Finite Group Databasebeta

← C23×#19047 #19166673

Subgroups — C23×#19047 #19166673

Order 5888.

814 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3563C211yesmaximal, characteristic
2C23×#3236C211yesmaximal, characteristic, complemented
3C23×#3181C211yesmaximal, characteristic
4C23×#3162C211yesmaximal, characteristic, complemented
5C23×#1664C212yesmaximal, complemented
6C23×#1663C212yesmaximal, complemented
7C23×#1663C212yesmaximal, complemented
8C23×#1660C212yesmaximal, complemented
9C23×#1532C211yesmaximal, characteristic
10C23×#1531C211yesmaximal, characteristic, complemented
11C2×C23×#330C211yesmaximal, characteristic, complemented
12C23×#564C2212yes—
13C23×#563C2212yes—
14C23×#557—21no—
15C23×#554—21no—
16C23×#552—21no—
17C23×#552—22no—
18C23×#552—21no—
19C23×#544—21no—
20C23×#544—21no—
21C23×#541—21no—
22C23×#541—22no—
23C23×#539—21no—
24C23×#533C2211yescharacteristic
25C23×#533C2211yescharacteristic
26C23×#529C2211yescharacteristic
27C2×C23×#125C2211yescharacteristic
28C23×#501—21no—
29C23×#501—21no—
30C23×#501C2211yescharacteristic
31C23×#501C2212yes—
32C23×#501C2212yes—
33C23×#501C2212yes—
34C23×#497C2211yescharacteristic
35C23×#491C2211yescharacteristic
36C23×#491C2212yes—
37C23×#489C2211yescharacteristic
38C23×#484—21no—
39C23×#483—21no—
40C23×#479—22no—
41C23×#467—21no—
42C23×#465—21no—
43C23×#441—21no—
44C23×#440—21no—
45C23×#432C2211yescharacteristic
46C23×#427—22no—
47C2×C23×#97C2211yescharacteristic
48C2×C23×#97C2211yescharacteristic
49C23×#330C2212yes—
50C23×#330C2212yes—
51C23×#330C2212yes—
52C23×#330C2212yes—
53C23×#330C2212yes—
54C23×#329C2212yes—
55C2×Q16×C23—21no—
56C2×SD16×C23—21no—
57C22×C8×C23—21no—
58C22×C8×C23—21no—
59C23×C4⋊Q8—21no—
60C23×C4⋊Q8—22no—
61C23×C4⋊Q8—22no—
62C23×C4⋊Q8D411yescharacteristic
63C23×C4⋊Q8D411yescharacteristic
64C23×C4⋊Q8C2311yescharacteristic
65C23×C4⋊Q8C2311yescharacteristic
66C23×#126—22no—
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—22no—
70C23×#126—21no—
71C23×#125—22no—
72C23×#125—21no—
73C23×#125—22no—
74C23×#125—22no—
75C23×#125—22no—
76C23×#125—22no—
77C23×#125—22no—
78C23×#125—22no—
79C23×#125—22no—
80C23×#125—21no—
81C23×#125—21no—
82C23×#125—21no—
83C23×#125D412yes—
84C23×#125C2312yes—
85C23×#125C2312yes—
86C23×#125C2311yescharacteristic
87C23×#124C2311yescharacteristic
88C23×#123—22no—
89C23×#123—22no—
90C23×#123—21no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8—21no—
96C23×C22⋊Q8—21no—
97C23×C22⋊Q8—22no—
98C23×C22⋊Q8—22no—
99C23×C22⋊Q8—21no—
100C23×C22⋊Q8—21no—
101C23×C22⋊Q8D411yescharacteristic
102C23×#121D411yescharacteristic
103C4×Q8×C23—21no—
104C4×Q8×C23—22no—
105C4×Q8×C23—21no—
106C4×Q8×C23—21no—
107C4×Q8×C23—21no—
108C4×Q8×C23—21no—
109C4×Q8×C23—21no—
110C4×Q8×C23D411yescharacteristic
111C4×D4×C23—22no—
112C4×D4×C23—21no—
113C4×D4×C23D411yescharacteristic
114C23×#117—22no—
115C23×#117—21no—
116C23×#117—22no—
117C23×#117—21no—
118C23×#117—21no—
119C23×#117D412yes—
120C2×C4⋊C4×C23—21no—
121C2×C4⋊C4×C23—21no—
122C2×C4⋊C4×C23—21no—
