Finite Group Databasebeta

← C109×#18206 #42136200

Subgroups — C109×#18206 #42136200

Order 27904.

1028 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3505C211yesmaximal, characteristic
2C2×#483×C109C211yesmaximal, characteristic, complemented
3C2×#480×C109C211yesmaximal, characteristic, complemented
4C109×#2486C211yesmaximal, characteristic, complemented
5C109×#1612C214yesmaximal, complemented
6C109×#1609C214yesmaximal, complemented
7C109×#1520C211yesmaximal, characteristic, complemented
8C109×#1519C211yesmaximal, characteristic, complemented
9C109×#1459C211yesmaximal, characteristic, complemented
10#553×C109—22no—
11#553×C109—21no—
12#552×C109—21no—
13#551×C109—22no—
14#548×C109—21no—
15#546×C109—21no—
16#539×C109C2212yes—
17#538×C109C2212yes—
18#534×C109—21no—
19#529×C109—21no—
20#525×C109—21no—
21C2×#123×C109—21no—
22C2×C22⋊Q8×C109C2211yescharacteristic
23C2×#121×C109C2211yescharacteristic
24#518×C109—21no—
25C2×C4×D4×C109C2211yescharacteristic
26C22×C4⋊C4×C109C2211yescharacteristic
27#483×C109—22no—
28#483×C109C2214yescomplemented
29#480×C109—22no—
30#480×C109C2214yes—
31C2×#107×C109—21no—
32C2×#106×C109—21no—
33C2×#103×C109—21no—
34C2×#102×C109—21no—
35C2×C22⋊C8×C109C2211yescharacteristic
36C2×C22⋊C8×C109C2211yescharacteristic
37#398×C109C2212yescomplemented
38#397×C109C2212yescomplemented
39#389×C109—21no—
40#381×C109—21no—
41#380×C109—21no—
42#380×C109C2211yescharacteristic
43#328×C109C2214yescomplemented
44#327×C109C2214yes—
45#324×C109C2212yescomplemented
46#323×C109C2212yescomplemented
47C2×#41×C109—21no—
48C22×Q8×C109D411yescharacteristic
49C22×D4×C109—21no—
50C22×D4×C109D411yescharacteristic
51C22×D4×C109D411yescharacteristic
52C23×C4×C109—21no—
53C23×C4×C109D411yescharacteristic
54C23×C4×C109C2311yescharacteristic
55C22×C8×C109—21no—
56C22×C8×C109—21no—
57#126×C109—22no—
58#126×C109—22no—
59#126×C109—22no—
60#125×C109—21no—
61#125×C109—22no—
62#124×C109—21no—
63#124×C109—22no—
64#123×C109—22no—
65#123×C109—22no—
66#123×C109—22no—
67#123×C109—22no—
68#123×C109—22no—
69#123×C109—44no—
70C22⋊Q8×C109—21no—
71C22⋊Q8×C109—22no—
72C22⋊Q8×C109—21no—
73C22⋊Q8×C109—21no—
74C22⋊Q8×C109D412yes—
75C22⋊Q8×C109C2312yes—
76#121×C109—21no—
77#121×C109—22no—
78#121×C109—22no—
79#121×C109—22no—
80#121×C109—21no—
81#121×C109—21no—
82#121×C109D412yes—
83#121×C109C2312yes—
84#120×C109—22no—
85#120×C109—22no—
86C4×Q8×C109—21no—
87C4×D4×C109—22no—
88C4×D4×C109—22no—
89C4×D4×C109—22no—
90C4×D4×C109—21no—
91C4×D4×C109—21no—
92C4×D4×C109—21no—
93C4×D4×C109—21no—
94C4×D4×C109—21no—
95C4×D4×C109—22no—
96C4×D4×C109—22no—
97#117×C109—22no—
98#117×C109—21no—
99#117×C109—22no—
100C2×C4⋊C4×C109—21no—
101C2×C4⋊C4×C109—21no—
102C2×C4⋊C4×C109—21no—
103C2×C4⋊C4×C109—21no—
104C2×C4⋊C4×C109—21no—
105C2×C4⋊C4×C109—21no—
106C2×C4⋊C4×C109—22no—
107C2×C4⋊C4×C109—21no—
