Finite Group Databasebeta

← C23×#13096 #19160722

Subgroups — C23×#13096 #19160722

Order 5888.

624 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3011C211yesmaximal, characteristic
2C23×#2889C211yesmaximal, characteristic
3C23×#2135C211yesmaximal, characteristic, complemented
4C23×#2123C211yesmaximal, characteristic, complemented
5C23×#1796C211yesmaximal, characteristic
6C23×#1395C211yesmaximal, characteristic, complemented
7C23×#1392C211yesmaximal, characteristic, complemented
8C2×C23×2−1+4C2211yescharacteristic
9C2×C23×#137C411yescharacteristic, complemented
10C22×Q16×C23C411yescharacteristic, complemented
11C23×#568—21no—
12C23×#536C2211yescharacteristic
13C23×#452—21no—
14C23×#452—21no—
15C23×#451—21no—
16C23×#451—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×#356—22no—
30C23×#352—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×C23C2×C411yescharacteristic
43C22×Q8×C23D411yescharacteristic
44C22×Q8×C23D411yescharacteristic
45C23×#137—42no—
46C2×Q16×C23—21no—
47C2×Q16×C23—21no—
48C2×Q16×C23—21no—
49C2×Q16×C23—42no—
50C2×SD16×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×C4⋊Q8—21no—
60C23×#124—21no—
61C23×#123—21no—
62C23×#123—21no—
63C23×C22⋊Q8—21no—
64C23×C22⋊Q8—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—
84#41×C23—22no—
85#41×C23—22no—
86#41×C23—22no—
87#41×C23—22no—
88#41×C23—41no—
89#41×C23—41no—
90#41×C23—41no—
91#41×C23—41no—
92#41×C23—44no—
93#41×C23—44no—
94#41×C23—44no—
95C2×Q8×C23—21no—
96C2×Q8×C23—21no—
97C2×Q8×C23—21no—
98C2×Q8×C23—22no—
99C2×Q8×C23—22no—
100C2×Q8×C23—21no—
101C2×Q8×C23—21no—
102C2×Q8×C23—41no—
103C2×Q8×C23—41no—
104C2×Q8×C23—41no—
105C2×Q8×C23—41no—
106C2×Q8×C23—44no—
107C2×Q8×C23—41no—
108C2×Q8×C23—41no—
109C2×Q8×C23C22⋊C411yescharacteristic
110C2×Q8×C23C22⋊C411yescharacteristic
111C2×Q8×C23C22⋊C411yescharacteristic
112C2×Q8×C23C2×D411yescharacteristic
113C2×D4×C23—21no—
114C2×D4×C23—21no—
115C2×D4×C23—21no—
116C2×D4×C23C22⋊C411yescharacteristic
117C2×D4×C23C2×D411yescharacteristic
118C2×D4×C23#4111yescharacteristic
119C2×D4×C23#4111yescharacteristic
120C22×C4×C23—21no—
121C22×C4×C23—21no—
122C22×C4×C23—21no—
123C22×C4×C23—21no—
124C22×C4×C23—21no—
125C22×C4×C23—21no—
126C22×C4×C23—21no—
127C22×C4×C23C22×C411yescharacteristic
128C22×C4×C23C2×D411yescharacteristic
129C22×C4×C23C2×D411yescharacteristic
130Q16×C23—42no—
131Q16×C23—42no—
132Q16×C23—42no—
133SD16×C23—42no—
134M16×C23—22no—
135M16×C23—22no—
136M16×C23—44no—
137M16×C23—41no—
138M16×C23—41no—
139M16×C23—41no—
140M16×C23—41no—
141M16×C23—41no—
142C2×C8×C23—21no—
143C2×C8×C23—21no—
144C2×C8×C23—44no—
145C2×C8×C23—41no—
146C2×C8×C23—41no—
147C2×C8×C23—41no—
148C2×C8×C23—41no—
149C4⋊C4×C23—21no—
150C4⋊C4×C23—41no—
151C4⋊C4×C23—41no—
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#13096C2311yesmaximal, characteristic, direct product factor, Sylow
160C23×C23—21no—
161C23×C23C4×D411yescharacteristic
162C23×C23#12111yescharacteristic
163C23×C23#12311yescharacteristic
164Q8×C23—21no—
165Q8×C23—21no—
166Q8×C23—21no—
167Q8×C23—21no—
168Q8×C23—21no—
