Finite Group Databasebeta

← C2×C23×#1859 #19168487

Subgroups — C2×C23×#1859 #19168487

Order 5888.

1566 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#520C211yesmaximal, characteristic
2C2×C23×#428C211yesmaximal, characteristic, complemented
3C22×C23×#104C211yesmaximal, characteristic
4C2×C23×#417C211yesmaximal, characteristic
5C2×C23×#384C211yesmaximal, characteristic, complemented
6C23×#1859C218yesmaximal, direct product factor
7C2×C23×#343C211yesmaximal, characteristic, complemented
8C2×C23×#339C211yesmaximal, characteristic, complemented
9C22×#41×C23C2211yescharacteristic
10C22×M16×C23C2211yescharacteristic
11C22×M16×C23C2211yescharacteristic
12C23×#520—22no—
13C23×#520—22no—
14C23×#520—22no—
15C23×#520C2214yes—
16C2×C4×Q8×C23—21no—
17C2×C4×D4×C23—21no—
18C2×C4×D4×C23—22no—
19C2×C23×#117—21no—
20C2×C23×#117C2211yescharacteristic
21C22×C4⋊C4×C23—21no—
22C22×C4⋊C4×C23—22no—
23C22×C4⋊C4×C23C2211yescharacteristic
24C22×C4⋊C4×C23C2211yescharacteristic
25C22×C42×C23C2211yescharacteristic
26C23×#428—22no—
27C23×#428C2214yescomplemented
28C2×C23×#107—21no—
29C2×C23×#106—21no—
30C2×C23×#104C418yescomplemented
31C2×C23×#104C2214yes—
32C23×#417—22no—
33C23×#417C2214yes—
34C2×C23×#103—21no—
35C2×C23×#102—21no—
36C23×#384—22no—
37C23×#384C2214yescomplemented
38C23×#343C2214yescomplemented
39C23×#339C2214yescomplemented
40C2×#41×C23—21no—
41C2×#41×C23—21no—
42C2×#41×C23—22no—
43C2×#41×C23—22no—
44C2×#41×C23C2×C414yes—
45C2×#41×C23D412yes—
46C2×#41×C23Q814yes—
47C2×#41×C23C2312yes—
48C22×Q8×C23D411yescharacteristic
49C22×D4×C23—21no—
50C22×D4×C23D411yescharacteristic
51C23×C4×C23—21no—
52C23×C4×C23—21no—
53C23×C4×C23—22no—
54C23×C4×C23—21no—
55C23×C4×C23D412yes—
56C23×C4×C23C2311yescharacteristic
57C2×M16×C23—21no—
58C2×M16×C23—22no—
59C2×M16×C23—21no—
60C2×M16×C23—22no—
61C2×M16×C23—22no—
62C2×M16×C23C2×C414yescomplemented
63C2×M16×C23C2312yes—
64C2×M16×C23C2312yes—
65C22×C8×C23—21no—
66C22×C8×C23—21no—
67C4×Q8×C23—24no—
68C4×Q8×C23—24no—
69C4×D4×C23—28no—
70C4×D4×C23—24no—
71C4×D4×C23—24no—
72C4×D4×C23—28no—
73C23×#117—24no—
74C23×#117—21no—
75C23×#117—21no—
76C23×#117—24no—
77C23×#117D412yes—
78C23×#117C2312yes—
79C2×C4⋊C4×C23—22no—
80C2×C4⋊C4×C23—24no—
81C2×C4⋊C4×C23—21no—
82C2×C4⋊C4×C23—21no—
83C2×C4⋊C4×C23—21no—
84C2×C4⋊C4×C23—21no—
85C2×C4⋊C4×C23—24no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23—21no—
88C2×C4⋊C4×C23—21no—
89C2×C4⋊C4×C23—21no—
90C2×C4⋊C4×C23—21no—
91C2×C4⋊C4×C23—28no—
92C2×C4⋊C4×C23—48no—
93C2×C4⋊C4×C23D412yes—
94C2×C4⋊C4×C23D411yescharacteristic
95C2×C4⋊C4×C23D411yescharacteristic
96C2×C4⋊C4×C23C2312yes—
97C2×C4⋊C4×C23C2312yes—
98C2×C22⋊C4×C23—22no—
99C2×C22⋊C4×C23—21no—
100C2×C42×C23—22no—
101C2×C42×C23—21no—
102C2×C42×C23—21no—
103C2×C42×C23—22no—
104C2×C42×C23C2×C414yescomplemented
105C2×C42×C23C2312yes—
106C23×#107—24no—
107C23×#106—24no—
108C23×#104C2×C4116yescomplemented
109C23×#103—24no—
110C23×#102—24no—