108C2×C4⋊C4×C109D411yescharacteristic
109C2×C4⋊C4×C109D411yescharacteristic
110C2×C4⋊C4×C109C2312yes—
111C2×C4⋊C4×C109C2311yescharacteristic
112C2×C22⋊C4×C109—22no—
113C2×C22⋊C4×C109—21no—
114C2×C22⋊C4×C109—21no—
115C2×C22⋊C4×C109—22no—
116C2×C22⋊C4×C109—21no—
117C2×C22⋊C4×C109D411yescharacteristic
118C2×C22⋊C4×C109C2311yescharacteristic
119C2×C42×C109D411yescharacteristic
120#107×C109—24no—
121#106×C109—24no—
122#103×C109—24no—
123#102×C109—24no—
124C22⋊C8×C109—21no—
125C22⋊C8×C109—21no—
126C22⋊C8×C109C2312yes—
127C22⋊C8×C109C2312yes—
128#95×C109—22no—
129#95×C109—22no—
130#95×C109C2312yes—
131C24×C109—21no—
132C24×C109C2×D411yescharacteristic
133C24×C109C2×D411yescharacteristic
134#41×C109—82no—
135C2×Q8×C109—21no—
136C2×Q8×C109—42no—
137C2×Q8×C109C2×D412yes—
138C2×D4×C109—21no—
139C2×D4×C109—21no—
140C2×D4×C109—22no—
141C2×D4×C109—22no—
142C2×D4×C109—22no—
143C2×D4×C109—22no—
144C2×D4×C109—22no—
145C2×D4×C109—22no—
146C2×D4×C109—21no—
147C2×D4×C109—42no—
148C2×D4×C109—42no—
149C2×D4×C109—42no—
150C2×D4×C109C2×D412yes—
151C22×C4×C109—21no—
152C22×C4×C109—22no—
153C22×C4×C109—22no—
154C22×C4×C109—21no—
155C22×C4×C109—21no—
156C22×C4×C109—21no—
157C22×C4×C109—21no—
158C22×C4×C109—22no—
159C22×C4×C109—22no—
160C22×C4×C109—42no—
161C22×C4×C109—42no—
162C22×C4×C109—41no—
163C22×C4×C109—41no—
164C22×C4×C109—42no—
165C22×C4×C109—42no—
166C22×C4×C109C2×D411yescharacteristic
167C22×C4×C109C2×D411yescharacteristic
168C22×C4×C109C2×D411yescharacteristic
169C22×C4×C109C2×D411yescharacteristic
170C22×C4×C109C2×D411yescharacteristic
171C22×C4×C109C2×D411yescharacteristic
172C22×C4×C109C2×D411yescharacteristic
173C22×C4×C109C2×D411yescharacteristic
174C22×C4×C109C2411yescharacteristic
175C2×C8×C109—22no—
176C2×C8×C109—22no—
177C2×C8×C109—42no—
178C2×C8×C109—42no—
179C4⋊C4×C109—22no—
180C4⋊C4×C109—22no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—21no—
185C4⋊C4×C109—21no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—21no—
188C4⋊C4×C109—21no—
189C4⋊C4×C109—22no—
190C4⋊C4×C109—22no—
191C4⋊C4×C109—42no—
192C4⋊C4×C109—44no—
193C4⋊C4×C109—41no—
194C4⋊C4×C109—41no—
195C4⋊C4×C109C2×D412yes—
196C4⋊C4×C109C2×D412yes—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—22no—
203C22⋊C4×C109—21no—
204C22⋊C4×C109—22no—
205C22⋊C4×C109—21no—
206C22⋊C4×C109—22no—
207C22⋊C4×C109—22no—
208C22⋊C4×C109—22no—
209C22⋊C4×C109—21no—
210C22⋊C4×C109—21no—
211C22⋊C4×C109—44no—
212C22⋊C4×C109—42no—
213C42×C109—22no—
214C42×C109—22no—
215C42×C109—21no—
216C42×C109—21no—
217C23×C109—21no—
218C23×C109—22no—
219C23×C109—21no—
220C23×C109—21no—
221C23×C109—41no—
222C23×C109—41no—
223C23×C109—41no—
224C23×C109—41no—
225C23×C109—42no—
226C23×C109—81no—
227C23×C109—81no—
228C23×C109#12011yescharacteristic
229C23×C109#12011yescharacteristic
230C23×C109#13512yes—
