Finite Group Databasebeta

← C23×#25517 #19173143

Subgroups — C23×#25517 #19173143

Order 5888.

1174 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic, complemented
2C23×#3055C212yesmaximal, complemented
3C23×#2501C211yesmaximal, characteristic, complemented
4C23×#2060C211yesmaximal, characteristic, complemented
5C23×#2039C211yesmaximal, characteristic, complemented
6C23×#2027C211yesmaximal, characteristic, complemented
7C23×#2006C211yesmaximal, characteristic, complemented
8C23×#1883C212yesmaximal, complemented
9C23×#1871C212yesmaximal, complemented
10C23×#1859C212yesmaximal, complemented
11C2×C23×#343C211yesmaximal, characteristic, complemented
12C2×C23×AGL(1,ℤ/8)C2212yes—
13C22×M16×C23C2212yes—
14C23×#554—22no—
15C23×#553—22no—
16C23×#552—22no—
17C23×#547—22no—
18C23×#544C2211yescharacteristic
19C23×#539—22no—
20C23×#537C2211yescharacteristic
21C23×#525—22no—
22C23×#525C2212yes—
23C2×C23×#123—22no—
24C2×C23×#121C2211yescharacteristic
25C23×#520C2212yes—
26C22×C4⋊C4×C23C2211yescharacteristic
27C23×#469—22no—
28C23×#468—22no—
29C23×#466—22no—
30C23×#465—22no—
31C23×#428C2212yes—
32C2×C23×#104C2212yes—
33C2×C23×#104C2212yes—
34C23×#418C2212yes—
35C23×#417C2212yes—
36C23×#399—22no—
37C23×#390C2211yescharacteristic
38C23×#390C2211yescharacteristic
39C23×#384C2211yescharacteristic
40C23×#384C2211yescharacteristic
41C4×C22⋊C4×C23C2211yescharacteristic
42C23×#343C2211yescharacteristic
43C23×#343C2211yescharacteristic
44C23×#343C2212yes—
45C23×#339C2212yes—
46C23×#339C2212yes—
47C2×#41×C23—22no—
48C2×#41×C23C2312yes—
49C22×D4×C23—21no—
50C22×D4×C23C2311yescharacteristic
51C23×C4×C23D412yes—
52C23×C4×C23C2311yescharacteristic
53C23×AGL(1,ℤ/8)—24no—
54C23×AGL(1,ℤ/8)—42no—
55C2×SD16×C23—22no—
56C2×D8×C23—22no—
57C2×M16×C23—22no—
58C2×M16×C23—24no—
59C2×M16×C23—42no—
60C2×M16×C23C2312yes—
61C2×M16×C23C2312yes—
62C22×C8×C23—22no—
63C23×C4⋊Q8—22no—
64C23×#126—24no—
65C23×#125—24no—
66C23×#125—22no—
67C23×#123—24no—
68C23×#123—24no—
69C23×#123—22no—
70C23×#123—48no—
71C23×C22⋊Q8—24no—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—22no—
74C23×C22⋊Q8—22no—
75C23×C22⋊Q8D412yes—
76C23×C22⋊Q8D412yes—
77C23×#121—22no—
78C23×#121—22no—
79C23×#121—22no—
80C23×#121—22no—
81C23×#121—22no—
82C23×#121D412yes—
83C23×#121D412yes—
84C23×#120—22no—
85C4×Q8×C23—22no—
86C4×D4×C23—24no—
87C4×D4×C23—22no—
88C4×D4×C23—22no—
89C4×D4×C23—22no—
90C4×D4×C23—24no—
91C23×#117—22no—
92C23×#117—22no—
93C23×#117C2312yes—
94C2×C4⋊C4×C23—22no—
95C2×C4⋊C4×C23—22no—
96C2×C4⋊C4×C23—22no—
97C2×C4⋊C4×C23—22no—
98C2×C4⋊C4×C23—22no—
99C2×C4⋊C4×C23D412yes—
100C2×C4⋊C4×C23C2311yescharacteristic
101C2×C4⋊C4×C23C2311yescharacteristic
102C2×C4⋊C4×C23C2311yescharacteristic
103C2×C22⋊C4×C23—21no—
104C2×C22⋊C4×C23—22no—
105C2×C22⋊C4×C23—22no—
106C2×C22⋊C4×C23—21no—
107C2×C42×C23C2311yescharacteristic
108C2×C42×C23C2311yescharacteristic
109C23×#107—22no—
110C23×#106—22no—
111C23×#104—24no—
112C23×#104—42no—
113C23×#103—22no—
114C23×#102—22no—
115C23×#102—22no—
116C23×#102—22no—
117C23×#95—21no—
118C24×C23—21no—
