Finite Group Databasebeta

← C23×#8510 #19156136

Subgroups — C23×#8510 #19156136

Order 5888.

808 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3558C211yesmaximal, characteristic
2C23×#1983C211yesmaximal, characteristic, complemented
3C23×#1980C211yesmaximal, characteristic, complemented
4C23×#1551C211yesmaximal, characteristic, complemented
5C23×#1527C211yesmaximal, characteristic, complemented
6C23×#1509C211yesmaximal, characteristic, complemented
7C23×#1483C211yesmaximal, characteristic, complemented
8C23×#562C412yescomplemented
9C23×H(3,𝔽4)C412yescomplemented
10C23×#559—21no—
11C23×#548—21no—
12C23×#546—22no—
13C23×#546—21no—
14D42×C23—21no—
15C23×#542—21no—
16C23×#538—22no—
17C23×#538—21no—
18C23×#535—21no—
19C23×#534—21no—
20C23×#532C2211yescharacteristic
21C23×#529C2211yescharacteristic
22C2×C23×#124C2211yescharacteristic
23C23×#432C2211yescharacteristic, complemented
24C2×C23×C4⋊C8C2211yescharacteristic, complemented
25C2×C23×#104—22no—
26C2×C23×#103—21no—
27C2×C23×#102—21no—
28C4×M16×C23C2211yescharacteristic, complemented
29C4×C4⋊C4×C23C2211yescharacteristic, complemented
30C23×#333—21no—
31C23×#331—21no—
32C23×#329—22no—
33C23×#328—21no—
34C23×#327—21no—
35C23×Q8⋊C8—21no—
36C23×D4⋊C8—21no—
37C2×#41×C23—21no—
38C2×#41×C23D412yes—
39C22×Q8×C23D411yescharacteristic
40C22×D4×C23—22no—
41C22×D4×C23—21no—
42C22×D4×C23D411yescharacteristic
43C2×M16×C23—21no—
44C22×C8×C23—21no—
45C23×C4⋊Q8D411yescharacteristic
46C23×#127—21no—
47C23×#127D411yescharacteristic
48C23×#126—22no—
49C23×#126—21no—
50C23×#125C2×C411yescharacteristic
51C23×#124—21no—
52C23×#124—22no—
53C23×#124—22no—
54C23×#124—22no—
55C23×#124—21no—
56C23×#124—21no—
57C23×#124C2×C411yescharacteristic
58C23×#124C2×C412yes—
59C23×#124C2×C412yes—
60C23×#124D412yes—
61C23×#123—22no—
62C23×#123—22no—
63C23×#123—21no—
64C23×#123—22no—
65C23×#123—22no—
66C23×C22⋊Q8—21no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—22no—
69C23×C22⋊Q8D411yescharacteristic
70C23×#121—22no—
71C23×#121—22no—
72C23×#121—22no—
73C23×#121—21no—
74C23×#121—22no—
75C23×#121—22no—
76C23×#121—21no—
77C23×#121—21no—
78C23×#121D411yescharacteristic
79C23×#120—22no—
80C23×#120—22no—
81C23×#120—22no—
82C23×#120—22no—
83C23×#120—22no—
84C4×Q8×C23—21no—
85C4×Q8×C23D411yescharacteristic
86C4×D4×C23—22no—
87C4×D4×C23—22no—
88C4×D4×C23—21no—
89C4×D4×C23—21no—
90C4×D4×C23—21no—
91C4×D4×C23—21no—
92C4×D4×C23D411yescharacteristic
93C23×#117—21no—
94C2×C4⋊C4×C23—21no—
95C2×C22⋊C4×C23—22no—
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—21no—
98C2×C42×C23—21no—
99C2×C42×C23C2311yescharacteristic
100C23×C4⋊C8—21no—
101C23×C4⋊C8—21no—
102C23×C4⋊C8—21no—
103C23×#104—44no—
104C23×#103—42no—
105C23×#102—42no—
106C23×C22⋊C8—21no—
107C23×#97—21no—
108C23×#97—21no—
109C4×C8×C23—21no—
110C23×#95—21no—
111C24×C23—22no—
112C24×C23—21no—
113#41×C23—42no—
114#41×C23—42no—
115#41×C23—81no—
116#41×C23—81no—
117C2×Q8×C23—21no—
118C2×Q8×C23—41no—
119C2×Q8×C23—41no—
120C2×Q8×C23C22⋊C411yescharacteristic
121C2×Q8×C23C22⋊C412yes—
122C2×Q8×C23C22⋊C412yes—
123C2×D4×C23—22no—
