Finite Group Databasebeta

← C23×#9113 #19156739

Subgroups — C23×#9113 #19156739

Order 5888.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×C22⋊C8C211yesmaximal, characteristic, complemented
2C23×#2847C211yesmaximal, characteristic
3C2×C23×#327C212yesmaximal, complemented
4C23×C22⋊#95C211yesmaximal, characteristic, complemented
5C23×#1294C212yesmaximal, complemented
6C24×C4×C23C2211yescharacteristic
7C23×C8×C23—21no—
8C2×C23×#121C411yescharacteristic, complemented
9C2×C23×#121C411yescharacteristic, complemented
10C2×C23×#121C2211yescharacteristic
11C2×C23×#120—21no—
12C22×C22⋊C4×C23—21no—
13C2×C23×C22⋊C8—24no—
14C2×C23×C22⋊C8—21no—
15C2×C23×C22⋊C8C2212yes—
16C23×#386—21no—
17C23×#386—21no—
18C23×#386—21no—
19C23×#380C411yescharacteristic, complemented
20C23×#380C411yescharacteristic, complemented
21C23×#380C2211yescharacteristic
22C2×C23×#95—21no—
23C2×C23×#95C2212yes—
24C23×#327—24no—
25C25×C23D411yescharacteristic
26C22×D4×C23—21no—
27C22×D4×C23—21no—
28C22×D4×C23—21no—
29C22×D4×C23C2×C411yescharacteristic
30C23×C4×C23—22no—
31C23×C4×C23—22no—
32C23×C4×C23—22no—
33C23×C4×C23—42no—
34C23×C4×C23—42no—
35C23×C4×C23C2×C411yescharacteristic
36C23×C4×C23C2×C411yescharacteristic
37C23×C4×C23D411yescharacteristic
38C23×C4×C23D411yescharacteristic
39C23×C4×C23D411yescharacteristic
40C23×C4×C23C2311yescharacteristic
41C22×C8×C23—24no—
42C22×C8×C23—22no—
43C22×C8×C23—42no—
44C22×C8×C23—42no—
45C23×#121—22no—
46C23×#121—22no—
47C23×#121—42no—
48C23×#121—42no—
49C23×#120—42no—
50C23×#120—42no—
51C2×C4⋊C4×C23C2×C411yescharacteristic
52C2×C4⋊C4×C23C2×C411yescharacteristic
53C2×C4⋊C4×C23C2×C411yescharacteristic
54C2×C22⋊C4×C23—24no—
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—21no—
61C2×C22⋊C4×C23—42no—
62C2×C22⋊C4×C23—42no—
63C23×C22⋊C8—24no—
64C23×C22⋊C8—22no—
65C23×C22⋊C8D414yescomplemented
66C23×#95—21no—
67C23×#95—21no—
68C23×#95—21no—
69C23×#95—22no—
70C23×#95—22no—
71C23×#95—42no—
72C24×C23—22no—
73C24×C23—22no—
74C24×C23—22no—
75C24×C23—22no—
76C24×C23—21no—
77C24×C23—41no—
78C24×C23—41no—
79C24×C23C22⋊C411yescharacteristic
80C24×C23C22⋊C411yescharacteristic
81C24×C23D812yes—
82C24×C23SD1612yes—
83C24×C23C2×D411yescharacteristic
84C2×D4×C23—22no—
85C2×D4×C23—41no—
86C2×D4×C23—41no—
87C2×D4×C23—42no—
88C2×D4×C23—42no—
89C2×D4×C23—42no—
90C2×D4×C23—42no—
91C2×D4×C23—42no—
92C2×D4×C23—42no—
93C2×D4×C23—42no—
94C2×D4×C23—41no—
95C2×D4×C23—41no—
96C2×D4×C23—42no—
97C22×C4×C23—22no—
98C22×C4×C23—22no—
99C22×C4×C23—21no—
100C22×C4×C23—21no—
101C22×C4×C23—22no—
102C22×C4×C23—22no—
103C22×C4×C23—22no—
104C22×C4×C23—22no—
105C22×C4×C23—22no—
106C22×C4×C23—22no—
107C22×C4×C23—21no—
108C22×C4×C23—21no—
109C22×C4×C23—24no—
110C22×C4×C23—21no—
111C22×C4×C23—21no—
112C22×C4×C23—21no—
113C22×C4×C23—41no—
114C22×C4×C23—41no—
