Finite Group Databasebeta

← C109×#18201 #42136195

Subgroups — C109×#18201 #42136195

Order 27904.

960 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3492C211yesmaximal, characteristic
2C109×#3091C212yesmaximal, complemented
3C109×#2484C211yesmaximal, characteristic, complemented
4C109×#1617C211yesmaximal, characteristic, complemented
5C109×#1616C211yesmaximal, characteristic, complemented
6C109×#1615C212yesmaximal, complemented
7C109×#1611C211yesmaximal, characteristic, complemented
8C109×#1610C211yesmaximal, characteristic, complemented
9C109×#1609C212yesmaximal, complemented
10C109×#1516C212yesmaximal, complemented
11C2×#324×C109C211yesmaximal, characteristic, complemented
12#554×C109—22no—
13#553×C109—22no—
14#552×C109—22no—
15#547×C109—22no—
16#544×C109C2211yescharacteristic
17#539×C109—21no—
18#539×C109C2212yes—
19#537×C109C2211yescharacteristic
20#525×C109—21no—
21#525×C109—21no—
22#525×C109C2212yes—
23#520×C109—21no—
24C2×#117×C109C2211yescharacteristic
25C22×C4⋊C4×C109C2211yescharacteristic
26#485×C109—22no—
27#484×C109C2212yes—
28#483×C109C2212yes—
29#482×C109—22no—
30#481×C109C2212yes—
31#480×C109C2212yes—
32#427×C109—22no—
33#416×C109—22no—
34C2×C22⋊C8×C109C2212yes—
35#400×C109C2211yescharacteristic
36#400×C109C2211yescharacteristic
37#398×C109C2211yescharacteristic
38#398×C109C2211yescharacteristic
39#387×C109—21no—
40#387×C109—21no—
41#385×C109—21no—
42#381×C109—21no—
43#381×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—
52C2×#41×C109—21no—
53C2×#41×C109D412yes—
54C23×C4×C109D411yescharacteristic
55C23×C4×C109D411yescharacteristic
56C23×C4×C109C2311yescharacteristic
57C22×C8×C109—22no—
58C4⋊Q8×C109—22no—
59#126×C109—22no—
60#126×C109—22no—
61#125×C109—22no—
62#125×C109—22no—
63#125×C109—22no—
64#123×C109—22no—
65#123×C109—22no—
66#123×C109—22no—
67#123×C109—22no—
68#123×C109—22no—
69C22⋊Q8×C109—21no—
70C22⋊Q8×C109—21no—
71C22⋊Q8×C109—21no—
72C22⋊Q8×C109—22no—
73C22⋊Q8×C109—22no—
74C22⋊Q8×C109—21no—
75C22⋊Q8×C109—22no—
76C22⋊Q8×C109D412yes—
77C22⋊Q8×C109C2312yes—
78#121×C109—21no—
79#121×C109—21no—
80#121×C109—21no—
81#121×C109—22no—
82#121×C109—21no—
83#121×C109D412yes—
84#121×C109C2312yes—
85#120×C109—22no—
86C4×Q8×C109—22no—
87C4×D4×C109—22no—
88C4×D4×C109—22no—
89C4×D4×C109—22no—
90C4×D4×C109—22no—
91C4×D4×C109—22no—
92C4×D4×C109—22no—
93C4×D4×C109—22no—
94#117×C109—22no—
95#117×C109—21no—
96#117×C109—21no—
97#117×C109—22no—
98#117×C109—22no—
99#117×C109D412yes—
100C2×C4⋊C4×C109—22no—
101C2×C4⋊C4×C109—22no—
102C2×C4⋊C4×C109—22no—
103C2×C4⋊C4×C109—21no—
104C2×C4⋊C4×C109—21no—
105C2×C4⋊C4×C109D411yescharacteristic
106C2×C4⋊C4×C109D411yescharacteristic
107C2×C4⋊C4×C109C2311yescharacteristic
108C2×C4⋊C4×C109C2311yescharacteristic
109C2×C4⋊C4×C109C2311yescharacteristic
110C2×C4⋊C4×C109C2311yescharacteristic
111C2×C22⋊C4×C109—21no—
112C2×C22⋊C4×C109—21no—
113C2×C42×C109—21no—
114#107×C109—22no—
115#107×C109—22no—
116#106×C109—22no—
117#106×C109—22no—
118#103×C109—22no—
119#103×C109—22no—
120#102×C109—22no—
121#102×C109—22no—
122C22⋊C8×C109—22no—
123C22⋊C8×C109C2312yes—
124C22⋊C8×C109C2312yes—
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
