Finite Group Databasebeta

← C23×#13082 #19160708

Subgroups — C23×#13082 #19160708

Order 5888.

1088 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#100C211yesmaximal, characteristic, complemented
2C23×#2403C211yesmaximal, characteristic
3C2×C23×AΣL(1,𝔽16)C212yesmaximal, complemented
4C23×#1781C211yesmaximal, characteristic, complemented
5C23×#1390C212yesmaximal, complemented
6C23×D4×C23C411yescharacteristic, complemented
7C23×D4×C23C2211yescharacteristic
8C22×M16×C23—21no—
9C2×C23×#120—21no—
10C2×C23×#120C411yescharacteristic, complemented
11C2×C23×#120C2211yescharacteristic
12C2×C23×#100—21no—
13C2×C23×#100—24no—
14C2×C23×#100C2212yes—
15C2×C23×#99—21no—
16C2×C23×#99C2212yes—
17C23×#392—21no—
18C23×#392—21no—
19C23×#386—21no—
20C23×#386C411yescharacteristic, complemented
21C23×#386C2211yescharacteristic
22C23×#379—21no—
23C23×#379C411yescharacteristic, complemented
24C23×AΣL(1,𝔽16)—24no—
25C25×C23C2×C411yescharacteristic
26C25×C23C2×C411yescharacteristic
27C25×C23D411yescharacteristic
28C22×D4×C23—21no—
29C22×D4×C23—22no—
30C22×D4×C23—22no—
31C22×D4×C23—21no—
32C22×D4×C23—21no—
33C22×D4×C23—21no—
34C22×D4×C23—42no—
35C22×D4×C23—41no—
36C22×D4×C23—41no—
37C22×D4×C23—44no—
38C22×D4×C23—41no—
39C22×D4×C23—41no—
40C22×D4×C23C2×C411yescharacteristic
41C22×D4×C23D411yescharacteristic
42C22×D4×C23D411yescharacteristic
43C22×D4×C23C2311yescharacteristic
44C23×C4×C23C2×C411yescharacteristic
45C23×C4×C23D411yescharacteristic
46C2×M16×C23—24no—
47C2×M16×C23—22no—
48C2×M16×C23—41no—
49C2×M16×C23—42no—
50C22×C8×C23—41no—
51C23×#120—22no—
52C23×#120—22no—
53C23×#120—44no—
54C23×#120—42no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—22no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—41no—
63C2×C22⋊C4×C23—41no—
64C2×C22⋊C4×C23C2×C411yescharacteristic
65C2×C22⋊C4×C23C2×C411yescharacteristic
66C23×#100—24no—
67C23×#100—22no—
68C23×#100D414yescomplemented
69C23×#99—22no—
70C23×#99—22no—
71C23×#99—42no—
72C23×#95—21no—
73C23×#95—21no—
74C24×C23—21no—
75C24×C23—21no—
76C24×C23—21no—
77C24×C23—21no—
78C24×C23—21no—
79C24×C23—22no—
80C24×C23—22no—
81C24×C23—22no—
82C24×C23—41no—
83C24×C23—41no—
84C24×C23—42no—
85C24×C23—41no—
86C24×C23—42no—
87C24×C23—42no—
88C24×C23—41no—
89C24×C23—81no—
90C24×C23—81no—
91C24×C23—81no—
92C24×C23C22⋊C411yescharacteristic
93C24×C23C22⋊C411yescharacteristic
94C24×C23C22⋊C411yescharacteristic
95C24×C23C22×C411yescharacteristic
96C24×C23C2×D411yescharacteristic
97C2×D4×C23—24no—
98C2×D4×C23—22no—
99C2×D4×C23—22no—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—21no—
103C2×D4×C23—21no—
104C2×D4×C23—41no—
105C2×D4×C23—41no—
106C2×D4×C23—41no—
107C2×D4×C23—41no—
108C2×D4×C23—41no—
109C2×D4×C23—41no—
110C2×D4×C23—44no—
111C2×D4×C23—44no—
112C2×D4×C23—44no—
113C2×D4×C23—44no—
114C2×D4×C23—42no—
115C2×D4×C23—42no—
116C2×D4×C23—42no—
117C2×D4×C23—42no—
118C2×D4×C23—44no—
119C2×D4×C23—44no—
120C2×D4×C23—44no—
121C2×D4×C23—41no—
122C2×D4×C23—41no—
