Finite Group Databasebeta

← C109×#18315 #42136309

Subgroups — C109×#18315 #42136309

Order 27904.

868 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3461C211yesmaximal, characteristic
2C109×#3232C212yesmaximal, complemented
3C109×#2634C211yesmaximal, characteristic, complemented
4C109×#1617C211yesmaximal, characteristic, complemented
5C109×#1616C211yesmaximal, characteristic
6C109×#1615C212yesmaximal, complemented
7C109×#1611C211yesmaximal, characteristic, complemented
8C109×#1610C211yesmaximal, characteristic
9C109×#1609C212yesmaximal, complemented
10C109×#1521C212yesmaximal, complemented
11C109×#1461C211yesmaximal, characteristic, complemented
12#544×C109C2211yescharacteristic
13#542×C109—22no—
14#541×C109—22no—
15#539×C109—21no—
16#539×C109C2212yes—
17#537×C109C2211yescharacteristic
18#536×C109—21no—
19#533×C109—22no—
20#529×C109—22no—
21#529×C109—21no—
22#529×C109C2212yes—
23C2×#125×C109C2211yescharacteristic
24C4×#41×C109—21no—
25C2×C4×Q8×C109C2211yescharacteristic
26#499×C109—22no—
27#488×C109—22no—
28#484×C109C2212yes—
29#483×C109C2212yes—
30#481×C109C2212yes—
31#480×C109C2212yes—
32#479×C109—22no—
33#478×C109—22no—
34#431×C109C2212yes—
35#400×C109—21no—
36#400×C109C2211yescharacteristic
37#400×C109C2211yescharacteristic
38#398×C109C2211yescharacteristic
39#398×C109C2211yescharacteristic
40#395×C109—21no—
41#387×C109—21no—
42#387×C109—21no—
43#383×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—
52C22×Q8×C109D411yescharacteristic
53C4⋊Q8×C109—22no—
54C4⋊Q8×C109—22no—
55#126×C109—22no—
56#126×C109—22no—
57#126×C109—22no—
58#126×C109—22no—
59#125×C109—21no—
60#125×C109—22no—
61#125×C109—22no—
62#125×C109—22no—
63#125×C109—22no—
64#125×C109—22no—
65#125×C109D412yes—
66#124×C109—22no—
67#124×C109—21no—
68#124×C109D412yes—
69#123×C109—22no—
70#123×C109—22no—
71#123×C109—22no—
72#123×C109—22no—
73C22⋊Q8×C109—22no—
74C22⋊Q8×C109—22no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109—22no—
77C22⋊Q8×C109—21no—
78C22⋊Q8×C109—22no—
79C22⋊Q8×C109C2312yes—
80#121×C109—22no—
81#121×C109—21no—
82#121×C109C2312yes—
83C4×Q8×C109—21no—
84C4×Q8×C109—21no—
85C4×Q8×C109—21no—
86C4×Q8×C109—22no—
87C4×Q8×C109—22no—
88C4×Q8×C109D412yes—
89C4×D4×C109—21no—
90C4×D4×C109—21no—
91C4×D4×C109—21no—
92C4×D4×C109—22no—
93C4×D4×C109—22no—
94C4×D4×C109D412yes—
95#117×C109—22no—
96#117×C109—22no—
97#117×C109—21no—
98#117×C109—22no—
99C2×C4⋊C4×C109—21no—
100C2×C4⋊C4×C109—21no—
101C2×C4⋊C4×C109—21no—
102C2×C4⋊C4×C109—21no—
103C2×C4⋊C4×C109D411yescharacteristic
104C2×C4⋊C4×C109C2311yescharacteristic
105C2×C4⋊C4×C109C2311yescharacteristic
106C2×C4⋊C4×C109C2311yescharacteristic
107C2×C4⋊C4×C109C2311yescharacteristic
108C2×C42×C109—22no—
109C2×C42×C109D411yescharacteristic
110C2×C42×C109D411yescharacteristic
111C2×C42×C109C2311yescharacteristic
112#107×C109—22no—
113#107×C109—22no—
114#106×C109—22no—
115#106×C109—22no—
116C4⋊C8×C109—22no—
117#103×C109—22no—
