Finite Group Databasebeta

← C109×#18157 #42136151

Subgroups — C109×#18157 #42136151

Order 27904.

926 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3455C211yesmaximal, characteristic
2C109×#3113C211yesmaximal, characteristic, complemented
3C109×#3112C211yesmaximal, characteristic, complemented
4C109×#2599C211yesmaximal, characteristic, complemented
5C109×#1612C212yesmaximal, complemented
6C109×#1609C212yesmaximal, complemented
7C109×#1606C212yesmaximal, complemented
8C109×#1603C212yesmaximal, complemented
9C109×#1515C211yesmaximal, characteristic, complemented
10C109×#1514C211yesmaximal, characteristic, complemented
11C109×#1464C211yesmaximal, characteristic, complemented
12#542×C109—21no—
13#542×C109—21no—
14#541×C109—21no—
15#540×C109—21no—
16#539×C109—21no—
17#539×C109C2212yes—
18#538×C109—21no—
19#538×C109C2212yes—
20#533×C109C2211yescharacteristic
21#532×C109C2211yescharacteristic
22#529×C109—22no—
23#529×C109—22no—
24#525×C109—21no—
25C4×#41×C109—21no—
26C2×C4×Q8×C109C2211yescharacteristic
27C2×#117×C109C2211yescharacteristic
28#483×C109C2212yes—
29#480×C109C2212yes—
30#477×C109—21no—
31#475×C109—21no—
32#462×C109—21no—
33#460×C109—21no—
34#451×C109C2212yes—
35#448×C109C2212yes—
36#441×C109—22no—
37#440×C109—22no—
38#432×C109C2211yescharacteristic
39#432×C109C2211yescharacteristic
40#398×C109C2211yescharacteristic
41#397×C109C2211yescharacteristic
42#395×C109C2211yescharacteristic
43#393×C109C2211yescharacteristic
44#388×C109—21no—
45#387×C109—21no—
46#387×C109—21no—
47#385×C109—21no—
48#383×C109C2211yescharacteristic
49#328×C109C2212yes—
50#328×C109C2212yes—
51#327×C109C2212yes—
52#327×C109C2212yes—
53#324×C109C2212yes—
54#323×C109C2212yes—
55C2×#41×C109—21no—
56C22×Q8×C109C2311yescharacteristic
57C23×C4×C109D411yescharacteristic
58C2×Q16×C109—22no—
59C2×SD16×C109—22no—
60C4⋊Q8×C109D411yescharacteristic
61C4⋊Q8×C109D411yescharacteristic
62C4⋊Q8×C109D411yescharacteristic
63#127×C109D411yescharacteristic
64#126×C109—22no—
65#126×C109—22no—
66#126×C109—22no—
67#126×C109—22no—
68#125×C109—22no—
69#125×C109—22no—
70#125×C109—21no—
71#124×C109—22no—
72#124×C109—22no—
73#124×C109—21no—
74#123×C109—22no—
75#123×C109—22no—
76#123×C109—22no—
77#123×C109—22no—
78#123×C109—22no—
79C22⋊Q8×C109—21no—
80C22⋊Q8×C109—21no—
81C22⋊Q8×C109—22no—
82C22⋊Q8×C109—22no—
83C22⋊Q8×C109—21no—
84C22⋊Q8×C109C2312yes—
85#121×C109—21no—
86#121×C109—21no—
87#121×C109—22no—
88#121×C109—22no—
89#121×C109—21no—
90#121×C109C2312yes—
91#120×C109—22no—
92C4×Q8×C109—21no—
93C4×Q8×C109—21no—
94C4×Q8×C109—21no—
95C4×Q8×C109—21no—
96C4×Q8×C109—21no—
97C4×Q8×C109—21no—
98C4×Q8×C109—21no—
99C4×Q8×C109D412yes—
100C4×D4×C109—21no—
101C4×D4×C109—21no—
102C4×D4×C109—22no—
103C4×D4×C109—22no—
104C4×D4×C109—22no—
105C4×D4×C109—22no—
106C4×D4×C109—21no—
