Finite Group Databasebeta

← C109×#18184 #42136178

Subgroups — C109×#18184 #42136178

Order 27904.

1028 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3493C211yesmaximal, characteristic, complemented
2C109×#3114C212yesmaximal, complemented
3C109×#2392C211yesmaximal, characteristic, complemented
4C109×#1612C212yesmaximal, complemented
5C109×#1611C211yesmaximal, characteristic, complemented
6C109×#1610C211yesmaximal, characteristic, complemented
7C109×#1600C212yesmaximal, complemented
8C109×#1599C211yesmaximal, characteristic, complemented
9C109×#1598C211yesmaximal, characteristic, complemented
10C109×#1509C212yesmaximal, complemented
11C109×#1464C211yesmaximal, characteristic, complemented
12#556×C109—22no—
13#555×C109—22no—
14#553×C109—22no—
15#552×C109—22no—
16#544×C109C2211yescharacteristic
17#538×C109—21no—
18#538×C109C2212yes—
19#537×C109C2211yescharacteristic
20#529×C109—21no—
21#529×C109C2212yes—
22#525×C109—21no—
23#520×C109—21no—
24C2×C4×D4×C109—21no—
25C2×C4×D4×C109C2211yescharacteristic
26C2×#117×C109C2211yescharacteristic
27#483×C109C2212yes—
28#480×C109C2212yes—
29#478×C109—22no—
30#463×C109—22no—
31#450×C109C2212yes—
32#447×C109C2212yes—
33#442×C109—22no—
34#440×C109—22no—
35#432×C109C2212yes—
36#398×C109C2211yescharacteristic
37#397×C109C2211yescharacteristic
38#396×C109—21no—
39#394×C109—21no—
40#394×C109C2211yescharacteristic
41#392×C109C2211yescharacteristic
42C4×C4⋊C4×C109C2211yescharacteristic
43C4×C22⋊C4×C109—21no—
44#328×C109C2212yes—
45#328×C109C2212yes—
46#327×C109C2212yes—
47#327×C109C2212yes—
48#324×C109C2211yescharacteristic
49#324×C109C2211yescharacteristic
50#323×C109C2212yes—
51C2×#41×C109—21no—
52C22×D4×C109—21no—
53C22×D4×C109C2311yescharacteristic
54C23×C4×C109—21no—
55C23×C4×C109D411yescharacteristic
56C2×SD16×C109—22no—
57C2×D8×C109—22no—
58C4⋊Q8×C109—22no—
59#126×C109—22no—
60#126×C109—22no—
61#126×C109—22no—
62#126×C109—22no—
63#125×C109—21no—
64#125×C109—22no—
65#125×C109D412yescomplemented
66#124×C109—21no—
67#124×C109—22no—
68#124×C109—22no—
69#124×C109D412yescomplemented
70#123×C109—22no—
71#123×C109—22no—
72#123×C109—22no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—21no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109C2312yes—
78#121×C109—21no—
79#121×C109—21no—
80#121×C109—22no—
81#121×C109—22no—
82#121×C109—22no—
83#121×C109—21no—
84#121×C109C2312yes—
85#120×C109—22no—
86C4×Q8×C109—22no—
87C4×Q8×C109—21no—
88C4×Q8×C109D412yescomplemented
89C4×D4×C109—22no—
90C4×D4×C109—22no—
91C4×D4×C109—22no—
92C4×D4×C109—21no—
93C4×D4×C109—22no—
94C4×D4×C109—22no—
95C4×D4×C109—22no—
96C4×D4×C109—22no—
97C4×D4×C109—44no—
98C4×D4×C109D412yescomplemented
99#117×C109—22no—
100#117×C109—22no—
101#117×C109—21no—
102#117×C109—22no—
103#117×C109—22no—
104#117×C109—22no—
105C2×C4⋊C4×C109—22no—
106C2×C4⋊C4×C109—21no—
107C2×C4⋊C4×C109—21no—
108C2×C4⋊C4×C109D411yescharacteristic
109C2×C4⋊C4×C109C2311yescharacteristic
110C2×C4⋊C4×C109C2311yescharacteristic
111C2×C22⋊C4×C109—21no—
112C2×C22⋊C4×C109—21no—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109—21no—
115C2×C22⋊C4×C109—21no—
116C2×C22⋊C4×C109D411yescharacteristic
117C2×C22⋊C4×C109C2311yescharacteristic
118C2×C42×C109—21no—
119C2×C42×C109D411yescharacteristic
120C2×C42×C109C2311yescharacteristic
121#107×C109—22no—
122#106×C109—22no—
123C4⋊C8×C109—22no—
124#103×C109—22no—
125#102×C109—22no—