118#103×C109—22no—
119#102×C109—22no—
120#102×C109—22no—
121C22⋊C8×C109C2312yes—
122C22⋊C8×C109C2312yes—
123C4×C8×C109—22no—
124#95×C109—21no—
125#95×C109—21no—
126#95×C109—21no—
127#95×C109—21no—
128#95×C109—21no—
129#95×C109—21no—
130#95×C109—21no—
131#95×C109—21no—
132#95×C109C2311yescharacteristic
133#95×C109C2311yescharacteristic
134#41×C109—42no—
135#41×C109—42no—
136C2×Q8×C109—21no—
137C2×Q8×C109—21no—
138C2×Q8×C109—22no—
139C2×Q8×C109—22no—
140C2×Q8×C109—41no—
141C2×Q8×C109C2×D412yes—
142C2×D4×C109—21no—
143C2×D4×C109—22no—
144C2×D4×C109—21no—
145C2×D4×C109C2×D412yes—
146C22×C4×C109—22no—
147C22×C4×C109—22no—
148C22×C4×C109—22no—
149C22×C4×C109—21no—
150C22×C4×C109—21no—
151C22×C4×C109—21no—
152C22×C4×C109—21no—
153C22×C4×C109C2×D411yescharacteristic
154C22×C4×C109C2×D411yescharacteristic
155C22×C4×C109C2×D411yescharacteristic
156C22×C4×C109C2×D411yescharacteristic
157C22×C4×C109C2×D411yescharacteristic
158C22×C4×C109C2×D411yescharacteristic
159C22×C4×C109C2411yescharacteristic
160C2×C8×C109—22no—
161C2×C8×C109—22no—
162C4⋊C4×C109—21no—
163C4⋊C4×C109—22no—
164C4⋊C4×C109—21no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—22no—
167C4⋊C4×C109—21no—
168C4⋊C4×C109—21no—
169C4⋊C4×C109—22no—
170C4⋊C4×C109—22no—
171C4⋊C4×C109—22no—
172C4⋊C4×C109—22no—
173C4⋊C4×C109—22no—
174C4⋊C4×C109—22no—
175C4⋊C4×C109—22no—
176C4⋊C4×C109—22no—
177C4⋊C4×C109—22no—
178C4⋊C4×C109—22no—
179C4⋊C4×C109—42no—
180C4⋊C4×C109—41no—
181C4⋊C4×C109—41no—
182C4⋊C4×C109—42no—
183C4⋊C4×C109—42no—
184C4⋊C4×C109C2×D412yes—
185C4⋊C4×C109C2×D412yes—
186C22⋊C4×C109—22no—
187C22⋊C4×C109—22no—
188C22⋊C4×C109—22no—
189C22⋊C4×C109—22no—
190C22⋊C4×C109—22no—
191C22⋊C4×C109—22no—
192C22⋊C4×C109—22no—
193C22⋊C4×C109—22no—
194C22⋊C4×C109—22no—
195C42×C109—22no—
196C42×C109—22no—
197C42×C109—21no—
198C42×C109—22no—
199C42×C109—21no—
200C42×C109—21no—
201C42×C109—21no—
202C42×C109C2×D412yes—
203C42×C109C2×D412yes—
204C23×C109—22no—
205C23×C109C22×D411yescharacteristic
206Q8×C109—44no—
207Q8×C109—42no—
208Q8×C109—42no—
209Q8×C109—81no—
210D4×C109—42no—
211D4×C109—42no—
212D4×C109—42no—
213D4×C109—42no—
214D4×C109—81no—
215C2×C4×C109—21no—
216C2×C4×C109—21no—
217C2×C4×C109—21no—
218C2×C4×C109—21no—
219C2×C4×C109—21no—
220C2×C4×C109—22no—
221C2×C4×C109—22no—
222C2×C4×C109—21no—
223C2×C4×C109—21no—
224C2×C4×C109—22no—
225C2×C4×C109—21no—
226C2×C4×C109—21no—
227C2×C4×C109—41no—
228C2×C4×C109—41no—
229C2×C4×C109—44no—
230C2×C4×C109—44no—
231C2×C4×C109—41no—
232C2×C4×C109—41no—
233C2×C4×C109—41no—
234C2×C4×C109—41no—
235C2×C4×C109—41no—
236C2×C4×C109—41no—
237C2×C4×C109—42no—
238C2×C4×C109—41no—
239C2×C4×C109—41no—
240C2×C4×C109—41no—
241C2×C4×C109—41no—
242C2×C4×C109—41no—
243C2×C4×C109—41no—
244C2×C4×C109—82no—
245C2×C4×C109#12011yescharacteristic
246C2×C4×C109#12011yescharacteristic
247C2×C4×C109#12012yes—
248C2×C4×C109#13512yes—
249C2×C4×C109C22×D412yes—
250C8×C109—42no—
251C8×C109—42no—
252C22×C109—21no—
