Finite Group Databasebeta

← C23×#14841 #19162467

Subgroups — C23×#14841 #19162467

Order 5888.

1534 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#400C211yesmaximal, characteristic, complemented
2C2×C23×#385C211yesmaximal, characteristic
3C2×C23×#380C211yesmaximal, characteristic
4C2×C4×C22⋊C4×C23C211yesmaximal, characteristic
5C23×#1445C211yesmaximal, characteristic, complemented
6C23×#1442C211yesmaximal, characteristic, complemented
7C23×#1441C211yesmaximal, characteristic, complemented
8C23×#1440C212yesmaximal
9C23×#1440C212yesmaximal
10C23×#1436C212yesmaximal, complemented
11C23×#1434C212yesmaximal, complemented
12C24×C4×C23C2211yescharacteristic
13C22×C4⋊C4×C23C2211yescharacteristic
14C22×C4⋊C4×C23C2211yescharacteristic
15C22×C4⋊C4×C23C2211yescharacteristic
16C22×C4⋊C4×C23C2211yescharacteristic
17C22×C22⋊C4×C23C2212yes—
18C22×C22⋊C4×C23C2212yes—
19C22×C22⋊C4×C23C2211yescharacteristic
20C22×C22⋊C4×C23C2211yescharacteristic
21C22×C42×C23C2212yes—
22C23×#400—24no—
23C23×#385—24no—
24C23×#380—24no—
25C4×C22⋊C4×C23—24no—
26C2×C23×#95C2211yescharacteristic
27C2×C23×#95C2212yes—
28C2×C23×#95C2212yes—
29C2×C23×#95C2212yes—
30C2×C23×#95C2212yes—
31C2×C23×#95C2212yes—
32C2×C23×#95C2212yes—
33C2×C23×#95C2212yes—
34C2×C23×#95C2211yescharacteristic
35C2×C23×#95C2211yescharacteristic
36C2×C23×#95C2212yes—
37C2×C23×#95C2212yes—
38C2×C23×#95C2211yescharacteristic
39C25×C23C2311yescharacteristic
40C23×C4×C23—22no—
41C23×C4×C23—22no—
42C23×C4×C23—24no—
43C23×C4×C23—42no—
44C23×C4×C23D414yes—
45C23×C4×C23C2311yescharacteristic
46C23×C4×C23C2312yes—
47C23×C4×C23C2312yes—
48C23×C4×C23C2311yescharacteristic
49C23×C4×C23C2312yes—
50C23×C4×C23C2312yes—
51C23×C4×C23C2311yescharacteristic
52C23×C4×C23C2311yescharacteristic
53C23×C4×C23C2311yescharacteristic
54C23×C4×C23C2311yescharacteristic
55C2×C4⋊C4×C23—22no—
56C2×C4⋊C4×C23—22no—
57C2×C4⋊C4×C23—22no—
58C2×C4⋊C4×C23—22no—
59C2×C4⋊C4×C23—22no—
60C2×C4⋊C4×C23—22no—
61C2×C4⋊C4×C23—22no—
62C2×C4⋊C4×C23—22no—
63C2×C4⋊C4×C23—24no—
64C2×C4⋊C4×C23—42no—
65C2×C22⋊C4×C23—22no—
66C2×C22⋊C4×C23—24no—
67C2×C22⋊C4×C23—24no—
68C2×C22⋊C4×C23—24no—
69C2×C22⋊C4×C23—22no—
70C2×C22⋊C4×C23—22no—
71C2×C22⋊C4×C23—24no—
72C2×C22⋊C4×C23—22no—
73C2×C22⋊C4×C23—24no—
74C2×C22⋊C4×C23—24no—
75C2×C22⋊C4×C23—42no—
76C2×C42×C23—24no—
77C2×C42×C23—24no—
78C2×C42×C23—42no—
79C23×#95—22no—
80C23×#95—24no—
81C23×#95—22no—
82C23×#95—22no—
83C23×#95—24no—
84C23×#95—24no—
85C23×#95—24no—
86C23×#95—24no—
87C23×#95—24no—
88C23×#95—24no—
89C23×#95—42no—
90C23×#95—42no—
91C23×#95D414yes—
92C24×C23—21no—
93C24×C23—21no—
94C24×C23—21no—
95C24×C23—22no—
96C24×C23—21no—
97C24×C23—22no—
98C24×C23—22no—
99C24×C23—21no—
100C24×C23—21no—
101C24×C23C2×D412yes—
102C24×C23#4114yes—
103C24×C23C2411yescharacteristic
104C22×C4×C23—21no—
105C22×C4×C23—21no—
106C22×C4×C23—22no—
107C22×C4×C23—22no—
108C22×C4×C23—22no—
109C22×C4×C23—22no—
110C22×C4×C23—21no—
111C22×C4×C23—21no—
112C22×C4×C23—21no—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—22no—
116C22×C4×C23—22no—
117C22×C4×C23—21no—
118C22×C4×C23—21no—
119C22×C4×C23—22no—
120C22×C4×C23—22no—
121C22×C4×C23—21no—
122C22×C4×C23—22no—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—22no—
