Finite Group Databasebeta

← C109×#14506 #42132500

Subgroups — C109×#14506 #42132500

Order 27904.

1962 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C109C211yesmaximal, characteristic
2C2×#388×C109C212yesmaximal, complemented
3C109×#1445C212yesmaximal, complemented
4C109×#1439C212yesmaximal, complemented
5C109×#1438C211yesmaximal, characteristic
6C109×#1438C211yesmaximal, characteristic
7C109×#1437C212yesmaximal, complemented
8C109×C4⋊#95C212yesmaximal, complemented
9C109×C22⋊#95C212yesmaximal, complemented
10C23×D4×C109C2211yescharacteristic
11C24×C4×C109C2211yescharacteristic
12C2×#123×C109—28no—
13C2×#123×C109C418yescomplemented
14C22×C4⋊C4×C109C2212yes—
15C22×C4⋊C4×C109C2212yes—
16C22×C22⋊C4×C109C2212yes—
17C22×C22⋊C4×C109C2212yes—
18C22×C22⋊C4×C109C2211yescharacteristic
19C22×C22⋊C4×C109C2212yes—
20C22×C22⋊C4×C109C2212yes—
21C22×C22⋊C4×C109C2211yescharacteristic
22C22×C22⋊C4×C109C2211yescharacteristic
23C22×C42×C109C2212yes—
24#388×C109—28no—
25C2×#95×C109C2212yes—
26C2×#95×C109C2212yes—
27C2×#95×C109C2212yes—
28C2×#95×C109C2212yes—
29C2×#95×C109C2212yes—
30C2×#95×C109C2212yes—
31C2×#95×C109C2211yescharacteristic
32C2×#95×C109C2212yes—
33C2×#95×C109C2211yescharacteristic
34C25×C109C2311yescharacteristic
35C25×C109C2311yescharacteristic
36C22×D4×C109—24no—
37C22×D4×C109—42no—
38C22×D4×C109—41no—
39C22×D4×C109—41no—
40C22×D4×C109C2×C414yes—
41C23×C4×C109—24no—
42C23×C4×C109—42no—
43C23×C4×C109—41no—
44C23×C4×C109—41no—
45C23×C4×C109C2×C414yes—
46C23×C4×C109C2312yes—
47C23×C4×C109C2312yes—
48C23×C4×C109C2312yes—
49C23×C4×C109C2311yescharacteristic
50C23×C4×C109C2312yes—
51C23×C4×C109C2312yes—
52C23×C4×C109C2312yes—
53#123×C109—232no—
54C2×C4⋊C4×C109—24no—
55C2×C4⋊C4×C109—24no—
56C2×C4⋊C4×C109—24no—
57C2×C4⋊C4×C109—24no—
58C2×C4⋊C4×C109—42no—
59C2×C4⋊C4×C109C2×C418yes—
60C2×C22⋊C4×C109—24no—
61C2×C22⋊C4×C109—24no—
62C2×C22⋊C4×C109—24no—
63C2×C22⋊C4×C109—24no—
64C2×C22⋊C4×C109—24no—
65C2×C22⋊C4×C109—24no—
66C2×C22⋊C4×C109—24no—
67C2×C22⋊C4×C109—24no—
68C2×C22⋊C4×C109—41no—
69C2×C22⋊C4×C109—42no—
70C2×C22⋊C4×C109—42no—
71C2×C22⋊C4×C109—42no—
72C2×C22⋊C4×C109—42no—
73C2×C22⋊C4×C109—41no—
74C2×C22⋊C4×C109—42no—
75C2×C22⋊C4×C109C2×C418yes—
76C2×C22⋊C4×C109C2×C414yes—
77C2×C22⋊C4×C109D418yes—
78C2×C42×C109—24no—
79C2×C42×C109—24no—
80C2×C42×C109—42no—
81#95×C109—24no—
82#95×C109—24no—
83#95×C109—41no—
84#95×C109—42no—
85#95×C109—42no—
86#95×C109—42no—
87#95×C109—42no—
88#95×C109—42no—
89#95×C109—41no—
90C24×C109—22no—
91C24×C109—22no—
92C24×C109—22no—
93C24×C109—22no—
94C24×C109—21no—
95C24×C109—21no—
96C24×C109—22no—
97C24×C109—22no—
