Finite Group Databasebeta

← C23×#12755 #19160381

Subgroups — C23×#12755 #19160381

Order 5888.

864 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3537C211yesmaximal, characteristic
2C23×#2134C211yesmaximal, characteristic, complemented
3C23×#2122C211yesmaximal, characteristic, complemented
4C23×#2120C211yesmaximal, characteristic, complemented
5C23×#1914C211yesmaximal, characteristic, complemented
6C23×#1910C211yesmaximal, characteristic, complemented
7C23×#1891C211yesmaximal, characteristic, complemented
8C23×#568C2211yescharacteristic, complemented
9C23×#560—21no—
10C23×#560C412yescomplemented
11C23×#553—21no—
12C23×#553—21no—
13D42×C23—21no—
14D42×C23—21no—
15D42×C23—21no—
16D42×C23C412yescomplemented
17C23×#540—21no—
18C23×#538—22no—
19C23×#535C2211yescharacteristic
20C23×#535C2211yescharacteristic
21C23×#534—22no—
22C23×#525C2211yescharacteristic
23C2×C23×#121—21no—
24C23×#418—21no—
25C23×#413C2211yescharacteristic, complemented
26C23×#413C2211yescharacteristic, complemented
27C2×C23×#100—21no—
28C23×#410C2211yescharacteristic, complemented
29C23×#407—21no—
30C23×#355—22no—
31C23×#352—21no—
32C23×AΣL(1,𝔽16)—21no—
33C23×AΣL(1,𝔽16)—22no—
34C23×#344—21no—
35C23×#344—21no—
36C23×H(3,ℤ/4)—21no—
37C2×#41×C23C2311yescharacteristic
38C22×D4×C23—21no—
39C22×D4×C23—22no—
40C22×D4×C23—22no—
41C22×D4×C23—22no—
42C22×D4×C23D411yescharacteristic
43C22×D4×C23D411yescharacteristic
44C23×C4×C23D411yescharacteristic
45C23×#131—42no—
46C2×M16×C23—21no—
47C2×M16×C23—21no—
48C2×M16×C23—21no—
49C2×M16×C23—21no—
50C2×M16×C23—21no—
51C23×#127—21no—
52C23×#127—21no—
53C23×#127C2×C412yes—
54C23×#126—22no—
55C23×#126—22no—
56C23×#125—21no—
57C23×#124—21no—
58C23×#124D412yes—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×#123—22no—
63C23×#123C2×C411yescharacteristic
64C23×#123D411yescharacteristic
65C23×C22⋊Q8—21no—
66C23×#121—21no—
67C23×#121—21no—
68C23×#121—22no—
69C23×#121—22no—
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—21no—
73C23×#121—22no—
74C23×#121—21no—
75C23×#121—21no—
76C23×#121—21no—
77C23×#121—21no—
78C23×#121—22no—
79C23×#121—22no—
80C23×#121—22no—
81C23×#120—22no—
82C23×#120—22no—
83C23×#120—22no—
84C23×#120—22no—
85C23×#120—21no—
86C23×#120—21no—
87C23×#120C2×C412yes—
88C23×#120C2×C411yescharacteristic
89C23×#120D411yescharacteristic
90C23×#120D412yes—
91C4×D4×C23—22no—
92C4×D4×C23—21no—
93C4×D4×C23—21no—
94C4×D4×C23—21no—
95C4×D4×C23—21no—
96C4×D4×C23—22no—
97C23×#117—21no—
98C23×#117D411yescharacteristic
99C23×#117D411yescharacteristic
100C23×#117D411yescharacteristic
101C2×C4⋊C4×C23—21no—
102C2×C22⋊C4×C23—22no—
103C23×#102—42no—
104C23×#101—21no—
105C23×#101—21no—
106C23×#100—21no—
107C23×#100—21no—
108C23×#100—42no—
109C23×#99—21no—
110C23×#99—21no—
111C23×C22⋊C8—42no—
112C24×C23—22no—
113C24×C23—21no—
114C24×C23C22⋊C411yescharacteristic
115C24×C23C22⋊C412yes—
116#41×C23—41no—
117#41×C23—41no—
118C2×Q8×C23C2×D411yescharacteristic
119C2×D4×C23—22no—
120C2×D4×C23—22no—
121C2×D4×C23—21no—
