Finite Group Databasebeta

← C109×#18170 #42136164

Subgroups — C109×#18170 #42136164

Order 27904.

1188 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3490C211yesmaximal, characteristic
2C109×#3092C212yesmaximal, complemented
3C2×#399×C109C211yesmaximal, characteristic, complemented
4C109×#1618C214yesmaximal, complemented
5C109×#1614C212yesmaximal, complemented
6C109×#1613C212yesmaximal, complemented
7C109×#1504C212yesmaximal, complemented
8C109×#1462C211yesmaximal, characteristic, complemented
9#551×C109—22no—
10#551×C109—22no—
11#546×C109—22no—
12#546×C109—22no—
13#543×C109C2211yescharacteristic
14#538×C109—21no—
15#538×C109C2212yes—
16#534×C109C2211yescharacteristic
17#525×C109—21no—
18#525×C109C2212yes—
19C2×#120×C109—21no—
20#518×C109—21no—
21C22×C4⋊C4×C109—21no—
22C22×C4⋊C4×C109C2211yescharacteristic, complemented
23C22×C22⋊C4×C109—21no—
24C22×C22⋊C4×C109C2211yescharacteristic
25#484×C109C2214yescomplemented
26#483×C109—22no—
27#481×C109C2214yescomplemented
28#480×C109—22no—
29#428×C109—22no—
30#417×C109—22no—
31#407×C109C2212yescomplemented
32#399×C109—21no—
33#399×C109—21no—
34#399×C109C2212yes—
35#399×C109C2212yescomplemented
36C2×#95×C109C2211yescharacteristic, complemented
37#328×C109C2214yescomplemented
38#327×C109C2214yescomplemented
39#323×C109C2212yes—
40#323×C109C2212yescomplemented
41C25×C109D411yescharacteristic
42C2×#41×C109—21no—
43C2×#41×C109D412yes—
44C22×D4×C109—21no—
45C22×D4×C109—21no—
46C22×D4×C109—21no—
47C23×C4×C109—21no—
48C23×C4×C109—21no—
49C23×C4×C109D411yescharacteristic
50C23×C4×C109C2311yescharacteristic
51C2×M16×C109—22no—
52#126×C109—22no—
53#126×C109—22no—
54#124×C109—22no—
55#124×C109—22no—
56#123×C109—22no—
57#123×C109—22no—
58#123×C109—22no—
59#123×C109—22no—
60#123×C109—22no—
61C22⋊Q8×C109—21no—
62C22⋊Q8×C109—22no—
63C22⋊Q8×C109—21no—
64C22⋊Q8×C109—22no—
65C22⋊Q8×C109D412yes—
66C22⋊Q8×C109C2312yes—
67#121×C109—21no—
68#121×C109—22no—
69#121×C109—22no—
70#121×C109—21no—
71#121×C109D412yes—
72#121×C109C2312yes—
73#120×C109—22no—
74#120×C109—22no—
75#120×C109—22no—
76#120×C109—22no—
77#120×C109—22no—
78C4×D4×C109—22no—
79C4×D4×C109—22no—
80C4×D4×C109—22no—
81C4×D4×C109—22no—
82C4×D4×C109—22no—
83C4×D4×C109—22no—
84#117×C109—22no—
85#117×C109—21no—
86#117×C109D412yes—
87C2×C4⋊C4×C109—22no—
88C2×C4⋊C4×C109—22no—
89C2×C4⋊C4×C109—22no—
90C2×C4⋊C4×C109—22no—
91C2×C4⋊C4×C109—21no—
92C2×C4⋊C4×C109—44no—
93C2×C4⋊C4×C109C2312yes—
94C2×C22⋊C4×C109—22no—
95C2×C22⋊C4×C109—21no—
96C2×C22⋊C4×C109—21no—
97C2×C22⋊C4×C109—21no—
98C2×C22⋊C4×C109—22no—
99C2×C22⋊C4×C109—22no—
100C2×C22⋊C4×C109—22no—
101C2×C22⋊C4×C109—22no—
102C2×C22⋊C4×C109—44no—
103C2×C22⋊C4×C109D411yescharacteristic
104C2×C22⋊C4×C109D411yescharacteristic
105C2×C22⋊C4×C109C2311yescharacteristic
106C2×C22⋊C4×C109C2311yescharacteristic
107#107×C109—24no—
108#106×C109—24no—
109#103×C109—24no—
110#102×C109—24no—
111C22⋊C8×C109—22no—
112C22⋊C8×C109C2314yes—
113#95×C109—21no—
114#95×C109C2312yes—
115C24×C109—21no—
116C24×C109—22no—
117C24×C109—21no—
118C24×C109—21no—
119C24×C109—41no—
120C24×C109—42no—
121C24×C109—42no—
122C24×C109C2×D411yescharacteristic
123C24×C109C2×D411yescharacteristic
124C24×C109C2×D411yescharacteristic
125#41×C109—44no—
126#41×C109—82no—
