Finite Group Databasebeta

← C109×#18135 #42136129

Subgroups — C109×#18135 #42136129

Order 27904.

1392 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×#3459C211yesmaximal, characteristic, complemented
2C109×#3003C212yesmaximal, complemented
3C109×#2408C211yesmaximal, characteristic, complemented
4C109×#1597C214yesmaximal, complemented
5C109×#1596C212yesmaximal, complemented
6C109×#1595C212yesmaximal, complemented
7C109×#1513C212yesmaximal, complemented
8C109×#1465C211yesmaximal, characteristic, complemented
9C23×D4×C109C2211yescharacteristic, complemented
10C2×AGL(1,ℤ/8)×C109—22no—
11#543×C109C2211yescharacteristic, complemented
12#539×C109—22no—
13#539×C109C2212yescomplemented
14#538×C109—22no—
15#538×C109—21no—
16#534×C109—21no—
17#534×C109C2211yescharacteristic, complemented
18#529×C109—22no—
19#529×C109—21no—
20#525×C109—22no—
21#525×C109C2212yes—
22C2×#121×C109—21no—
23C2×#120×C109—21no—
24C2×C4×D4×C109—21no—
25C2×C4×D4×C109C2211yescharacteristic
26#450×C109—22no—
27#450×C109C2214yescomplemented
28#447×C109—22no—
29#447×C109C2214yes—
30#418×C109—22no—
31#407×C109C2212yescomplemented
32#394×C109C2212yescomplemented
33#392×C109C2212yes—
34#390×C109—21no—
35#380×C109C2211yescharacteristic, complemented
36#379×C109—21no—
37#328×C109C2214yescomplemented
38#327×C109C2214yescomplemented
39#324×C109C2212yescomplemented
40#323×C109C2212yescomplemented
41C25×C109—21no—
42C2×#41×C109D412yescomplemented
43C22×D4×C109—21no—
44C22×D4×C109—22no—
45C22×D4×C109—22no—
46C22×D4×C109—21no—
47C22×D4×C109—22no—
48C22×D4×C109—44no—
49C22×D4×C109D411yescharacteristic
50C22×D4×C109C2312yes—
51C23×C4×C109—21no—
52C23×C4×C109D411yescharacteristic
53C23×C4×C109C2311yescharacteristic
54AGL(1,ℤ/8)×C109—48no—
55C2×SD16×C109—24no—
56C2×D8×C109—24no—
57C2×M16×C109—22no—
58#126×C109—22no—
59#126×C109—22no—
60#126×C109—22no—
61#125×C109—21no—
62#125×C109—22no—
63#124×C109—22no—
64#124×C109—21no—
65#123×C109—22no—
66#123×C109—22no—
67#123×C109—22no—
68#123×C109—22no—
69#123×C109—22no—
70#123×C109—22no—
71C22⋊Q8×C109—22no—
72C22⋊Q8×C109—22no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109—22no—
75C22⋊Q8×C109D412yescomplemented
76C22⋊Q8×C109C2312yes—
77#121×C109—22no—
78#121×C109—22no—
79#121×C109—22no—
80#121×C109—21no—
81#121×C109—44no—
82#121×C109D412yescomplemented
83#121×C109C2312yes—
84#120×C109—22no—
85#120×C109—22no—
86#120×C109—22no—
87#120×C109—44no—
88C4×Q8×C109—21no—
89C4×D4×C109—22no—
90C4×D4×C109—22no—
91C4×D4×C109—21no—
92C4×D4×C109—22no—
93C4×D4×C109—22no—
94C4×D4×C109—22no—
95C4×D4×C109—22no—
96C4×D4×C109—22no—
97#117×C109—22no—
98#117×C109D412yescomplemented
99C2×C4⋊C4×C109—21no—
100C2×C4⋊C4×C109—21no—
101C2×C4⋊C4×C109C2311yescharacteristic
102C2×C22⋊C4×C109—21no—
103C2×C22⋊C4×C109—21no—
104C2×C22⋊C4×C109—21no—
105C2×C22⋊C4×C109—22no—
106C2×C22⋊C4×C109—22no—
107C2×C22⋊C4×C109—22no—
