Finite Group Databasebeta

← C23×#13041 #19160667

Subgroups — C23×#13041 #19160667

Order 5888.

964 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×B(4,𝔽2)C211yesmaximal, characteristic
2C23×#2358C211yesmaximal, characteristic, complemented
3C2×C23×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C23×#1792C211yesmaximal, characteristic, complemented
5C23×#1790C211yesmaximal, characteristic, complemented
6C23×#1393C211yesmaximal, characteristic, complemented
7C23×#1390C211yesmaximal, characteristic, complemented
8C2×C23×H(5,𝔽2)C2211yescharacteristic
9C2×C23×#120—21no—
10C2×C23×#120C2211yescharacteristic
11C2×C4×D4×C23—21no—
12C22×C22⋊C4×C23—21no—
13C23×B(4,𝔽2)—22no—
14C23×B(4,𝔽2)C414yescomplemented
15C2×C23×#104—21no—
16C23×#417—21no—
17C2×C23×#100C2211yescharacteristic
18C2×C23×#99—21no—
19C2×C23×#99—21no—
20C2×C23×#99C2211yescharacteristic
21C2×C23×#99C2211yescharacteristic
22C23×#390C2211yescharacteristic
23C23×#388—21no—
24C23×#386—21no—
25C23×#386—21no—
26C23×#386C2211yescharacteristic
27C23×#385—21no—
28C4×C22⋊C4×C23—21no—
29C23×AΣL(1,𝔽16)—22no—
30C25×C23D411yescharacteristic
31C23×H(5,𝔽2)—21no—
32C23×H(5,𝔽2)—21no—
33C23×H(5,𝔽2)—21no—
34C23×H(5,𝔽2)—42no—
35C23×H(5,𝔽2)C2×C412yescomplemented
36C2×#41×C23—21no—
37C2×#41×C23—41no—
38C22×D4×C23—21no—
39C22×D4×C23—21no—
40C22×D4×C23—21no—
41C22×D4×C23—41no—
42C22×D4×C23—41no—
43C22×D4×C23C2311yescharacteristic
44C23×C4×C23—21no—
45C23×C4×C23—21no—
46C23×C4×C23D411yescharacteristic, complemented
47C2×M16×C23—21no—
48C23×#120—21no—
49C23×#120—21no—
50C23×#120—24no—
51C23×#120—42no—
52C23×#120C2×C412yes—
53C23×#120D412yes—
54C4×D4×C23—44no—
55C2×C4⋊C4×C23—21no—
56C2×C4⋊C4×C23D411yescharacteristic, complemented
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—21no—
65C2×C22⋊C4×C23—44no—
66C2×C22⋊C4×C23—41no—
67C2×C22⋊C4×C23—41no—
68C2×C22⋊C4×C23D411yescharacteristic
69C2×C22⋊C4×C23D411yescharacteristic, complemented
70C2×C42×C23—21no—
71C2×C42×C23D411yescharacteristic, complemented
72C23×#104—42no—
73C23×#103—41no—
74C23×#102—41no—
75C23×#100—21no—
76C23×#100—21no—
77C23×#99—21no—
78C23×#99—24no—
79C23×#99—21no—
80C23×#99—21no—
81C23×#99—42no—
82C23×#99C2×C412yescomplemented
83C23×#95—21no—
84C23×#95—21no—
85C23×#95—21no—
86C23×#95—41no—
87C24×C23—21no—
88C24×C23—21no—
89C24×C23—21no—
90C24×C23—21no—
91C24×C23—42no—
92C24×C23—41no—
93C24×C23—42no—
94C24×C23—41no—
95C24×C23C22⋊C412yescomplemented
96C24×C23C2×D411yescharacteristic
97C24×C23C2×D411yescharacteristic
98#41×C23—22no—
99#41×C23—22no—
100#41×C23—42no—
101#41×C23—41no—
102#41×C23—42no—
103#41×C23—42no—
104#41×C23—42no—
105#41×C23—41no—
106C2×Q8×C23—21no—
107C2×D4×C23—21no—
108C2×D4×C23—22no—
109C2×D4×C23—22no—
110C2×D4×C23—22no—
111C2×D4×C23—22no—
112C2×D4×C23—22no—
113C2×D4×C23—22no—
114C2×D4×C23—41no—
115C2×D4×C23—41no—
116C2×D4×C23—42no—
117C2×D4×C23—42no—
118C2×D4×C23—42no—
119C2×D4×C23—42no—
120C2×D4×C23—41no—
121C2×D4×C23—41no—
122C2×D4×C23—41no—
123C2×D4×C23—41no—
124C2×D4×C23—42no—
125C2×D4×C23—42no—
126C2×D4×C23—41no—
127C2×D4×C23—41no—
128C2×D4×C23—41no—
129C2×D4×C23—44no—
130C2×D4×C23—82no—
131C2×D4×C23—82no—
132C2×D4×C23C22×C411yescharacteristic
133C2×D4×C23C2×D411yescharacteristic
134C2×D4×C23#4111yescharacteristic
