Finite Group Databasebeta

← C109×#18199 #42136193

Subgroups — C109×#18199 #42136193

Order 27904.

1200 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3505C211yesmaximal, characteristic
2C2×#451×C109C211yesmaximal, characteristic, complemented
3C2×#448×C109C211yesmaximal, characteristic, complemented
4C109×#2409C211yesmaximal, characteristic, complemented
5C109×#1606C214yesmaximal, complemented
6C109×#1603C214yesmaximal, complemented
7C109×#1520C211yesmaximal, characteristic, complemented
8C109×#1519C211yesmaximal, characteristic, complemented
9C109×#1459C211yesmaximal, characteristic, complemented
10C23×Q8×C109C2211yescharacteristic
11C22×Q16×C109—21no—
12C22×SD16×C109—21no—
13#553×C109—22no—
14#553×C109—21no—
15#552×C109—21no—
16#551×C109—22no—
17#548×C109—21no—
18#546×C109—21no—
19#539×C109C2212yes—
20#538×C109C2212yes—
21#534×C109—21no—
22#529×C109—21no—
23#525×C109—21no—
24C2×C22⋊Q8×C109—21no—
25C2×C22⋊Q8×C109C2211yescharacteristic
26C2×#121×C109C2211yescharacteristic
27#518×C109—21no—
28C2×C4×D4×C109C2211yescharacteristic
29#451×C109—22no—
30#451×C109C2214yescomplemented
31#448×C109—22no—
32#448×C109C2214yes—
33C2×#103×C109—21no—
34C2×#103×C109—21no—
35C2×C22⋊C8×C109C2211yescharacteristic
36C2×C22⋊C8×C109C2211yescharacteristic
37#395×C109C2212yescomplemented
38#393×C109C2212yescomplemented
39#391×C109—21no—
40#386×C109—21no—
41#385×C109—21no—
42#380×C109C2211yescharacteristic
43#328×C109C2214yescomplemented
44#327×C109C2214yes—
45#324×C109C2212yescomplemented
46#323×C109C2212yescomplemented
47C2×#41×C109—21no—
48C22×Q8×C109—21no—
49C22×Q8×C109—21no—
50C22×Q8×C109—21no—
51C22×Q8×C109—21no—
52C22×Q8×C109—21no—
53C22×Q8×C109—44no—
54C22×Q8×C109D411yescharacteristic
55C22×Q8×C109C2312yes—
56C22×D4×C109—21no—
57C22×D4×C109D411yescharacteristic
58C22×D4×C109D411yescharacteristic
59C23×C4×C109—21no—
60C23×C4×C109D411yescharacteristic
61C23×C4×C109C2311yescharacteristic
62C2×Q16×C109—24no—
63C2×Q16×C109—44no—
64C2×SD16×C109—24no—
65C2×SD16×C109—44no—
66C22×C8×C109—21no—
67C22×C8×C109—21no—
68#126×C109—22no—
69#126×C109—22no—
70#126×C109—22no—
71#125×C109—21no—
72#125×C109—22no—
73#124×C109—21no—
74#124×C109—22no—
75#123×C109—22no—
76#123×C109—22no—
77#123×C109—22no—
78#123×C109—22no—
79#123×C109—22no—
80C22⋊Q8×C109—21no—
81C22⋊Q8×C109—22no—
82C22⋊Q8×C109—21no—
83C22⋊Q8×C109—21no—
84C22⋊Q8×C109—44no—
85C22⋊Q8×C109D412yes—
86C22⋊Q8×C109C2312yes—
87#121×C109—21no—
88#121×C109—22no—
89#121×C109—22no—
90#121×C109—22no—
91#121×C109—21no—
92#121×C109—21no—
93#121×C109D412yes—
94#121×C109C2312yes—
95#120×C109—22no—
96#120×C109—22no—
97C4×Q8×C109—21no—
98C4×D4×C109—22no—
99C4×D4×C109—22no—
100C4×D4×C109—22no—
101C4×D4×C109—21no—
102C4×D4×C109—21no—
103C4×D4×C109—21no—
104C4×D4×C109—21no—
105C4×D4×C109—21no—
106C4×D4×C109—22no—
107C4×D4×C109—22no—
108#117×C109—22no—
109#117×C109—21no—
110#117×C109—22no—
111C2×C4⋊C4×C109—21no—
112C2×C4⋊C4×C109—21no—
113C2×C4⋊C4×C109—22no—
114C2×C4⋊C4×C109D411yescharacteristic