253C22×C109—21no—
254C22×C109—42no—
255C22×C109—42no—
256C22×C109—42no—
257C22×C109C2×#12011yescharacteristic
258C22×C109C2×#13511yescharacteristic
259C22×C109#57511yescharacteristic
260C4×C109—22no—
261C4×C109—22no—
262C4×C109—21no—
263C4×C109—21no—
264C4×C109—22no—
265C4×C109—41no—
266C4×C109—42no—
267C4×C109—42no—
268C4×C109—41no—
269C4×C109—41no—
270C4×C109—42no—
271C4×C109—81no—
272C4×C109—81no—
273C4×C109—81no—
274C4×C109—81no—
275#18315C10911yesmaximal, characteristic, direct product factor, Sylow
276C2×C109—21no—
277C2×C109—21no—
278C2×C109—82no—
279C2×C109#300211yescharacteristic
280C2×C109#302011yescharacteristic
281C2×C109#302811yescharacteristic
282#3461C2×C10911yescharacteristic
283#3232C2×C10912yescomplemented
284#2634C2×C10911yescharacteristic, complemented
285#1617C2×C10911yescharacteristic, complemented
286#1616C2×C10911yescharacteristic
287#1615C2×C10912yescomplemented
288#1611C2×C10911yescharacteristic, complemented
289#1610C2×C10911yescharacteristic
290#1609C2×C10912yescomplemented
291#1521C2×C10912yescomplemented
292#1461C2×C10911yescharacteristic, complemented
293C109#1831511yescharacteristic, direct product factor, Sylow
294#544C22×C10911yescharacteristic
295#542—22no—
296#541—22no—
297#539—21no—
298#539C22×C10912yes—
299#537C22×C10911yescharacteristic
300#536—21no—
301#533—22no—
302#529—22no—
303#529—21no—
304#529C22×C10912yes—
305C2×#125C22×C10911yescharacteristic
306C4×#41—21no—
307C2×C4×Q8C22×C10911yescharacteristic
308#499—22no—
309#488—22no—
310#484C22×C10912yes—
311#483C22×C10912yes—
312#481C22×C10912yes—
313#480C22×C10912yes—
314#479—22no—
315#478—22no—
316#431C22×C10912yes—
317#400—21no—
318#400C22×C10911yescharacteristic
319#400C22×C10911yescharacteristic
320#398C22×C10911yescharacteristic
321#398C22×C10911yescharacteristic
322#395—21no—
323#387—21no—
324#387—21no—
325#383C22×C10911yescharacteristic
326#328C22×C10912yes—
327#328C22×C10912yes—
328#327C22×C10912yes—
329#327C22×C10912yes—
330#324C22×C10912yes—
331#324C22×C10911yescharacteristic
332#324C22×C10911yescharacteristic
333C2×#41—21no—
334C22×Q8D4×C10911yescharacteristic
335C4⋊Q8—22no—
336C4⋊Q8—22no—
337#126—22no—
338#126—22no—
339#126—22no—
340#126—22no—
341#125—21no—
342#125—22no—
343#125—22no—
344#125—22no—
345#125—22no—
346#125—22no—
347#125D4×C10912yes—
348#124—22no—
349#124—21no—
350#124D4×C10912yes—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—22no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—21no—
360C22⋊Q8—22no—
361C22⋊Q8C23×C10912yes—
362#121—22no—
363#121—21no—
364#121C23×C10912yes—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—21no—
368C4×Q8—22no—
369C4×Q8—22no—
370C4×Q8D4×C10912yes—
371C4×D4—21no—
372C4×D4—21no—
373C4×D4—21no—
374C4×D4—22no—
375C4×D4—22no—
376C4×D4D4×C10912yes—
377#117—22no—
378#117—22no—
379#117—21no—
380#117—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4D4×C10911yescharacteristic
386C2×C4⋊C4C23×C10911yescharacteristic