135C2×D4×C23#4111yescharacteristic
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—21no—
140C22×C4×C23—21no—
141C22×C4×C23—21no—
142C22×C4×C23—21no—
143C22×C4×C23—41no—
144C22×C4×C23—42no—
145C22×C4×C23—42no—
146C22×C4×C23—42no—
147C22×C4×C23—41no—
148C22×C4×C23—41no—
149C22×C4×C23—42no—
150C22×C4×C23—42no—
151C22×C4×C23—41no—
152C22×C4×C23—81no—
153C22×C4×C23—81no—
154C22×C4×C23C2×D411yescharacteristic
155M16×C23—41no—
156M16×C23—41no—
157C2×C8×C23—41no—
158C4⋊C4×C23—41no—
159C4⋊C4×C23—42no—
160C22⋊C4×C23—24no—
161C22⋊C4×C23—21no—
162C22⋊C4×C23—22no—
163C22⋊C4×C23—22no—
164C22⋊C4×C23—22no—
165C22⋊C4×C23—42no—
166C22⋊C4×C23—42no—
167C22⋊C4×C23—42no—
168C22⋊C4×C23—44no—
169C22⋊C4×C23—42no—
170C22⋊C4×C23—41no—
171C22⋊C4×C23—41no—
172C22⋊C4×C23—41no—
173C22⋊C4×C23—41no—
174C22⋊C4×C23—82no—
175C22⋊C4×C23C22⋊C412yescomplemented
176C42×C23—41no—
177C42×C23—42no—
178#13041C2311yesmaximal, characteristic, direct product factor, Sylow
179C23×C23—22no—
180C23×C23—22no—
181C23×C23—22no—
182C23×C23—21no—
183C23×C23—21no—
184C23×C23—21no—
185C23×C23—21no—
186C23×C23—22no—
187C23×C23—22no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23—41no—
192C23×C23—42no—
193C23×C23—41no—
194C23×C23—44no—
195C23×C23—41no—
196C23×C23—44no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—41no—
200C23×C23—41no—
201C23×C23—42no—
202C23×C23—42no—
203C23×C23—41no—
204C23×C23—81no—
205C23×C23—84no—
206C23×C23—81no—
207C23×C23—81no—
208C23×C23—81no—
209C23×C23—81no—
210C23×C23—81no—
211C23×C23—81no—
212C23×C23C2×C22⋊C411yescharacteristic
213C23×C23C4×D411yescharacteristic
214C23×C23#12011yescharacteristic
215C23×C23#12111yescharacteristic
216C23×C23#12311yescharacteristic
217Q8×C23—22no—
218Q8×C23—41no—
219D4×C23—22no—
220D4×C23—41no—
221D4×C23—41no—
222D4×C23—41no—
223D4×C23—41no—
224D4×C23—41no—
225D4×C23—44no—
226D4×C23—44no—
227D4×C23—42no—
228D4×C23—42no—
229D4×C23—42no—
230D4×C23—42no—
231D4×C23—42no—
232D4×C23—42no—
233D4×C23—88no—
234C2×C4×C23—21no—
235C2×C4×C23—21no—
236C2×C4×C23—22no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—22no—
240C2×C4×C23—22no—
241C2×C4×C23—42no—
242C2×C4×C23—42no—
243C2×C4×C23—41no—
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—41no—
251C2×C4×C23—41no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—41no—
255C2×C4×C23—41no—
256C2×C4×C23—41no—
257C2×C4×C23—41no—
258C2×C4×C23—41no—
259C2×C4×C23—41no—
260C2×C4×C23—81no—
261C2×C4×C23—84no—
262C2×C4×C23—84no—
263C2×C4×C23—81no—
264C2×C4×C23—81no—
265C2×C4×C23—81no—
266C2×C4×C23—81no—
267C2×C4×C23—81no—
268C2×C4×C23C4×D411yescharacteristic
269C2×C4×C23#12111yescharacteristic
270C8×C23—81no—
271C2×B(4,𝔽2)C2×C2311yescharacteristic
272#2358C2×C2311yescharacteristic, complemented
273C2×AΣL(1,𝔽16)C2×C2311yescharacteristic, complemented
274#1792C2×C2311yescharacteristic, complemented
275#1790C2×C2311yescharacteristic, complemented
276#1393C2×C2311yescharacteristic, complemented
277#1390C2×C2311yescharacteristic, complemented
278C22×C23—21no—
279C22×C23—21no—
280C22×C23—24no—
281C22×C23—21no—
282C22×C23—21no—
283C22×C23—21no—
284C22×C23—21no—
285C22×C23—41no—
286C22×C23—41no—
287C22×C23—41no—
288C22×C23—41no—
289C22×C23—42no—
290C22×C23—42no—
291C22×C23—42no—
