Finite Group Databasebeta

← C109×#9113 #42127107

Subgroups — C109×#9113 #42127107

Order 27904.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊C8×C109C211yesmaximal, characteristic, complemented
2C109×#2847C211yesmaximal, characteristic
3C2×#327×C109C212yesmaximal, complemented
4C109×C22⋊#95C211yesmaximal, characteristic, complemented
5C109×#1294C212yesmaximal, complemented
6C24×C4×C109C2211yescharacteristic
7C23×C8×C109—21no—
8C2×#121×C109C411yescharacteristic, complemented
9C2×#121×C109C411yescharacteristic, complemented
10C2×#121×C109C2211yescharacteristic
11C2×#120×C109—21no—
12C22×C22⋊C4×C109—21no—
13C2×C22⋊C8×C109—21no—
14C2×C22⋊C8×C109—24no—
15C2×C22⋊C8×C109C2212yes—
16#386×C109—21no—
17#386×C109—21no—
18#386×C109—21no—
19#380×C109C411yescharacteristic, complemented
20#380×C109C411yescharacteristic, complemented
21#380×C109C2211yescharacteristic
22C2×#95×C109—21no—
23C2×#95×C109C2212yes—
24#327×C109—24no—
25C25×C109D411yescharacteristic
26C22×D4×C109—21no—
27C22×D4×C109—21no—
28C22×D4×C109—21no—
29C22×D4×C109C2×C411yescharacteristic
30C23×C4×C109—22no—
31C23×C4×C109—22no—
32C23×C4×C109—22no—
33C23×C4×C109—42no—
34C23×C4×C109—42no—
35C23×C4×C109C2×C411yescharacteristic
36C23×C4×C109C2×C411yescharacteristic
37C23×C4×C109D411yescharacteristic
38C23×C4×C109D411yescharacteristic
39C23×C4×C109D411yescharacteristic
40C23×C4×C109C2311yescharacteristic
41C22×C8×C109—24no—
42C22×C8×C109—22no—
43C22×C8×C109—42no—
44C22×C8×C109—42no—
45#121×C109—22no—
46#121×C109—22no—
47#121×C109—42no—
48#121×C109—42no—
49#120×C109—42no—
50#120×C109—42no—
51C2×C4⋊C4×C109C2×C411yescharacteristic
52C2×C4⋊C4×C109C2×C411yescharacteristic
53C2×C4⋊C4×C109C2×C411yescharacteristic
54C2×C22⋊C4×C109—24no—
55C2×C22⋊C4×C109—21no—
56C2×C22⋊C4×C109—21no—
57C2×C22⋊C4×C109—21no—
58C2×C22⋊C4×C109—21no—
59C2×C22⋊C4×C109—21no—
60C2×C22⋊C4×C109—21no—
61C2×C22⋊C4×C109—42no—
62C2×C22⋊C4×C109—42no—
63C22⋊C8×C109—24no—
64C22⋊C8×C109—22no—
65C22⋊C8×C109D414yescomplemented
66#95×C109—21no—
67#95×C109—21no—
68#95×C109—21no—
69#95×C109—22no—
70#95×C109—22no—
71#95×C109—42no—
72C24×C109—22no—
73C24×C109—22no—
74C24×C109—22no—
75C24×C109—22no—
76C24×C109—21no—
77C24×C109—41no—
78C24×C109—41no—
79C24×C109C22⋊C411yescharacteristic
80C24×C109C22⋊C411yescharacteristic
81C24×C109D812yes—
82C24×C109SD1612yes—
83C24×C109C2×D411yescharacteristic
84C2×D4×C109—22no—
85C2×D4×C109—41no—
86C2×D4×C109—41no—
87C2×D4×C109—42no—
88C2×D4×C109—42no—
89C2×D4×C109—42no—
90C2×D4×C109—42no—
91C2×D4×C109—42no—
92C2×D4×C109—42no—
93C2×D4×C109—42no—
94C2×D4×C109—41no—
95C2×D4×C109—41no—
96C2×D4×C109—42no—
97C22×C4×C109—22no—
98C22×C4×C109—22no—
99C22×C4×C109—21no—
100C22×C4×C109—21no—
101C22×C4×C109—22no—
102C22×C4×C109—22no—
103C22×C4×C109—22no—
104C22×C4×C109—22no—
105C22×C4×C109—22no—
106C22×C4×C109—22no—
107C22×C4×C109—21no—
