Finite Group Databasebeta

← C23×#19033 #19166659

Subgroups — C23×#19033 #19166659

Order 5888.

948 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3561C211yesmaximal, characteristic
2C23×#3239C211yesmaximal, characteristic, complemented
3C23×#3165C211yesmaximal, characteristic, complemented
4C2×C23×#497C211yesmaximal, characteristic, complemented
5C23×#1661C214yesmaximal, complemented
6C23×#1660C214yesmaximal, complemented
7C23×#1531C211yesmaximal, characteristic, complemented
8C23×#1527C211yesmaximal, characteristic, complemented
9C23×#1524C211yesmaximal, characteristic, complemented
10C22×Q16×C23—21no—
11C22×SD16×C23—21no—
12C23×#563C2212yes—
13C23×#562C2212yes—
14C23×#556—21no—
15C23×#551—22no—
16D4×Q8×C23—21no—
17C23×#548—21no—
18C23×#546—21no—
19C23×#543—21no—
20C23×#542—21no—
21C23×#542—22no—
22C23×#538—21no—
23C23×#536—21no—
24C23×#533C2211yescharacteristic
25C2×C23×C4⋊Q8C2211yescharacteristic
26C23×#529C2211yescharacteristic
27C2×C23×#124C2211yescharacteristic
28C23×#501C2214yescomplemented
29C23×#501C2212yescomplemented
30C23×#500—21no—
31C23×#498—21no—
32C23×#497—22no—
33C23×#497C2214yes—
34C23×#489C2212yescomplemented
35C23×#484—21no—
36C23×#481—21no—
37C23×#475—21no—
38C23×#474—21no—
39C23×#432C2211yescharacteristic
40C23×#428—22no—
41C4×M16×C23C2211yescharacteristic
42C2×C23×#97C2211yescharacteristic
43C23×#330C2212yescomplemented
44C23×#330C2214yescomplemented
45C23×#329C2214yes—
46C23×#329C2212yescomplemented
47C2×#41×C23—21no—
48C22×Q8×C23—21no—
49C22×Q8×C23—21no—
50C22×Q8×C23D411yescharacteristic
51C22×D4×C23D411yescharacteristic
52C2×Q16×C23—24no—
53C2×Q16×C23—44no—
54C2×SD16×C23—24no—
55C2×SD16×C23—44no—
56C2×M16×C23—21no—
57C22×C8×C23—21no—
58C23×C4⋊Q8—21no—
59C23×C4⋊Q8—22no—
60C23×C4⋊Q8—21no—
61C23×C4⋊Q8—21no—
62C23×C4⋊Q8—21no—
63C23×C4⋊Q8D411yescharacteristic
64C23×C4⋊Q8D411yescharacteristic
65C23×C4⋊Q8C2312yes—
66C23×#126—22no—
67C23×#126—22no—
68C23×#126—22no—
69C23×#126—21no—
70C23×#125C2312yes—
71C23×#125C2311yescharacteristic
72C23×#124—22no—
73C23×#124—21no—
74C23×#124—22no—
75C23×#124—21no—
76C23×#124—21no—
77C23×#124D412yes—
78C23×#124C2311yescharacteristic
79C23×#124C2312yes—
80C23×#123—22no—
81C23×#123—22no—
82C23×#123—21no—
83C23×#123—22no—
84C23×#123—22no—
85C23×C22⋊Q8—21no—
86C23×C22⋊Q8—22no—
87C23×C22⋊Q8—22no—
88C23×C22⋊Q8—22no—
89C23×C22⋊Q8—22no—
90C23×C22⋊Q8—22no—
91C23×C22⋊Q8—21no—
92C23×C22⋊Q8—21no—
93C23×C22⋊Q8—22no—
94C23×C22⋊Q8—22no—
95C23×C22⋊Q8D411yescharacteristic
96C23×#121—21no—
97C23×#121—22no—
98C23×#121—22no—
99C23×#121D411yescharacteristic
100C23×#120—22no—
101C4×Q8×C23—21no—
102C4×Q8×C23—21no—
103C4×Q8×C23—21no—
104C4×Q8×C23D411yescharacteristic
105C4×D4×C23—22no—
106C4×D4×C23—22no—
107C4×D4×C23—21no—
108C4×D4×C23—21no—
109C4×D4×C23D411yescharacteristic
110C23×#117—22no—
111C23×#117—21no—
112C23×#117D412yes—
113C2×C4⋊C4×C23—21no—
114C2×C4⋊C4×C23—21no—
115C2×C22⋊C4×C23—22no—
116C2×C22⋊C4×C23—21no—
117C2×C22⋊C4×C23—21no—
118C2×C42×C23C2311yescharacteristic
119C23×#107—24no—
120C23×#107—22no—
121C23×#106—24no—
122C23×#106—22no—
123C23×C4⋊C8—21no—
124C23×#103—22no—
