Finite Group Databasebeta

← C109×#18193 #42136187

Subgroups — C109×#18193 #42136187

Order 27904.

1200 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3492C211yesmaximal, characteristic, complemented
2C109×#3002C212yesmaximal, complemented
3C109×#2406C211yesmaximal, characteristic, complemented
4C109×#1605C211yesmaximal, characteristic, complemented
5C109×#1604C211yesmaximal, characteristic, complemented
6C109×#1603C212yesmaximal, complemented
7C109×#1599C211yesmaximal, characteristic, complemented
8C109×#1598C211yesmaximal, characteristic, complemented
9C109×#1597C212yesmaximal, complemented
10C109×#1516C212yesmaximal, complemented
11C2×#324×C109C211yesmaximal, characteristic, complemented
12C22×#41×C109C2211yescharacteristic
13C2×#135×C109—22no—
14#554×C109—22no—
15#553×C109—22no—
16#552×C109—22no—
17#547×C109—22no—
18#544×C109C2211yescharacteristic
19#539×C109—21no—
20#539×C109C2212yes—
21#537×C109C2211yescharacteristic
22#525×C109—21no—
23#525×C109—21no—
24#525×C109C2212yes—
25C2×#123×C109—21no—
26#520×C109—21no—
27C22×C4⋊C4×C109C2211yescharacteristic
28#452×C109—22no—
29#451×C109C2212yes—
30#450×C109C2212yes—
31#449×C109—22no—
32#448×C109C2212yes—
33#447×C109C2212yes—
34#416×C109—22no—
35C2×C22⋊C8×C109C2212yes—
36#395×C109C2211yescharacteristic
37#395×C109C2211yescharacteristic
38#394×C109C2211yescharacteristic
39#394×C109C2211yescharacteristic
40#386×C109—21no—
41#385×C109—21no—
42#385×C109—21no—
43#381×C109C2211yescharacteristic
44#328×C109C2212yes—
45#328×C109C2212yes—
46#327×C109C2212yes—
47#327×C109C2212yes—
48#324×C109C2212yes—
49#324×C109C2211yescharacteristic
50#324×C109C2211yescharacteristic
51C2×#41×C109—21no—
52C2×#41×C109—21no—
53C2×#41×C109—22no—
54C2×#41×C109—22no—
55C2×#41×C109—44no—
56C2×#41×C109D412yes—
57C22×Q8×C109C2311yescharacteristic
58C22×D4×C109—21no—
59C22×D4×C109C2311yescharacteristic
60C23×C4×C109—21no—
61C23×C4×C109D411yescharacteristic
62C23×C4×C109D411yescharacteristic
63C23×C4×C109C2311yescharacteristic
64#135×C109—48no—
65C2×Q16×C109—22no—
66C2×SD16×C109—22no—
67C2×SD16×C109—22no—
68C2×D8×C109—22no—
69C22×C8×C109—22no—
70C4⋊Q8×C109—22no—
71#126×C109—22no—
72#126×C109—22no—
73#125×C109—22no—
74#125×C109—22no—
75#125×C109—22no—
76#123×C109—22no—
77#123×C109—22no—
78#123×C109—22no—
79#123×C109—22no—
80#123×C109—22no—
81#123×C109—44no—
82C22⋊Q8×C109—21no—
83C22⋊Q8×C109—21no—
84C22⋊Q8×C109—21no—
85C22⋊Q8×C109—22no—
86C22⋊Q8×C109—22no—
87C22⋊Q8×C109—21no—
88C22⋊Q8×C109—22no—
89C22⋊Q8×C109D412yes—
90C22⋊Q8×C109C2312yes—
91#121×C109—21no—
92#121×C109—21no—
93#121×C109—21no—
94#121×C109—22no—
95#121×C109—21no—
96#121×C109D412yes—
97#121×C109C2312yes—
98#120×C109—22no—
99C4×Q8×C109—22no—
100C4×D4×C109—22no—
101C4×D4×C109—22no—
102C4×D4×C109—22no—
103C4×D4×C109—22no—
104C4×D4×C109—22no—
105C4×D4×C109—22no—
106C4×D4×C109—22no—
107#117×C109—22no—
108#117×C109—21no—
109#117×C109—21no—
