Finite Group Databasebeta

← C23×#12767 #19160393

Subgroups — C23×#12767 #19160393

Order 5888.

878 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3543C211yesmaximal, characteristic
2C2×C23×#353C211yesmaximal, characteristic, complemented
3C23×#2121C212yesmaximal, complemented
4C23×#1913C212yesmaximal, complemented
5C23×#1887C211yesmaximal, characteristic, complemented
6C23×#560—21no—
7C23×#560C412yescomplemented
8C23×#559C411yescharacteristic, complemented
9C23×#550—21no—
10C23×#546—22no—
11D42×C23—22no—
12D42×C23C411yescharacteristic, complemented
13C23×#540—22no—
14C23×#535—22no—
15C23×#534—21no—
16C23×#534C2212yes—
17C2×C23×#127C2211yescharacteristic
18C23×#529—21no—
19C23×#518C2211yescharacteristic, complemented
20C23×#418—21no—
21C2×C23×#100C2211yescharacteristic, complemented
22C2×C23×#99—22no—
23C2×C23×#99C2212yescomplemented
24C23×#353—21no—
25C23×#353—21no—
26C23×#352—22no—
27C23×AΣL(1,𝔽16)—22no—
28C23×#342—22no—
29C23×#339—21no—
30C2×#41×C23—22no—
31C22×D4×C23—21no—
32C22×D4×C23—22no—
33C22×D4×C23—22no—
34C22×D4×C23D412yes—
35C22×D4×C23D411yescharacteristic
36C22×D4×C23C2311yescharacteristic
37C2×M16×C23—21no—
38C23×#127—22no—
39C23×#127—22no—
40C23×#127C2×C411yescharacteristic
41C23×#127C2×C411yescharacteristic
42C23×#127D411yescharacteristic
43C23×#127D411yescharacteristic
44C23×#126—22no—
45C23×#125—21no—
46C23×#124—21no—
47C23×#124—22no—
48C23×#123—22no—
49C23×#123—21no—
50C23×#123—21no—
51C23×#123—22no—
52C23×#123—22no—
53C23×#123C2×C412yes—
54C23×#123D412yes—
55C23×C22⋊Q8—21no—
56C23×#121—22no—
57C23×#121—22no—
58C23×#121—22no—
59C23×#121—21no—
60C23×#121—22no—
61C23×#121—22no—
62C23×#121—22no—
63C23×#121—22no—
64C23×#121—22no—
65C23×#121—22no—
66C23×#121—22no—
67C23×#120—22no—
68C23×#120—21no—
69C23×#120—22no—
70C23×#120—22no—
71C23×#120—22no—
72C23×#120—21no—
73C23×#120C2×C412yes—
74C23×#120D412yes—
75C4×Q8×C23—21no—
76C4×D4×C23—22no—
77C4×D4×C23—22no—
78C4×D4×C23—22no—
79C4×D4×C23—21no—
80C4×D4×C23—42no—
81C23×#117—22no—
82C23×#117—21no—
83C2×C22⋊C4×C23—21no—
84C2×C22⋊C4×C23—22no—
85C2×C22⋊C4×C23—22no—
86C2×C22⋊C4×C23D412yes—
87C2×C42×C23D411yescharacteristic
88C23×#102—42no—
89C23×#100—22no—
90C23×#99—22no—
91C23×#99—22no—
92C23×#99—44no—
93C24×C23—21no—
94C24×C23—21no—
95C24×C23C22⋊C412yes—
96C24×C23C2×D412yes—
97#41×C23—84no—
98C2×Q8×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—22no—
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—22no—
108C2×D4×C23—22no—
109C2×D4×C23—22no—
110C2×D4×C23—22no—
111C2×D4×C23—21no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C2×D4×C23—41no—
115C2×D4×C23—41no—
116C2×D4×C23—42no—
117C2×D4×C23—42no—
118C2×D4×C23—84no—
119C2×D4×C23—84no—
120C2×D4×C23—81no—
121C2×D4×C23—81no—
122C2×D4×C23C22⋊C411yescharacteristic
123C2×D4×C23C22⋊C411yescharacteristic
124C2×D4×C23C22×C411yescharacteristic
125C2×D4×C23C2×D411yescharacteristic
126C2×D4×C23C2×D412yes—
127C22×C4×C23—22no—
128C22×C4×C23—21no—
129C22×C4×C23—22no—
130C22×C4×C23—21no—
131C22×C4×C23—22no—
132C22×C4×C23—42no—
133C22×C4×C23—42no—
134C22×C4×C23C22⋊C412yes—
135C22×C4×C23C2×D411yescharacteristic
136M16×C23—41no—
137M16×C23—41no—
138C2×C8×C23—41no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—21no—
141C4⋊C4×C23—21no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—22no—
145C4⋊C4×C23—22no—
146C22⋊C4×C23—21no—
147C22⋊C4×C23—21no—
148C22⋊C4×C23—22no—
