Finite Group Databasebeta

← C23×#25519 #19173145

Subgroups — C23×#25519 #19173145

Order 5888.

1092 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic
2C23×#3056C212yesmaximal, complemented
3C23×#2502C211yesmaximal, characteristic, complemented
4C23×#2060C211yesmaximal, characteristic, complemented
5C23×#2039C211yesmaximal, characteristic, complemented
6C23×#2028C211yesmaximal, characteristic, complemented
7C23×#2007C211yesmaximal, characteristic, complemented
8C23×#1882C212yesmaximal, complemented
9C23×#1871C212yesmaximal, complemented
10C23×#1859C212yesmaximal, complemented
11C2×C23×#343C211yesmaximal, characteristic, complemented
12C2×C23×#137C2212yes—
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×C22⋊Q8C2211yescharacteristic
24C23×#520C2212yes—
25C22×C4⋊C4×C23C2211yescharacteristic
26C23×#470—22no—
27C23×#468—22no—
28C23×#467—22no—
29C23×#465—22no—
30C23×#428C2212yes—
31C2×C23×#104C2212yes—
32C2×C23×#104C2212yes—
33C23×#419C2212yes—
34C23×#417C2212yes—
35C23×#398—22no—
36C23×#391C2211yescharacteristic
37C23×#391C2211yescharacteristic
38C23×#385—22no—
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×Q8×C23C2311yescharacteristic
50C23×C4×C23D412yes—
51C23×C4×C23C2311yescharacteristic
52C23×#137—24no—
53C23×#137—42no—
54C2×Q16×C23—22no—
55C2×SD16×C23—22no—
56C2×M16×C23—22no—
57C2×M16×C23—24no—
58C2×M16×C23—42no—
59C2×M16×C23C2312yes—
60C2×M16×C23C2312yes—
61C22×C8×C23—22no—
62C23×C4⋊Q8—22no—
63C23×#126—24no—
64C23×#125—24no—
65C23×#125—22no—
66C23×#123—24no—
67C23×#123—24no—
68C23×#123—22no—
69C23×C22⋊Q8—24no—
70C23×C22⋊Q8—22no—
71C23×C22⋊Q8—22no—
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×#121D412yes—
81C23×#121D412yes—
82C23×#120—22no—
83C4×Q8×C23—22no—
84C4×D4×C23—24no—
85C4×D4×C23—22no—
86C4×D4×C23—22no—
87C4×D4×C23—22no—
88C4×D4×C23—24no—
89C23×#117—22no—
90C23×#117—22no—
91C23×#117C2312yes—
92C2×C4⋊C4×C23—22no—
93C2×C4⋊C4×C23—22no—
94C2×C4⋊C4×C23—22no—
95C2×C4⋊C4×C23—21no—
96C2×C4⋊C4×C23—22no—
97C2×C4⋊C4×C23—22no—
98C2×C4⋊C4×C23D412yes—
99C2×C4⋊C4×C23C2311yescharacteristic
100C2×C4⋊C4×C23C2311yescharacteristic
101C2×C4⋊C4×C23C2311yescharacteristic
102C2×C22⋊C4×C23—21no—
103C2×C22⋊C4×C23—21no—
104C2×C42×C23C2311yescharacteristic
105C2×C42×C23C2311yescharacteristic
106C23×#107—22no—
107C23×#106—22no—
108C23×#104—24no—
109C23×#104—42no—
110C23×#103—22no—
111C23×#103—22no—
112C23×#103—22no—
113C23×#102—22no—
114C23×#95—21no—
115C23×#95—22no—
116C23×#95—22no—
117C24×C23C2×D411yescharacteristic
118#41×C23—22no—
119#41×C23—22no—
120#41×C23—22no—
121#41×C23—22no—
122#41×C23—84no—
123C2×Q8×C23—22no—
124C2×Q8×C23—22no—
125C2×Q8×C23—22no—
126C2×Q8×C23—21no—
127C2×Q8×C23—21no—
128C2×Q8×C23C2×D412yes—
129C2×D4×C23—22no—
130C2×D4×C23—22no—
131C2×D4×C23—22no—
132C2×D4×C23—22no—
133C2×D4×C23C2×D412yes—
134C22×C4×C23—22no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—22no—
138C22×C4×C23—21no—
139C22×C4×C23—21no—
140C22×C4×C23—24no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—21no—
144C22×C4×C23—21no—
145C22×C4×C23—82no—
146C22×C4×C23C2×D411yescharacteristic
147C22×C4×C23C2×D412yes—
148C22×C4×C23C2×D412yes—
149C22×C4×C23C2×D412yes—
150C22×C4×C23C2×D412yes—
151C22×C4×C23C2411yescharacteristic
152Q16×C23—44no—
153SD16×C23—44no—
154M16×C23—22no—
155M16×C23—22no—
156M16×C23—22no—
157M16×C23—44no—
158M16×C23#4114yescomplemented
159C2×C8×C23—22no—
160C2×C8×C23—22no—
161C2×C8×C23—44no—
162C4⋊C4×C23—24no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—24no—
166C4⋊C4×C23—24no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—22no—
172C4⋊C4×C23—42no—
173C4⋊C4×C23—82no—
174C4⋊C4×C23—82no—
175C4⋊C4×C23C2×D412yes—
176C4⋊C4×C23C2×D412yes—
177C22⋊C4×C23—24no—
178C22⋊C4×C23—24no—
