Finite Group Databasebeta

← C23×#12725 #19160351

Subgroups — C23×#12725 #19160351

Order 5888.

846 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3490C211yesmaximal, characteristic
2C23×#2121C212yesmaximal, complemented
3C2×C23×AΣL(1,𝔽16)C212yesmaximal, complemented
4C23×#1829C211yesmaximal, characteristic, complemented
5C23×#1828C211yesmaximal, characteristic, complemented
6C23×#551—22no—
7C23×#551—22no—
8C23×#546—22no—
9C23×#546—22no—
10C23×#543C411yescharacteristic, complemented
11C23×#538—22no—
12C23×#534C2211yescharacteristic
13C23×#525—22no—
14C2×C23×#120C411yescharacteristic, complemented
15C2×C23×#120C2211yescharacteristic
16C23×#518C411yescharacteristic, complemented
17C23×#518C2211yescharacteristic
18C22×C22⋊C4×C23C411yescharacteristic, complemented
19C2×C23×#100C2212yes—
20C2×C23×#99C2212yes—
21C23×#352—22no—
22C23×AΣL(1,𝔽16)—22no—
23C23×AΣL(1,𝔽16)—22no—
24C23×AΣL(1,𝔽16)—22no—
25C23×#344—22no—
26C23×#342—22no—
27C25×C23C2×C411yescharacteristic
28C2×#41×C23—22no—
29C22×D4×C23—21no—
30C22×D4×C23—21no—
31C22×D4×C23C2×C411yescharacteristic
32C22×D4×C23C2311yescharacteristic
33C23×C4×C23—21no—
34C2×M16×C23—22no—
35C23×#126—22no—
36C23×#126—22no—
37C23×#124—22no—
38C23×#124—22no—
39C23×#123—22no—
40C23×#123—22no—
41C23×#123—22no—
42C23×#123—22no—
43C23×#123—21no—
44C23×#123D412yes—
45C23×C22⋊Q8—22no—
46C23×C22⋊Q8—22no—
47C23×C22⋊Q8—22no—
48C23×C22⋊Q8—22no—
49C23×#121—22no—
50C23×#121—22no—
51C23×#121—22no—
52C23×#121—22no—
53C23×#120—21no—
54C23×#120—21no—
55C23×#120—21no—
56C23×#120—22no—
57C23×#120—22no—
58C23×#120D412yes—
59C23×#120D412yes—
60C23×#120D412yes—
61C4×D4×C23—22no—
62C4×D4×C23—22no—
63C4×D4×C23—22no—
64C4×D4×C23—22no—
65C4×D4×C23—22no—
66C4×D4×C23—22no—
67C23×#117—22no—
68C23×#117—21no—
69C23×#117D412yes—
70C2×C22⋊C4×C23—21no—
71C2×C22⋊C4×C23—21no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—21no—
74C2×C22⋊C4×C23—22no—
75C2×C22⋊C4×C23—22no—
76C2×C22⋊C4×C23—22no—
77C2×C22⋊C4×C23C2×C411yescharacteristic
78C2×C22⋊C4×C23C2×C411yescharacteristic
79C2×C22⋊C4×C23C2×C411yescharacteristic
80C2×C22⋊C4×C23C2×C411yescharacteristic
81C2×C22⋊C4×C23D412yes—
82C23×#100—22no—
83C23×#100—22no—
84C23×#99—22no—
85C23×#99—22no—
86C24×C23—21no—
87C24×C23—21no—
88C24×C23—42no—
89C24×C23—82no—
90C24×C23C22⋊C411yescharacteristic
91C24×C23C22⋊C411yescharacteristic
92C24×C23C22⋊C411yescharacteristic
93C24×C23C22×C411yescharacteristic
94C24×C23C2×D411yescharacteristic
95#41×C23—84no—
96C2×Q8×C23—22no—
97C2×D4×C23—22no—
98C2×D4×C23—21no—
99C2×D4×C23—21no—
100C2×D4×C23—22no—
101C2×D4×C23—22no—
102C2×D4×C23—22no—
103C2×D4×C23—22no—
104C2×D4×C23—42no—
105C2×D4×C23—42no—
106C2×D4×C23—82no—
107C2×D4×C23—82no—
108C2×D4×C23C2×D412yes—
109C2×D4×C23C2×D412yes—
110C22×C4×C23—22no—
111C22×C4×C23—22no—
112C22×C4×C23—22no—
113C22×C4×C23—21no—
114C22×C4×C23—21no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—22no—
118C22×C4×C23—42no—
119C22×C4×C23—82no—
120C22×C4×C23C22⋊C411yescharacteristic
121C22×C4×C23C2×D411yescharacteristic
122M16×C23—42no—
123M16×C23—42no—
124C2×C8×C23—42no—
125C4⋊C4×C23—22no—
126C4⋊C4×C23—22no—
127C4⋊C4×C23—22no—
128C4⋊C4×C23—22no—
129C4⋊C4×C23—22no—
130C4⋊C4×C23—22no—
131C22⋊C4×C23—22no—
132C22⋊C4×C23—22no—
133C22⋊C4×C23—22no—
134C22⋊C4×C23—22no—
135C22⋊C4×C23—22no—
