Finite Group Databasebeta

← C109×#9117 #42127111

Subgroups — C109×#9117 #42127111

Order 27904.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#407×C109C211yesmaximal, characteristic, complemented
2C109×#2847C211yesmaximal, characteristic
3C2×#327×C109C212yesmaximal, complemented
4C109×C22⋊#95C211yesmaximal, characteristic, complemented
5C109×#1294C212yesmaximal, complemented
6C24×C4×C109C2211yescharacteristic
7C22×M16×C109—21no—
8C2×#121×C109C411yescharacteristic, complemented
9C2×#121×C109C411yescharacteristic, complemented
10C2×#121×C109C2211yescharacteristic
11C2×#120×C109—21no—
12C22×C22⋊C4×C109—21no—
13#407×C109—24no—
14C2×C22⋊C8×C109—21no—
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
41C2×M16×C109—24no—
42C2×M16×C109—42no—
43C2×M16×C109—42no—
44C22×C8×C109—22no—
45#121×C109—22no—
46#121×C109—22no—
47#121×C109—42no—
48#121×C109—42no—
49#120×C109—44no—
50C2×C4⋊C4×C109C2×C411yescharacteristic
51C2×C4⋊C4×C109C2×C411yescharacteristic
52C2×C4⋊C4×C109C2×C411yescharacteristic
53C2×C22⋊C4×C109—24no—
54C2×C22⋊C4×C109—21no—
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—42no—
61C2×C22⋊C4×C109—42no—
62C22⋊C8×C109—24no—
63C22⋊C8×C109—22no—
64C22⋊C8×C109D414yescomplemented
65#95×C109—21no—
66#95×C109—21no—
67#95×C109—21no—
68#95×C109—22no—
69#95×C109—22no—
70#95×C109—42no—
71C24×C109—22no—
72C24×C109—22no—
73C24×C109—21no—
74C24×C109—21no—
75C24×C109—22no—
76C24×C109—22no—
77C24×C109—21no—
78C24×C109—41no—
79C24×C109—41no—
80C24×C109C22⋊C411yescharacteristic
81C24×C109C22⋊C411yescharacteristic
82C24×C109C2×D411yescharacteristic
83C2×D4×C109—22no—
84C2×D4×C109—41no—
85C2×D4×C109—41no—
86C2×D4×C109—42no—
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—
96C22×C4×C109—22no—
97C22×C4×C109—22no—
98C22×C4×C109—21no—
99C22×C4×C109—21no—
100C22×C4×C109—22no—
101C22×C4×C109—22no—
102C22×C4×C109—22no—
103C22×C4×C109—22no—
104C22×C4×C109—22no—
105C22×C4×C109—22no—
106C22×C4×C109—21no—
107C22×C4×C109—21no—
108C22×C4×C109—24no—
109C22×C4×C109—21no—
110C22×C4×C109—21no—
111C22×C4×C109—21no—
112C22×C4×C109—41no—
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—48no—
120C22×C4×C109—48no—
121C22×C4×C109—42no—
122C22×C4×C109—42no—
123C22×C4×C109—42no—
124C22×C4×C109—42no—
125C22×C4×C109—42no—
126C22×C4×C109—42no—
127C22×C4×C109C22⋊C411yescharacteristic
128C22×C4×C109C22⋊C411yescharacteristic
129C22×C4×C109C22×C411yescharacteristic
130C22×C4×C109C2×D411yescharacteristic
131C22×C4×C109C2×D412yes—
132C22×C4×C109#4112yes—
133M16×C109—48no—
134C2×C8×C109—24no—
135C2×C8×C109—42no—
136C2×C8×C109—42no—
137C4⋊C4×C109—22no—
138C4⋊C4×C109—41no—
139C4⋊C4×C109—41no—
140C22⋊C4×C109—42no—
141C22⋊C4×C109—48no—
142C22⋊C4×C109—42no—
143C22⋊C4×C109—42no—
144C22⋊C4×C109—42no—
145C22⋊C4×C109—42no—
146C22⋊C4×C109—41no—
147C22⋊C4×C109—41no—
148C23×C109—21no—
149C23×C109—21no—
150C23×C109—21no—
151C23×C109—21no—
152C23×C109—24no—
153C23×C109—24no—
154C23×C109—21no—
155C23×C109—21no—
156C23×C109—21no—
157C23×C109—21no—
158C23×C109—21no—
159C23×C109—21no—
160C23×C109—21no—
161C23×C109—21no—
162C23×C109—21no—
163C23×C109—21no—
164C23×C109—21no—
165C23×C109—21no—
166C23×C109—44no—
167C23×C109—44no—
168C23×C109—44no—
169C23×C109—44no—
170C23×C109—42no—
171C23×C109—42no—
