Finite Group Databasebeta

← C23×#18320 #19165946

Subgroups — C23×#18320 #19165946

Order 5888.

1008 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
1C2×C23×#539C211yesmaximal, characteristic, complemented
2C23×#3193C211yesmaximal, characteristic, complemented
3C23×#3192C211yesmaximal, characteristic
4C23×#2591C211yesmaximal, characteristic, complemented
5C23×#1615C214yesmaximal, complemented
6C23×#1597C214yesmaximal, complemented
7C23×#1508C211yesmaximal, characteristic, complemented
8C23×#1507C211yesmaximal, characteristic, complemented
9C23×#1463C211yesmaximal, characteristic, complemented
10C23×#539—22no—
11C23×#539—22no—
12C23×#539—22no—
13C23×#539C2214yes—
14C2×C23×#126—21no—
15C2×C23×#125—21no—
16C2×C23×#123—21no—
17C2×C23×#123—21no—
18C2×C23×C22⋊Q8—21no—
19C2×C23×C22⋊Q8C2211yescharacteristic
20C2×C23×#121C2211yescharacteristic
21C2×C23×#121C2211yescharacteristic
22C2×C4×D4×C23—21no—
23C22×C4⋊C4×C23C2211yescharacteristic
24C23×#484C2212yescomplemented
25C23×#481—21no—
26C23×#481C2212yes—
27C23×#480—21no—
28C23×#469—22no—
29C23×#468—22no—
30C23×#450C2212yescomplemented
31C23×#448—21no—
32C23×#447—21no—
33C23×#447C2212yes—
34C23×#407C2211yescharacteristic
35C23×#407C2211yescharacteristic
36C23×#400C2212yescomplemented
37C23×#394C2212yescomplemented
38C23×#388—22no—
39C23×#386—21no—
40C23×#386—21no—
41C4×C22⋊C4×C23C2211yescharacteristic
42C23×#328C2214yescomplemented
43C23×#327C2214yes—
44C23×#324C2214yescomplemented
45C22×Q8×C23D411yescharacteristic
46C22×D4×C23—21no—
47C22×D4×C23—21no—
48C22×D4×C23D411yescharacteristic
49C22×D4×C23C2311yescharacteristic
50C23×C4×C23—21no—
51C23×C4×C23D411yescharacteristic
52C23×C4×C23D411yescharacteristic
53C23×C4×C23C2311yescharacteristic
54C2×SD16×C23—22no—
55C2×D8×C23—22no—
56C2×M16×C23—21no—
57C2×M16×C23—21no—
58C23×#126—24no—
59C23×#126—24no—
60C23×#125—24no—
61C23×#125—24no—
62C23×#123—24no—
63C23×#123—24no—
64C23×#123—24no—
65C23×#123—24no—
66C23×C22⋊Q8—24no—
67C23×C22⋊Q8—24no—
68C23×C22⋊Q8—21no—
69C23×C22⋊Q8—21no—
70C23×C22⋊Q8D412yes—
71C23×C22⋊Q8C2312yes—
72C23×#121—21no—
73C23×#121—21no—
74C23×#121—21no—
75C23×#121—21no—
76C23×#121—21no—
77C23×#121—21no—
78C23×#121D412yes—
79C23×#121C2312yes—
80C4×D4×C23—24no—
81C4×D4×C23—24no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23—21no—
84C2×C4⋊C4×C23—21no—
85C2×C4⋊C4×C23—21no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—21no—
89C2×C4⋊C4×C23—21no—
90C2×C4⋊C4×C23D412yes—
91C2×C4⋊C4×C23D411yescharacteristic
92C2×C4⋊C4×C23D411yescharacteristic
93C2×C4⋊C4×C23C2311yescharacteristic
94C2×C4⋊C4×C23C2312yes—
95C2×C22⋊C4×C23—21no—
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C22⋊C4×C23—21no—
99C2×C22⋊C4×C23—21no—
100C2×C22⋊C4×C23—21no—
101C2×C22⋊C4×C23—22no—
102C2×C22⋊C4×C23—21no—
103C2×C22⋊C4×C23—22no—
104C2×C42×C23—21no—
105C2×C42×C23—21no—
106C23×#107—22no—
107C23×#106—22no—
108C23×#103—22no—
109C23×#102—22no—
110C23×#102—22no—
111C23×#102—22no—
112C23×C22⋊C8—21no—