107C4×D4×C109D412yes—
108#117×C109—22no—
109#117×C109—21no—
110#117×C109—21no—
111#117×C109—21no—
112#117×C109—21no—
113#117×C109—21no—
114C2×C4⋊C4×C109—21no—
115C2×C4⋊C4×C109—21no—
116C2×C4⋊C4×C109—21no—
117C2×C4⋊C4×C109D411yescharacteristic
118C2×C4⋊C4×C109C2311yescharacteristic
119C2×C4⋊C4×C109C2311yescharacteristic
120C2×C22⋊C4×C109—21no—
121C2×C22⋊C4×C109—21no—
122C2×C22⋊C4×C109D411yescharacteristic
123C2×C22⋊C4×C109C2311yescharacteristic
124C2×C42×C109—22no—
125C2×C42×C109—21no—
126C2×C42×C109D411yescharacteristic
127C2×C42×C109C2311yescharacteristic
128#107×C109—22no—
129#106×C109—22no—
130C4⋊C8×C109—21no—
131C4⋊C8×C109—21no—
132#103×C109—22no—
133#103×C109—22no—
134#103×C109—22no—
135#102×C109—22no—
136C22⋊C8×C109C2312yes—
137C22⋊C8×C109C2312yes—
138#97×C109—21no—
139#97×C109—21no—
140#95×C109—21no—
141#95×C109—21no—
142#95×C109—21no—
143#95×C109—21no—
144#95×C109—21no—
145#95×C109—21no—
146#95×C109C2311yescharacteristic
147#95×C109C2311yescharacteristic
148C24×C109C2×D411yescharacteristic
149#41×C109—42no—
150#41×C109—42no—
151C2×Q8×C109—21no—
152C2×Q8×C109—21no—
153C2×Q8×C109—21no—
154C2×Q8×C109—21no—
155C2×Q8×C109—42no—
156C2×Q8×C109C2×D412yes—
157C2×D4×C109—22no—
158C2×D4×C109—22no—
159C2×D4×C109—22no—
160C2×D4×C109—21no—
161C2×D4×C109—21no—
162C2×D4×C109C2×D412yes—
163C22×C4×C109—21no—
164C22×C4×C109—22no—
165C22×C4×C109—22no—
166C22×C4×C109—22no—
167C22×C4×C109—21no—
168C22×C4×C109—21no—
169C22×C4×C109—21no—
170C22×C4×C109—21no—
171C22×C4×C109—21no—
172C22×C4×C109—41no—
173C22×C4×C109—41no—
174C22×C4×C109C2×D411yescharacteristic
175C22×C4×C109C2×D411yescharacteristic
176C22×C4×C109C2×D411yescharacteristic
177C22×C4×C109C2×D411yescharacteristic
178C22×C4×C109C2×D411yescharacteristic
179C22×C4×C109C2411yescharacteristic
180Q16×C109—44no—
181SD16×C109—44no—
182C2×C8×C109—22no—
183C2×C8×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—21no—
189C4⋊C4×C109—21no—
190C4⋊C4×C109—21no—
191C4⋊C4×C109—21no—
192C4⋊C4×C109—21no—
193C4⋊C4×C109—21no—
194C4⋊C4×C109—22no—
195C4⋊C4×C109—21no—
196C4⋊C4×C109—21no—
197C4⋊C4×C109—22no—
198C4⋊C4×C109—21no—
199C4⋊C4×C109—21no—
200C4⋊C4×C109—42no—
201C4⋊C4×C109—42no—
202C4⋊C4×C109—41no—
203C4⋊C4×C109—41no—
204C4⋊C4×C109C2×D412yes—
205C4⋊C4×C109C2×D412yes—
206C22⋊C4×C109—22no—
207C22⋊C4×C109—22no—
208C22⋊C4×C109—22no—
209C22⋊C4×C109—22no—
210C22⋊C4×C109—22no—
211C22⋊C4×C109—22no—
212C22⋊C4×C109—22no—
213C22⋊C4×C109—21no—
214C22⋊C4×C109—21no—
215C22⋊C4×C109—22no—
216C22⋊C4×C109—21no—
217C22⋊C4×C109—21no—
218C22⋊C4×C109—22no—
219C22⋊C4×C109—42no—
220C22⋊C4×C109—42no—
221C42×C109—22no—
222C42×C109—22no—
223C42×C109—21no—
224C42×C109—22no—
225C42×C109—21no—
226C42×C109—41no—
227C42×C109—41no—
228C42×C109C2×D411yescharacteristic
229C42×C109C2×D411yescharacteristic
230C42×C109C2×D411yescharacteristic
