Finite Group Databasebeta

← C109×#18269 #42136263

Subgroups — C109×#18269 #42136263

Order 27904.

924 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3538C211yesmaximal, characteristic
2C109×#3092C211yesmaximal, characteristic
3C2×#484×C109C211yesmaximal, characteristic, complemented
4C109×#2503C211yesmaximal, characteristic, complemented
5C109×#1617C214yesmaximal, complemented
6C109×#1615C214yesmaximal, complemented
7C109×#1508C211yesmaximal, characteristic, complemented
8C109×#1501C211yesmaximal, characteristic
9C109×#1472C211yesmaximal, characteristic, complemented
10#563×C109—22no—
11#552×C109—21no—
12#552×C109—21no—
13#551×C109—22no—
14D4×Q8×C109—21no—
15D4×Q8×C109—21no—
16#544×C109—21no—
17#544×C109C2212yes—
18#541×C109—21no—
19#539×C109—21no—
20#539×C109C2212yes—
21#536×C109—21no—
22#525×C109C2211yescharacteristic
23C2×C22⋊Q8×C109C2211yescharacteristic
24C2×C22⋊Q8×C109C2211yescharacteristic
25C22×C4⋊C4×C109C2211yescharacteristic
26#484×C109—22no—
27#484×C109C2214yes—
28#484×C109C2212yes—
29#483×C109—21no—
30#481×C109C2212yes—
31#480×C109—21no—
32#428×C109—21no—
33C2×#107×C109—21no—
34#417×C109—21no—
35C2×#103×C109—21no—
36#407×C109C2211yescharacteristic
37C2×C22⋊C8×C109C2211yescharacteristic
38#400×C109C2214yes—
39#391×C109—21no—
40#382×C109—21no—
41#381×C109—22no—
42C4×C22⋊C4×C109C2211yescharacteristic
43#328×C109C2212yes—
44#328×C109C2214yes—
45#327×C109C2212yes—
46#324×C109C2214yes—
47C2×#41×C109D411yescharacteristic
48C22×Q8×C109—21no—
49C22×Q8×C109D411yescharacteristic
50C22×Q8×C109D411yescharacteristic
51C23×C4×C109—21no—
52C23×C4×C109C2311yescharacteristic
53C2×M16×C109—21no—
54C22×C8×C109—21no—
55C4⋊Q8×C109—21no—
56C4⋊Q8×C109—22no—
57C4⋊Q8×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×C109—22no—
66#125×C109—22no—
67#125×C109—22no—
68#124×C109—22no—
69#123×C109—22no—
70#123×C109—22no—
71#123×C109—22no—
72#123×C109—21no—
73#123×C109—22no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—21no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109—22no—
78C22⋊Q8×C109—22no—
79C22⋊Q8×C109—21no—
80C22⋊Q8×C109—21no—
81C22⋊Q8×C109—21no—
82C22⋊Q8×C109—21no—
83C22⋊Q8×C109—21no—
84C22⋊Q8×C109—21no—
85C22⋊Q8×C109—22no—
86C22⋊Q8×C109D412yes—
87C22⋊Q8×C109D411yescharacteristic
88C22⋊Q8×C109C2312yes—
89C22⋊Q8×C109C2311yescharacteristic
90#121×C109D411yescharacteristic
91#121×C109C2311yescharacteristic
92#120×C109—21no—
93C4×Q8×C109—21no—
94C4×Q8×C109—21no—
95C4×D4×C109—22no—
96C4×D4×C109—21no—
97C4×D4×C109—21no—
98C4×D4×C109—22no—
99#117×C109—21no—
100#117×C109—22no—
101#117×C109—22no—
102#117×C109D411yescharacteristic
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×C109—21no—
109C2×C4⋊C4×C109D411yescharacteristic
110C2×C4⋊C4×C109D411yescharacteristic
111C2×C4⋊C4×C109C2312yes—
112C2×C4⋊C4×C109C2312yes—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109—21no—
115C2×C22⋊C4×C109D412yes—
116C2×C42×C109—21no—
117#107×C109—24no—
118#107×C109—22no—
119#106×C109—22no—
120#103×C109—24no—
121#103×C109—22no—
