Finite Group Databasebeta

← C23×#13080 #19160706

Subgroups — C23×#13080 #19160706

Order 5888.

864 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C23×#100C211yesmaximal, characteristic
2C23×#2405C211yesmaximal, characteristic
3C2×C23×#352C212yesmaximal, complemented
4C23×#1781C211yesmaximal, characteristic
5C23×#1390C212yesmaximal, complemented
6C23×D4×C23C2211yescharacteristic
7C22×M16×C23—21no—
8C2×C23×#123C411yescharacteristic, complemented
9C2×C23×#123C2211yescharacteristic
10C22×C22⋊C4×C23C411yescharacteristic, complemented
11C2×C23×#100—21no—
12C2×C23×#100—24no—
13C2×C23×#100C2212yes—
14C2×C23×#99—21no—
15C2×C23×#99C2212yes—
16C23×#394—21no—
17C23×#394—21no—
18C23×#386C411yescharacteristic, complemented
19C23×#386C2211yescharacteristic
20C23×#385—21no—
21C23×#379—21no—
22C23×#379—21no—
23C23×#379C411yescharacteristic, complemented
24C23×#352—24no—
25C25×C23C2×C411yescharacteristic
26C25×C23D411yescharacteristic
27C22×D4×C23—22no—
28C22×D4×C23—22no—
29C22×D4×C23—42no—
30C22×D4×C23—41no—
31C22×D4×C23—41no—
32C22×D4×C23C2×C411yescharacteristic
33C22×D4×C23D411yescharacteristic
34C22×D4×C23D411yescharacteristic
35C22×D4×C23C2311yescharacteristic
36C23×C4×C23C2×C411yescharacteristic
37C23×C4×C23C2×C411yescharacteristic
38C23×C4×C23D411yescharacteristic
39C2×M16×C23—24no—
40C2×M16×C23—22no—
41C2×M16×C23—41no—
42C2×M16×C23—42no—
43C22×C8×C23—41no—
44C23×#123—22no—
45C23×#123—22no—
46C23×#123—42no—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23—21no—
49C2×C22⋊C4×C23—21no—
50C2×C22⋊C4×C23—21no—
51C2×C22⋊C4×C23—21no—
52C2×C22⋊C4×C23—21no—
53C2×C22⋊C4×C23—21no—
54C2×C22⋊C4×C23—22no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—41no—
59C2×C22⋊C4×C23—41no—
60C2×C22⋊C4×C23—41no—
61C2×C22⋊C4×C23—41no—
62C2×C22⋊C4×C23C2×C411yescharacteristic
63C2×C22⋊C4×C23C2×C411yescharacteristic
64C23×#100—24no—
65C23×#100—22no—
66C23×#100D414yes—
67C23×#99—22no—
68C23×#99—22no—
69C23×#99—42no—
70C23×#95—21no—
71C23×#95—21no—
72C24×C23—21no—
73C24×C23—21no—
74C24×C23—21no—
75C24×C23—21no—
76C24×C23—22no—
77C24×C23—22no—
78C24×C23—41no—
79C24×C23—41no—
80C24×C23—42no—
81C24×C23—42no—
82C24×C23—81no—
83C24×C23—81no—
84C24×C23C22⋊C411yescharacteristic
85C24×C23C22⋊C411yescharacteristic
86C24×C23C22⋊C411yescharacteristic
87C24×C23C22×C411yescharacteristic
88C24×C23C2×D411yescharacteristic
89C2×D4×C23—22no—
90C2×D4×C23—22no—
91C2×D4×C23—22no—
92C2×D4×C23—22no—
93C2×D4×C23—21no—
94C2×D4×C23—21no—
95C2×D4×C23—41no—
96C2×D4×C23—41no—
97C2×D4×C23—41no—
98C2×D4×C23—41no—
99C2×D4×C23—41no—
100C2×D4×C23—41no—
101C2×D4×C23—44no—
102C2×D4×C23—44no—
103C2×D4×C23—44no—
104C2×D4×C23—44no—
105C2×D4×C23C2×D412yes—
106C2×D4×C23#4112yes—
107C22×C4×C23—21no—
108C22×C4×C23—21no—
109C22×C4×C23—22no—
110C22×C4×C23—21no—
111C22×C4×C23—21no—
112C22×C4×C23—21no—
113C22×C4×C23—21no—
114C22×C4×C23—21no—
115C22×C4×C23—21no—
116C22×C4×C23—22no—
117C22×C4×C23—41no—
118C22×C4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23—42no—
121C22×C4×C23—81no—
122C22×C4×C23C22⋊C411yescharacteristic
123C22×C4×C23C2×D411yescharacteristic
124M16×C23—24no—
125M16×C23—44no—
126M16×C23—42no—
127C2×C8×C23—44no—
128C2×C8×C23—42no—
129C4⋊C4×C23—42no—
130C4⋊C4×C23—41no—
131C4⋊C4×C23—41no—
132C22⋊C4×C23—22no—
133C22⋊C4×C23—42no—
134C22⋊C4×C23—42no—
135C22⋊C4×C23—42no—
136C22⋊C4×C23—41no—
137C22⋊C4×C23—41no—
138C22⋊C4×C23—42no—
139C22⋊C4×C23—41no—
140C22⋊C4×C23—42no—
141C22⋊C4×C23—42no—
142C22⋊C4×C23—81no—
143C22⋊C4×C23—81no—
144C22⋊C4×C23—81no—
145C22⋊C4×C23—82no—
146C22⋊C4×C23—82no—
147#13080C2311yesmaximal, characteristic, direct product factor, Sylow
148C23×C23—22no—