123C2×C4⋊C4×C23D411yescharacteristic
124C2×C4⋊C4×C23D411yescharacteristic
125C2×C42×C23C2311yescharacteristic
126C23×#107—22no—
127C23×#107—22no—
128C23×#107—21no—
129C23×#107—22no—
130C23×#106—22no—
131C23×#106—22no—
132C23×#106—21no—
133C23×#106—22no—
134C23×C4⋊C8—21no—
135C23×#103—21no—
136C23×#103—21no—
137C23×#103—21no—
138C23×#103—22no—
139C23×#103—22no—
140C23×#102—21no—
141C23×C22⋊C8—21no—
142C23×#97—21no—
143C23×#97—21no—
144C23×#97C2312yes—
145C23×#97C2312yes—
146C23×#97C2311yescharacteristic
147C23×#97C2311yescharacteristic
148C2×Q8×C23—22no—
149C2×Q8×C23—21no—
150C2×Q8×C23—21no—
151C2×Q8×C23—21no—
152C2×Q8×C23—21no—
153C2×Q8×C23C2×D411yescharacteristic
154C2×D4×C23—21no—
155C2×D4×C23C2×D411yescharacteristic
156C22×C4×C23—21no—
157C22×C4×C23—22no—
158C22×C4×C23—21no—
159C22×C4×C23—21no—
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163Q16×C23—42no—
164SD16×C23—42no—
165C2×C8×C23—24no—
166C2×C8×C23—22no—
167C2×C8×C23—21no—
168C2×C8×C23—21no—
169C2×C8×C23—22no—
170C2×C8×C23—42no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—22no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—22no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—21no—
184C4⋊C4×C23—21no—
185C4⋊C4×C23—21no—
186C4⋊C4×C23—21no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—22no—
189C4⋊C4×C23—21no—
190C4⋊C4×C23—21no—
191C4⋊C4×C23—22no—
192C4⋊C4×C23—22no—
193C4⋊C4×C23—21no—
194C4⋊C4×C23—21no—
195C4⋊C4×C23—21no—
196C4⋊C4×C23—21no—
197C4⋊C4×C23—41no—
198C4⋊C4×C23—41no—
199C4⋊C4×C23C2×D412yes—
200C4⋊C4×C23C2×D411yescharacteristic
201C4⋊C4×C23C2×D412yes—
202C4⋊C4×C23C2×D411yescharacteristic
203C4⋊C4×C23C2×D412yes—
204C4⋊C4×C23C2×D412yes—
205C22⋊C4×C23—22no—
206C22⋊C4×C23—22no—
207C22⋊C4×C23—22no—
208C22⋊C4×C23—21no—
209C22⋊C4×C23—22no—
210C22⋊C4×C23—21no—
211C22⋊C4×C23—21no—
212C22⋊C4×C23—21no—
213C22⋊C4×C23—21no—
214C22⋊C4×C23—41no—
215C42×C23—22no—
216C42×C23—22no—
217C42×C23—22no—
218C42×C23—22no—
219C42×C23—21no—
220C42×C23—41no—
221C42×C23C2×D411yescharacteristic
222C42×C23C2×D411yescharacteristic
223C42×C23C2×D411yescharacteristic
224C42×C23C2411yescharacteristic
225#19047C2311yesmaximal, characteristic, direct product factor, Sylow
226C23×C23—21no—
227C23×C23#12011yescharacteristic
228Q8×C23—41no—
229Q8×C23—42no—
230Q8×C23—41no—
231Q8×C23—41no—
232Q8×C23—42no—
233Q8×C23—81no—
234Q8×C23—82no—
235D4×C23—41no—
236D4×C23—42no—
237C2×C4×C23—22no—
238C2×C4×C23—22no—
239C2×C4×C23—21no—
240C2×C4×C23—21no—
241C2×C4×C23—22no—
242C2×C4×C23—21no—
243C2×C4×C23—22no—
244C2×C4×C23—22no—
245C2×C4×C23—41no—
246C2×C4×C23—41no—
247C2×C4×C23—42no—
248C2×C4×C23—42no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—81no—
252C2×C4×C23—82no—
253C2×C4×C23#12011yescharacteristic
254C2×C4×C23#12011yescharacteristic
255C2×C4×C23#12011yescharacteristic
256C2×C4×C23C22×D411yescharacteristic
257C2×C4×C23C22×D411yescharacteristic
258C2×C4×C23C22×D411yescharacteristic
259C8×C23—24no—
260C8×C23—22no—
261C8×C23—42no—
262C8×C23—41no—
263#3563C2×C2311yescharacteristic
