Finite Group Databasebeta

← C109×#24018 #42142012

Subgroups — C109×#24018 #42142012

Order 27904.

1578 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3484C211yesmaximal, characteristic
2C109×#3092C212yesmaximal, complemented
3C2×#390×C109C211yesmaximal, characteristic, complemented
4C109×#2037C214yesmaximal, complemented
5C109×#1896C212yesmaximal, complemented
6C109×#1856C212yesmaximal, complemented
7C109×#1792C212yesmaximal, complemented
8C109×#1783C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C109C2211yescharacteristic
10C23×D4×C109C2211yescharacteristic
11#546×C109—22no—
12#546×C109—22no—
13D42×C109—22no—
14D42×C109—22no—
15#535×C109C2212yes—
16#534×C109—22no—
17#525×C109C2212yes—
18C2×#120×C109—22no—
19#518×C109C2211yescharacteristic
20C22×C4⋊C4×C109C2211yescharacteristic, complemented
21C22×C22⋊C4×C109—22no—
22C22×C42×C109C2211yescharacteristic, complemented
23#484×C109—22no—
24#483×C109—22no—
25#481×C109—22no—
26#480×C109—22no—
27#428×C109C2214yescomplemented
28C2×#104×C109C2214yescomplemented
29#417×C109C2214yescomplemented
30C2×#100×C109C2212yes—
31C2×#100×C109C2212yescomplemented
32#407×C109C2212yescomplemented
33#390×C109—22no—
34#390×C109C2212yes—
35#390×C109C2212yescomplemented
36#339×C109C2214yescomplemented
37C25×C109D412yes—
38H(5,𝔽2)×C109—24no—
39H(5,𝔽2)×C109—42no—
40C2×#41×C109—22no—
41C2×#41×C109C2312yes—
42C22×D4×C109—22no—
43C22×D4×C109—22no—
44C22×D4×C109—22no—
45C22×D4×C109—22no—
46C22×D4×C109—24no—
47C22×D4×C109—22no—
48C22×D4×C109—22no—
49C22×D4×C109—42no—
50C22×D4×C109D412yes—
51C22×D4×C109C2311yescharacteristic
52C22×D4×C109C2311yescharacteristic
53C23×C4×C109—21no—
54C23×C4×C109—21no—
55C23×C4×C109C2311yescharacteristic
56C2×M16×C109C2314yes—
57#127×C109—22no—
58#124×C109—22no—
59#123×C109—24no—
60#123×C109—22no—
61C22⋊Q8×C109D412yes—
62C22⋊Q8×C109D412yes—
63#121×C109—24no—
64#121×C109—24no—
65#121×C109—22no—
66#121×C109—22no—
67#121×C109D412yes—
68#121×C109D412yes—
69#120×C109—24no—
70#120×C109—24no—
71#120×C109—24no—
72#120×C109—22no—
73#120×C109—24no—
74C4×D4×C109—22no—
75C4×D4×C109—22no—
76C4×D4×C109—22no—
77C4×D4×C109—22no—
78#117×C109C2312yes—
79C2×C4⋊C4×C109—22no—
80C2×C4⋊C4×C109—22no—
81C2×C4⋊C4×C109—21no—
82C2×C4⋊C4×C109C2312yes—
83C2×C22⋊C4×C109—22no—
84C2×C22⋊C4×C109—22no—
85C2×C22⋊C4×C109—22no—
86C2×C22⋊C4×C109—24no—
87C2×C22⋊C4×C109—24no—
88C2×C22⋊C4×C109—48no—
89C2×C42×C109—21no—
90C2×C42×C109—42no—
91C2×C42×C109C2312yes—
92#107×C109—24no—
93#106×C109—24no—
94#104×C109—44no—
95#103×C109—24no—
96#102×C109—24no—
97#100×C109—24no—
98#100×C109—42no—
99C22⋊C8×C109—22no—
100C22⋊C8×C109—22no—
101C24×C109—22no—
102C24×C109—24no—
103C24×C109—22no—
104C24×C109—22no—
105C24×C109—42no—
106C24×C109—44no—
107C24×C109—44no—
108C24×C109C2×D411yescharacteristic
109C24×C109C2×D412yes—
110C24×C109C2×D412yes—
111#41×C109—24no—
112#41×C109—24no—
113#41×C109—48no—
114C2×Q8×C109C2×D412yes—
115C2×D4×C109—24no—
116C2×D4×C109—22no—
117C2×D4×C109—22no—
118C2×D4×C109—24no—
119C2×D4×C109—22no—
120C2×D4×C109—24no—
121C2×D4×C109—24no—