149C22⋊C4×C23—22no—
150C22⋊C4×C23—22no—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—22no—
156C22⋊C4×C23—22no—
157C22⋊C4×C23—24no—
158C22⋊C4×C23—41no—
159C22⋊C4×C23—42no—
160C22⋊C4×C23—42no—
161C22⋊C4×C23C22⋊C412yes—
162C22⋊C4×C23C22⋊C412yes—
163C42×C23—22no—
164C42×C23—21no—
165C42×C23—41no—
166C42×C23C22⋊C411yescharacteristic
167C42×C23C22⋊C411yescharacteristic
168#12767C2311yesmaximal, characteristic, direct product factor, Sylow
169C23×C23—22no—
170C23×C23—22no—
171C23×C23—22no—
172C23×C23—21no—
173C23×C23—21no—
174C23×C23—42no—
175C23×C23—41no—
176C23×C23—41no—
177C23×C23—82no—
178C23×C23—82no—
179C23×C23—82no—
180C23×C23—82no—
181C23×C23—81no—
182C23×C23C2×C22⋊C412yes—
183C23×C23#12012yes—
184C23×C23#12011yescharacteristic
185Q8×C23—42no—
186D4×C23—41no—
187D4×C23—42no—
188D4×C23—41no—
189D4×C23—41no—
190D4×C23—41no—
191D4×C23—44no—
192D4×C23—42no—
193D4×C23—44no—
194D4×C23—42no—
195D4×C23—42no—
196D4×C23—81no—
197D4×C23—82no—
198D4×C23—82no—
199D4×C23—82no—
200D4×C23—82no—
201D4×C23—82no—
202D4×C23—82no—
203D4×C23—82no—
204D4×C23—82no—
205C2×C4×C23—21no—
206C2×C4×C23—21no—
207C2×C4×C23—21no—
208C2×C4×C23—21no—
209C2×C4×C23—22no—
210C2×C4×C23—22no—
211C2×C4×C23—22no—
212C2×C4×C23—42no—
213C2×C4×C23—44no—
214C2×C4×C23—42no—
215C2×C4×C23—42no—
216C2×C4×C23—81no—
217C2×C4×C23—82no—
218C2×C4×C23—82no—
219C2×C4×C23—82no—
220C2×C4×C23—162no—
221C2×C4×C23—162no—
222C2×C4×C23C2×C22⋊C411yescharacteristic
223C2×C4×C23#12011yescharacteristic
224C8×C23—81no—
225#3543C2×C2311yescharacteristic
226C2×#353C2×C2311yescharacteristic, complemented
227#2121C2×C2312yescomplemented
228#1913C2×C2312yescomplemented
229#1887C2×C2311yescharacteristic, complemented
230C22×C23—22no—
231C22×C23—22no—
232C22×C23—41no—
233C22×C23—42no—
234C22×C23—42no—
235C22×C23—42no—
236C22×C23—42no—
237C22×C23—42no—
238C22×C23—42no—
239C22×C23—41no—
240C22×C23—42no—
241C22×C23—41no—
242C22×C23—82no—
243C22×C23—82no—
244C22×C23—161no—
245C22×C23—161no—
246C22×C23—162no—
247C22×C23—162no—
248C22×C23#37911yescharacteristic
249C22×C23B(4,𝔽2)11yescharacteristic
250C22×C23#45811yescharacteristic
251C4×C23—22no—
252C4×C23—21no—
253C4×C23—21no—
254C4×C23—42no—
255C4×C23—41no—
256C4×C23—42no—
257C4×C23—42no—
258C4×C23—82no—
259C4×C23—162no—
260#560—21no—
261#560C4×C2312yescomplemented
262#559C4×C2311yescharacteristic, complemented
263#550—21no—
264#546—22no—
265D42—22no—
266D42C4×C2311yescharacteristic, complemented
267#540—22no—
268#535—22no—
269#534—21no—
270#534C22×C2312yes—
271C2×#127C22×C2311yescharacteristic
272#529—21no—
273#518C22×C2311yescharacteristic, complemented
274#418—21no—
275C2×#100C22×C2311yescharacteristic, complemented
276C2×#99—22no—
277C2×#99C22×C2312yescomplemented
278#353—21no—
279#353—21no—
280#352—22no—
281AΣL(1,𝔽16)—22no—
282#342—22no—
283#339—21no—
284C2×C23—21no—
285C2×C23—21no—
286C2×C23—21no—
287C2×C23—42no—
288C2×C23—42no—
289C2×C23—81no—
290C2×C23—82no—
291C2×C23—81no—
292C2×C23#179011yescharacteristic
293C2×#41—22no—
294C22×D4—21no—
295C22×D4—22no—
296C22×D4—22no—
297C22×D4D4×C2312yes—
298C22×D4D4×C2311yescharacteristic
299C22×D4C23×C2311yescharacteristic
300C2×M16—21no—
301#127—22no—
302#127—22no—
303#127C2×C4×C2311yescharacteristic
304#127C2×C4×C2311yescharacteristic
305#127D4×C2311yescharacteristic
306#127D4×C2311yescharacteristic
307#126—22no—
308#125—21no—
309#124—21no—
310#124—22no—
311#123—21no—
