Finite Group Databasebeta

← C23×#13097 #19160723

Subgroups — C23×#13097 #19160723

Order 5888.

650 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×#3009C211yesmaximal, characteristic
2C23×#2888C211yesmaximal, characteristic
3C23×#2135C211yesmaximal, characteristic, complemented
4C23×#2123C211yesmaximal, characteristic, complemented
5C23×#1795C211yesmaximal, characteristic
6C23×#1395C211yesmaximal, characteristic, complemented
7C23×#1392C211yesmaximal, characteristic, complemented
8C2×C23×H(5,𝔽2)C2211yescharacteristic
9C2×C23×#137C411yescharacteristic, complemented
10C22×SD16×C23C411yescharacteristic, complemented
11C23×#568—21no—
12C23×#536C2211yescharacteristic
13C23×#452—21no—
14C23×#452—21no—
15C23×#450—21no—
16C23×#450—21no—
17C23×#421—21no—
18C23×#418—21no—
19C23×#418C411yescharacteristic, complemented
20C23×#416C411yescharacteristic, complemented
21C23×#413—22no—
22C23×#413C2211yescharacteristic
23C23×#413C2211yescharacteristic
24C23×#413C2211yescharacteristic
25C2×C23×#100—21no—
26C2×C23×#100—22no—
27C23×#410C2211yescharacteristic
28C23×#408C2211yescharacteristic
29C23×#356—22no—
30C23×#352—22no—
31C23×H(5,𝔽2)—42no—
32C23×H(5,𝔽2)—42no—
33C2×#41×C23—21no—
34C2×#41×C23C2×C411yescharacteristic
35C2×#41×C23D411yescharacteristic
36C2×#41×C23D411yescharacteristic
37C2×#41×C23C2311yescharacteristic
38C22×Q8×C23C2×C411yescharacteristic
39C22×D4×C23—21no—
40C22×D4×C23—21no—
41C22×D4×C23—21no—
42C22×D4×C23C2×C411yescharacteristic
43C22×D4×C23D411yescharacteristic
44C22×D4×C23D411yescharacteristic
45C23×#137—42no—
46C2×Q16×C23—21no—
47C2×SD16×C23—21no—
48C2×SD16×C23—21no—
49C2×SD16×C23—21no—
50C2×SD16×C23—42no—
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×#102—21no—
70C23×#102—21no—
71C23×#102—21no—
72C23×#102—42no—
73C23×#101—21no—
74C23×#101—21no—
75C23×#101D412yes—
76C23×#100—24no—
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—
85C24×C23—21no—
86C24×C23—21no—
87#41×C23—22no—
88#41×C23—22no—
89#41×C23—22no—
90#41×C23—22no—
91#41×C23—41no—
92#41×C23—41no—
93#41×C23—41no—
94#41×C23—41no—
95#41×C23—44no—
96C2×Q8×C23—21no—
97C2×Q8×C23—21no—
98C2×Q8×C23—41no—
99C2×Q8×C23—41no—
100C2×Q8×C23C22⋊C411yescharacteristic
101C2×Q8×C23C2×D411yescharacteristic
102C2×D4×C23—21no—
103C2×D4×C23—22no—
104C2×D4×C23—22no—
105C2×D4×C23—21no—
106C2×D4×C23—21no—
107C2×D4×C23—21no—
108C2×D4×C23—21no—
109C2×D4×C23—21no—
110C2×D4×C23—41no—
111C2×D4×C23—41no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C2×D4×C23—44no—
115C2×D4×C23—44no—
116C2×D4×C23—44no—
117C2×D4×C23C22⋊C411yescharacteristic
118C2×D4×C23C22⋊C411yescharacteristic
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×C23C22×C411yescharacteristic
128C22×C4×C23C2×D411yescharacteristic
129C22×C4×C23C2×D411yescharacteristic
130Q16×C23—42no—
131SD16×C23—42no—
132SD16×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#13097C2311yesmaximal, characteristic, direct product factor, Sylow
160C23×C23—21no—
161C23×C23—21no—
162C23×C23—21no—
163C23×C23—42no—
164C23×C23—42no—
165C23×C23—42no—
166C23×C23—161no—
167C23×C23—161no—
168C23×C23—161no—