111C24×C23—21no—
112C24×C23C2×D411yescharacteristic
113#41×C23—24no—
114#41×C23—24no—
115#41×C23—24no—
116#41×C23—24no—
117#41×C23—24no—
118#41×C23—22no—
119#41×C23—22no—
120#41×C23C4⋊C418yescomplemented
121#41×C23C22×C414yes—
122#41×C23C2×Q814yes—
123C2×Q8×C23—21no—
124C2×Q8×C23—22no—
125C2×Q8×C23C4⋊C414yescomplemented
126C2×Q8×C23C2×D412yes—
127C2×D4×C23—24no—
128C2×D4×C23—22no—
129C2×D4×C23—24no—
130C2×D4×C23—22no—
131C2×D4×C23—21no—
132C2×D4×C23—22no—
133C2×D4×C23C4⋊C414yescomplemented
134C2×D4×C23C2×D412yes—
135C22×C4×C23—24no—
136C22×C4×C23—24no—
137C22×C4×C23—24no—
138C22×C4×C23—21no—
139C22×C4×C23—22no—
140C22×C4×C23—24no—
141C22×C4×C23—22no—
142C22×C4×C23—24no—
143C22×C4×C23—22no—
144C22×C4×C23—21no—
145C22×C4×C23—22no—
146C22×C4×C23—21no—
147C22×C4×C23—28no—
148C22×C4×C23—22no—
149C22×C4×C23—22no—
150C22×C4×C23—44no—
151C22×C4×C23—44no—
152C22×C4×C23—41no—
153C22×C4×C23—41no—
154C22×C4×C23—84no—
155C22×C4×C23C22×C412yes—
156C22×C4×C23C2×D414yes—
157C22×C4×C23C2×D411yescharacteristic
158C22×C4×C23C2×D411yescharacteristic
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×Q812yes—
163C22×C4×C23#4112yes—
164C22×C4×C23#4112yes—
165C22×C4×C23C2411yescharacteristic
166M16×C23—24no—
167M16×C23—22no—
168M16×C23—22no—
169M16×C23—22no—
170M16×C23—42no—
171M16×C23C22×C414yescomplemented
172C2×C8×C23—24no—
173C2×C8×C23—22no—
174C2×C8×C23—22no—
175C2×C8×C23—42no—
176C4⋊C4×C23—28no—
177C4⋊C4×C23—22no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—22no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—22no—
182C4⋊C4×C23—22no—
183C4⋊C4×C23—22no—
184C4⋊C4×C23—22no—
185C4⋊C4×C23—416no—
186C4⋊C4×C23—41no—
187C4⋊C4×C23—41no—
188C4⋊C4×C23C2×D412yes—
189C4⋊C4×C23C2×D412yes—
190C22⋊C4×C23—28no—
191C22⋊C4×C23—22no—
192C22⋊C4×C23—22no—
193C42×C23—28no—
194C42×C23—24no—
195C42×C23—22no—
196C42×C23—22no—
197C42×C23—22no—
198C42×C23C22×C414yescomplemented
199C2×#1859C2311yesmaximal, characteristic, direct product factor, Sylow
200C23×C23—22no—
201C23×C23—22no—
202C23×C23—22no—
203C23×C23—24no—
204C23×C23—42no—
205C23×C23—42no—
206C23×C23C4×D412yes—
207C23×C23#12011yescharacteristic
208C23×C23#12011yescharacteristic
209C23×C23C22⋊Q812yes—
210C23×C23C22×D411yescharacteristic
211Q8×C23—22no—
212Q8×C23—24no—
213Q8×C23C2×C4⋊C414yescomplemented
214D4×C23—24no—
215D4×C23—24no—
216D4×C23—28no—
217D4×C23—22no—
218D4×C23—24no—
219D4×C23C2×C4⋊C414yescomplemented
220C2×C4×C23—24no—
221C2×C4×C23—216no—
222C2×C4×C23—24no—
223C2×C4×C23—24no—
224C2×C4×C23—22no—
225C2×C4×C23—24no—
226C2×C4×C23—24no—
227C2×C4×C23—24no—
228C2×C4×C23—28no—
229C2×C4×C23—22no—
230C2×C4×C23—24no—
231C2×C4×C23—22no—
232C2×C4×C23—44no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—44no—
236C2×C4×C23—44no—
237C2×C4×C23—42no—
238C2×C4×C23—82no—
239C2×C4×C23—88no—
240C2×C4×C23—82no—
241C2×C4×C23—82no—
242C2×C4×C23C2×C4⋊C412yes—
243C2×C4×C23C2×C4⋊C412yes—
244C2×C4×C23C4×D412yes—
