Finite Group Databasebeta

← C23×#24079 #19171705

Subgroups — C23×#24079 #19171705

Order 5888.

1030 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3492C211yesmaximal, characteristic
2C23×#3093C212yesmaximal, complemented
3C23×#2317C211yesmaximal, characteristic, complemented
4C23×#2061C211yesmaximal, characteristic, complemented
5C23×#2060C211yesmaximal, characteristic, complemented
6C23×#2040C211yesmaximal, characteristic, complemented
7C23×#2039C211yesmaximal, characteristic, complemented
8C23×#1897C212yesmaximal, complemented
9C23×#1860C212yesmaximal, complemented
10C23×#1859C212yesmaximal, complemented
11C23×#1821C211yesmaximal, characteristic, complemented
12C23×#554—22no—
13C23×#553—22no—
14C23×#552—22no—
15C23×#547—22no—
16C23×#544C2211yescharacteristic
17C23×#539—22no—
18C23×#537C2211yescharacteristic
19C23×#525—22no—
20C23×#525C2212yes—
21C23×#520C2212yes—
22C2×C23×#117C2211yescharacteristic
23C2×C23×#117C2211yescharacteristic
24C22×C4⋊C4×C23C2211yescharacteristic
25C23×#485—22no—
26C23×#485—22no—
27C23×#482—22no—
28C23×#482—22no—
29C23×#428C2212yes—
30C23×#428C2212yes—
31C2×C23×#104C2212yes—
32C2×C23×#104C2212yes—
33C23×#417C2212yes—
34C23×#417C2212yes—
35C23×#407C2212yes—
36C23×#389—22no—
37C23×#385—22no—
38C23×#384C2211yescharacteristic
39C23×#384C2211yescharacteristic
40C23×#382C2211yescharacteristic
41C23×#382C2211yescharacteristic
42C23×#343C2211yescharacteristic
43C23×#343C2211yescharacteristic
44C23×#343C2212yes—
45C23×#339C2212yes—
46C23×#339C2212yes—
47C2×#41×C23—22no—
48C2×#41×C23C2312yes—
49C23×C4×C23D412yes—
50C23×C4×C23C2311yescharacteristic
51C2×M16×C23C2312yes—
52C2×M16×C23C2312yes—
53C23×C4⋊Q8—22no—
54C23×#126—24no—
55C23×#125—24no—
56C23×#125—22no—
57C23×#123—24no—
58C23×#123—24no—
59C23×#123—22no—
60C23×C22⋊Q8—24no—
61C23×C22⋊Q8—22no—
62C23×C22⋊Q8—22no—
63C23×C22⋊Q8—22no—
64C23×C22⋊Q8D412yes—
65C23×C22⋊Q8D412yes—
66C23×#121—22no—
67C23×#121—22no—
68C23×#121—22no—
69C23×#121D412yes—
70C23×#121D412yes—
71C23×#120—22no—
72C4×Q8×C23—22no—
73C4×D4×C23—24no—
74C4×D4×C23—22no—
75C4×D4×C23—22no—
76C4×D4×C23—22no—
77C4×D4×C23—24no—
78C23×#117—22no—
79C23×#117—22no—
80C23×#117—22no—
81C23×#117—22no—
82C23×#117—22no—
83C23×#117—22no—
84C23×#117C2312yes—
85C2×C4⋊C4×C23—22no—
86C2×C4⋊C4×C23—22no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—22no—
89C2×C4⋊C4×C23—22no—
90C2×C4⋊C4×C23—22no—
91C2×C4⋊C4×C23D412yes—
92C2×C4⋊C4×C23C2311yescharacteristic
93C2×C4⋊C4×C23C2311yescharacteristic
94C2×C4⋊C4×C23C2311yescharacteristic
95C2×C4⋊C4×C23C2311yescharacteristic
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C42×C23—21no—
99C2×C42×C23C2311yescharacteristic
100C2×C42×C23C2311yescharacteristic
101C23×#107—22no—
102C23×#107—22no—
103C23×#106—22no—
104C23×#106—22no—
105C23×#104—24no—
106C23×#104—24no—
107C23×#103—22no—
108C23×#103—22no—