115C22×C4×C23—41no—
116C22×C4×C23—41no—
117C22×C4×C23—41no—
118C22×C4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23—48no—
121C22×C4×C23—48no—
122C22×C4×C23—42no—
123C22×C4×C23—42no—
124C22×C4×C23—42no—
125C22×C4×C23—42no—
126C22×C4×C23—42no—
127C22×C4×C23—42no—
128C22×C4×C23C22⋊C411yescharacteristic
129C22×C4×C23C22⋊C411yescharacteristic
130C22×C4×C23C22×C411yescharacteristic
131C22×C4×C23C2×D411yescharacteristic
132C22×C4×C23C2×D412yes—
133C22×C4×C23#4112yes—
134C2×C8×C23—24no—
135C2×C8×C23—48no—
136C2×C8×C23—42no—
137C2×C8×C23—42no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—41no—
140C4⋊C4×C23—41no—
141C22⋊C4×C23—42no—
142C22⋊C4×C23—48no—
143C22⋊C4×C23—42no—
144C22⋊C4×C23—42no—
145C22⋊C4×C23—41no—
146C22⋊C4×C23—41no—
147C22⋊C4×C23—42no—
148C22⋊C4×C23—41no—
149C22⋊C4×C23—41no—
150#9113C2311yesmaximal, characteristic, direct product factor, Sylow
151C23×C23—21no—
152C23×C23—21no—
153C23×C23—21no—
154C23×C23—21no—
155C23×C23—21no—
156C23×C23—21no—
157C23×C23—21no—
158C23×C23—21no—
159C23×C23—24no—
160C23×C23—24no—
161C23×C23—21no—
162C23×C23—21no—
163C23×C23—21no—
164C23×C23—21no—
165C23×C23—21no—
166C23×C23—21no—
167C23×C23—24no—
168C23×C23—24no—
169C23×C23—42no—
170C23×C23—42no—
171C23×C23—44no—
172C23×C23—44no—
173C23×C23—44no—
174C23×C23—44no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—41no—
178C23×C23—41no—
179C23×C23—41no—
180C23×C23—41no—
181C23×C23—41no—
182C23×C23#9912yes—
183C23×C23#10212yes—
184C23×C23#10212yes—
185C23×C23#10412yes—
186C23×C23C2×C22⋊C411yescharacteristic
187C23×C23#12112yes—
188C23×C23C2×D811yescharacteristic
189C23×C23C2×SD1611yescharacteristic
190C23×C23#13512yes—
191D4×C23—82no—
192D4×C23—84no—
193D4×C23—84no—
194D4×C23—82no—
195D4×C23—82no—
196C2×C4×C23—24no—
197C2×C4×C23—24no—
198C2×C4×C23—21no—
199C2×C4×C23—21no—
200C2×C4×C23—21no—
201C2×C4×C23—21no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—21no—
205C2×C4×C23—21no—
206C2×C4×C23—21no—
207C2×C4×C23—42no—
208C2×C4×C23—42no—
209C2×C4×C23—44no—
210C2×C4×C23—44no—
211C2×C4×C23—44no—
212C2×C4×C23—42no—
213C2×C4×C23—42no—
214C2×C4×C23—44no—
215C2×C4×C23—44no—
216C2×C4×C23—44no—
217C2×C4×C23—41no—
218C2×C4×C23—41no—
219C2×C4×C23—41no—
220C2×C4×C23—41no—
221C2×C4×C23—41no—
222C2×C4×C23—41no—
223C2×C4×C23—41no—
224C2×C4×C23—41no—
225C2×C4×C23—41no—
226C2×C4×C23—82no—
227C2×C4×C23—82no—
228C2×C4×C23—82no—
229C2×C4×C23—82no—
230C2×C4×C23#9912yes—
231C2×C4×C23C4×D412yes—
232C2×C4×C23#12011yescharacteristic
233C2×C4×C23#12311yescharacteristic
234C8×C23—44no—
235C22×C22⋊C8C2×C2311yescharacteristic, complemented
236#2847C2×C2311yescharacteristic
237C2×#327C2×C2312yescomplemented
238C22⋊#95C2×C2311yescharacteristic, complemented
239#1294C2×C2312yescomplemented
240C22×C23—22no—
241C22×C23—22no—
242C22×C23—22no—