122C2×D4×C109—24no—
123C2×D4×C109—22no—
124C2×D4×C109—22no—
125C2×D4×C109—22no—
126C2×D4×C109—22no—
127C2×D4×C109—22no—
128C2×D4×C109—48no—
129C2×D4×C109—48no—
130C2×D4×C109—42no—
131C2×D4×C109—48no—
132C2×D4×C109—44no—
133C2×D4×C109—44no—
134C2×D4×C109—84no—
135C2×D4×C109—84no—
136C2×D4×C109C2×D412yes—
137C2×D4×C109#4114yes—
138C22×C4×C109—22no—
139C22×C4×C109—22no—
140C22×C4×C109—22no—
141C22×C4×C109—24no—
142C22×C4×C109—22no—
143C22×C4×C109—22no—
144C22×C4×C109—22no—
145C22×C4×C109—22no—
146C22×C4×C109—22no—
147C22×C4×C109—42no—
148C22×C4×C109—42no—
149C22×C4×C109—44no—
150C22×C4×C109—41no—
151C22×C4×C109—41no—
152C22×C4×C109C2×D411yescharacteristic
153C22×C4×C109C2×D412yes—
154C22×C4×C109C2×D412yes—
155C22×C4×C109C2411yescharacteristic
156M16×C109—24no—
157M16×C109—24no—
158C2×C8×C109—24no—
159C4⋊C4×C109—22no—
160C4⋊C4×C109—22no—
161C4⋊C4×C109—24no—
162C4⋊C4×C109—42no—
163C4⋊C4×C109C2×D412yes—
164C4⋊C4×C109C2×D412yes—
165C22⋊C4×C109—24no—
166C22⋊C4×C109—24no—
167C22⋊C4×C109—24no—
168C22⋊C4×C109—22no—
169C22⋊C4×C109—22no—
170C22⋊C4×C109—22no—
171C22⋊C4×C109—22no—
172C22⋊C4×C109—22no—
173C22⋊C4×C109—48no—
174C22⋊C4×C109—48no—
175C42×C109—22no—
176C42×C109—24no—
177C42×C109—42no—
178C23×C109—21no—
179C23×C109—21no—
180C23×C109—22no—
181C23×C109—22no—
182C23×C109—22no—
183C23×C109—22no—
184C23×C109—22no—
185C23×C109—22no—
186C23×C109—22no—
187C23×C109—22no—
188C23×C109—44no—
189C23×C109—44no—
190C23×C109—42no—
191C23×C109—42no—
192C23×C109—42no—
193C23×C109—42no—
194C23×C109—416no—
195C23×C109—44no—
196C23×C109—44no—
197C23×C109—44no—
198C23×C109—82no—
199C23×C109—82no—
200C23×C109—88no—
201C23×C109—82no—
202C23×C109—82no—
203C23×C109#12011yescharacteristic
204C23×C109#12011yescharacteristic
205C23×C109#12314yes—
206C23×C109C22×D411yescharacteristic
207Q8×C109—24no—
208D4×C109—28no—
209D4×C109—24no—
210D4×C109—48no—
211D4×C109—48no—
212D4×C109—42no—
213D4×C109—42no—
214D4×C109—44no—
215D4×C109—44no—
216D4×C109—44no—
217D4×C109—84no—
218D4×C109—84no—
219C2×C4×C109—21no—
220C2×C4×C109—21no—
221C2×C4×C109—24no—
222C2×C4×C109—24no—
223C2×C4×C109—24no—
224C2×C4×C109—22no—
225C2×C4×C109—22no—
226C2×C4×C109—22no—
227C2×C4×C109—48no—
228C2×C4×C109—42no—
229C2×C4×C109—44no—
230C2×C4×C109—42no—
231C2×C4×C109—44no—
232C2×C4×C109—44no—
233C2×C4×C109—42no—
234C2×C4×C109—82no—
235C2×C4×C109—82no—
236C2×C4×C109—82no—
237C2×C4×C109—82no—
238C2×C4×C109#12012yes—
239C2×C4×C109C22×D412yes—
240C2×C4×C109C2×#4112yes—
241C2×C4×C109H(5,𝔽2)12yes—
242C8×C109—44no—
243C22×C109—24no—
244C22×C109—24no—
245C22×C109—21no—
246C22×C109—21no—
247C22×C109—21no—
248C22×C109—21no—
249C22×C109—21no—
250C22×C109—21no—
251C22×C109—24no—
252C22×C109—44no—
253C22×C109—44no—
254C22×C109—42no—
255C22×C109—42no—
256C22×C109—42no—
257C22×C109—42no—
258C22×C109—42no—
259C22×C109—42no—
260C22×C109—42no—
261C22×C109—42no—
262C22×C109—82no—
263C22×C109—88no—
264C22×C109—88no—
265C22×C109—82no—
266C22×C109—164no—
267C22×C109#45312yes—
268C22×C109#45512yes—
269C22×C109C2×#12011yescharacteristic
