Finite Group Databasebeta

← C109×#23987 #42141981

Subgroups — C109×#23987 #42141981

Order 27904.

1514 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3490C211yesmaximal, characteristic, complemented
2C109×#2613C212yesmaximal, complemented
3C109×#2308C211yesmaximal, characteristic, complemented
4C109×#2030C211yesmaximal, characteristic, complemented
5C109×#2029C211yesmaximal, characteristic, complemented
6C109×#2022C211yesmaximal, characteristic, complemented
7C109×#2021C211yesmaximal, characteristic, complemented
8C109×#1898C212yesmaximal, complemented
9C109×#1890C212yesmaximal, complemented
10C109×#1889C212yesmaximal, complemented
11C109×#1819C211yesmaximal, characteristic, complemented
12C23×D4×C109C2211yescharacteristic
13#551×C109—24no—
14#546×C109—24no—
15#543×C109C2211yescharacteristic
16#538×C109—22no—
17#534×C109C2211yescharacteristic
18#525×C109—22no—
19C2×#123×C109C2212yes—
20C2×#120×C109C2212yes—
21C2×#120×C109C2212yes—
22#518×C109C2212yes—
23C22×C22⋊C4×C109C2211yescharacteristic
24C22×C22⋊C4×C109C2211yescharacteristic
25C2×#99×C109C2212yes—
26C2×#99×C109C2212yes—
27#394×C109—24no—
28#392×C109—24no—
29#386×C109—22no—
30#386×C109—22no—
31#381×C109C2211yescharacteristic
32#381×C109C2212yes—
33#381×C109C2211yescharacteristic
34#379×C109C2211yescharacteristic
35#379×C109C2212yes—
36#379×C109C2211yescharacteristic
37#379×C109C2212yes—
38#342×C109C2211yescharacteristic
39#342×C109C2211yescharacteristic
40#342×C109C2212yes—
41#342×C109C2212yes—
42#342×C109C2212yes—
43C25×C109C2311yescharacteristic
44C25×C109C2311yescharacteristic
45C2×#41×C109—22no—
46C22×D4×C109—21no—
47C22×D4×C109—21no—
48C22×D4×C109—22no—
49C22×D4×C109—24no—
50C22×D4×C109—22no—
51C22×D4×C109—22no—
52C22×D4×C109—42no—
53C22×D4×C109D414yescomplemented
54C22×D4×C109C2312yes—
55C23×C4×C109—21no—
56C23×C4×C109D412yescomplemented
57C23×C4×C109C2311yescharacteristic
58#126×C109—24no—
59#124×C109—24no—
60#123×C109—24no—
61#123×C109—24no—
62#123×C109—22no—
63#123×C109—24no—
64#123×C109—42no—
65C22⋊Q8×C109—22no—
66C22⋊Q8×C109—22no—
67C22⋊Q8×C109—24no—
68#121×C109—22no—
69#121×C109—24no—
70#121×C109—22no—
71#120×C109—22no—
72#120×C109—22no—
73#120×C109—24no—
74#120×C109—22no—
75#120×C109—24no—
76#120×C109—42no—
77C4×D4×C109—24no—
78C4×D4×C109—24no—
79C4×D4×C109—24no—
80#117×C109—22no—
81#117×C109—22no—
82C2×C4⋊C4×C109—22no—
83C2×C22⋊C4×C109—24no—
84C2×C22⋊C4×C109—22no—
85C2×C22⋊C4×C109—21no—
86C2×C22⋊C4×C109—22no—
87C2×C22⋊C4×C109—22no—
88C2×C22⋊C4×C109—24no—
89C2×C22⋊C4×C109—22no—
90C2×C22⋊C4×C109—22no—
91C2×C22⋊C4×C109—24no—
92C2×C22⋊C4×C109—42no—
93C2×C22⋊C4×C109D412yescomplemented
94C2×C22⋊C4×C109D414yescomplemented
95C2×C22⋊C4×C109C2311yescharacteristic
96C2×C22⋊C4×C109C2311yescharacteristic
97C2×C22⋊C4×C109C2312yes—
98C2×C22⋊C4×C109C2312yes—
99C2×C22⋊C4×C109C2311yescharacteristic
100C2×C22⋊C4×C109C2311yescharacteristic
101C2×C22⋊C4×C109C2312yes—
102#99×C109—42no—
103#99×C109—42no—
104#95×C109—22no—
105#95×C109—22no—
106#95×C109—24no—
107C24×C109—21no—
108C24×C109—21no—
109C24×C109—22no—
110C24×C109—21no—
111C24×C109—22no—
112C24×C109—21no—
113C24×C109—22no—
114C24×C109—82no—
115C24×C109—82no—
116C24×C109C2×D412yes—
117C24×C109C2×D414yes—
118C24×C109C2×D411yescharacteristic
119C24×C109C2×D411yescharacteristic
120C24×C109#4112yescomplemented
121C24×C109C2411yescharacteristic
122#41×C109—84no—
123C2×Q8×C109—22no—
124C2×D4×C109—24no—