115C2×C4⋊C4×C109D411yescharacteristic
116C2×C4⋊C4×C109C2311yescharacteristic
117C2×C22⋊C4×C109—22no—
118C2×C22⋊C4×C109—21no—
119C2×C22⋊C4×C109—21no—
120C2×C22⋊C4×C109—22no—
121C2×C22⋊C4×C109—21no—
122C2×C22⋊C4×C109D411yescharacteristic
123C2×C22⋊C4×C109C2311yescharacteristic
124C2×C42×C109D411yescharacteristic
125#103×C109—24no—
126#103×C109—24no—
127C22⋊C8×C109—21no—
128C22⋊C8×C109—21no—
129C22⋊C8×C109C2312yes—
130C22⋊C8×C109C2312yes—
131#95×C109—22no—
132#95×C109—22no—
133#95×C109C2312yes—
134C24×C109—21no—
135C24×C109C2×D411yescharacteristic
136C24×C109C2×D411yescharacteristic
137#41×C109—82no—
138C2×Q8×C109—24no—
139C2×Q8×C109—21no—
140C2×Q8×C109—24no—
141C2×Q8×C109—24no—
142C2×Q8×C109—22no—
143C2×Q8×C109—22no—
144C2×Q8×C109—44no—
145C2×Q8×C109—42no—
146C2×Q8×C109—42no—
147C2×Q8×C109—42no—
148C2×Q8×C109—42no—
149C2×Q8×C109—44no—
150C2×Q8×C109—44no—
151C2×Q8×C109—41no—
152C2×Q8×C109—41no—
153C2×Q8×C109C2×D412yes—
154C2×D4×C109—21no—
155C2×D4×C109—21no—
156C2×D4×C109—22no—
157C2×D4×C109—22no—
158C2×D4×C109—22no—
159C2×D4×C109—22no—
160C2×D4×C109—22no—
161C2×D4×C109—22no—
162C2×D4×C109—21no—
163C2×D4×C109—42no—
164C2×D4×C109—42no—
165C2×D4×C109—42no—
166C2×D4×C109C2×D412yes—
167C22×C4×C109—21no—
168C22×C4×C109—22no—
169C22×C4×C109—22no—
170C22×C4×C109—22no—
171C22×C4×C109—21no—
172C22×C4×C109—21no—
173C22×C4×C109—21no—
174C22×C4×C109—21no—
175C22×C4×C109—22no—
176C22×C4×C109—42no—
177C22×C4×C109—44no—
178C22×C4×C109—42no—
179C22×C4×C109—41no—
180C22×C4×C109—41no—
181C22×C4×C109C2×D411yescharacteristic
182C22×C4×C109C2×D411yescharacteristic
183C22×C4×C109C2×D411yescharacteristic
184C22×C4×C109C2×D411yescharacteristic
185C22×C4×C109C2×D411yescharacteristic
186C22×C4×C109C2×D411yescharacteristic
187C22×C4×C109C2×D411yescharacteristic
188C22×C4×C109C2×D411yescharacteristic
189C22×C4×C109C2411yescharacteristic
190Q16×C109—48no—
191SD16×C109—48no—
192C2×C8×C109—22no—
193C2×C8×C109—22no—
194C2×C8×C109—42no—
195C2×C8×C109—42no—
196C4⋊C4×C109—22no—
197C4⋊C4×C109—22no—
198C4⋊C4×C109—22no—
199C4⋊C4×C109—22no—
200C4⋊C4×C109—22no—
201C4⋊C4×C109—21no—
202C4⋊C4×C109—21no—
203C4⋊C4×C109—22no—
204C4⋊C4×C109—21no—
205C4⋊C4×C109—21no—
206C4⋊C4×C109—42no—
207C4⋊C4×C109—44no—
208C4⋊C4×C109C2×D412yes—
209C4⋊C4×C109C2×D412yes—
210C22⋊C4×C109—22no—
211C22⋊C4×C109—22no—
212C22⋊C4×C109—22no—
213C22⋊C4×C109—22no—
214C22⋊C4×C109—22no—
215C22⋊C4×C109—22no—
216C22⋊C4×C109—21no—
217C22⋊C4×C109—22no—
218C22⋊C4×C109—21no—
219C22⋊C4×C109—22no—
220C22⋊C4×C109—22no—
221C22⋊C4×C109—22no—
222C22⋊C4×C109—21no—
223C22⋊C4×C109—21no—
224C22⋊C4×C109—44no—
225C22⋊C4×C109—42no—
226C42×C109—22no—
227C42×C109—22no—
228C42×C109—21no—
229C42×C109—21no—
230C23×C109—21no—
231C23×C109—22no—
232C23×C109—21no—
233C23×C109—21no—
234C23×C109—41no—
235C23×C109—41no—
236C23×C109—41no—
237C23×C109—41no—
238C23×C109—42no—
239C23×C109—81no—
240C23×C109—81no—
241C23×C109#12011yescharacteristic
242C23×C109#12011yescharacteristic
243C23×C109#13512yes—
244C23×C109#13712yes—
245C23×C109C22×D411yescharacteristic