119C24×C23C2×D411yescharacteristic
120#41×C23—22no—
121#41×C23—22no—
122#41×C23—22no—
123#41×C23—22no—
124#41×C23—84no—
125C2×Q8×C23—22no—
126C2×Q8×C23C2×D412yes—
127C2×D4×C23—22no—
128C2×D4×C23—22no—
129C2×D4×C23—22no—
130C2×D4×C23—22no—
131C2×D4×C23—22no—
132C2×D4×C23—22no—
133C2×D4×C23—21no—
134C2×D4×C23—21no—
135C2×D4×C23—44no—
136C2×D4×C23—41no—
137C2×D4×C23—41no—
138C2×D4×C23C2×D412yes—
139C22×C4×C23—22no—
140C22×C4×C23—21no—
141C22×C4×C23—21no—
142C22×C4×C23—22no—
143C22×C4×C23—21no—
144C22×C4×C23—21no—
145C22×C4×C23—24no—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—21no—
149C22×C4×C23—44no—
150C22×C4×C23C2×D411yescharacteristic
151C22×C4×C23C2×D412yes—
152C22×C4×C23C2×D412yes—
153C22×C4×C23C2×D412yes—
154C22×C4×C23C2×D412yes—
155C22×C4×C23C2411yescharacteristic
156SD16×C23—44no—
157D8×C23—44no—
158M16×C23—22no—
159M16×C23—22no—
160M16×C23—22no—
161M16×C23—44no—
162M16×C23#4114yescomplemented
163C2×C8×C23—22no—
164C2×C8×C23—22no—
165C2×C8×C23—44no—
166C4⋊C4×C23—24no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—24no—
170C4⋊C4×C23—24no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—44no—
177C4⋊C4×C23—44no—
178C4⋊C4×C23C2×D412yes—
179C4⋊C4×C23C2×D412yes—
180C22⋊C4×C23—24no—
181C22⋊C4×C23—24no—
182C22⋊C4×C23—24no—
183C22⋊C4×C23—22no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—22no—
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—44no—
189C22⋊C4×C23—44no—
190C22⋊C4×C23—42no—
191C42×C23—24no—
192C42×C23—22no—
193C42×C23—22no—
194C42×C23—21no—
195C42×C23—21no—
196#25517C2311yesmaximal, characteristic, direct product factor, Sylow
197C23×C23—21no—
198C23×C23—21no—
199C23×C23—22no—
200C23×C23—22no—
201C23×C23—41no—
202C23×C23—42no—
203C23×C23—42no—
204C23×C23—41no—
205C23×C23—41no—
206C23×C23—41no—
207C23×C23#12011yescharacteristic
208C23×C23#12011yescharacteristic
209C23×C23C22×D411yescharacteristic
210Q8×C23—22no—
211Q8×C23—22no—
212Q8×C23—82no—
213D4×C23—22no—
214D4×C23—22no—
215D4×C23—44no—
216D4×C23—42no—
217D4×C23—42no—
218D4×C23—44no—
219D4×C23—42no—
220D4×C23—42no—
221D4×C23—82no—
222D4×C23—82no—
223D4×C23—82no—
224C2×C4×C23—21no—
225C2×C4×C23—21no—
226C2×C4×C23—22no—
227C2×C4×C23—22no—
228C2×C4×C23—22no—
229C2×C4×C23—22no—
230C2×C4×C23—22no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—22no—
235C2×C4×C23—22no—
236C2×C4×C23—22no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—41no—
240C2×C4×C23—41no—
241C2×C4×C23—44no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—41no—
248C2×C4×C23—82no—
249C2×C4×C23—84no—
250C2×C4×C23#12012yes—
251C2×C4×C23C22×D412yes—
252C2×C4×C23C2×#4112yes—
253C2×C4×C23H(5,𝔽2)11yescharacteristic
254C2×C4×C232−1+411yescharacteristic
255C8×C23—24no—
256C8×C23—42no—
257#3492C2×C2311yescharacteristic, complemented
258#3055C2×C2312yescomplemented
259#2501C2×C2311yescharacteristic, complemented
260#2060C2×C2311yescharacteristic, complemented
261#2039C2×C2311yescharacteristic, complemented
262#2027C2×C2311yescharacteristic, complemented
263#2006C2×C2311yescharacteristic, complemented