110#117×C109D412yes—
111C2×C4⋊C4×C109—22no—
112C2×C4⋊C4×C109—22no—
113C2×C4⋊C4×C109—22no—
114C2×C4⋊C4×C109—21no—
115C2×C4⋊C4×C109—21no—
116C2×C4⋊C4×C109D411yescharacteristic
117C2×C4⋊C4×C109D411yescharacteristic
118C2×C4⋊C4×C109C2311yescharacteristic
119C2×C4⋊C4×C109C2311yescharacteristic
120C2×C22⋊C4×C109—21no—
121C2×C22⋊C4×C109—21no—
122C2×C22⋊C4×C109—21no—
123C2×C22⋊C4×C109—21no—
124C2×C22⋊C4×C109—21no—
125#103×C109—22no—
126#103×C109—22no—
127#102×C109—22no—
128#102×C109—22no—
129C22⋊C8×C109—22no—
130C22⋊C8×C109C2312yes—
131C22⋊C8×C109C2312yes—
132#95×C109—21no—
133#95×C109—21no—
134#95×C109—21no—
135#95×C109C2311yescharacteristic
136#95×C109C2311yescharacteristic
137C24×C109—21no—
138C24×C109C2×D411yescharacteristic
139#41×C109—28no—
140#41×C109—42no—
141#41×C109—44no—
142#41×C109—44no—
143#41×C109—44no—
144#41×C109—44no—
145#41×C109—42no—
146#41×C109—82no—
147C2×Q8×C109—21no—
148C2×Q8×C109—21no—
149C2×Q8×C109—22no—
150C2×Q8×C109—42no—
151C2×Q8×C109C2×D412yes—
152C2×D4×C109—21no—
153C2×D4×C109—24no—
154C2×D4×C109—22no—
155C2×D4×C109—22no—
156C2×D4×C109—22no—
157C2×D4×C109—21no—
158C2×D4×C109—22no—
159C2×D4×C109—42no—
160C2×D4×C109—42no—
161C2×D4×C109—41no—
162C2×D4×C109—41no—
163C2×D4×C109C2×D412yes—
164C22×C4×C109—22no—
165C22×C4×C109—21no—
166C22×C4×C109—21no—
167C22×C4×C109—21no—
168C22×C4×C109—21no—
169C22×C4×C109—22no—
170C22×C4×C109—21no—
171C22×C4×C109—22no—
172C22×C4×C109—22no—
173C22×C4×C109—21no—
174C22×C4×C109—21no—
175C22×C4×C109—41no—
176C22×C4×C109—42no—
177C22×C4×C109—42no—
178C22×C4×C109—42no—
179C22×C4×C109—42no—
180C22×C4×C109—42no—
181C22×C4×C109C2×D411yescharacteristic
182C22×C4×C109C2×D412yes—
183C22×C4×C109C2×D411yescharacteristic
184C22×C4×C109C2×D411yescharacteristic
185C22×C4×C109C2×D411yescharacteristic
186C22×C4×C109C2×D411yescharacteristic
187C22×C4×C109C2×D412yes—
188C22×C4×C109C2411yescharacteristic
189Q16×C109—44no—
190SD16×C109—44no—
191SD16×C109—44no—
192D8×C109—44no—
193C2×C8×C109—22no—
194C2×C8×C109—22no—
195C2×C8×C109—44no—
196C4⋊C4×C109—21no—
197C4⋊C4×C109—22no—
198C4⋊C4×C109—21no—
199C4⋊C4×C109—21no—
200C4⋊C4×C109—21no—
201C4⋊C4×C109—22no—
202C4⋊C4×C109—22no—
203C4⋊C4×C109—22no—
204C4⋊C4×C109—22no—
205C4⋊C4×C109—22no—
206C4⋊C4×C109—22no—
207C4⋊C4×C109—22no—
208C4⋊C4×C109—22no—
209C4⋊C4×C109—22no—
210C4⋊C4×C109—42no—
211C4⋊C4×C109—42no—
212C4⋊C4×C109C2×D412yes—
213C4⋊C4×C109C2×D412yes—
214C22⋊C4×C109—22no—
215C22⋊C4×C109—22no—
216C22⋊C4×C109—22no—
217C22⋊C4×C109—22no—
218C22⋊C4×C109—22no—
219C22⋊C4×C109—22no—
220C22⋊C4×C109—22no—
221C22⋊C4×C109—22no—
222C22⋊C4×C109—22no—
223C22⋊C4×C109—42no—
224C22⋊C4×C109—42no—
225C22⋊C4×C109—42no—
226C22⋊C4×C109—42no—
227C22⋊C4×C109—42no—
228C42×C109—22no—
229C42×C109—22no—
230C42×C109—22no—
231C23×C109—21no—
232C23×C109—21no—
233C23×C109—22no—
234C23×C109—22no—
235C23×C109—22no—
236C23×C109—41no—
237C23×C109—41no—
238C23×C109—41no—