169C23×C23C4×D411yescharacteristic
170C23×C23#12111yescharacteristic
171C23×C23#12311yescharacteristic
172Q8×C23—21no—
173Q8×C23—41no—
174Q8×C23—41no—
175Q8×C23—41no—
176Q8×C23—81no—
177Q8×C23#9912yes—
178D4×C23—24no—
179D4×C23—21no—
180D4×C23—21no—
181D4×C23—21no—
182D4×C23—21no—
183D4×C23—21no—
184D4×C23—41no—
185D4×C23—41no—
186D4×C23—41no—
187D4×C23—42no—
188D4×C23—42no—
189D4×C23—42no—
190D4×C23—42no—
191D4×C23—42no—
192D4×C23#9912yes—
193C2×C4×C23—24no—
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—21no—
201C2×C4×C23—42no—
202C2×C4×C23—42no—
203C2×C4×C23—41no—
204C2×C4×C23—41no—
205C2×C4×C23—81no—
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#3009C2×C2311yescharacteristic
217#2888C2×C2311yescharacteristic
218#2135C2×C2311yescharacteristic, complemented
219#2123C2×C2311yescharacteristic, complemented
220#1795C2×C2311yescharacteristic
221#1395C2×C2311yescharacteristic, complemented
222#1392C2×C2311yescharacteristic, complemented
223C22×C23—21no—
224C22×C23—21no—
225C22×C23—21no—
226C22×C23—22no—
227C22×C23—41no—
228C22×C23—41no—
229C22×C23—41no—
230C22×C23—41no—
231C22×C23—41no—
232C22×C23—81no—
233C22×C23—81no—
234C22×C23—81no—
235C22×C23—162no—
236C22×C23—162no—
237C22×C23—162no—
238C22×C23#38611yescharacteristic
239C4×C23—21no—
240C4×C23—21no—
241C4×C23—22no—
242C4×C23—21no—
243C4×C23—81no—
244C4×C23—81no—
245C4×C23—161no—
246C4×C23C2×#9911yescharacteristic
247C4×C23C2×#9911yescharacteristic
248C2×H(5,𝔽2)C22×C2311yescharacteristic
249C2×#137C4×C2311yescharacteristic, complemented
250C22×SD16C4×C2311yescharacteristic, complemented
251#568—21no—
252#536C22×C2311yescharacteristic
253#452—21no—
254#452—21no—
255#450—21no—
256#450—21no—
257#421—21no—
258#418—21no—
259#418C4×C2311yescharacteristic, complemented
260#416C4×C2311yescharacteristic, complemented
261#413—22no—
262#413C22×C2311yescharacteristic
263#413C22×C2311yescharacteristic
264#413C22×C2311yescharacteristic
265C2×#100—21no—
266C2×#100—22no—
267#410C22×C2311yescharacteristic
268#408C22×C2311yescharacteristic
269#356—22no—
270#352—22no—
271C2×C23—21no—
272C2×C23—41no—
273C2×C23—41no—
274C2×C23—41no—
275C2×C23—42no—
276C2×C23—81no—
277C2×C23—81no—
278C2×C23#189811yescharacteristic
279H(5,𝔽2)—42no—
280H(5,𝔽2)—42no—
281C2×#41—21no—
282C2×#41C2×C4×C2311yescharacteristic
283C2×#41D4×C2311yescharacteristic
284C2×#41D4×C2311yescharacteristic
285C2×#41C23×C2311yescharacteristic
286C22×Q8C2×C4×C2311yescharacteristic
287C22×D4—21no—
288C22×D4—21no—
289C22×D4—21no—
290C22×D4C2×C4×C2311yescharacteristic
291C22×D4D4×C2311yescharacteristic
292C22×D4D4×C2311yescharacteristic
293#137—42no—
294C2×Q16—21no—
295C2×SD16—21no—
296C2×SD16—21no—
297C2×SD16—21no—
298C2×SD16—42no—
299#131—42no—
300#131—42no—
301C2×M16—24no—
302C2×M16—21no—
303C2×M16—21no—
304C2×M16—21no—
305C2×M16C2×C4×C2311yescharacteristic
306C22×C8C2×C4×C2311yescharacteristic
307C4⋊Q8—21no—
308#124—21no—
309#123—21no—
310#123—21no—
311C22⋊Q8—21no—
312C22⋊Q8—21no—
313#117—21no—
314#117C2×C4×C2311yescharacteristic
315#104—42no—
316#103—21no—
317#102—21no—
318#102—21no—
