Finite Group Databasebeta

← C23×#13043 #19160669

Subgroups — C23×#13043 #19160669

Order 5888.

844 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×B(4,𝔽2)C211yesmaximal, characteristic
2C23×#2375C211yesmaximal, characteristic, complemented
3C2×C23×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
4C23×#1792C211yesmaximal, characteristic, complemented
5C23×#1790C211yesmaximal, characteristic, complemented
6C23×#1393C211yesmaximal, characteristic, complemented
7C23×#1390C211yesmaximal, characteristic, complemented
8C2×C23×H(5,𝔽2)C2211yescharacteristic
9C2×C23×#120—21no—
10C2×C23×#120C2211yescharacteristic
11C23×B(4,𝔽2)—22no—
12C23×B(4,𝔽2)C414yescomplemented
13C2×C23×#104—21no—
14C23×#417—21no—
15C2×C23×#100C2211yescharacteristic
16C2×C23×#99—21no—
17C2×C23×#99—21no—
18C2×C23×#99C2211yescharacteristic
19C2×C23×#99C2211yescharacteristic
20C23×#390—21no—
21C23×#390C2211yescharacteristic
22C23×#388—21no—
23C23×#388—21no—
24C23×#388—21no—
25C23×#386—21no—
26C23×#386—21no—
27C23×#386C2211yescharacteristic
28C23×#379—21no—
29C23×AΣL(1,𝔽16)—22no—
30C25×C23D411yescharacteristic
31C23×H(5,𝔽2)—21no—
32C23×H(5,𝔽2)—21no—
33C23×H(5,𝔽2)—21no—
34C23×H(5,𝔽2)—42no—
35C23×H(5,𝔽2)C2×C412yes—
36C2×#41×C23—21no—
37C2×#41×C23—41no—
38C22×D4×C23—21no—
39C22×D4×C23—21no—
40C22×D4×C23—21no—
41C22×D4×C23—41no—
42C22×D4×C23—41no—
43C22×D4×C23C2311yescharacteristic
44C23×C4×C23D411yescharacteristic, complemented
45C2×M16×C23—21no—
46C23×#120—21no—
47C23×#120—21no—
48C23×#120—24no—
49C23×#120—42no—
50C23×#120C2×C412yes—
51C23×#120D412yes—
52C2×C4⋊C4×C23—21no—
53C2×C4⋊C4×C23D411yescharacteristic, complemented
54C2×C22⋊C4×C23—21no—
55C2×C22⋊C4×C23—21no—
56C2×C22⋊C4×C23—21no—
57C2×C22⋊C4×C23—21no—
58C2×C22⋊C4×C23—21no—
59C2×C22⋊C4×C23—21no—
60C2×C22⋊C4×C23—21no—
61C2×C22⋊C4×C23—21no—
62C2×C22⋊C4×C23—21no—
63C2×C22⋊C4×C23—21no—
64C2×C22⋊C4×C23—41no—
65C2×C22⋊C4×C23—41no—
66C2×C22⋊C4×C23D411yescharacteristic
67C2×C22⋊C4×C23D411yescharacteristic, complemented
68C2×C42×C23—21no—
69C2×C42×C23D411yescharacteristic, complemented
70C23×#104—42no—
71C23×#103—41no—
72C23×#102—41no—
73C23×#100—21no—
74C23×#100—21no—
75C23×#99—21no—
76C23×#99—24no—
77C23×#99—21no—
78C23×#99—21no—
79C23×#99—42no—
80C23×#99C2×C412yes—
81C23×#95—21no—
82C23×#95—21no—
83C23×#95—21no—
84C23×#95—41no—
85C24×C23—21no—
86C24×C23—21no—
87C24×C23—21no—
88C24×C23—21no—
89C24×C23—41no—
90C24×C23—41no—
91C24×C23—81no—
92C24×C23—81no—
93C24×C23C22⋊C412yescomplemented
94C24×C23C2×D411yescharacteristic
95C24×C23C2×D411yescharacteristic
96#41×C23—22no—
97#41×C23—22no—
98#41×C23—42no—
99#41×C23—41no—
100#41×C23—42no—
101#41×C23—42no—
102#41×C23—42no—
103#41×C23—41no—
104C2×Q8×C23—21no—
105C2×D4×C23—21no—
106C2×D4×C23—22no—
107C2×D4×C23—22no—
108C2×D4×C23—22no—
109C2×D4×C23—22no—
110C2×D4×C23—22no—
111C2×D4×C23—22no—
112C2×D4×C23—41no—
113C2×D4×C23—41no—
114C2×D4×C23—42no—
115C2×D4×C23—42no—
116C2×D4×C23—41no—
117C2×D4×C23—41no—
118C2×D4×C23—41no—
119C2×D4×C23—41no—
120C2×D4×C23—42no—
121C2×D4×C23—42no—
122C2×D4×C23—41no—
123C2×D4×C23—41no—
124C2×D4×C23—41no—