231C42×C109C2×D411yescharacteristic
232C23×C109—21no—
233C23×C109—22no—
234C23×C109—21no—
235C23×C109—21no—
236C23×C109—41no—
237C23×C109—41no—
238C23×C109C22×D411yescharacteristic
239Q8×C109—22no—
240Q8×C109—22no—
241Q8×C109—42no—
242Q8×C109—41no—
243Q8×C109—41no—
244Q8×C109—42no—
245Q8×C109—81no—
246D4×C109—42no—
247D4×C109—42no—
248D4×C109—42no—
249D4×C109—42no—
250D4×C109—81no—
251D4×C109—82no—
252D4×C109—82no—
253C2×C4×C109—21no—
254C2×C4×C109—21no—
255C2×C4×C109—22no—
256C2×C4×C109—21no—
257C2×C4×C109—21no—
258C2×C4×C109—22no—
259C2×C4×C109—22no—
260C2×C4×C109—22no—
261C2×C4×C109—21no—
262C2×C4×C109—21no—
263C2×C4×C109—21no—
264C2×C4×C109—41no—
265C2×C4×C109—41no—
266C2×C4×C109—42no—
267C2×C4×C109—42no—
268C2×C4×C109—44no—
269C2×C4×C109—41no—
270C2×C4×C109—41no—
271C2×C4×C109—41no—
272C2×C4×C109—41no—
273C2×C4×C109—41no—
274C2×C4×C109—41no—
275C2×C4×C109—41no—
276C2×C4×C109—41no—
277C2×C4×C109—41no—
278C2×C4×C109—41no—
279C2×C4×C109—41no—
280C2×C4×C109—41no—
281C2×C4×C109—41no—
282C2×C4×C109—82no—
283C2×C4×C109—82no—
284C2×C4×C109#12011yescharacteristic
285C2×C4×C109#12011yescharacteristic
286C2×C4×C109#12011yescharacteristic
287C2×C4×C109#12011yescharacteristic
288C2×C4×C109C22×D411yescharacteristic
289C2×C4×C109C22×D411yescharacteristic
290C8×C109—42no—
291C8×C109—42no—
292C22×C109—21no—
293C22×C109—21no—
294C22×C109—42no—
295C22×C109—41no—
296C22×C109—42no—
297C22×C109—42no—
298C22×C109—41no—
299C22×C109—42no—
300C22×C109—41no—
301C22×C109—81no—
302C22×C109C2×#12011yescharacteristic
303C22×C109#57511yescharacteristic
304C22×C109#57511yescharacteristic
305C4×C109—21no—
306C4×C109—21no—
307C4×C109—21no—
308C4×C109—21no—
309C4×C109—21no—
310C4×C109—21no—
311C4×C109—41no—
312C4×C109—42no—
313C4×C109—41no—
314C4×C109—42no—
315C4×C109—42no—
316C4×C109—41no—
317C4×C109—42no—
318C4×C109—81no—
319C4×C109—81no—
320C4×C109—81no—
321#18157C10911yesmaximal, characteristic, direct product factor, Sylow
322C2×C109—21no—
323C2×C109—21no—
324C2×C109—82no—
325C2×C109—81no—
326C2×C109#300511yescharacteristic
327C2×C109#302211yescharacteristic
328C2×C109#302511yescharacteristic
329#3455C2×C10911yescharacteristic
330#3113C2×C10911yescharacteristic, complemented
331#3112C2×C10911yescharacteristic, complemented
332#2599C2×C10911yescharacteristic, complemented
333#1612C2×C10912yescomplemented
334#1609C2×C10912yescomplemented
335#1606C2×C10912yescomplemented
336#1603C2×C10912yescomplemented
337#1515C2×C10911yescharacteristic, complemented
338#1514C2×C10911yescharacteristic, complemented
339#1464C2×C10911yescharacteristic, complemented
340C109#1815711yescharacteristic, direct product factor, Sylow
341#542—21no—
342#542—21no—
343#541—21no—
344#540—21no—
345#539—21no—
346#539C22×C10912yes—
347#538—21no—
348#538C22×C10912yes—