264#3236C2×C2311yescharacteristic, complemented
265#3181C2×C2311yescharacteristic
266#3162C2×C2311yescharacteristic, complemented
267#1664C2×C2312yescomplemented
268#1663C2×C2312yescomplemented
269#1663C2×C2312yescomplemented
270#1660C2×C2312yescomplemented
271#1532C2×C2311yescharacteristic
272#1531C2×C2311yescharacteristic, complemented
273C2×#330C2×C2311yescharacteristic, complemented
274C22×C23—21no—
275C22×C23—21no—
276C22×C23—41no—
277C22×C23—41no—
278C22×C23—41no—
279C22×C23#45612yes—
280C22×C23C2×#12011yescharacteristic
281C4×C23—21no—
282C4×C23—21no—
283C4×C23—21no—
284C4×C23—21no—
285C4×C23—21no—
286C4×C23—21no—
287C4×C23—42no—
288C4×C23—41no—
289C4×C23—42no—
290C4×C23—42no—
291C4×C23—41no—
292C4×C23—41no—
293C4×C23—42no—
294C4×C23—81no—
295C4×C23—81no—
296#564C22×C2312yes—
297#563C22×C2312yes—
298#557—21no—
299#554—21no—
300#552—21no—
301#552—22no—
302#552—21no—
303#544—21no—
304#544—21no—
305#541—21no—
306#541—22no—
307#539—21no—
308#533C22×C2311yescharacteristic
309#533C22×C2311yescharacteristic
310#529C22×C2311yescharacteristic
311C2×#125C22×C2311yescharacteristic
312#501—21no—
313#501—21no—
314#501C22×C2312yes—
315#501C22×C2312yes—
316#501C22×C2311yescharacteristic
317#501C22×C2312yes—
318#497C22×C2311yescharacteristic
319#491C22×C2311yescharacteristic
320#491C22×C2312yes—
321#489C22×C2311yescharacteristic
322#484—21no—
323#483—21no—
324#479—22no—
325#467—21no—
326#465—21no—
327#441—21no—
328#440—21no—
329#432C22×C2311yescharacteristic
330#427—22no—
331C2×#97C22×C2311yescharacteristic
332C2×#97C22×C2311yescharacteristic
333#330C22×C2312yes—
334#330C22×C2312yes—
335#330C22×C2312yes—
336#330C22×C2312yes—
337#330C22×C2312yes—
338#329C22×C2312yes—
339C2×C23—22no—
340C2×C23—81no—
341C2×C23C2×#45611yescharacteristic
342C2×C23#301811yescharacteristic
343C2×C23#302011yescharacteristic
344C2×Q16—21no—
345C2×SD16—21no—
346C22×C8—21no—
347C22×C8—21no—
348C4⋊Q8—22no—
349C4⋊Q8—21no—
350C4⋊Q8—22no—
351C4⋊Q8D4×C2311yescharacteristic
352C4⋊Q8D4×C2311yescharacteristic
353C4⋊Q8C23×C2311yescharacteristic
354C4⋊Q8C23×C2311yescharacteristic
355#126—22no—
356#126—22no—
357#126—22no—
358#126—22no—
359#126—21no—
360#125—22no—
361#125—22no—
362#125—21no—
363#125—22no—
364#125—22no—
365#125—22no—
366#125—22no—
367#125—22no—
368#125—22no—
369#125—21no—
370#125—21no—
371#125—21no—
372#125D4×C2312yes—
373#125C23×C2312yes—
374#125C23×C2311yescharacteristic
375#125C23×C2312yes—
376#124C23×C2311yescharacteristic
377#123—22no—
378#123—22no—
379#123—21no—
380C22⋊Q8—22no—
381C22⋊Q8—21no—
382C22⋊Q8—21no—
383C22⋊Q8—22no—
384C22⋊Q8—21no—
385C22⋊Q8—21no—
386C22⋊Q8—22no—
387C22⋊Q8—22no—
388C22⋊Q8—21no—
389C22⋊Q8—21no—
390C22⋊Q8D4×C2311yescharacteristic
391#121D4×C2311yescharacteristic
392C4×Q8—21no—
393C4×Q8—22no—
394C4×Q8—21no—
395C4×Q8—21no—
396C4×Q8—21no—
397C4×Q8—21no—
398C4×Q8—21no—
399C4×Q8D4×C2311yescharacteristic
400C4×D4—22no—
401C4×D4—21no—
402C4×D4D4×C2311yescharacteristic
403#117—22no—
404#117—21no—
405#117—22no—
406#117—21no—
407#117—21no—