292C22×C23—44no—
293C22×C23—42no—
294C22×C23—41no—
295C22×C23—42no—
296C22×C23—41no—
297C22×C23—41no—
298C22×C23—41no—
299C22×C23—42no—
300C22×C23—42no—
301C22×C23—81no—
302C22×C23—81no—
303C22×C23—82no—
304C22×C23—82no—
305C22×C23—82no—
306C22×C23—82no—
307C22×C23—82no—
308C22×C23—81no—
309C22×C23—81no—
310C22×C23—161no—
311C22×C23—161no—
312C22×C23#38611yescharacteristic
313C22×C23B(4,𝔽2)11yescharacteristic
314C22×C23B(4,𝔽2)11yescharacteristic
315C4×C23—21no—
316C4×C23—21no—
317C4×C23—42no—
318C4×C23—42no—
319C4×C23—42no—
320C4×C23—42no—
321C4×C23—81no—
322C4×C23—82no—
323C4×C23—81no—
324C4×C23—161no—
325C2×H(5,𝔽2)C22×C2311yescharacteristic
326C2×#120—21no—
327C2×#120C22×C2311yescharacteristic
328C2×C4×D4—21no—
329C22×C22⋊C4—21no—
330B(4,𝔽2)—22no—
331B(4,𝔽2)C4×C2314yescomplemented
332C2×#104—21no—
333#417—21no—
334C2×#100C22×C2311yescharacteristic
335C2×#99—21no—
336C2×#99—21no—
337C2×#99C22×C2311yescharacteristic
338C2×#99C22×C2311yescharacteristic
339#390C22×C2311yescharacteristic
340#388—21no—
341#386—21no—
342#386—21no—
343#386C22×C2311yescharacteristic
344#385—21no—
345C4×C22⋊C4—21no—
346AΣL(1,𝔽16)—22no—
347C2×C23—21no—
348C2×C23—21no—
349C2×C23—41no—
350C2×C23—41no—
351C2×C23—44no—
352C2×C23—41no—
353C2×C23—41no—
354C2×C23—82no—
355C2×C23#188911yescharacteristic
356C2×C23#219611yescharacteristic
357C2×C23#219711yescharacteristic
358C25D4×C2311yescharacteristic
359H(5,𝔽2)—21no—
360H(5,𝔽2)—21no—
361H(5,𝔽2)—21no—
362H(5,𝔽2)—42no—
363H(5,𝔽2)C2×C4×C2312yescomplemented
364C2×#41—21no—
365C2×#41—41no—
366C22×D4—21no—
367C22×D4—21no—
368C22×D4—21no—
369C22×D4—41no—
370C22×D4—41no—
371C22×D4C23×C2311yescharacteristic
372C23×C4—21no—
373C23×C4—21no—
374C23×C4D4×C2311yescharacteristic, complemented
375C2×M16—21no—
376#120—21no—
377#120—21no—
378#120—24no—
379#120—42no—
380#120C2×C4×C2312yes—
381#120D4×C2312yes—
382C4×D4—44no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4D4×C2311yescharacteristic, complemented
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—44no—
394C2×C22⋊C4—41no—
395C2×C22⋊C4—41no—
396C2×C22⋊C4D4×C2311yescharacteristic
397C2×C22⋊C4D4×C2311yescharacteristic, complemented
398C2×C42—21no—
399C2×C42D4×C2311yescharacteristic, complemented
400#104—42no—
401#103—41no—
402#102—41no—
403#100—21no—
404#100—21no—
405#99—21no—
406#99—24no—
407#99—21no—
408#99—21no—
409#99—42no—
410#99C2×C4×C2312yescomplemented
411#95—21no—
412#95—21no—
413#95—21no—
414#95—41no—
415C23#1304111yescharacteristic, direct product factor, Sylow
416C24—21no—
417C24—21no—
418C24—21no—
419C24—21no—
420C24—42no—
421C24—41no—
422C24—42no—
423C24—41no—
424C24C22⋊C4×C2312yescomplemented
425C24C2×D4×C2311yescharacteristic
426C24C2×D4×C2311yescharacteristic
427#41—22no—
428#41—22no—
429#41—42no—
430#41—41no—
431#41—42no—
432#41—42no—
433#41—42no—
434#41—41no—
435C2×Q8—21no—
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—41no—
451C2×D4—41no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—41no—
456C2×D4—41no—
457C2×D4—41no—
458C2×D4—44no—
459C2×D4—82no—
460C2×D4—82no—
461C2×D4C22×C4×C2311yescharacteristic
462C2×D4C2×D4×C2311yescharacteristic
463C2×D4#41×C2311yescharacteristic
464C2×D4#41×C2311yescharacteristic