108C2×C22⋊C4×C109D411yescharacteristic
109C2×C22⋊C4×C109C2311yescharacteristic
110C2×C42×C109—21no—
111C2×C42×C109D411yescharacteristic
112#102×C109—24no—
113#102×C109—24no—
114C22⋊C8×C109—22no—
115C22⋊C8×C109C2314yes—
116#95×C109C2312yes—
117C24×C109—21no—
118C24×C109—22no—
119C24×C109—44no—
120C24×C109—42no—
121C24×C109—42no—
122C24×C109—42no—
123C24×C109—82no—
124C24×C109C2×D411yescharacteristic
125C24×C109C2×D411yescharacteristic
126#41×C109—44no—
127C2×Q8×C109—21no—
128C2×Q8×C109C2×D412yes—
129C2×D4×C109—28no—
130C2×D4×C109—22no—
131C2×D4×C109—24no—
132C2×D4×C109—22no—
133C2×D4×C109—22no—
134C2×D4×C109—22no—
135C2×D4×C109—21no—
136C2×D4×C109—22no—
137C2×D4×C109—22no—
138C2×D4×C109—24no—
139C2×D4×C109—41no—
140C2×D4×C109—42no—
141C2×D4×C109—44no—
142C2×D4×C109—44no—
143C2×D4×C109—48no—
144C2×D4×C109—42no—
145C2×D4×C109—42no—
146C2×D4×C109—44no—
147C2×D4×C109—44no—
148C2×D4×C109—41no—
149C2×D4×C109—41no—
150C2×D4×C109—82no—
151C2×D4×C109C2×D412yes—
152C22×C4×C109—21no—
153C22×C4×C109—21no—
154C22×C4×C109—21no—
155C22×C4×C109—22no—
156C22×C4×C109—21no—
157C22×C4×C109—22no—
158C22×C4×C109—21no—
159C22×C4×C109—22no—
160C22×C4×C109—42no—
161C22×C4×C109—42no—
162C22×C4×C109—82no—
163C22×C4×C109C2×D411yescharacteristic
164C22×C4×C109C2×D411yescharacteristic
165C22×C4×C109C2×D412yes—
166C22×C4×C109C2×D411yescharacteristic
167C22×C4×C109C2×D411yescharacteristic
168C22×C4×C109C2×D412yes—
169C22×C4×C109C2411yescharacteristic
170SD16×C109—48no—
171D8×C109—48no—
172M16×C109—44no—
173C2×C8×C109—24no—
174C4⋊C4×C109—22no—
175C4⋊C4×C109—22no—
176C4⋊C4×C109—22no—
177C4⋊C4×C109—21no—
178C4⋊C4×C109—21no—
179C4⋊C4×C109—22no—
180C4⋊C4×C109—22no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—42no—
184C4⋊C4×C109C2×D412yes—
185C4⋊C4×C109C2×D412yes—
186C22⋊C4×C109—22no—
187C22⋊C4×C109—22no—
188C22⋊C4×C109—22no—
189C22⋊C4×C109—22no—
190C22⋊C4×C109—22no—
191C22⋊C4×C109—22no—
192C22⋊C4×C109—22no—
193C22⋊C4×C109—22no—
194C22⋊C4×C109—22no—
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—44no—
199C22⋊C4×C109—42no—
200C22⋊C4×C109—44no—
201C22⋊C4×C109—82no—
202C42×C109—22no—
203C42×C109—22no—
204C42×C109—21no—
205C42×C109—21no—
206C23×C109—21no—
207C23×C109—21no—
208C23×C109—21no—
209C23×C109—21no—
210C23×C109—24no—
211C23×C109—22no—
212C23×C109—41no—
213C23×C109—41no—
214C23×C109—41no—
215C23×C109—41no—
216C23×C109—42no—
217C23×C109—48no—
218C23×C109—42no—
219C23×C109—42no—
220C23×C109—42no—
221C23×C109—42no—
222C23×C109—42no—
223C23×C109—42no—
224C23×C109—84no—
225C23×C109—84no—
226C23×C109—81no—
227C23×C109—81no—
228C23×C109—88no—
229C23×C109—88no—
230C23×C109#12012yes—
231C23×C109AGL(1,ℤ/8)12yescomplemented
232C23×C109C22×D411yescharacteristic
233Q8×C109—42no—
234Q8×C109—81no—
235D4×C109—28no—
236D4×C109—42no—
237D4×C109—44no—
238D4×C109—44no—
239D4×C109—44no—
240D4×C109—44no—
241D4×C109—42no—
242D4×C109—42no—