125C2×D4×C23—44no—
126C2×D4×C23—81no—
127C2×D4×C23—81no—
128C2×D4×C23—82no—
129C2×D4×C23—82no—
130C2×D4×C23C22×C411yescharacteristic
131C2×D4×C23C2×D411yescharacteristic
132C2×D4×C23#4111yescharacteristic
133C2×D4×C23#4111yescharacteristic
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—21no—
138C22×C4×C23—21no—
139C22×C4×C23—21no—
140C22×C4×C23—21no—
141C22×C4×C23—41no—
142C22×C4×C23—41no—
143C22×C4×C23—41no—
144C22×C4×C23—41no—
145C22×C4×C23—81no—
146C22×C4×C23C2×D411yescharacteristic
147M16×C23—41no—
148M16×C23—41no—
149C2×C8×C23—41no—
150C4⋊C4×C23—41no—
151C4⋊C4×C23—81no—
152C22⋊C4×C23—21no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—22no—
156C22⋊C4×C23—44no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23—41no—
159C22⋊C4×C23—41no—
160C22⋊C4×C23—41no—
161C22⋊C4×C23—81no—
162C22⋊C4×C23—81no—
163C22⋊C4×C23—81no—
164C22⋊C4×C23—81no—
165C22⋊C4×C23—81no—
166C22⋊C4×C23—81no—
167C22⋊C4×C23—81no—
168C22⋊C4×C23—82no—
169C22⋊C4×C23C22⋊C412yescomplemented
170C42×C23—41no—
171C42×C23—81no—
172#13043C2311yesmaximal, characteristic, direct product factor, Sylow
173C23×C23—21no—
174C23×C23—21no—
175C23×C23—21no—
176C23×C23—21no—
177C23×C23—22no—
178C23×C23—22no—
179C23×C23—22no—
180C23×C23—42no—
181C23×C23—41no—
182C23×C23—41no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—41no—
186C23×C23—41no—
187C23×C23—41no—
188C23×C23—41no—
189C23×C23—41no—
190C23×C23—41no—
191C23×C23—42no—
192C23×C23—42no—
193C23×C23—41no—
194C23×C23—41no—
195C23×C23—41no—
196C23×C23—81no—
197C23×C23—82no—
198C23×C23—82no—
199C23×C23—81no—
200C23×C23—81no—
201C23×C23—84no—
202C23×C23—81no—
203C23×C23—81no—
204C23×C23—81no—
205C23×C23—81no—
206C23×C23C2×C22⋊C411yescharacteristic
207C23×C23C4×D411yescharacteristic
208C23×C23#12011yescharacteristic
209C23×C23#12111yescharacteristic
210C23×C23#12311yescharacteristic
211Q8×C23—22no—
212Q8×C23—41no—
213D4×C23—22no—
214D4×C23—41no—
215D4×C23—41no—
216D4×C23—41no—
217D4×C23—41no—
218D4×C23—41no—
219D4×C23—42no—
220D4×C23—42no—
221D4×C23—42no—
222D4×C23—42no—
223D4×C23—42no—
224D4×C23—42no—
225D4×C23—82no—
226D4×C23—82no—
227D4×C23—88no—
228C2×C4×C23—21no—
229C2×C4×C23—21no—
230C2×C4×C23—22no—
231C2×C4×C23—22no—
232C2×C4×C23—22no—
233C2×C4×C23—42no—
234C2×C4×C23—42no—
235C2×C4×C23—41no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—41no—
240C2×C4×C23—41no—
241C2×C4×C23—41no—
242C2×C4×C23—42no—
243C2×C4×C23—41no—
244C2×C4×C23—41no—
245C2×C4×C23—41no—
246C2×C4×C23—41no—
247C2×C4×C23—41no—
248C2×C4×C23—41no—
249C2×C4×C23—42no—
250C2×C4×C23—41no—
251C2×C4×C23—41no—
252C2×C4×C23—41no—
253C2×C4×C23—41no—
254C2×C4×C23—81no—
255C2×C4×C23—81no—
256C2×C4×C23—81no—
257C2×C4×C23—81no—
258C2×C4×C23—81no—
259C2×C4×C23—81no—
260C2×C4×C23C4×D411yescharacteristic
261C2×C4×C23#12111yescharacteristic
262C8×C23—81no—
263C2×B(4,𝔽2)C2×C2311yescharacteristic
264#2375C2×C2311yescharacteristic, complemented
265C2×AΣL(1,𝔽16)C2×C2311yescharacteristic, complemented
266#1792C2×C2311yescharacteristic, complemented
267#1790C2×C2311yescharacteristic, complemented
268#1393C2×C2311yescharacteristic, complemented