125C2×D4×C109—24no—
126C2×D4×C109—24no—
127C2×D4×C109—22no—
128C2×D4×C109—22no—
129C2×D4×C109—22no—
130C2×D4×C109—22no—
131C2×D4×C109—22no—
132C2×D4×C109—22no—
133C2×D4×C109—22no—
134C2×D4×C109—24no—
135C2×D4×C109—48no—
136C2×D4×C109—48no—
137C2×D4×C109—42no—
138C2×D4×C109—42no—
139C2×D4×C109—42no—
140C2×D4×C109—42no—
141C2×D4×C109—44no—
142C2×D4×C109—84no—
143C2×D4×C109—84no—
144C22×C4×C109—22no—
145C22×C4×C109—21no—
146C22×C4×C109—22no—
147C22×C4×C109—21no—
148C22×C4×C109—22no—
149C22×C4×C109—24no—
150C22×C4×C109—22no—
151C22×C4×C109—21no—
152C22×C4×C109—21no—
153C22×C4×C109—22no—
154C22×C4×C109—22no—
155C22×C4×C109—42no—
156C22×C4×C109—82no—
157C22×C4×C109—82no—
158C22×C4×C109C2×D412yes—
159C22×C4×C109C2×D412yes—
160C22×C4×C109C2×D414yes—
161C22×C4×C109C2×D412yes—
162C22×C4×C109#4112yescomplemented
163C4⋊C4×C109—24no—
164C4⋊C4×C109—22no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—24no—
167C4⋊C4×C109—44no—
168C22⋊C4×C109—24no—
169C22⋊C4×C109—24no—
170C22⋊C4×C109—24no—
171C22⋊C4×C109—24no—
172C22⋊C4×C109—24no—
173C22⋊C4×C109—22no—
174C22⋊C4×C109—22no—
175C22⋊C4×C109—22no—
176C22⋊C4×C109—22no—
177C22⋊C4×C109—22no—
178C22⋊C4×C109—22no—
179C22⋊C4×C109—22no—
180C22⋊C4×C109—22no—
181C22⋊C4×C109—22no—
182C22⋊C4×C109—22no—
183C22⋊C4×C109—44no—
184C22⋊C4×C109—44no—
185C22⋊C4×C109—42no—
186C22⋊C4×C109—82no—
187C22⋊C4×C109—82no—
188C22⋊C4×C109—84no—
189C22⋊C4×C109—84no—
190C42×C109—24no—
191C23×C109—21no—
192C23×C109—21no—
193C23×C109—21no—
194C23×C109—21no—
195C23×C109—24no—
196C23×C109—24no—
197C23×C109—22no—
198C23×C109—22no—
199C23×C109—24no—
200C23×C109—22no—
201C23×C109—22no—
202C23×C109—42no—
203C23×C109—42no—
204C23×C109—42no—
205C23×C109—44no—
206C23×C109—44no—
207C23×C109—42no—
208C23×C109—42no—
209C23×C109—44no—
210C23×C109—44no—
211C23×C109—44no—
212C23×C109—41no—
213C23×C109—41no—
214C23×C109—42no—
215C23×C109—82no—
216C23×C109—82no—
217C23×C109—82no—
218C23×C109—82no—
219C23×C109—84no—
220C23×C109—84no—
221C23×C109#12012yes—
222C23×C109#12012yes—
223C23×C109C22×D411yescharacteristic
224C23×C109C22×D412yes—
225C23×C109C2×#4111yescharacteristic
226C23×C109C2×#4111yescharacteristic
227C23×C109H(5,𝔽2)12yes—
228Q8×C109—82no—
229D4×C109—48no—
230D4×C109—48no—
231D4×C109—48no—
232D4×C109—42no—
233D4×C109—42no—
234D4×C109—42no—
235D4×C109—42no—
236D4×C109—84no—
237D4×C109—82no—
238D4×C109—88no—
239C2×C4×C109—22no—
240C2×C4×C109—22no—
241C2×C4×C109—24no—
242C2×C4×C109—22no—
243C2×C4×C109—22no—
244C2×C4×C109—22no—
245C2×C4×C109—24no—
246C2×C4×C109—22no—
247C2×C4×C109—44no—
248C2×C4×C109—41no—
249C2×C4×C109—41no—
250C2×C4×C109—42no—
251C2×C4×C109—42no—
252C2×C4×C109—42no—
253C2×C4×C109—44no—
254C2×C4×C109—44no—
255C2×C4×C109—42no—
256C2×C4×C109—42no—
257C2×C4×C109—44no—
258C2×C4×C109—41no—
259C2×C4×C109—41no—
260C2×C4×C109—42no—
261C2×C4×C109—42no—
262C2×C4×C109—82no—
263C2×C4×C109—84no—
264C2×C4×C109—84no—
265C22×C109—22no—
266C22×C109—24no—
267C22×C109—22no—
268C22×C109—22no—
269C22×C109—22no—
270C22×C109—22no—
271C22×C109—24no—
272C22×C109—41no—
273C22×C109—41no—
274C22×C109—42no—
275C22×C109—42no—
276C22×C109—42no—
277C22×C109—42no—
278C22×C109—42no—
279C22×C109—44no—
280C22×C109—44no—
281C22×C109—42no—
282C22×C109—42no—
283C22×C109—42no—
284C22×C109—42no—
285C22×C109—44no—
286C22×C109—84no—
287C22×C109—84no—
288C22×C109—82no—
289C22×C109—82no—
290C22×C109—82no—
291C22×C109—84no—
292C22×C109—82no—
