Finite Group Databasebeta

← C109×#23901 #42141895

Subgroups — C109×#23901 #42141895

Order 27904.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3484C211yesmaximal, characteristic
2C22×#123×C109C211yesmaximal, characteristic, complemented
3C2×#458×C109C212yesmaximal, complemented
4C109×#2849C212yesmaximal, complemented
5C109×#2022C214yesmaximal, complemented
6C109×#1898C212yesmaximal, complemented
7C109×#1890C212yesmaximal, complemented
8C109×#1781C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C109C2211yescharacteristic
10C23×D4×C109C2211yescharacteristic
11C24×C4×C109C2211yescharacteristic, complemented
12#546×C109—24no—
13D42×C109—24no—
14#535×C109—21no—
15#534×C109—21no—
16#534×C109C2212yes—
17#525×C109—21no—
18C2×#123×C109—21no—
19C2×#123×C109—21no—
20C2×#123×C109—28no—
21C2×#123×C109C2212yes—
22C2×#123×C109C2212yescomplemented
23C2×#123×C109C2214yescomplemented
24C2×#120×C109—21no—
25C2×#120×C109C2212yes—
26#518×C109C2211yescharacteristic
27C22×C4⋊C4×C109—21no—
28C22×C22⋊C4×C109—21no—
29C22×C22⋊C4×C109C2211yescharacteristic, complemented
30#458×C109—28no—
31C2×#99×C109C2212yes—
32C2×#99×C109C2214yescomplemented
33C2×#99×C109C2212yescomplemented
34#386×C109—24no—
35#385×C109—24no—
36#381×C109C2214yescomplemented
37#379×C109C2212yescomplemented
38#342×C109C2214yescomplemented
39C25×C109D411yescharacteristic
40C25×C109C2311yescharacteristic
41H(5,𝔽2)×C109—24no—
42H(5,𝔽2)×C109—42no—
43C2×#41×C109—21no—
44C2×#41×C109—22no—
45C22×D4×C109—21no—
46C22×D4×C109—22no—
47C22×D4×C109—24no—
48C22×D4×C109—22no—
49C22×D4×C109—24no—
50C22×D4×C109—42no—
51C22×D4×C109D411yescharacteristic
52C22×D4×C109D411yescharacteristic
53C22×D4×C109D414yes—
54C22×D4×C109C2312yes—
55C22×D4×C109C2311yescharacteristic
56C22×D4×C109C2311yescharacteristic
57C23×C4×C109—21no—
58C23×C4×C109—21no—
59C23×C4×C109—21no—
60C23×C4×C109—24no—
61C23×C4×C109—24no—
62C23×C4×C109—42no—
63C23×C4×C109D411yescharacteristic
64C23×C4×C109C2312yes—
65#127×C109—22no—
66#124×C109—22no—
67#123×C109—28no—
68#123×C109—28no—
69#123×C109—22no—
70#123×C109—24no—
71#123×C109—22no—
72#123×C109—22no—
73#123×C109—24no—
74#123×C109—24no—
75#123×C109—28no—
76#123×C109—44no—
77#123×C109D418yescomplemented
78C22⋊Q8×C109—22no—
79#121×C109—22no—
80#121×C109—24no—
81#121×C109—24no—
82#121×C109—24no—
83#120×C109—24no—
84#120×C109—22no—
85#120×C109—22no—
86#120×C109—22no—
87#120×C109—22no—
88#120×C109—24no—
89#120×C109—22no—
90C4×D4×C109—24no—
91C4×D4×C109—24no—
92#117×C109—21no—
93C2×C4⋊C4×C109—28no—
94C2×C4⋊C4×C109—22no—
95C2×C4⋊C4×C109—22no—
96C2×C4⋊C4×C109—24no—
97C2×C22⋊C4×C109—28no—
98C2×C22⋊C4×C109—22no—
99C2×C22⋊C4×C109—22no—
100C2×C22⋊C4×C109—22no—
101C2×C22⋊C4×C109—21no—
102C2×C22⋊C4×C109—21no—
103C2×C22⋊C4×C109—24no—
104C2×C22⋊C4×C109—24no—
105C2×C22⋊C4×C109—24no—
106C2×C22⋊C4×C109D414yes—
107C2×C22⋊C4×C109C2314yes—
108C2×C22⋊C4×C109C2312yes—
109C2×C22⋊C4×C109C2312yes—
110#99×C109—24no—
111#99×C109—28no—
112#99×C109—42no—
113#95×C109—28no—
114C24×C109—22no—
115C24×C109—22no—
116C24×C109—21no—
117C24×C109—22no—
118C24×C109—22no—
119C24×C109—44no—
120C24×C109—41no—
121C24×C109—44no—
122C24×C109C2×D411yescharacteristic
123C24×C109C2×D414yes—
124C24×C109C2×D411yescharacteristic
125C24×C109C2×D412yes—
126C24×C109C2×D411yescharacteristic
127C24×C109C2×D411yescharacteristic
128C24×C109C2411yescharacteristic
