Finite Group Databasebeta

← C109×#18167 #42136161

Subgroups — C109×#18167 #42136161

Order 27904.

1218 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3501C211yesmaximal, characteristic, complemented
2C109×#3007C212yesmaximal, complemented
3C109×#2388C211yesmaximal, characteristic
4C109×#1605C212yesmaximal, complemented
5C109×#1602C212yesmaximal, complemented
6C109×#1599C212yesmaximal, complemented
7C109×#1596C212yesmaximal, complemented
8C109×#1508C212yesmaximal, complemented
9C109×#1465C211yesmaximal, characteristic, complemented
10C22×#41×C109C2211yescharacteristic
11C2×#137×C109—22no—
12#554×C109—24no—
13#554×C109—22no—
14D4×Q8×C109—22no—
15#544×C109C2212yes—
16#543×C109—21no—
17#543×C109C2212yes—
18#533×C109—21no—
19C2×C22⋊Q8×C109—21no—
20C2×C22⋊Q8×C109C2212yes—
21#518×C109—21no—
22C2×C4×D4×C109—21no—
23C2×C4×D4×C109C2211yescharacteristic
24#452×C109—24no—
25#451×C109C2214yes—
26#450×C109C2214yes—
27#417×C109—22no—
28#407×C109C2212yes—
29#395×C109C2211yescharacteristic
30#394×C109—21no—
31#394×C109C2211yescharacteristic
32#393×C109C2211yescharacteristic
33#392×C109—21no—
34#392×C109C2211yescharacteristic
35C4×C22⋊C4×C109—21no—
36C4×C22⋊C4×C109C2211yescharacteristic
37#328×C109C2214yes—
38#328×C109C2214yes—
39#324×C109C2212yes—
40#323×C109C2212yes—
41C2×#41×C109—22no—
42C2×#41×C109—22no—
43C2×#41×C109—44no—
44C22×Q8×C109—21no—
45C22×Q8×C109D412yescomplemented
46C22×Q8×C109C2311yescharacteristic
47C22×D4×C109—21no—
48C22×D4×C109—21no—
49C22×D4×C109D411yescharacteristic
50C22×D4×C109C2311yescharacteristic
51C23×C4×C109—21no—
52C23×C4×C109D411yescharacteristic
53C23×C4×C109C2311yescharacteristic
54#137×C109—48no—
55C2×Q16×C109—24no—
56C2×SD16×C109—24no—
57C2×M16×C109—22no—
58C4⋊Q8×C109—21no—
59C4⋊Q8×C109—21no—
60C4⋊Q8×C109—24no—
61#125×C109—22no—
62#125×C109—24no—
63C22⋊Q8×C109—22no—
64C22⋊Q8×C109—22no—
65C22⋊Q8×C109—24no—
66C22⋊Q8×C109—24no—
67C22⋊Q8×C109—24no—
68C22⋊Q8×C109—22no—
69C22⋊Q8×C109—22no—
70C22⋊Q8×C109—22no—
71C22⋊Q8×C109—22no—
72C22⋊Q8×C109—24no—
73C22⋊Q8×C109D414yescomplemented
74C22⋊Q8×C109C2314yes—
75C4×Q8×C109—22no—
76C4×Q8×C109—22no—
77C4×D4×C109—24no—
78C4×D4×C109—24no—
79C4×D4×C109—22no—
80C4×D4×C109—22no—
81C4×D4×C109—44no—
82#117×C109—22no—
83C2×C4⋊C4×C109—21no—
84C2×C4⋊C4×C109—22no—
85C2×C4⋊C4×C109—22no—
86C2×C4⋊C4×C109—21no—
87C2×C4⋊C4×C109D412yescomplemented
88C2×C4⋊C4×C109C2311yescharacteristic
89C2×C22⋊C4×C109—22no—
90C2×C22⋊C4×C109—22no—
91C2×C22⋊C4×C109—21no—
92C2×C22⋊C4×C109—21no—
93C2×C22⋊C4×C109—21no—
94C2×C22⋊C4×C109—21no—
95C2×C22⋊C4×C109—21no—
96C2×C22⋊C4×C109D411yescharacteristic
97C2×C22⋊C4×C109C2311yescharacteristic
98C2×C42×C109—21no—
99C2×C42×C109D411yescharacteristic
100#103×C109—24no—
101#102×C109—24no—
102C22⋊C8×C109—22no—
103C22⋊C8×C109C2314yes—
104#95×C109—21no—
105#95×C109—21no—
106#95×C109C2311yescharacteristic
107#95×C109C2311yescharacteristic
108C24×C109—21no—
109C24×C109C2×D411yescharacteristic
110C24×C109C2×D411yescharacteristic
111#41×C109—28no—
112#41×C109—44no—