269#1390C2×C2311yescharacteristic, complemented
270C22×C23—21no—
271C22×C23—21no—
272C22×C23—21no—
273C22×C23—21no—
274C22×C23—21no—
275C22×C23—21no—
276C22×C23—41no—
277C22×C23—41no—
278C22×C23—41no—
279C22×C23—41no—
280C22×C23—42no—
281C22×C23—42no—
282C22×C23—41no—
283C22×C23—41no—
284C22×C23—41no—
285C22×C23—41no—
286C22×C23—44no—
287C22×C23—42no—
288C22×C23—42no—
289C22×C23—81no—
290C22×C23—81no—
291C22×C23—81no—
292C22×C23—81no—
293C22×C23—82no—
294C22×C23—82no—
295C22×C23—82no—
296C22×C23—82no—
297C22×C23—82no—
298C22×C23—81no—
299C22×C23—81no—
300C22×C23—81no—
301C22×C23—81no—
302C22×C23—161no—
303C22×C23—161no—
304C22×C23#38611yescharacteristic
305C22×C23B(4,𝔽2)11yescharacteristic
306C22×C23B(4,𝔽2)11yescharacteristic
307C4×C23—21no—
308C4×C23—21no—
309C4×C23—42no—
310C4×C23—42no—
311C4×C23—81no—
312C4×C23—81no—
313C4×C23—82no—
314C4×C23—81no—
315C4×C23—81no—
316C4×C23—161no—
317C2×H(5,𝔽2)C22×C2311yescharacteristic
318C2×#120—21no—
319C2×#120C22×C2311yescharacteristic
320B(4,𝔽2)—22no—
321B(4,𝔽2)C4×C2314yescomplemented
322C2×#104—21no—
323#417—21no—
324C2×#100C22×C2311yescharacteristic
325C2×#99—21no—
326C2×#99—21no—
327C2×#99C22×C2311yescharacteristic
328C2×#99C22×C2311yescharacteristic
329#390—21no—
330#390C22×C2311yescharacteristic
331#388—21no—
332#388—21no—
333#388—21no—
334#386—21no—
335#386—21no—
336#386C22×C2311yescharacteristic
337#379—21no—
338AΣL(1,𝔽16)—22no—
339C2×C23—21no—
340C2×C23—21no—
341C2×C23—41no—
342C2×C23—41no—
343C2×C23—41no—
344C2×C23—41no—
345C2×C23—82no—
346C2×C23—82no—
347C2×C23#188911yescharacteristic
348C2×C23#219911yescharacteristic
349C2×C23#220011yescharacteristic
350C25D4×C2311yescharacteristic
351H(5,𝔽2)—21no—
352H(5,𝔽2)—21no—
353H(5,𝔽2)—21no—
354H(5,𝔽2)—42no—
355H(5,𝔽2)C2×C4×C2312yes—
356C2×#41—21no—
357C2×#41—41no—
358C22×D4—21no—
359C22×D4—21no—
360C22×D4—21no—
361C22×D4—41no—
362C22×D4—41no—
363C22×D4C23×C2311yescharacteristic
364C23×C4D4×C2311yescharacteristic, complemented
365C2×M16—21no—
366#120—21no—
367#120—21no—
368#120—24no—
369#120—42no—
370#120C2×C4×C2312yes—
371#120D4×C2312yes—
372C2×C4⋊C4—21no—
373C2×C4⋊C4D4×C2311yescharacteristic, complemented
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—41no—
385C2×C22⋊C4—41no—
386C2×C22⋊C4D4×C2311yescharacteristic
387C2×C22⋊C4D4×C2311yescharacteristic, complemented
388C2×C42—21no—
389C2×C42D4×C2311yescharacteristic, complemented
390#104—42no—
391#103—41no—
392#102—41no—
393#100—21no—
394#100—21no—
395#99—21no—
396#99—24no—
397#99—21no—
398#99—21no—
399#99—42no—
400#99C2×C4×C2312yes—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—41no—
405C23#1304311yescharacteristic, direct product factor, Sylow
406C24—21no—
407C24—21no—
408C24—21no—
409C24—21no—
410C24—41no—
411C24—41no—
412C24—81no—
413C24—81no—
414C24C22⋊C4×C2312yescomplemented
415C24C2×D4×C2311yescharacteristic
416C24C2×D4×C2311yescharacteristic
417#41—22no—
418#41—22no—
419#41—42no—
420#41—41no—
421#41—42no—
422#41—42no—
423#41—42no—
424#41—41no—
425C2×Q8—21no—
426C2×D4—21no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—22no—