408#117D4×C2312yes—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4D4×C2311yescharacteristic
413C2×C4⋊C4D4×C2311yescharacteristic
414C2×C42C23×C2311yescharacteristic
415#107—22no—
416#107—22no—
417#107—21no—
418#107—22no—
419#106—22no—
420#106—22no—
421#106—21no—
422#106—22no—
423C4⋊C8—21no—
424#103—21no—
425#103—21no—
426#103—21no—
427#103—22no—
428#103—22no—
429#102—21no—
430C22⋊C8—21no—
431#97—21no—
432#97—21no—
433#97C23×C2312yes—
434#97C23×C2312yes—
435#97C23×C2311yescharacteristic
436#97C23×C2311yescharacteristic
437C23#1904711yescharacteristic, direct product factor, Sylow
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8C2×D4×C2311yescharacteristic
444C2×D4—21no—
445C2×D4C2×D4×C2311yescharacteristic
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4C2×D4×C2311yescharacteristic
451C22×C4C2×D4×C2311yescharacteristic
452C22×C4C2×D4×C2311yescharacteristic
453Q16—42no—
454SD16—42no—
455C2×C8—24no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—42no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4C2×D4×C2312yes—
490C4⋊C4C2×D4×C2311yescharacteristic
491C4⋊C4C2×D4×C2312yes—
492C4⋊C4C2×D4×C2311yescharacteristic
493C4⋊C4C2×D4×C2312yes—
494C4⋊C4C2×D4×C2312yes—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—41no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—21no—
510C42—41no—
511C42C2×D4×C2311yescharacteristic
512C42C2×D4×C2311yescharacteristic
513C42C2×D4×C2311yescharacteristic
514C42C24×C2311yescharacteristic
515C23—21no—
516C23C23×#12011yescharacteristic
517Q8—41no—
518Q8—42no—
519Q8—41no—
520Q8—41no—
521Q8—42no—
522Q8—81no—
523Q8—82no—
524D4—41no—
525D4—42no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—81no—
541C2×C4—82no—
542C2×C4C23×#12011yescharacteristic
543C2×C4C23×#12011yescharacteristic
544C2×C4C23×#12011yescharacteristic
545C2×C4C22×D4×C2311yescharacteristic
546C2×C4C22×D4×C2311yescharacteristic
547C2×C4C22×D4×C2311yescharacteristic
548C8—24no—
549C8—22no—
550C8—41no—
551C8—42no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C23×#45612yes—
558C22C2×C23×#12011yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—42no—
566C4—41no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—81no—
573C4—81no—
574C2—22no—
575C2—81no—
576C2C2×C23×#45611yescharacteristic
577C2C23×#301811yescharacteristic
578C2C23×#302011yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3236C2semidirect
4. C23×#3162C2semidirect
5. C23×#1664C2semidirect
6. C23×#1663C2semidirect
7. C23×#1663C2semidirect
8. C23×#1660C2semidirect
10. C23×#1531C2semidirect
11. C2×C23×#330C2semidirect
225. #19047C23direct (N × H)
264. #3236C2×C23semidirect
266. #3162C2×C23semidirect
267. #1664C2×C23semidirect
268. #1663C2×C23semidirect
269. #1663C2×C23semidirect
270. #1660C2×C23semidirect
272. #1531C2×C23semidirect
273. C2×#330C2×C23semidirect
437. C23#19047direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35632
C23×#32362
C23×#31812
C23×#31622
C23×#16642
C23×#16632
C23×#16602
C23×#15322
C23×#15312
C2×C23×#3302
#1904723