113#41×C109—44no—
114#41×C109—44no—
115C2×Q8×C109—22no—
116C2×Q8×C109—22no—
117C2×Q8×C109—22no—
118C2×Q8×C109—22no—
119C2×Q8×C109—42no—
120C2×Q8×C109—44no—
121C2×Q8×C109—82no—
122C2×Q8×C109C2×D414yes—
123C2×D4×C109—22no—
124C2×D4×C109—22no—
125C2×D4×C109—22no—
126C2×D4×C109—22no—
127C2×D4×C109—42no—
128C2×D4×C109—42no—
129C2×D4×C109—42no—
130C2×D4×C109—42no—
131C2×D4×C109—42no—
132C2×D4×C109—42no—
133C2×D4×C109—42no—
134C22×C4×C109—24no—
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—21no—
142C22×C4×C109—21no—
143C22×C4×C109—21no—
144C22×C4×C109—42no—
145C22×C4×C109—42no—
146C22×C4×C109—42no—
147C22×C4×C109—41no—
148C22×C4×C109—41no—
149C22×C4×C109C2×D411yescharacteristic
150C22×C4×C109C2×D412yes—
151C22×C4×C109C2×D411yescharacteristic
152C22×C4×C109C2×D411yescharacteristic
153C22×C4×C109C2×D411yescharacteristic
154C22×C4×C109C2×D412yes—
155C22×C4×C109C2411yescharacteristic
156Q16×C109—48no—
157SD16×C109—48no—
158M16×C109—44no—
159C2×C8×C109—24no—
160C4⋊C4×C109—22no—
161C4⋊C4×C109—24no—
162C4⋊C4×C109—22no—
163C4⋊C4×C109—22no—
164C4⋊C4×C109—24no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—24no—
167C4⋊C4×C109—22no—
168C4⋊C4×C109—24no—
169C4⋊C4×C109—22no—
170C4⋊C4×C109—42no—
171C4⋊C4×C109C2×D414yes—
172C22⋊C4×C109—24no—
173C22⋊C4×C109—24no—
174C22⋊C4×C109—22no—
175C22⋊C4×C109—24no—
176C22⋊C4×C109—22no—
177C22⋊C4×C109—42no—
178C22⋊C4×C109—42no—
179C22⋊C4×C109—42no—
180C22⋊C4×C109—42no—
181C22⋊C4×C109—42no—
182C42×C109—24no—
183C42×C109—21no—
184C42×C109—21no—
185C42×C109—42no—
186C23×C109—21no—
187C23×C109—21no—
188C23×C109—21no—
189C23×C109—21no—
190C23×C109—22no—
191C23×C109—41no—
192C23×C109—41no—
193C23×C109—41no—
194C23×C109—41no—
195C23×C109—41no—
196C23×C109—41no—
197C23×C109—41no—
198C23×C109—41no—
199C23×C109—41no—
200C23×C109—41no—
201C23×C109#12012yes—
202C23×C109#13712yescomplemented
203C23×C109C22×D411yescharacteristic
204Q8×C109—24no—
205Q8×C109—44no—
206Q8×C109—42no—
207Q8×C109—44no—
208Q8×C109—82no—
209Q8×C109—82no—
210D4×C109—28no—
211D4×C109—24no—
212D4×C109—42no—
213D4×C109—48no—
214D4×C109—48no—
215C2×C4×C109—21no—
216C2×C4×C109—21no—
217C2×C4×C109—22no—
218C2×C4×C109—24no—
219C2×C4×C109—22no—
220C2×C4×C109—22no—
221C2×C4×C109—21no—
222C2×C4×C109—24no—
223C2×C4×C109—24no—
224C2×C4×C109—21no—
225C2×C4×C109—42no—
226C2×C4×C109—41no—
227C2×C4×C109—44no—
228C2×C4×C109—41no—
229C2×C4×C109—41no—
230C2×C4×C109—41no—
231C2×C4×C109—41no—
232C2×C4×C109—44no—
233C2×C4×C109—42no—
234C2×C4×C109—41no—
235C2×C4×C109—41no—
236C2×C4×C109—41no—
237C2×C4×C109—41no—
238C2×C4×C109—41no—
239C2×C4×C109—81no—
240C2×C4×C109—82no—
241C2×C4×C109—82no—
242C2×C4×C109—82no—
243C2×C4×C109—82no—
244C2×C4×C109#12011yescharacteristic
245C2×C4×C109#12011yescharacteristic
246C2×C4×C109#13712yescomplemented
247C2×C4×C109C22×D412yes—
248C8×C109—44no—
249C22×C109—21no—
250C22×C109—21no—
251C22×C109—22no—
252C22×C109—22no—
253C22×C109—22no—
254C22×C109—22no—
255C22×C109—42no—
256C22×C109—42no—
257C22×C109—42no—
258C22×C109—41no—
259C22×C109—41no—