172C23×C109—42no—
173C23×C109—42no—
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#10412yes—
183C23×C109C2×C22⋊C411yescharacteristic
184C23×C109#12112yes—
185C23×C109AGL(1,ℤ/8)11yescharacteristic
186C23×C109AGL(1,ℤ/8)12yes—
187C23×C109#13711yescharacteristic
188D4×C109—82no—
189D4×C109—84no—
190D4×C109—84no—
191D4×C109—82no—
192D4×C109—82no—
193C2×C4×C109—24no—
194C2×C4×C109—24no—
195C2×C4×C109—21no—
196C2×C4×C109—21no—
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—42no—
205C2×C4×C109—42no—
206C2×C4×C109—44no—
207C2×C4×C109—44no—
208C2×C4×C109—44no—
209C2×C4×C109—42no—
210C2×C4×C109—42no—
211C2×C4×C109—44no—
212C2×C4×C109—44no—
213C2×C4×C109—44no—
214C2×C4×C109—41no—
215C2×C4×C109—41no—
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—82no—
224C2×C4×C109—82no—
225C2×C4×C109—82no—
226C2×C4×C109—82no—
227C2×C4×C109#9912yes—
228C2×C4×C109C4×D412yes—
229C2×C4×C109#12011yescharacteristic
230C2×C4×C109#12311yescharacteristic
231C8×C109—44no—
232C22×C109—22no—
233C22×C109—22no—
234C22×C109—21no—
235C22×C109—21no—
236C22×C109—22no—
237C22×C109—22no—
238C22×C109—21no—
239C22×C109—21no—
240C22×C109—22no—
241C22×C109—22no—
242C22×C109—22no—
243C22×C109—22no—
244C22×C109—22no—
245C22×C109—22no—
246C22×C109—48no—
247C22×C109—48no—
248C22×C109—41no—
249C22×C109—41no—
250C22×C109—41no—
251C22×C109—41no—
252C22×C109—41no—
253C22×C109—41no—
254C22×C109—41no—
255C22×C109—41no—
256C22×C109—81no—
257C22×C109—81no—
258C22×C109—81no—
259C22×C109—81no—
260C22×C109—81no—
261C22×C109—81no—
262C22×C109—81no—
263C22×C109#38611yescharacteristic
264C22×C109C2×#9911yescharacteristic
265C22×C109C2×#9911yescharacteristic
266C22×C109#41711yescharacteristic
267C22×C109#41811yescharacteristic
268C22×C109C2×#10411yescharacteristic
269C22×C109#42111yescharacteristic
270C22×C109#46812yes—
271C22×C109#46912yes—
272C4×C109—21no—
273C4×C109—22no—
274C4×C109—21no—
275C4×C109—22no—
276C4×C109—41no—
277C4×C109—84no—
278C4×C109—81no—
279C4×C109—81no—
280C4×C109—81no—
281#9117C10911yesmaximal, characteristic, direct product factor, Sylow
282C2×C109—22no—
283C2×C109—22no—
284C2×C109—42no—
285C2×C109—42no—
286C2×C109—161no—
287C2×C109#150411yescharacteristic
288C2×C109#150711yescharacteristic
289C2×C109#151011yescharacteristic
290C2×C109#151311yescharacteristic
291C2×C109#187111yescharacteristic
292C2×C109#189511yescharacteristic
293C2×C109#189811yescharacteristic
294C2×#407C2×C10911yescharacteristic, complemented
295#2847C2×C10911yescharacteristic
296C2×#327C2×C10912yescomplemented
297C22⋊#95C2×C10911yescharacteristic, complemented
298#1294C2×C10912yescomplemented
299C109#911711yescharacteristic, direct product factor, Sylow
300C24×C4C22×C10911yescharacteristic
301C22×M16—21no—
302C2×#121C4×C10911yescharacteristic, complemented
303C2×#121C4×C10911yescharacteristic, complemented
304C2×#121C22×C10911yescharacteristic
305C2×#120—21no—
306C22×C22⋊C4—21no—
307#407—24no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8C22×C10912yes—
310#386—21no—
311#386—21no—
312#386—21no—
313#380C4×C10911yescharacteristic, complemented
314#380C4×C10911yescharacteristic, complemented
315#380C22×C10911yescharacteristic
316C2×#95—21no—
317C2×#95C22×C10912yes—
318#327—24no—
319C25D4×C10911yescharacteristic