122C2×D4×C23—21no—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—22no—
128C2×D4×C23—22no—
129C2×D4×C23—22no—
130C2×D4×C23—21no—
131C2×D4×C23—21no—
132C2×D4×C23—21no—
133C2×D4×C23—41no—
134C2×D4×C23—41no—
135C2×D4×C23—41no—
136C2×D4×C23—41no—
137C2×D4×C23—81no—
138C2×D4×C23—81no—
139C2×D4×C23—84no—
140C2×D4×C23—84no—
141C2×D4×C23—84no—
142C2×D4×C23C22⋊C412yes—
143C2×D4×C23C22×C411yescharacteristic
144C2×D4×C23C2×D411yescharacteristic
145C2×D4×C23C2×D411yescharacteristic
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—21no—
150C22×C4×C23—21no—
151C22×C4×C23—41no—
152C22×C4×C23—41no—
153C22×C4×C23C22⋊C411yescharacteristic
154C22×C4×C23C2×D411yescharacteristic
155C22×C4×C23C2×D411yescharacteristic
156C22×C4×C23C2×D411yescharacteristic
157M16×C23—22no—
158M16×C23—41no—
159M16×C23—41no—
160M16×C23—41no—
161M16×C23—41no—
162M16×C23—41no—
163M16×C23—41no—
164M16×C23—41no—
165C2×C8×C23—22no—
166C2×C8×C23—22no—
167C2×C8×C23—41no—
168C2×C8×C23—41no—
169C2×C8×C23—41no—
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—41no—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C22⋊C4×C23—21no—
184C22⋊C4×C23—21no—
185C22⋊C4×C23—21no—
186C22⋊C4×C23—21no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—21no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—41no—
193C22⋊C4×C23C22⋊C412yes—
194C22⋊C4×C23C22⋊C411yescharacteristic
195C22⋊C4×C23C22⋊C411yescharacteristic
196C42×C23—21no—
197C42×C23—21no—
198C42×C23—41no—
199C42×C23C22⋊C412yes—
200#12755C2311yesmaximal, characteristic, direct product factor, Sylow
201C23×C23—21no—
202C23×C23—22no—
203C23×C23—22no—
204C23×C23—41no—
205C23×C23—42no—
206C23×C23—44no—
207C23×C23—42no—
208C23×C23—81no—
209C23×C23—82no—
210C23×C23—82no—
211C23×C23—82no—
212C23×C23—82no—
213C23×C23C2×C22⋊C411yescharacteristic
214C23×C23C2×C22⋊C411yescharacteristic
215C23×C23#12011yescharacteristic
216Q8×C23—41no—
217D4×C23—41no—
218D4×C23—41no—
219D4×C23—41no—
220D4×C23—42no—
221D4×C23—42no—
222D4×C23—42no—
223D4×C23—44no—
224D4×C23—44no—
225D4×C23—41no—
226D4×C23—41no—
227D4×C23—41no—
228D4×C23—41no—
229D4×C23—81no—
230D4×C23—82no—
231D4×C23—82no—
232D4×C23—82no—
233D4×C23—82no—
234D4×C23—82no—
235D4×C23—82no—
236D4×C23—82no—
237C2×C4×C23—21no—
238C2×C4×C23—21no—
239C2×C4×C23—21no—
240C2×C4×C23—21no—
241C2×C4×C23—21no—
242C2×C4×C23—21no—
243C2×C4×C23—22no—
244C2×C4×C23—21no—
245C2×C4×C23—21no—
246C2×C4×C23—22no—
247C2×C4×C23—21no—
248C2×C4×C23—41no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—81no—
252C2×C4×C23—82no—
253C2×C4×C23—82no—
254C2×C4×C23—82no—
255C2×C4×C23—81no—
256C2×C4×C23C2×C22⋊C411yescharacteristic
257C2×C4×C23#12011yescharacteristic
258C2×C4×C23#12011yescharacteristic
259C2×C4×C23#12011yescharacteristic
260C8×C23—42no—
261C8×C23—81no—
262C8×C23—81no—
263#3537C2×C2311yescharacteristic
264#2134C2×C2311yescharacteristic, complemented
265#2122C2×C2311yescharacteristic, complemented
266#2120C2×C2311yescharacteristic, complemented
