Finite Group Databasebeta

← C23×#13098 #19160724

Subgroups — C23×#13098 #19160724

Order 5888.

640 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3010C211yesmaximal, characteristic
2C23×#2889C211yesmaximal, characteristic, complemented
3C23×#2134C211yesmaximal, characteristic, complemented
4C23×#2122C211yesmaximal, characteristic, complemented
5C23×#1796C211yesmaximal, characteristic, complemented
6C23×#1395C211yesmaximal, characteristic, complemented
7C23×#1392C211yesmaximal, characteristic, complemented
8C2×C23×2−1+4C2211yescharacteristic
9C2×C23×AGL(1,ℤ/8)C411yescharacteristic, complemented
10C22×SD16×C23C411yescharacteristic, complemented
11C23×#568—21no—
12C23×#535C2211yescharacteristic
13C23×#449—21no—
14C23×#449—21no—
15C23×#448—21no—
16C23×#448—21no—
17C23×#421—21no—
18C23×#419—21no—
19C23×#419C411yescharacteristic, complemented
20C23×#416C411yescharacteristic, complemented
21C23×#413—22no—
22C23×#413C2211yescharacteristic
23C23×#413C2211yescharacteristic
24C23×#413C2211yescharacteristic
25C2×C23×#101—21no—
26C2×C23×#101—22no—
27C23×#410C2211yescharacteristic
28C23×#408C2211yescharacteristic
29C23×#355—22no—
30C23×AΣL(1,𝔽16)—22no—
31C23×2−1+4—42no—
32C23×2−1+4—42no—
33C2×#41×C23—21no—
34C2×#41×C23—21no—
35C2×#41×C23—21no—
36C2×#41×C23C2×C411yescharacteristic
37C2×#41×C23D411yescharacteristic
38C2×#41×C23D411yescharacteristic
39C2×#41×C23C2311yescharacteristic
40C22×Q8×C23—21no—
41C22×Q8×C23C2×C411yescharacteristic
42C22×Q8×C23D411yescharacteristic
43C22×Q8×C23D411yescharacteristic
44C22×D4×C23C2×C411yescharacteristic
45C23×AGL(1,ℤ/8)—42no—
46C2×SD16×C23—21no—
47C2×SD16×C23—21no—
48C2×SD16×C23—21no—
49C2×SD16×C23—42no—
50C2×D8×C23—21no—
51C23×#131—42no—
52C23×#131—42no—
53C2×M16×C23—24no—
54C2×M16×C23—21no—
55C2×M16×C23—21no—
56C2×M16×C23—21no—
57C2×M16×C23C2×C411yescharacteristic
58C22×C8×C23C2×C411yescharacteristic
59C23×#127—21no—
60C23×#124—21no—
61C23×#121—21no—
62C23×#121—21no—
63C23×#120—21no—
64C23×#120—21no—
65C23×#117—21no—
66C23×#117C2×C411yescharacteristic
67C23×#104—42no—
68C23×#103—21no—
69C23×#103—21no—
70C23×#103—21no—
71C23×#103—42no—
72C23×#102—21no—
73C23×#101—24no—
74C23×#101—21no—
75C23×#101—21no—
76C23×#101D412yes—
77C23×#100—21no—
78C23×#100—21no—
79C23×#100D412yes—
80C23×#99—21no—
81C23×#99—21no—
82C23×C22⋊C8—21no—
83C23×C22⋊C8—21no—
84C24×C23—21no—
85#41×C23—22no—
86#41×C23—22no—
87#41×C23—22no—
88#41×C23—22no—
89#41×C23—41no—
90#41×C23—41no—
91#41×C23—41no—
92#41×C23—41no—
93#41×C23—44no—
94#41×C23—44no—
95#41×C23—44no—
96C2×Q8×C23—21no—
97C2×Q8×C23—21no—
98C2×Q8×C23—21no—
99C2×Q8×C23—22no—
100C2×Q8×C23—22no—
101C2×Q8×C23—41no—
102C2×Q8×C23—41no—
103C2×Q8×C23—41no—
104C2×Q8×C23—41no—
105C2×Q8×C23—44no—
106C2×Q8×C23C22⋊C411yescharacteristic
107C2×Q8×C23C22⋊C411yescharacteristic
108C2×Q8×C23C22⋊C411yescharacteristic
109C2×Q8×C23C2×D411yescharacteristic
110C2×D4×C23—21no—
111C2×D4×C23—21no—
112C2×D4×C23—21no—
113C2×D4×C23—21no—
114C2×D4×C23—21no—
115C2×D4×C23—41no—
116C2×D4×C23—41no—
117C2×D4×C23—41no—
118C2×D4×C23—41no—
119C2×D4×C23C22⋊C411yescharacteristic
120C2×D4×C23C2×D411yescharacteristic
121C2×D4×C23#4111yescharacteristic
122C2×D4×C23#4111yescharacteristic
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—21no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—21no—
129C22×C4×C23C22×C411yescharacteristic
130C22×C4×C23C2×D411yescharacteristic
131C22×C4×C23C2×D411yescharacteristic
132SD16×C23—42no—
133SD16×C23—42no—
134SD16×C23—42no—
135D8×C23—42no—
136M16×C23—22no—
137M16×C23—22no—
