Finite Group Databasebeta

← C23×#24097 #19171723

Subgroups — C23×#24097 #19171723

Order 5888.

1138 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#554C211yesmaximal, characteristic
2C2×C23×#483C211yesmaximal, characteristic, complemented
3C22×C23×#106C211yesmaximal, characteristic, complemented
4C23×#2486C211yesmaximal, characteristic
5C23×#2058C212yesmaximal, complemented
6C23×#2035C212yesmaximal, complemented
7C23×#1892C211yesmaximal, characteristic, complemented
8C23×#1869C211yesmaximal, characteristic
9C23×#1865C212yesmaximal, complemented
10C23×#1853C212yesmaximal, complemented
11C23×#1824C211yesmaximal, characteristic, complemented
12C23×C8×C23C2211yescharacteristic
13C23×#554—24no—
14C23×#554—24no—
15C2×C23×C4⋊Q8C2211yescharacteristic
16C2×C23×#125—21no—
17C2×C23×#123—21no—
18C2×C23×C22⋊Q8—21no—
19C2×C23×C22⋊Q8—21no—
20C2×C23×C22⋊Q8C2211yescharacteristic
21C2×C23×C22⋊Q8C2211yescharacteristic
22C2×C4×Q8×C23C2211yescharacteristic
23C2×C4×D4×C23—21no—
24C2×C4×D4×C23C2211yescharacteristic
25C22×C4⋊C4×C23C2211yescharacteristic
26C22×C4⋊C4×C23C2211yescharacteristic
27C22×C4⋊C4×C23C2211yescharacteristic
28C23×#483—24no—
29C2×C23×#106—24no—
30C2×C23×#106C2212yes—
31C2×C23×#106C2212yes—
32C2×C23×#106C2212yes—
33C2×C23×#103C2212yes—
34C2×C23×#103C2212yes—
35C2×C23×C22⋊C8C2212yes—
36C2×C23×C22⋊C8C2211yescharacteristic
37C23×#398—21no—
38C23×#397—21no—
39C23×#389C2212yes—
40C23×#381—21no—
41C23×#380C2211yescharacteristic
42C23×#380C2212yes—
43C23×#340C2212yes—
44C23×#340C2212yes—
45C23×#340C2212yes—
46C23×#336C2212yes—
47C22×Q8×C23—21no—
48C22×Q8×C23C2311yescharacteristic
49C22×D4×C23D411yescharacteristic
50C23×C4×C23—21no—
51C23×C4×C23—21no—
52C23×C4×C23—21no—
53C23×C4×C23C2311yescharacteristic
54C23×C4×C23C2311yescharacteristic
55C22×C8×C23—22no—
56C22×C8×C23—24no—
57C22×C8×C23C2312yes—
58C22×C8×C23C2312yes—
59C23×C4⋊Q8—22no—
60C23×C4⋊Q8—22no—
61C23×#125—28no—
62C23×#123—44no—
63C23×C22⋊Q8—28no—
64C23×C22⋊Q8—28no—
65C23×C22⋊Q8—22no—
66C23×C22⋊Q8—22no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8D414yes—
69C4×Q8×C23—22no—
70C4×Q8×C23—22no—
71C4×D4×C23—28no—
72C4×D4×C23—22no—
73C4×D4×C23—22no—
74C2×C4⋊C4×C23—22no—
75C2×C4⋊C4×C23—21no—
76C2×C4⋊C4×C23—21no—
77C2×C4⋊C4×C23—21no—
78C2×C4⋊C4×C23—21no—
79C2×C4⋊C4×C23—21no—
80C2×C4⋊C4×C23—21no—
81C2×C4⋊C4×C23—22no—
82C2×C4⋊C4×C23—22no—
83C2×C4⋊C4×C23—22no—
84C2×C4⋊C4×C23—22no—
85C2×C4⋊C4×C23—22no—
86C2×C4⋊C4×C23—21no—
87C2×C4⋊C4×C23D411yescharacteristic
88C2×C4⋊C4×C23D414yes—
89C2×C4⋊C4×C23C2312yes—
90C2×C4⋊C4×C23C2312yes—
91C2×C4⋊C4×C23C2311yescharacteristic
92C2×C4⋊C4×C23C2311yescharacteristic
93C2×C4⋊C4×C23C2311yescharacteristic
94C2×C22⋊C4×C23—21no—
95C2×C22⋊C4×C23—21no—
96C2×C22⋊C4×C23—21no—
97C2×C22⋊C4×C23—22no—
98C2×C22⋊C4×C23D411yescharacteristic
99C2×C22⋊C4×C23D411yescharacteristic
100C2×C42×C23—21no—
101C2×C42×C23C2311yescharacteristic
102C23×#106—24no—
103C23×#106—24no—
104C23×#106—24no—
105C23×#103—24no—
106C23×#103—24no—
107C23×C22⋊C8—22no—
108C23×C22⋊C8—24no—
109C23×#95—22no—
110C23×#95—21no—
111C23×#95—22no—
112C24×C23C2×D411yescharacteristic
113C24×C23C2×D411yescharacteristic
114C2×Q8×C23—24no—
115C2×Q8×C23—21no—
116C2×Q8×C23—22no—
117C2×Q8×C23—22no—
118C2×Q8×C23—82no—
119C2×Q8×C23C2×D412yes—
