
63#123×C109—22no—
64#123×C109—22no—
65#123×C109—22no—
66#123×C109—21no—
67#123×C109—22no—
68C22⋊Q8×C109—21no—
69C22⋊Q8×C109—22no—
70C22⋊Q8×C109—22no—
71C22⋊Q8×C109—22no—
72C22⋊Q8×C109—22no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—21no—
76C22⋊Q8×C109—21no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109—22no—
79C22⋊Q8×C109D411yescharacteristic
80C22⋊Q8×C109C2311yescharacteristic
81#121×C109—21no—
82#121×C109—22no—
83#121×C109—22no—
84#121×C109—22no—
85#121×C109—22no—
86#121×C109—21no—
87#121×C109—21no—
88#121×C109—22no—
89#121×C109D412yes—
90#121×C109D411yescharacteristic
91#121×C109C2312yes—
92#121×C109C2311yescharacteristic
93#120×C109—22no—
94#120×C109—21no—
95C4×Q8×C109—21no—
96C4×D4×C109—22no—
97C4×D4×C109—21no—
98C4×D4×C109—21no—
99C4×D4×C109—21no—
100C4×D4×C109—22no—
101#117×C109D411yescharacteristic
102C2×C4⋊C4×C109—21no—
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×C109D411yescharacteristic
109C2×C4⋊C4×C109D411yescharacteristic
110C2×C4⋊C4×C109C2312yes—
111C2×C4⋊C4×C109C2312yes—
112C2×C22⋊C4×C109—22no—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109—21no—
115C2×C22⋊C4×C109D412yes—
116C2×C42×C109—21no—
117#107×C109—22no—
118#106×C109—24no—
119#106×C109—22no—
120#103×C109—22no—
121#102×C109—24no—
122#102×C109—22no—
123C22⋊C8×C109—21no—
124C22⋊C8×C109—21no—
125C22⋊C8×C109C2312yes—
126C22⋊C8×C109C2312yes—
127#95×C109—24no—
128#95×C109—24no—
129#95×C109C2312yes—
130C24×C109—21no—
131C24×C109C2×D411yescharacteristic
132#41×C109—42no—
133#41×C109—84no—
134#41×C109—82no—
135C2×Q8×C109—22no—
136C2×Q8×C109—21no—
137C2×Q8×C109—21no—
138C2×Q8×C109—42no—
139C2×Q8×C109C2×D411yescharacteristic
140C2×D4×C109—21no—
141C2×D4×C109—21no—
142C2×D4×C109—22no—
143C2×D4×C109—22no—
144C2×D4×C109—22no—
145C2×D4×C109—21no—
146C2×D4×C109—22no—
147C2×D4×C109—21no—
148C2×D4×C109—42no—
149C2×D4×C109—82no—
150C2×D4×C109C2×D411yescharacteristic
151C2×D4×C109C2×D412yes—
152C22×C4×C109—22no—
153C22×C4×C109—22no—
154C22×C4×C109—22no—
155C22×C4×C109—21no—
156C22×C4×C109—21no—
157C22×C4×C109—22no—
158C22×C4×C109—22no—
159C22×C4×C109—42no—
160C22×C4×C109—44no—
161C22×C4×C109—41no—
162C22×C4×C109—41no—
163C22×C4×C109C2×D412yes—
164C22×C4×C109C2×D411yescharacteristic
165C22×C4×C109C2×D411yescharacteristic
166C22×C4×C109C2×D411yescharacteristic
167C22×C4×C109C2×D412yes—
168C22×C4×C109C2×D411yescharacteristic
169C22×C4×C109C2×D411yescharacteristic
170C22×C4×C109C2411yescharacteristic
171M16×C109—42no—
172C2×C8×C109—22no—
173C2×C8×C109—22no—
174C2×C8×C109—42no—
175C4⋊C4×C109—22no—
176C4⋊C4×C109—22no—
177C4⋊C4×C109—22no—
178C4⋊C4×C109—22no—
179C4⋊C4×C109—21no—
180C4⋊C4×C109—21no—
181C4⋊C4×C109—21no—
182C4⋊C4×C109—21no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—21no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—41no—
189C4⋊C4×C109—41no—
190C4⋊C4×C109C2×D412yes—
191C4⋊C4×C109C2×D411yescharacteristic
192C4⋊C4×C109C2×D411yescharacteristic
193C22⋊C4×C109—22no—
194C22⋊C4×C109—22no—
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—21no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—21no—
202C22⋊C4×C109—21no—
203C22⋊C4×C109—22no—
204C22⋊C4×C109—21no—
205C22⋊C4×C109—22no—
206C22⋊C4×C109—21no—
207C22⋊C4×C109—42no—
208C42×C109—22no—
209C42×C109—21no—
210C42×C109—42no—
211C23×C109—22no—
212C23×C109—22no—
213C23×C109—22no—
214C23×C109—21no—
215C23×C109—41no—
216C23×C109—42no—
217C23×C109—81no—
218C23×C109—81no—
219C23×C109#12011yescharacteristic
220C23×C109#12011yescharacteristic
221C23×C109C22×D411yescharacteristic
222Q8×C109—41no—
223Q8×C109—42no—
224Q8×C109—81no—
225Q8×C109—82no—
226D4×C109—42no—
227D4×C109—41no—
228D4×C109—44no—
229D4×C109—42no—
230D4×C109—42no—
231D4×C109—42no—
232D4×C109—82no—
233D4×C109—81no—
234D4×C109—82no—
235D4×C109—82no—
