Finite Group Databasebeta

← C23×#8479 #19156105

Subgroups — C23×#8479 #19156105

Order 5888.

912 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
1C23×#3537C211yesmaximal, characteristic
2C23×#1964C211yesmaximal, characteristic, complemented
3C23×#1900C211yesmaximal, characteristic, complemented
4C23×#1895C211yesmaximal, characteristic, complemented
5C23×#1871C211yesmaximal, characteristic, complemented
6C23×#1513C211yesmaximal, characteristic, complemented
7C23×#1507C211yesmaximal, characteristic, complemented
8C22×M16×C23C2211yescharacteristic, complemented
9C23×#560—22no—
10C23×#553—21no—
11C23×#553—21no—
12D42×C23—22no—
13D42×C23—21no—
14D42×C23—21no—
15C23×#540—21no—
16C23×#538—21no—
17C23×#538C412yescomplemented
18C23×#535—21no—
19C23×#534—21no—
20C23×#534C412yescomplemented
21C23×#525C2211yescharacteristic
22C2×C23×#121C2211yescharacteristic
23C2×C23×#121C2211yescharacteristic
24C2×C23×#104—21no—
25C2×C23×#102—21no—
26C2×C23×#100—21no—
27C2×C23×#99—22no—
28C23×#407C2211yescharacteristic, complemented
29C23×#380C2211yescharacteristic, complemented
30C4×C22⋊C4×C23C2211yescharacteristic, complemented
31C23×#343—21no—
32C23×#340—21no—
33C23×#339—21no—
34C23×#328—21no—
35C23×#327—22no—
36C23×#327—21no—
37C2×#41×C23D411yescharacteristic
38C22×D4×C23—22no—
39C22×D4×C23—21no—
40C22×D4×C23—22no—
41C22×D4×C23—21no—
42C22×D4×C23D412yes—
43C22×D4×C23D411yescharacteristic
44C22×D4×C23D411yescharacteristic
45C23×C4×C23C2311yescharacteristic
46C2×M16×C23—21no—
47C2×M16×C23—21no—
48C2×M16×C23—21no—
49C2×M16×C23—42no—
50C22×C8×C23—21no—
51C23×#127—22no—
52C23×#127—21no—
53C23×#126—22no—
54C23×#126—22no—
55C23×#125—21no—
56C23×#124—22no—
57C23×#123—22no—
58C23×#123—21no—
59C23×#123—22no—
60C23×#123—22no—
61C23×#123—22no—
62C23×C22⋊Q8C2×C411yescharacteristic
63C23×C22⋊Q8D411yescharacteristic
64C23×#121—22no—
65C23×#121—21no—
66C23×#121—21no—
67C23×#121—22no—
68C23×#121—22no—
69C23×#121—22no—
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—21no—
73C23×#121—21no—
74C23×#121—21no—
75C23×#121—21no—
76C23×#121—21no—
77C23×#121—21no—
78C23×#121C2×C412yes—
79C23×#121C2×C411yescharacteristic
80C23×#121D412yes—
81C23×#121D411yescharacteristic
82C23×#120—22no—
83C23×#120—22no—
84C23×#120—22no—
85C23×#120—22no—
86C23×#120—21no—
87C23×#120—22no—
88C23×#120—22no—
89C4×D4×C23—22no—
90C4×D4×C23—21no—
91C4×D4×C23—21no—
92C4×D4×C23—22no—
93C4×D4×C23—21no—
94C4×D4×C23—21no—
95C23×#117—21no—
96C23×#117D411yescharacteristic
97C2×C4⋊C4×C23—21no—
98C2×C4⋊C4×C23D411yescharacteristic
99C2×C4⋊C4×C23D411yescharacteristic
100C2×C22⋊C4×C23—21no—
101C2×C22⋊C4×C23—21no—
102C2×C22⋊C4×C23—21no—
103C2×C22⋊C4×C23C2×C412yes—
104C2×C42×C23—21no—
