Finite Group Databasebeta

← C23×#23901 #19171527

Subgroups — C23×#23901 #19171527

Order 5888.

2056 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3484C211yesmaximal, characteristic
2C22×C23×#123C211yesmaximal, characteristic, complemented
3C2×C23×#458C212yesmaximal, complemented
4C23×#2849C212yesmaximal, complemented
5C23×#2022C214yesmaximal, complemented
6C23×#1898C212yesmaximal, complemented
7C23×#1890C212yesmaximal, complemented
8C23×#1781C211yesmaximal, characteristic, complemented
9C2×C23×H(5,𝔽2)C2211yescharacteristic
10C23×D4×C23C2211yescharacteristic
11C24×C4×C23C2211yescharacteristic, complemented
12C23×#546—24no—
13D42×C23—24no—
14C23×#535—21no—
15C23×#534—21no—
16C23×#534C2212yes—
17C23×#525—21no—
18C2×C23×#123—21no—
19C2×C23×#123—21no—
20C2×C23×#123—28no—
21C2×C23×#123C2212yes—
22C2×C23×#123C2212yescomplemented
23C2×C23×#123C2214yescomplemented
24C2×C23×#120—21no—
25C2×C23×#120C2212yes—
26C23×#518C2211yescharacteristic
27C22×C4⋊C4×C23—21no—
28C22×C22⋊C4×C23—21no—
29C22×C22⋊C4×C23C2211yescharacteristic, complemented
30C23×#458—28no—
31C2×C23×#99C2212yes—
32C2×C23×#99C2214yescomplemented
33C2×C23×#99C2212yescomplemented
34C23×#386—24no—
35C23×#385—24no—
36C23×#381C2214yescomplemented
37C23×#379C2212yescomplemented
38C23×#342C2214yescomplemented
39C25×C23D411yescharacteristic
40C25×C23C2311yescharacteristic
41C23×H(5,𝔽2)—24no—
42C23×H(5,𝔽2)—42no—
43C2×#41×C23—21no—
44C2×#41×C23—22no—
45C22×D4×C23—24no—
46C22×D4×C23—21no—
47C22×D4×C23—22no—
48C22×D4×C23—22no—
49C22×D4×C23—24no—
50C22×D4×C23—42no—
51C22×D4×C23D411yescharacteristic
52C22×D4×C23D411yescharacteristic
53C22×D4×C23D414yes—
54C22×D4×C23C2311yescharacteristic
55C22×D4×C23C2312yes—
56C22×D4×C23C2311yescharacteristic
57C23×C4×C23—21no—
58C23×C4×C23—21no—
59C23×C4×C23—21no—
60C23×C4×C23—24no—
61C23×C4×C23—24no—
62C23×C4×C23—42no—
63C23×C4×C23D411yescharacteristic
64C23×C4×C23C2312yes—
65C23×#127—22no—
66C23×#124—22no—
67C23×#123—28no—
68C23×#123—28no—
69C23×#123—22no—
70C23×#123—24no—
71C23×#123—22no—
72C23×#123—22no—
73C23×#123—24no—
74C23×#123—24no—
75C23×#123—28no—
76C23×#123—44no—
77C23×#123D418yescomplemented
78C23×C22⋊Q8—22no—
79C23×#121—22no—
80C23×#121—24no—
81C23×#121—24no—
82C23×#121—24no—
83C23×#120—24no—
84C23×#120—22no—
85C23×#120—22no—
86C23×#120—22no—
87C23×#120—22no—
88C23×#120—24no—
89C23×#120—22no—
90C4×D4×C23—24no—
91C4×D4×C23—24no—
92C23×#117—21no—
93C2×C4⋊C4×C23—28no—
94C2×C4⋊C4×C23—22no—
95C2×C4⋊C4×C23—22no—
96C2×C4⋊C4×C23—24no—
97C2×C22⋊C4×C23—28no—
98C2×C22⋊C4×C23—22no—
99C2×C22⋊C4×C23—22no—
100C2×C22⋊C4×C23—22no—
101C2×C22⋊C4×C23—21no—
102C2×C22⋊C4×C23—21no—
103C2×C22⋊C4×C23—24no—
104C2×C22⋊C4×C23—24no—
105C2×C22⋊C4×C23—24no—
106C2×C22⋊C4×C23D414yes—
107C2×C22⋊C4×C23C2314yes—
108C2×C22⋊C4×C23C2312yes—
109C2×C22⋊C4×C23C2312yes—
110C23×#99—24no—
111C23×#99—28no—
112C23×#99—42no—
113C23×#95—28no—
114C24×C23—22no—
115C24×C23—22no—
116C24×C23—21no—