349#533C22×C10911yescharacteristic
350#532C22×C10911yescharacteristic
351#529—22no—
352#529—22no—
353#525—21no—
354C4×#41—21no—
355C2×C4×Q8C22×C10911yescharacteristic
356C2×#117C22×C10911yescharacteristic
357#483C22×C10912yes—
358#480C22×C10912yes—
359#477—21no—
360#475—21no—
361#462—21no—
362#460—21no—
363#451C22×C10912yes—
364#448C22×C10912yes—
365#441—22no—
366#440—22no—
367#432C22×C10911yescharacteristic
368#432C22×C10911yescharacteristic
369#398C22×C10911yescharacteristic
370#397C22×C10911yescharacteristic
371#395C22×C10911yescharacteristic
372#393C22×C10911yescharacteristic
373#388—21no—
374#387—21no—
375#387—21no—
376#385—21no—
377#383C22×C10911yescharacteristic
378#328C22×C10912yes—
379#328C22×C10912yes—
380#327C22×C10912yes—
381#327C22×C10912yes—
382#324C22×C10912yes—
383#323C22×C10912yes—
384C2×#41—21no—
385C22×Q8C23×C10911yescharacteristic
386C23×C4D4×C10911yescharacteristic
387C2×Q16—22no—
388C2×SD16—22no—
389C4⋊Q8D4×C10911yescharacteristic
390C4⋊Q8D4×C10911yescharacteristic
391C4⋊Q8D4×C10911yescharacteristic
392#127D4×C10911yescharacteristic
393#126—22no—
394#126—22no—
395#126—22no—
396#126—22no—
397#125—22no—
398#125—22no—
399#125—21no—
400#124—22no—
401#124—22no—
402#124—21no—
403#123—22no—
404#123—22no—
405#123—22no—
406#123—22no—
407#123—22no—
408C22⋊Q8—21no—
409C22⋊Q8—21no—
410C22⋊Q8—22no—
411C22⋊Q8—22no—
412C22⋊Q8—21no—
413C22⋊Q8C23×C10912yes—
414#121—21no—
415#121—21no—
416#121—22no—
417#121—22no—
418#121—21no—
419#121C23×C10912yes—
420#120—22no—
421C4×Q8—21no—
422C4×Q8—21no—
423C4×Q8—21no—
424C4×Q8—21no—
425C4×Q8—21no—
426C4×Q8—21no—
427C4×Q8—21no—
428C4×Q8D4×C10912yes—
429C4×D4—21no—
430C4×D4—21no—
431C4×D4—22no—
432C4×D4—22no—
433C4×D4—22no—
434C4×D4—22no—
435C4×D4—21no—
436C4×D4D4×C10912yes—
437#117—22no—
438#117—21no—
439#117—21no—
440#117—21no—
441#117—21no—
442#117—21no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—21no—
446C2×C4⋊C4D4×C10911yescharacteristic
447C2×C4⋊C4C23×C10911yescharacteristic
448C2×C4⋊C4C23×C10911yescharacteristic
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4D4×C10911yescharacteristic
452C2×C22⋊C4C23×C10911yescharacteristic
453C2×C42—22no—
454C2×C42—21no—
455C2×C42D4×C10911yescharacteristic
456C2×C42C23×C10911yescharacteristic
457#107—22no—
458#106—22no—
459C4⋊C8—21no—
460C4⋊C8—21no—
461#103—22no—
462#103—22no—
463#103—22no—
464#102—22no—
465C22⋊C8C23×C10912yes—
466C22⋊C8C23×C10912yes—
467#97—21no—
468#97—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—21no—
475#95C23×C10911yescharacteristic
476#95C23×C10911yescharacteristic
477C24C2×D4×C10911yescharacteristic
478#41—42no—
479#41—42no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—42no—
485C2×Q8C2×D4×C10912yes—
486C2×D4—22no—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4C2×D4×C10912yes—