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×C109C2×D411yescharacteristic
131#41×C109—42no—
132C2×Q8×C109—22no—
133C2×Q8×C109—21no—
134C2×Q8×C109—21no—
135C2×Q8×C109—21no—
136C2×Q8×C109—42no—
137C2×Q8×C109—42no—
138C2×Q8×C109—82no—
139C2×Q8×C109C2×D412yes—
140C2×Q8×C109C2×D411yescharacteristic
141C2×D4×C109—21no—
142C2×D4×C109—21no—
143C2×D4×C109—21no—
144C2×D4×C109C2×D411yescharacteristic
145C22×C4×C109—21no—
146C22×C4×C109—22no—
147C22×C4×C109—21no—
148C22×C4×C109—21no—
149C22×C4×C109—22no—
150C22×C4×C109—22no—
151C22×C4×C109—42no—
152C22×C4×C109—44no—
153C22×C4×C109—41no—
154C22×C4×C109—41no—
155C22×C4×C109C2×D412yes—
156C22×C4×C109C2×D411yescharacteristic
157C22×C4×C109C2×D411yescharacteristic
158C22×C4×C109C2×D411yescharacteristic
159C22×C4×C109C2×D412yes—
160C22×C4×C109C2×D411yescharacteristic
161C22×C4×C109C2×D411yescharacteristic
162C22×C4×C109C2411yescharacteristic
163M16×C109—42no—
164C2×C8×C109—22no—
165C2×C8×C109—22no—
166C2×C8×C109—42no—
167C4⋊C4×C109—22no—
168C4⋊C4×C109—22no—
169C4⋊C4×C109—22no—
170C4⋊C4×C109—22no—
171C4⋊C4×C109—22no—
172C4⋊C4×C109—22no—
173C4⋊C4×C109—22no—
174C4⋊C4×C109—22no—
175C4⋊C4×C109—22no—
176C4⋊C4×C109—21no—
177C4⋊C4×C109—21no—
178C4⋊C4×C109—21no—
179C4⋊C4×C109—21no—
180C4⋊C4×C109—22no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—21no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—41no—
187C4⋊C4×C109—41no—
188C4⋊C4×C109C2×D411yescharacteristic
189C4⋊C4×C109C2×D412yes—
190C4⋊C4×C109C2×D411yescharacteristic
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—21no—
198C22⋊C4×C109—21no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—21no—
201C22⋊C4×C109—21no—
202C22⋊C4×C109—21no—
203C22⋊C4×C109—42no—
204C42×C109—22no—
205C42×C109—22no—
206C42×C109—22no—
207C42×C109—21no—
208C42×C109—42no—
209C23×C109—22no—
210C23×C109—22no—
211C23×C109—21no—
212C23×C109—41no—
213C23×C109#12011yescharacteristic
214C23×C109#12011yescharacteristic
215C23×C109C22×D411yescharacteristic
216Q8×C109—42no—
217Q8×C109—41no—
218Q8×C109—42no—
219Q8×C109—42no—
220Q8×C109—81no—
221Q8×C109—82no—
222D4×C109—41no—
223D4×C109—42no—
224D4×C109—42no—
225D4×C109—81no—
226C2×C4×C109—22no—
227C2×C4×C109—22no—
228C2×C4×C109—22no—
229C2×C4×C109—21no—
230C2×C4×C109—21no—
231C2×C4×C109—22no—
232C2×C4×C109—22no—
233C2×C4×C109—21no—
234C2×C4×C109—22no—
235C2×C4×C109—22no—
236C2×C4×C109—24no—
237C2×C4×C109—41no—
238C2×C4×C109—41no—
239C2×C4×C109—41no—
240C2×C4×C109—42no—
241C2×C4×C109—42no—
242C2×C4×C109—42no—
243C2×C4×C109—42no—
244C2×C4×C109—42no—
245C2×C4×C109—44no—
246C2×C4×C109—42no—
247C2×C4×C109—42no—
248C2×C4×C109—81no—
249C2×C4×C109—82no—
250C2×C4×C109—82no—
251C2×C4×C109—82no—
252C2×C4×C109#12011yescharacteristic
253C2×C4×C109#12011yescharacteristic
254C2×C4×C109C22×D411yescharacteristic
255C2×C4×C109C22×D411yescharacteristic
256C8×C109—42no—
257C8×C109—42no—
258C22×C109—21no—
259C22×C109—21no—
260C22×C109—21no—
261C22×C109—21no—
262C22×C109—21no—
263C22×C109—21no—
264C22×C109—21no—
265C22×C109—44no—
266C22×C109—41no—