149C23×C23—21no—
150C23×C23—21no—
151C23×C23—21no—
152C23×C23—21no—
153C23×C23—21no—
154C23×C23—21no—
155C23×C23—21no—
156C23×C23—21no—
157C23×C23—21no—
158C23×C23—21no—
159C23×C23—21no—
160C23×C23—21no—
161C23×C23—41no—
162C23×C23—42no—
163C23×C23—42no—
164C23×C23—42no—
165C23×C23—42no—
166C23×C23—41no—
167C23×C23—41no—
168C23×C23—41no—
169C23×C23—41no—
170C23×C23—41no—
171C23×C23—41no—
172C23×C23—42no—
173C23×C23—42no—
174C23×C23—41no—
175C23×C23—41no—
176C23×C23—42no—
177C23×C23—42no—
178C23×C23—81no—
179C23×C23—81no—
180C23×C23—81no—
181C23×C23—81no—
182C23×C23—84no—
183C23×C23—84no—
184C23×C23—84no—
185C23×C23—84no—
186C23×C23#9911yescharacteristic
187C23×C23#9911yescharacteristic
188C23×C23C2×C22⋊C411yescharacteristic
189C23×C23C4×D412yes—
190C23×C23#12112yes—
191D4×C23—42no—
192D4×C23—42no—
193D4×C23—44no—
194D4×C23—44no—
195D4×C23—42no—
196D4×C23—42no—
197C2×C4×C23—22no—
198C2×C4×C23—22no—
199C2×C4×C23—22no—
200C2×C4×C23—22no—
201C2×C4×C23—21no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—21no—
205C2×C4×C23—41no—
206C2×C4×C23—41no—
207C2×C4×C23—41no—
208C2×C4×C23—41no—
209C2×C4×C23—41no—
210C2×C4×C23—41no—
211C2×C4×C23—41no—
212C2×C4×C23—41no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—41no—
216C2×C4×C23—41no—
217C2×C4×C23—82no—
218C2×C4×C23—82no—
219C2×C4×C23—82no—
220C2×C4×C23—82no—
221C2×C4×C23—82no—
222C2×C4×C23#9912yes—
223C2×C4×C23#12011yescharacteristic
224C2×C4×C23#12311yescharacteristic
225C8×C23—44no—
226C22×#100C2×C2311yescharacteristic
227#2405C2×C2311yescharacteristic
228C2×#352C2×C2312yescomplemented
229#1781C2×C2311yescharacteristic
230#1390C2×C2312yescomplemented
231C22×C23—21no—
232C22×C23—21no—
233C22×C23—21no—
234C22×C23—21no—
235C22×C23—21no—
236C22×C23—21no—
237C22×C23—21no—
238C22×C23—22no—
239C22×C23—22no—
240C22×C23—42no—
241C22×C23—42no—
242C22×C23—42no—
243C22×C23—42no—
244C22×C23—42no—
245C22×C23—42no—
246C22×C23—41no—
247C22×C23—41no—
248C22×C23—41no—
249C22×C23—41no—
250C22×C23—41no—
251C22×C23—41no—
252C22×C23—41no—
253C22×C23—41no—
254C22×C23—41no—
255C22×C23—41no—
256C22×C23—81no—
257C22×C23—84no—
258C22×C23—84no—
259C22×C23—81no—
260C22×C23—81no—
261C22×C23—81no—
262C22×C23—84no—
263C22×C23—84no—
264C22×C23#35212yescomplemented
265C22×C23#38611yescharacteristic
266C22×C23C2×#9911yescharacteristic
267C22×C23C2×#9911yescharacteristic
268C4×C23—21no—
269C4×C23—22no—
270C4×C23—21no—
271C4×C23—81no—
272C4×C23—81no—
273C4×C23—81no—
274C4×C23—81no—
275C4×C23—162no—
276C23×D4C22×C2311yescharacteristic
277C22×M16—21no—
278C2×#123C4×C2311yescharacteristic, complemented
279C2×#123C22×C2311yescharacteristic
280C22×C22⋊C4C4×C2311yescharacteristic, complemented
281C2×#100—21no—
282C2×#100—24no—
283C2×#100C22×C2312yes—
284C2×#99—21no—
285C2×#99C22×C2312yes—
286#394—21no—
287#394—21no—
288#386C4×C2311yescharacteristic, complemented
289#386C22×C2311yescharacteristic
290#385—21no—
291#379—21no—
292#379—21no—
293#379C4×C2311yescharacteristic, complemented
294#352—24no—
295C2×C23—21no—
296C2×C23—21no—
297C2×C23—22no—
298C2×C23—42no—
299C2×C23—41no—
300C2×C23—42no—
301C2×C23—81no—
302C2×C23—81no—
303C2×C23#189811yescharacteristic
304C2×C23C2×#35211yescharacteristic
305C2×C23#212111yescharacteristic
306C25C2×C4×C2311yescharacteristic
307C25D4×C2311yescharacteristic
308C22×D4—22no—
309C22×D4—22no—
310C22×D4—42no—
311C22×D4—41no—
312C22×D4—41no—
313C22×D4C2×C4×C2311yescharacteristic
314C22×D4D4×C2311yescharacteristic
315C22×D4D4×C2311yescharacteristic
316C22×D4C23×C2311yescharacteristic
317C23×C4C2×C4×C2311yescharacteristic
318C23×C4C2×C4×C2311yescharacteristic
319C23×C4D4×C2311yescharacteristic