293C22×C109C2×#12011yescharacteristic
294C22×C109#52511yescharacteristic
295C22×C109#53511yescharacteristic
296C22×C109#54612yes—
297C22×C109#54612yes—
298C4×C109—42no—
299C4×C109—42no—
300C4×C109—42no—
301C4×C109—42no—
302C4×C109—44no—
303C4×C109—82no—
304C4×C109—81no—
305C4×C109—81no—
306C4×C109—82no—
307#23987C10911yesmaximal, characteristic, direct product factor, Sylow
308C2×C109—21no—
309C2×C109—22no—
310C2×C109—22no—
311C2×C109—21no—
312C2×C109—42no—
313C2×C109—42no—
314C2×C109—82no—
315C2×C109—82no—
316C2×C109#240411yescharacteristic
317C2×C109#302612yes—
318#3490C2×C10911yescharacteristic, complemented
319#2613C2×C10912yescomplemented
320#2308C2×C10911yescharacteristic, complemented
321#2030C2×C10911yescharacteristic, complemented
322#2029C2×C10911yescharacteristic, complemented
323#2022C2×C10911yescharacteristic, complemented
324#2021C2×C10911yescharacteristic, complemented
325#1898C2×C10912yescomplemented
326#1890C2×C10912yescomplemented
327#1889C2×C10912yescomplemented
328#1819C2×C10911yescharacteristic, complemented
329C109#2398711yescharacteristic, direct product factor, Sylow
330C23×D4C22×C10911yescharacteristic
331#551—24no—
332#546—24no—
333#543C22×C10911yescharacteristic
334#538—22no—
335#534C22×C10911yescharacteristic
336#525—22no—
337C2×#123C22×C10912yes—
338C2×#120C22×C10912yes—
339C2×#120C22×C10912yes—
340#518C22×C10912yes—
341C22×C22⋊C4C22×C10911yescharacteristic
342C22×C22⋊C4C22×C10911yescharacteristic
343C2×#99C22×C10912yes—
344C2×#99C22×C10912yes—
345#394—24no—
346#392—24no—
347#386—22no—
348#386—22no—
349#381C22×C10911yescharacteristic
350#381C22×C10911yescharacteristic
351#381C22×C10912yes—
352#379C22×C10911yescharacteristic
353#379C22×C10911yescharacteristic
354#379C22×C10912yes—
355#379C22×C10912yes—
356#342C22×C10912yes—
357#342C22×C10911yescharacteristic
358#342C22×C10911yescharacteristic
359#342C22×C10912yes—
360#342C22×C10912yes—
361C25C23×C10911yescharacteristic
362C25C23×C10911yescharacteristic
363C2×#41—22no—
364C22×D4—21no—
365C22×D4—21no—
366C22×D4—22no—
367C22×D4—24no—
368C22×D4—22no—
369C22×D4—22no—
370C22×D4—42no—
371C22×D4D4×C10914yescomplemented
372C22×D4C23×C10912yes—
373C23×C4—21no—
374C23×C4D4×C10912yescomplemented
375C23×C4C23×C10911yescharacteristic
376#126—24no—
377#124—24no—
378#123—24no—
379#123—24no—
380#123—22no—
381#123—24no—
382#123—42no—
383C22⋊Q8—22no—
384C22⋊Q8—22no—
385C22⋊Q8—24no—
386#121—22no—
387#121—24no—
388#121—22no—
389#120—22no—
390#120—22no—
391#120—24no—
392#120—22no—
393#120—24no—
394#120—42no—
395C4×D4—24no—
396C4×D4—24no—
397C4×D4—24no—
398#117—22no—
399#117—22no—
400C2×C4⋊C4—22no—
401C2×C22⋊C4—24no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—42no—
411C2×C22⋊C4D4×C10912yescomplemented
412C2×C22⋊C4D4×C10914yescomplemented
413C2×C22⋊C4C23×C10911yescharacteristic
414C2×C22⋊C4C23×C10911yescharacteristic
415C2×C22⋊C4C23×C10912yes—
416C2×C22⋊C4C23×C10912yes—
417C2×C22⋊C4C23×C10911yescharacteristic
418C2×C22⋊C4C23×C10911yescharacteristic
419C2×C22⋊C4C23×C10912yes—
420#99—42no—
421#99—42no—
422#95—22no—
423#95—22no—
424#95—24no—
425C24—21no—
426C24—21no—
427C24—22no—
428C24—21no—
429C24—22no—
430C24—21no—
431C24—22no—
432C24—82no—
433C24—82no—
434C24C2×D4×C10912yes—
435C24C2×D4×C10914yes—
436C24C2×D4×C10911yescharacteristic
437C24C2×D4×C10911yescharacteristic
438C24#41×C10912yescomplemented
439C24C24×C10911yescharacteristic
440#41—84no—