267C22×C109—41no—
268C22×C109—41no—
269C22×C109#45412yes—
270C22×C109C2×#12011yescharacteristic
271C22×C109#57812yes—
272C4×C109—21no—
273C4×C109—21no—
274C4×C109—21no—
275C4×C109—42no—
276C4×C109—41no—
277C4×C109—42no—
278C4×C109—41no—
279C4×C109—41no—
280C4×C109—42no—
281C4×C109—42no—
282C4×C109—82no—
283C4×C109—82no—
284C4×C109#45812yes—
285#18269C10911yesmaximal, characteristic, direct product factor, Sylow
286C2×C109—22no—
287C2×C109—22no—
288C2×C109—41no—
289C2×C109—81no—
290C2×C109#301111yescharacteristic
291C2×C109C2×#45411yescharacteristic
292C2×C109C2×#45811yescharacteristic
293#3538C2×C10911yescharacteristic
294#3092C2×C10911yescharacteristic
295C2×#484C2×C10911yescharacteristic, complemented
296#2503C2×C10911yescharacteristic, complemented
297#1617C2×C10914yescomplemented
298#1615C2×C10914yescomplemented
299#1508C2×C10911yescharacteristic, complemented
300#1501C2×C10911yescharacteristic
301#1472C2×C10911yescharacteristic, complemented
302C109#1826911yescharacteristic, direct product factor, Sylow
303#563—22no—
304#552—21no—
305#552—21no—
306#551—22no—
307D4×Q8—21no—
308D4×Q8—21no—
309#544—21no—
310#544C22×C10912yes—
311#541—21no—
312#539—21no—
313#539C22×C10912yes—
314#536—21no—
315#525C22×C10911yescharacteristic
316C2×C22⋊Q8C22×C10911yescharacteristic
317C2×C22⋊Q8C22×C10911yescharacteristic
318C22×C4⋊C4C22×C10911yescharacteristic
319#484—22no—
320#484C22×C10914yes—
321#484C22×C10912yes—
322#483—21no—
323#481C22×C10912yes—
324#480—21no—
325#428—21no—
326C2×#107—21no—
327#417—21no—
328C2×#103—21no—
329#407C22×C10911yescharacteristic
330C2×C22⋊C8C22×C10911yescharacteristic
331#400C22×C10914yes—
332#391—21no—
333#382—21no—
334#381—22no—
335C4×C22⋊C4C22×C10911yescharacteristic
336#328C22×C10914yes—
337#328C22×C10912yes—
338#327C22×C10912yes—
339#324C22×C10914yes—
340C2×#41D4×C10911yescharacteristic
341C22×Q8—21no—
342C22×Q8D4×C10911yescharacteristic
343C22×Q8D4×C10911yescharacteristic
344C23×C4—21no—
345C23×C4C23×C10911yescharacteristic
346C2×M16—21no—
347C22×C8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—22no—
350C4⋊Q8—22no—
351C4⋊Q8—22no—
352#126—22no—
353#126—22no—
354#126—22no—
355#126—22no—
356#125—21no—
357#125—22no—
358#125—22no—
359#125—22no—
360#125—22no—
361#124—22no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123—21no—
366#123—22no—
367C22⋊Q8—21no—
368C22⋊Q8—21no—
369C22⋊Q8—22no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8—21no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—22no—
379C22⋊Q8D4×C10912yes—
380C22⋊Q8D4×C10911yescharacteristic
381C22⋊Q8C23×C10912yes—
382C22⋊Q8C23×C10911yescharacteristic
383#121D4×C10911yescharacteristic
384#121C23×C10911yescharacteristic
385#120—21no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×D4—22no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4—22no—
392#117—21no—
393#117—22no—
394#117—22no—
395#117D4×C10911yescharacteristic
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4D4×C10911yescharacteristic
403C2×C4⋊C4D4×C10911yescharacteristic