433C2×D4—41no—
434C2×D4—41no—
435C2×D4—42no—
436C2×D4—42no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—42no—
443C2×D4—41no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—44no—
447C2×D4—81no—
448C2×D4—81no—
449C2×D4—82no—
450C2×D4—82no—
451C2×D4C22×C4×C2311yescharacteristic
452C2×D4C2×D4×C2311yescharacteristic
453C2×D4#41×C2311yescharacteristic
454C2×D4#41×C2311yescharacteristic
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—81no—
467C22×C4C2×D4×C2311yescharacteristic
468M16—41no—
469M16—41no—
470C2×C8—41no—
471C4⋊C4—41no—
472C4⋊C4—81no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—44no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C22⋊C4C22⋊C4×C2312yescomplemented
491C42—41no—
492C42—81no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—81no—
517C23—82no—
518C23—82no—
519C23—81no—
520C23—81no—
521C23—84no—
522C23—81no—
523C23—81no—
524C23—81no—
525C23—81no—
526C23C2×C22⋊C4×C2311yescharacteristic
527C23C4×D4×C2311yescharacteristic
528C23C23×#12011yescharacteristic
529C23C23×#12111yescharacteristic
530C23C23×#12311yescharacteristic
531Q8—22no—
532Q8—41no—
533D4—22no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—42no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—42no—
545D4—82no—
546D4—82no—
547D4—88no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4C4×D4×C2311yescharacteristic
581C2×C4C23×#12111yescharacteristic
582C8—81no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—82no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—161no—
616C22—161no—
617C22C23×#38611yescharacteristic
618C22C23×B(4,𝔽2)11yescharacteristic
619C22C23×B(4,𝔽2)11yescharacteristic
620C4—21no—
621C4—21no—
622C4—42no—
623C4—42no—
624C4—81no—
625C4—81no—
626C4—82no—
627C4—81no—
628C4—81no—
629C4—161no—
630C2—21no—
631C2—21no—
632C2—41no—
633C2—41no—
634C2—41no—
635C2—41no—
636C2—82no—
637C2—82no—
638C2C23×#188911yescharacteristic
639C2C23×#219911yescharacteristic
640C2C23×#220011yescharacteristic
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×B(4,𝔽2)C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C23 ◁ C22×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2375C2semidirect
3. C2×C23×AΣL(1,𝔽16)C2semidirect
4. C23×#1792C2semidirect
5. C23×#1790C2semidirect
6. C23×#1393C2semidirect
7. C23×#1390C2semidirect
12. C23×B(4,𝔽2)C4semidirect
44. C23×C4×C23D4semidirect
53. C2×C4⋊C4×C23D4semidirect
67. C2×C22⋊C4×C23D4semidirect
69. C2×C42×C23D4semidirect
93. C24×C23C22⋊C4semidirect
169. C22⋊C4×C23C22⋊C4semidirect
172. #13043C23direct (N × H)
264. #2375C2×C23semidirect
265. C2×AΣL(1,𝔽16)C2×C23semidirect
266. #1792C2×C23semidirect
267. #1790C2×C23semidirect
268. #1393C2×C23semidirect
269. #1390C2×C23semidirect
321. B(4,𝔽2)C4×C23semidirect
364. C23×C4D4×C23semidirect
373. C2×C4⋊C4D4×C23semidirect
387. C2×C22⋊C4D4×C23semidirect
389. C2×C42D4×C23semidirect
405. C23#13043direct (N × H)
414. C24C22⋊C4×C23semidirect
490. C22⋊C4C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×B(4,𝔽2)2
C23×#23752
C2×C23×AΣL(1,𝔽16)2
C23×#17922
C23×#17902
C23×#13932
C23×#13902
#1304323