236C2×C4×C109—22no—
237C2×C4×C109—22no—
238C2×C4×C109—22no—
239C2×C4×C109—21no—
240C2×C4×C109—22no—
241C2×C4×C109—21no—
242C2×C4×C109—22no—
243C2×C4×C109—21no—
244C2×C4×C109—22no—
245C2×C4×C109—24no—
246C2×C4×C109—41no—
247C2×C4×C109—41no—
248C2×C4×C109—41no—
249C2×C4×C109—42no—
250C2×C4×C109—42no—
251C2×C4×C109—42no—
252C2×C4×C109—42no—
253C2×C4×C109—44no—
254C2×C4×C109—42no—
255C2×C4×C109—42no—
256C2×C4×C109—81no—
257C2×C4×C109—82no—
258C2×C4×C109—82no—
259C2×C4×C109—82no—
260C2×C4×C109—82no—
261C2×C4×C109—82no—
262C2×C4×C109#12011yescharacteristic
263C2×C4×C109#12011yescharacteristic
264C2×C4×C109C22×D411yescharacteristic
265C2×C4×C109C22×D411yescharacteristic
266C8×C109—42no—
267C8×C109—42no—
268C22×C109—21no—
269C22×C109—21no—
270C22×C109—21no—
271C22×C109—21no—
272C22×C109—21no—
273C22×C109—21no—
274C22×C109—21no—
275C22×C109—44no—
276C22×C109—41no—
277C22×C109—42no—
278C22×C109—41no—
279C22×C109—42no—
280C22×C109—41no—
281C22×C109—42no—
282C22×C109—162no—
283C22×C109#45412yes—
284C22×C109C2×#12011yescharacteristic
285C22×C109#57712yes—
286C4×C109—21no—
287C4×C109—21no—
288C4×C109—21no—
289C4×C109—42no—
290C4×C109—42no—
291C4×C109—41no—
292C4×C109—41no—
293C4×C109—41no—
294C4×C109—42no—
295C4×C109—82no—
296C4×C109—82no—
297C4×C109#45812yes—
298#18271C10911yesmaximal, characteristic, direct product factor, Sylow
299C2×C109—22no—
300C2×C109—22no—
301C2×C109—41no—
302C2×C109—81no—
303C2×C109—82no—
304C2×C109#300911yescharacteristic
305C2×C109C2×#45411yescharacteristic
306C2×C109C2×#45811yescharacteristic
307#3536C2×C10911yescharacteristic
308#3092C2×C10911yescharacteristic, complemented
309C2×#481C2×C10911yescharacteristic, complemented
310#2503C2×C10911yescharacteristic, complemented
311#1616C2×C10914yescomplemented
312#1615C2×C10914yescomplemented
313#1507C2×C10911yescharacteristic, complemented
314#1501C2×C10911yescharacteristic, complemented
315#1472C2×C10911yescharacteristic, complemented
316C109#1827111yescharacteristic, direct product factor, Sylow
317#562—22no—
318#554—21no—
319#548—21no—
320#547—21no—
321#546—21no—
322#546—22no—
323#543—21no—
324#542—21no—
325#539C22×C10912yes—
326#538—21no—
327#537—21no—
328#537C22×C10912yes—
329#525C22×C10911yescharacteristic
330C2×C22⋊Q8C22×C10911yescharacteristic
331C2×#121C22×C10911yescharacteristic
332C22×C4⋊C4C22×C10911yescharacteristic
333#484C22×C10912yescomplemented
334#483—21no—
335#481—22no—
336#481C22×C10914yes—
337#481C22×C10912yescomplemented
338#480—21no—
339#428—21no—
340C2×#106—21no—
341#417—21no—
342C2×#102—21no—
343#407C22×C10911yescharacteristic
344C2×C22⋊C8C22×C10911yescharacteristic
345#400C22×C10914yescomplemented
346#391—21no—
347#382—21no—
348#381—22no—
349C4×C22⋊C4C22×C10911yescharacteristic
350#328C22×C10912yescomplemented
351#327C22×C10914yes—
352#327C22×C10912yescomplemented
353#324C22×C10914yescomplemented
354C2×#41—21no—
355C2×#41—22no—
356C2×#41D4×C10911yescharacteristic
357C22×Q8D4×C10911yescharacteristic
358C22×D4—21no—
359C22×D4D4×C10911yescharacteristic
360C23×C4—21no—
361C23×C4C23×C10911yescharacteristic
362C2×M16—21no—
363C22×C8—21no—
364C4⋊Q8—21no—
365#126—22no—
366#125—22no—
367#124—22no—
368#124—21no—
369#124—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—21no—
374#123—22no—
375C22⋊Q8—21no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—22no—
380C22⋊Q8—21no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—21no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8D4×C10911yescharacteristic
387C22⋊Q8C23×C10911yescharacteristic
388#121—21no—
389#121—22no—
390#121—22no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—21no—
395#121—22no—
396#121D4×C10912yes—
397#121D4×C10911yescharacteristic
398#121C23×C10912yes—
399#121C23×C10911yescharacteristic
400#120—22no—
401#120—21no—