267#1914C2×C2311yescharacteristic, complemented
268#1910C2×C2311yescharacteristic, complemented
269#1891C2×C2311yescharacteristic, complemented
270C22×C23—22no—
271C22×C23—21no—
272C22×C23—21no—
273C22×C23—21no—
274C22×C23—41no—
275C22×C23—41no—
276C22×C23—41no—
277C22×C23—41no—
278C22×C23—42no—
279C22×C23—42no—
280C22×C23—42no—
281C22×C23—42no—
282C22×C23—42no—
283C22×C23—41no—
284C22×C23—42no—
285C22×C23—42no—
286C22×C23—81no—
287C22×C23—162no—
288C22×C23—164no—
289C22×C23—162no—
290C22×C23#37911yescharacteristic
291C4×C23—21no—
292C4×C23—21no—
293C4×C23—21no—
294C4×C23—42no—
295C4×C23—41no—
296C4×C23—41no—
297C4×C23—42no—
298C4×C23—41no—
299C4×C23—161no—
300C4×C23B(4,𝔽2)11yescharacteristic
301C4×C23#45811yescharacteristic
302#568C22×C2311yescharacteristic, complemented
303#560—21no—
304#560C4×C2312yescomplemented
305#553—21no—
306#553—21no—
307D42—21no—
308D42—21no—
309D42—21no—
310D42C4×C2312yescomplemented
311#540—21no—
312#538—22no—
313#535C22×C2311yescharacteristic
314#535C22×C2311yescharacteristic
315#534—22no—
316#525C22×C2311yescharacteristic
317C2×#121—21no—
318#418—21no—
319#413C22×C2311yescharacteristic, complemented
320#413C22×C2311yescharacteristic, complemented
321C2×#100—21no—
322#410C22×C2311yescharacteristic, complemented
323#407—21no—
324#355—22no—
325#352—21no—
326AΣL(1,𝔽16)—21no—
327AΣL(1,𝔽16)—22no—
328#344—21no—
329#344—21no—
330H(3,ℤ/4)—21no—
331C2×C23—21no—
332C2×C23—41no—
333C2×C23—41no—
334C2×C23—42no—
335C2×C23—41no—
336C2×C23—82no—
337C2×C23—82no—
338C2×C23#179011yescharacteristic
339C2×#41C23×C2311yescharacteristic
340C22×D4—21no—
341C22×D4—22no—
342C22×D4—22no—
343C22×D4—22no—
344C22×D4D4×C2311yescharacteristic
345C22×D4D4×C2311yescharacteristic
346C23×C4D4×C2311yescharacteristic
347#131—42no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353#127—21no—
354#127—21no—
355#127C2×C4×C2312yes—
356#126—22no—
357#126—22no—
358#125—21no—
359#124—21no—
360#124D4×C2312yes—
361#123—22no—
362#123—22no—
363#123—22no—
364#123—22no—
365#123C2×C4×C2311yescharacteristic
366#123D4×C2311yescharacteristic
367C22⋊Q8—21no—
368#121—22no—
369#121—21no—
370#121—21no—
371#121—22no—
372#121—22no—
373#121—22no—
374#121—21no—
375#121—22no—
376#121—21no—
377#121—21no—
378#121—21no—
379#121—21no—
380#121—22no—
381#121—22no—
382#121—22no—
383#120—22no—
384#120—22no—
385#120—22no—
386#120—22no—
387#120—21no—
388#120—21no—
389#120C2×C4×C2311yescharacteristic
390#120C2×C4×C2312yes—
391#120D4×C2311yescharacteristic
392#120D4×C2312yes—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—22no—
399#117—21no—
400#117D4×C2311yescharacteristic
401#117D4×C2311yescharacteristic
402#117D4×C2311yescharacteristic
403C2×C4⋊C4—21no—
404C2×C22⋊C4—22no—
405#102—42no—
406#101—21no—
407#101—21no—
408#100—21no—
409#100—21no—
410#100—42no—
411#99—21no—
412#99—21no—
413C22⋊C8—42no—
414C23#1275511yescharacteristic, direct product factor, Sylow
415C24—22no—
416C24—21no—
417C24C22⋊C4×C2311yescharacteristic
418C24C22⋊C4×C2312yes—
419#41—41no—