320C2×M16—24no—
321C2×M16—22no—
322C2×M16—41no—
323C2×M16—42no—
324C22×C8—41no—
325#123—22no—
326#123—22no—
327#123—42no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—41no—
340C2×C22⋊C4—41no—
341C2×C22⋊C4—41no—
342C2×C22⋊C4—41no—
343C2×C22⋊C4C2×C4×C2311yescharacteristic
344C2×C22⋊C4C2×C4×C2311yescharacteristic
345#100—24no—
346#100—22no—
347#100D4×C2314yes—
348#99—22no—
349#99—22no—
350#99—42no—
351#95—21no—
352#95—21no—
353C23#1308011yescharacteristic, direct product factor, Sylow
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—41no—
361C24—41no—
362C24—42no—
363C24—42no—
364C24—81no—
365C24—81no—
366C24C22⋊C4×C2311yescharacteristic
367C24C22⋊C4×C2311yescharacteristic
368C24C22⋊C4×C2311yescharacteristic
369C24C22×C4×C2311yescharacteristic
370C24C2×D4×C2311yescharacteristic
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—21no—
376C2×D4—21no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—41no—
382C2×D4—41no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4C2×D4×C2312yes—
388C2×D4#41×C2312yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—42no—
403C22×C4—81no—
404C22×C4C22⋊C4×C2311yescharacteristic
405C22×C4C2×D4×C2311yescharacteristic
406M16—24no—
407M16—44no—
408M16—42no—
409C2×C8—44no—
410C2×C8—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C22⋊C4—22no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—42no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—42no—
421C22⋊C4—41no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—81no—
425C22⋊C4—81no—
426C22⋊C4—81no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—81no—
460C23—81no—
461C23—81no—
462C23—81no—
463C23—84no—
464C23—84no—
465C23—84no—
466C23—84no—
467C23C23×#9911yescharacteristic
468C23C23×#9911yescharacteristic
469C23C2×C22⋊C4×C2311yescharacteristic
470C23C4×D4×C2312yes—
471C23C23×#12112yes—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—44no—
476D4—42no—
477D4—42no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4C23×#9912yes—
504C2×C4C23×#12011yescharacteristic
505C2×C4C23×#12311yescharacteristic
506C8—44no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—22no—
513C22—22no—
514C22—21no—
515C22—21no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—81no—
533C22—84no—
534C22—84no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—84no—
539C22—84no—
540C22C23×#35212yescomplemented
541C22C23×#38611yescharacteristic
542C22C2×C23×#9911yescharacteristic
543C22C2×C23×#9911yescharacteristic
544C4—21no—
545C4—22no—
546C4—21no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—162no—
552C2—21no—
553C2—21no—
554C2—22no—
555C2—42no—
556C2—41no—
557C2—42no—
558C2—81no—
559C2—81no—
560C2C23×#189811yescharacteristic
561C2C2×C23×#35211yescharacteristic
562C2C23×#212111yescharacteristic
subgroupG/H
center, socleC22×C23C2×#99
commutatorC24C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×D4×C23C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C24×C23 ◁ C23×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C23×#352C2semidirect
5. C23×#1390C2semidirect
8. C2×C23×#123C4semidirect
10. C22×C22⋊C4×C23C4semidirect
18. C23×#386C4semidirect
23. C23×#379C4semidirect
147. #13080C23direct (N × H)
228. C2×#352C2×C23semidirect
230. #1390C2×C23semidirect
264. C22×C23#352semidirect
278. C2×#123C4×C23semidirect
280. C22×C22⋊C4C4×C23semidirect
288. #386C4×C23semidirect
293. #379C4×C23semidirect
353. C23#13080direct (N × H)
540. C22C23×#352semidirect

Maximal subgroups

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

subgroupindex
C22×C23×#1002
C23×#24052
C2×C23×#3522
C23×#17812
C23×#13902
#1308023