246Q8×C109—24no—
247Q8×C109—24no—
248Q8×C109—42no—
249Q8×C109—42no—
250Q8×C109—42no—
251Q8×C109—44no—
252Q8×C109—44no—
253Q8×C109—42no—
254D4×C109—42no—
255D4×C109—42no—
256D4×C109—42no—
257D4×C109—42no—
258D4×C109—42no—
259D4×C109—44no—
260D4×C109—42no—
261D4×C109—82no—
262D4×C109—82no—
263C2×C4×C109—22no—
264C2×C4×C109—22no—
265C2×C4×C109—22no—
266C2×C4×C109—24no—
267C2×C4×C109—21no—
268C2×C4×C109—21no—
269C2×C4×C109—22no—
270C2×C4×C109—21no—
271C2×C4×C109—22no—
272C2×C4×C109—22no—
273C2×C4×C109—21no—
274C2×C4×C109—21no—
275C2×C4×C109—41no—
276C2×C4×C109—41no—
277C2×C4×C109—41no—
278C2×C4×C109—42no—
279C2×C4×C109—44no—
280C2×C4×C109—44no—
281C2×C4×C109—42no—
282C2×C4×C109—42no—
283C2×C4×C109—42no—
284C2×C4×C109—41no—
285C2×C4×C109—41no—
286C2×C4×C109—42no—
287C2×C4×C109—42no—
288C2×C4×C109—42no—
289C2×C4×C109—81no—
290C2×C4×C109—81no—
291C2×C4×C109—81no—
292C2×C4×C109—82no—
293C2×C4×C109—82no—
294C2×C4×C109#12011yescharacteristic
295C2×C4×C109#12011yescharacteristic
296C2×C4×C109C22×D411yescharacteristic
297C2×C4×C109C22×D411yescharacteristic
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—42no—
307C22×C109—42no—
308C22×C109—42no—
309C22×C109—41no—
310C22×C109—42no—
311C22×C109—42no—
312C22×C109—41no—
313C22×C109—41no—
314C22×C109—42no—
315C22×C109—81no—
316C22×C109—162no—
317C22×C109#45212yes—
318C22×C109#45212yes—
319C22×C109C2×#12011yescharacteristic
320C22×C109C2×#13511yescharacteristic
321C22×C109C2×#13711yescharacteristic
322C4×C109—21no—
323C4×C109—21no—
324C4×C109—21no—
325C4×C109—41no—
326C4×C109—42no—
327C4×C109—41no—
328C4×C109—42no—
329C4×C109—44no—
330C4×C109—41no—
331C4×C109—42no—
332C4×C109—82no—
333C4×C109B(4,𝔽2)12yes—
334#18199C10911yesmaximal, characteristic, direct product factor, Sylow
335C2×C109—21no—
336C2×C109—21no—
337C2×C109—22no—
338C2×C109—22no—
339C2×C109—81no—
340C2×C109—82no—
341C2×C109C2×#45211yescharacteristic
342C2×C109#302211yescharacteristic
343C2×C109C2×B(4,𝔽2)11yescharacteristic
344#3505C2×C10911yescharacteristic
345C2×#451C2×C10911yescharacteristic, complemented
346C2×#448C2×C10911yescharacteristic, complemented
347#2409C2×C10911yescharacteristic, complemented
348#1606C2×C10914yescomplemented
349#1603C2×C10914yescomplemented
350#1520C2×C10911yescharacteristic, complemented
351#1519C2×C10911yescharacteristic, complemented
352#1459C2×C10911yescharacteristic, complemented
353C109#1819911yescharacteristic, direct product factor, Sylow
354C23×Q8C22×C10911yescharacteristic
355C22×Q16—21no—
356C22×SD16—21no—
357#553—22no—
358#553—21no—
359#552—21no—
360#551—22no—
361#548—21no—
362#546—21no—
363#539C22×C10912yes—
364#538C22×C10912yes—
365#534—21no—
366#529—21no—
367#525—21no—
368C2×C22⋊Q8—21no—
369C2×C22⋊Q8C22×C10911yescharacteristic
370C2×#121C22×C10911yescharacteristic
371#518—21no—
372C2×C4×D4C22×C10911yescharacteristic
373#451—22no—
374#451C22×C10914yescomplemented
375#448—22no—
376#448C22×C10914yes—
377C2×#103—21no—
378C2×#103—21no—
379C2×C22⋊C8C22×C10911yescharacteristic