108C22×C4×C109—21no—
109C22×C4×C109—24no—
110C22×C4×C109—21no—
111C22×C4×C109—21no—
112C22×C4×C109—21no—
113C22×C4×C109—41no—
114C22×C4×C109—41no—
115C22×C4×C109—41no—
116C22×C4×C109—41no—
117C22×C4×C109—41no—
118C22×C4×C109—41no—
119C22×C4×C109—41no—
120C22×C4×C109—48no—
121C22×C4×C109—48no—
122C22×C4×C109—42no—
123C22×C4×C109—42no—
124C22×C4×C109—42no—
125C22×C4×C109—42no—
126C22×C4×C109—42no—
127C22×C4×C109—42no—
128C22×C4×C109C22⋊C411yescharacteristic
129C22×C4×C109C22⋊C411yescharacteristic
130C22×C4×C109C22×C411yescharacteristic
131C22×C4×C109C2×D411yescharacteristic
132C22×C4×C109C2×D412yes—
133C22×C4×C109#4112yes—
134C2×C8×C109—24no—
135C2×C8×C109—48no—
136C2×C8×C109—42no—
137C2×C8×C109—42no—
138C4⋊C4×C109—22no—
139C4⋊C4×C109—41no—
140C4⋊C4×C109—41no—
141C22⋊C4×C109—42no—
142C22⋊C4×C109—48no—
143C22⋊C4×C109—42no—
144C22⋊C4×C109—42no—
145C22⋊C4×C109—41no—
146C22⋊C4×C109—41no—
147C22⋊C4×C109—42no—
148C22⋊C4×C109—41no—
149C22⋊C4×C109—41no—
150C23×C109—21no—
151C23×C109—21no—
152C23×C109—21no—
153C23×C109—21no—
154C23×C109—21no—
155C23×C109—21no—
156C23×C109—21no—
157C23×C109—21no—
158C23×C109—24no—
159C23×C109—24no—
160C23×C109—21no—
161C23×C109—21no—
162C23×C109—21no—
163C23×C109—21no—
164C23×C109—21no—
165C23×C109—21no—
166C23×C109—24no—
167C23×C109—24no—
168C23×C109—42no—
169C23×C109—42no—
170C23×C109—44no—
171C23×C109—44no—
172C23×C109—44no—
173C23×C109—44no—
174C23×C109—41no—
175C23×C109—41no—
176C23×C109—41no—
177C23×C109—41no—
178C23×C109—41no—
179C23×C109—41no—
180C23×C109—41no—
181C23×C109#9912yes—
182C23×C109#10212yes—
183C23×C109#10212yes—
184C23×C109#10412yes—
185C23×C109C2×C22⋊C411yescharacteristic
186C23×C109#12112yes—
187C23×C109C2×D811yescharacteristic
188C23×C109C2×SD1611yescharacteristic
189C23×C109#13512yes—
190D4×C109—82no—
191D4×C109—84no—
192D4×C109—84no—
193D4×C109—82no—
194D4×C109—82no—
195C2×C4×C109—24no—
196C2×C4×C109—24no—
197C2×C4×C109—21no—
198C2×C4×C109—21no—
199C2×C4×C109—21no—
200C2×C4×C109—21no—
201C2×C4×C109—21no—
202C2×C4×C109—21no—
203C2×C4×C109—21no—
204C2×C4×C109—21no—
205C2×C4×C109—21no—
206C2×C4×C109—42no—
207C2×C4×C109—42no—
208C2×C4×C109—44no—
209C2×C4×C109—44no—
210C2×C4×C109—44no—
211C2×C4×C109—42no—
212C2×C4×C109—42no—
213C2×C4×C109—44no—
214C2×C4×C109—44no—
215C2×C4×C109—44no—
216C2×C4×C109—41no—
217C2×C4×C109—41no—
218C2×C4×C109—41no—
219C2×C4×C109—41no—
220C2×C4×C109—41no—
221C2×C4×C109—41no—
222C2×C4×C109—41no—
223C2×C4×C109—41no—
224C2×C4×C109—41no—
225C2×C4×C109—82no—
226C2×C4×C109—82no—
227C2×C4×C109—82no—
228C2×C4×C109—82no—
229C2×C4×C109#9912yes—
230C2×C4×C109C4×D412yes—
231C2×C4×C109#12011yescharacteristic
232C2×C4×C109#12311yescharacteristic
233C8×C109—44no—
234C22×C109—22no—
235C22×C109—22no—
236C22×C109—22no—
237C22×C109—22no—
238C22×C109—22no—
239C22×C109—21no—
240C22×C109—21no—
241C22×C109—22no—
242C22×C109—22no—