134C24×C109C2×D411yescharacteristic
135#41×C109—42no—
136#41×C109—44no—
137#41×C109—42no—
138#41×C109—82no—
139C2×Q8×C109—21no—
140C2×Q8×C109—21no—
141C2×Q8×C109C2×D412yes—
142C2×D4×C109—21no—
143C2×D4×C109—22no—
144C2×D4×C109—22no—
145C2×D4×C109—22no—
146C2×D4×C109—21no—
147C2×D4×C109C2×D412yes—
148C22×C4×C109—22no—
149C22×C4×C109—21no—
150C22×C4×C109—21no—
151C22×C4×C109—21no—
152C22×C4×C109—22no—
153C22×C4×C109—21no—
154C22×C4×C109—22no—
155C22×C4×C109—22no—
156C22×C4×C109—21no—
157C22×C4×C109—21no—
158C22×C4×C109—21no—
159C22×C4×C109—21no—
160C22×C4×C109—41no—
161C22×C4×C109—42no—
162C22×C4×C109—42no—
163C22×C4×C109C2×D411yescharacteristic
164C22×C4×C109C2×D412yes—
165C22×C4×C109C2×D411yescharacteristic
166C22×C4×C109C2×D411yescharacteristic
167C22×C4×C109C2×D411yescharacteristic
168C22×C4×C109C2×D411yescharacteristic
169C22×C4×C109C2×D412yes—
170C22×C4×C109C2411yescharacteristic
171C2×C8×C109—22no—
172C2×C8×C109—22no—
173C2×C8×C109—44no—
174C4⋊C4×C109—21no—
175C4⋊C4×C109—22no—
176C4⋊C4×C109—21no—
177C4⋊C4×C109—21no—
178C4⋊C4×C109—21no—
179C4⋊C4×C109—22no—
180C4⋊C4×C109—22no—
181C4⋊C4×C109—22no—
182C4⋊C4×C109—22no—
183C4⋊C4×C109—22no—
184C4⋊C4×C109—22no—
185C4⋊C4×C109—22no—
186C4⋊C4×C109—22no—
187C4⋊C4×C109—22no—
188C4⋊C4×C109—22no—
189C4⋊C4×C109—22no—
190C4⋊C4×C109—42no—
191C4⋊C4×C109—42no—
192C4⋊C4×C109C2×D412yes—
193C4⋊C4×C109C2×D412yes—
194C22⋊C4×C109—22no—
195C22⋊C4×C109—22no—
196C22⋊C4×C109—22no—
197C22⋊C4×C109—22no—
198C22⋊C4×C109—22no—
199C22⋊C4×C109—22no—
200C22⋊C4×C109—22no—
201C22⋊C4×C109—22no—
202C22⋊C4×C109—22no—
203C22⋊C4×C109—42no—
204C22⋊C4×C109—41no—
205C22⋊C4×C109—41no—
206C42×C109—22no—
207C42×C109—22no—
208C42×C109—22no—
209C42×C109—42no—
210C23×C109—21no—
211C23×C109—21no—
212C23×C109—22no—
213C23×C109—22no—
214C23×C109—41no—
215C23×C109#12011yescharacteristic
216C23×C109#12011yescharacteristic
217C23×C109C22×D411yescharacteristic
218Q8×C109—42no—
219Q8×C109—42no—
220Q8×C109—81no—
221D4×C109—42no—
222D4×C109—42no—
223D4×C109—44no—
224D4×C109—42no—
225D4×C109—42no—
226D4×C109—44no—
227D4×C109—81no—
228C2×C4×C109—21no—
229C2×C4×C109—21no—
230C2×C4×C109—24no—
231C2×C4×C109—22no—
232C2×C4×C109—22no—
233C2×C4×C109—21no—
234C2×C4×C109—22no—
235C2×C4×C109—21no—
236C2×C4×C109—21no—
237C2×C4×C109—21no—
238C2×C4×C109—21no—
239C2×C4×C109—42no—
240C2×C4×C109—41no—
241C2×C4×C109—41no—
242C2×C4×C109—42no—
243C2×C4×C109—42no—
244C2×C4×C109—44no—
245C2×C4×C109—42no—
246C2×C4×C109—41no—
247C2×C4×C109—41no—
248C2×C4×C109—41no—
249C2×C4×C109—41no—
250C2×C4×C109—41no—
251C2×C4×C109—41no—
252C2×C4×C109—42no—
253C2×C4×C109—41no—
254C2×C4×C109—41no—
255C2×C4×C109—41no—
256C2×C4×C109—41no—
257C2×C4×C109—41no—
258C2×C4×C109—41no—
259C2×C4×C109—82no—
260C2×C4×C109—81no—
261C2×C4×C109—82no—
262C2×C4×C109#12012yes—
263C2×C4×C109#13512yes—
264C2×C4×C109C22×D412yes—
265C8×C109—42no—
266C8×C109—42no—
267C22×C109—21no—
268C22×C109—21no—
269C22×C109—21no—
270C22×C109—21no—
271C22×C109—21no—
272C22×C109—21no—
273C22×C109—21no—
274C22×C109—42no—
275C22×C109—42no—
276C22×C109—42no—
277C22×C109—42no—
278C22×C109—42no—
279C22×C109#45812yes—
280C22×C109C2×#12011yescharacteristic