127C22×C4×C23—22no—
128C22×C4×C23—22no—
129C22×C4×C23—21no—
130C22×C4×C23—21no—
131C22×C4×C23—28no—
132C22×C4×C23—28no—
133C22×C4×C23—21no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—21no—
137C22×C4×C23—22no—
138C22×C4×C23—21no—
139C22×C4×C23—22no—
140C22×C4×C23—22no—
141C22×C4×C23—22no—
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—21no—
145C22×C4×C23—22no—
146C22×C4×C23—22no—
147C22×C4×C23—22no—
148C22×C4×C23—22no—
149C22×C4×C23—22no—
150C22×C4×C23—22no—
151C22×C4×C23—22no—
152C22×C4×C23—22no—
153C22×C4×C23—22no—
154C22×C4×C23—22no—
155C22×C4×C23—21no—
156C22×C4×C23—21no—
157C22×C4×C23—44no—
158C22×C4×C23—48no—
159C22×C4×C23—48no—
160C22×C4×C23C2×D414yes—
161C22×C4×C23C2×D412yes—
162C22×C4×C23C2×D412yes—
163C22×C4×C23C2×D412yes—
164C22×C4×C23#4112yes—
165C22×C4×C23#4114yes—
166C22×C4×C23#4112yes—
167C22×C4×C23#4112yes—
168C22×C4×C23#4114yes—
169C4⋊C4×C23—44no—
170C4⋊C4×C23—44no—
171C4⋊C4×C23—44no—
172C4⋊C4×C23—44no—
173C22⋊C4×C23—216no—
174C22⋊C4×C23—28no—
175C22⋊C4×C23—44no—
176C22⋊C4×C23—48no—
177C42×C23—48no—
178#14841C2311yesmaximal, characteristic, direct product factor, Sylow
179C23×C23—22no—
180C23×C23—22no—
181C23×C23—22no—
182C23×C23—24no—
183C23×C23—24no—
184C23×C23—22no—
185C23×C23—22no—
186C23×C23—22no—
187C23×C23—24no—
188C23×C23—24no—
189C23×C23—24no—
190C23×C23—24no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—42no—
194C23×C23—42no—
195C23×C23—42no—
196C23×C23—41no—
197C23×C23—41no—
198C23×C23—42no—
199C23×C23—41no—
200C23×C23—41no—
201C23×C23#12414yes—
202C23×C23C22×D411yescharacteristic
203C23×C23C22×D411yescharacteristic
204C23×C23C2×#4111yescharacteristic
205C23×C23C2×#4112yes—
206C23×C23C2×#4112yes—
207C23×C23C2×#4112yes—
208C23×C23C2×#4111yescharacteristic
209C23×C23C2×#4111yescharacteristic
210C23×C23H(5,𝔽2)12yes—
211C23×C232−1+411yescharacteristic
212C23×C232−1+411yescharacteristic
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—22no—
216C2×C4×C23—22no—
217C2×C4×C23—24no—
218C2×C4×C23—24no—
219C2×C4×C23—24no—
220C2×C4×C23—24no—
221C2×C4×C23—24no—
222C2×C4×C23—22no—
223C2×C4×C23—22no—
224C2×C4×C23—24no—
225C2×C4×C23—24no—
226C2×C4×C23—24no—
227C2×C4×C23—24no—
228C2×C4×C23—24no—
229C2×C4×C23—24no—
230C2×C4×C23—22no—
231C2×C4×C23—22no—
232C2×C4×C23—416no—
233C2×C4×C23—41no—
234C2×C4×C23—41no—
235C2×C4×C23—42no—
236C2×C4×C23—42no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—42no—
240C2×C4×C23—42no—
241C2×C4×C23—41no—
242C2×C4×C23—42no—
243C2×C4×C23—42no—
244C2×C4×C23—42no—
245C2×C4×C23—42no—
246C2×C4×C23—42no—
247C2×C4×C23—41no—
248C2×C4×C23—42no—
249C2×C4×C23—41no—
250C2×C4×C23—41no—
251C2×C4×C23—42no—
252C2×C4×C23—42no—
253C2×C4×C23—42no—
254C2×C4×C23—42no—
255C2×C4×C23—42no—
256C2×C4×C23—42no—
257C2×C4×C23—42no—
258C2×C4×C23—42no—
259C2×C4×C23—41no—
260C2×C4×C23—41no—
261C2×C4×C23—41no—
262C2×C4×C23—42no—
263C2×C4×C23—41no—
264C2×C4×C23—42no—
265C2×C4×C23—42no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23—42no—
269C2×C4×C23—42no—
270C2×C4×C23—42no—
271C2×C4×C23—42no—
272C2×C4×C23#12114yes—
273C2×C4×C23#12314yes—
274C2×#400C2×C2311yescharacteristic, complemented
275C2×#385C2×C2311yescharacteristic
276C2×#380C2×C2311yescharacteristic
277C2×C4×C22⋊C4C2×C2311yescharacteristic