264#1883C2×C2312yescomplemented
265#1871C2×C2312yescomplemented
266#1859C2×C2312yescomplemented
267C2×#343C2×C2311yescharacteristic, complemented
268C22×C23—21no—
269C22×C23—21no—
270C22×C23—21no—
271C22×C23—21no—
272C22×C23—21no—
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—42no—
276C22×C23—42no—
277C22×C23—42no—
278C22×C23—42no—
279C22×C23—42no—
280C22×C23—82no—
281C22×C23—82no—
282C22×C23—81no—
283C22×C23—81no—
284C22×C23—81no—
285C22×C23#45312yes—
286C22×C23C2×#12011yescharacteristic
287C22×C23#53211yescharacteristic
288C22×C23#53711yescharacteristic
289C4×C23—22no—
290C4×C23—42no—
291C4×C23—42no—
292C4×C23—42no—
293C4×C23—42no—
294C4×C23—42no—
295C4×C23—42no—
296C4×C23—42no—
297C4×C23—81no—
298C4×C23#54612yes—
299C4×C23#54712yes—
300C2×AGL(1,ℤ/8)C22×C2312yes—
301C22×M16C22×C2312yes—
302#554—22no—
303#553—22no—
304#552—22no—
305#547—22no—
306#544C22×C2311yescharacteristic
307#539—22no—
308#537C22×C2311yescharacteristic
309#525—22no—
310#525C22×C2312yes—
311C2×#123—22no—
312C2×#121C22×C2311yescharacteristic
313#520C22×C2312yes—
314C22×C4⋊C4C22×C2311yescharacteristic
315#469—22no—
316#468—22no—
317#466—22no—
318#465—22no—
319#428C22×C2312yes—
320C2×#104C22×C2312yes—
321C2×#104C22×C2312yes—
322#418C22×C2312yes—
323#417C22×C2312yes—
324#399—22no—
325#390C22×C2311yescharacteristic
326#390C22×C2311yescharacteristic
327#384C22×C2311yescharacteristic
328#384C22×C2311yescharacteristic
329C4×C22⋊C4C22×C2311yescharacteristic
330#343C22×C2311yescharacteristic
331#343C22×C2312yes—
332#343C22×C2311yescharacteristic
333#339C22×C2312yes—
334#339C22×C2312yes—
335C2×C23—21no—
336C2×C23—21no—
337C2×C23—22no—
338C2×C23—41no—
339C2×C23—82no—
340C2×C23—161no—
341C2×C23#243011yescharacteristic
342C2×C23C2×#45311yescharacteristic
343C2×C23#302011yescharacteristic
344C2×#41—22no—
345C2×#41C23×C2312yes—
346C22×D4—21no—
347C22×D4C23×C2311yescharacteristic
348C23×C4D4×C2312yes—
349C23×C4C23×C2311yescharacteristic
350AGL(1,ℤ/8)—24no—
351AGL(1,ℤ/8)—42no—
352C2×SD16—22no—
353C2×D8—22no—
354C2×M16—22no—
355C2×M16—24no—
356C2×M16—42no—
357C2×M16C23×C2312yes—
358C2×M16C23×C2312yes—
359C22×C8—22no—
360C4⋊Q8—22no—
361#126—24no—
362#125—24no—
363#125—22no—
364#123—24no—
365#123—24no—
366#123—22no—
367#123—48no—
368C22⋊Q8—22no—
369C22⋊Q8—24no—
370C22⋊Q8—22no—
371C22⋊Q8—22no—
372C22⋊Q8D4×C2312yes—
373C22⋊Q8D4×C2312yes—
374#121—22no—
375#121—22no—
376#121—22no—
377#121—22no—
378#121—22no—
379#121D4×C2312yes—
380#121D4×C2312yes—
381#120—22no—
382C4×Q8—22no—
383C4×D4—24no—
384C4×D4—22no—
385C4×D4—24no—
386C4×D4—22no—
387C4×D4—22no—
388#117—22no—
389#117—22no—
390#117C23×C2312yes—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4D4×C2312yes—
397C2×C4⋊C4C23×C2311yescharacteristic
398C2×C4⋊C4C23×C2311yescharacteristic
399C2×C4⋊C4C23×C2311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—22no—
404C2×C42C23×C2311yescharacteristic
405C2×C42C23×C2311yescharacteristic
406#107—22no—
407#106—22no—
408#104—24no—
409#104—42no—
410#103—22no—
411#102—22no—
412#102—22no—
413#102—22no—
414#95—21no—