243C22×C23—22no—
244C22×C23—22no—
245C22×C23—21no—
246C22×C23—21no—
247C22×C23—22no—
248C22×C23—22no—
249C22×C23—22no—
250C22×C23—22no—
251C22×C23—21no—
252C22×C23—22no—
253C22×C23—21no—
254C22×C23—22no—
255C22×C23—22no—
256C22×C23—22no—
257C22×C23—48no—
258C22×C23—48no—
259C22×C23—41no—
260C22×C23—41no—
261C22×C23—41no—
262C22×C23—41no—
263C22×C23—81no—
264C22×C23—81no—
265C22×C23—81no—
266C22×C23—81no—
267C22×C23—81no—
268C22×C23—81no—
269C22×C23—81no—
270C22×C23#32714yescomplemented
271C22×C23#38611yescharacteristic
272C22×C23C2×#9911yescharacteristic
273C22×C23C2×#9911yescharacteristic
274C22×C23C2×#10211yescharacteristic
275C22×C23#41611yescharacteristic
276C22×C23C2×#10411yescharacteristic
277C22×C23#42111yescharacteristic
278C22×C23#46512yes—
279C22×C23#46612yes—
280C4×C23—21no—
281C4×C23—22no—
282C4×C23—21no—
283C4×C23—22no—
284C4×C23—41no—
285C4×C23—84no—
286C4×C23—81no—
287C4×C23—81no—
288C4×C23—81no—
289C24×C4C22×C2311yescharacteristic
290C23×C8—21no—
291C2×#121C4×C2311yescharacteristic, complemented
292C2×#121C4×C2311yescharacteristic, complemented
293C2×#121C22×C2311yescharacteristic
294C2×#120—21no—
295C22×C22⋊C4—21no—
296C2×C22⋊C8—21no—
297C2×C22⋊C8—24no—
298C2×C22⋊C8C22×C2312yes—
299#386—21no—
300#386—21no—
301#386—21no—
302#380C4×C2311yescharacteristic, complemented
303#380C4×C2311yescharacteristic, complemented
304#380C22×C2311yescharacteristic
305C2×#95—21no—
306C2×#95C22×C2312yes—
307#327—24no—
308C2×C23—22no—
309C2×C23—24no—
310C2×C23—22no—
311C2×C23—42no—
312C2×C23—161no—
313C2×C23C2×#32711yescharacteristic
314C2×C23#150111yescharacteristic
315C2×C23#151611yescharacteristic
316C2×C23#151911yescharacteristic
317C2×C23#187111yescharacteristic
318C2×C23#189311yescharacteristic
319C2×C23#189811yescharacteristic
320C25D4×C2311yescharacteristic
321C22×D4—21no—
322C22×D4—21no—
323C22×D4—21no—
324C22×D4C2×C4×C2311yescharacteristic
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4C2×C4×C2311yescharacteristic
331C23×C4C2×C4×C2311yescharacteristic
332C23×C4D4×C2311yescharacteristic
333C23×C4D4×C2311yescharacteristic
334C23×C4D4×C2311yescharacteristic
335C23×C4C23×C2311yescharacteristic
336C22×C8—24no—
337C22×C8—22no—
338C22×C8—42no—
339C22×C8—42no—
340#121—22no—
341#121—22no—
342#121—42no—
343#121—42no—
344#120—42no—
345#120—42no—
346C2×C4⋊C4C2×C4×C2311yescharacteristic
347C2×C4⋊C4C2×C4×C2311yescharacteristic
348C2×C4⋊C4C2×C4×C2311yescharacteristic
349C2×C22⋊C4—24no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—42no—
357C2×C22⋊C4—42no—
358C22⋊C8—24no—
359C22⋊C8—22no—
360C22⋊C8D4×C2314yescomplemented
361#95—21no—
362#95—21no—
363#95—21no—
364#95—22no—
365#95—22no—
366#95—42no—
367C23#911311yescharacteristic, direct product factor, Sylow
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—41no—
374C24—41no—