278#1445C2×C2311yescharacteristic, complemented
279#1442C2×C2311yescharacteristic, complemented
280#1441C2×C2311yescharacteristic, complemented
281#1440C2×C2312yes—
282#1440C2×C2312yes—
283#1436C2×C2312yescomplemented
284#1434C2×C2312yescomplemented
285C22×C23—24no—
286C22×C23—24no—
287C22×C23—24no—
288C22×C23—44no—
289C22×C23—42no—
290C22×C23—42no—
291C22×C23—42no—
292C22×C23—42no—
293C22×C23—44no—
294C22×C23—44no—
295C22×C23—44no—
296C22×C23C2×#12111yescharacteristic
297C22×C23C2×#12311yescharacteristic
298C22×C23#52511yescharacteristic
299C22×C23#52512yes—
300C22×C23C2×#12411yescharacteristic
301C22×C23#52912yes—
302C22×C23#52912yes—
303C22×C23#52912yes—
304C22×C23#53211yescharacteristic
305C22×C23#53411yescharacteristic
306C22×C23#53611yescharacteristic
307C22×C23#53912yes—
308C22×C23#54212yes—
309C22×C23#54612yes—
310C22×C23#54711yescharacteristic
311C22×C23#54811yescharacteristic
312C22×C23#55112yes—
313C22×C23#55212yes—
314C22×C23#55211yescharacteristic
315C22×C23#55212yes—
316C22×C23#55312yes—
317C22×C23#55312yes—
318C22×C23#55611yescharacteristic
319C4×C23—24no—
320C4×C23—24no—
321C4×C23—42no—
322C4×C23—42no—
323C4×C23—44no—
324C4×C23—44no—
325C4×C23—82no—
326C4×C23—82no—
327C4×C23—81no—
328C4×C23—81no—
329C24×C4C22×C2311yescharacteristic
330C22×C4⋊C4C22×C2311yescharacteristic
331C22×C4⋊C4C22×C2311yescharacteristic
332C22×C4⋊C4C22×C2311yescharacteristic
333C22×C4⋊C4C22×C2311yescharacteristic
334C22×C22⋊C4C22×C2312yes—
335C22×C22⋊C4C22×C2312yes—
336C22×C22⋊C4C22×C2311yescharacteristic
337C22×C22⋊C4C22×C2311yescharacteristic
338C22×C42C22×C2312yes—
339#400—24no—
340#385—24no—
341#380—24no—
342C4×C22⋊C4—24no—
343C2×#95C22×C2312yes—
344C2×#95C22×C2312yes—
345C2×#95C22×C2311yescharacteristic
346C2×#95C22×C2312yes—
347C2×#95C22×C2312yes—
348C2×#95C22×C2312yes—
349C2×#95C22×C2312yes—
350C2×#95C22×C2312yes—
351C2×#95C22×C2311yescharacteristic
352C2×#95C22×C2311yescharacteristic
353C2×#95C22×C2312yes—
354C2×#95C22×C2312yes—
355C2×#95C22×C2311yescharacteristic
356C2×C23—44no—
357C2×C23#239511yescharacteristic
358C2×C23#241211yescharacteristic
359C2×C23#241511yescharacteristic
360C2×C23#246012yes—
361C2×C23#246812yes—
362C2×C23#250311yescharacteristic
363C2×C23#253012yes—
364C2×C23#257712yes—
365C2×C23#259111yescharacteristic
366C2×C23#266311yescharacteristic
367C2×C23#285011yescharacteristic
368C25C23×C2311yescharacteristic
369C23×C4—22no—
370C23×C4—22no—
371C23×C4—24no—
372C23×C4—42no—
373C23×C4D4×C2314yes—
374C23×C4C23×C2311yescharacteristic
375C23×C4C23×C2312yes—
376C23×C4C23×C2312yes—
377C23×C4C23×C2311yescharacteristic
378C23×C4C23×C2312yes—
379C23×C4C23×C2311yescharacteristic
380C23×C4C23×C2311yescharacteristic
381C23×C4C23×C2312yes—
382C23×C4C23×C2311yescharacteristic
383C23×C4C23×C2311yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—22no—
388C2×C4⋊C4—22no—
389C2×C4⋊C4—22no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—22no—
392C2×C4⋊C4—24no—
393C2×C4⋊C4—42no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—24no—
396C2×C22⋊C4—24no—
397C2×C22⋊C4—24no—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—24no—
401C2×C22⋊C4—22no—
402C2×C22⋊C4—24no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—42no—
405C2×C42—24no—