98C24×C109—22no—
99C24×C109—22no—
100C24×C109—22no—
101C24×C109—22no—
102C24×C109—21no—
103C24×C109—21no—
104C24×C109—22no—
105C24×C109C22×C412yes—
106C24×C109C22×C412yes—
107C24×C109C2×D414yes—
108C24×C109C2411yescharacteristic
109C2×D4×C109—28no—
110C2×D4×C109—44no—
111C2×D4×C109—44no—
112C2×D4×C109—48no—
113C2×D4×C109—48no—
114C22×C4×C109—22no—
115C22×C4×C109—22no—
116C22×C4×C109—22no—
117C22×C4×C109—22no—
118C22×C4×C109—28no—
119C22×C4×C109—22no—
120C22×C4×C109—22no—
121C22×C4×C109—22no—
122C22×C4×C109—22no—
123C22×C4×C109—22no—
124C22×C4×C109—22no—
125C22×C4×C109—22no—
126C22×C4×C109—22no—
127C22×C4×C109—22no—
128C22×C4×C109—22no—
129C22×C4×C109—22no—
130C22×C4×C109—22no—
131C22×C4×C109—22no—
132C22×C4×C109—22no—
133C22×C4×C109—22no—
134C22×C4×C109—22no—
135C22×C4×C109—21no—
136C22×C4×C109—21no—
137C22×C4×C109—22no—
138C22×C4×C109—22no—
139C22×C4×C109—22no—
140C22×C4×C109—22no—
141C22×C4×C109—22no—
142C22×C4×C109—22no—
143C22×C4×C109—22no—
144C22×C4×C109—22no—
145C22×C4×C109—22no—
146C22×C4×C109—22no—
147C22×C4×C109—22no—
148C22×C4×C109—22no—
149C22×C4×C109—22no—
150C22×C4×C109—22no—
151C22×C4×C109—22no—
152C22×C4×C109—22no—
153C22×C4×C109—22no—
154C22×C4×C109—22no—
155C22×C4×C109—44no—
156C22×C4×C109—44no—
157C22×C4×C109—48no—
158C22×C4×C109—48no—
159C22×C4×C109C22×C414yes—
160C22×C4×C109C22×C412yes—
161C22×C4×C109C22×C414yes—
162C22×C4×C109C2×D414yes—
163C22×C4×C109C2×D414yes—
164C22×C4×C109#4114yes—
165C22×C4×C109#4114yes—
166C4⋊C4×C109—216no—
167C4⋊C4×C109—48no—
168C22⋊C4×C109—216no—
169C22⋊C4×C109—44no—
170C22⋊C4×C109—48no—
171C22⋊C4×C109—48no—
172C22⋊C4×C109—48no—
173C22⋊C4×C109—44no—
174C22⋊C4×C109C22⋊C4116yescomplemented
175C42×C109—48no—
176C23×C109—24no—
177C23×C109—22no—
178C23×C109—22no—
179C23×C109—24no—
180C23×C109—24no—
181C23×C109—24no—
182C23×C109—24no—
183C23×C109—24no—
184C23×C109—24no—
185C23×C109—24no—
186C23×C109—22no—
187C23×C109—24no—
188C23×C109—22no—
189C23×C109—22no—
190C23×C109—24no—
191C23×C109—42no—
192C23×C109—42no—
193C23×C109—42no—
194C23×C109—42no—
195C23×C109—41no—
196C23×C109—42no—
197C23×C109—41no—
198C23×C109—41no—
199C23×C109—42no—
200C23×C109—41no—
201C23×C109—42no—
202C23×C109—42no—
203C23×C109—42no—
204C23×C109—42no—
205C23×C109—42no—
206C23×C109—42no—
207C23×C109—42no—
208C23×C109—42no—
209C23×C109—42no—
210C23×C109—42no—
211C23×C109—41no—
212C23×C109—42no—
213C23×C109—41no—
214C23×C109—41no—
215C23×C109—42no—
216C23×C109—41no—
217C23×C109C2×C22⋊C414yes—
218C23×C109C23×C411yescharacteristic
219C23×C109C22×D412yes—
220C23×C109C2×#4112yes—
221C23×C109C2×#4112yes—
222C23×C109H(5,𝔽2)12yes—