420#41—41no—
421C2×Q8C2×D4×C2311yescharacteristic
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—81no—
441C2×D4—81no—
442C2×D4—84no—
443C2×D4—84no—
444C2×D4—84no—
445C2×D4C22⋊C4×C2312yes—
446C2×D4C22×C4×C2311yescharacteristic
447C2×D4C2×D4×C2311yescharacteristic
448C2×D4C2×D4×C2311yescharacteristic
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4C22⋊C4×C2311yescharacteristic
457C22×C4C2×D4×C2311yescharacteristic
458C22×C4C2×D4×C2311yescharacteristic
459C22×C4C2×D4×C2311yescharacteristic
460M16—22no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—21no—
494C22⋊C4—22no—
495C22⋊C4—41no—
496C22⋊C4C22⋊C4×C2312yes—
497C22⋊C4C22⋊C4×C2311yescharacteristic
498C22⋊C4C22⋊C4×C2311yescharacteristic
499C42—21no—
500C42—21no—
501C42—41no—
502C42C22⋊C4×C2312yes—
503C23—21no—
504C23—22no—
505C23—22no—
506C23—41no—
507C23—42no—
508C23—44no—
509C23—42no—
510C23—81no—
511C23—82no—
512C23—82no—
513C23—82no—
514C23—82no—
515C23C2×C22⋊C4×C2311yescharacteristic
516C23C2×C22⋊C4×C2311yescharacteristic
517C23C23×#12011yescharacteristic
518Q8—41no—
519D4—41no—
520D4—41no—
521D4—41no—
522D4—42no—
523D4—42no—
524D4—42no—
525D4—44no—
526D4—44no—
527D4—41no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—81no—
532D4—82no—
533D4—82no—
534D4—82no—
535D4—82no—
536D4—82no—
537D4—82no—
538D4—82no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—81no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4C2×C22⋊C4×C2311yescharacteristic
559C2×C4C23×#12011yescharacteristic
560C2×C4C23×#12011yescharacteristic
561C2×C4C23×#12011yescharacteristic
562C8—42no—
563C8—81no—
564C8—81no—
565C22—22no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—42no—
580C22—42no—
581C22—81no—
582C22—162no—
583C22—164no—
584C22—162no—
585C22C23×#37911yescharacteristic
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—41no—
594C4—161no—
595C4C23×B(4,𝔽2)11yescharacteristic
596C4C23×#45811yescharacteristic
597C2—21no—
598C2—41no—
599C2—42no—
600C2—41no—
601C2—41no—
602C2—82no—
603C2—82no—
604C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×D4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3537C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2134C2semidirect
3. C23×#2122C2semidirect
4. C23×#2120C2semidirect
5. C23×#1914C2semidirect
6. C23×#1910C2semidirect
7. C23×#1891C2semidirect
8. C23×#568C22semidirect
10. C23×#560C4semidirect
16. D42×C23C4semidirect
25. C23×#413C22semidirect
26. C23×#413C22semidirect
28. C23×#410C22semidirect
200. #12755C23direct (N × H)
264. #2134C2×C23semidirect
265. #2122C2×C23semidirect
266. #2120C2×C23semidirect
267. #1914C2×C23semidirect
268. #1910C2×C23semidirect
269. #1891C2×C23semidirect
302. #568C22×C23semidirect
304. #560C4×C23semidirect
310. D42C4×C23semidirect
319. #413C22×C23semidirect
320. #413C22×C23semidirect
322. #410C22×C23semidirect
414. C23#12755direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35372
C23×#21342
C23×#21222
C23×#21202
C23×#19142
C23×#19102
C23×#18912
#1275523