270C22×C109C2×#12311yescharacteristic
271C22×C109#53411yescharacteristic
272C4×C109—22no—
273C4×C109—44no—
274C4×C109—42no—
275C4×C109—42no—
276C4×C109—42no—
277C4×C109—82no—
278C4×C109#54614yes—
279#24018C10911yesmaximal, characteristic, direct product factor, Sylow
280C2×C109—21no—
281C2×C109—21no—
282C2×C109—22no—
283C2×C109—22no—
284C2×C109—44no—
285C2×C109—82no—
286C2×C109—82no—
287C2×C109#240511yescharacteristic
288C2×C109C2×#45311yescharacteristic
289C2×C109C2×#45511yescharacteristic
290#3484C2×C10911yescharacteristic
291#3092C2×C10912yescomplemented
292C2×#390C2×C10911yescharacteristic, complemented
293#2037C2×C10914yescomplemented
294#1896C2×C10912yescomplemented
295#1856C2×C10912yescomplemented
296#1792C2×C10912yescomplemented
297#1783C2×C10911yescharacteristic, complemented
298C109#2401811yescharacteristic, direct product factor, Sylow
299C2×H(5,𝔽2)C22×C10911yescharacteristic
300C23×D4C22×C10911yescharacteristic
301#546—22no—
302#546—22no—
303D42—22no—
304D42—22no—
305#535C22×C10912yes—
306#534—22no—
307#525C22×C10912yes—
308C2×#120—22no—
309#518C22×C10911yescharacteristic
310C22×C4⋊C4C22×C10911yescharacteristic, complemented
311C22×C22⋊C4—22no—
312C22×C42C22×C10911yescharacteristic, complemented
313#484—22no—
314#483—22no—
315#481—22no—
316#480—22no—
317#428C22×C10914yescomplemented
318C2×#104C22×C10914yescomplemented
319#417C22×C10914yescomplemented
320C2×#100C22×C10912yes—
321C2×#100C22×C10912yescomplemented
322#407C22×C10912yescomplemented
323#390—22no—
324#390C22×C10912yes—
325#390C22×C10912yescomplemented
326#339C22×C10914yescomplemented
327C25D4×C10912yes—
328H(5,𝔽2)—24no—
329H(5,𝔽2)—42no—
330C2×#41—22no—
331C2×#41C23×C10912yes—
332C22×D4—22no—
333C22×D4—22no—
334C22×D4—22no—
335C22×D4—22no—
336C22×D4—24no—
337C22×D4—22no—
338C22×D4—22no—
339C22×D4—42no—
340C22×D4D4×C10912yes—
341C22×D4C23×C10911yescharacteristic
342C22×D4C23×C10911yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4C23×C10911yescharacteristic
346C2×M16C23×C10914yes—
347#127—22no—
348#124—22no—
349#123—24no—
350#123—22no—
351C22⋊Q8D4×C10912yes—
352C22⋊Q8D4×C10912yes—
353#121—24no—
354#121—24no—
355#121—22no—
356#121—22no—
357#121D4×C10912yes—
358#121D4×C10912yes—
359#120—24no—
360#120—24no—
361#120—24no—
362#120—22no—
363#120—24no—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368#117C23×C10912yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4C23×C10912yes—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—24no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—48no—
379C2×C42—21no—
380C2×C42—42no—
381C2×C42C23×C10912yes—
382#107—24no—
383#106—24no—
384#104—44no—
385#103—24no—
386#102—24no—
387#100—24no—
388#100—42no—
389C22⋊C8—22no—
390C22⋊C8—22no—
391C24—22no—
392C24—24no—
393C24—22no—
394C24—22no—
395C24—42no—
396C24—44no—
397C24—44no—
398C24C2×D4×C10911yescharacteristic
399C24C2×D4×C10912yes—
400C24C2×D4×C10912yes—
401#41—24no—
402#41—24no—
403#41—48no—
404C2×Q8C2×D4×C10912yes—
405C2×D4—24no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—24no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—42no—