260C22×C109—42no—
261C22×C109—81no—
262C22×C109—81no—
263C22×C109—81no—
264C22×C109—81no—
265C22×C109—81no—
266C22×C109—81no—
267C22×C109—81no—
268C22×C109C2×#12011yescharacteristic
269C22×C109C2×#13711yescharacteristic
270C22×C109C2×#13711yescharacteristic
271C4×C109—22no—
272C4×C109—22no—
273C4×C109—21no—
274C4×C109—42no—
275C4×C109—41no—
276C4×C109—44no—
277C4×C109—42no—
278C4×C109—41no—
279C4×C109—44no—
280C4×C109—81no—
281C4×C109B(4,𝔽2)12yes—
282#18167C10911yesmaximal, characteristic, direct product factor, Sylow
283C2×C109—21no—
284C2×C109—21no—
285C2×C109—42no—
286C2×C109—42no—
287C2×C109—82no—
288C2×C109#300411yescharacteristic
289C2×C109#301811yescharacteristic
290C2×C109C2×B(4,𝔽2)11yescharacteristic
291#3501C2×C10911yescharacteristic, complemented
292#3007C2×C10912yescomplemented
293#2388C2×C10911yescharacteristic
294#1605C2×C10912yescomplemented
295#1602C2×C10912yescomplemented
296#1599C2×C10912yescomplemented
297#1596C2×C10912yescomplemented
298#1508C2×C10912yescomplemented
299#1465C2×C10911yescharacteristic, complemented
300C109#1816711yescharacteristic, direct product factor, Sylow
301C22×#41C22×C10911yescharacteristic
302C2×#137—22no—
303#554—24no—
304#554—22no—
305D4×Q8—22no—
306#544C22×C10912yes—
307#543—21no—
308#543C22×C10912yes—
309#533—21no—
310C2×C22⋊Q8—21no—
311C2×C22⋊Q8C22×C10912yes—
312#518—21no—
313C2×C4×D4—21no—
314C2×C4×D4C22×C10911yescharacteristic
315#452—24no—
316#451C22×C10914yes—
317#450C22×C10914yes—
318#417—22no—
319#407C22×C10912yes—
320#395C22×C10911yescharacteristic
321#394—21no—
322#394C22×C10911yescharacteristic
323#393C22×C10911yescharacteristic
324#392—21no—
325#392C22×C10911yescharacteristic
326C4×C22⋊C4—21no—
327C4×C22⋊C4C22×C10911yescharacteristic
328#328C22×C10914yes—
329#328C22×C10914yes—
330#324C22×C10912yes—
331#323C22×C10912yes—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—44no—
335C22×Q8—21no—
336C22×Q8D4×C10912yescomplemented
337C22×Q8C23×C10911yescharacteristic
338C22×D4—21no—
339C22×D4—21no—
340C22×D4D4×C10911yescharacteristic
341C22×D4C23×C10911yescharacteristic
342C23×C4—21no—
343C23×C4D4×C10911yescharacteristic
344C23×C4C23×C10911yescharacteristic
345#137—48no—
346C2×Q16—24no—
347C2×SD16—24no—
348C2×M16—22no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—24no—
352#125—22no—
353#125—24no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—24no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8—22no—
363C22⋊Q8—24no—
364C22⋊Q8D4×C10914yescomplemented
365C22⋊Q8C23×C10914yes—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—22no—
371C4×D4—22no—
372C4×D4—44no—
373#117—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4D4×C10912yescomplemented
379C2×C4⋊C4C23×C10911yescharacteristic
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4D4×C10911yescharacteristic
388C2×C22⋊C4C23×C10911yescharacteristic
389C2×C42—21no—
390C2×C42D4×C10911yescharacteristic
391#103—24no—
392#102—24no—
393C22⋊C8—22no—
394C22⋊C8C23×C10914yes—
395#95—21no—
396#95—21no—