231C23×C109AGL(1,ℤ/8)12yes—
232C23×C109C22×D411yescharacteristic
233Q8×C109—42no—
234Q8×C109—42no—
235D4×C109—42no—
236D4×C109—42no—
237D4×C109—42no—
238D4×C109—42no—
239D4×C109—42no—
240D4×C109—44no—
241D4×C109—42no—
242D4×C109—82no—
243D4×C109—82no—
244C2×C4×C109—22no—
245C2×C4×C109—22no—
246C2×C4×C109—22no—
247C2×C4×C109—21no—
248C2×C4×C109—21no—
249C2×C4×C109—22no—
250C2×C4×C109—21no—
251C2×C4×C109—22no—
252C2×C4×C109—22no—
253C2×C4×C109—21no—
254C2×C4×C109—21no—
255C2×C4×C109—41no—
256C2×C4×C109—41no—
257C2×C4×C109—41no—
258C2×C4×C109—42no—
259C2×C4×C109—42no—
260C2×C4×C109—41no—
261C2×C4×C109—41no—
262C2×C4×C109—42no—
263C2×C4×C109—42no—
264C2×C4×C109—42no—
265C2×C4×C109—42no—
266C2×C4×C109—42no—
267C2×C4×C109—42no—
268C2×C4×C109—81no—
269C2×C4×C109—81no—
270C2×C4×C109—81no—
271C2×C4×C109—82no—
272C2×C4×C109—82no—
273C2×C4×C109—82no—
274C2×C4×C109—82no—
275C2×C4×C109#12011yescharacteristic
276C2×C4×C109#12011yescharacteristic
277C2×C4×C109C22×D411yescharacteristic
278C2×C4×C109C22×D411yescharacteristic
279C8×C109—42no—
280C8×C109—42no—
281C22×C109—21no—
282C22×C109—21no—
283C22×C109—21no—
284C22×C109—21no—
285C22×C109—21no—
286C22×C109—21no—
287C22×C109—42no—
288C22×C109—42no—
289C22×C109—42no—
290C22×C109—41no—
291C22×C109—42no—
292C22×C109—42no—
293C22×C109—41no—
294C22×C109—41no—
295C22×C109—42no—
296C22×C109—81no—
297C22×C109—162no—
298C22×C109#44912yes—
299C22×C109#44912yes—
300C22×C109C2×#12011yescharacteristic
301C22×C109C2×#13511yescharacteristic
302C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C109—21no—
304C4×C109—21no—
305C4×C109—21no—
306C4×C109—41no—
307C4×C109—42no—
308C4×C109—41no—
309C4×C109—42no—
310C4×C109—41no—
311C4×C109—42no—
312C4×C109—82no—
313C4×C109—82no—
314C4×C109#45812yes—
315#18206C10911yesmaximal, characteristic, direct product factor, Sylow
316C2×C109—21no—
317C2×C109—21no—
318C2×C109—22no—
319C2×C109—22no—
320C2×C109—81no—
321C2×C109—82no—
322C2×C109C2×#44911yescharacteristic
323C2×C109#302211yescharacteristic
324C2×C109C2×#45811yescharacteristic
325#3505C2×C10911yescharacteristic
326C2×#483C2×C10911yescharacteristic, complemented
327C2×#480C2×C10911yescharacteristic, complemented
328#2486C2×C10911yescharacteristic, complemented
329#1612C2×C10914yescomplemented
330#1609C2×C10914yescomplemented
331#1520C2×C10911yescharacteristic, complemented
332#1519C2×C10911yescharacteristic, complemented
333#1459C2×C10911yescharacteristic, complemented
334C109#1820611yescharacteristic, direct product factor, Sylow
335#553—22no—
336#553—21no—
337#552—21no—
338#551—22no—
339#548—21no—
340#546—21no—
341#539C22×C10912yes—
342#538C22×C10912yes—
343#534—21no—
344#529—21no—
345#525—21no—
346C2×#123—21no—
347C2×C22⋊Q8C22×C10911yescharacteristic
348C2×#121C22×C10911yescharacteristic
349#518—21no—
350C2×C4×D4C22×C10911yescharacteristic