223C23×C109H(5,𝔽2)11yescharacteristic
224C23×C109H(5,𝔽2)11yescharacteristic
225C23×C109H(5,𝔽2)12yes—
226C23×C1092−1+412yes—
227D4×C109—416no—
228C2×C4×C109—24no—
229C2×C4×C109—24no—
230C2×C4×C109—24no—
231C2×C4×C109—24no—
232C2×C4×C109—24no—
233C2×C4×C109—24no—
234C2×C4×C109—24no—
235C2×C4×C109—22no—
236C2×C4×C109—24no—
237C2×C4×C109—24no—
238C2×C4×C109—22no—
239C2×C4×C109—22no—
240C2×C4×C109—24no—
241C2×C4×C109—24no—
242C2×C4×C109—24no—
243C2×C4×C109—24no—
244C2×C4×C109—24no—
245C2×C4×C109—24no—
246C2×C4×C109—416no—
247C2×C4×C109—42no—
248C2×C4×C109—42no—
249C2×C4×C109—42no—
250C2×C4×C109—42no—
251C2×C4×C109—42no—
252C2×C4×C109—42no—
253C2×C4×C109—42no—
254C2×C4×C109—42no—
255C2×C4×C109—42no—
256C2×C4×C109—42no—
257C2×C4×C109—42no—
258C2×C4×C109—42no—
259C2×C4×C109—42no—
260C2×C4×C109—42no—
261C2×C4×C109—42no—
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—42no—
269C2×C4×C109—42no—
270C2×C4×C109—42no—
271C2×C4×C109—42no—
272C2×C4×C109—42no—
273C2×C4×C109—42no—
274C2×C4×C109—42no—
275C2×C4×C109C2×C22⋊C418yes—
276C22×C109—24no—
277C22×C109—28no—
278C22×C109—44no—
279C22×C109—42no—
280C22×C109—42no—
281C22×C109—44no—
282C22×C109—44no—
283C22×C109—44no—
284C22×C109—44no—
285C22×C109—42no—
286C22×C109—44no—
287C22×C109—44no—
288C22×C109—42no—
289C22×C109—42no—
290C22×C109—82no—
291C22×C109—82no—
292C22×C109—82no—
293C22×C109—82no—
294C22×C109C22×C22⋊C411yescharacteristic
295C22×C109C4×#4112yes—
296C22×C109#51811yescharacteristic
297C22×C109#51811yescharacteristic
298C22×C109#52012yes—
299C22×C109#53512yes—
300C22×C109#53512yes—
301C22×C109#53812yes—
302C22×C109#53912yes—
303C22×C109#54012yes—
304C22×C109#54212yes—
305C22×C109#54612yes—
306C22×C109#54612yes—
307C22×C109#54712yes—
308C22×C109#54712yes—
309C22×C109#55111yescharacteristic
310C22×C109#55312yes—
311C22×C109#55911yescharacteristic
312C22×C109#56011yescharacteristic
313C22×C109#56212yes—
314C22×C109#56311yescharacteristic
315C4×C109—28no—
316C4×C109—44no—
317C4×C109—42no—
318C4×C109—44no—
319C4×C109—44no—
320C4×C109—82no—
321C4×C109—82no—
322#14506C10911yesmaximal, characteristic, direct product factor, Sylow
323C2×C109—44no—
324C2×C109—82no—
325C2×C109#230911yescharacteristic
326C2×C109#233512yes—
327C2×C109#233912yes—
328C2×C109#237411yescharacteristic
329C2×C109#237611yescharacteristic
330C2×C109#270312yes—
331C2×C109#270612yes—
332C2×C109#271012yes—
333C2×C109#271812yes—
334C22×#123C2×C10911yescharacteristic
335C2×#388C2×C10912yescomplemented
336#1445C2×C10912yescomplemented
337#1439C2×C10912yescomplemented
338#1438C2×C10911yescharacteristic
339#1438C2×C10911yescharacteristic