125C23×#102—22no—
126C23×C22⋊C8—21no—
127C23×#97—21no—
128C23×#97C2312yes—
129C23×#97C2312yes—
130C23×#97C2312yes—
131C4×C8×C23—21no—
132C24×C23—21no—
133#41×C23—82no—
134C2×Q8×C23—22no—
135C2×Q8×C23—21no—
136C2×Q8×C23—21no—
137C2×Q8×C23—21no—
138C2×Q8×C23—22no—
139C2×Q8×C23—42no—
140C2×Q8×C23—42no—
141C2×Q8×C23—42no—
142C2×Q8×C23—41no—
143C2×Q8×C23—82no—
144C2×Q8×C23C2×D411yescharacteristic
145C2×Q8×C23C2×D412yes—
146C2×D4×C23—21no—
147C2×D4×C23—22no—
148C2×D4×C23—22no—
149C2×D4×C23—21no—
150C2×D4×C23—42no—
151C2×D4×C23C2×D411yescharacteristic
152C2×D4×C23C2×D412yes—
153C22×C4×C23—22no—
154C22×C4×C23—22no—
155C22×C4×C23—22no—
156C22×C4×C23—21no—
157C22×C4×C23—21no—
158C22×C4×C23—41no—
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162Q16×C23—48no—
163SD16×C23—48no—
164M16×C23—24no—
165C2×C8×C23—24no—
166C2×C8×C23—22no—
167C2×C8×C23—22no—
168C2×C8×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—22no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—21no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—22no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—41no—
184C4⋊C4×C23—41no—
185C4⋊C4×C23C2×D412yes—
186C4⋊C4×C23C2×D411yescharacteristic
187C4⋊C4×C23C2×D412yes—
188C4⋊C4×C23C2×D411yescharacteristic
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—21no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—22no—
198C22⋊C4×C23—21no—
199C22⋊C4×C23—21no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—21no—
203C22⋊C4×C23—21no—
204C42×C23—22no—
205C42×C23—22no—
206C42×C23—21no—
207C42×C23C2×D411yescharacteristic
208C42×C23C2×D411yescharacteristic
209C42×C23C2×D411yescharacteristic
210C42×C23C2411yescharacteristic
211#19033C2311yesmaximal, characteristic, direct product factor, Sylow
212C23×C23—22no—
213C23×C23—21no—
214C23×C23—42no—
215C23×C23—81no—
216C23×C23—81no—
217C23×C23#12011yescharacteristic
218Q8×C23—42no—
219Q8×C23—41no—
220Q8×C23—42no—
221Q8×C23—42no—
222Q8×C23—42no—
223Q8×C23—42no—
224Q8×C23—81no—
225Q8×C23—82no—
226Q8×C23—82no—
227D4×C23—42no—
228D4×C23—41no—
229D4×C23—42no—
230D4×C23—42no—
231D4×C23—82no—
232D4×C23—82no—
233C2×C4×C23—22no—
234C2×C4×C23—22no—
235C2×C4×C23—21no—
236C2×C4×C23—22no—
237C2×C4×C23—21no—
238C2×C4×C23—22no—
239C2×C4×C23—22no—
240C2×C4×C23—21no—
241C2×C4×C23—41no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—81no—
245C2×C4×C23—82no—
246C2×C4×C23—82no—
247C2×C4×C23—82no—
248C2×C4×C23—82no—
249C2×C4×C23#12011yescharacteristic
250C2×C4×C23#12011yescharacteristic
251C2×C4×C23#12011yescharacteristic
252C2×C4×C23C22×D411yescharacteristic
253C2×C4×C23C22×D411yescharacteristic
254C2×C4×C23C22×D411yescharacteristic
255C8×C23—24no—
256C8×C23—24no—
257C8×C23—42no—
258#3561C2×C2311yescharacteristic
259#3239C2×C2311yescharacteristic, complemented
260#3165C2×C2311yescharacteristic, complemented
261C2×#497C2×C2311yescharacteristic, complemented
262#1661C2×C2314yescomplemented
263#1660C2×C2314yescomplemented
264#1531C2×C2311yescharacteristic, complemented
265#1527C2×C2311yescharacteristic, complemented
266#1524C2×C2311yescharacteristic, complemented
267C22×C23—21no—