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—41no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—41no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4C2×D4×C2311yescharacteristic
484M16—41no—
485M16—41no—
486C2×C8—41no—
487C4⋊C4—41no—
488C4⋊C4—42no—
489C22⋊C4—24no—
490C22⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—42no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—82no—
504C22⋊C4C22⋊C4×C2312yescomplemented
505C42—41no—
506C42—42no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—22no—
515C23—22no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—44no—
523C23—41no—
524C23—44no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—81no—
533C23—84no—
534C23—81no—
535C23—81no—
536C23—81no—
537C23—81no—
538C23—81no—
539C23—81no—
540C23C2×C22⋊C4×C2311yescharacteristic
541C23C4×D4×C2311yescharacteristic
542C23C23×#12011yescharacteristic
543C23C23×#12111yescharacteristic
544C23C23×#12311yescharacteristic
545Q8—22no—
546Q8—41no—
547D4—22no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—41no—
552D4—41no—
553D4—42no—
554D4—42no—
555D4—42no—
556D4—42no—
557D4—44no—
558D4—44no—
559D4—42no—
560D4—42no—
561D4—88no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—81no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—81no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—81no—
596C2×C4C4×D4×C2311yescharacteristic
597C2×C4C23×#12111yescharacteristic
598C8—81no—
599C22—21no—
600C22—21no—
601C22—24no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—42no—
611C22—42no—
612C22—42no—
613C22—44no—
614C22—42no—
615C22—41no—
616C22—42no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—81no—
630C22—81no—
631C22—161no—
632C22—161no—
633C22C23×#38611yescharacteristic
634C22C23×B(4,𝔽2)11yescharacteristic
635C22C23×B(4,𝔽2)11yescharacteristic
636C4—21no—
637C4—21no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—82no—
644C4—81no—
645C4—161no—
646C2—21no—
647C2—21no—
648C2—41no—
649C2—41no—
650C2—44no—
651C2—41no—
652C2—41no—
653C2—82no—
654C2C23×#188911yescharacteristic
655C2C23×#219611yescharacteristic
656C2C23×#219711yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×B(4,𝔽2)C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2358C2semidirect
3. C2×C23×AΣL(1,𝔽16)C2semidirect
4. C23×#1792C2semidirect
5. C23×#1790C2semidirect
6. C23×#1393C2semidirect
7. C23×#1390C2semidirect
14. C23×B(4,𝔽2)C4semidirect
35. C23×H(5,𝔽2)C2×C4semidirect
46. C23×C4×C23D4semidirect
56. C2×C4⋊C4×C23D4semidirect
69. C2×C22⋊C4×C23D4semidirect
71. C2×C42×C23D4semidirect
82. C23×#99C2×C4semidirect
95. C24×C23C22⋊C4semidirect
175. C22⋊C4×C23C22⋊C4semidirect
178. #13041C23direct (N × H)
272. #2358C2×C23semidirect
273. C2×AΣL(1,𝔽16)C2×C23semidirect
274. #1792C2×C23semidirect
275. #1790C2×C23semidirect
276. #1393C2×C23semidirect
277. #1390C2×C23semidirect
331. B(4,𝔽2)C4×C23semidirect
363. H(5,𝔽2)C2×C4×C23semidirect
374. C23×C4D4×C23semidirect
384. C2×C4⋊C4D4×C23semidirect
397. C2×C22⋊C4D4×C23semidirect
399. C2×C42D4×C23semidirect
410. #99C2×C4×C23semidirect
415. C23#13041direct (N × H)
424. C24C22⋊C4×C23semidirect
504. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×B(4,𝔽2)2
C23×#23582
C2×C23×AΣL(1,𝔽16)2
C23×#17922
C23×#17902
C23×#13932
C23×#13902
#1304123