138M16×C23—44no—
139M16×C23—41no—
140M16×C23—41no—
141M16×C23—41no—
142M16×C23—41no—
143M16×C23—41no—
144C2×C8×C23—21no—
145C2×C8×C23—21no—
146C2×C8×C23—44no—
147C2×C8×C23—41no—
148C2×C8×C23—41no—
149C2×C8×C23—41no—
150C2×C8×C23—41no—
151C4⋊C4×C23—21no—
152C4⋊C4×C23—41no—
153C22⋊C4×C23—21no—
154C22⋊C4×C23—41no—
155C22⋊C4×C23—41no—
156C22⋊C4×C23—41no—
157C42×C23—21no—
158C42×C23—41no—
159#13098C2311yesmaximal, characteristic, direct product factor, Sylow
160C23×C23—21no—
161C23×C23—41no—
162C23×C23—81no—
163C23×C23—161no—
164C23×C23C4×D411yescharacteristic
165C23×C23#12111yescharacteristic
166C23×C23#12311yescharacteristic
167Q8×C23—21no—
168Q8×C23—21no—
169Q8×C23—21no—
170Q8×C23—21no—
171Q8×C23—21no—
172Q8×C23—41no—
173Q8×C23—42no—
174Q8×C23—42no—
175Q8×C23—42no—
176Q8×C23#9912yes—
177D4×C23—24no—
178D4×C23—21no—
179D4×C23—41no—
180D4×C23—41no—
181D4×C23—41no—
182D4×C23—42no—
183D4×C23—42no—
184D4×C23—41no—
185D4×C23—41no—
186D4×C23—81no—
187D4×C23—161no—
188D4×C23—161no—
189D4×C23#9912yes—
190C2×C4×C23—24no—
191C2×C4×C23—21no—
192C2×C4×C23—21no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—21no—
197C2×C4×C23—21no—
198C2×C4×C23—21no—
199C2×C4×C23—21no—
200C2×C4×C23—42no—
201C2×C4×C23—42no—
202C2×C4×C23—42no—
203C2×C4×C23—42no—
204C2×C4×C23—42no—
205C2×C4×C23—41no—
206C2×C4×C23—81no—
207C2×C4×C23C2×C22⋊C411yescharacteristic
208C2×C4×C23C4×D411yescharacteristic
209C2×C4×C23#12011yescharacteristic
210C2×C4×C23#12111yescharacteristic
211C8×C23—42no—
212C8×C23—42no—
213C8×C23—41no—
214C8×C23—41no—
215C8×C23—81no—
216#3010C2×C2311yescharacteristic
217#2889C2×C2311yescharacteristic, complemented
218#2134C2×C2311yescharacteristic, complemented
219#2122C2×C2311yescharacteristic, complemented
220#1796C2×C2311yescharacteristic, complemented
221#1395C2×C2311yescharacteristic, complemented
222#1392C2×C2311yescharacteristic, complemented
223C22×C23—21no—
224C22×C23—21no—
225C22×C23—22no—
226C22×C23—21no—
227C22×C23—41no—
228C22×C23—41no—
229C22×C23—41no—
230C22×C23—81no—
231C22×C23—81no—
232C22×C23—81no—
233C22×C23—81no—
234C22×C23—162no—
235C22×C23#38611yescharacteristic
236C4×C23—21no—
237C4×C23—21no—
238C4×C23—22no—
239C4×C23—21no—
240C4×C23—41no—
241C4×C23—41no—
242C4×C23—81no—
243C4×C23—161no—
244C4×C23C2×#9911yescharacteristic
245C4×C23C2×#9911yescharacteristic
246C2×2−1+4C22×C2311yescharacteristic
247C2×AGL(1,ℤ/8)C4×C2311yescharacteristic, complemented
248C22×SD16C4×C2311yescharacteristic, complemented
249#568—21no—
250#535C22×C2311yescharacteristic
251#449—21no—
252#449—21no—
253#448—21no—
254#448—21no—
255#421—21no—
256#419—21no—
257#419C4×C2311yescharacteristic, complemented
258#416C4×C2311yescharacteristic, complemented
259#413—22no—
260#413C22×C2311yescharacteristic
261#413C22×C2311yescharacteristic
262#413C22×C2311yescharacteristic
263C2×#101—21no—
264C2×#101—22no—
265#410C22×C2311yescharacteristic
266#408C22×C2311yescharacteristic
267#355—22no—
268AΣL(1,𝔽16)—22no—
269C2×C23—21no—
270C2×C23—41no—
271C2×C23—41no—
272C2×C23—41no—
273C2×C23—42no—
274C2×C23—161no—
275C2×C23#189811yescharacteristic
2762−1+4—42no—
2772−1+4—42no—
278C2×#41—21no—
279C2×#41—21no—
280C2×#41—21no—
281C2×#41C2×C4×C2311yescharacteristic
282C2×#41D4×C2311yescharacteristic
283C2×#41D4×C2311yescharacteristic
284C2×#41C23×C2311yescharacteristic
285C22×Q8—21no—
286C22×Q8C2×C4×C2311yescharacteristic
287C22×Q8D4×C2311yescharacteristic
288C22×Q8D4×C2311yescharacteristic
289C22×D4C2×C4×C2311yescharacteristic
290AGL(1,ℤ/8)—42no—