124C2×D4×C23—21no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23—22no—
128C2×D4×C23—22no—
129C2×D4×C23—22no—
130C2×D4×C23—21no—
131C2×D4×C23—21no—
132C2×D4×C23—21no—
133C2×D4×C23—41no—
134C2×D4×C23—41no—
135C2×D4×C23—81no—
136C2×D4×C23—81no—
137C2×D4×C23—82no—
138C2×D4×C23—82no—
139C2×D4×C23C22⋊C411yescharacteristic
140C2×D4×C23C22⋊C412yes—
141C2×D4×C23C22⋊C412yes—
142C22×C4×C23—22no—
143C22×C4×C23—22no—
144C22×C4×C23—21no—
145C22×C4×C23—21no—
146C22×C4×C23—22no—
147C22×C4×C23—41no—
148C22×C4×C23C2×D411yescharacteristic
149C22×C4×C23C2×D411yescharacteristic
150C22×C4×C23C2×D411yescharacteristic
151M16×C23—22no—
152M16×C23—41no—
153C2×C8×C23—41no—
154C2×C8×C23—41no—
155C2×C8×C23—41no—
156C2×C8×C23—41no—
157C2×C8×C23—41no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—21no—
161C4⋊C4×C23—21no—
162C4⋊C4×C23—21no—
163C4⋊C4×C23—21no—
164C4⋊C4×C23—21no—
165C4⋊C4×C23—21no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—41no—
168C4⋊C4×C23C22⋊C411yescharacteristic
169C4⋊C4×C23C22⋊C411yescharacteristic
170C22⋊C4×C23—22no—
171C22⋊C4×C23—22no—
172C22⋊C4×C23—22no—
173C22⋊C4×C23—22no—
174C22⋊C4×C23—22no—
175C22⋊C4×C23—21no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—21no—
179C22⋊C4×C23—21no—
180C22⋊C4×C23—22no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—22no—
183C22⋊C4×C23—21no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—21no—
186C42×C23—22no—
187C42×C23—21no—
188C42×C23—42no—
189C42×C23C22×C411yescharacteristic
190C42×C23C2×D411yescharacteristic
191C42×C23C2×D411yescharacteristic
192C42×C23C2×D411yescharacteristic
193#8510C2311yesmaximal, characteristic, direct product factor, Sylow
194C23×C23—21no—
195C23×C23—22no—
196C23×C23—22no—
197C23×C23—41no—
198C23×C23—41no—
199C23×C23—81no—
200C23×C23—81no—
201C23×C23—82no—
202C23×C23—82no—
203C23×C23—82no—
204C23×C23#12011yescharacteristic
205Q8×C23—42no—
206Q8×C23—41no—
207Q8×C23—42no—
208Q8×C23—41no—
209Q8×C23—41no—
210D4×C23—42no—
211D4×C23—41no—
212D4×C23—42no—
213D4×C23—41no—
214D4×C23—42no—
215D4×C23—41no—
216D4×C23—42no—
217D4×C23—41no—
218D4×C23—41no—
219D4×C23—41no—
220D4×C23—41no—
221D4×C23—81no—
222D4×C23—82no—
223D4×C23—82no—
224D4×C23—82no—
225D4×C23—82no—
226D4×C23—82no—
227C2×C4×C23—21no—
228C2×C4×C23—21no—
229C2×C4×C23—21no—
230C2×C4×C23—21no—
231C2×C4×C23—21no—
232C2×C4×C23—22no—
233C2×C4×C23—22no—
234C2×C4×C23—21no—
235C2×C4×C23—21no—
236C2×C4×C23—41no—
237C2×C4×C23—41no—
238C2×C4×C23—41no—
239C2×C4×C23—42no—
240C2×C4×C23—82no—
241C2×C4×C23—81no—
242C2×C4×C23—82no—
243C2×C4×C23—82no—
244C2×C4×C23—81no—
245C2×C4×C23—81no—
246C2×C4×C23C2×C22⋊C411yescharacteristic
247C2×C4×C23C2×C22⋊C411yescharacteristic
248C2×C4×C23C2×C22⋊C411yescharacteristic
249C2×C4×C23#12011yescharacteristic
250C2×C4×C23#12011yescharacteristic
251C2×C4×C23#12011yescharacteristic
252C8×C23—42no—
253C8×C23—81no—
254C8×C23—81no—
255#3558C2×C2311yescharacteristic
256#1983C2×C2311yescharacteristic, complemented
257#1980C2×C2311yescharacteristic, complemented
258#1551C2×C2311yescharacteristic, complemented
259#1527C2×C2311yescharacteristic, complemented
260#1509C2×C2311yescharacteristic, complemented