127C2×Q8×C109—21no—
128C2×Q8×C109C2×D412yes—
129C2×D4×C109—22no—
130C2×D4×C109—22no—
131C2×D4×C109—22no—
132C2×D4×C109—22no—
133C2×D4×C109—21no—
134C2×D4×C109—22no—
135C2×D4×C109—22no—
136C2×D4×C109—42no—
137C2×D4×C109—42no—
138C2×D4×C109—82no—
139C2×D4×C109—82no—
140C2×D4×C109C2×D412yes—
141C22×C4×C109—21no—
142C22×C4×C109—24no—
143C22×C4×C109—21no—
144C22×C4×C109—22no—
145C22×C4×C109—21no—
146C22×C4×C109—21no—
147C22×C4×C109—22no—
148C22×C4×C109—21no—
149C22×C4×C109—22no—
150C22×C4×C109—22no—
151C22×C4×C109—42no—
152C22×C4×C109—42no—
153C22×C4×C109—42no—
154C22×C4×C109—42no—
155C22×C4×C109—42no—
156C22×C4×C109—42no—
157C22×C4×C109—41no—
158C22×C4×C109—41no—
159C22×C4×C109C2×D412yes—
160C22×C4×C109C2×D411yescharacteristic
161C22×C4×C109C2×D411yescharacteristic
162C22×C4×C109C2×D411yescharacteristic
163C22×C4×C109C2×D412yes—
164C22×C4×C109C2411yescharacteristic
165M16×C109—44no—
166C2×C8×C109—24no—
167C4⋊C4×C109—22no—
168C4⋊C4×C109—21no—
169C4⋊C4×C109—21no—
170C4⋊C4×C109—22no—
171C4⋊C4×C109—22no—
172C4⋊C4×C109—22no—
173C4⋊C4×C109—24no—
174C4⋊C4×C109—44no—
175C4⋊C4×C109—44no—
176C4⋊C4×C109—42no—
177C4⋊C4×C109C2×D412yes—
178C4⋊C4×C109C2×D412yes—
179C22⋊C4×C109—22no—
180C22⋊C4×C109—22no—
181C22⋊C4×C109—22no—
182C22⋊C4×C109—22no—
183C22⋊C4×C109—22no—
184C22⋊C4×C109—22no—
185C22⋊C4×C109—22no—
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—44no—
194C22⋊C4×C109—44no—
195C42×C109—22no—
196C42×C109—22no—
197C23×C109—21no—
198C23×C109—21no—
199C23×C109—22no—
200C23×C109—22no—
201C23×C109—21no—
202C23×C109—21no—
203C23×C109—21no—
204C23×C109—22no—
205C23×C109—21no—
206C23×C109—21no—
207C23×C109—42no—
208C23×C109—42no—
209C23×C109—41no—
210C23×C109—41no—
211C23×C109—41no—
212C23×C109—41no—
213C23×C109—48no—
214C23×C109—42no—
215C23×C109—42no—
216C23×C109—42no—
217C23×C109—81no—
218C23×C109—81no—
219C23×C109—84no—
220C23×C109—81no—
221C23×C109—81no—
222C23×C109#12011yescharacteristic
223C23×C109#12011yescharacteristic
224C23×C109#13712yes—
225C23×C109C22×D411yescharacteristic
226Q8×C109—42no—
227Q8×C109—81no—
228D4×C109—42no—
229D4×C109—44no—
230D4×C109—42no—
231D4×C109—42no—
232D4×C109—44no—
233D4×C109—44no—
234D4×C109—82no—
235D4×C109—82no—
236D4×C109—81no—
237C2×C4×C109—24no—
238C2×C4×C109—24no—
239C2×C4×C109—22no—
240C2×C4×C109—21no—
241C2×C4×C109—22no—
242C2×C4×C109—22no—
243C2×C4×C109—21no—
244C2×C4×C109—42no—
245C2×C4×C109—41no—
246C2×C4×C109—41no—
247C2×C4×C109—48no—
248C2×C4×C109—42no—
249C2×C4×C109—42no—
250C2×C4×C109—42no—
251C2×C4×C109—42no—
252C2×C4×C109—41no—
253C2×C4×C109—41no—
254C2×C4×C109—42no—
255C2×C4×C109—42no—
256C2×C4×C109—42no—
257C2×C4×C109—82no—
258C2×C4×C109—81no—
259C2×C4×C109—81no—
260C2×C4×C109—81no—
261C2×C4×C109—82no—
262C2×C4×C109—82no—
263C2×C4×C109—82no—
264C2×C4×C109#12012yes—
265C2×C4×C109C22×D412yes—
266C8×C109—44no—
267C22×C109—24no—
268C22×C109—24no—
269C22×C109—21no—
270C22×C109—21no—
271C22×C109—21no—
272C22×C109—21no—
273C22×C109—21no—
274C22×C109—21no—
275C22×C109—22no—
276C22×C109—42no—
277C22×C109—42no—
278C22×C109—41no—
279C22×C109—41no—
280C22×C109—42no—
281C22×C109—41no—
282C22×C109—42no—
283C22×C109—41no—
284C22×C109—42no—
285C22×C109—41no—
286C22×C109—81no—
287C22×C109—84no—
288C22×C109—84no—