415C23#2551711yescharacteristic, direct product factor, Sylow
416C24—21no—
417C24C2×D4×C2311yescharacteristic
418#41—22no—
419#41—22no—
420#41—22no—
421#41—22no—
422#41—84no—
423C2×Q8—22no—
424C2×Q8C2×D4×C2312yes—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—44no—
434C2×D4—41no—
435C2×D4—41no—
436C2×D4C2×D4×C2312yes—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—44no—
448C22×C4C2×D4×C2311yescharacteristic
449C22×C4C2×D4×C2312yes—
450C22×C4C2×D4×C2312yes—
451C22×C4C2×D4×C2312yes—
452C22×C4C2×D4×C2312yes—
453C22×C4C24×C2311yescharacteristic
454SD16—44no—
455D8—44no—
456M16—22no—
457M16—22no—
458M16—22no—
459M16—44no—
460M16#41×C2314yescomplemented
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—24no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4C2×D4×C2312yes—
477C4⋊C4C2×D4×C2312yes—
478C22⋊C4—24no—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—44no—
487C22⋊C4—44no—
488C22⋊C4—42no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—21no—
493C42—21no—
494C23—22no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23C23×#12011yescharacteristic
505C23C23×#12011yescharacteristic
506C23C22×D4×C2311yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—82no—
510D4—22no—
511D4—22no—
512D4—44no—
513D4—44no—
514D4—42no—
515D4—42no—
516D4—42no—
517D4—42no—
518D4—82no—
519D4—82no—
520D4—82no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4C23×#12012yes—
548C2×C4C22×D4×C2312yes—
549C2×C4C2×#41×C2312yes—
550C2×C4C23×H(5,𝔽2)11yescharacteristic
551C2×C4C23×2−1+411yescharacteristic
552C8—24no—
553C8—42no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—42no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—82no—
567C22—82no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C23×#45312yes—
572C22C2×C23×#12011yescharacteristic
573C22C23×#53211yescharacteristic
574C22C23×#53711yescharacteristic
575C4—22no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4C23×#54612yes—
585C4C23×#54712yes—
586C2—21no—
587C2—21no—
588C2—22no—
589C2—41no—
590C2—82no—
591C2—161no—
592C2C23×#243011yescharacteristic
593C2C2×C23×#45311yescharacteristic
594C2C23×#302011yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#3492C2semidirect
2. C23×#3055C2semidirect
3. C23×#2501C2semidirect
4. C23×#2060C2semidirect
5. C23×#2039C2semidirect
6. C23×#2027C2semidirect
7. C23×#2006C2semidirect
8. C23×#1883C2semidirect
9. C23×#1871C2semidirect
10. C23×#1859C2semidirect
11. C2×C23×#343C2semidirect
162. M16×C23#41semidirect
196. #25517C23direct (N × H)
257. #3492C2×C23semidirect
258. #3055C2×C23semidirect
259. #2501C2×C23semidirect
260. #2060C2×C23semidirect
261. #2039C2×C23semidirect
262. #2027C2×C23semidirect
263. #2006C2×C23semidirect
264. #1883C2×C23semidirect
265. #1871C2×C23semidirect
266. #1859C2×C23semidirect
267. C2×#343C2×C23semidirect
415. C23#25517direct (N × H)
460. M16#41×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#34922
C23×#30552
C23×#25012
C23×#20602
C23×#20392
C23×#20272
C23×#20062
C23×#18832
C23×#18712
C23×#18592
C2×C23×#3432
#2551723