492C22×C4—21no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4C2×D4×C10911yescharacteristic
504C22×C4C2×D4×C10911yescharacteristic
505C22×C4C2×D4×C10911yescharacteristic
506C22×C4C2×D4×C10911yescharacteristic
507C22×C4C2×D4×C10911yescharacteristic
508C22×C4C24×C10911yescharacteristic
509Q16—44no—
510SD16—44no—
511C2×C8—22no—
512C2×C8—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—22no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4C2×D4×C10912yes—
534C4⋊C4C2×D4×C10912yes—
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—21no—
543C22⋊C4—21no—
544C22⋊C4—22no—
545C22⋊C4—21no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C42—22no—
551C42—22no—
552C42—21no—
553C42—22no—
554C42—21no—
555C42—41no—
556C42—41no—
557C42C2×D4×C10911yescharacteristic
558C42C2×D4×C10911yescharacteristic
559C42C2×D4×C10911yescharacteristic
560C42C2×D4×C10911yescharacteristic
561C23—21no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—41no—
566C23—41no—
567C23C22×D4×C10911yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—42no—
571Q8—41no—
572Q8—41no—
573Q8—42no—
574Q8—81no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—42no—
579D4—81no—
580D4—82no—
581D4—82no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4#120×C10911yescharacteristic
614C2×C4#120×C10911yescharacteristic
615C2×C4#120×C10911yescharacteristic
616C2×C4#120×C10911yescharacteristic
617C2×C4C22×D4×C10911yescharacteristic
618C2×C4C22×D4×C10911yescharacteristic
619C8—42no—
620C8—42no—
621C22—21no—
622C22—21no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—41no—
628C22—42no—
629C22—41no—
630C22—81no—
631C22C2×#120×C10911yescharacteristic
632C22#575×C10911yescharacteristic
633C22#575×C10911yescharacteristic
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—41no—
646C4—42no—
647C4—81no—
648C4—81no—
649C4—81no—
650C2—21no—
651C2—21no—
652C2—81no—
653C2—82no—
654C2C109×#300511yescharacteristic
655C2C109×#302211yescharacteristic
656C2C109×#302511yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3113C2semidirect
3. C109×#3112C2semidirect
4. C109×#2599C2semidirect
5. C109×#1612C2semidirect
6. C109×#1609C2semidirect
7. C109×#1606C2semidirect
8. C109×#1603C2semidirect
9. C109×#1515C2semidirect
10. C109×#1514C2semidirect
11. C109×#1464C2semidirect
321. #18157C109direct (N × H)
330. #3113C2×C109semidirect
331. #3112C2×C109semidirect
332. #2599C2×C109semidirect
333. #1612C2×C109semidirect
334. #1609C2×C109semidirect
335. #1606C2×C109semidirect
336. #1603C2×C109semidirect
337. #1515C2×C109semidirect
338. #1514C2×C109semidirect
339. #1464C2×C109semidirect
340. C109#18157direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#34552
C109×#31132
C109×#31122
C109×#25992
C109×#16122
C109×#16092
C109×#16062
C109×#16032
C109×#15152
C109×#15142
C109×#14642
#18157109