120C2×D4×C23—22no—
121C2×D4×C23—22no—
122C2×D4×C23SD1614yes—
123C22×C4×C23—22no—
124C22×C4×C23—22no—
125C22×C4×C23—21no—
126C22×C4×C23—21no—
127C22×C4×C23—21no—
128C22×C4×C23—21no—
129C22×C4×C23—24no—
130C22×C4×C23—24no—
131C22×C4×C23—21no—
132C22×C4×C23—21no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—22no—
136C22×C4×C23—22no—
137C22×C4×C23—42no—
138C22×C4×C23—42no—
139C22×C4×C23—42no—
140C22×C4×C23—82no—
141C22×C4×C23—82no—
142C22×C4×C23C2×D411yescharacteristic
143C22×C4×C23C2×D411yescharacteristic
144C22×C4×C23C2×D411yescharacteristic
145C22×C4×C23C2×D412yes—
146C22×C4×C23C2×D412yes—
147C22×C4×C23C2×D411yescharacteristic
148C22×C4×C23C2411yescharacteristic
149C2×C8×C23—28no—
150C2×C8×C23—22no—
151C2×C8×C23—22no—
152C2×C8×C23—22no—
153C2×C8×C23—22no—
154C2×C8×C23—22no—
155C2×C8×C23#4114yes—
156C4⋊C4×C23—24no—
157C4⋊C4×C23—24no—
158C4⋊C4×C23—24no—
159C4⋊C4×C23—24no—
160C4⋊C4×C23—24no—
161C4⋊C4×C23—24no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—21no—
164C4⋊C4×C23—21no—
165C4⋊C4×C23—21no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—22no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—42no—
171C4⋊C4×C23—44no—
172C4⋊C4×C23C2×D412yes—
173C4⋊C4×C23C2×D412yes—
174C4⋊C4×C23C2×D412yes—
175C22⋊C4×C23—24no—
176C22⋊C4×C23—24no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—42no—
180C22⋊C4×C23—44no—
181C42×C23—24no—
182C42×C23—21no—
183C42×C23—21no—
184#24097C2311yesmaximal, characteristic, direct product factor, Sylow
185C23×C23—22no—
186C23×C23—22no—
187C23×C23—21no—
188C23×C23—22no—
189C23×C23—21no—
190C23×C23—22no—
191C23×C23#12012yes—
192C23×C23C2×SD1612yes—
193C23×C23C2×SD1612yes—
194C23×C23#13712yes—
195C23×C23C22×D411yescharacteristic
196Q8×C23—24no—
197Q8×C23—41no—
198Q8×C23—41no—
199Q8×C23—84no—
200D4×C23—28no—
201C2×C4×C23—21no—
202C2×C4×C23—22no—
203C2×C4×C23—21no—
204C2×C4×C23—22no—
205C2×C4×C23—22no—
206C2×C4×C23—24no—
207C2×C4×C23—22no—
208C2×C4×C23—21no—
209C2×C4×C23—22no—
210C2×C4×C23—21no—
211C2×C4×C23—22no—
212C2×C4×C23—22no—
213C2×C4×C23—22no—
214C2×C4×C23—22no—
215C2×C4×C23—42no—
216C2×C4×C23—42no—
217C2×C4×C23—42no—
218C2×C4×C23—42no—
219C2×C4×C23—41no—
220C2×C4×C23—41no—
221C2×C4×C23—42no—
222C2×C4×C23—42no—
223C2×C4×C23—42no—
224C2×C4×C23—42no—
225C2×C4×C23—42no—
226C2×C4×C23—42no—
227C2×C4×C23—42no—
228C2×C4×C23—84no—
229C2×C4×C23—84no—
230C2×C4×C23—84no—
231C2×C4×C23#12012yes—
232C2×C4×C23C2×SD1612yes—
233C2×C4×C23C22×D411yescharacteristic
234C2×C4×C23C22×D411yescharacteristic
235C2×C4×C23C2×#4112yes—
236C2×C4×C232−1+412yes—
237C8×C23—24no—
238C8×C23—42no—
239C2×#554C2×C2311yescharacteristic
240C2×#483C2×C2311yescharacteristic, complemented
241C22×#106C2×C2311yescharacteristic, complemented
242#2486C2×C2311yescharacteristic
243#2058C2×C2312yescomplemented
244#2035C2×C2312yescomplemented
245#1892C2×C2311yescharacteristic, complemented
246#1869C2×C2311yescharacteristic
247#1865C2×C2312yescomplemented
248#1853C2×C2312yescomplemented
249#1824C2×C2311yescharacteristic, complemented
250C22×C23—24no—
251C22×C23—24no—
252C22×C23—22no—
253C22×C23—22no—
254C22×C23—24no—
255C22×C23—22no—
256C22×C23—42no—
257C22×C23—41no—
258C22×C23—41no—
259C22×C23#44814yes—
260C22×C23C2×#12011yescharacteristic
261C22×C23#54712yes—