113C23×C22⋊C8—21no—
114C23×C22⋊C8C2312yes—
115C23×C22⋊C8C2312yes—
116C23×#95—22no—
117C23×#95—22no—
118C23×#95C2312yes—
119C24×C23—21no—
120C24×C23—21no—
121C24×C23C2×D411yescharacteristic
122C2×Q8×C23—21no—
123C2×Q8×C23—81no—
124C2×Q8×C23C2×D412yes—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—21no—
128C2×D4×C23—21no—
129C2×D4×C23—21no—
130C2×D4×C23—44no—
131C2×D4×C23—44no—
132C2×D4×C23—41no—
133C2×D4×C23—41no—
134C2×D4×C23—81no—
135C2×D4×C23—81no—
136C2×D4×C23C2×D412yes—
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—24no—
144C22×C4×C23—21no—
145C22×C4×C23—22no—
146C22×C4×C23—21no—
147C22×C4×C23—42no—
148C22×C4×C23—81no—
149C22×C4×C23—81no—
150C22×C4×C23—82no—
151C22×C4×C23C2×D412yes—
152C22×C4×C23C2×D412yes—
153C22×C4×C23C2×D411yescharacteristic
154C22×C4×C23C2×D411yescharacteristic
155C22×C4×C23C2×D411yescharacteristic
156C22×C4×C23C2×D411yescharacteristic
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2411yescharacteristic
159SD16×C23—82no—
160D8×C23—82no—
161M16×C23—81no—
162M16×C23—81no—
163C2×C8×C23—22no—
164C2×C8×C23—22no—
165C4⋊C4×C23—24no—
166C4⋊C4×C23—24no—
167C4⋊C4×C23—24no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—22no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
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—44no—
188C22⋊C4×C23—42no—
189C22⋊C4×C23—41no—
190C22⋊C4×C23—41no—
191C22⋊C4×C23—82no—
192C42×C23—24no—
193C42×C23—81no—
194#18320C2311yesmaximal, characteristic, direct product factor, Sylow
195C23×C23—22no—
196C23×C23—22no—
197C23×C23—22no—
198C23×C23—41no—
199C23×C23—42no—
200C23×C23—42no—
201C23×C23—42no—
202C23×C23—42no—
203C23×C23—42no—
204C23×C23—41no—
205C23×C23—42no—
206C23×C23#12011yescharacteristic
207C23×C23#12011yescharacteristic
208C23×C23C22×D411yescharacteristic
209Q8×C23—42no—
210Q8×C23—81no—
211D4×C23—42no—
212D4×C23—42no—
213D4×C23—42no—
214D4×C23—44no—
215D4×C23—44no—
216D4×C23—81no—
217D4×C23—82no—
218D4×C23—82no—
219C2×C4×C23—22no—
220C2×C4×C23—22no—
221C2×C4×C23—22no—
222C2×C4×C23—22no—
223C2×C4×C23—22no—
224C2×C4×C23—22no—
225C2×C4×C23—22no—
226C2×C4×C23—41no—
227C2×C4×C23—41no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—41no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—42no—
237C2×C4×C23—42no—
238C2×C4×C23—42no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—42no—
242C2×C4×C23—41no—
243C2×C4×C23—42no—
244C2×C4×C23—81no—
245C2×C4×C23—84no—
246C2×C4×C23#12011yescharacteristic
247C2×C4×C23#12011yescharacteristic
248C2×C4×C23C22×D411yescharacteristic
249C2×C4×C23C22×D411yescharacteristic
250C8×C23—42no—
251C8×C23—42no—
252C2×#539C2×C2311yescharacteristic, complemented
253#3193C2×C2311yescharacteristic, complemented
254#3192C2×C2311yescharacteristic
255#2591C2×C2311yescharacteristic, complemented
256#1615C2×C2314yescomplemented
257#1597C2×C2314yescomplemented
258#1508C2×C2311yescharacteristic, complemented
259#1507C2×C2311yescharacteristic, complemented
260#1463C2×C2311yescharacteristic, complemented