179C22⋊C4×C23—24no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C22⋊C4×C23—22no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—41no—
186C22⋊C4×C23—41no—
187C42×C23—24no—
188C42×C23—22no—
189C42×C23—22no—
190C42×C23—21no—
191C42×C23—21no—
192#25519C2311yesmaximal, characteristic, direct product factor, Sylow
193C23×C23—21no—
194C23×C23—21no—
195C23×C23—22no—
196C23×C23—22no—
197C23×C23—41no—
198C23×C23#12011yescharacteristic
199C23×C23#12011yescharacteristic
200C23×C23C22×D411yescharacteristic
201Q8×C23—22no—
202Q8×C23—22no—
203Q8×C23—42no—
204Q8×C23—42no—
205Q8×C23—82no—
206D4×C23—22no—
207D4×C23—22no—
208D4×C23—44no—
209D4×C23—42no—
210D4×C23—42no—
211D4×C23—44no—
212D4×C23—82no—
213C2×C4×C23—21no—
214C2×C4×C23—21no—
215C2×C4×C23—22no—
216C2×C4×C23—22no—
217C2×C4×C23—22no—
218C2×C4×C23—22no—
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—44no—
229C2×C4×C23—41no—
230C2×C4×C23—41no—
231C2×C4×C23—42no—
232C2×C4×C23—42no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—42no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—41no—
240C2×C4×C23—82no—
241C2×C4×C23—84no—
242C2×C4×C23#12012yes—
243C2×C4×C23C22×D412yes—
244C2×C4×C23C2×#4112yes—
245C2×C4×C23H(5,𝔽2)11yescharacteristic
246C2×C4×C232−1+411yescharacteristic
247C8×C23—24no—
248C8×C23—42no—
249#3492C2×C2311yescharacteristic
250#3056C2×C2312yescomplemented
251#2502C2×C2311yescharacteristic, complemented
252#2060C2×C2311yescharacteristic, complemented
253#2039C2×C2311yescharacteristic, complemented
254#2028C2×C2311yescharacteristic, complemented
255#2007C2×C2311yescharacteristic, complemented
256#1882C2×C2312yescomplemented
257#1871C2×C2312yescomplemented
258#1859C2×C2312yescomplemented
259C2×#343C2×C2311yescharacteristic, complemented
260C22×C23—21no—
261C22×C23—21no—
262C22×C23—21no—
263C22×C23—21no—
264C22×C23—21no—
265C22×C23—21no—
266C22×C23—21no—
267C22×C23—42no—
268C22×C23—42no—
269C22×C23—42no—
270C22×C23—42no—
271C22×C23—42no—
272C22×C23#45612yes—
273C22×C23C2×#12011yescharacteristic
274C22×C23#53211yescharacteristic
275C22×C23#53711yescharacteristic
276C4×C23—22no—
277C4×C23—42no—
278C4×C23—42no—
279C4×C23—42no—
280C4×C23—42no—
281C4×C23—42no—
282C4×C23—42no—
283C4×C23—42no—
284C4×C23—81no—
285C4×C23—81no—
286C4×C23#54612yes—
287C4×C23#54712yes—
288C2×#137C22×C2312yes—
289C22×M16C22×C2312yes—
290#554—22no—
291#553—22no—
292#552—22no—
293#547—22no—
294#544C22×C2311yescharacteristic
295#539—22no—
296#537C22×C2311yescharacteristic
297#525—22no—
298#525C22×C2312yes—
299C2×C22⋊Q8C22×C2311yescharacteristic
300#520C22×C2312yes—
301C22×C4⋊C4C22×C2311yescharacteristic
302#470—22no—
303#468—22no—
304#467—22no—
305#465—22no—
306#428C22×C2312yes—
307C2×#104C22×C2312yes—
308C2×#104C22×C2312yes—
309#419C22×C2312yes—
310#417C22×C2312yes—
311#398—22no—
312#391C22×C2311yescharacteristic
313#391C22×C2311yescharacteristic
314#385—22no—
315#384C22×C2311yescharacteristic
316#384C22×C2311yescharacteristic
317C4×C22⋊C4C22×C2311yescharacteristic
318#343C22×C2311yescharacteristic
319#343C22×C2312yes—
320#343C22×C2311yescharacteristic
321#339C22×C2312yes—
322#339C22×C2312yes—
323C2×C23—21no—
324C2×C23—21no—
325C2×C23—22no—
326C2×C23—41no—
327C2×C23—82no—
328C2×C23#243011yescharacteristic
329C2×C23C2×#45611yescharacteristic
330C2×C23#301911yescharacteristic
331C2×#41—22no—
332C2×#41C23×C2312yes—
333C22×Q8C23×C2311yescharacteristic
334C23×C4D4×C2312yes—
335C23×C4C23×C2311yescharacteristic
336#137—24no—
337#137—42no—
338C2×Q16—22no—
339C2×SD16—22no—
340C2×M16—22no—
341C2×M16—24no—
342C2×M16—42no—
343C2×M16C23×C2312yes—
344C2×M16C23×C2312yes—
345C22×C8—22no—
346C4⋊Q8—22no—
347#126—24no—
348#125—24no—
349#125—22no—
350#123—24no—
351#123—24no—