123C2×D4×C23—44no—
124C2×D4×C23—42no—
125C2×D4×C23—42no—
126C2×D4×C23—42no—
127C2×D4×C23—42no—
128C2×D4×C23—81no—
129C2×D4×C23—81no—
130C2×D4×C23C2×D412yes—
131C2×D4×C23#4112yes—
132C22×C4×C23—21no—
133C22×C4×C23—21no—
134C22×C4×C23—22no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—21no—
140C22×C4×C23—42no—
141C22×C4×C23—41no—
142C22×C4×C23—42no—
143C22×C4×C23C22⋊C411yescharacteristic
144C22×C4×C23C2×D411yescharacteristic
145M16×C23—24no—
146M16×C23—44no—
147M16×C23—42no—
148C2×C8×C23—44no—
149C2×C8×C23—42no—
150C22⋊C4×C23—22no—
151C22⋊C4×C23—42no—
152C22⋊C4×C23—42no—
153C22⋊C4×C23—42no—
154C22⋊C4×C23—42no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23—41no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23—42no—
159C22⋊C4×C23—42no—
160C22⋊C4×C23—81no—
161C22⋊C4×C23—82no—
162C22⋊C4×C23—82no—
163#13082C2311yesmaximal, characteristic, direct product factor, Sylow
164C23×C23—22no—
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—21no—
168C23×C23—21no—
169C23×C23—21no—
170C23×C23—21no—
171C23×C23—22no—
172C23×C23—21no—
173C23×C23—21no—
174C23×C23—21no—
175C23×C23—21no—
176C23×C23—21no—
177C23×C23—21no—
178C23×C23—41no—
179C23×C23—42no—
180C23×C23—42no—
181C23×C23—42no—
182C23×C23—42no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—41no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—42no—
190C23×C23—41no—
191C23×C23—42no—
192C23×C23—41no—
193C23×C23—41no—
194C23×C23—41no—
195C23×C23—41no—
196C23×C23—42no—
197C23×C23—42no—
198C23×C23—42no—
199C23×C23—42no—
200C23×C23—41no—
201C23×C23—41no—
202C23×C23—41no—
203C23×C23—81no—
204C23×C23—81no—
205C23×C23—81no—
206C23×C23—81no—
207C23×C23—82no—
208C23×C23—84no—
209C23×C23—84no—
210C23×C23—84no—
211C23×C23—84no—
212C23×C23—84no—
213C23×C23—84no—
214C23×C23—82no—
215C23×C23#9911yescharacteristic
216C23×C23#9911yescharacteristic
217C23×C23C2×C22⋊C411yescharacteristic
218C23×C23C4×D412yes—
219C23×C23#12112yes—
220D4×C23—42no—
221D4×C23—42no—
222D4×C23—44no—
223D4×C23—44no—
224D4×C23—42no—
225D4×C23—42no—
226D4×C23—44no—
227D4×C23—48no—
228D4×C23—44no—
229D4×C23—84no—
230D4×C23—84no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—22no—
235C2×C4×C23—21no—
236C2×C4×C23—21no—
237C2×C4×C23—21no—
238C2×C4×C23—21no—
239C2×C4×C23—22no—
240C2×C4×C23—41no—
241C2×C4×C23—42no—
242C2×C4×C23—41no—
243C2×C4×C23—42no—
244C2×C4×C23—41no—
245C2×C4×C23—41no—
246C2×C4×C23—41no—
247C2×C4×C23—41no—
248C2×C4×C23—41no—
249C2×C4×C23—41no—
250C2×C4×C23—82no—
251C2×C4×C23—82no—
252C2×C4×C23—82no—
253C2×C4×C23#9912yes—
254C2×C4×C23#12011yescharacteristic
255C2×C4×C23#12311yescharacteristic
256C8×C23—44no—
257C22×#100C2×C2311yescharacteristic, complemented
258#2403C2×C2311yescharacteristic
259C2×AΣL(1,𝔽16)C2×C2312yescomplemented
260#1781C2×C2311yescharacteristic, complemented
261#1390C2×C2312yescomplemented
262C22×C23—21no—
263C22×C23—21no—
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—21no—
267C22×C23—21no—
268C22×C23—21no—