289C22×C109—81no—
290C22×C109—162no—
291C22×C109#45312yes—
292C22×C109#45812yes—
293C22×C109C2×#12011yescharacteristic
294C22×C109C2×#13711yescharacteristic
295C22×C109#57511yescharacteristic
296C4×C109—22no—
297C4×C109—22no—
298C4×C109—44no—
299C4×C109—42no—
300C4×C109—41no—
301C4×C109—41no—
302C4×C109—42no—
303C4×C109—42no—
304C4×C109—82no—
305#18170C10911yesmaximal, characteristic, direct product factor, Sylow
306C2×C109—21no—
307C2×C109—21no—
308C2×C109—22no—
309C2×C109—22no—
310C2×C109—42no—
311C2×C109—81no—
312C2×C109—82no—
313C2×C109#300711yescharacteristic
314C2×C109C2×#45311yescharacteristic
315C2×C109C2×#45811yescharacteristic
316#3490C2×C10911yescharacteristic
317#3092C2×C10912yescomplemented
318C2×#399C2×C10911yescharacteristic, complemented
319#1618C2×C10914yescomplemented
320#1614C2×C10912yescomplemented
321#1613C2×C10912yescomplemented
322#1504C2×C10912yescomplemented
323#1462C2×C10911yescharacteristic, complemented
324C109#1817011yescharacteristic, direct product factor, Sylow
325#551—22no—
326#551—22no—
327#546—22no—
328#546—22no—
329#543C22×C10911yescharacteristic
330#538—21no—
331#538C22×C10912yes—
332#534C22×C10911yescharacteristic
333#525—21no—
334#525C22×C10912yes—
335C2×#120—21no—
336#518—21no—
337C22×C4⋊C4—21no—
338C22×C4⋊C4C22×C10911yescharacteristic, complemented
339C22×C22⋊C4—21no—
340C22×C22⋊C4C22×C10911yescharacteristic
341#484C22×C10914yescomplemented
342#483—22no—
343#481C22×C10914yescomplemented
344#480—22no—
345#428—22no—
346#417—22no—
347#407C22×C10912yescomplemented
348#399—21no—
349#399—21no—
350#399C22×C10912yes—
351#399C22×C10912yescomplemented
352C2×#95C22×C10911yescharacteristic, complemented
353#328C22×C10914yescomplemented
354#327C22×C10914yescomplemented
355#323C22×C10912yes—
356#323C22×C10912yescomplemented
357C25D4×C10911yescharacteristic
358C2×#41—21no—
359C2×#41D4×C10912yes—
360C22×D4—21no—
361C22×D4—21no—
362C22×D4—21no—
363C23×C4—21no—
364C23×C4—21no—
365C23×C4D4×C10911yescharacteristic
366C23×C4C23×C10911yescharacteristic
367C2×M16—22no—
368#126—22no—
369#126—22no—
370#124—22no—
371#124—22no—
372#123—22no—
373#123—22no—
374#123—22no—
375#123—22no—
376#123—22no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8D4×C10912yes—
382C22⋊Q8C23×C10912yes—
383#121—21no—
384#121—22no—
385#121—22no—
386#121—21no—
387#121D4×C10912yes—
388#121C23×C10912yes—
389#120—22no—
390#120—22no—
391#120—22no—
392#120—22no—
393#120—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399C4×D4—22no—
400#117—22no—
401#117—21no—
402#117D4×C10912yes—
403C2×C4⋊C4—22no—
404C2×C4⋊C4—22no—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—44no—
409C2×C4⋊C4C23×C10912yes—
410C2×C22⋊C4—22no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—22no—
415C2×C22⋊C4—22no—
416C2×C22⋊C4—22no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—44no—
419C2×C22⋊C4D4×C10911yescharacteristic
420C2×C22⋊C4D4×C10911yescharacteristic
421C2×C22⋊C4C23×C10911yescharacteristic
422C2×C22⋊C4C23×C10911yescharacteristic
423#107—24no—
424#106—24no—
425#103—24no—
426#102—24no—
427C22⋊C8—22no—
428C22⋊C8C23×C10914yes—
429#95—21no—
430#95C23×C10912yes—
431C24—21no—
432C24—22no—
433C24—21no—
434C24—21no—
435C24—41no—
436C24—42no—
437C24—42no—