136C22⋊C4×C23—21no—
137C22⋊C4×C23—21no—
138C22⋊C4×C23—21no—
139C22⋊C4×C23—21no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—22no—
142C22⋊C4×C23—22no—
143C22⋊C4×C23—22no—
144C22⋊C4×C23—22no—
145C22⋊C4×C23—42no—
146C22⋊C4×C23—42no—
147C22⋊C4×C23C22⋊C412yes—
148C22⋊C4×C23C22⋊C412yes—
149C22⋊C4×C23C22⋊C412yes—
150C22⋊C4×C23C22⋊C412yes—
151C42×C23—22no—
152C42×C23—22no—
153#12725C2311yesmaximal, characteristic, direct product factor, Sylow
154C23×C23—21no—
155C23×C23—22no—
156C23×C23—22no—
157C23×C23—21no—
158C23×C23—21no—
159C23×C23—42no—
160C23×C23—41no—
161C23×C23—42no—
162C23×C23—42no—
163C23×C23—42no—
164C23×C23—41no—
165C23×C23—41no—
166C23×C23—42no—
167C23×C23—81no—
168C23×C23—81no—
169C23×C23—81no—
170C23×C23—81no—
171C23×C23—81no—
172C23×C23—81no—
173C23×C23—84no—
174C23×C23—84no—
175C23×C23C2×C22⋊C411yescharacteristic
176C23×C23C2×C22⋊C411yescharacteristic
177C23×C23C2×C22⋊C411yescharacteristic
178C23×C23#12012yes—
179Q8×C23—82no—
180D4×C23—42no—
181D4×C23—42no—
182D4×C23—42no—
183D4×C23—44no—
184D4×C23—44no—
185D4×C23—42no—
186D4×C23—82no—
187D4×C23—82no—
188D4×C23—82no—
189C2×C4×C23—22no—
190C2×C4×C23—21no—
191C2×C4×C23—22no—
192C2×C4×C23—22no—
193C2×C4×C23—22no—
194C2×C4×C23—22no—
195C2×C4×C23—44no—
196C2×C4×C23—41no—
197C2×C4×C23—81no—
198C2×C4×C23—82no—
199C2×C4×C23—81no—
200C2×C4×C23—81no—
201C2×C4×C23—81no—
202C2×C4×C23—82no—
203C2×C4×C23—82no—
204C2×C4×C23—82no—
205C2×C4×C23—162no—
206C2×C4×C23#12012yes—
207C8×C23—82no—
208#3490C2×C2311yescharacteristic
209#2121C2×C2312yescomplemented
210C2×AΣL(1,𝔽16)C2×C2312yescomplemented
211#1829C2×C2311yescharacteristic, complemented
212#1828C2×C2311yescharacteristic, complemented
213C22×C23—22no—
214C22×C23—21no—
215C22×C23—22no—
216C22×C23—21no—
217C22×C23—41no—
218C22×C23—41no—
219C22×C23—42no—
220C22×C23—41no—
221C22×C23—41no—
222C22×C23—42no—
223C22×C23—41no—
224C22×C23—41no—
225C22×C23—42no—
226C22×C23—42no—
227C22×C23—41no—
228C22×C23—41no—
229C22×C23—41no—
230C22×C23—82no—
231C22×C23—82no—
232C22×C23—84no—
233C22×C23—84no—
234C22×C23—82no—
235C22×C23—161no—
236C22×C23#37911yescharacteristic
237C22×C23#41012yes—
238C4×C23—22no—
239C4×C23—42no—
240C4×C23—42no—
241C4×C23—42no—
242C4×C23—42no—
243C4×C23—41no—
244C4×C23—162no—
245#551—22no—
246#551—22no—
247#546—22no—
248#546—22no—
249#543C4×C2311yescharacteristic, complemented
250#538—22no—
251#534C22×C2311yescharacteristic
252#525—22no—
253C2×#120C4×C2311yescharacteristic, complemented
254C2×#120C22×C2311yescharacteristic
255#518C4×C2311yescharacteristic, complemented
256#518C22×C2311yescharacteristic
257C22×C22⋊C4C4×C2311yescharacteristic, complemented
258C2×#100C22×C2312yes—
259C2×#99C22×C2312yes—
260#352—22no—
261AΣL(1,𝔽16)—22no—
262AΣL(1,𝔽16)—22no—
263AΣL(1,𝔽16)—22no—
264#344—22no—
265#342—22no—
266C2×C23—21no—
267C2×C23—22no—
268C2×C23—42no—
269C2×C23—42no—
270C2×C23—41no—
271C2×C23—41no—
272C2×C23—81no—
273C2×C23—81no—
274C2×C23#178211yescharacteristic
275C25C2×C4×C2311yescharacteristic
276C2×#41—22no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2×C4×C2311yescharacteristic
280C22×D4C23×C2311yescharacteristic
281C23×C4—21no—
282C2×M16—22no—
283#126—22no—
284#126—22no—
285#124—22no—
286#124—22no—
287#123—22no—
288#123—22no—
289#123—22no—
290#123—22no—
291#123—21no—