269C22×C23—22no—
270C22×C23—22no—
271C22×C23—42no—
272C22×C23—42no—
273C22×C23—42no—
274C22×C23—42no—
275C22×C23—42no—
276C22×C23—42no—
277C22×C23—42no—
278C22×C23—42no—
279C22×C23—41no—
280C22×C23—41no—
281C22×C23—41no—
282C22×C23—41no—
283C22×C23—42no—
284C22×C23—41no—
285C22×C23—41no—
286C22×C23—41no—
287C22×C23—41no—
288C22×C23—41no—
289C22×C23—41no—
290C22×C23—81no—
291C22×C23—84no—
292C22×C23—84no—
293C22×C23—81no—
294C22×C23—81no—
295C22×C23—81no—
296C22×C23—81no—
297C22×C23—81no—
298C22×C23—84no—
299C22×C23—84no—
300C22×C23—84no—
301C22×C23—84no—
302C22×C23—81no—
303C22×C23—81no—
304C22×C23AΣL(1,𝔽16)12yescomplemented
305C22×C23#38611yescharacteristic
306C22×C23C2×#9911yescharacteristic
307C22×C23C2×#9911yescharacteristic
308C4×C23—21no—
309C4×C23—22no—
310C4×C23—21no—
311C4×C23—42no—
312C4×C23—81no—
313C4×C23—81no—
314C4×C23—162no—
315C23×D4C4×C2311yescharacteristic, complemented
316C23×D4C22×C2311yescharacteristic
317C22×M16—21no—
318C2×#120—21no—
319C2×#120C4×C2311yescharacteristic, complemented
320C2×#120C22×C2311yescharacteristic
321C2×#100—21no—
322C2×#100—24no—
323C2×#100C22×C2312yes—
324C2×#99—21no—
325C2×#99C22×C2312yes—
326#392—21no—
327#392—21no—
328#386—21no—
329#386C4×C2311yescharacteristic, complemented
330#386C22×C2311yescharacteristic
331#379—21no—
332#379C4×C2311yescharacteristic, complemented
333AΣL(1,𝔽16)—24no—
334C2×C23—21no—
335C2×C23—21no—
336C2×C23—22no—
337C2×C23—42no—
338C2×C23—41no—
339C2×C23—42no—
340C2×C23—81no—
341C2×C23—82no—
342C2×C23—81no—
343C2×C23#189811yescharacteristic
344C2×C23C2×AΣL(1,𝔽16)11yescharacteristic
345C2×C23#212111yescharacteristic
346C25C2×C4×C2311yescharacteristic
347C25C2×C4×C2311yescharacteristic
348C25D4×C2311yescharacteristic
349C22×D4—21no—
350C22×D4—22no—
351C22×D4—22no—
352C22×D4—21no—
353C22×D4—21no—
354C22×D4—21no—
355C22×D4—42no—
356C22×D4—41no—
357C22×D4—41no—
358C22×D4—44no—
359C22×D4—41no—
360C22×D4—41no—
361C22×D4C2×C4×C2311yescharacteristic
362C22×D4D4×C2311yescharacteristic
363C22×D4D4×C2311yescharacteristic
364C22×D4C23×C2311yescharacteristic
365C23×C4C2×C4×C2311yescharacteristic
366C23×C4D4×C2311yescharacteristic
367C2×M16—24no—
368C2×M16—22no—
369C2×M16—41no—
370C2×M16—42no—
371C22×C8—41no—
372#120—22no—
373#120—22no—
374#120—44no—
375#120—42no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—41no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4C2×C4×C2311yescharacteristic
386C2×C22⋊C4C2×C4×C2311yescharacteristic
387#100—24no—
388#100—22no—
389#100D4×C2314yescomplemented
390#99—22no—
391#99—22no—
392#99—42no—
393#95—21no—
394#95—21no—
395C23#1308211yescharacteristic, direct product factor, Sylow
396C24—21no—
397C24—21no—
398C24—21no—
399C24—21no—
400C24—21no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—41no—
405C24—41no—
406C24—42no—
407C24—41no—
408C24—42no—
409C24—42no—
410C24—41no—
411C24—81no—
412C24—81no—
413C24—81no—
414C24C22⋊C4×C2311yescharacteristic
415C24C22⋊C4×C2311yescharacteristic