105C23×#104—42no—
106C23×#102—42no—
107C23×#100—42no—
108C23×#99—44no—
109C23×C22⋊C8—21no—
110C23×C22⋊C8—21no—
111C23×#95—21no—
112C23×#95—21no—
113C24×C23—22no—
114C24×C23—21no—
115C24×C23C22⋊C412yes—
116C24×C23C2×D411yescharacteristic
117#41×C23—41no—
118#41×C23—41no—
119C2×Q8×C23C22⋊C411yescharacteristic
120C2×D4×C23—22no—
121C2×D4×C23—21no—
122C2×D4×C23—22no—
123C2×D4×C23—21no—
124C2×D4×C23—22no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—22no—
128C2×D4×C23—22no—
129C2×D4×C23—21no—
130C2×D4×C23—22no—
131C2×D4×C23—22no—
132C2×D4×C23—21no—
133C2×D4×C23—21no—
134C2×D4×C23—21no—
135C2×D4×C23—41no—
136C2×D4×C23—41no—
137C2×D4×C23—41no—
138C2×D4×C23—41no—
139C2×D4×C23—42no—
140C2×D4×C23—42no—
141C2×D4×C23—81no—
142C2×D4×C23—81no—
143C2×D4×C23—82no—
144C2×D4×C23—84no—
145C2×D4×C23—82no—
146C2×D4×C23—82no—
147C2×D4×C23C22⋊C411yescharacteristic
148C2×D4×C23C22⋊C412yes—
149C22×C4×C23—22no—
150C22×C4×C23—22no—
151C22×C4×C23—21no—
152C22×C4×C23—21no—
153C22×C4×C23—41no—
154C22×C4×C23—41no—
155C22×C4×C23—41no—
156C22×C4×C23—41no—
157C22×C4×C23C22⋊C412yes—
158C22×C4×C23C22×C411yescharacteristic
159C22×C4×C23C2×D411yescharacteristic
160C22×C4×C23C2×D411yescharacteristic
161C22×C4×C23C2×D411yescharacteristic
162C22×C4×C23C2×D411yescharacteristic
163C22×C4×C23C2×D411yescharacteristic
164M16×C23—22no—
165M16×C23—22no—
166M16×C23—41no—
167M16×C23—41no—
168M16×C23—41no—
169M16×C23—41no—
170C2×C8×C23—22no—
171C2×C8×C23—41no—
172C2×C8×C23—41no—
173C2×C8×C23—41no—
174C4⋊C4×C23—22no—
175C4⋊C4×C23—22no—
176C4⋊C4×C23—21no—
177C4⋊C4×C23—21no—
178C4⋊C4×C23—21no—
179C4⋊C4×C23—21no—
180C4⋊C4×C23—21no—
181C4⋊C4×C23—41no—
182C4⋊C4×C23—41no—
183C4⋊C4×C23C22⋊C412yes—
184C4⋊C4×C23C22⋊C411yescharacteristic
185C4⋊C4×C23C22⋊C411yescharacteristic
186C22⋊C4×C23—22no—
187C22⋊C4×C23—22no—
188C22⋊C4×C23—22no—
189C22⋊C4×C23—22no—
190C22⋊C4×C23—22no—
191C22⋊C4×C23—21no—
192C22⋊C4×C23—21no—
193C22⋊C4×C23—21no—
194C22⋊C4×C23—22no—
195C22⋊C4×C23—22no—
196C22⋊C4×C23—22no—
197C22⋊C4×C23—21no—
198C22⋊C4×C23—22no—
199C22⋊C4×C23—21no—
200C22⋊C4×C23—22no—
201C22⋊C4×C23—41no—
202C22⋊C4×C23—42no—
203C42×C23—22no—
204C42×C23—21no—
205C42×C23—41no—
206C42×C23—41no—
207#8479C2311yesmaximal, characteristic, direct product factor, Sylow
208C23×C23—21no—
209C23×C23—21no—
210C23×C23—22no—
211C23×C23—22no—
212C23×C23—21no—
213C23×C23—41no—
214C23×C23—41no—
215C23×C23—42no—
216C23×C23—42no—
217C23×C23—42no—
218C23×C23—41no—
219C23×C23—41no—
220C23×C23—81no—
221C23×C23—81no—
222C23×C23—82no—
223C23×C23—82no—
224C23×C23—82no—
225C23×C23C2×C22⋊C411yescharacteristic
226C23×C23#12011yescharacteristic
227C23×C23#12011yescharacteristic
228Q8×C23—41no—
229D4×C23—42no—