261#1483C2×C2311yescharacteristic, complemented
262C22×C23—21no—
263C22×C23—21no—
264C22×C23—41no—
265C22×C23—42no—
266C22×C23—42no—
267C22×C23—42no—
268C22×C23—42no—
269C22×C23—41no—
270C22×C23—41no—
271C22×C23—42no—
272C22×C23—42no—
273C22×C23—161no—
274C22×C23—161no—
275C22×C23—164no—
276C22×C23#37911yescharacteristic
277C22×C23#45311yescharacteristic
278C22×C23#45511yescharacteristic
279C4×C23—21no—
280C4×C23—21no—
281C4×C23—21no—
282C4×C23—21no—
283C4×C23—21no—
284C4×C23—41no—
285C4×C23—41no—
286C4×C23—42no—
287C4×C23—42no—
288C4×C23—41no—
289C4×C23—82no—
290C4×C23#45311yescharacteristic
291C4×C23#45511yescharacteristic
292#562C4×C2312yescomplemented
293H(3,𝔽4)C4×C2312yescomplemented
294#559—21no—
295#548—21no—
296#546—22no—
297#546—21no—
298D42—21no—
299#542—21no—
300#538—22no—
301#538—21no—
302#535—21no—
303#534—21no—
304#532C22×C2311yescharacteristic
305#529C22×C2311yescharacteristic
306C2×#124C22×C2311yescharacteristic
307#432C22×C2311yescharacteristic, complemented
308C2×C4⋊C8C22×C2311yescharacteristic, complemented
309C2×#104—22no—
310C2×#103—21no—
311C2×#102—21no—
312C4×M16C22×C2311yescharacteristic, complemented
313C4×C4⋊C4C22×C2311yescharacteristic, complemented
314#333—21no—
315#331—21no—
316#329—22no—
317#328—21no—
318#327—21no—
319Q8⋊C8—21no—
320D4⋊C8—21no—
321C2×C23—21no—
322C2×C23—21no—
323C2×C23—82no—
324C2×C23—81no—
325C2×C23—82no—
326C2×C23#179211yescharacteristic
327C2×C23#179211yescharacteristic
328C2×C23#179411yescharacteristic
329C2×#41—21no—
330C2×#41D4×C2312yes—
331C22×Q8D4×C2311yescharacteristic
332C22×D4—22no—
333C22×D4—21no—
334C22×D4D4×C2311yescharacteristic
335C2×M16—21no—
336C22×C8—21no—
337C4⋊Q8D4×C2311yescharacteristic
338#127—21no—
339#127D4×C2311yescharacteristic
340#126—22no—
341#126—21no—
342#125C2×C4×C2311yescharacteristic
343#124—21no—
344#124—22no—
345#124—22no—
346#124—22no—
347#124—21no—
348#124—21no—
349#124C2×C4×C2312yes—
350#124C2×C4×C2311yescharacteristic
351#124C2×C4×C2312yes—
352#124D4×C2312yes—
353#123—22no—
354#123—22no—
355#123—21no—
356#123—22no—
357#123—22no—
358C22⋊Q8—21no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8D4×C2311yescharacteristic
362#121—22no—
363#121—22no—
364#121—22no—
365#121—21no—
366#121—22no—
367#121—22no—
368#121—21no—
369#121—21no—
370#121D4×C2311yescharacteristic
371#120—22no—
372#120—22no—
373#120—22no—
374#120—22no—
375#120—22no—
376C4×Q8—21no—
377C4×Q8D4×C2311yescharacteristic
378C4×D4—22no—
379C4×D4—22no—
380C4×D4—21no—
381C4×D4—21no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4D4×C2311yescharacteristic
385#117—21no—
386C2×C4⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—21no—
391C2×C42C23×C2311yescharacteristic
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—21no—
395#104—44no—
396#103—42no—
397#102—42no—
398C22⋊C8—21no—
399#97—21no—
400#97—21no—
401C4×C8—21no—
402#95—21no—
403C23#851011yescharacteristic, direct product factor, Sylow
404C24—22no—
405C24—21no—
406#41—42no—
407#41—42no—
408#41—81no—
409#41—81no—
410C2×Q8—21no—
411C2×Q8—41no—
412C2×Q8—41no—
413C2×Q8C22⋊C4×C2311yescharacteristic
414C2×Q8C22⋊C4×C2312yes—