126#102×C109—22no—
127#102×C109—22no—
128C22⋊C8×C109C2312yes—
129C22⋊C8×C109C2312yes—
130#97×C109—22no—
131#95×C109—21no—
132#95×C109—21no—
133#95×C109C2311yescharacteristic
134#95×C109C2311yescharacteristic
135C24×C109—21no—
136C24×C109C2×D411yescharacteristic
137#41×C109—42no—
138#41×C109—42no—
139C2×Q8×C109—21no—
140C2×Q8×C109—21no—
141C2×Q8×C109C2×D412yes—
142C2×D4×C109—21no—
143C2×D4×C109—22no—
144C2×D4×C109—22no—
145C2×D4×C109—22no—
146C2×D4×C109—21no—
147C2×D4×C109—22no—
148C2×D4×C109—42no—
149C2×D4×C109—42no—
150C2×D4×C109—42no—
151C2×D4×C109—42no—
152C2×D4×C109C2×D412yes—
153C22×C4×C109—22no—
154C22×C4×C109—21no—
155C22×C4×C109—22no—
156C22×C4×C109—21no—
157C22×C4×C109—21no—
158C22×C4×C109—22no—
159C22×C4×C109—22no—
160C22×C4×C109—21no—
161C22×C4×C109—21no—
162C22×C4×C109—41no—
163C22×C4×C109—42no—
164C22×C4×C109—42no—
165C22×C4×C109—41no—
166C22×C4×C109—41no—
167C22×C4×C109C2×D411yescharacteristic
168C22×C4×C109C2×D411yescharacteristic
169C22×C4×C109C2×D411yescharacteristic
170C22×C4×C109C2×D411yescharacteristic
171C22×C4×C109C2×D411yescharacteristic
172C22×C4×C109C2411yescharacteristic
173SD16×C109—44no—
174D8×C109—44no—
175C2×C8×C109—22no—
176C2×C8×C109—22no—
177C4⋊C4×C109—21no—
178C4⋊C4×C109—22no—
179C4⋊C4×C109—21no—
180C4⋊C4×C109—22no—
181C4⋊C4×C109—21no—
182C4⋊C4×C109—21no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—22no—
189C4⋊C4×C109—22no—
190C4⋊C4×C109—42no—
191C4⋊C4×C109—42no—
192C4⋊C4×C109C2×D412yes—
193C4⋊C4×C109C2×D412yes—
194C22⋊C4×C109—22no—
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—22no—
203C22⋊C4×C109—22no—
204C22⋊C4×C109—22no—
205C22⋊C4×C109—42no—
206C22⋊C4×C109—42no—
207C22⋊C4×C109—42no—
208C22⋊C4×C109—42no—
209C22⋊C4×C109—42no—
210C42×C109—22no—
211C42×C109—22no—
212C42×C109—21no—
213C42×C109—21no—
214C42×C109—22no—
215C42×C109—42no—
216C42×C109C2×D412yes—
217C42×C109C2×D412yes—
218C23×C109—22no—
219C23×C109—21no—
220C23×C109—22no—
221C23×C109—21no—
222C23×C109—21no—
223C23×C109—41no—
224C23×C109—41no—
225C23×C109—41no—
226C23×C109—41no—
227C23×C109—41no—
228C23×C109—41no—
229C23×C109—41no—
230C23×C109—41no—
231C23×C109C22×D411yescharacteristic
232Q8×C109—42no—
233Q8×C109—42no—
234Q8×C109—81no—
235D4×C109—24no—
236D4×C109—42no—
237D4×C109—42no—
238D4×C109—42no—
239D4×C109—48no—
240D4×C109—42no—
241D4×C109—42no—
242D4×C109—82no—
243D4×C109—81no—
244D4×C109—82no—
245C2×C4×C109—21no—
246C2×C4×C109—21no—
247C2×C4×C109—21no—
248C2×C4×C109—21no—
249C2×C4×C109—22no—
250C2×C4×C109—21no—
251C2×C4×C109—22no—
252C2×C4×C109—21no—
253C2×C4×C109—22no—
254C2×C4×C109—22no—
255C2×C4×C109—41no—
256C2×C4×C109—41no—
257C2×C4×C109—42no—
258C2×C4×C109—42no—
259C2×C4×C109—44no—
260C2×C4×C109—41no—
261C2×C4×C109—41no—
262C2×C4×C109—41no—
263C2×C4×C109—41no—
264C2×C4×C109—41no—
265C2×C4×C109—41no—
266C2×C4×C109—41no—
267C2×C4×C109—41no—
268C2×C4×C109—41no—
269C2×C4×C109—41no—
270C2×C4×C109—82no—
271C2×C4×C109—82no—
272C2×C4×C109—82no—
273C2×C4×C109—82no—
274C2×C4×C109#12012yes—
275C2×C4×C109#12011yescharacteristic
276C2×C4×C109#12011yescharacteristic
277C2×C4×C109AGL(1,ℤ/8)12yes—
278C2×C4×C109C22×D412yes—
279C8×C109—42no—
280C8×C109—42no—
281C22×C109—21no—
282C22×C109—21no—
283C22×C109—42no—
284C22×C109—42no—
285C22×C109—42no—
286C22×C109—41no—
287C22×C109—42no—
288C22×C109—42no—
289C22×C109—41no—
290C22×C109—42no—
291C22×C109—42no—
292C22×C109—41no—
293C22×C109—81no—
294C22×C109—81no—