243D4×C109—44no—
244D4×C109—82no—
245D4×C109—82no—
246D4×C109—81no—
247D4×C109—88no—
248C2×C4×C109—22no—
249C2×C4×C109—24no—
250C2×C4×C109—21no—
251C2×C4×C109—22no—
252C2×C4×C109—21no—
253C2×C4×C109—22no—
254C2×C4×C109—21no—
255C2×C4×C109—22no—
256C2×C4×C109—21no—
257C2×C4×C109—21no—
258C2×C4×C109—42no—
259C2×C4×C109—41no—
260C2×C4×C109—42no—
261C2×C4×C109—42no—
262C2×C4×C109—44no—
263C2×C4×C109—41no—
264C2×C4×C109—41no—
265C2×C4×C109—42no—
266C2×C4×C109—42no—
267C2×C4×C109—42no—
268C2×C4×C109—81no—
269C2×C4×C109—81no—
270C2×C4×C109—82no—
271C2×C4×C109—81no—
272C2×C4×C109#12011yescharacteristic
273C2×C4×C109#12011yescharacteristic
274C2×C4×C109C22×D412yes—
275C8×C109—44no—
276C22×C109—21no—
277C22×C109—21no—
278C22×C109—22no—
279C22×C109—22no—
280C22×C109—22no—
281C22×C109—22no—
282C22×C109—42no—
283C22×C109—42no—
284C22×C109—42no—
285C22×C109—44no—
286C22×C109—44no—
287C22×C109—41no—
288C22×C109—41no—
289C22×C109—42no—
290C22×C109—42no—
291C22×C109—42no—
292C22×C109—44no—
293C22×C109—81no—
294C22×C109—82no—
295C22×C109—82no—
296C22×C109—84no—
297C22×C109—816no—
298C22×C109—82no—
299C22×C109—82no—
300C22×C109C2×#12011yescharacteristic
301C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C109#57511yescharacteristic
303C4×C109—22no—
304C4×C109—21no—
305C4×C109—41no—
306C4×C109—42no—
307C4×C109—41no—
308C4×C109—42no—
309C4×C109—41no—
310C4×C109—42no—
311C4×C109—82no—
312C4×C109B(4,𝔽2)12yes—
313#18135C10911yesmaximal, characteristic, direct product factor, Sylow
314C2×C109—21no—
315C2×C109—21no—
316C2×C109—42no—
317C2×C109—42no—
318C2×C109—82no—
319C2×C109—84no—
320C2×C109#300611yescharacteristic
321C2×C109#302111yescharacteristic
322C2×C109C2×B(4,𝔽2)11yescharacteristic
323#3459C2×C10911yescharacteristic, complemented
324#3003C2×C10912yescomplemented
325#2408C2×C10911yescharacteristic, complemented
326#1597C2×C10914yescomplemented
327#1596C2×C10912yescomplemented
328#1595C2×C10912yescomplemented
329#1513C2×C10912yescomplemented
330#1465C2×C10911yescharacteristic, complemented
331C109#1813511yescharacteristic, direct product factor, Sylow
332C23×D4C22×C10911yescharacteristic, complemented
333C2×AGL(1,ℤ/8)—22no—
334#543C22×C10911yescharacteristic, complemented
335#539—22no—
336#539C22×C10912yescomplemented
337#538—22no—
338#538—21no—
339#534—21no—
340#534C22×C10911yescharacteristic, complemented
341#529—22no—
342#529—21no—
343#525—22no—
344#525C22×C10912yes—
345C2×#121—21no—
346C2×#120—21no—
347C2×C4×D4—21no—
348C2×C4×D4C22×C10911yescharacteristic
349#450—22no—
350#450C22×C10914yescomplemented
351#447—22no—
352#447C22×C10914yes—
353#418—22no—
354#407C22×C10912yescomplemented
355#394C22×C10912yescomplemented
356#392C22×C10912yes—
357#390—21no—
358#380C22×C10911yescharacteristic, complemented
359#379—21no—
360#328C22×C10914yescomplemented
361#327C22×C10914yescomplemented
362#324C22×C10912yescomplemented