117C24×C23—22no—
118C24×C23—22no—
119C24×C23—44no—
120C24×C23—41no—
121C24×C23—44no—
122C24×C23C2×D411yescharacteristic
123C24×C23C2×D414yes—
124C24×C23C2×D411yescharacteristic
125C24×C23C2×D412yes—
126C24×C23C2×D411yescharacteristic
127C24×C23C2×D411yescharacteristic
128C24×C23C2411yescharacteristic
129#41×C23—48no—
130#41×C23—42no—
131#41×C23—42no—
132C2×Q8×C23—21no—
133C2×D4×C23—22no—
134C2×D4×C23—22no—
135C2×D4×C23—22no—
136C2×D4×C23—21no—
137C2×D4×C23—24no—
138C2×D4×C23—22no—
139C2×D4×C23—22no—
140C2×D4×C23—22no—
141C2×D4×C23—24no—
142C2×D4×C23—22no—
143C2×D4×C23—22no—
144C2×D4×C23—22no—
145C2×D4×C23—22no—
146C2×D4×C23—22no—
147C2×D4×C23—24no—
148C2×D4×C23—24no—
149C2×D4×C23—24no—
150C2×D4×C23—22no—
151C2×D4×C23—41no—
152C2×D4×C23—41no—
153C2×D4×C23—416no—
154C2×D4×C23—48no—
155C2×D4×C23—42no—
156C2×D4×C23—42no—
157C2×D4×C23—48no—
158C2×D4×C23—84no—
159C2×D4×C23C2×D414yes—
160C22×C4×C23—24no—
161C22×C4×C23—24no—
162C22×C4×C23—21no—
163C22×C4×C23—22no—
164C22×C4×C23—24no—
165C22×C4×C23—22no—
166C22×C4×C23—22no—
167C22×C4×C23—22no—
168C22×C4×C23—24no—
169C22×C4×C23—22no—
170C22×C4×C23—24no—
171C22×C4×C23—22no—
172C22×C4×C23—22no—
173C22×C4×C23—24no—
174C22×C4×C23—41no—
175C22×C4×C23—416no—
176C22×C4×C23—44no—
177C22×C4×C23—42no—
178C22×C4×C23—42no—
179C22×C4×C23—42no—
180C22×C4×C23—42no—
181C22×C4×C23C2×D412yes—
182C22×C4×C23C2×D411yescharacteristic
183C22×C4×C23C2×D411yescharacteristic
184C22×C4×C23C2×D414yes—
185C22×C4×C23C2×D414yescomplemented
186C4⋊C4×C23—28no—
187C4⋊C4×C23—22no—
188C4⋊C4×C23—24no—
189C4⋊C4×C23—28no—
190C4⋊C4×C23—48no—
191C22⋊C4×C23—24no—
192C22⋊C4×C23—28no—
193C22⋊C4×C23—22no—
194C22⋊C4×C23—24no—
195C22⋊C4×C23—24no—
196C22⋊C4×C23—28no—
197C22⋊C4×C23—24no—
198C22⋊C4×C23—24no—
199C22⋊C4×C23—22no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—22no—
202C22⋊C4×C23—24no—
203C22⋊C4×C23—24no—
204C22⋊C4×C23—48no—
205C22⋊C4×C23—48no—
206C22⋊C4×C23C2×D414yescomplemented
207C42×C23—22no—
208#23901C2311yesmaximal, characteristic, direct product factor, Sylow
209C23×C23—21no—
210C23×C23—21no—
211C23×C23—24no—
212C23×C23—21no—
213C23×C23—21no—
214C23×C23—21no—
215C23×C23—28no—
216C23×C23—28no—
217C23×C23—21no—
218C23×C23—21no—
219C23×C23—21no—
220C23×C23—22no—
221C23×C23—21no—
222C23×C23—24no—
223C23×C23—44no—
224C23×C23—42no—
225C23×C23—42no—
226C23×C23—44no—
227C23×C23—48no—
228C23×C23—48no—
229C23×C23—42no—
230C23×C23—42no—
231C23×C23—48no—
232C23×C23—42no—
233C23×C23—44no—
234C23×C23—44no—
235C23×C23—82no—
236C23×C23—84no—
237C23×C23—82no—
238C23×C23#12012yes—
239C23×C23#12012yes—
240C23×C23#12012yes—
241C23×C23C22×D411yescharacteristic
242C23×C23C22×D412yes—
243C23×C23C22×D412yes—
244C23×C23H(5,𝔽2)12yes—
245Q8×C23—42no—
246D4×C23—28no—
247D4×C23—44no—
248D4×C23—42no—
249D4×C23—44no—
250D4×C23—44no—
251D4×C23—42no—
252D4×C23—42no—
253D4×C23—44no—
254D4×C23—44no—
255D4×C23—48no—
256D4×C23—84no—
257D4×C23—84no—
258C2×C4×C23—21no—