352#123—22no—
353C22⋊Q8—22no—
354C22⋊Q8—24no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8D4×C2312yes—
360C22⋊Q8D4×C2312yes—
361#121—22no—
362#121—22no—
363#121—22no—
364#121D4×C2312yes—
365#121D4×C2312yes—
366#120—22no—
367C4×Q8—22no—
368C4×D4—24no—
369C4×D4—22no—
370C4×D4—24no—
371C4×D4—22no—
372C4×D4—22no—
373#117—22no—
374#117—22no—
375#117C23×C2312yes—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4D4×C2312yes—
383C2×C4⋊C4C23×C2311yescharacteristic
384C2×C4⋊C4C23×C2311yescharacteristic
385C2×C4⋊C4C23×C2311yescharacteristic
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C42C23×C2311yescharacteristic
389C2×C42C23×C2311yescharacteristic
390#107—22no—
391#106—22no—
392#104—24no—
393#104—42no—
394#103—22no—
395#103—22no—
396#103—22no—
397#102—22no—
398#95—21no—
399#95—22no—
400#95—22no—
401C23#2551911yescharacteristic, direct product factor, Sylow
402C24C2×D4×C2311yescharacteristic
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—84no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8C2×D4×C2312yes—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4C2×D4×C2312yes—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—82no—
431C22×C4C2×D4×C2311yescharacteristic
432C22×C4C2×D4×C2312yes—
433C22×C4C2×D4×C2312yes—
434C22×C4C2×D4×C2312yes—
435C22×C4C2×D4×C2312yes—
436C22×C4C24×C2311yescharacteristic
437Q16—44no—
438SD16—44no—
439M16—22no—
440M16—22no—
441M16—22no—
442M16—44no—
443M16#41×C2314yescomplemented
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—44no—
447C4⋊C4—24no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—42no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C2×D4×C2312yes—
461C4⋊C4C2×D4×C2312yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—41no—
471C22⋊C4—41no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—41no—
482C23C23×#12011yescharacteristic
483C23C23×#12011yescharacteristic
484C23C22×D4×C2311yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—42no—
488Q8—42no—
489Q8—82no—
490D4—22no—
491D4—22no—
492D4—44no—
493D4—44no—
494D4—42no—
495D4—42no—
496D4—82no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—44no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4C23×#12012yes—
527C2×C4C22×D4×C2312yes—
528C2×C4C2×#41×C2312yes—
529C2×C4C23×H(5,𝔽2)11yescharacteristic
530C2×C4C23×2−1+411yescharacteristic
531C8—24no—
532C8—42no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22C23×#45612yes—
546C22C2×C23×#12011yescharacteristic
547C22C23×#53211yescharacteristic
548C22C23×#53711yescharacteristic
549C4—22no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—81no—
558C4—81no—
559C4C23×#54612yes—
560C4C23×#54712yes—
561C2—21no—
562C2—21no—
563C2—22no—
564C2—41no—
565C2—82no—
566C2C23×#243011yescharacteristic
567C2C2×C23×#45611yescharacteristic
568C2C23×#301911yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3492C2
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
2. C23×#3056C2semidirect
3. C23×#2502C2semidirect
4. C23×#2060C2semidirect
5. C23×#2039C2semidirect
6. C23×#2028C2semidirect
7. C23×#2007C2semidirect
8. C23×#1882C2semidirect
9. C23×#1871C2semidirect
10. C23×#1859C2semidirect
11. C2×C23×#343C2semidirect
158. M16×C23#41semidirect
192. #25519C23direct (N × H)
250. #3056C2×C23semidirect
251. #2502C2×C23semidirect
252. #2060C2×C23semidirect
253. #2039C2×C23semidirect
254. #2028C2×C23semidirect
255. #2007C2×C23semidirect
256. #1882C2×C23semidirect
257. #1871C2×C23semidirect
258. #1859C2×C23semidirect
259. C2×#343C2×C23semidirect
401. C23#25519direct (N × H)
443. 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×#30562
C23×#25022
C23×#20602
C23×#20392
C23×#20282
C23×#20072
C23×#18822
C23×#18712
C23×#18592
C2×C23×#3432
#2551923