380C2×C22⋊C8C22×C10911yescharacteristic
381#395C22×C10912yescomplemented
382#393C22×C10912yescomplemented
383#391—21no—
384#386—21no—
385#385—21no—
386#380C22×C10911yescharacteristic
387#328C22×C10914yescomplemented
388#327C22×C10914yes—
389#324C22×C10912yescomplemented
390#323C22×C10912yescomplemented
391C2×#41—21no—
392C22×Q8—21no—
393C22×Q8—21no—
394C22×Q8—21no—
395C22×Q8—21no—
396C22×Q8—21no—
397C22×Q8—44no—
398C22×Q8D4×C10911yescharacteristic
399C22×Q8C23×C10912yes—
400C22×D4—21no—
401C22×D4D4×C10911yescharacteristic
402C22×D4D4×C10911yescharacteristic
403C23×C4—21no—
404C23×C4D4×C10911yescharacteristic
405C23×C4C23×C10911yescharacteristic
406C2×Q16—24no—
407C2×Q16—44no—
408C2×SD16—24no—
409C2×SD16—44no—
410C22×C8—21no—
411C22×C8—21no—
412#126—22no—
413#126—22no—
414#126—22no—
415#125—21no—
416#125—22no—
417#124—21no—
418#124—22no—
419#123—22no—
420#123—22no—
421#123—22no—
422#123—22no—
423#123—22no—
424C22⋊Q8—21no—
425C22⋊Q8—22no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—44no—
429C22⋊Q8D4×C10912yes—
430C22⋊Q8C23×C10912yes—
431#121—21no—
432#121—22no—
433#121—22no—
434#121—22no—
435#121—21no—
436#121—21no—
437#121D4×C10912yes—
438#121C23×C10912yes—
439#120—22no—
440#120—22no—
441C4×Q8—21no—
442C4×D4—22no—
443C4×D4—22no—
444C4×D4—22no—
445C4×D4—21no—
446C4×D4—21no—
447C4×D4—21no—
448C4×D4—21no—
449C4×D4—21no—
450C4×D4—22no—
451C4×D4—22no—
452#117—22no—
453#117—21no—
454#117—22no—
455C2×C4⋊C4—21no—
456C2×C4⋊C4—21no—
457C2×C4⋊C4—22no—
458C2×C4⋊C4D4×C10911yescharacteristic
459C2×C4⋊C4D4×C10911yescharacteristic
460C2×C4⋊C4C23×C10911yescharacteristic
461C2×C22⋊C4—22no—
462C2×C22⋊C4—21no—
463C2×C22⋊C4—21no—
464C2×C22⋊C4—22no—
465C2×C22⋊C4—21no—
466C2×C22⋊C4D4×C10911yescharacteristic
467C2×C22⋊C4C23×C10911yescharacteristic
468C2×C42D4×C10911yescharacteristic
469#103—24no—
470#103—24no—
471C22⋊C8—21no—
472C22⋊C8—21no—
473C22⋊C8C23×C10912yes—
474C22⋊C8C23×C10912yes—
475#95—22no—
476#95—22no—
477#95C23×C10912yes—
478C24—21no—
479C24C2×D4×C10911yescharacteristic
480C24C2×D4×C10911yescharacteristic
481#41—82no—
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—24no—
485C2×Q8—24no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—44no—
489C2×Q8—42no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—42no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—41no—
496C2×Q8—41no—
497C2×Q8C2×D4×C10912yes—
498C2×D4—21no—
499C2×D4—21no—
500C2×D4—22no—
501C2×D4—22no—
502C2×D4—22no—
503C2×D4—22no—
504C2×D4—22no—
505C2×D4—22no—
506C2×D4—21no—
507C2×D4—42no—
508C2×D4—42no—
509C2×D4—42no—
510C2×D4C2×D4×C10912yes—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—22no—
520C22×C4—42no—
521C22×C4—44no—
522C22×C4—42no—
523C22×C4—41no—
524C22×C4—41no—
525C22×C4C2×D4×C10911yescharacteristic
526C22×C4C2×D4×C10911yescharacteristic
527C22×C4C2×D4×C10911yescharacteristic
528C22×C4C2×D4×C10911yescharacteristic
529C22×C4C2×D4×C10911yescharacteristic
530C22×C4C2×D4×C10911yescharacteristic
531C22×C4C2×D4×C10911yescharacteristic