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4C2×C4×C10911yescharacteristic
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C2×C4×C10911yescharacteristic
330C23×C4C2×C4×C10911yescharacteristic
331C23×C4D4×C10911yescharacteristic
332C23×C4D4×C10911yescharacteristic
333C23×C4D4×C10911yescharacteristic
334C23×C4C23×C10911yescharacteristic
335C2×M16—24no—
336C2×M16—42no—
337C2×M16—42no—
338C22×C8—22no—
339#121—22no—
340#121—22no—
341#121—42no—
342#121—42no—
343#120—44no—
344C2×C4⋊C4C2×C4×C10911yescharacteristic
345C2×C4⋊C4C2×C4×C10911yescharacteristic
346C2×C4⋊C4C2×C4×C10911yescharacteristic
347C2×C22⋊C4—24no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—42no—
356C22⋊C8—24no—
357C22⋊C8—22no—
358C22⋊C8D4×C10914yescomplemented
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—42no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—41no—
373C24—41no—
374C24C22⋊C4×C10911yescharacteristic
375C24C22⋊C4×C10911yescharacteristic
376C24C2×D4×C10911yescharacteristic
377C2×D4—22no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—41no—
389C2×D4—41no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4C22⋊C4×C10911yescharacteristic
422C22×C4C22⋊C4×C10911yescharacteristic
423C22×C4C22×C4×C10911yescharacteristic
424C22×C4C2×D4×C10911yescharacteristic
425C22×C4C2×D4×C10912yes—
426C22×C4#41×C10912yes—
427M16—48no—
428C2×C8—24no—
429C2×C8—42no—
430C2×C8—42no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C22⋊C4—42no—
435C22⋊C4—48no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23#99×C10912yes—
476C23#104×C10912yes—
477C23C2×C22⋊C4×C10911yescharacteristic
478C23#121×C10912yes—
479C23AGL(1,ℤ/8)×C10911yescharacteristic
480C23AGL(1,ℤ/8)×C10912yes—
481C23#137×C10911yescharacteristic
482D4—82no—
483D4—84no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4#99×C10912yes—
522C2×C4C4×D4×C10912yes—
523C2×C4#120×C10911yescharacteristic
524C2×C4#123×C10911yescharacteristic
525C8—44no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#386×C10911yescharacteristic
558C22C2×#99×C10911yescharacteristic
559C22C2×#99×C10911yescharacteristic
560C22#417×C10911yescharacteristic
561C22#418×C10911yescharacteristic
562C22C2×#104×C10911yescharacteristic
563C22#421×C10911yescharacteristic
564C22#468×C10912yes—
565C22#469×C10912yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—81no—
573C4—81no—
574C4—81no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—161no—
580C2C109×#150411yescharacteristic
581C2C109×#150711yescharacteristic
582C2C109×#151011yescharacteristic
583C2C109×#151311yescharacteristic
584C2C109×#187111yescharacteristic
585C2C109×#189511yescharacteristic
586C2C109×#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. C2×#407×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
64. C22⋊C8×C109D4semidirect
281. #9117C109direct (N × H)
294. C2×#407C2×C109semidirect
296. C2×#327C2×C109semidirect
297. C22⋊#95C2×C109semidirect
298. #1294C2×C109semidirect
299. C109#9117direct (N × H)
302. C2×#121C4×C109semidirect
303. C2×#121C4×C109semidirect
313. #380C4×C109semidirect
314. #380C4×C109semidirect
358. C22⋊C8D4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#407×C1092
C109×#28472
C2×#327×C1092
C109×C22⋊#952
C109×#12942
#9117109