243C22×C109—22no—
244C22×C109—22no—
245C22×C109—21no—
246C22×C109—22no—
247C22×C109—21no—
248C22×C109—22no—
249C22×C109—22no—
250C22×C109—22no—
251C22×C109—48no—
252C22×C109—48no—
253C22×C109—41no—
254C22×C109—41no—
255C22×C109—41no—
256C22×C109—41no—
257C22×C109—81no—
258C22×C109—81no—
259C22×C109—81no—
260C22×C109—81no—
261C22×C109—81no—
262C22×C109—81no—
263C22×C109—81no—
264C22×C109#32714yescomplemented
265C22×C109#38611yescharacteristic
266C22×C109C2×#9911yescharacteristic
267C22×C109C2×#9911yescharacteristic
268C22×C109C2×#10211yescharacteristic
269C22×C109#41611yescharacteristic
270C22×C109C2×#10411yescharacteristic
271C22×C109#42111yescharacteristic
272C22×C109#46512yes—
273C22×C109#46612yes—
274C4×C109—21no—
275C4×C109—22no—
276C4×C109—21no—
277C4×C109—22no—
278C4×C109—41no—
279C4×C109—84no—
280C4×C109—81no—
281C4×C109—81no—
282C4×C109—81no—
283#9113C10911yesmaximal, characteristic, direct product factor, Sylow
284C2×C109—22no—
285C2×C109—24no—
286C2×C109—22no—
287C2×C109—42no—
288C2×C109—161no—
289C2×C109C2×#32711yescharacteristic
290C2×C109#150111yescharacteristic
291C2×C109#151611yescharacteristic
292C2×C109#151911yescharacteristic
293C2×C109#187111yescharacteristic
294C2×C109#189311yescharacteristic
295C2×C109#189811yescharacteristic
296C22×C22⋊C8C2×C10911yescharacteristic, complemented
297#2847C2×C10911yescharacteristic
298C2×#327C2×C10912yescomplemented
299C22⋊#95C2×C10911yescharacteristic, complemented
300#1294C2×C10912yescomplemented
301C109#911311yescharacteristic, direct product factor, Sylow
302C24×C4C22×C10911yescharacteristic
303C23×C8—21no—
304C2×#121C4×C10911yescharacteristic, complemented
305C2×#121C4×C10911yescharacteristic, complemented
306C2×#121C22×C10911yescharacteristic
307C2×#120—21no—
308C22×C22⋊C4—21no—
309C2×C22⋊C8—21no—
310C2×C22⋊C8—24no—
311C2×C22⋊C8C22×C10912yes—
312#386—21no—
313#386—21no—
314#386—21no—
315#380C4×C10911yescharacteristic, complemented
316#380C4×C10911yescharacteristic, complemented
317#380C22×C10911yescharacteristic
318C2×#95—21no—
319C2×#95C22×C10912yes—
320#327—24no—
321C25D4×C10911yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—21no—
325C22×D4C2×C4×C10911yescharacteristic
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C2×C4×C10911yescharacteristic
332C23×C4C2×C4×C10911yescharacteristic
333C23×C4D4×C10911yescharacteristic
334C23×C4D4×C10911yescharacteristic
335C23×C4D4×C10911yescharacteristic
336C23×C4C23×C10911yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—22no—
342#121—22no—
343#121—42no—
344#121—42no—
345#120—42no—
346#120—42no—
347C2×C4⋊C4C2×C4×C10911yescharacteristic
348C2×C4⋊C4C2×C4×C10911yescharacteristic
349C2×C4⋊C4C2×C4×C10911yescharacteristic
350C2×C22⋊C4—24no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—42no—
358C2×C22⋊C4—42no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8D4×C10914yescomplemented