402C4×Q8—21no—
403C4×D4—22no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—21no—
407C4×D4—22no—
408#117D4×C10911yescharacteristic
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4D4×C10911yescharacteristic
416C2×C4⋊C4D4×C10911yescharacteristic
417C2×C4⋊C4C23×C10912yes—
418C2×C4⋊C4C23×C10912yes—
419C2×C22⋊C4—22no—
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4D4×C10912yes—
423C2×C42—21no—
424#107—22no—
425#106—24no—
426#106—22no—
427#103—22no—
428#102—24no—
429#102—22no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8C23×C10912yes—
433C22⋊C8C23×C10912yes—
434#95—24no—
435#95—24no—
436#95C23×C10912yes—
437C24—21no—
438C24C2×D4×C10911yescharacteristic
439#41—42no—
440#41—84no—
441#41—82no—
442C2×Q8—22no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—42no—
446C2×Q8C2×D4×C10911yescharacteristic
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—42no—
456C2×D4—82no—
457C2×D4C2×D4×C10911yescharacteristic
458C2×D4C2×D4×C10912yes—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4C2×D4×C10912yes—
471C22×C4C2×D4×C10911yescharacteristic
472C22×C4C2×D4×C10911yescharacteristic
473C22×C4C2×D4×C10911yescharacteristic
474C22×C4C2×D4×C10912yes—
475C22×C4C2×D4×C10911yescharacteristic
476C22×C4C2×D4×C10911yescharacteristic
477C22×C4C24×C10911yescharacteristic
478M16—42no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—42no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4C2×D4×C10912yes—
498C4⋊C4C2×D4×C10911yescharacteristic
499C4⋊C4C2×D4×C10911yescharacteristic
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—22no—
511C22⋊C4—21no—
512C22⋊C4—22no—
513C22⋊C4—21no—
514C22⋊C4—42no—
515C42—22no—
516C42—21no—
517C42—42no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—21no—
522C23—41no—
523C23—42no—
524C23—81no—
525C23—81no—
526C23#120×C10911yescharacteristic
527C23#120×C10911yescharacteristic
528C23C22×D4×C10911yescharacteristic
529Q8—41no—
530Q8—42no—
531Q8—81no—
532Q8—82no—
533D4—42no—
534D4—41no—
535D4—44no—
536D4—42no—
537D4—42no—
538D4—42no—
539D4—82no—
540D4—81no—
541D4—82no—
542D4—82no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#120×C10911yescharacteristic
570C2×C4#120×C10911yescharacteristic
571C2×C4C22×D4×C10911yescharacteristic
572C2×C4C22×D4×C10911yescharacteristic
573C8—42no—
574C8—42no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—44no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22—42no—
587C22—41no—
588C22—42no—
589C22—162no—
590C22#454×C10912yes—
591C22C2×#120×C10911yescharacteristic
592C22#577×C10912yes—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—42no—
602C4—82no—
603C4—82no—
604C4#458×C10912yes—
605C2—22no—
606C2—22no—
607C2—41no—
608C2—82no—
609C2—81no—
610C2C109×#300911yescharacteristic
611C2C2×#454×C10911yescharacteristic
612C2C2×#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×#481×C109C2semidirect
4. C109×#2503C2semidirect
5. C109×#1616C2semidirect
6. C109×#1615C2semidirect
7. C109×#1507C2semidirect
8. C109×#1501C2semidirect
9. C109×#1472C2semidirect
26. #484×C109C22semidirect
30. #481×C109C22semidirect
38. #400×C109C22semidirect
43. #328×C109C22semidirect
45. #327×C109C22semidirect
46. #324×C109C22semidirect
298. #18271C109direct (N × H)
308. #3092C2×C109semidirect
309. C2×#481C2×C109semidirect
310. #2503C2×C109semidirect
311. #1616C2×C109semidirect
312. #1615C2×C109semidirect
313. #1507C2×C109semidirect
314. #1501C2×C109semidirect
315. #1472C2×C109semidirect
316. C109#18271direct (N × H)
333. #484C22×C109semidirect
337. #481C22×C109semidirect
345. #400C22×C109semidirect
350. #328C22×C109semidirect
352. #327C22×C109semidirect
353. #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×#30922
C2×#481×C1092
C109×#25032
C109×#16162
C109×#16152
C109×#15072
C109×#15012
C109×#14722
#18271109