441C2×Q8—22no—
442C2×D4—24no—
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—22no—
452C2×D4—24no—
453C2×D4—48no—
454C2×D4—48no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—44no—
460C2×D4—84no—
461C2×D4—84no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—42no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4C2×D4×C10912yes—
477C22×C4C2×D4×C10912yes—
478C22×C4C2×D4×C10914yes—
479C22×C4C2×D4×C10912yes—
480C22×C4#41×C10912yescomplemented
481C4⋊C4—24no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—24no—
485C4⋊C4—44no—
486C22⋊C4—24no—
487C22⋊C4—24no—
488C22⋊C4—24no—
489C22⋊C4—24no—
490C22⋊C4—24no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—42no—
504C22⋊C4—82no—
505C22⋊C4—82no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C42—24no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—82no—
534C23—82no—
535C23—82no—
536C23—82no—
537C23—84no—
538C23—84no—
539C23#120×C10912yes—
540C23#120×C10912yes—
541C23C22×D4×C10911yescharacteristic
542C23C22×D4×C10912yes—
543C23C2×#41×C10911yescharacteristic
544C23C2×#41×C10911yescharacteristic
545C23H(5,𝔽2)×C10912yes—
546Q8—82no—
547D4—48no—
548D4—48no—
549D4—48no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—82no—
556D4—88no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—82no—
581C2×C4—84no—
582C2×C4—84no—
583C22—22no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—24no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—42no—
600C22—42no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22—84no—
605C22—84no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—84no—
610C22—82no—
611C22C2×#120×C10911yescharacteristic
612C22#525×C10911yescharacteristic
613C22#535×C10911yescharacteristic
614C22#546×C10912yes—
615C22#546×C10912yes—
616C4—42no—
617C4—42no—
618C4—44no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—81no—
623C4—82no—
624C4—82no—
625C2—21no—
626C2—22no—
627C2—22no—
628C2—21no—
629C2—42no—
630C2—42no—
631C2—82no—
632C2—82no—
633C2C109×#240411yescharacteristic
634C2C109×#302612yes—
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC24C24×C109
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C25×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#3490C2semidirect
2. C109×#2613C2semidirect
3. C109×#2308C2semidirect
4. C109×#2030C2semidirect
5. C109×#2029C2semidirect
6. C109×#2022C2semidirect
7. C109×#2021C2semidirect
8. C109×#1898C2semidirect
9. C109×#1890C2semidirect
10. C109×#1889C2semidirect
11. C109×#1819C2semidirect
53. C22×D4×C109D4semidirect
56. C23×C4×C109D4semidirect
93. C2×C22⋊C4×C109D4semidirect
94. C2×C22⋊C4×C109D4semidirect
120. C24×C109#41semidirect
162. C22×C4×C109#41semidirect
307. #23987C109direct (N × H)
318. #3490C2×C109semidirect
319. #2613C2×C109semidirect
320. #2308C2×C109semidirect
321. #2030C2×C109semidirect
322. #2029C2×C109semidirect
323. #2022C2×C109semidirect
324. #2021C2×C109semidirect
325. #1898C2×C109semidirect
326. #1890C2×C109semidirect
327. #1889C2×C109semidirect
328. #1819C2×C109semidirect
329. C109#23987direct (N × H)
371. C22×D4D4×C109semidirect
374. C23×C4D4×C109semidirect
411. C2×C22⋊C4D4×C109semidirect
412. C2×C22⋊C4D4×C109semidirect
438. C24#41×C109semidirect
480. C22×C4#41×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34902
C109×#26132
C109×#23082
C109×#20302
C109×#20292
C109×#20222
C109×#20212
C109×#18982
C109×#18902
C109×#18892
C109×#18192
#23987109