129#41×C109—48no—
130#41×C109—42no—
131#41×C109—42no—
132C2×Q8×C109—21no—
133C2×D4×C109—22no—
134C2×D4×C109—22no—
135C2×D4×C109—22no—
136C2×D4×C109—21no—
137C2×D4×C109—24no—
138C2×D4×C109—22no—
139C2×D4×C109—22no—
140C2×D4×C109—22no—
141C2×D4×C109—24no—
142C2×D4×C109—22no—
143C2×D4×C109—22no—
144C2×D4×C109—22no—
145C2×D4×C109—22no—
146C2×D4×C109—22no—
147C2×D4×C109—24no—
148C2×D4×C109—24no—
149C2×D4×C109—24no—
150C2×D4×C109—22no—
151C2×D4×C109—41no—
152C2×D4×C109—41no—
153C2×D4×C109—416no—
154C2×D4×C109—48no—
155C2×D4×C109—42no—
156C2×D4×C109—42no—
157C2×D4×C109—48no—
158C2×D4×C109—84no—
159C2×D4×C109C2×D414yes—
160C22×C4×C109—24no—
161C22×C4×C109—24no—
162C22×C4×C109—21no—
163C22×C4×C109—22no—
164C22×C4×C109—24no—
165C22×C4×C109—22no—
166C22×C4×C109—24no—
167C22×C4×C109—22no—
168C22×C4×C109—22no—
169C22×C4×C109—22no—
170C22×C4×C109—24no—
171C22×C4×C109—22no—
172C22×C4×C109—22no—
173C22×C4×C109—24no—
174C22×C4×C109—41no—
175C22×C4×C109—416no—
176C22×C4×C109—44no—
177C22×C4×C109—42no—
178C22×C4×C109—42no—
179C22×C4×C109—42no—
180C22×C4×C109—42no—
181C22×C4×C109C2×D412yes—
182C22×C4×C109C2×D411yescharacteristic
183C22×C4×C109C2×D411yescharacteristic
184C22×C4×C109C2×D414yes—
185C22×C4×C109C2×D414yescomplemented
186C4⋊C4×C109—28no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—24no—
189C4⋊C4×C109—28no—
190C4⋊C4×C109—48no—
191C22⋊C4×C109—24no—
192C22⋊C4×C109—28no—
193C22⋊C4×C109—22no—
194C22⋊C4×C109—24no—
195C22⋊C4×C109—24no—
196C22⋊C4×C109—28no—
197C22⋊C4×C109—24no—
198C22⋊C4×C109—24no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—24no—
203C22⋊C4×C109—24no—
204C22⋊C4×C109—48no—
205C22⋊C4×C109—48no—
206C22⋊C4×C109C2×D414yescomplemented
207C42×C109—22no—
208C23×C109—21no—
209C23×C109—21no—
210C23×C109—24no—
211C23×C109—21no—
212C23×C109—21no—
213C23×C109—21no—
214C23×C109—28no—
215C23×C109—28no—
216C23×C109—21no—
217C23×C109—21no—
218C23×C109—21no—
219C23×C109—22no—
220C23×C109—21no—
221C23×C109—24no—
222C23×C109—44no—
223C23×C109—42no—
224C23×C109—42no—
225C23×C109—44no—
226C23×C109—48no—
227C23×C109—48no—
228C23×C109—42no—
229C23×C109—42no—
230C23×C109—48no—
231C23×C109—42no—
232C23×C109—44no—
233C23×C109—44no—
234C23×C109—82no—
235C23×C109—84no—
236C23×C109—82no—
237C23×C109#12012yes—
238C23×C109#12012yes—
239C23×C109#12012yes—
240C23×C109C22×D411yescharacteristic
241C23×C109C22×D412yes—
242C23×C109C22×D412yes—
243C23×C109H(5,𝔽2)12yes—
244Q8×C109—42no—
245D4×C109—28no—
246D4×C109—44no—
247D4×C109—42no—
248D4×C109—44no—
249D4×C109—44no—
250D4×C109—42no—
251D4×C109—42no—
252D4×C109—44no—
253D4×C109—44no—
254D4×C109—48no—
255D4×C109—84no—
256D4×C109—84no—
257C2×C4×C109—21no—
258C2×C4×C109—21no—
259C2×C4×C109—24no—
260C2×C4×C109—24no—
261C2×C4×C109—21no—
262C2×C4×C109—24no—
263C2×C4×C109—22no—
264C2×C4×C109—21no—
265C2×C4×C109—21no—
266C2×C4×C109—28no—
267C2×C4×C109—24no—
268C2×C4×C109—44no—
269C2×C4×C109—44no—
270C2×C4×C109—48no—
271C2×C4×C109—44no—
272C2×C4×C109—48no—
273C2×C4×C109—48no—
274C2×C4×C109—42no—
275C2×C4×C109—42no—
276C2×C4×C109—42no—
277C2×C4×C109—44no—
278C2×C4×C109—44no—
279C2×C4×C109—48no—
280C2×C4×C109—44no—
281C2×C4×C109—82no—
282C2×C4×C109#12012yes—
283C22×C109—24no—
284C22×C109—24no—
285C22×C109—22no—
286C22×C109—24no—
287C22×C109—22no—
288C22×C109—22no—
289C22×C109—22no—
290C22×C109—22no—
291C22×C109—24no—
292C22×C109—42no—
293C22×C109—42no—