312#123—22no—
313#123—21no—
314#123—22no—
315#123—22no—
316#123C2×C4×C2312yes—
317#123D4×C2312yes—
318C22⋊Q8—21no—
319#121—21no—
320#121—22no—
321#121—22no—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—22no—
326#121—22no—
327#121—22no—
328#121—22no—
329#121—22no—
330#120—21no—
331#120—22no—
332#120—22no—
333#120—22no—
334#120—22no—
335#120—21no—
336#120C2×C4×C2312yes—
337#120D4×C2312yes—
338C4×Q8—21no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4—22no—
342C4×D4—21no—
343C4×D4—42no—
344#117—22no—
345#117—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4D4×C2312yes—
350C2×C42D4×C2311yescharacteristic
351#102—42no—
352#100—22no—
353#99—22no—
354#99—22no—
355#99—44no—
356C23#1276711yescharacteristic, direct product factor, Sylow
357C24—21no—
358C24—21no—
359C24C22⋊C4×C2312yes—
360C24C2×D4×C2312yes—
361#41—84no—
362C2×Q8—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4C22⋊C4×C2311yescharacteristic
387C2×D4C22⋊C4×C2311yescharacteristic
388C2×D4C22×C4×C2311yescharacteristic
389C2×D4C2×D4×C2311yescharacteristic
390C2×D4C2×D4×C2312yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4C22⋊C4×C2312yes—
399C22×C4C2×D4×C2311yescharacteristic
400M16—41no—
401M16—41no—
402C2×C8—41no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—24no—
422C22⋊C4—41no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4C22⋊C4×C2312yes—
426C22⋊C4C22⋊C4×C2312yes—
427C42—22no—
428C42—21no—
429C42—41no—
430C42C22⋊C4×C2311yescharacteristic
431C42C22⋊C4×C2311yescharacteristic
432C23—22no—
433C23—22no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23—81no—
445C23C2×C22⋊C4×C2312yes—
446C23C23×#12012yes—
447C23C23×#12011yescharacteristic
448Q8—42no—
449D4—41no—
450D4—42no—
451D4—41no—
452D4—41no—
453D4—41no—
454D4—44no—
455D4—42no—
456D4—44no—
457D4—42no—
458D4—42no—
459D4—81no—
460D4—82no—
461D4—82no—
462D4—82no—
463D4—82no—
464D4—82no—
465D4—82no—
466D4—82no—
467D4—82no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—162no—
484C2×C4—162no—
485C2×C4C2×C22⋊C4×C2311yescharacteristic
486C2×C4C23×#12011yescharacteristic
487C8—81no—
488C22—22no—
489C22—22no—
490C22—41no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—82no—
501C22—82no—
502C22—161no—
503C22—161no—
504C22—162no—
505C22—162no—
506C22C23×#37911yescharacteristic
507C22C23×B(4,𝔽2)11yescharacteristic
508C22C23×#45811yescharacteristic
509C4—22no—
510C4—21no—
511C4—21no—
512C4—42no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—162no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—42no—
522C2—42no—
523C2—81no—
524C2—82no—
525C2—81no—
526C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#3543C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#353C2semidirect
3. C23×#2121C2semidirect
4. C23×#1913C2semidirect
5. C23×#1887C2semidirect
7. C23×#560C4semidirect
8. C23×#559C4semidirect
12. D42×C23C4semidirect
19. C23×#518C22semidirect
21. C2×C23×#100C22semidirect
23. C2×C23×#99C22semidirect
168. #12767C23direct (N × H)
226. C2×#353C2×C23semidirect
227. #2121C2×C23semidirect
228. #1913C2×C23semidirect
229. #1887C2×C23semidirect
261. #560C4×C23semidirect
262. #559C4×C23semidirect
266. D42C4×C23semidirect
273. #518C22×C23semidirect
275. C2×#100C22×C23semidirect
277. C2×#99C22×C23semidirect
356. C23#12767direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35432
C2×C23×#3532
C23×#21212
C23×#19132
C23×#18872
#1276723