421C2×D4—48no—
422C2×D4—44no—
423C2×D4—44no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4C2×D4×C10912yes—
427C2×D4#41×C10914yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—44no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4C2×D4×C10911yescharacteristic
443C22×C4C2×D4×C10912yes—
444C22×C4C2×D4×C10912yes—
445C22×C4C24×C10911yescharacteristic
446M16—24no—
447M16—24no—
448C2×C8—24no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—42no—
453C4⋊C4C2×D4×C10912yes—
454C4⋊C4C2×D4×C10912yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C42—22no—
466C42—24no—
467C42—42no—
468C23—21no—
469C23—21no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—416no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—82no—
489C23—82no—
490C23—88no—
491C23—82no—
492C23—82no—
493C23#120×C10911yescharacteristic
494C23#120×C10911yescharacteristic
495C23#123×C10914yes—
496C23C22×D4×C10911yescharacteristic
497Q8—24no—
498D4—28no—
499D4—24no—
500D4—48no—
501D4—48no—
502D4—42no—
503D4—42no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#120×C10912yes—
529C2×C4C22×D4×C10912yes—
530C2×C4C2×#41×C10912yes—
531C2×C4H(5,𝔽2)×C10912yes—
532C8—44no—
533C22—24no—
534C22—24no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—24no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—88no—
554C22—88no—
555C22—82no—
556C22—164no—
557C22#453×C10912yes—
558C22#455×C10912yes—
559C22C2×#120×C10911yescharacteristic
560C22C2×#123×C10911yescharacteristic
561C22#534×C10911yescharacteristic
562C4—22no—
563C4—44no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4#546×C10914yes—
569C2—21no—
570C2—21no—
571C2—22no—
572C2—22no—
573C2—44no—
574C2—82no—
575C2—82no—
576C2C109×#240511yescharacteristic
577C2C2×#453×C10911yescharacteristic
578C2C2×#455×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3484C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3092C2semidirect
3. C2×#390×C109C2semidirect
4. C109×#2037C2semidirect
5. C109×#1896C2semidirect
6. C109×#1856C2semidirect
7. C109×#1792C2semidirect
8. C109×#1783C2semidirect
20. C22×C4⋊C4×C109C22semidirect
22. C22×C42×C109C22semidirect
27. #428×C109C22semidirect
28. C2×#104×C109C22semidirect
29. #417×C109C22semidirect
31. C2×#100×C109C22semidirect
32. #407×C109C22semidirect
35. #390×C109C22semidirect
36. #339×C109C22semidirect
279. #24018C109direct (N × H)
291. #3092C2×C109semidirect
292. C2×#390C2×C109semidirect
293. #2037C2×C109semidirect
294. #1896C2×C109semidirect
295. #1856C2×C109semidirect
296. #1792C2×C109semidirect
297. #1783C2×C109semidirect
298. C109#24018direct (N × H)
310. C22×C4⋊C4C22×C109semidirect
312. C22×C42C22×C109semidirect
317. #428C22×C109semidirect
318. C2×#104C22×C109semidirect
319. #417C22×C109semidirect
321. C2×#100C22×C109semidirect
322. #407C22×C109semidirect
325. #390C22×C109semidirect
326. #339C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34842
C109×#30922
C2×#390×C1092
C109×#20372
C109×#18962
C109×#18562
C109×#17922
C109×#17832
#24018109