259C2×C4×C23—21no—
260C2×C4×C23—24no—
261C2×C4×C23—24no—
262C2×C4×C23—21no—
263C2×C4×C23—24no—
264C2×C4×C23—22no—
265C2×C4×C23—21no—
266C2×C4×C23—21no—
267C2×C4×C23—28no—
268C2×C4×C23—24no—
269C2×C4×C23—44no—
270C2×C4×C23—44no—
271C2×C4×C23—48no—
272C2×C4×C23—44no—
273C2×C4×C23—48no—
274C2×C4×C23—48no—
275C2×C4×C23—42no—
276C2×C4×C23—42no—
277C2×C4×C23—42no—
278C2×C4×C23—44no—
279C2×C4×C23—44no—
280C2×C4×C23—48no—
281C2×C4×C23—44no—
282C2×C4×C23—82no—
283C2×C4×C23#12012yes—
284#3484C2×C2311yescharacteristic
285C22×#123C2×C2311yescharacteristic, complemented
286C2×#458C2×C2312yescomplemented
287#2849C2×C2312yescomplemented
288#2022C2×C2314yescomplemented
289#1898C2×C2312yescomplemented
290#1890C2×C2312yescomplemented
291#1781C2×C2311yescharacteristic, complemented
292C22×C23—24no—
293C22×C23—24no—
294C22×C23—22no—
295C22×C23—24no—
296C22×C23—22no—
297C22×C23—22no—
298C22×C23—22no—
299C22×C23—22no—
300C22×C23—24no—
301C22×C23—42no—
302C22×C23—42no—
303C22×C23—416no—
304C22×C23—42no—
305C22×C23—42no—
306C22×C23—42no—
307C22×C23—42no—
308C22×C23—42no—
309C22×C23—42no—
310C22×C23—41no—
311C22×C23—44no—
312C22×C23—42no—
313C22×C23—81no—
314C22×C23—88no—
315C22×C23—84no—
316C22×C23—81no—
317C22×C23—162no—
318C22×C23#45814yescomplemented
319C22×C23C2×#12011yescharacteristic
320C22×C23C2×#12011yescharacteristic
321C22×C23#53411yescharacteristic
322C22×C23D4214yes—
323C4×C23—22no—
324C4×C23—44no—
325C4×C23—44no—
326C4×C23—41no—
327C4×C23—42no—
328C4×C23—44no—
329C4×C23—84no—
330C2×H(5,𝔽2)C22×C2311yescharacteristic
331C23×D4C22×C2311yescharacteristic
332C24×C4C22×C2311yescharacteristic, complemented
333#546—24no—
334D42—24no—
335#535—21no—
336#534—21no—
337#534C22×C2312yes—
338#525—21no—
339C2×#123—21no—
340C2×#123—21no—
341C2×#123—28no—
342C2×#123C22×C2312yes—
343C2×#123C22×C2312yescomplemented
344C2×#123C22×C2314yescomplemented
345C2×#120—21no—
346C2×#120C22×C2312yes—
347#518C22×C2311yescharacteristic
348C22×C4⋊C4—21no—
349C22×C22⋊C4—21no—
350C22×C22⋊C4C22×C2311yescharacteristic, complemented
351#458—28no—
352C2×#99C22×C2312yes—
353C2×#99C22×C2314yescomplemented
354C2×#99C22×C2312yescomplemented
355#386—24no—
356#385—24no—
357#381C22×C2314yescomplemented
358#379C22×C2312yescomplemented
359#342C22×C2314yescomplemented
360C2×C23—21no—
361C2×C23—24no—
362C2×C23—21no—
363C2×C23—24no—
364C2×C23—42no—
365C2×C23—42no—
366C2×C23—81no—
367C2×C23—82no—
368C2×C23#240311yescharacteristic
369C2×C23C2×#45812yes—
370C25D4×C2311yescharacteristic
371C25C23×C2311yescharacteristic
372H(5,𝔽2)—24no—
373H(5,𝔽2)—42no—
374C2×#41—21no—
375C2×#41—22no—
376C22×D4—21no—
377C22×D4—22no—
378C22×D4—24no—
379C22×D4—22no—
380C22×D4—24no—
381C22×D4—42no—
382C22×D4D4×C2311yescharacteristic
383C22×D4D4×C2311yescharacteristic
384C22×D4D4×C2314yes—
385C22×D4C23×C2312yes—
386C22×D4C23×C2311yescharacteristic
387C22×D4C23×C2311yescharacteristic
388C23×C4—21no—
389C23×C4—21no—