109C23×#102—22no—
110C23×#102—22no—
111C23×C22⋊C8—22no—
112C23×C22⋊C8—22no—
113C23×#95—22no—
114C23×#95—22no—
115C24×C23C2×D411yescharacteristic
116#41×C23—22no—
117#41×C23—22no—
118#41×C23—22no—
119#41×C23—22no—
120#41×C23—84no—
121C2×Q8×C23—22no—
122C2×Q8×C23C2×D412yes—
123C2×D4×C23—22no—
124C2×D4×C23—22no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23C2×D412yes—
128C22×C4×C23—21no—
129C22×C4×C23—21no—
130C22×C4×C23—22no—
131C22×C4×C23—22no—
132C22×C4×C23—21no—
133C22×C4×C23—21no—
134C22×C4×C23—24no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—82no—
140C22×C4×C23C2×D411yescharacteristic
141C22×C4×C23C2×D412yes—
142C22×C4×C23C2×D412yes—
143C22×C4×C23C2×D412yes—
144C22×C4×C23C2×D412yes—
145C22×C4×C23C2411yescharacteristic
146M16×C23—22no—
147M16×C23—22no—
148M16×C23—22no—
149M16×C23—22no—
150C2×C8×C23—22no—
151C2×C8×C23—22no—
152C4⋊C4×C23—24no—
153C4⋊C4×C23—24no—
154C4⋊C4×C23—22no—
155C4⋊C4×C23—22no—
156C4⋊C4×C23—24no—
157C4⋊C4×C23—24no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—82no—
165C4⋊C4×C23—82no—
166C4⋊C4×C23C2×D412yes—
167C4⋊C4×C23C2×D412yes—
168C22⋊C4×C23—24no—
169C22⋊C4×C23—24no—
170C22⋊C4×C23—24no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—22no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—22no—
176C22⋊C4×C23—41no—
177C22⋊C4×C23—41no—
178C42×C23—24no—
179C42×C23—22no—
180C42×C23—21no—
181C42×C23—21no—
182C42×C23—42no—
183C42×C23#4112yescomplemented
184C42×C23#4112yescomplemented
185#24079C2311yesmaximal, characteristic, direct product factor, Sylow
186C23×C23—21no—
187C23×C23—21no—
188C23×C23—22no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23#12011yescharacteristic
192C23×C23#12011yescharacteristic
193C23×C23C22×D411yescharacteristic
194Q8×C23—22no—
195Q8×C23—22no—
196Q8×C23—82no—
197D4×C23—22no—
198D4×C23—22no—
199D4×C23—44no—
200D4×C23—44no—
201D4×C23—42no—
202D4×C23—42no—
203D4×C23—82no—
204C2×C4×C23—21no—
205C2×C4×C23—21no—
206C2×C4×C23—22no—
207C2×C4×C23—22no—
208C2×C4×C23—22no—
209C2×C4×C23—22no—
210C2×C4×C23—22no—
211C2×C4×C23—22no—
212C2×C4×C23—22no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—22no—
216C2×C4×C23—22no—
217C2×C4×C23—41no—
218C2×C4×C23—41no—
219C2×C4×C23—41no—
220C2×C4×C23—41no—
221C2×C4×C23—44no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—42no—
228C2×C4×C23—41no—
229C2×C4×C23—41no—
230C2×C4×C23—82no—
231C2×C4×C23—84no—
232C2×C4×C23#12012yes—
233C2×C4×C23C22×D412yes—
234C2×C4×C23C2×#4111yescharacteristic
235C2×C4×C23C2×#4111yescharacteristic
236C2×C4×C232−1+412yes—
237C8×C23—42no—
238C8×C23—42no—
239#3492C2×C2311yescharacteristic
240#3093C2×C2312yescomplemented
241#2317C2×C2311yescharacteristic, complemented
242#2061C2×C2311yescharacteristic, complemented
243#2060C2×C2311yescharacteristic, complemented
244#2040C2×C2311yescharacteristic, complemented
245#2039C2×C2311yescharacteristic, complemented