230D4×C23—41no—
231D4×C23—44no—
232D4×C23—42no—
233D4×C23—42no—
234D4×C23—41no—
235D4×C23—42no—
236D4×C23—41no—
237D4×C23—41no—
238D4×C23—41no—
239D4×C23—41no—
240D4×C23—41no—
241D4×C23—81no—
242D4×C23—82no—
243D4×C23—82no—
244D4×C23—81no—
245D4×C23—82no—
246D4×C23—82no—
247D4×C23—82no—
248D4×C23—82no—
249D4×C23—82no—
250C2×C4×C23—21no—
251C2×C4×C23—21no—
252C2×C4×C23—21no—
253C2×C4×C23—21no—
254C2×C4×C23—21no—
255C2×C4×C23—21no—
256C2×C4×C23—21no—
257C2×C4×C23—22no—
258C2×C4×C23—21no—
259C2×C4×C23—22no—
260C2×C4×C23—21no—
261C2×C4×C23—41no—
262C2×C4×C23—41no—
263C2×C4×C23—41no—
264C2×C4×C23—41no—
265C2×C4×C23—41no—
266C2×C4×C23—42no—
267C2×C4×C23—42no—
268C2×C4×C23—81no—
269C2×C4×C23—82no—
270C2×C4×C23—82no—
271C2×C4×C23—81no—
272C2×C4×C23—81no—
273C2×C4×C23—81no—
274C2×C4×C23—81no—
275C2×C4×C23—81no—
276C2×C4×C23C2×C22⋊C411yescharacteristic
277C2×C4×C23C2×C22⋊C411yescharacteristic
278C2×C4×C23#12011yescharacteristic
279C2×C4×C23#12011yescharacteristic
280C8×C23—42no—
281C8×C23—81no—
282#3537C2×C2311yescharacteristic
283#1964C2×C2311yescharacteristic, complemented
284#1900C2×C2311yescharacteristic, complemented
285#1895C2×C2311yescharacteristic, complemented
286#1871C2×C2311yescharacteristic, complemented
287#1513C2×C2311yescharacteristic, complemented
288#1507C2×C2311yescharacteristic, complemented
289C22×C23—21no—
290C22×C23—21no—
291C22×C23—21no—
292C22×C23—21no—
293C22×C23—21no—
294C22×C23—21no—
295C22×C23—21no—
296C22×C23—21no—
297C22×C23—42no—
298C22×C23—42no—
299C22×C23—41no—
300C22×C23—42no—
301C22×C23—42no—
302C22×C23—41no—
303C22×C23—41no—
304C22×C23—42no—
305C22×C23—42no—
306C22×C23—81no—
307C22×C23—81no—
308C22×C23—161no—
309C22×C23—161no—
310C22×C23—162no—
311C22×C23—164no—
312C22×C23#37911yescharacteristic
313C22×C23#45311yescharacteristic
314C22×C23#45511yescharacteristic
315C4×C23—21no—
316C4×C23—21no—
317C4×C23—21no—
318C4×C23—42no—
319C4×C23—41no—
320C4×C23—42no—
321C4×C23—41no—
322C4×C23—41no—
323C4×C23—82no—
324C4×C23—161no—
325C4×C23B(4,𝔽2)11yescharacteristic
326C4×C23#45811yescharacteristic
327C22×M16C22×C2311yescharacteristic, complemented
328#560—22no—
329#553—21no—
330#553—21no—
331D42—22no—
332D42—21no—
333D42—21no—
334#540—21no—
335#538—21no—
336#538C4×C2312yescomplemented
337#535—21no—
338#534—21no—
339#534C4×C2312yescomplemented
340#525C22×C2311yescharacteristic
341C2×#121C22×C2311yescharacteristic
342C2×#121C22×C2311yescharacteristic
343C2×#104—21no—
344C2×#102—21no—
345C2×#100—21no—
346C2×#99—22no—
347#407C22×C2311yescharacteristic, complemented
348#380C22×C2311yescharacteristic, complemented
349C4×C22⋊C4C22×C2311yescharacteristic, complemented
350#343—21no—
351#340—21no—
352#339—21no—