268C22×C23—21no—
269C22×C23—41no—
270C22×C23—42no—
271C22×C23—41no—
272C22×C23—42no—
273C22×C23—42no—
274C22×C23—41no—
275C22×C23—162no—
276C22×C23#45612yes—
277C22×C23C2×#12011yescharacteristic
278C4×C23—21no—
279C4×C23—21no—
280C4×C23—21no—
281C4×C23—21no—
282C4×C23—21no—
283C4×C23—41no—
284C4×C23—42no—
285C4×C23—42no—
286C4×C23—41no—
287C4×C23—41no—
288C4×C23—42no—
289C4×C23—82no—
290C4×C23#45312yes—
291C22×Q16—21no—
292C22×SD16—21no—
293#563C22×C2312yes—
294#562C22×C2312yes—
295#556—21no—
296#551—22no—
297D4×Q8—21no—
298#548—21no—
299#546—21no—
300#543—21no—
301#542—21no—
302#542—22no—
303#538—21no—
304#536—21no—
305#533C22×C2311yescharacteristic
306C2×C4⋊Q8C22×C2311yescharacteristic
307#529C22×C2311yescharacteristic
308C2×#124C22×C2311yescharacteristic
309#501C22×C2314yescomplemented
310#501C22×C2312yescomplemented
311#500—21no—
312#498—21no—
313#497—22no—
314#497C22×C2314yes—
315#489C22×C2312yescomplemented
316#484—21no—
317#481—21no—
318#475—21no—
319#474—21no—
320#432C22×C2311yescharacteristic
321#428—22no—
322C4×M16C22×C2311yescharacteristic
323C2×#97C22×C2311yescharacteristic
324#330C22×C2312yescomplemented
325#330C22×C2314yescomplemented
326#329C22×C2314yes—
327#329C22×C2312yescomplemented
328C2×C23—22no—
329C2×C23—81no—
330C2×C23—82no—
331C2×C23C2×#45311yescharacteristic
332C2×C23C2×#45611yescharacteristic
333C2×C23#302211yescharacteristic
334C2×#41—21no—
335C22×Q8—21no—
336C22×Q8—21no—
337C22×Q8D4×C2311yescharacteristic
338C22×D4D4×C2311yescharacteristic
339C2×Q16—24no—
340C2×Q16—44no—
341C2×SD16—24no—
342C2×SD16—44no—
343C2×M16—21no—
344C22×C8—21no—
345C4⋊Q8—21no—
346C4⋊Q8—22no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8D4×C2311yescharacteristic
351C4⋊Q8D4×C2311yescharacteristic
352C4⋊Q8C23×C2312yes—
353#126—22no—
354#126—22no—
355#126—22no—
356#126—21no—
357#125C23×C2311yescharacteristic
358#125C23×C2312yes—
359#124—21no—
360#124—22no—
361#124—22no—
362#124—21no—
363#124—21no—
364#124D4×C2312yes—
365#124C23×C2312yes—
366#124C23×C2311yescharacteristic
367#123—22no—
368#123—22no—
369#123—21no—
370#123—22no—
371#123—22no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—21no—
375C22⋊Q8—22no—
376C22⋊Q8—22no—
377C22⋊Q8—22no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8D4×C2311yescharacteristic
383#121—21no—
384#121—22no—
385#121—22no—
386#121D4×C2311yescharacteristic
387#120—22no—
388C4×Q8—21no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8D4×C2311yescharacteristic
392C4×D4—22no—
393C4×D4—21no—
394C4×D4—22no—
395C4×D4—21no—
396C4×D4D4×C2311yescharacteristic
397#117—22no—
398#117—21no—
399#117D4×C2312yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C22⋊C4—22no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C42C23×C2311yescharacteristic
406#107—24no—
407#107—22no—
408#106—24no—
409#106—22no—
410C4⋊C8—21no—
411#103—22no—
412#102—22no—
413C22⋊C8—21no—
414#97—21no—
415#97C23×C2312yes—
416#97C23×C2312yes—
417#97C23×C2312yes—
418C4×C8—21no—
419C23#1903311yescharacteristic, direct product factor, Sylow
420C24—21no—
421#41—82no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—22no—