169Q8×C23—41no—
170Q8×C23—42no—
171Q8×C23—42no—
172Q8×C23—42no—
173Q8×C23—41no—
174Q8×C23—41no—
175Q8×C23—81no—
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#9912yes—
185C2×C4×C23—24no—
186C2×C4×C23—21no—
187C2×C4×C23—21no—
188C2×C4×C23—21no—
189C2×C4×C23—21no—
190C2×C4×C23—21no—
191C2×C4×C23—21no—
192C2×C4×C23—21no—
193C2×C4×C23—21no—
194C2×C4×C23—21no—
195C2×C4×C23—42no—
196C2×C4×C23—42no—
197C2×C4×C23—42no—
198C2×C4×C23—42no—
199C2×C4×C23—42no—
200C2×C4×C23—41no—
201C2×C4×C23—41no—
202C2×C4×C23—81no—
203C2×C4×C23—81no—
204C2×C4×C23C2×C22⋊C411yescharacteristic
205C2×C4×C23C4×D411yescharacteristic
206C2×C4×C23#12011yescharacteristic
207C2×C4×C23#12111yescharacteristic
208C8×C23—42no—
209C8×C23—42no—
210C8×C23—41no—
211C8×C23—41no—
212C8×C23—81no—
213#3011C2×C2311yescharacteristic
214#2889C2×C2311yescharacteristic
215#2135C2×C2311yescharacteristic, complemented
216#2123C2×C2311yescharacteristic, complemented
217#1796C2×C2311yescharacteristic
218#1395C2×C2311yescharacteristic, complemented
219#1392C2×C2311yescharacteristic, complemented
220C22×C23—21no—
221C22×C23—21no—
222C22×C23—21no—
223C22×C23—22no—
224C22×C23—41no—
225C22×C23—41no—
226C22×C23—41no—
227C22×C23—81no—
228C22×C23#38611yescharacteristic
229C4×C23—21no—
230C4×C23—21no—
231C4×C23—22no—
232C4×C23—21no—
233C4×C23—41no—
234C4×C23—41no—
235C4×C23—81no—
236C4×C23—81no—
237C4×C23—161no—
238C4×C23C2×#9911yescharacteristic
239C4×C23C2×#9911yescharacteristic
240C2×2−1+4C22×C2311yescharacteristic
241C2×#137C4×C2311yescharacteristic, complemented
242C22×Q16C4×C2311yescharacteristic, complemented
243#568—21no—
244#536C22×C2311yescharacteristic
245#452—21no—
246#452—21no—
247#451—21no—
248#451—21no—
249#421—21no—
250#419—21no—
251#419C4×C2311yescharacteristic, complemented
252#416C4×C2311yescharacteristic, complemented
253#413—22no—
254#413C22×C2311yescharacteristic
255#413C22×C2311yescharacteristic
256#413C22×C2311yescharacteristic
257C2×#101—21no—
258C2×#101—22no—
259#410C22×C2311yescharacteristic
260#408C22×C2311yescharacteristic
261#356—22no—
262#352—22no—
263C2×C23—21no—
264C2×C23—41no—
265C2×C23—41no—
266C2×C23—41no—
267C2×C23—42no—
268C2×C23#189811yescharacteristic
2692−1+4—42no—
2702−1+4—42no—
271C2×#41—21no—
272C2×#41—21no—
273C2×#41—21no—
274C2×#41C2×C4×C2311yescharacteristic
275C2×#41D4×C2311yescharacteristic
276C2×#41D4×C2311yescharacteristic
277C2×#41C23×C2311yescharacteristic
278C22×Q8—21no—
279C22×Q8C2×C4×C2311yescharacteristic
280C22×Q8C2×C4×C2311yescharacteristic
281C22×Q8D4×C2311yescharacteristic
282C22×Q8D4×C2311yescharacteristic
283#137—42no—
284C2×Q16—21no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16—42no—
288C2×SD16—21no—
289#131—42no—
290#131—42no—
291C2×M16—24no—
292C2×M16—21no—
293C2×M16—21no—
294C2×M16—21no—
295C2×M16C2×C4×C2311yescharacteristic
296C22×C8C2×C4×C2311yescharacteristic
297C4⋊Q8—21no—
298#124—21no—
299#123—21no—
300#123—21no—
301C22⋊Q8—21no—
302C22⋊Q8—21no—
303#117—21no—
304#117C2×C4×C2311yescharacteristic
305#104—42no—
306#103—21no—
307#103—21no—
308#103—21no—
309#103—42no—