532C22×C4C2×D4×C10911yescharacteristic
533C22×C4C24×C10911yescharacteristic
534Q16—48no—
535SD16—48no—
536C2×C8—22no—
537C2×C8—22no—
538C2×C8—42no—
539C2×C8—42no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—22no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4C2×D4×C10912yes—
553C4⋊C4C2×D4×C10912yes—
554C22⋊C4—22no—
555C22⋊C4—22no—
556C22⋊C4—22no—
557C22⋊C4—22no—
558C22⋊C4—22no—
559C22⋊C4—22no—
560C22⋊C4—21no—
561C22⋊C4—22no—
562C22⋊C4—21no—
563C22⋊C4—22no—
564C22⋊C4—22no—
565C22⋊C4—22no—
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—44no—
569C22⋊C4—42no—
570C42—22no—
571C42—22no—
572C42—21no—
573C42—21no—
574C23—21no—
575C23—22no—
576C23—21no—
577C23—21no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—42no—
583C23—81no—
584C23—81no—
585C23#120×C10911yescharacteristic
586C23#120×C10911yescharacteristic
587C23#135×C10912yes—
588C23#137×C10912yes—
589C23C22×D4×C10911yescharacteristic
590Q8—24no—
591Q8—24no—
592Q8—42no—
593Q8—42no—
594Q8—42no—
595Q8—44no—
596Q8—44no—
597Q8—42no—
598D4—42no—
599D4—42no—
600D4—42no—
601D4—42no—
602D4—42no—
603D4—44no—
604D4—42no—
605D4—82no—
606D4—82no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4#120×C10911yescharacteristic
639C2×C4#120×C10911yescharacteristic
640C2×C4C22×D4×C10911yescharacteristic
641C2×C4C22×D4×C10911yescharacteristic
642C8—42no—
643C8—42no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—41no—
654C22—42no—
655C22—42no—
656C22—41no—
657C22—41no—
658C22—42no—
659C22—81no—
660C22—162no—
661C22#452×C10912yes—
662C22#452×C10912yes—
663C22C2×#120×C10911yescharacteristic
664C22C2×#135×C10911yescharacteristic
665C22C2×#137×C10911yescharacteristic
666C4—21no—
667C4—21no—
668C4—21no—
669C4—41no—
670C4—42no—
671C4—41no—
672C4—42no—
673C4—44no—
674C4—41no—
675C4—42no—
676C4—82no—
677C4B(4,𝔽2)×C10912yes—
678C2—21no—
679C2—21no—
680C2—22no—
681C2—22no—
682C2—81no—
683C2—82no—
684C2C2×#452×C10911yescharacteristic
685C2C109×#302211yescharacteristic
686C2C2×B(4,𝔽2)×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×#451×C109C2semidirect
3. C2×#448×C109C2semidirect
4. C109×#2409C2semidirect
5. C109×#1606C2semidirect
6. C109×#1603C2semidirect
7. C109×#1520C2semidirect
8. C109×#1519C2semidirect
9. C109×#1459C2semidirect
30. #451×C109C22semidirect
37. #395×C109C22semidirect
38. #393×C109C22semidirect
43. #328×C109C22semidirect
45. #324×C109C22semidirect
46. #323×C109C22semidirect
334. #18199C109direct (N × H)
345. C2×#451C2×C109semidirect
346. C2×#448C2×C109semidirect
347. #2409C2×C109semidirect
348. #1606C2×C109semidirect
349. #1603C2×C109semidirect
350. #1520C2×C109semidirect
351. #1519C2×C109semidirect
352. #1459C2×C109semidirect
353. C109#18199direct (N × H)
374. #451C22×C109semidirect
381. #395C22×C109semidirect
382. #393C22×C109semidirect
387. #328C22×C109semidirect
389. #324C22×C109semidirect
390. #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×#451×C1092
C2×#448×C1092
C109×#24092
C109×#16062
C109×#16032
C109×#15202
C109×#15192
C109×#14592
#18199109