363#323C22×C10912yescomplemented
364C25—21no—
365C2×#41D4×C10912yescomplemented
366C22×D4—21no—
367C22×D4—22no—
368C22×D4—22no—
369C22×D4—21no—
370C22×D4—22no—
371C22×D4—44no—
372C22×D4D4×C10911yescharacteristic
373C22×D4C23×C10912yes—
374C23×C4—21no—
375C23×C4D4×C10911yescharacteristic
376C23×C4C23×C10911yescharacteristic
377AGL(1,ℤ/8)—48no—
378C2×SD16—24no—
379C2×D8—24no—
380C2×M16—22no—
381#126—22no—
382#126—22no—
383#126—22no—
384#125—21no—
385#125—22no—
386#124—22no—
387#124—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394C22⋊Q8—22no—
395C22⋊Q8—22no—
396C22⋊Q8—21no—
397C22⋊Q8—22no—
398C22⋊Q8D4×C10912yescomplemented
399C22⋊Q8C23×C10912yes—
400#121—22no—
401#121—22no—
402#121—22no—
403#121—21no—
404#121—44no—
405#121D4×C10912yescomplemented
406#121C23×C10912yes—
407#120—22no—
408#120—22no—
409#120—22no—
410#120—44no—
411C4×Q8—21no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—21no—
415C4×D4—22no—
416C4×D4—22no—
417C4×D4—22no—
418C4×D4—22no—
419C4×D4—22no—
420#117—22no—
421#117D4×C10912yescomplemented
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4C23×C10911yescharacteristic
425C2×C22⋊C4—21no—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—21no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4D4×C10911yescharacteristic
432C2×C22⋊C4C23×C10911yescharacteristic
433C2×C42—21no—
434C2×C42D4×C10911yescharacteristic
435#102—24no—
436#102—24no—
437C22⋊C8—22no—
438C22⋊C8C23×C10914yes—
439#95C23×C10912yes—
440C24—21no—
441C24—22no—
442C24—44no—
443C24—42no—
444C24—42no—
445C24—42no—
446C24—82no—
447C24C2×D4×C10911yescharacteristic
448C24C2×D4×C10911yescharacteristic
449#41—44no—
450C2×Q8—21no—
451C2×Q8C2×D4×C10912yes—
452C2×D4—28no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—24no—
462C2×D4—41no—
463C2×D4—42no—
464C2×D4—44no—
465C2×D4—44no—
466C2×D4—48no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—44no—
470C2×D4—44no—
471C2×D4—41no—
472C2×D4—41no—
473C2×D4—82no—
474C2×D4C2×D4×C10912yes—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—22no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—82no—
486C22×C4C2×D4×C10911yescharacteristic
487C22×C4C2×D4×C10911yescharacteristic
488C22×C4C2×D4×C10912yes—
489C22×C4C2×D4×C10911yescharacteristic
490C22×C4C2×D4×C10911yescharacteristic
491C22×C4C2×D4×C10912yes—
492C22×C4C24×C10911yescharacteristic
493SD16—48no—
494D8—48no—
495M16—44no—
496C2×C8—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—42no—
507C4⋊C4C2×D4×C10912yes—
508C4⋊C4C2×D4×C10912yes—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—44no—
522C22⋊C4—42no—
523C22⋊C4—44no—
524C22⋊C4—82no—
525C42—22no—
526C42—22no—
527C42—21no—
528C42—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—24no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—48no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—84no—
548C23—84no—