353#328—21no—
354#327—22no—
355#327—21no—
356C2×C23—21no—
357C2×C23—21no—
358C2×C23—41no—
359C2×C23—41no—
360C2×C23—82no—
361C2×C23—82no—
362C2×C23—81no—
363C2×C23#179011yescharacteristic
364C2×C23#179211yescharacteristic
365C2×C23#179511yescharacteristic
366C2×#41D4×C2311yescharacteristic
367C22×D4—22no—
368C22×D4—21no—
369C22×D4—22no—
370C22×D4—21no—
371C22×D4D4×C2312yes—
372C22×D4D4×C2311yescharacteristic
373C22×D4D4×C2311yescharacteristic
374C23×C4C23×C2311yescharacteristic
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—42no—
379C22×C8—21no—
380#127—22no—
381#127—21no—
382#126—22no—
383#126—22no—
384#125—21no—
385#124—22no—
386#123—22no—
387#123—21no—
388#123—22no—
389#123—22no—
390#123—22no—
391C22⋊Q8C2×C4×C2311yescharacteristic
392C22⋊Q8D4×C2311yescharacteristic
393#121—22no—
394#121—21no—
395#121—21no—
396#121—22no—
397#121—22no—
398#121—22no—
399#121—22no—
400#121—22no—
401#121—21no—
402#121—21no—
403#121—21no—
404#121—21no—
405#121—21no—
406#121—21no—
407#121C2×C4×C2312yes—
408#121C2×C4×C2311yescharacteristic
409#121D4×C2312yes—
410#121D4×C2311yescharacteristic
411#120—22no—
412#120—22no—
413#120—22no—
414#120—22no—
415#120—21no—
416#120—22no—
417#120—22no—
418C4×D4—22no—
419C4×D4—21no—
420C4×D4—21no—
421C4×D4—22no—
422C4×D4—21no—
423C4×D4—21no—
424#117—21no—
425#117D4×C2311yescharacteristic
426C2×C4⋊C4—21no—
427C2×C4⋊C4D4×C2311yescharacteristic
428C2×C4⋊C4D4×C2311yescharacteristic
429C2×C22⋊C4—21no—
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4C2×C4×C2312yes—
433C2×C42—21no—
434#104—42no—
435#102—42no—
436#100—42no—
437#99—44no—
438C22⋊C8—21no—
439C22⋊C8—21no—
440#95—21no—
441#95—21no—
442C23#847911yescharacteristic, direct product factor, Sylow
443C24—22no—
444C24—21no—
445C24C22⋊C4×C2312yes—
446C24C2×D4×C2311yescharacteristic
447#41—41no—
448#41—41no—
449C2×Q8C22⋊C4×C2311yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—41no—
466C2×D4—41no—
467C2×D4—41no—
468C2×D4—41no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—81no—
472C2×D4—81no—
473C2×D4—82no—
474C2×D4—84no—
475C2×D4—82no—
476C2×D4—82no—
477C2×D4C22⋊C4×C2311yescharacteristic
478C2×D4C22⋊C4×C2312yes—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4C22⋊C4×C2312yes—
488C22×C4C22×C4×C2311yescharacteristic
489C22×C4C2×D4×C2311yescharacteristic
490C22×C4C2×D4×C2311yescharacteristic
491C22×C4C2×D4×C2311yescharacteristic
492C22×C4C2×D4×C2311yescharacteristic
493C22×C4C2×D4×C2311yescharacteristic
494M16—22no—
495M16—22no—
496M16—41no—
497M16—41no—
498M16—41no—
499M16—41no—
500C2×C8—22no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—41no—
513C4⋊C4C22⋊C4×C2312yes—
514C4⋊C4C22⋊C4×C2311yescharacteristic
515C4⋊C4C22⋊C4×C2311yescharacteristic