246#1897C2×C2312yescomplemented
247#1860C2×C2312yescomplemented
248#1859C2×C2312yescomplemented
249#1821C2×C2311yescharacteristic, complemented
250C22×C23—21no—
251C22×C23—21no—
252C22×C23—21no—
253C22×C23—21no—
254C22×C23—21no—
255C22×C23—21no—
256C22×C23—21no—
257C22×C23—21no—
258C22×C23—42no—
259C22×C23—42no—
260C22×C23—42no—
261C22×C23—42no—
262C22×C23—42no—
263C22×C23C2×#12011yescharacteristic
264C22×C23#52511yescharacteristic
265C22×C23#53611yescharacteristic
266C4×C23—22no—
267C4×C23—42no—
268C4×C23—42no—
269C4×C23—42no—
270C4×C23—42no—
271C4×C23—42no—
272C4×C23—42no—
273C4×C23—42no—
274C4×C23—81no—
275C4×C23—81no—
276C4×C23#54712yes—
277C4×C23#54712yes—
278#554—22no—
279#553—22no—
280#552—22no—
281#547—22no—
282#544C22×C2311yescharacteristic
283#539—22no—
284#537C22×C2311yescharacteristic
285#525—22no—
286#525C22×C2312yes—
287#520C22×C2312yes—
288C2×#117C22×C2311yescharacteristic
289C2×#117C22×C2311yescharacteristic
290C22×C4⋊C4C22×C2311yescharacteristic
291#485—22no—
292#485—22no—
293#482—22no—
294#482—22no—
295#428C22×C2312yes—
296#428C22×C2312yes—
297C2×#104C22×C2312yes—
298C2×#104C22×C2312yes—
299#417C22×C2312yes—
300#417C22×C2312yes—
301#407C22×C2312yes—
302#389—22no—
303#385—22no—
304#384C22×C2311yescharacteristic
305#384C22×C2311yescharacteristic
306#382C22×C2311yescharacteristic
307#382C22×C2311yescharacteristic
308#343C22×C2311yescharacteristic
309#343C22×C2311yescharacteristic
310#343C22×C2312yes—
311#339C22×C2312yes—
312#339C22×C2312yes—
313C2×C23—21no—
314C2×C23—21no—
315C2×C23—41no—
316C2×C23—41no—
317C2×C23—82no—
318C2×C23#240611yescharacteristic
319C2×C23#301911yescharacteristic
320C2×C23#302011yescharacteristic
321C2×#41—22no—
322C2×#41C23×C2312yes—
323C23×C4D4×C2312yes—
324C23×C4C23×C2311yescharacteristic
325C2×M16C23×C2312yes—
326C2×M16C23×C2312yes—
327C4⋊Q8—22no—
328#126—24no—
329#125—24no—
330#125—22no—
331#123—24no—
332#123—24no—
333#123—22no—
334C22⋊Q8—22no—
335C22⋊Q8—24no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8D4×C2312yes—
339C22⋊Q8D4×C2312yes—
340#121—22no—
341#121—22no—
342#121—22no—
343#121D4×C2312yes—
344#121D4×C2312yes—
345#120—22no—
346C4×Q8—22no—
347C4×D4—24no—
348C4×D4—22no—
349C4×D4—24no—
350C4×D4—22no—
351C4×D4—22no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117—22no—
356#117—22no—
357#117—22no—
358#117C23×C2312yes—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4D4×C2312yes—
366C2×C4⋊C4C23×C2311yescharacteristic
367C2×C4⋊C4C23×C2311yescharacteristic
368C2×C4⋊C4C23×C2311yescharacteristic
369C2×C4⋊C4C23×C2311yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C42—21no—
373C2×C42C23×C2311yescharacteristic
374C2×C42C23×C2311yescharacteristic
375#107—22no—
376#107—22no—
377#106—22no—
378#106—22no—
379#104—24no—
380#104—24no—
381#103—22no—
382#103—22no—
383#102—22no—
384#102—22no—