549C23—81no—
550C23—81no—
551C23—88no—
552C23—88no—
553C23#120×C10912yes—
554C23AGL(1,ℤ/8)×C10912yescomplemented
555C23C22×D4×C10911yescharacteristic
556Q8—42no—
557Q8—81no—
558D4—28no—
559D4—42no—
560D4—44no—
561D4—44no—
562D4—44no—
563D4—44no—
564D4—42no—
565D4—42no—
566D4—44no—
567D4—82no—
568D4—82no—
569D4—81no—
570D4—88no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—81no—
592C2×C4—82no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4#120×C10911yescharacteristic
596C2×C4#120×C10911yescharacteristic
597C2×C4C22×D4×C10912yes—
598C8—44no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—44no—
609C22—44no—
610C22—41no—
611C22—41no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—81no—
617C22—82no—
618C22—82no—
619C22—816no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22C2×#120×C10911yescharacteristic
624C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
625C22#575×C10911yescharacteristic
626C4—22no—
627C4—21no—
628C4—41no—
629C4—42no—
630C4—41no—
631C4—42no—
632C4—41no—
633C4—42no—
634C4—82no—
635C4B(4,𝔽2)×C10912yes—
636C2—21no—
637C2—21no—
638C2—42no—
639C2—42no—
640C2—84no—
641C2—82no—
642C2C109×#300611yescharacteristic
643C2C109×#302111yescharacteristic
644C2C2×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×#3459C2semidirect
2. C109×#3003C2semidirect
3. C109×#2408C2semidirect
4. C109×#1597C2semidirect
5. C109×#1596C2semidirect
6. C109×#1595C2semidirect
7. C109×#1513C2semidirect
8. C109×#1465C2semidirect
9. C23×D4×C109C22semidirect
11. #543×C109C22semidirect
13. #539×C109C22semidirect
17. #534×C109C22semidirect
27. #450×C109C22semidirect
31. #407×C109C22semidirect
32. #394×C109C22semidirect
35. #380×C109C22semidirect
37. #328×C109C22semidirect
38. #327×C109C22semidirect
39. #324×C109C22semidirect
40. #323×C109C22semidirect
42. C2×#41×C109D4semidirect
75. C22⋊Q8×C109D4semidirect
82. #121×C109D4semidirect
98. #117×C109D4semidirect
231. C23×C109AGL(1,ℤ/8)semidirect
313. #18135C109direct (N × H)
323. #3459C2×C109semidirect
324. #3003C2×C109semidirect
325. #2408C2×C109semidirect
326. #1597C2×C109semidirect
327. #1596C2×C109semidirect
328. #1595C2×C109semidirect
329. #1513C2×C109semidirect
330. #1465C2×C109semidirect
331. C109#18135direct (N × H)
332. C23×D4C22×C109semidirect
334. #543C22×C109semidirect
336. #539C22×C109semidirect
340. #534C22×C109semidirect
350. #450C22×C109semidirect
354. #407C22×C109semidirect
355. #394C22×C109semidirect
358. #380C22×C109semidirect
360. #328C22×C109semidirect
361. #327C22×C109semidirect
362. #324C22×C109semidirect
363. #323C22×C109semidirect
365. C2×#41D4×C109semidirect
398. C22⋊Q8D4×C109semidirect
405. #121D4×C109semidirect
421. #117D4×C109semidirect
554. C23AGL(1,ℤ/8)×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34592
C109×#30032
C109×#24082
C109×#15972
C109×#15962
C109×#15952
C109×#15132
C109×#14652
#18135109