351C22×C4⋊C4C22×C10911yescharacteristic
352#483—22no—
353#483C22×C10914yescomplemented
354#480—22no—
355#480C22×C10914yes—
356C2×#107—21no—
357C2×#106—21no—
358C2×#103—21no—
359C2×#102—21no—
360C2×C22⋊C8C22×C10911yescharacteristic
361C2×C22⋊C8C22×C10911yescharacteristic
362#398C22×C10912yescomplemented
363#397C22×C10912yescomplemented
364#389—21no—
365#381—21no—
366#380—21no—
367#380C22×C10911yescharacteristic
368#328C22×C10914yescomplemented
369#327C22×C10914yes—
370#324C22×C10912yescomplemented
371#323C22×C10912yescomplemented
372C2×#41—21no—
373C22×Q8D4×C10911yescharacteristic
374C22×D4—21no—
375C22×D4D4×C10911yescharacteristic
376C22×D4D4×C10911yescharacteristic
377C23×C4—21no—
378C23×C4D4×C10911yescharacteristic
379C23×C4C23×C10911yescharacteristic
380C22×C8—21no—
381C22×C8—21no—
382#126—22no—
383#126—22no—
384#126—22no—
385#125—21no—
386#125—22no—
387#124—21no—
388#124—22no—
389#123—22no—
390#123—22no—
391#123—22no—
392#123—22no—
393#123—22no—
394#123—44no—
395C22⋊Q8—21no—
396C22⋊Q8—22no—
397C22⋊Q8—21no—
398C22⋊Q8—21no—
399C22⋊Q8D4×C10912yes—
400C22⋊Q8C23×C10912yes—
401#121—21no—
402#121—22no—
403#121—22no—
404#121—22no—
405#121—21no—
406#121—21no—
407#121D4×C10912yes—
408#121C23×C10912yes—
409#120—22no—
410#120—22no—
411C4×Q8—21no—
412C4×D4—22no—
413C4×D4—22no—
414C4×D4—22no—
415C4×D4—21no—
416C4×D4—21no—
417C4×D4—21no—
418C4×D4—21no—
419C4×D4—21no—
420C4×D4—22no—
421C4×D4—22no—
422#117—22no—
423#117—21no—
424#117—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—22no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4D4×C10911yescharacteristic
434C2×C4⋊C4D4×C10911yescharacteristic
435C2×C4⋊C4C23×C10912yes—
436C2×C4⋊C4C23×C10911yescharacteristic
437C2×C22⋊C4—22no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—22no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4D4×C10911yescharacteristic
443C2×C22⋊C4C23×C10911yescharacteristic
444C2×C42D4×C10911yescharacteristic
445#107—24no—
446#106—24no—
447#103—24no—
448#102—24no—
449C22⋊C8—21no—
450C22⋊C8—21no—
451C22⋊C8C23×C10912yes—
452C22⋊C8C23×C10912yes—
453#95—22no—
454#95—22no—
455#95C23×C10912yes—
456C24—21no—
457C24C2×D4×C10911yescharacteristic
458C24C2×D4×C10911yescharacteristic
459#41—82no—
460C2×Q8—21no—
461C2×Q8—42no—
462C2×Q8C2×D4×C10912yes—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—21no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×D4×C10912yes—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4C2×D4×C10911yescharacteristic
492C22×C4C2×D4×C10911yescharacteristic
493C22×C4C2×D4×C10911yescharacteristic
494C22×C4C2×D4×C10911yescharacteristic
495C22×C4C2×D4×C10911yescharacteristic
496C22×C4C2×D4×C10911yescharacteristic
497C22×C4C2×D4×C10911yescharacteristic
498C22×C4C2×D4×C10911yescharacteristic
499C22×C4C24×C10911yescharacteristic