362#95—21no—
363#95—21no—
364#95—21no—
365#95—22no—
366#95—22no—
367#95—42no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24C22⋊C4×C10911yescharacteristic
376C24C22⋊C4×C10911yescharacteristic
377C24D8×C10912yes—
378C24SD16×C10912yes—
379C24C2×D4×C10911yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22⋊C4×C10911yescharacteristic
425C22×C4C22⋊C4×C10911yescharacteristic
426C22×C4C22×C4×C10911yescharacteristic
427C22×C4C2×D4×C10911yescharacteristic
428C22×C4C2×D4×C10912yes—
429C22×C4#41×C10912yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23#99×C10912yes—
478C23#102×C10912yes—
479C23#102×C10912yes—
480C23#104×C10912yes—
481C23C2×C22⋊C4×C10911yescharacteristic
482C23#121×C10912yes—
483C23C2×D8×C10911yescharacteristic
484C23C2×SD16×C10911yescharacteristic
485C23#135×C10912yes—
486D4—82no—
487D4—84no—
488D4—84no—
489D4—82no—
490D4—82no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#99×C10912yes—
526C2×C4C4×D4×C10912yes—
527C2×C4#120×C10911yescharacteristic
528C2×C4#123×C10911yescharacteristic
529C8—44no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22#327×C10914yescomplemented
561C22#386×C10911yescharacteristic
562C22C2×#99×C10911yescharacteristic
563C22C2×#99×C10911yescharacteristic
564C22C2×#102×C10911yescharacteristic
565C22#416×C10911yescharacteristic
566C22C2×#104×C10911yescharacteristic
567C22#421×C10911yescharacteristic
568C22#465×C10912yes—
569C22#466×C10912yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—81no—
577C4—81no—
578C4—81no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—161no—
584C2C2×#327×C10911yescharacteristic
585C2C109×#150111yescharacteristic
586C2C109×#151611yescharacteristic
587C2C109×#151911yescharacteristic
588C2C109×#187111yescharacteristic
589C2C109×#189311yescharacteristic
590C2C109×#189811yescharacteristic
subgroupG/H
center, socleC23×C109C2×C22⋊C4
commutatorC22×C4C22×C4×C109
Frattini, cosocleC23×C4C23×C109
prime radicalC109×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C109 ◁ C24×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊C8×C109C2semidirect
3. C2×#327×C109C2semidirect
4. C109×C22⋊#95C2semidirect
5. C109×#1294C2semidirect
8. C2×#121×C109C4semidirect
9. C2×#121×C109C4semidirect
19. #380×C109C4semidirect
20. #380×C109C4semidirect
65. C22⋊C8×C109D4semidirect
264. C22×C109#327semidirect
283. #9113C109direct (N × H)
296. C22×C22⋊C8C2×C109semidirect
298. C2×#327C2×C109semidirect
299. C22⋊#95C2×C109semidirect
300. #1294C2×C109semidirect
301. C109#9113direct (N × H)
304. C2×#121C4×C109semidirect
305. C2×#121C4×C109semidirect
315. #380C4×C109semidirect
316. #380C4×C109semidirect
361. C22⋊C8D4×C109semidirect
560. C22#327×C109semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊C8×C1092
C109×#28472
C2×#327×C1092
C109×C22⋊#952
C109×#12942
#9113109