385C22⋊C8—22no—
386C22⋊C8—22no—
387#95—22no—
388#95—22no—
389C23#2407911yescharacteristic, direct product factor, Sylow
390C24C2×D4×C2311yescharacteristic
391#41—22no—
392#41—22no—
393#41—22no—
394#41—22no—
395#41—84no—
396C2×Q8—22no—
397C2×Q8C2×D4×C2312yes—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4C2×D4×C2312yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—82no—
415C22×C4C2×D4×C2311yescharacteristic
416C22×C4C2×D4×C2312yes—
417C22×C4C2×D4×C2312yes—
418C22×C4C2×D4×C2312yes—
419C22×C4C2×D4×C2312yes—
420C22×C4C24×C2311yescharacteristic
421M16—22no—
422M16—22no—
423M16—22no—
424M16—22no—
425C2×C8—22no—
426C2×C8—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4C2×D4×C2312yes—
442C4⋊C4C2×D4×C2312yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—24no—
454C42—22no—
455C42—21no—
456C42—21no—
457C42—42no—
458C42#41×C2312yescomplemented
459C42#41×C2312yescomplemented
460C23—21no—
461C23—21no—
462C23—22no—
463C23—41no—
464C23—41no—
465C23C23×#12011yescharacteristic
466C23C23×#12011yescharacteristic
467C23C22×D4×C2311yescharacteristic
468Q8—22no—
469Q8—22no—
470Q8—82no—
471D4—22no—
472D4—22no—
473D4—44no—
474D4—44no—
475D4—42no—
476D4—42no—
477D4—82no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—82no—
505C2×C4—84no—
506C2×C4C23×#12012yes—
507C2×C4C22×D4×C2312yes—
508C2×C4C2×#41×C2311yescharacteristic
509C2×C4C2×#41×C2311yescharacteristic
510C2×C4C23×2−1+412yes—
511C8—42no—
512C8—42no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22C2×C23×#12011yescharacteristic
527C22C23×#52511yescharacteristic
528C22C23×#53611yescharacteristic
529C4—22no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—81no—
538C4—81no—
539C4C23×#54712yes—
540C4C23×#54712yes—
541C2—21no—
542C2—21no—
543C2—41no—
544C2—41no—
545C2—82no—
546C2C23×#240611yescharacteristic
547C2C23×#301911yescharacteristic
548C2C23×#302011yescharacteristic
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC23×#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#3093C2semidirect
3. C23×#2317C2semidirect
4. C23×#2061C2semidirect
5. C23×#2060C2semidirect
6. C23×#2040C2semidirect
7. C23×#2039C2semidirect
8. C23×#1897C2semidirect
9. C23×#1860C2semidirect
10. C23×#1859C2semidirect
11. C23×#1821C2semidirect
183. C42×C23#41semidirect
184. C42×C23#41semidirect
185. #24079C23direct (N × H)
240. #3093C2×C23semidirect
241. #2317C2×C23semidirect
242. #2061C2×C23semidirect
243. #2060C2×C23semidirect
244. #2040C2×C23semidirect
245. #2039C2×C23semidirect
246. #1897C2×C23semidirect
247. #1860C2×C23semidirect
248. #1859C2×C23semidirect
249. #1821C2×C23semidirect
389. C23#24079direct (N × H)
458. C42#41×C23semidirect
459. C42#41×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#34922
C23×#30932
C23×#23172
C23×#20612
C23×#20602
C23×#20402
C23×#20392
C23×#18972
C23×#18602
C23×#18592
C23×#18212
#2407923