340#1437C2×C10912yescomplemented
341C4⋊#95C2×C10912yescomplemented
342C22⋊#95C2×C10912yescomplemented
343C109#1450611yescharacteristic, direct product factor, Sylow
344C23×D4C22×C10911yescharacteristic
345C24×C4C22×C10911yescharacteristic
346C2×#123—28no—
347C2×#123C4×C10918yescomplemented
348C22×C4⋊C4C22×C10912yes—
349C22×C4⋊C4C22×C10912yes—
350C22×C22⋊C4C22×C10912yes—
351C22×C22⋊C4C22×C10912yes—
352C22×C22⋊C4C22×C10911yescharacteristic
353C22×C22⋊C4C22×C10912yes—
354C22×C22⋊C4C22×C10912yes—
355C22×C22⋊C4C22×C10911yescharacteristic
356C22×C22⋊C4C22×C10911yescharacteristic
357C22×C42C22×C10912yes—
358#388—28no—
359C2×#95C22×C10912yes—
360C2×#95C22×C10912yes—
361C2×#95C22×C10912yes—
362C2×#95C22×C10912yes—
363C2×#95C22×C10912yes—
364C2×#95C22×C10912yes—
365C2×#95C22×C10911yescharacteristic
366C2×#95C22×C10912yes—
367C2×#95C22×C10911yescharacteristic
368C25C23×C10911yescharacteristic
369C25C23×C10911yescharacteristic
370C22×D4—24no—
371C22×D4—42no—
372C22×D4—41no—
373C22×D4—41no—
374C22×D4C2×C4×C10914yes—
375C23×C4—24no—
376C23×C4—42no—
377C23×C4—41no—
378C23×C4—41no—
379C23×C4C2×C4×C10914yes—
380C23×C4C23×C10912yes—
381C23×C4C23×C10912yes—
382C23×C4C23×C10912yes—
383C23×C4C23×C10911yescharacteristic
384C23×C4C23×C10912yes—
385C23×C4C23×C10912yes—
386C23×C4C23×C10912yes—
387#123—232no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—24no—
390C2×C4⋊C4—24no—
391C2×C4⋊C4—24no—
392C2×C4⋊C4—42no—
393C2×C4⋊C4C2×C4×C10918yes—
394C2×C22⋊C4—24no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—24no—
399C2×C22⋊C4—24no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—41no—
403C2×C22⋊C4—42no—
404C2×C22⋊C4—42no—
405C2×C22⋊C4—42no—
406C2×C22⋊C4—42no—
407C2×C22⋊C4—41no—
408C2×C22⋊C4—42no—
409C2×C22⋊C4C2×C4×C10918yes—
410C2×C22⋊C4C2×C4×C10914yes—
411C2×C22⋊C4D4×C10918yes—
412C2×C42—24no—
413C2×C42—24no—
414C2×C42—42no—
415#95—24no—
416#95—24no—
417#95—41no—
418#95—42no—
419#95—42no—
420#95—42no—
421#95—42no—
422#95—42no—
423#95—41no—
424C24—22no—
425C24—22no—
426C24—22no—
427C24—22no—
428C24—21no—
429C24—21no—
430C24—22no—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—21no—
437C24—21no—
438C24—22no—
439C24C22×C4×C10912yes—
440C24C22×C4×C10912yes—
441C24C2×D4×C10914yes—
442C24C24×C10911yescharacteristic
443C2×D4—28no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—48no—
447C2×D4—48no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4C22×C4×C10914yes—
494C22×C4C22×C4×C10912yes—
495C22×C4C22×C4×C10914yes—
496C22×C4C2×D4×C10914yes—
497C22×C4C2×D4×C10914yes—
498C22×C4#41×C10914yes—
499C22×C4#41×C10914yes—
500C4⋊C4—216no—