245C2×C4×C23C4×D412yes—
246C2×C4×C23C4×D412yes—
247C2×C4×C23#12011yescharacteristic
248C2×C4×C23#12011yescharacteristic
249C2×C4×C23C22⋊Q812yes—
250C2×C4×C23C22⋊Q812yes—
251C2×C4×C23C22⋊Q812yes—
252C2×C4×C23#12312yes—
253C2×C4×C23#12312yes—
254C2×C4×C23C23×C411yescharacteristic
255C2×C4×C23C22×D411yescharacteristic
256C2×C4×C23C22×D411yescharacteristic
257C2×C4×C23C22×Q811yescharacteristic
258C2×C4×C23C2×#4111yescharacteristic
259C2×C4×C23C2×#4111yescharacteristic
260C8×C23—24no—
261C8×C23—42no—
262C2×#520C2×C2311yescharacteristic
263C2×#428C2×C2311yescharacteristic, complemented
264C22×#104C2×C2311yescharacteristic
265C2×#417C2×C2311yescharacteristic
266C2×#384C2×C2311yescharacteristic, complemented
267#1859C2×C2318yesdirect product factor
268C2×#343C2×C2311yescharacteristic, complemented
269C2×#339C2×C2311yescharacteristic, complemented
270C22×C23—24no—
271C22×C23—24no—
272C22×C23—21no—
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—21no—
276C22×C23—24no—
277C22×C23—48no—
278C22×C23—42no—
279C22×C23—42no—
280C22×C23#38512yes—
281C22×C23#38512yes—
282C22×C23#45312yes—
283C22×C23#45612yes—
284C22×C23C2×C4×D411yescharacteristic
285C22×C23C2×#12011yescharacteristic
286C22×C23C2×C22⋊Q811yescharacteristic
287C4×C23—28no—
288C4×C23—24no—
289C4×C23—44no—
290C4×C23—42no—
291C4×C23—42no—
292C4×C23—82no—
293C4×C23#38512yes—
294C4×C23#38512yes—
295C4×C23C22×C4⋊C411yescharacteristic
296C4×C23C2×C4×D411yescharacteristic
297C4×C23C2×C22⋊Q811yescharacteristic
298C4×C23C2×#12311yescharacteristic
299C22×#41C22×C2311yescharacteristic
300C22×M16C22×C2311yescharacteristic
301C22×M16C22×C2311yescharacteristic
302#520—22no—
303#520—22no—
304#520—22no—
305#520C22×C2314yes—
306C2×C4×Q8—21no—
307C2×C4×D4—21no—
308C2×C4×D4—22no—
309C2×#117—21no—
310C2×#117C22×C2311yescharacteristic
311C22×C4⋊C4—21no—
312C22×C4⋊C4—22no—
313C22×C4⋊C4C22×C2311yescharacteristic
314C22×C4⋊C4C22×C2311yescharacteristic
315C22×C42C22×C2311yescharacteristic
316#428—22no—
317#428C22×C2314yescomplemented
318C2×#107—21no—
319C2×#106—21no—
320C2×#104C4×C2318yescomplemented
321C2×#104C22×C2314yes—
322#417—22no—
323#417C22×C2314yes—
324C2×#103—21no—
325C2×#102—21no—
326#384—22no—
327#384C22×C2314yescomplemented
328#343C22×C2314yescomplemented
329#339C22×C2314yescomplemented
330C2×C23—21no—
331C2×C23—21no—
332C2×C23—22no—
333C2×C23—44no—
334C2×C23#185914yesdirect product factor
335C2×C23C2×#38511yescharacteristic
336C2×C23C2×#45311yescharacteristic
337C2×C23C2×#45611yescharacteristic
338C2×#41—21no—
339C2×#41—21no—
340C2×#41—22no—
341C2×#41—22no—
342C2×#41C2×C4×C2314yes—
343C2×#41D4×C2312yes—
344C2×#41Q8×C2314yes—
345C2×#41C23×C2312yes—
346C22×Q8D4×C2311yescharacteristic
347C22×D4—21no—
348C22×D4D4×C2311yescharacteristic
349C23×C4—21no—
350C23×C4—21no—
351C23×C4—22no—
352C23×C4—21no—
353C23×C4D4×C2312yes—
354C23×C4C23×C2311yescharacteristic
355C2×M16—21no—
356C2×M16—22no—
357C2×M16—21no—
358C2×M16—22no—