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—22no—
525C22⋊C4—22no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—22no—
529C22⋊C4—21no—
530C22⋊C4—22no—
531C22⋊C4—41no—
532C22⋊C4—42no—
533C42—22no—
534C42—21no—
535C42—41no—
536C42—41no—
537C23—21no—
538C23—21no—
539C23—22no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—41no—
548C23—41no—
549C23—81no—
550C23—81no—
551C23—82no—
552C23—82no—
553C23—82no—
554C23C2×C22⋊C4×C2311yescharacteristic
555C23C23×#12011yescharacteristic
556C23C23×#12011yescharacteristic
557Q8—41no—
558D4—42no—
559D4—41no—
560D4—44no—
561D4—42no—
562D4—42no—
563D4—41no—
564D4—42no—
565D4—41no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—81no—
571D4—82no—
572D4—82no—
573D4—81no—
574D4—82no—
575D4—82no—
576D4—82no—
577D4—82no—
578D4—82no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—81no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4C2×C22⋊C4×C2311yescharacteristic
606C2×C4C2×C22⋊C4×C2311yescharacteristic
607C2×C4C23×#12011yescharacteristic
608C2×C4C23×#12011yescharacteristic
609C8—42no—
610C8—81no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—41no—
625C22—42no—
626C22—42no—
627C22—41no—
628C22—81no—
629C22—81no—
630C22—161no—
631C22—161no—
632C22—162no—
633C22—164no—
634C22C23×#37911yescharacteristic
635C22C23×#45311yescharacteristic
636C22C23×#45511yescharacteristic
637C4—21no—
638C4—21no—
639C4—21no—
640C4—41no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—41no—
645C4—82no—
646C4—161no—
647C4C23×B(4,𝔽2)11yescharacteristic
648C4C23×#45811yescharacteristic
649C2—21no—
650C2—21no—
651C2—41no—
652C2—41no—
653C2—81no—
654C2—82no—
655C2—82no—
656C2C23×#179011yescharacteristic
657C2C23×#179211yescharacteristic
658C2C23×#179511yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#3537C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1964C2semidirect
3. C23×#1900C2semidirect
4. C23×#1895C2semidirect
5. C23×#1871C2semidirect
6. C23×#1513C2semidirect
7. C23×#1507C2semidirect
8. C22×M16×C23C22semidirect
17. C23×#538C4semidirect
20. C23×#534C4semidirect
28. C23×#407C22semidirect
29. C23×#380C22semidirect
30. C4×C22⋊C4×C23C22semidirect
207. #8479C23direct (N × H)
283. #1964C2×C23semidirect
284. #1900C2×C23semidirect
285. #1895C2×C23semidirect
286. #1871C2×C23semidirect
287. #1513C2×C23semidirect
288. #1507C2×C23semidirect
327. C22×M16C22×C23semidirect
336. #538C4×C23semidirect
339. #534C4×C23semidirect
347. #407C22×C23semidirect
348. #380C22×C23semidirect
349. C4×C22⋊C4C22×C23semidirect
442. C23#8479direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35372
C23×#19642
C23×#19002
C23×#18952
C23×#18712
C23×#15132
C23×#15072
#847923