291C2×SD16—21no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—42no—
295C2×D8—21no—
296#131—42no—
297#131—42no—
298C2×M16—24no—
299C2×M16—21no—
300C2×M16—21no—
301C2×M16—21no—
302C2×M16C2×C4×C2311yescharacteristic
303C22×C8C2×C4×C2311yescharacteristic
304#127—21no—
305#124—21no—
306#121—21no—
307#121—21no—
308#120—21no—
309#120—21no—
310#117—21no—
311#117C2×C4×C2311yescharacteristic
312#104—42no—
313#103—21no—
314#103—21no—
315#103—21no—
316#103—42no—
317#102—21no—
318#101—24no—
319#101—21no—
320#101—21no—
321#101D4×C2312yes—
322#100—21no—
323#100—21no—
324#100D4×C2312yes—
325#99—21no—
326#99—21no—
327C22⋊C8—21no—
328C22⋊C8—21no—
329C23#1309811yescharacteristic, direct product factor, Sylow
330C24—21no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—44no—
352C2×Q8C22⋊C4×C2311yescharacteristic
353C2×Q8C22⋊C4×C2311yescharacteristic
354C2×Q8C22⋊C4×C2311yescharacteristic
355C2×Q8C2×D4×C2311yescharacteristic
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4C22⋊C4×C2311yescharacteristic
366C2×D4C2×D4×C2311yescharacteristic
367C2×D4#41×C2311yescharacteristic
368C2×D4#41×C2311yescharacteristic
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C22×C4×C2311yescharacteristic
376C22×C4C2×D4×C2311yescharacteristic
377C22×C4C2×D4×C2311yescharacteristic
378SD16—42no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—22no—
383M16—22no—
384M16—44no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—44no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C22⋊C4—21no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C42—21no—
404C42—41no—
405C23—21no—
406C23—41no—
407C23—81no—
408C23—161no—
409C23C4×D4×C2311yescharacteristic
410C23C23×#12111yescharacteristic
411C23C23×#12311yescharacteristic
412Q8—21no—
413Q8—21no—
414Q8—21no—
415Q8—21no—
416Q8—21no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8C23×#9912yes—
422D4—24no—
423D4—21no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—42no—
428D4—42no—
429D4—41no—
430D4—41no—
431D4—81no—
432D4—161no—
433D4—161no—
434D4C23×#9912yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4C2×C22⋊C4×C2311yescharacteristic
453C2×C4C4×D4×C2311yescharacteristic
454C2×C4C23×#12011yescharacteristic
455C2×C4C23×#12111yescharacteristic
456C8—42no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—81no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—162no—
473C22C23×#38611yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—161no—
482C4C2×C23×#9911yescharacteristic
483C4C2×C23×#9911yescharacteristic
484C2—21no—
485C2—41no—
486C2—41no—
487C2—41no—
488C2—42no—
489C2—161no—
490C2C23×#189811yescharacteristic
subgroupG/H
center, socleC2×C23#1898
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3010C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×C23×2−1+4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2889C2semidirect
3. C23×#2134C2semidirect
4. C23×#2122C2semidirect
5. C23×#1796C2semidirect
6. C23×#1395C2semidirect
7. C23×#1392C2semidirect
9. C2×C23×AGL(1,ℤ/8)C4semidirect
10. C22×SD16×C23C4semidirect
19. C23×#419C4semidirect
20. C23×#416C4semidirect
159. #13098C23direct (N × H)
217. #2889C2×C23semidirect
218. #2134C2×C23semidirect
219. #2122C2×C23semidirect
220. #1796C2×C23semidirect
221. #1395C2×C23semidirect
222. #1392C2×C23semidirect
247. C2×AGL(1,ℤ/8)C4×C23semidirect
248. C22×SD16C4×C23semidirect
257. #419C4×C23semidirect
258. #416C4×C23semidirect
329. C23#13098direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30102
C23×#28892
C23×#21342
C23×#21222
C23×#17962
C23×#13952
C23×#13922
#1309823