438C24C2×D4×C10911yescharacteristic
439C24C2×D4×C10911yescharacteristic
440C24C2×D4×C10911yescharacteristic
441#41—44no—
442#41—82no—
443C2×Q8—21no—
444C2×Q8C2×D4×C10912yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—21no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4C2×D4×C10912yes—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4C2×D4×C10912yes—
476C22×C4C2×D4×C10911yescharacteristic
477C22×C4C2×D4×C10911yescharacteristic
478C22×C4C2×D4×C10911yescharacteristic
479C22×C4C2×D4×C10912yes—
480C22×C4C24×C10911yescharacteristic
481M16—44no—
482C2×C8—24no—
483C4⋊C4—22no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—42no—
493C4⋊C4C2×D4×C10912yes—
494C4⋊C4C2×D4×C10912yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C42—22no—
512C42—22no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—22no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—48no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—81no—
534C23—81no—
535C23—84no—
536C23—81no—
537C23—81no—
538C23#120×C10911yescharacteristic
539C23#120×C10911yescharacteristic
540C23#137×C10912yes—
541C23C22×D4×C10911yescharacteristic
542Q8—42no—
543Q8—81no—
544D4—42no—
545D4—44no—
546D4—42no—
547D4—42no—
548D4—44no—
549D4—44no—
550D4—82no—
551D4—82no—
552D4—81no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—48no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4#120×C10912yes—
581C2×C4C22×D4×C10912yes—
582C8—44no—
583C22—24no—
584C22—24no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—41no—
598C22—42no—
599C22—41no—
600C22—42no—
601C22—41no—
602C22—81no—
603C22—84no—
604C22—84no—
605C22—81no—
606C22—162no—
607C22#453×C10912yes—
608C22#458×C10912yes—
609C22C2×#120×C10911yescharacteristic
610C22C2×#137×C10911yescharacteristic
611C22#575×C10911yescharacteristic
612C4—22no—
613C4—22no—
614C4—44no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—82no—
621C2—21no—
622C2—21no—
623C2—22no—
624C2—22no—
625C2—42no—
626C2—81no—
627C2—82no—
628C2C109×#300711yescharacteristic
629C2C2×#453×C10911yescharacteristic
630C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3490C2
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×#399×C109C2semidirect
4. C109×#1618C2semidirect
5. C109×#1614C2semidirect
6. C109×#1613C2semidirect
7. C109×#1504C2semidirect
8. C109×#1462C2semidirect
22. C22×C4⋊C4×C109C22semidirect
25. #484×C109C22semidirect
27. #481×C109C22semidirect
31. #407×C109C22semidirect
35. #399×C109C22semidirect
36. C2×#95×C109C22semidirect
37. #328×C109C22semidirect
38. #327×C109C22semidirect
40. #323×C109C22semidirect
305. #18170C109direct (N × H)
317. #3092C2×C109semidirect
318. C2×#399C2×C109semidirect
319. #1618C2×C109semidirect
320. #1614C2×C109semidirect
321. #1613C2×C109semidirect
322. #1504C2×C109semidirect
323. #1462C2×C109semidirect
324. C109#18170direct (N × H)
338. C22×C4⋊C4C22×C109semidirect
341. #484C22×C109semidirect
343. #481C22×C109semidirect
347. #407C22×C109semidirect
351. #399C22×C109semidirect
352. C2×#95C22×C109semidirect
353. #328C22×C109semidirect
354. #327C22×C109semidirect
356. #323C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34902
C109×#30922
C2×#399×C1092
C109×#16182
C109×#16142
C109×#16132
C109×#15042
C109×#14622
#18170109