359C2×M16—22no—
360C2×M16C2×C4×C2314yescomplemented
361C2×M16C23×C2312yes—
362C2×M16C23×C2312yes—
363C22×C8—21no—
364C22×C8—21no—
365C4×Q8—24no—
366C4×Q8—24no—
367C4×D4—28no—
368C4×D4—24no—
369C4×D4—24no—
370C4×D4—28no—
371#117—24no—
372#117—21no—
373#117—21no—
374#117—24no—
375#117D4×C2312yes—
376#117C23×C2312yes—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—24no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—28no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—48no—
391C2×C4⋊C4D4×C2312yes—
392C2×C4⋊C4D4×C2311yescharacteristic
393C2×C4⋊C4D4×C2311yescharacteristic
394C2×C4⋊C4C23×C2312yes—
395C2×C4⋊C4C23×C2312yes—
396C2×C22⋊C4—22no—
397C2×C22⋊C4—21no—
398C2×C42—22no—
399C2×C42—21no—
400C2×C42—21no—
401C2×C42—22no—
402C2×C42C2×C4×C2314yescomplemented
403C2×C42C23×C2312yes—
404#107—24no—
405#106—24no—
406#104C2×C4×C23116yescomplemented
407#103—24no—
408#102—24no—
409C23C2×#185911yescharacteristic, direct product factor, Sylow
410C24—21no—
411C24C2×D4×C2311yescharacteristic
412#41—24no—
413#41—24no—
414#41—24no—
415#41—24no—
416#41—24no—
417#41—22no—
418#41—22no—
419#41C4⋊C4×C2318yescomplemented
420#41C22×C4×C2314yes—
421#41C2×Q8×C2314yes—
422C2×Q8—21no—
423C2×Q8—22no—
424C2×Q8C4⋊C4×C2314yescomplemented
425C2×Q8C2×D4×C2312yes—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—21no—
431C2×D4—22no—
432C2×D4C4⋊C4×C2314yescomplemented
433C2×D4C2×D4×C2312yes—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—28no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—84no—
454C22×C4C22×C4×C2312yes—
455C22×C4C2×D4×C2314yes—
456C22×C4C2×D4×C2311yescharacteristic
457C22×C4C2×D4×C2311yescharacteristic
458C22×C4C2×D4×C2311yescharacteristic
459C22×C4C2×D4×C2311yescharacteristic
460C22×C4C2×D4×C2311yescharacteristic
461C22×C4C2×Q8×C2312yes—
462C22×C4#41×C2312yes—
463C22×C4#41×C2312yes—
464C22×C4C24×C2311yescharacteristic
465M16—24no—
466M16—22no—
467M16—22no—
468M16—22no—
469M16—42no—
470M16C22×C4×C2314yescomplemented
471C2×C8—24no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—42no—
475C4⋊C4—28no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—416no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4C2×D4×C2312yes—
488C4⋊C4C2×D4×C2312yes—
489C22⋊C4—28no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C42—28no—
493C42—24no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42C22×C4×C2314yescomplemented
498C23—22no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—42no—
503C23—42no—
504C23C4×D4×C2312yes—
505C23C23×#12011yescharacteristic
506C23C23×#12011yescharacteristic
507C23C23×C22⋊Q812yes—
508C23C22×D4×C2311yescharacteristic
509Q8—24no—
510Q8—22no—
511Q8C2×C4⋊C4×C2314yescomplemented
512D4—28no—
513D4—24no—
514D4—24no—
515D4—24no—
516D4—22no—
517D4C2×C4⋊C4×C2314yescomplemented
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—216no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—82no—