294C22×C109—416no—
295C22×C109—42no—
296C22×C109—42no—
297C22×C109—42no—
298C22×C109—42no—
299C22×C109—42no—
300C22×C109—42no—
301C22×C109—41no—
302C22×C109—44no—
303C22×C109—42no—
304C22×C109—81no—
305C22×C109—88no—
306C22×C109—84no—
307C22×C109—81no—
308C22×C109—162no—
309C22×C109#45814yescomplemented
310C22×C109C2×#12011yescharacteristic
311C22×C109C2×#12011yescharacteristic
312C22×C109#53411yescharacteristic
313C22×C109D4214yes—
314C4×C109—22no—
315C4×C109—44no—
316C4×C109—44no—
317C4×C109—41no—
318C4×C109—42no—
319C4×C109—44no—
320C4×C109—84no—
321#23901C10911yesmaximal, characteristic, direct product factor, Sylow
322C2×C109—21no—
323C2×C109—24no—
324C2×C109—21no—
325C2×C109—24no—
326C2×C109—42no—
327C2×C109—42no—
328C2×C109—81no—
329C2×C109—82no—
330C2×C109#240311yescharacteristic
331C2×C109C2×#45812yes—
332#3484C2×C10911yescharacteristic
333C22×#123C2×C10911yescharacteristic, complemented
334C2×#458C2×C10912yescomplemented
335#2849C2×C10912yescomplemented
336#2022C2×C10914yescomplemented
337#1898C2×C10912yescomplemented
338#1890C2×C10912yescomplemented
339#1781C2×C10911yescharacteristic, complemented
340C109#2390111yescharacteristic, direct product factor, Sylow
341C2×H(5,𝔽2)C22×C10911yescharacteristic
342C23×D4C22×C10911yescharacteristic
343C24×C4C22×C10911yescharacteristic, complemented
344#546—24no—
345D42—24no—
346#535—21no—
347#534—21no—
348#534C22×C10912yes—
349#525—21no—
350C2×#123—21no—
351C2×#123—21no—
352C2×#123—28no—
353C2×#123C22×C10912yes—
354C2×#123C22×C10912yescomplemented
355C2×#123C22×C10914yescomplemented
356C2×#120—21no—
357C2×#120C22×C10912yes—
358#518C22×C10911yescharacteristic
359C22×C4⋊C4—21no—
360C22×C22⋊C4—21no—
361C22×C22⋊C4C22×C10911yescharacteristic, complemented
362#458—28no—
363C2×#99C22×C10912yes—
364C2×#99C22×C10914yescomplemented
365C2×#99C22×C10912yescomplemented
366#386—24no—
367#385—24no—
368#381C22×C10914yescomplemented
369#379C22×C10912yescomplemented
370#342C22×C10914yescomplemented
371C25D4×C10911yescharacteristic
372C25C23×C10911yescharacteristic
373H(5,𝔽2)—24no—
374H(5,𝔽2)—42no—
375C2×#41—21no—
376C2×#41—22no—
377C22×D4—21no—
378C22×D4—22no—
379C22×D4—24no—
380C22×D4—22no—
381C22×D4—24no—
382C22×D4—42no—
383C22×D4D4×C10911yescharacteristic
384C22×D4D4×C10911yescharacteristic
385C22×D4D4×C10914yes—
386C22×D4C23×C10912yes—
387C22×D4C23×C10911yescharacteristic
388C22×D4C23×C10911yescharacteristic
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—21no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—42no—
395C23×C4D4×C10911yescharacteristic
396C23×C4C23×C10912yes—
397#127—22no—
398#124—22no—
399#123—28no—
400#123—28no—
401#123—22no—
402#123—24no—
403#123—22no—
404#123—22no—
405#123—24no—
406#123—24no—
407#123—28no—
408#123—44no—
409#123D4×C10918yescomplemented
410C22⋊Q8—22no—
411#121—22no—
412#121—24no—
413#121—24no—
414#121—24no—
415#120—24no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—22no—
420#120—24no—
421#120—22no—
422C4×D4—24no—
423C4×D4—24no—
424#117—21no—
425C2×C4⋊C4—28no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—24no—
429C2×C22⋊C4—28no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—22no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—21no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4—24no—
438C2×C22⋊C4D4×C10914yes—
439C2×C22⋊C4C23×C10914yes—