261C22×C23—21no—
262C22×C23—21no—
263C22×C23—21no—
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—21no—
267C22×C23—21no—
268C22×C23—44no—
269C22×C23—42no—
270C22×C23—42no—
271C22×C23—42no—
272C22×C23—84no—
273C22×C23—84no—
274C22×C23—81no—
275C22×C23—81no—
276C22×C23—81no—
277C22×C23#45412yes—
278C22×C23C2×#12011yescharacteristic
279C22×C23#57512yes—
280C4×C23—21no—
281C4×C23—21no—
282C4×C23—21no—
283C4×C23—21no—
284C4×C23—42no—
285C4×C23—42no—
286C4×C23—42no—
287C4×C23—42no—
288C4×C23—42no—
289C4×C23—82no—
290C4×C23—81no—
291#539—22no—
292#539—22no—
293#539—22no—
294#539C22×C2314yes—
295C2×#126—21no—
296C2×#125—21no—
297C2×#123—21no—
298C2×#123—21no—
299C2×C22⋊Q8—21no—
300C2×C22⋊Q8C22×C2311yescharacteristic
301C2×#121C22×C2311yescharacteristic
302C2×#121C22×C2311yescharacteristic
303C2×C4×D4—21no—
304C22×C4⋊C4C22×C2311yescharacteristic
305#484C22×C2312yescomplemented
306#481—21no—
307#481C22×C2312yes—
308#480—21no—
309#469—22no—
310#468—22no—
311#450C22×C2312yescomplemented
312#448—21no—
313#447—21no—
314#447C22×C2312yes—
315#407C22×C2311yescharacteristic
316#407C22×C2311yescharacteristic
317#400C22×C2312yescomplemented
318#394C22×C2312yescomplemented
319#388—22no—
320#386—21no—
321#386—21no—
322C4×C22⋊C4C22×C2311yescharacteristic
323#328C22×C2314yescomplemented
324#327C22×C2314yes—
325#324C22×C2314yescomplemented
326C2×C23—22no—
327C2×C23—22no—
328C2×C23—41no—
329C2×C23—82no—
330C2×C23—161no—
331C2×C23#300511yescharacteristic
332C2×C23C2×#45411yescharacteristic
333C2×C23#302611yescharacteristic
334C22×Q8D4×C2311yescharacteristic
335C22×D4—21no—
336C22×D4—21no—
337C22×D4D4×C2311yescharacteristic
338C22×D4C23×C2311yescharacteristic
339C23×C4—21no—
340C23×C4D4×C2311yescharacteristic
341C23×C4D4×C2311yescharacteristic
342C23×C4C23×C2311yescharacteristic
343C2×SD16—22no—
344C2×D8—22no—
345C2×M16—21no—
346C2×M16—21no—
347#126—24no—
348#126—24no—
349#125—24no—
350#125—24no—
351#123—24no—
352#123—24no—
353#123—24no—
354#123—24no—
355C22⋊Q8—24no—
356C22⋊Q8—24no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8D4×C2312yes—
360C22⋊Q8C23×C2312yes—
361#121—21no—
362#121—21no—
363#121—21no—
364#121—21no—
365#121—21no—
366#121—21no—
367#121D4×C2312yes—
368#121C23×C2312yes—
369C4×D4—24no—
370C4×D4—24no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4D4×C2312yes—
380C2×C4⋊C4D4×C2311yescharacteristic
381C2×C4⋊C4D4×C2311yescharacteristic
382C2×C4⋊C4C23×C2311yescharacteristic
383C2×C4⋊C4C23×C2312yes—
384C2×C22⋊C4—21no—
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—22no—
392C2×C22⋊C4—22no—
393C2×C42—21no—
394C2×C42—21no—
395#107—22no—
396#106—22no—
397#103—22no—
398#102—22no—
399#102—22no—
400#102—22no—
401C22⋊C8—21no—
402C22⋊C8—21no—
403C22⋊C8C23×C2312yes—
404C22⋊C8C23×C2312yes—
405#95—22no—
406#95—22no—
407#95C23×C2312yes—
408C23#1832011yescharacteristic, direct product factor, Sylow
409C24—21no—
410C24—21no—
411C24C2×D4×C2311yescharacteristic
412C2×Q8—21no—
413C2×Q8—81no—