390C23×C4—21no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4D4×C2311yescharacteristic
395C23×C4C23×C2312yes—
396#127—22no—
397#124—22no—
398#123—28no—
399#123—28no—
400#123—22no—
401#123—24no—
402#123—22no—
403#123—22no—
404#123—24no—
405#123—24no—
406#123—28no—
407#123—44no—
408#123D4×C2318yescomplemented
409C22⋊Q8—22no—
410#121—22no—
411#121—24no—
412#121—24no—
413#121—24no—
414#120—24no—
415#120—22no—
416#120—22no—
417#120—22no—
418#120—22no—
419#120—24no—
420#120—22no—
421C4×D4—24no—
422C4×D4—24no—
423#117—21no—
424C2×C4⋊C4—28no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C4⋊C4—24no—
428C2×C22⋊C4—28no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—21no—
433C2×C22⋊C4—21no—
434C2×C22⋊C4—24no—
435C2×C22⋊C4—24no—
436C2×C22⋊C4—24no—
437C2×C22⋊C4D4×C2314yes—
438C2×C22⋊C4C23×C2314yes—
439C2×C22⋊C4C23×C2312yes—
440C2×C22⋊C4C23×C2312yes—
441#99—24no—
442#99—28no—
443#99—42no—
444#95—28no—
445C23#2390111yescharacteristic, direct product factor, Sylow
446C24—22no—
447C24—22no—
448C24—21no—
449C24—22no—
450C24—22no—
451C24—44no—
452C24—41no—
453C24—44no—
454C24C2×D4×C2311yescharacteristic
455C24C2×D4×C2314yes—
456C24C2×D4×C2311yescharacteristic
457C24C2×D4×C2312yes—
458C24C2×D4×C2311yescharacteristic
459C24C2×D4×C2311yescharacteristic
460C24C24×C2311yescharacteristic
461#41—48no—
462#41—42no—
463#41—42no—
464C2×Q8—21no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—21no—
469C2×D4—24no—
470C2×D4—22no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—22no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4—24no—
480C2×D4—24no—
481C2×D4—24no—
482C2×D4—22no—
483C2×D4—41no—
484C2×D4—41no—
485C2×D4—416no—
486C2×D4—48no—
487C2×D4—42no—
488C2×D4—42no—
489C2×D4—48no—
490C2×D4—84no—
491C2×D4C2×D4×C2314yes—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—21no—
495C22×C4—22no—
496C22×C4—24no—
497C22×C4—22no—
498C22×C4—24no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—24no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—24no—
506C22×C4—41no—
507C22×C4—416no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—44no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4C2×D4×C2312yes—
514C22×C4C2×D4×C2311yescharacteristic
515C22×C4C2×D4×C2311yescharacteristic
516C22×C4C2×D4×C2314yes—
517C22×C4C2×D4×C2314yescomplemented
518C4⋊C4—28no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—28no—
522C4⋊C4—48no—
523C22⋊C4—24no—
524C22⋊C4—28no—
525C22⋊C4—22no—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—28no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4C2×D4×C2314yescomplemented
539C42—22no—
540C23—24no—
541C23—21no—
542C23—21no—
543C23—28no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—28no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—21no—
553C23—24no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—48no—
559C23—48no—
560C23—42no—
561C23—42no—
562C23—48no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—82no—