440C2×C22⋊C4C23×C10912yes—
441C2×C22⋊C4C23×C10912yes—
442#99—24no—
443#99—28no—
444#99—42no—
445#95—28no—
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C10911yescharacteristic
455C24C2×D4×C10914yes—
456C24C2×D4×C10911yescharacteristic
457C24C2×D4×C10912yes—
458C24C2×D4×C10911yescharacteristic
459C24C2×D4×C10911yescharacteristic
460C24C24×C10911yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C10914yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—44no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C10912yes—
514C22×C4C2×D4×C10911yescharacteristic
515C22×C4C2×D4×C10911yescharacteristic
516C22×C4C2×D4×C10914yes—
517C22×C4C2×D4×C10914yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C10914yescomplemented
539C42—22no—
540C23—21no—
541C23—21no—
542C23—24no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—28no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23#120×C10912yes—
570C23#120×C10912yes—
571C23#120×C10912yes—
572C23C22×D4×C10911yescharacteristic
573C23C22×D4×C10912yes—
574C23C22×D4×C10912yes—
575C23H(5,𝔽2)×C10912yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—44no—
586D4—48no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4#120×C10912yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—24no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—41no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22#458×C10914yescomplemented
642C22C2×#120×C10911yescharacteristic
643C22C2×#120×C10911yescharacteristic
644C22#534×C10911yescharacteristic
645C22D42×C10914yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—44no—
650C4—41no—
651C4—42no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—21no—
656C2—24no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C109×#240311yescharacteristic
662C2C2×#458×C10912yes—
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC24C24×C109
prime radicalC109×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C25×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×#123×C109C2semidirect
3. C2×#458×C109C2semidirect
4. C109×#2849C2semidirect
5. C109×#2022C2semidirect
6. C109×#1898C2semidirect
7. C109×#1890C2semidirect
8. C109×#1781C2semidirect
11. C24×C4×C109C22semidirect
22. C2×#123×C109C22semidirect
23. C2×#123×C109C22semidirect
29. C22×C22⋊C4×C109C22semidirect
32. C2×#99×C109C22semidirect
33. C2×#99×C109C22semidirect
36. #381×C109C22semidirect
37. #379×C109C22semidirect
38. #342×C109C22semidirect
77. #123×C109D4semidirect
185. C22×C4×C109C2×D4semidirect
206. C22⋊C4×C109C2×D4semidirect
309. C22×C109#458semidirect
321. #23901C109direct (N × H)
333. C22×#123C2×C109semidirect
334. C2×#458C2×C109semidirect
335. #2849C2×C109semidirect
336. #2022C2×C109semidirect
337. #1898C2×C109semidirect
338. #1890C2×C109semidirect
339. #1781C2×C109semidirect
340. C109#23901direct (N × H)
343. C24×C4C22×C109semidirect
354. C2×#123C22×C109semidirect
355. C2×#123C22×C109semidirect
361. C22×C22⋊C4C22×C109semidirect
364. C2×#99C22×C109semidirect
365. C2×#99C22×C109semidirect
368. #381C22×C109semidirect
369. #379C22×C109semidirect
370. #342C22×C109semidirect
409. #123D4×C109semidirect
517. C22×C4C2×D4×C109semidirect
538. C22⋊C4C2×D4×C109semidirect
641. C22#458×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34842
C22×#123×C1092
C2×#458×C1092
C109×#28492
C109×#20222
C109×#18982
C109×#18902
C109×#17812
#23901109