295C22×C109—81no—
296C22×C109—81no—
297C22×C109—81no—
298C22×C109C2×#12011yescharacteristic
299C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C109#57511yescharacteristic
301C4×C109—22no—
302C4×C109—21no—
303C4×C109—21no—
304C4×C109—22no—
305C4×C109—22no—
306C4×C109—42no—
307C4×C109—42no—
308C4×C109—41no—
309C4×C109—42no—
310C4×C109—41no—
311C4×C109—42no—
312C4×C109—81no—
313C4×C109—81no—
314#18184C10911yesmaximal, characteristic, direct product factor, Sylow
315C2×C109—21no—
316C2×C109—21no—
317C2×C109—82no—
318C2×C109—81no—
319C2×C109—82no—
320C2×C109#300611yescharacteristic
321C2×C109#301811yescharacteristic
322C2×C109#302511yescharacteristic
323#3493C2×C10911yescharacteristic, complemented
324#3114C2×C10912yescomplemented
325#2392C2×C10911yescharacteristic, complemented
326#1612C2×C10912yescomplemented
327#1611C2×C10911yescharacteristic, complemented
328#1610C2×C10911yescharacteristic, complemented
329#1600C2×C10912yescomplemented
330#1599C2×C10911yescharacteristic, complemented
331#1598C2×C10911yescharacteristic, complemented
332#1509C2×C10912yescomplemented
333#1464C2×C10911yescharacteristic, complemented
334C109#1818411yescharacteristic, direct product factor, Sylow
335#556—22no—
336#555—22no—
337#553—22no—
338#552—22no—
339#544C22×C10911yescharacteristic
340#538—21no—
341#538C22×C10912yes—
342#537C22×C10911yescharacteristic
343#529—21no—
344#529C22×C10912yes—
345#525—21no—
346#520—21no—
347C2×C4×D4—21no—
348C2×C4×D4C22×C10911yescharacteristic
349C2×#117C22×C10911yescharacteristic
350#483C22×C10912yes—
351#480C22×C10912yes—
352#478—22no—
353#463—22no—
354#450C22×C10912yes—
355#447C22×C10912yes—
356#442—22no—
357#440—22no—
358#432C22×C10912yes—
359#398C22×C10911yescharacteristic
360#397C22×C10911yescharacteristic
361#396—21no—
362#394—21no—
363#394C22×C10911yescharacteristic
364#392C22×C10911yescharacteristic
365C4×C4⋊C4C22×C10911yescharacteristic
366C4×C22⋊C4—21no—
367#328C22×C10912yes—
368#328C22×C10912yes—
369#327C22×C10912yes—
370#327C22×C10912yes—
371#324C22×C10911yescharacteristic
372#324C22×C10911yescharacteristic
373#323C22×C10912yes—
374C2×#41—21no—
375C22×D4—21no—
376C22×D4C23×C10911yescharacteristic
377C23×C4—21no—
378C23×C4D4×C10911yescharacteristic
379C2×SD16—22no—
380C2×D8—22no—
381C4⋊Q8—22no—
382#126—22no—
383#126—22no—
384#126—22no—
385#126—22no—
386#125—21no—
387#125—22no—
388#125D4×C10912yescomplemented
389#124—21no—
390#124—22no—
391#124—22no—
392#124D4×C10912yescomplemented
393#123—22no—
394#123—22no—
395#123—22no—
396C22⋊Q8—21no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8—22no—
400C22⋊Q8C23×C10912yes—
401#121—21no—
402#121—21no—
403#121—22no—
404#121—22no—
405#121—22no—
406#121—21no—
407#121C23×C10912yes—
408#120—22no—
409C4×Q8—22no—
410C4×Q8—21no—
411C4×Q8D4×C10912yescomplemented
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420C4×D4—44no—
421C4×D4D4×C10912yescomplemented
422#117—22no—
423#117—22no—
424#117—21no—
425#117—22no—
426#117—22no—
427#117—22no—
428C2×C4⋊C4—22no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4D4×C10911yescharacteristic
432C2×C4⋊C4C23×C10911yescharacteristic
433C2×C4⋊C4C23×C10911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4D4×C10911yescharacteristic
440C2×C22⋊C4C23×C10911yescharacteristic
441C2×C42—21no—
442C2×C42D4×C10911yescharacteristic
443C2×C42C23×C10911yescharacteristic
444#107—22no—
445#106—22no—