239C23×C109—41no—
240C23×C109—41no—
241C23×C109—41no—
242C23×C109—41no—
243C23×C109#12011yescharacteristic
244C23×C109#12011yescharacteristic
245C23×C109C22×D411yescharacteristic
246Q8×C109—24no—
247Q8×C109—42no—
248Q8×C109—42no—
249Q8×C109—42no—
250Q8×C109—81no—
251D4×C109—24no—
252D4×C109—42no—
253D4×C109—42no—
254D4×C109—42no—
255D4×C109—44no—
256D4×C109—44no—
257D4×C109—42no—
258D4×C109—42no—
259D4×C109—44no—
260D4×C109—44no—
261D4×C109—81no—
262C2×C4×C109—21no—
263C2×C4×C109—21no—
264C2×C4×C109—24no—
265C2×C4×C109—22no—
266C2×C4×C109—22no—
267C2×C4×C109—22no—
268C2×C4×C109—21no—
269C2×C4×C109—22no—
270C2×C4×C109—21no—
271C2×C4×C109—21no—
272C2×C4×C109—21no—
273C2×C4×C109—21no—
274C2×C4×C109—41no—
275C2×C4×C109—42no—
276C2×C4×C109—41no—
277C2×C4×C109—42no—
278C2×C4×C109—42no—
279C2×C4×C109—44no—
280C2×C4×C109—48no—
281C2×C4×C109—41no—
282C2×C4×C109—41no—
283C2×C4×C109—42no—
284C2×C4×C109—41no—
285C2×C4×C109—41no—
286C2×C4×C109—41no—
287C2×C4×C109—41no—
288C2×C4×C109—41no—
289C2×C4×C109—41no—
290C2×C4×C109—41no—
291C2×C4×C109—41no—
292C2×C4×C109—82no—
293C2×C4×C109—81no—
294C2×C4×C109—82no—
295C2×C4×C109#12012yes—
296C2×C4×C109#13512yescomplemented
297C2×C4×C109C22×D412yes—
298C8×C109—42no—
299C8×C109—42no—
300C22×C109—21no—
301C22×C109—21no—
302C22×C109—21no—
303C22×C109—21no—
304C22×C109—21no—
305C22×C109—21no—
306C22×C109—21no—
307C22×C109—42no—
308C22×C109—42no—
309C22×C109—42no—
310C22×C109—42no—
311C22×C109—42no—
312C22×C109—42no—
313C22×C109—42no—
314C22×C109—42no—
315C22×C109—81no—
316C22×C109—81no—
317C22×C109—81no—
318C22×C109—81no—
319C22×C109B(4,𝔽2)12yes—
320C22×C109C2×#12011yescharacteristic
321C22×C109C2×#13511yescharacteristic
322C22×C109#57511yescharacteristic
323C4×C109—22no—
324C4×C109—22no—
325C4×C109—22no—
326C4×C109—41no—
327C4×C109—42no—
328C4×C109—42no—
329C4×C109—42no—
330C4×C109—41no—
331C4×C109—41no—
332C4×C109—42no—
333C4×C109—81no—
334C4×C109—81no—
335#18193C10911yesmaximal, characteristic, direct product factor, Sylow
336C2×C109—21no—
337C2×C109—21no—
338C2×C109—22no—
339C2×C109—41no—
340C2×C109—82no—
341C2×C109—82no—
342C2×C109#300211yescharacteristic
343C2×C109#302011yescharacteristic
344C2×C109C2×B(4,𝔽2)11yescharacteristic
345#3492C2×C10911yescharacteristic, complemented
346#3002C2×C10912yescomplemented
347#2406C2×C10911yescharacteristic, complemented
348#1605C2×C10911yescharacteristic, complemented
349#1604C2×C10911yescharacteristic, complemented
350#1603C2×C10912yescomplemented
351#1599C2×C10911yescharacteristic, complemented
352#1598C2×C10911yescharacteristic, complemented
353#1597C2×C10912yescomplemented
354#1516C2×C10912yescomplemented
355C2×#324C2×C10911yescharacteristic, complemented
356C109#1819311yescharacteristic, direct product factor, Sylow
357C22×#41C22×C10911yescharacteristic
358C2×#135—22no—
359#554—22no—
360#553—22no—
361#552—22no—
362#547—22no—
363#544C22×C10911yescharacteristic