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—42no—
503C2×C8—42no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—41no—
519C4⋊C4—41no—
520C4⋊C4C2×D4×C10912yes—
521C4⋊C4C2×D4×C10912yes—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—21no—
529C22⋊C4—22no—
530C22⋊C4—21no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—21no—
535C22⋊C4—21no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C42—22no—
539C42—22no—
540C42—21no—
541C42—21no—
542C23—21no—
543C23—22no—
544C23—21no—
545C23—21no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—42no—
551C23—81no—
552C23—81no—
553C23#120×C10911yescharacteristic
554C23#120×C10911yescharacteristic
555C23#135×C10912yes—
556C23AGL(1,ℤ/8)×C10912yes—
557C23C22×D4×C10911yescharacteristic
558Q8—42no—
559Q8—42no—
560D4—42no—
561D4—42no—
562D4—42no—
563D4—42no—
564D4—42no—
565D4—44no—
566D4—42no—
567D4—82no—
568D4—82no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4#120×C10911yescharacteristic
601C2×C4#120×C10911yescharacteristic
602C2×C4C22×D4×C10911yescharacteristic
603C2×C4C22×D4×C10911yescharacteristic
604C8—42no—
605C8—42no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—42no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—81no—
622C22—162no—
623C22#449×C10912yes—
624C22#449×C10912yes—
625C22C2×#120×C10911yescharacteristic
626C22C2×#135×C10911yescharacteristic
627C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
628C4—21no—
629C4—21no—
630C4—21no—
631C4—41no—
632C4—42no—
633C4—41no—
634C4—42no—
635C4—41no—
636C4—42no—
637C4—82no—
638C4—82no—
639C4#458×C10912yes—
640C2—21no—
641C2—21no—
642C2—22no—
643C2—22no—
644C2—81no—
645C2—82no—
646C2C2×#449×C10911yescharacteristic
647C2C109×#302211yescharacteristic
648C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3505C2
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. C2×#483×C109C2semidirect
3. C2×#480×C109C2semidirect
4. C109×#2486C2semidirect
5. C109×#1612C2semidirect
6. C109×#1609C2semidirect
7. C109×#1520C2semidirect
8. C109×#1519C2semidirect
9. C109×#1459C2semidirect
28. #483×C109C22semidirect
37. #398×C109C22semidirect
38. #397×C109C22semidirect
43. #328×C109C22semidirect
45. #324×C109C22semidirect
46. #323×C109C22semidirect
315. #18206C109direct (N × H)
326. C2×#483C2×C109semidirect
327. C2×#480C2×C109semidirect
328. #2486C2×C109semidirect
329. #1612C2×C109semidirect
330. #1609C2×C109semidirect
331. #1520C2×C109semidirect
332. #1519C2×C109semidirect
333. #1459C2×C109semidirect
334. C109#18206direct (N × H)
353. #483C22×C109semidirect
362. #398C22×C109semidirect
363. #397C22×C109semidirect
368. #328C22×C109semidirect
370. #324C22×C109semidirect
371. #323C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35052
C2×#483×C1092
C2×#480×C1092
C109×#24862
C109×#16122
C109×#16092
C109×#15202
C109×#15192
C109×#14592
#18206109