310#102—21no—
311#101—24no—
312#101—21no—
313#101—21no—
314#101D4×C2312yes—
315#100—21no—
316#100—21no—
317#100D4×C2312yes—
318#99—21no—
319#99—21no—
320C22⋊C8—21no—
321C22⋊C8—21no—
322C23#1309611yescharacteristic, direct product factor, Sylow
323#41—22no—
324#41—22no—
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—44no—
332#41—44no—
333#41—44no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8C22⋊C4×C2311yescharacteristic
349C2×Q8C22⋊C4×C2311yescharacteristic
350C2×Q8C22⋊C4×C2311yescharacteristic
351C2×Q8C2×D4×C2311yescharacteristic
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4C22⋊C4×C2311yescharacteristic
356C2×D4C2×D4×C2311yescharacteristic
357C2×D4#41×C2311yescharacteristic
358C2×D4#41×C2311yescharacteristic
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4C22×C4×C2311yescharacteristic
367C22×C4C2×D4×C2311yescharacteristic
368C22×C4C2×D4×C2311yescharacteristic
369Q16—42no—
370Q16—42no—
371Q16—42no—
372SD16—42no—
373M16—22no—
374M16—22no—
375M16—44no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—44no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C4⋊C4—21no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C22⋊C4—21no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C42—21no—
397C42—41no—
398C23—21no—
399C23C4×D4×C2311yescharacteristic
400C23C23×#12111yescharacteristic
401C23C23×#12311yescharacteristic
402Q8—21no—
403Q8—21no—
404Q8—21no—
405Q8—21no—
406Q8—21no—
407Q8—41no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—41no—
412Q8—41no—
413Q8—81no—
414Q8C23×#9912yes—
415D4—24no—
416D4—21no—
417D4—41no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—42no—
422D4C23×#9912yes—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4C2×C22⋊C4×C2311yescharacteristic
443C2×C4C4×D4×C2311yescharacteristic
444C2×C4C23×#12011yescharacteristic
445C2×C4C23×#12111yescharacteristic
446C8—42no—
447C8—42no—
448C8—41no—
449C8—41no—
450C8—81no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—81no—
459C22C23×#38611yescharacteristic
460C4—21no—
461C4—21no—
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—81no—
467C4—81no—
468C4—161no—
469C4C2×C23×#9911yescharacteristic
470C4C2×C23×#9911yescharacteristic
471C2—21no—
472C2—41no—
473C2—41no—
474C2—41no—
475C2—42no—
476C2C23×#189811yescharacteristic
subgroupG/H
center, socleC2×C23#1898
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC2×C23×2−1+4C22
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
3. C23×#2135C2semidirect
4. C23×#2123C2semidirect
6. C23×#1395C2semidirect
7. C23×#1392C2semidirect
9. C2×C23×#137C4semidirect
10. C22×Q16×C23C4semidirect
19. C23×#419C4semidirect
20. C23×#416C4semidirect
159. #13096C23direct (N × H)
215. #2135C2×C23semidirect
216. #2123C2×C23semidirect
218. #1395C2×C23semidirect
219. #1392C2×C23semidirect
241. C2×#137C4×C23semidirect
242. C22×Q16C4×C23semidirect
251. #419C4×C23semidirect
252. #416C4×C23semidirect
322. C23#13096direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30112
C23×#28892
C23×#21352
C23×#21232
C23×#17962
C23×#13952
C23×#13922
#1309623