404C2×C4⋊C4C23×C10912yes—
405C2×C4⋊C4C23×C10912yes—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C22⋊C4D4×C10912yes—
409C2×C42—21no—
410#107—24no—
411#107—22no—
412#106—22no—
413#103—24no—
414#103—22no—
415#102—22no—
416C22⋊C8—21no—
417C22⋊C8—21no—
418C22⋊C8C23×C10912yes—
419C22⋊C8C23×C10912yes—
420#95—24no—
421#95—24no—
422#95C23×C10912yes—
423C24C2×D4×C10911yescharacteristic
424#41—42no—
425C2×Q8—22no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—82no—
432C2×Q8C2×D4×C10912yes—
433C2×Q8C2×D4×C10911yescharacteristic
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4C2×D4×C10911yescharacteristic
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—44no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4C2×D4×C10912yes—
449C22×C4C2×D4×C10911yescharacteristic
450C22×C4C2×D4×C10911yescharacteristic
451C22×C4C2×D4×C10911yescharacteristic
452C22×C4C2×D4×C10912yes—
453C22×C4C2×D4×C10911yescharacteristic
454C22×C4C2×D4×C10911yescharacteristic
455C22×C4C24×C10911yescharacteristic
456M16—42no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—42no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4C2×D4×C10911yescharacteristic
482C4⋊C4C2×D4×C10912yes—
483C4⋊C4C2×D4×C10911yescharacteristic
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C22⋊C4—21no—
496C22⋊C4—42no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—21no—
501C42—42no—
502C23—22no—
503C23—22no—
504C23—21no—
505C23—41no—
506C23#120×C10911yescharacteristic
507C23#120×C10911yescharacteristic
508C23C22×D4×C10911yescharacteristic
509Q8—42no—
510Q8—41no—
511Q8—42no—
512Q8—42no—
513Q8—81no—
514Q8—82no—
515D4—41no—
516D4—42no—
517D4—42no—
518D4—81no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4#120×C10911yescharacteristic
546C2×C4#120×C10911yescharacteristic
547C2×C4C22×D4×C10911yescharacteristic
548C2×C4C22×D4×C10911yescharacteristic
549C8—42no—
550C8—42no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—44no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22#454×C10912yes—
563C22C2×#120×C10911yescharacteristic
564C22#578×C10912yes—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—41no—
573C4—42no—
574C4—42no—
575C4—82no—
576C4—82no—
577C4#458×C10912yes—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—81no—
582C2C109×#301111yescharacteristic
583C2C2×#454×C10911yescharacteristic
584C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radical#525×C109C22
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
3. C2×#484×C109C2semidirect
4. C109×#2503C2semidirect
5. C109×#1617C2semidirect
6. C109×#1615C2semidirect
7. C109×#1508C2semidirect
9. C109×#1472C2semidirect
285. #18269C109direct (N × H)
295. C2×#484C2×C109semidirect
296. #2503C2×C109semidirect
297. #1617C2×C109semidirect
298. #1615C2×C109semidirect
299. #1508C2×C109semidirect
301. #1472C2×C109semidirect
302. C109#18269direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#35382
C109×#30922
C2×#484×C1092
C109×#25032
C109×#16172
C109×#16152
C109×#15082
C109×#15012
C109×#14722
#18269109