387C2×C4⋊C4C23×C10911yescharacteristic
388C2×C4⋊C4C23×C10911yescharacteristic
389C2×C4⋊C4C23×C10911yescharacteristic
390C2×C42—22no—
391C2×C42D4×C10911yescharacteristic
392C2×C42D4×C10911yescharacteristic
393C2×C42C23×C10911yescharacteristic
394#107—22no—
395#107—22no—
396#106—22no—
397#106—22no—
398C4⋊C8—22no—
399#103—22no—
400#103—22no—
401#102—22no—
402#102—22no—
403C22⋊C8C23×C10912yes—
404C22⋊C8C23×C10912yes—
405C4×C8—22no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—21no—
414#95C23×C10911yescharacteristic
415#95C23×C10911yescharacteristic
416#41—42no—
417#41—42no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—41no—
423C2×Q8C2×D4×C10912yes—
424C2×D4—21no—
425C2×D4—22no—
426C2×D4—21no—
427C2×D4C2×D4×C10912yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4C2×D4×C10911yescharacteristic
436C22×C4C2×D4×C10911yescharacteristic
437C22×C4C2×D4×C10911yescharacteristic
438C22×C4C2×D4×C10911yescharacteristic
439C22×C4C2×D4×C10911yescharacteristic
440C22×C4C2×D4×C10911yescharacteristic
441C22×C4C24×C10911yescharacteristic
442C2×C8—22no—
443C2×C8—22no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4C2×D4×C10912yes—
467C4⋊C4C2×D4×C10912yes—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42C2×D4×C10912yes—
485C42C2×D4×C10912yes—
486C23—22no—
487C23C22×D4×C10911yescharacteristic
488Q8—44no—
489Q8—42no—
490Q8—42no—
491Q8—81no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—81no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—82no—
527C2×C4#120×C10911yescharacteristic
528C2×C4#120×C10911yescharacteristic
529C2×C4#120×C10912yes—
530C2×C4#135×C10912yes—
531C2×C4C22×D4×C10912yes—
532C8—42no—
533C8—42no—
534C22—21no—
535C22—21no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22C2×#120×C10911yescharacteristic
540C22C2×#135×C10911yescharacteristic
541C22#575×C10911yescharacteristic
542C4—22no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—41no—
551C4—41no—
552C4—42no—
553C4—81no—
554C4—81no—
555C4—81no—
556C4—81no—
557C2—21no—
558C2—21no—
559C2—82no—
560C2C109×#300211yescharacteristic
561C2C109×#302011yescharacteristic
562C2C109×#302811yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radical#537×C109C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3232C2semidirect
3. C109×#2634C2semidirect
4. C109×#1617C2semidirect
6. C109×#1615C2semidirect
7. C109×#1611C2semidirect
9. C109×#1609C2semidirect
10. C109×#1521C2semidirect
11. C109×#1461C2semidirect
275. #18315C109direct (N × H)
283. #3232C2×C109semidirect
284. #2634C2×C109semidirect
285. #1617C2×C109semidirect
287. #1615C2×C109semidirect
288. #1611C2×C109semidirect
290. #1609C2×C109semidirect
291. #1521C2×C109semidirect
292. #1461C2×C109semidirect
293. C109#18315direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#34612
C109×#32322
C109×#26342
C109×#16172
C109×#16162
C109×#16152
C109×#16112
C109×#16102
C109×#16092
C109×#15212
C109×#14612
#18315109