281C22×C109C2×#13511yescharacteristic
282C22×C109#57511yescharacteristic
283C4×C109—22no—
284C4×C109—22no—
285C4×C109—22no—
286C4×C109—41no—
287C4×C109—42no—
288C4×C109—42no—
289C4×C109—41no—
290C4×C109—41no—
291C4×C109—42no—
292C4×C109—81no—
293C4×C109—81no—
294C4×C109—81no—
295C4×C109—81no—
296#18201C10911yesmaximal, characteristic, direct product factor, Sylow
297C2×C109—21no—
298C2×C109—21no—
299C2×C109—22no—
300C2×C109—41no—
301C2×C109—82no—
302C2×C109#300211yescharacteristic
303C2×C109#302011yescharacteristic
304C2×C109C2×#45811yescharacteristic
305#3492C2×C10911yescharacteristic
306#3091C2×C10912yescomplemented
307#2484C2×C10911yescharacteristic, complemented
308#1617C2×C10911yescharacteristic, complemented
309#1616C2×C10911yescharacteristic, complemented
310#1615C2×C10912yescomplemented
311#1611C2×C10911yescharacteristic, complemented
312#1610C2×C10911yescharacteristic, complemented
313#1609C2×C10912yescomplemented
314#1516C2×C10912yescomplemented
315C2×#324C2×C10911yescharacteristic, complemented
316C109#1820111yescharacteristic, direct product factor, Sylow
317#554—22no—
318#553—22no—
319#552—22no—
320#547—22no—
321#544C22×C10911yescharacteristic
322#539—21no—
323#539C22×C10912yes—
324#537C22×C10911yescharacteristic
325#525—21no—
326#525—21no—
327#525C22×C10912yes—
328#520—21no—
329C2×#117C22×C10911yescharacteristic
330C22×C4⋊C4C22×C10911yescharacteristic
331#485—22no—
332#484C22×C10912yes—
333#483C22×C10912yes—
334#482—22no—
335#481C22×C10912yes—
336#480C22×C10912yes—
337#427—22no—
338#416—22no—
339C2×C22⋊C8C22×C10912yes—
340#400C22×C10911yescharacteristic
341#400C22×C10911yescharacteristic
342#398C22×C10911yescharacteristic
343#398C22×C10911yescharacteristic
344#387—21no—
345#387—21no—
346#385—21no—
347#381—21no—
348#381C22×C10911yescharacteristic
349#328C22×C10912yes—
350#328C22×C10912yes—
351#327C22×C10912yes—
352#327C22×C10912yes—
353#324C22×C10912yes—
354#324C22×C10911yescharacteristic
355#324C22×C10911yescharacteristic
356C2×#41—21no—
357C2×#41—21no—
358C2×#41D4×C10912yes—
359C23×C4D4×C10911yescharacteristic
360C23×C4D4×C10911yescharacteristic
361C23×C4C23×C10911yescharacteristic
362C22×C8—22no—
363C4⋊Q8—22no—
364#126—22no—
365#126—22no—
366#125—22no—
367#125—22no—
368#125—22no—
369#123—22no—
370#123—22no—
371#123—22no—
372#123—22no—
373#123—22no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377C22⋊Q8—22no—
378C22⋊Q8—22no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8D4×C10912yes—
382C22⋊Q8C23×C10912yes—
383#121—21no—
384#121—21no—
385#121—21no—
386#121—22no—
387#121—21no—
388#121D4×C10912yes—
389#121C23×C10912yes—
390#120—22no—
391C4×Q8—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—22no—
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—22no—
398C4×D4—22no—
399#117—22no—
400#117—21no—
401#117—21no—
402#117—22no—
403#117—22no—
404#117D4×C10912yes—
405C2×C4⋊C4—22no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—22no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4D4×C10911yescharacteristic
411C2×C4⋊C4D4×C10911yescharacteristic
412C2×C4⋊C4C23×C10911yescharacteristic
413C2×C4⋊C4C23×C10911yescharacteristic