319#102—21no—
320#102—42no—
321#101—21no—
322#101—21no—
323#101D4×C2312yes—
324#100—24no—
325#100—21no—
326#100—21no—
327#100D4×C2312yes—
328#99—21no—
329#99—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C23#1309711yescharacteristic, direct product factor, Sylow
333C24—21no—
334C24—21no—
335C24—21no—
336#41—22no—
337#41—22no—
338#41—22no—
339#41—22no—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—44no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8C22⋊C4×C2311yescharacteristic
350C2×Q8C2×D4×C2311yescharacteristic
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4C22⋊C4×C2311yescharacteristic
367C2×D4C22⋊C4×C2311yescharacteristic
368C2×D4C22⋊C4×C2311yescharacteristic
369C2×D4C2×D4×C2311yescharacteristic
370C2×D4#41×C2311yescharacteristic
371C2×D4#41×C2311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4C22×C4×C2311yescharacteristic
377C22×C4C2×D4×C2311yescharacteristic
378C22×C4C2×D4×C2311yescharacteristic
379Q16—42no—
380SD16—42no—
381SD16—42no—
382SD16—42no—
383M16—22no—
384M16—22no—
385M16—44no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—44no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C4⋊C4—21no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C22⋊C4—21no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C42—21no—
407C42—41no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—161no—
415C23—161no—
416C23—161no—
417C23C4×D4×C2311yescharacteristic
418C23C23×#12111yescharacteristic
419C23C23×#12311yescharacteristic
420Q8—21no—
421Q8—41no—
422Q8—41no—
423Q8—41no—
424Q8—81no—
425Q8C23×#9912yes—
426D4—24no—
427D4—21no—
428D4—21no—
429D4—21no—
430D4—21no—
431D4—21no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4C23×#9912yes—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4C2×C22⋊C4×C2311yescharacteristic
456C2×C4C4×D4×C2311yescharacteristic
457C2×C4C23×#12011yescharacteristic
458C2×C4C23×#12111yescharacteristic
459C8—42no—
460C8—42no—
461C8—41no—
462C8—41no—
463C8—81no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—162no—
477C22—162no—
478C22—162no—
479C22C23×#38611yescharacteristic
480C4—21no—
481C4—21no—
482C4—22no—
483C4—21no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4C2×C23×#9911yescharacteristic
488C4C2×C23×#9911yescharacteristic
489C2—21no—
490C2—41no—
491C2—41no—
492C2—41no—
493C2—42no—
494C2—81no—
495C2—81no—
496C2C23×#189811yescharacteristic
subgroupG/H
center, socleC2×C23#1898
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC2×C23×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×C23×H(5,𝔽2) ◁ 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×SD16×C23C4semidirect
19. C23×#418C4semidirect
20. C23×#416C4semidirect
159. #13097C23direct (N × H)
218. #2135C2×C23semidirect
219. #2123C2×C23semidirect
221. #1395C2×C23semidirect
222. #1392C2×C23semidirect
249. C2×#137C4×C23semidirect
250. C22×SD16C4×C23semidirect
259. #418C4×C23semidirect
260. #416C4×C23semidirect
332. C23#13097direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30092
C23×#28882
C23×#21352
C23×#21232
C23×#17952
C23×#13952
C23×#13922
#1309723