537C2×C4—88no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4C2×C4⋊C4×C2312yes—
541C2×C4C2×C4⋊C4×C2312yes—
542C2×C4C4×D4×C2312yes—
543C2×C4C4×D4×C2312yes—
544C2×C4C4×D4×C2312yes—
545C2×C4C23×#12011yescharacteristic
546C2×C4C23×#12011yescharacteristic
547C2×C4C23×C22⋊Q812yes—
548C2×C4C23×C22⋊Q812yes—
549C2×C4C23×C22⋊Q812yes—
550C2×C4C23×#12312yes—
551C2×C4C23×#12312yes—
552C2×C4C23×C4×C2311yescharacteristic
553C2×C4C22×D4×C2311yescharacteristic
554C2×C4C22×D4×C2311yescharacteristic
555C2×C4C22×Q8×C2311yescharacteristic
556C2×C4C2×#41×C2311yescharacteristic
557C2×C4C2×#41×C2311yescharacteristic
558C8—24no—
559C8—42no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—48no—
570C22C23×#38512yes—
571C22C23×#38512yes—
572C22C23×#45312yes—
573C22C23×#45612yes—
574C22C2×C4×D4×C2311yescharacteristic
575C22C2×C23×#12011yescharacteristic
576C22C2×C23×C22⋊Q811yescharacteristic
577C4—28no—
578C4—24no—
579C4—44no—
580C4—42no—
581C4—42no—
582C4—82no—
583C4C23×#38512yes—
584C4C23×#38512yes—
585C4C22×C4⋊C4×C2311yescharacteristic
586C4C2×C4×D4×C2311yescharacteristic
587C4C2×C23×C22⋊Q811yescharacteristic
588C4C2×C23×#12311yescharacteristic
589C2—21no—
590C2—22no—
591C2—21no—
592C2—44no—
593C2C23×#185914yesdirect product factor
594C2C2×C23×#38511yescharacteristic
595C2C2×C23×#45311yescharacteristic
596C2C2×C23×#45611yescharacteristic
subgroupG/H
center, socleC23×C23#120
commutatorC2×C4C23×C4×C23
Frattini, cosocleC22×C4C24×C23
prime radicalC22×#41×C23C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#428C2semidirect
5. C2×C23×#384C2semidirect
6. C23×#1859C2direct (N × H)
7. C2×C23×#343C2semidirect
8. C2×C23×#339C2semidirect
27. C23×#428C22semidirect
30. C2×C23×#104C4semidirect
37. C23×#384C22semidirect
38. C23×#343C22semidirect
39. C23×#339C22semidirect
62. C2×M16×C23C2×C4semidirect
104. C2×C42×C23C2×C4semidirect
108. C23×#104C2×C4semidirect
120. #41×C23C4⋊C4semidirect
125. C2×Q8×C23C4⋊C4semidirect
133. C2×D4×C23C4⋊C4semidirect
171. M16×C23C22×C4semidirect
198. C42×C23C22×C4semidirect
199. C2×#1859C23direct (N × H)
213. Q8×C23C2×C4⋊C4semidirect
219. D4×C23C2×C4⋊C4semidirect
263. C2×#428C2×C23semidirect
266. C2×#384C2×C23semidirect
267. #1859C2×C23direct (N × H)
268. C2×#343C2×C23semidirect
269. C2×#339C2×C23semidirect
317. #428C22×C23semidirect
320. C2×#104C4×C23semidirect
327. #384C22×C23semidirect
328. #343C22×C23semidirect
329. #339C22×C23semidirect
334. C2×C23#1859direct (N × H)
360. C2×M16C2×C4×C23semidirect
402. C2×C42C2×C4×C23semidirect
406. #104C2×C4×C23semidirect
409. C23C2×#1859direct (N × H)
419. #41C4⋊C4×C23semidirect
424. C2×Q8C4⋊C4×C23semidirect
432. C2×D4C4⋊C4×C23semidirect
470. M16C22×C4×C23semidirect
497. C42C22×C4×C23semidirect
511. Q8C2×C4⋊C4×C23semidirect
517. D4C2×C4⋊C4×C23semidirect
593. C2C23×#1859direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5202
C2×C23×#4282
C22×C23×#1042
C2×C23×#4172
C2×C23×#3842
C23×#18592
C2×C23×#3432
C2×C23×#3392
C2×#185923