414C2×C4⋊C4C23×C10911yescharacteristic
415C2×C4⋊C4C23×C10911yescharacteristic
416C2×C22⋊C4—21no—
417C2×C22⋊C4—21no—
418C2×C42—21no—
419#107—22no—
420#107—22no—
421#106—22no—
422#106—22no—
423#103—22no—
424#103—22no—
425#102—22no—
426#102—22no—
427C22⋊C8—22no—
428C22⋊C8C23×C10912yes—
429C22⋊C8C23×C10912yes—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95C23×C10911yescharacteristic
438#95C23×C10911yescharacteristic
439C24C2×D4×C10911yescharacteristic
440#41—42no—
441#41—44no—
442#41—42no—
443#41—82no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8C2×D4×C10912yes—
447C2×D4—21no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4C2×D4×C10912yes—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4C2×D4×C10911yescharacteristic
469C22×C4C2×D4×C10912yes—
470C22×C4C2×D4×C10911yescharacteristic
471C22×C4C2×D4×C10911yescharacteristic
472C22×C4C2×D4×C10911yescharacteristic
473C22×C4C2×D4×C10911yescharacteristic
474C22×C4C2×D4×C10912yes—
475C22×C4C24×C10911yescharacteristic
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—44no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4C2×D4×C10912yes—
498C4⋊C4C2×D4×C10912yes—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—22no—
506C22⋊C4—22no—
507C22⋊C4—22no—
508C22⋊C4—42no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—42no—
515C23—21no—
516C23—21no—
517C23—22no—
518C23—22no—
519C23—41no—
520C23#120×C10911yescharacteristic
521C23#120×C10911yescharacteristic
522C23C22×D4×C10911yescharacteristic
523Q8—42no—
524Q8—42no—
525Q8—81no—
526D4—42no—
527D4—42no—
528D4—44no—
529D4—42no—
530D4—42no—
531D4—44no—
532D4—81no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—82no—
565C2×C4—81no—
566C2×C4—82no—
567C2×C4#120×C10912yes—
568C2×C4#135×C10912yes—
569C2×C4C22×D4×C10912yes—
570C8—42no—
571C8—42no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22#458×C10912yes—
585C22C2×#120×C10911yescharacteristic
586C22C2×#135×C10911yescharacteristic
587C22#575×C10911yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—41no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4—81no—
601C2—21no—
602C2—21no—
603C2—22no—
604C2—41no—
605C2—82no—
606C2C109×#300211yescharacteristic
607C2C109×#302011yescharacteristic
608C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3492C2
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. C109×#3091C2semidirect
3. C109×#2484C2semidirect
4. C109×#1617C2semidirect
5. C109×#1616C2semidirect
6. C109×#1615C2semidirect
7. C109×#1611C2semidirect
8. C109×#1610C2semidirect
9. C109×#1609C2semidirect
10. C109×#1516C2semidirect
11. C2×#324×C109C2semidirect
296. #18201C109direct (N × H)
306. #3091C2×C109semidirect
307. #2484C2×C109semidirect
308. #1617C2×C109semidirect
309. #1616C2×C109semidirect
310. #1615C2×C109semidirect
311. #1611C2×C109semidirect
312. #1610C2×C109semidirect
313. #1609C2×C109semidirect
314. #1516C2×C109semidirect
315. C2×#324C2×C109semidirect
316. C109#18201direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#34922
C109×#30912
C109×#24842
C109×#16172
C109×#16162
C109×#16152
C109×#16112
C109×#16102
C109×#16092
C109×#15162
C2×#324×C1092
#18201109