375C24C22⋊C4×C2311yescharacteristic
376C24C22⋊C4×C2311yescharacteristic
377C24D8×C2312yes—
378C24SD16×C2312yes—
379C24C2×D4×C2311yescharacteristic
380C2×D4—22no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—42no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4C22⋊C4×C2311yescharacteristic
425C22×C4C22⋊C4×C2311yescharacteristic
426C22×C4C22×C4×C2311yescharacteristic
427C22×C4C2×D4×C2311yescharacteristic
428C22×C4C2×D4×C2312yes—
429C22×C4#41×C2312yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—42no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C23×#9912yes—
478C23C23×#10212yes—
479C23C23×#10212yes—
480C23C23×#10412yes—
481C23C2×C22⋊C4×C2311yescharacteristic
482C23C23×#12112yes—
483C23C2×D8×C2311yescharacteristic
484C23C2×SD16×C2311yescharacteristic
485C23C23×#13512yes—
486D4—82no—
487D4—84no—
488D4—84no—
489D4—82no—
490D4—82no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4C23×#9912yes—
526C2×C4C4×D4×C2312yes—
527C2×C4C23×#12011yescharacteristic
528C2×C4C23×#12311yescharacteristic
529C8—44no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—21no—
540C22—22no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C23×#32714yescomplemented
561C22C23×#38611yescharacteristic
562C22C2×C23×#9911yescharacteristic
563C22C2×C23×#9911yescharacteristic
564C22C2×C23×#10211yescharacteristic
565C22C23×#41611yescharacteristic
566C22C2×C23×#10411yescharacteristic
567C22C23×#42111yescharacteristic
568C22C23×#46512yes—
569C22C23×#46612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—81no—
577C4—81no—
578C4—81no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—161no—
584C2C2×C23×#32711yescharacteristic
585C2C23×#150111yescharacteristic
586C2C23×#151611yescharacteristic
587C2C23×#151911yescharacteristic
588C2C23×#187111yescharacteristic
589C2C23×#189311yescharacteristic
590C2C23×#189811yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#2847C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C24×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C23×C22⋊C8C2semidirect
3. C2×C23×#327C2semidirect
4. C23×C22⋊#95C2semidirect
5. C23×#1294C2semidirect
8. C2×C23×#121C4semidirect
9. C2×C23×#121C4semidirect
19. C23×#380C4semidirect
20. C23×#380C4semidirect
65. C23×C22⋊C8D4semidirect
150. #9113C23direct (N × H)
235. C22×C22⋊C8C2×C23semidirect
237. C2×#327C2×C23semidirect
238. C22⋊#95C2×C23semidirect
239. #1294C2×C23semidirect
270. C22×C23#327semidirect
291. C2×#121C4×C23semidirect
292. C2×#121C4×C23semidirect
302. #380C4×C23semidirect
303. #380C4×C23semidirect
360. C22⋊C8D4×C23semidirect
367. C23#9113direct (N × H)
560. C22C23×#327semidirect

Maximal subgroups

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

subgroupindex
C22×C23×C22⋊C82
C23×#28472
C2×C23×#3272
C23×C22⋊#952
C23×#12942
#911323