567C23—84no—
568C23—82no—
569C23C23×#12012yes—
570C23C23×#12012yes—
571C23C23×#12012yes—
572C23C22×D4×C2311yescharacteristic
573C23C22×D4×C2312yes—
574C23C22×D4×C2312yes—
575C23C23×H(5,𝔽2)12yes—
576Q8—42no—
577D4—28no—
578D4—44no—
579D4—42no—
580D4—44no—
581D4—44no—
582D4—48no—
583D4—42no—
584D4—42no—
585D4—44no—
586D4—44no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—28no—
599C2×C4—24no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—48no—
603C2×C4—48no—
604C2×C4—48no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—48no—
612C2×C4—44no—
613C2×C4—82no—
614C2×C4C23×#12012yes—
615C22—24no—
616C22—24no—
617C22—22no—
618C22—24no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—24no—
624C22—42no—
625C22—42no—
626C22—416no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—41no—
634C22—44no—
635C22—42no—
636C22—81no—
637C22—88no—
638C22—81no—
639C22—84no—
640C22—162no—
641C22C23×#45814yescomplemented
642C22C2×C23×#12011yescharacteristic
643C22C2×C23×#12011yescharacteristic
644C22C23×#53411yescharacteristic
645C22D42×C2314yes—
646C4—22no—
647C4—44no—
648C4—44no—
649C4—41no—
650C4—42no—
651C4—44no—
652C4—84no—
653C2—21no—
654C2—24no—
655C2—24no—
656C2—21no—
657C2—42no—
658C2—42no—
659C2—81no—
660C2—82no—
661C2C23×#240311yescharacteristic
662C2C2×C23×#45812yes—
subgroupG/H
center, socleC22×C23C2×#120
commutator, Frattini, cosocleC24C24×C23
prime radicalC23×#3484C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C25×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C23×#123C2semidirect
3. C2×C23×#458C2semidirect
4. C23×#2849C2semidirect
5. C23×#2022C2semidirect
6. C23×#1898C2semidirect
7. C23×#1890C2semidirect
8. C23×#1781C2semidirect
11. C24×C4×C23C22semidirect
22. C2×C23×#123C22semidirect
23. C2×C23×#123C22semidirect
29. C22×C22⋊C4×C23C22semidirect
32. C2×C23×#99C22semidirect
33. C2×C23×#99C22semidirect
36. C23×#381C22semidirect
37. C23×#379C22semidirect
38. C23×#342C22semidirect
77. C23×#123D4semidirect
185. C22×C4×C23C2×D4semidirect
206. C22⋊C4×C23C2×D4semidirect
208. #23901C23direct (N × H)
285. C22×#123C2×C23semidirect
286. C2×#458C2×C23semidirect
287. #2849C2×C23semidirect
288. #2022C2×C23semidirect
289. #1898C2×C23semidirect
290. #1890C2×C23semidirect
291. #1781C2×C23semidirect
318. C22×C23#458semidirect
332. C24×C4C22×C23semidirect
343. C2×#123C22×C23semidirect
344. C2×#123C22×C23semidirect
350. C22×C22⋊C4C22×C23semidirect
353. C2×#99C22×C23semidirect
354. C2×#99C22×C23semidirect
357. #381C22×C23semidirect
358. #379C22×C23semidirect
359. #342C22×C23semidirect
408. #123D4×C23semidirect
445. C23#23901direct (N × H)
517. C22×C4C2×D4×C23semidirect
538. C22⋊C4C2×D4×C23semidirect
641. C22C23×#458semidirect

Maximal subgroups

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

subgroupindex
C23×#34842
C22×C23×#1232
C2×C23×#4582
C23×#28492
C23×#20222
C23×#18982
C23×#18902
C23×#17812
#2390123