397#95C23×C10911yescharacteristic
398#95C23×C10911yescharacteristic
399C24—21no—
400C24C2×D4×C10911yescharacteristic
401C24C2×D4×C10911yescharacteristic
402#41—28no—
403#41—44no—
404#41—44no—
405#41—44no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—42no—
411C2×Q8—44no—
412C2×Q8—82no—
413C2×Q8C2×D4×C10914yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—42no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—42no—
423C2×D4—42no—
424C2×D4—42no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4C2×D4×C10911yescharacteristic
441C22×C4C2×D4×C10912yes—
442C22×C4C2×D4×C10911yescharacteristic
443C22×C4C2×D4×C10911yescharacteristic
444C22×C4C2×D4×C10911yescharacteristic
445C22×C4C2×D4×C10912yes—
446C22×C4C24×C10911yescharacteristic
447Q16—48no—
448SD16—48no—
449M16—44no—
450C2×C8—24no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—42no—
462C4⋊C4C2×D4×C10914yes—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—24no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C42—24no—
474C42—21no—
475C42—21no—
476C42—42no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23#120×C10912yes—
493C23#137×C10912yescomplemented
494C23C22×D4×C10911yescharacteristic
495Q8—24no—
496Q8—44no—
497Q8—42no—
498Q8—44no—
499Q8—82no—
500Q8—82no—
501D4—28no—
502D4—24no—
503D4—42no—
504D4—48no—
505D4—48no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#120×C10911yescharacteristic
536C2×C4#120×C10911yescharacteristic
537C2×C4#137×C10912yescomplemented
538C2×C4C22×D4×C10912yes—
539C8—44no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#120×C10911yescharacteristic
560C22C2×#137×C10911yescharacteristic
561C22C2×#137×C10911yescharacteristic
562C4—22no—
563C4—22no—
564C4—21no—
565C4—42no—
566C4—41no—
567C4—44no—
568C4—42no—
569C4—41no—
570C4—44no—
571C4—81no—
572C4B(4,𝔽2)×C10912yes—
573C2—21no—
574C2—21no—
575C2—42no—
576C2—42no—
577C2—82no—
578C2C109×#300411yescharacteristic
579C2C109×#301811yescharacteristic
580C2C2×B(4,𝔽2)×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#2388C2
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
1. C109×#3501C2semidirect
2. C109×#3007C2semidirect
4. C109×#1605C2semidirect
5. C109×#1602C2semidirect
6. C109×#1599C2semidirect
7. C109×#1596C2semidirect
8. C109×#1508C2semidirect
9. C109×#1465C2semidirect
45. C22×Q8×C109D4semidirect
73. C22⋊Q8×C109D4semidirect
87. C2×C4⋊C4×C109D4semidirect
202. C23×C109#137semidirect
246. C2×C4×C109#137semidirect
282. #18167C109direct (N × H)
291. #3501C2×C109semidirect
292. #3007C2×C109semidirect
294. #1605C2×C109semidirect
295. #1602C2×C109semidirect
296. #1599C2×C109semidirect
297. #1596C2×C109semidirect
298. #1508C2×C109semidirect
299. #1465C2×C109semidirect
300. C109#18167direct (N × H)
336. C22×Q8D4×C109semidirect
364. C22⋊Q8D4×C109semidirect
378. C2×C4⋊C4D4×C109semidirect
493. C23#137×C109semidirect
537. C2×C4#137×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#35012
C109×#30072
C109×#23882
C109×#16052
C109×#16022
C109×#15992
C109×#15962
C109×#15082
C109×#14652
#18167109