416C24C22⋊C4×C2311yescharacteristic
417C24C22×C4×C2311yescharacteristic
418C24C2×D4×C2311yescharacteristic
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—41no—
429C2×D4—41no—
430C2×D4—41no—
431C2×D4—41no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—42no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—44no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—81no—
451C2×D4—81no—
452C2×D4C2×D4×C2312yes—
453C2×D4#41×C2312yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—41no—
464C22×C4—42no—
465C22×C4C22⋊C4×C2311yescharacteristic
466C22×C4C2×D4×C2311yescharacteristic
467M16—24no—
468M16—44no—
469M16—42no—
470C2×C8—44no—
471C2×C8—42no—
472C22⋊C4—22no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—81no—
483C22⋊C4—82no—
484C22⋊C4—82no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—82no—
529C23—84no—
530C23—84no—
531C23—84no—
532C23—84no—
533C23—84no—
534C23—84no—
535C23—82no—
536C23C23×#9911yescharacteristic
537C23C23×#9911yescharacteristic
538C23C2×C22⋊C4×C2311yescharacteristic
539C23C4×D4×C2312yes—
540C23C23×#12112yes—
541D4—42no—
542D4—42no—
543D4—44no—
544D4—44no—
545D4—42no—
546D4—42no—
547D4—44no—
548D4—48no—
549D4—44no—
550D4—84no—
551D4—84no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4C23×#9912yes—
575C2×C4C23×#12011yescharacteristic
576C2×C4C23×#12311yescharacteristic
577C8—44no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—22no—
584C22—22no—
585C22—21no—
586C22—21no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—81no—
607C22—84no—
608C22—84no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—84no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—81no—
619C22—81no—
620C22C23×AΣL(1,𝔽16)12yescomplemented
621C22C23×#38611yescharacteristic
622C22C2×C23×#9911yescharacteristic
623C22C2×C23×#9911yescharacteristic
624C4—21no—
625C4—22no—
626C4—21no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—162no—
631C2—21no—
632C2—21no—
633C2—22no—
634C2—42no—
635C2—41no—
636C2—42no—
637C2—81no—
638C2—82no—
639C2—81no—
640C2C23×#189811yescharacteristic
641C2C2×C23×AΣL(1,𝔽16)11yescharacteristic
642C2C23×#212111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#2403C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C23×#100C2semidirect
3. C2×C23×AΣL(1,𝔽16)C2semidirect
4. C23×#1781C2semidirect
5. C23×#1390C2semidirect
6. C23×D4×C23C4semidirect
10. C2×C23×#120C4semidirect
20. C23×#386C4semidirect
23. C23×#379C4semidirect
68. C23×#100D4semidirect
163. #13082C23direct (N × H)
257. C22×#100C2×C23semidirect
259. C2×AΣL(1,𝔽16)C2×C23semidirect
260. #1781C2×C23semidirect
261. #1390C2×C23semidirect
304. C22×C23AΣL(1,𝔽16)semidirect
315. C23×D4C4×C23semidirect
319. C2×#120C4×C23semidirect
329. #386C4×C23semidirect
332. #379C4×C23semidirect
389. #100D4×C23semidirect
395. C23#13082direct (N × H)
620. C22C23×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1002
C23×#24032
C2×C23×AΣL(1,𝔽16)2
C23×#17812
C23×#13902
#1308223