501C4⋊C4—48no—
502C22⋊C4—216no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—44no—
508C22⋊C4C22⋊C4×C109116yescomplemented
509C42—48no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—41no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23C2×C22⋊C4×C10914yes—
552C23C23×C4×C10911yescharacteristic
553C23C22×D4×C10912yes—
554C23C2×#41×C10912yes—
555C23C2×#41×C10912yes—
556C23H(5,𝔽2)×C10912yes—
557C23H(5,𝔽2)×C10911yescharacteristic
558C23H(5,𝔽2)×C10911yescharacteristic
559C23H(5,𝔽2)×C10912yes—
560C232−1+4×C10912yes—
561D4—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—416no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4C2×C22⋊C4×C10918yes—
610C22—24no—
611C22—28no—
612C22—44no—
613C22—44no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—44no—
622C22—42no—
623C22—42no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22C22×C22⋊C4×C10911yescharacteristic
629C22C4×#41×C10912yes—
630C22#518×C10911yescharacteristic
631C22#518×C10911yescharacteristic
632C22#520×C10912yes—
633C22#535×C10912yes—
634C22#535×C10912yes—
635C22#538×C10912yes—
636C22#539×C10912yes—
637C22#540×C10912yes—
638C22#542×C10912yes—
639C22#546×C10912yes—
640C22#546×C10912yes—
641C22#547×C10912yes—
642C22#547×C10912yes—
643C22#551×C10911yescharacteristic
644C22#553×C10912yes—
645C22#559×C10911yescharacteristic
646C22#560×C10911yescharacteristic
647C22#562×C10912yes—
648C22#563×C10911yescharacteristic
649C4—28no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—82no—
655C4—82no—
656C2—44no—
657C2—82no—
658C2C109×#230911yescharacteristic
659C2C109×#233512yes—
660C2C109×#233912yes—
661C2C109×#237411yescharacteristic
662C2C109×#237611yescharacteristic
663C2C109×#270312yes—
664C2C109×#270612yes—
665C2C109×#271012yes—
666C2C109×#271812yes—
subgroupG/H
center, socleC24×C109C24
commutatorC23C23×C4×C109
Frattini, cosocleC24C24×C109
prime radicalC23×D4×C109C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#388×C109C2semidirect
3. C109×#1445C2semidirect
4. C109×#1439C2semidirect
7. C109×#1437C2semidirect
8. C109×C4⋊#95C2semidirect
9. C109×C22⋊#95C2semidirect
13. C2×#123×C109C4semidirect
174. C22⋊C4×C109C22⋊C4semidirect
322. #14506C109direct (N × H)
335. C2×#388C2×C109semidirect
336. #1445C2×C109semidirect
337. #1439C2×C109semidirect
340. #1437C2×C109semidirect
341. C4⋊#95C2×C109semidirect
342. C22⋊#95C2×C109semidirect
343. C109#14506direct (N × H)
347. C2×#123C4×C109semidirect
508. C22⋊C4C22⋊C4×C109semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C1092
C2×#388×C1092
C109×#14452
C109×#14392
C109×#14382
C109×#14372
C109×C4⋊#952
C109×C22⋊#952
#14506109