292#123D4×C2312yes—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297#121—22no—
298#121—22no—
299#121—22no—
300#121—22no—
301#120—21no—
302#120—21no—
303#120—21no—
304#120—22no—
305#120—22no—
306#120D4×C2312yes—
307#120D4×C2312yes—
308#120D4×C2312yes—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315#117—22no—
316#117—21no—
317#117D4×C2312yes—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4C2×C4×C2311yescharacteristic
326C2×C22⋊C4C2×C4×C2311yescharacteristic
327C2×C22⋊C4C2×C4×C2311yescharacteristic
328C2×C22⋊C4C2×C4×C2311yescharacteristic
329C2×C22⋊C4D4×C2312yes—
330#100—22no—
331#100—22no—
332#99—22no—
333#99—22no—
334C23#1272511yescharacteristic, direct product factor, Sylow
335C24—21no—
336C24—21no—
337C24—42no—
338C24—82no—
339C24C22⋊C4×C2311yescharacteristic
340C24C22⋊C4×C2311yescharacteristic
341C24C22⋊C4×C2311yescharacteristic
342C24C22×C4×C2311yescharacteristic
343C24C2×D4×C2311yescharacteristic
344#41—84no—
345C2×Q8—22no—
346C2×D4—22no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—82no—
356C2×D4—82no—
357C2×D4C2×D4×C2312yes—
358C2×D4C2×D4×C2312yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—82no—
369C22×C4C22⋊C4×C2311yescharacteristic
370C22×C4C2×D4×C2311yescharacteristic
371M16—42no—
372M16—42no—
373C2×C8—42no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4C22⋊C4×C2312yes—
397C22⋊C4C22⋊C4×C2312yes—
398C22⋊C4C22⋊C4×C2312yes—
399C22⋊C4C22⋊C4×C2312yes—
400C42—22no—
401C42—22no—
402C23—21no—
403C23—22no—
404C23—22no—
405C23—21no—
406C23—21no—
407C23—42no—
408C23—41no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—42no—
415C23—81no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—84no—
422C23—84no—
423C23C2×C22⋊C4×C2311yescharacteristic
424C23C2×C22⋊C4×C2311yescharacteristic
425C23C2×C22⋊C4×C2311yescharacteristic
426C23C23×#12012yes—
427Q8—82no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—44no—
432D4—44no—
433D4—42no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—162no—
454C2×C4C23×#12012yes—
455C8—82no—
456C22—22no—
457C22—21no—
458C22—22no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—82no—
474C22—82no—
475C22—84no—
476C22—84no—
477C22—82no—
478C22—161no—
479C22C23×#37911yescharacteristic
480C22C23×#41012yes—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—41no—
487C4—162no—
488C2—21no—
489C2—22no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—41no—
494C2—81no—
495C2—81no—
496C2C23×#178211yescharacteristic
subgroupG/H
center, socleC2×C23#1782
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#3490C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C23×#518 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2121C2semidirect
3. C2×C23×AΣL(1,𝔽16)C2semidirect
4. C23×#1829C2semidirect
5. C23×#1828C2semidirect
10. C23×#543C4semidirect
14. C2×C23×#120C4semidirect
16. C23×#518C4semidirect
18. C22×C22⋊C4×C23C4semidirect
153. #12725C23direct (N × H)
209. #2121C2×C23semidirect
210. C2×AΣL(1,𝔽16)C2×C23semidirect
211. #1829C2×C23semidirect
212. #1828C2×C23semidirect
249. #543C4×C23semidirect
253. C2×#120C4×C23semidirect
255. #518C4×C23semidirect
257. C22×C22⋊C4C4×C23semidirect
334. C23#12725direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34902
C23×#21212
C2×C23×AΣL(1,𝔽16)2
C23×#18292
C23×#18282
#1272523