446C4⋊C8—22no—
447#103—22no—
448#102—22no—
449#102—22no—
450#102—22no—
451C22⋊C8C23×C10912yes—
452C22⋊C8C23×C10912yes—
453#97—22no—
454#95—21no—
455#95—21no—
456#95C23×C10911yescharacteristic
457#95C23×C10911yescharacteristic
458C24—21no—
459C24C2×D4×C10911yescharacteristic
460#41—42no—
461#41—42no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8C2×D4×C10912yes—
465C2×D4—21no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—21no—
470C2×D4—22no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C10912yes—
476C22×C4—22no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—41no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4C2×D4×C10911yescharacteristic
491C22×C4C2×D4×C10911yescharacteristic
492C22×C4C2×D4×C10911yescharacteristic
493C22×C4C2×D4×C10911yescharacteristic
494C22×C4C2×D4×C10911yescharacteristic
495C22×C4C24×C10911yescharacteristic
496SD16—44no—
497D8—44no—
498C2×C8—22no—
499C2×C8—22no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4C2×D4×C10912yes—
516C4⋊C4C2×D4×C10912yes—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C42—22no—
534C42—22no—
535C42—21no—
536C42—21no—
537C42—22no—
538C42—42no—
539C42C2×D4×C10912yes—
540C42C2×D4×C10912yes—
541C23—22no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23C22×D4×C10911yescharacteristic
555Q8—42no—
556Q8—42no—
557Q8—81no—
558D4—24no—
559D4—42no—
560D4—42no—
561D4—42no—
562D4—48no—
563D4—42no—
564D4—42no—
565D4—82no—
566D4—81no—
567D4—82no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#120×C10912yes—
598C2×C4#120×C10911yescharacteristic
599C2×C4#120×C10911yescharacteristic
600C2×C4AGL(1,ℤ/8)×C10912yes—
601C2×C4C22×D4×C10912yes—
602C8—42no—
603C8—42no—
604C22—21no—
605C22—21no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—41no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×#120×C10911yescharacteristic
622C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
623C22#575×C10911yescharacteristic
624C4—22no—
625C4—21no—
626C4—21no—
627C4—22no—
628C4—22no—
629C4—42no—
630C4—42no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C2—21no—
638C2—21no—
639C2—81no—
640C2—82no—
641C2—82no—
642C2C109×#300611yescharacteristic
643C2C109×#301811yescharacteristic
644C2C109×#302511yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
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
1. C109×#3493C2semidirect
2. C109×#3114C2semidirect
3. C109×#2392C2semidirect
4. C109×#1612C2semidirect
5. C109×#1611C2semidirect
6. C109×#1610C2semidirect
7. C109×#1600C2semidirect
8. C109×#1599C2semidirect
9. C109×#1598C2semidirect
10. C109×#1509C2semidirect
11. C109×#1464C2semidirect
65. #125×C109D4semidirect
69. #124×C109D4semidirect
88. C4×Q8×C109D4semidirect
98. C4×D4×C109D4semidirect
314. #18184C109direct (N × H)
323. #3493C2×C109semidirect
324. #3114C2×C109semidirect
325. #2392C2×C109semidirect
326. #1612C2×C109semidirect
327. #1611C2×C109semidirect
328. #1610C2×C109semidirect
329. #1600C2×C109semidirect
330. #1599C2×C109semidirect
331. #1598C2×C109semidirect
332. #1509C2×C109semidirect
333. #1464C2×C109semidirect
334. C109#18184direct (N × H)
388. #125D4×C109semidirect
392. #124D4×C109semidirect
411. C4×Q8D4×C109semidirect
421. C4×D4D4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34932
C109×#31142
C109×#23922
C109×#16122
C109×#16112
C109×#16102
C109×#16002
C109×#15992
C109×#15982
C109×#15092
C109×#14642
#18184109