364#539—21no—
365#539C22×C10912yes—
366#537C22×C10911yescharacteristic
367#525—21no—
368#525—21no—
369#525C22×C10912yes—
370C2×#123—21no—
371#520—21no—
372C22×C4⋊C4C22×C10911yescharacteristic
373#452—22no—
374#451C22×C10912yes—
375#450C22×C10912yes—
376#449—22no—
377#448C22×C10912yes—
378#447C22×C10912yes—
379#416—22no—
380C2×C22⋊C8C22×C10912yes—
381#395C22×C10911yescharacteristic
382#395C22×C10911yescharacteristic
383#394C22×C10911yescharacteristic
384#394C22×C10911yescharacteristic
385#386—21no—
386#385—21no—
387#385—21no—
388#381C22×C10911yescharacteristic
389#328C22×C10912yes—
390#328C22×C10912yes—
391#327C22×C10912yes—
392#327C22×C10912yes—
393#324C22×C10912yes—
394#324C22×C10911yescharacteristic
395#324C22×C10911yescharacteristic
396C2×#41—21no—
397C2×#41—21no—
398C2×#41—22no—
399C2×#41—22no—
400C2×#41—44no—
401C2×#41D4×C10912yes—
402C22×Q8C23×C10911yescharacteristic
403C22×D4—21no—
404C22×D4C23×C10911yescharacteristic
405C23×C4—21no—
406C23×C4D4×C10911yescharacteristic
407C23×C4D4×C10911yescharacteristic
408C23×C4C23×C10911yescharacteristic
409#135—48no—
410C2×Q16—22no—
411C2×SD16—22no—
412C2×SD16—22no—
413C2×D8—22no—
414C22×C8—22no—
415C4⋊Q8—22no—
416#126—22no—
417#126—22no—
418#125—22no—
419#125—22no—
420#125—22no—
421#123—22no—
422#123—22no—
423#123—22no—
424#123—22no—
425#123—22no—
426#123—44no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—21no—
430C22⋊Q8—22no—
431C22⋊Q8—22no—
432C22⋊Q8—21no—
433C22⋊Q8—22no—
434C22⋊Q8D4×C10912yes—
435C22⋊Q8C23×C10912yes—
436#121—21no—
437#121—21no—
438#121—21no—
439#121—22no—
440#121—21no—
441#121D4×C10912yes—
442#121C23×C10912yes—
443#120—22no—
444C4×Q8—22no—
445C4×D4—22no—
446C4×D4—22no—
447C4×D4—22no—
448C4×D4—22no—
449C4×D4—22no—
450C4×D4—22no—
451C4×D4—22no—
452#117—22no—
453#117—21no—
454#117—21no—
455#117D4×C10912yes—
456C2×C4⋊C4—22no—
457C2×C4⋊C4—22no—
458C2×C4⋊C4—22no—
459C2×C4⋊C4—21no—
460C2×C4⋊C4—21no—
461C2×C4⋊C4D4×C10911yescharacteristic
462C2×C4⋊C4D4×C10911yescharacteristic
463C2×C4⋊C4C23×C10911yescharacteristic
464C2×C4⋊C4C23×C10911yescharacteristic
465C2×C22⋊C4—21no—
466C2×C22⋊C4—21no—
467C2×C22⋊C4—21no—
468C2×C22⋊C4—21no—
469C2×C22⋊C4—21no—
470#103—22no—
471#103—22no—
472#102—22no—
473#102—22no—
474C22⋊C8—22no—
475C22⋊C8C23×C10912yes—
476C22⋊C8C23×C10912yes—
477#95—21no—
478#95—21no—
479#95—21no—
480#95C23×C10911yescharacteristic
481#95C23×C10911yescharacteristic
482C24—21no—
483C24C2×D4×C10911yescharacteristic
484#41—28no—
485#41—42no—
486#41—44no—
487#41—44no—
488#41—44no—
489#41—44no—
490#41—42no—
491#41—82no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—22no—
495C2×Q8—42no—
496C2×Q8C2×D4×C10912yes—
497C2×D4—21no—
498C2×D4—24no—
499C2×D4—22no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—21no—
503C2×D4—22no—
504C2×D4—42no—
505C2×D4—42no—
506C2×D4—41no—
507C2×D4—41no—
508C2×D4C2×D4×C10912yes—