415C2×Q8C22⋊C4×C2312yes—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—41no—
427C2×D4—41no—
428C2×D4—81no—
429C2×D4—81no—
430C2×D4—82no—
431C2×D4—82no—
432C2×D4C22⋊C4×C2311yescharacteristic
433C2×D4C22⋊C4×C2312yes—
434C2×D4C22⋊C4×C2312yes—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—41no—
441C22×C4C2×D4×C2311yescharacteristic
442C22×C4C2×D4×C2311yescharacteristic
443C22×C4C2×D4×C2311yescharacteristic
444M16—22no—
445M16—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—41no—
461C4⋊C4C22⋊C4×C2311yescharacteristic
462C4⋊C4C22⋊C4×C2311yescharacteristic
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—21no—
477C22⋊C4—22no—
478C22⋊C4—21no—
479C42—22no—
480C42—21no—
481C42—42no—
482C42C22×C4×C2311yescharacteristic
483C42C2×D4×C2311yescharacteristic
484C42C2×D4×C2311yescharacteristic
485C42C2×D4×C2311yescharacteristic
486C23—21no—
487C23—22no—
488C23—22no—
489C23—41no—
490C23—41no—
491C23—81no—
492C23—81no—
493C23—82no—
494C23—82no—
495C23—82no—
496C23C23×#12011yescharacteristic
497Q8—42no—
498Q8—42no—
499Q8—41no—
500Q8—41no—
501Q8—41no—
502D4—42no—
503D4—42no—
504D4—41no—
505D4—41no—
506D4—42no—
507D4—41no—
508D4—42no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—41no—
513D4—81no—
514D4—82no—
515D4—82no—
516D4—82no—
517D4—82no—
518D4—82no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4C2×C22⋊C4×C2311yescharacteristic
539C2×C4C2×C22⋊C4×C2311yescharacteristic
540C2×C4C2×C22⋊C4×C2311yescharacteristic
541C2×C4C23×#12011yescharacteristic
542C2×C4C23×#12011yescharacteristic
543C2×C4C23×#12011yescharacteristic
544C8—42no—
545C8—81no—
546C8—81no—
547C22—21no—
548C22—21no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—161no—
559C22—161no—
560C22—164no—
561C22C23×#37911yescharacteristic
562C22C23×#45311yescharacteristic
563C22C23×#45511yescharacteristic
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—42no—
573C4—41no—
574C4—82no—
575C4C23×#45311yescharacteristic
576C4C23×#45511yescharacteristic
577C2—21no—
578C2—21no—
579C2—81no—
580C2—82no—
581C2—82no—
582C2C23×#179211yescharacteristic
583C2C23×#179211yescharacteristic
584C2C23×#179411yescharacteristic
subgroupG/H
center, socleC22×C23#379
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3558C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#1983C2semidirect
3. C23×#1980C2semidirect
4. C23×#1551C2semidirect
5. C23×#1527C2semidirect
6. C23×#1509C2semidirect
7. C23×#1483C2semidirect
8. C23×#562C4semidirect
9. C23×H(3,𝔽4)C4semidirect
23. C23×#432C22semidirect
24. C2×C23×C4⋊C8C22semidirect
28. C4×M16×C23C22semidirect
29. C4×C4⋊C4×C23C22semidirect
193. #8510C23direct (N × H)
256. #1983C2×C23semidirect
257. #1980C2×C23semidirect
258. #1551C2×C23semidirect
259. #1527C2×C23semidirect
260. #1509C2×C23semidirect
261. #1483C2×C23semidirect
292. #562C4×C23semidirect
293. H(3,𝔽4)C4×C23semidirect
307. #432C22×C23semidirect
308. C2×C4⋊C8C22×C23semidirect
312. C4×M16C22×C23semidirect
313. C4×C4⋊C4C22×C23semidirect
403. C23#8510direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35582
C23×#19832
C23×#19802
C23×#15512
C23×#15272
C23×#15092
C23×#14832
#851023