427C2×Q8—42no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—41no—
431C2×Q8—82no—
432C2×Q8C2×D4×C2311yescharacteristic
433C2×Q8C2×D4×C2312yes—
434C2×D4—21no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4C2×D4×C2311yescharacteristic
440C2×D4C2×D4×C2312yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—41no—
447C22×C4C2×D4×C2311yescharacteristic
448C22×C4C2×D4×C2311yescharacteristic
449C22×C4C2×D4×C2311yescharacteristic
450Q16—48no—
451SD16—48no—
452M16—24no—
453C2×C8—24no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4C2×D4×C2312yes—
474C4⋊C4C2×D4×C2311yescharacteristic
475C4⋊C4C2×D4×C2312yes—
476C4⋊C4C2×D4×C2311yescharacteristic
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—21no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—21no—
491C22⋊C4—21no—
492C42—22no—
493C42—22no—
494C42—21no—
495C42C2×D4×C2311yescharacteristic
496C42C2×D4×C2311yescharacteristic
497C42C2×D4×C2311yescharacteristic
498C42C24×C2311yescharacteristic
499C23—22no—
500C23—21no—
501C23—42no—
502C23—81no—
503C23—81no—
504C23C23×#12011yescharacteristic
505Q8—42no—
506Q8—41no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—81no—
512Q8—82no—
513Q8—82no—
514D4—42no—
515D4—41no—
516D4—42no—
517D4—42no—
518D4—82no—
519D4—82no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4C23×#12011yescharacteristic
537C2×C4C23×#12011yescharacteristic
538C2×C4C23×#12011yescharacteristic
539C2×C4C22×D4×C2311yescharacteristic
540C2×C4C22×D4×C2311yescharacteristic
541C2×C4C22×D4×C2311yescharacteristic
542C8—24no—
543C8—24no—
544C8—42no—
545C22—21no—
546C22—21no—
547C22—41no—
548C22—42no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—162no—
554C22C23×#45612yes—
555C22C2×C23×#12011yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—41no—
562C4—42no—
563C4—42no—
564C4—41no—
565C4—41no—
566C4—42no—
567C4—82no—
568C4C23×#45312yes—
569C2—22no—
570C2—82no—
571C2—81no—
572C2C2×C23×#45311yescharacteristic
573C2C2×C23×#45611yescharacteristic
574C2C23×#302211yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC42C24×C23
prime radicalC23×#3561C2
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×#3239C2semidirect
3. C23×#3165C2semidirect
4. C2×C23×#497C2semidirect
5. C23×#1661C2semidirect
6. C23×#1660C2semidirect
7. C23×#1531C2semidirect
8. C23×#1527C2semidirect
9. C23×#1524C2semidirect
28. C23×#501C22semidirect
29. C23×#501C22semidirect
34. C23×#489C22semidirect
43. C23×#330C22semidirect
44. C23×#330C22semidirect
46. C23×#329C22semidirect
211. #19033C23direct (N × H)
259. #3239C2×C23semidirect
260. #3165C2×C23semidirect
261. C2×#497C2×C23semidirect
262. #1661C2×C23semidirect
263. #1660C2×C23semidirect
264. #1531C2×C23semidirect
265. #1527C2×C23semidirect
266. #1524C2×C23semidirect
309. #501C22×C23semidirect
310. #501C22×C23semidirect
315. #489C22×C23semidirect
324. #330C22×C23semidirect
325. #330C22×C23semidirect
327. #329C22×C23semidirect
419. C23#19033direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35612
C23×#32392
C23×#31652
C2×C23×#4972
C23×#16612
C23×#16602
C23×#15312
C23×#15272
C23×#15242
#1903323