509C22×C4—22no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—41no—
521C22×C4—42no—
522C22×C4—42no—
523C22×C4—42no—
524C22×C4—42no—
525C22×C4—42no—
526C22×C4C2×D4×C10911yescharacteristic
527C22×C4C2×D4×C10912yes—
528C22×C4C2×D4×C10911yescharacteristic
529C22×C4C2×D4×C10911yescharacteristic
530C22×C4C2×D4×C10911yescharacteristic
531C22×C4C2×D4×C10911yescharacteristic
532C22×C4C2×D4×C10912yes—
533C22×C4C24×C10911yescharacteristic
534Q16—44no—
535SD16—44no—
536SD16—44no—
537D8—44no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—44no—
541C4⋊C4—21no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4C2×D4×C10912yes—
558C4⋊C4C2×D4×C10912yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—42no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C42—22no—
574C42—22no—
575C42—22no—
576C23—21no—
577C23—21no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23#120×C10911yescharacteristic
589C23#120×C10911yescharacteristic
590C23C22×D4×C10911yescharacteristic
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—81no—
596D4—24no—
597D4—42no—
598D4—42no—
599D4—42no—
600D4—44no—
601D4—44no—
602D4—42no—
603D4—42no—
604D4—44no—
605D4—44no—
606D4—81no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—48no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—82no—
638C2×C4—81no—
639C2×C4—82no—
640C2×C4#120×C10912yes—
641C2×C4#135×C10912yescomplemented
642C2×C4C22×D4×C10912yes—
643C8—42no—
644C8—42no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—81no—
664C22B(4,𝔽2)×C10912yes—
665C22C2×#120×C10911yescharacteristic
666C22C2×#135×C10911yescharacteristic
667C22#575×C10911yescharacteristic
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—41no—
676C4—41no—
677C4—42no—
678C4—81no—
679C4—81no—
680C2—21no—
681C2—21no—
682C2—22no—
683C2—41no—
684C2—82no—
685C2—82no—
686C2C109×#300211yescharacteristic
687C2C109×#302011yescharacteristic
688C2C2×B(4,𝔽2)×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
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
1. C109×#3492C2semidirect
2. C109×#3002C2semidirect
3. C109×#2406C2semidirect
4. C109×#1605C2semidirect
5. C109×#1604C2semidirect
6. C109×#1603C2semidirect
7. C109×#1599C2semidirect
8. C109×#1598C2semidirect
9. C109×#1597C2semidirect
10. C109×#1516C2semidirect
11. C2×#324×C109C2semidirect
296. C2×C4×C109#135semidirect
335. #18193C109direct (N × H)
345. #3492C2×C109semidirect
346. #3002C2×C109semidirect
347. #2406C2×C109semidirect
348. #1605C2×C109semidirect
349. #1604C2×C109semidirect
350. #1603C2×C109semidirect
351. #1599C2×C109semidirect
352. #1598C2×C109semidirect
353. #1597C2×C109semidirect
354. #1516C2×C109semidirect
355. C2×#324C2×C109semidirect
356. C109#18193direct (N × H)
641. C2×C4#135×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34922
C109×#30022
C109×#24062
C109×#16052
C109×#16042
C109×#16032
C109×#15992
C109×#15982
C109×#15972
C109×#15162
C2×#324×C1092
#18193109