414C2×Q8C2×D4×C2312yes—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—81no—
425C2×D4—81no—
426C2×D4C2×D4×C2312yes—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—42no—
438C22×C4—81no—
439C22×C4—81no—
440C22×C4—82no—
441C22×C4C2×D4×C2312yes—
442C22×C4C2×D4×C2312yes—
443C22×C4C2×D4×C2311yescharacteristic
444C22×C4C2×D4×C2311yescharacteristic
445C22×C4C2×D4×C2311yescharacteristic
446C22×C4C2×D4×C2311yescharacteristic
447C22×C4C2×D4×C2311yescharacteristic
448C22×C4C24×C2311yescharacteristic
449SD16—82no—
450D8—82no—
451M16—81no—
452M16—81no—
453C2×C8—22no—
454C2×C8—22no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4C2×D4×C2312yes—
469C4⋊C4C2×D4×C2312yes—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—44no—
478C22⋊C4—42no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—82no—
482C42—24no—
483C42—81no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—41no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—41no—
495C23C23×#12011yescharacteristic
496C23C23×#12011yescharacteristic
497C23C22×D4×C2311yescharacteristic
498Q8—42no—
499Q8—81no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—44no—
504D4—44no—
505D4—81no—
506D4—82no—
507D4—82no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—81no—
534C2×C4—84no—
535C2×C4C23×#12011yescharacteristic
536C2×C4C23×#12011yescharacteristic
537C2×C4C22×D4×C2311yescharacteristic
538C2×C4C22×D4×C2311yescharacteristic
539C8—42no—
540C8—42no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—44no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—84no—
553C22—84no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22C23×#45412yes—
558C22C2×C23×#12011yescharacteristic
559C22C23×#57512yes—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—82no—
570C4—81no—
571C2—22no—
572C2—22no—
573C2—41no—
574C2—82no—
575C2—161no—
576C2C23×#300511yescharacteristic
577C2C2×C23×#45411yescharacteristic
578C2C23×#302611yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3192C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#539C2semidirect
2. C23×#3193C2semidirect
4. C23×#2591C2semidirect
5. C23×#1615C2semidirect
6. C23×#1597C2semidirect
7. C23×#1508C2semidirect
8. C23×#1507C2semidirect
9. C23×#1463C2semidirect
24. C23×#484C22semidirect
30. C23×#450C22semidirect
36. C23×#400C22semidirect
37. C23×#394C22semidirect
42. C23×#328C22semidirect
44. C23×#324C22semidirect
194. #18320C23direct (N × H)
252. C2×#539C2×C23semidirect
253. #3193C2×C23semidirect
255. #2591C2×C23semidirect
256. #1615C2×C23semidirect
257. #1597C2×C23semidirect
258. #1508C2×C23semidirect
259. #1507C2×C23semidirect
260. #1463C2×C23semidirect
305. #484C22×C23semidirect
311. #450C22×C23semidirect
317. #400C22×C23semidirect
318. #394C22×C23semidirect
323. #328C22×C23semidirect
325. #324C22×C23semidirect
408. C23#18320direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5392
C23×#31932
C23×#31922
C23×#25912
C23×#16152
C23×#15972
C23×#15082
C23×#15072
C23×#14632
#1832023