406C2×C42—24no—
407C2×C42—42no—
408#95—22no—
409#95—24no—
410#95—22no—
411#95—22no—
412#95—24no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—42no—
419#95—42no—
420#95D4×C2314yes—
421C23#1484111yescharacteristic, direct product factor, Sylow
422C24—21no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24C2×D4×C2312yes—
432C24#41×C2314yes—
433C24C24×C2311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4C2×D4×C2314yes—
491C22×C4C2×D4×C2312yes—
492C22×C4C2×D4×C2312yes—
493C22×C4C2×D4×C2312yes—
494C22×C4#41×C2312yes—
495C22×C4#41×C2312yes—
496C22×C4#41×C2314yes—
497C22×C4#41×C2312yes—
498C22×C4#41×C2314yes—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C22⋊C4—216no—
504C22⋊C4—28no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C42—48no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23C23×#12414yes—
531C23C22×D4×C2311yescharacteristic
532C23C22×D4×C2311yescharacteristic
533C23C2×#41×C2311yescharacteristic
534C23C2×#41×C2311yescharacteristic
535C23C2×#41×C2311yescharacteristic
536C23C2×#41×C2312yes—
537C23C2×#41×C2312yes—
538C23C2×#41×C2312yes—
539C23C23×H(5,𝔽2)12yes—
540C23C23×2−1+411yescharacteristic
541C23C23×2−1+411yescharacteristic
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—416no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4C23×#12114yes—
602C2×C4C23×#12314yes—
603C22—24no—
604C22—24no—
605C22—24no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22C2×C23×#12111yescharacteristic
615C22C2×C23×#12311yescharacteristic
616C22C23×#52511yescharacteristic
617C22C23×#52512yes—
618C22C2×C23×#12411yescharacteristic
619C22C23×#52912yes—
620C22C23×#52912yes—
621C22C23×#52912yes—
622C22C23×#53211yescharacteristic
623C22C23×#53411yescharacteristic
624C22C23×#53611yescharacteristic
625C22C23×#53912yes—
626C22C23×#54212yes—
627C22C23×#54612yes—
628C22C23×#54711yescharacteristic
629C22C23×#54811yescharacteristic
630C22C23×#55112yes—
631C22C23×#55211yescharacteristic
632C22C23×#55212yes—
633C22C23×#55212yes—
634C22C23×#55312yes—
635C22C23×#55312yes—
636C22C23×#55611yescharacteristic
637C4—24no—
638C4—24no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—42no—
643C4—81no—
644C4—81no—
645C4—82no—
646C4—82no—
647C2—44no—
648C2C23×#239511yescharacteristic
649C2C23×#241211yescharacteristic
650C2C23×#241511yescharacteristic
651C2C23×#246012yes—
652C2C23×#246812yes—
653C2C23×#250311yescharacteristic
654C2C23×#253012yes—
655C2C23×#257712yes—
656C2C23×#259111yescharacteristic
657C2C23×#266311yescharacteristic
658C2C23×#285011yescharacteristic
subgroupG/H
center, socleC24×C23C24
commutator, Frattini, cosocleC24C24×C23
prime radicalC25×C23C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C23×#400C2semidirect
5. C23×#1445C2semidirect
6. C23×#1442C2semidirect
7. C23×#1441C2semidirect
10. C23×#1436C2semidirect
11. C23×#1434C2semidirect
178. #14841C23direct (N × H)
274. C2×#400C2×C23semidirect
278. #1445C2×C23semidirect
279. #1442C2×C23semidirect
280. #1441C2×C23semidirect
283. #1436C2×C23semidirect
284. #1434C2×C23semidirect
421. C23#14841direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4002
C2×C23×#3852
C2×C23×#3802
C2×C4×C22⋊C4×C232
C23×#14452
C23×#14422
C23×#14412
C23×#14402
C23×#14362
C23×#14342
#1484123