262C22×C23C22×SD1611yescharacteristic
263C22×C23C2×#13711yescharacteristic
264C22×C23#57712yes—
265C4×C23—22no—
266C4×C23—22no—
267C4×C23—41no—
268C4×C23—42no—
269C4×C23—41no—
270C4×C23—42no—
271C4×C23—42no—
272C4×C23—42no—
273C4×C23—82no—
274C4×C23—82no—
275C4×C23#52511yescharacteristic
276C4×C23#53611yescharacteristic
277C4×C23#54712yes—
278C23×C8C22×C2311yescharacteristic
279#554—24no—
280#554—24no—
281C2×C4⋊Q8C22×C2311yescharacteristic
282C2×#125—21no—
283C2×#123—21no—
284C2×C22⋊Q8—21no—
285C2×C22⋊Q8—21no—
286C2×C22⋊Q8C22×C2311yescharacteristic
287C2×C22⋊Q8C22×C2311yescharacteristic
288C2×C4×Q8C22×C2311yescharacteristic
289C2×C4×D4—21no—
290C2×C4×D4C22×C2311yescharacteristic
291C22×C4⋊C4C22×C2311yescharacteristic
292C22×C4⋊C4C22×C2311yescharacteristic
293C22×C4⋊C4C22×C2311yescharacteristic
294#483—24no—
295C2×#106—24no—
296C2×#106C22×C2312yes—
297C2×#106C22×C2312yes—
298C2×#106C22×C2312yes—
299C2×#103C22×C2312yes—
300C2×#103C22×C2312yes—
301C2×C22⋊C8C22×C2311yescharacteristic
302C2×C22⋊C8C22×C2312yes—
303#398—21no—
304#397—21no—
305#389C22×C2312yes—
306#381—21no—
307#380C22×C2311yescharacteristic
308#380C22×C2312yes—
309#340C22×C2312yes—
310#340C22×C2312yes—
311#340C22×C2312yes—
312#336C22×C2312yes—
313C2×C23—24no—
314C2×C23—42no—
315C2×C23#240611yescharacteristic
316C2×C23C2×#44811yescharacteristic
317C2×C23#300911yescharacteristic
318C2×C23#333512yes—
319C2×C23#334412yes—
320C22×Q8—21no—
321C22×Q8C23×C2311yescharacteristic
322C22×D4D4×C2311yescharacteristic
323C23×C4—21no—
324C23×C4—21no—
325C23×C4—21no—
326C23×C4C23×C2311yescharacteristic
327C23×C4C23×C2311yescharacteristic
328C22×C8—22no—
329C22×C8—24no—
330C22×C8C23×C2312yes—
331C22×C8C23×C2312yes—
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334#125—28no—
335#123—44no—
336C22⋊Q8—28no—
337C22⋊Q8—28no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8D4×C2314yes—
342C4×Q8—22no—
343C4×Q8—22no—
344C4×D4—28no—
345C4×D4—22no—
346C4×D4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4D4×C2311yescharacteristic
361C2×C4⋊C4D4×C2314yes—
362C2×C4⋊C4C23×C2312yes—
363C2×C4⋊C4C23×C2312yes—
364C2×C4⋊C4C23×C2311yescharacteristic
365C2×C4⋊C4C23×C2311yescharacteristic
366C2×C4⋊C4C23×C2311yescharacteristic
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4D4×C2311yescharacteristic
372C2×C22⋊C4D4×C2311yescharacteristic
373C2×C42—21no—
374C2×C42C23×C2311yescharacteristic
375#106—24no—
376#106—24no—
377#106—24no—
378#103—24no—
379#103—24no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382#95—22no—
383#95—21no—
384#95—22no—
385C23#2409711yescharacteristic, direct product factor, Sylow
386C24C2×D4×C2311yescharacteristic
387C24C2×D4×C2311yescharacteristic
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—82no—
393C2×Q8C2×D4×C2312yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4SD16×C2314yes—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C2×D4×C2311yescharacteristic
417C22×C4C2×D4×C2311yescharacteristic
418C22×C4C2×D4×C2311yescharacteristic
419C22×C4C2×D4×C2312yes—
420C22×C4C2×D4×C2312yes—
421C22×C4C2×D4×C2311yescharacteristic
422C22×C4C24×C2311yescharacteristic
423C2×C8—28no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8#41×C2314yes—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4C2×D4×C2312yes—
447C4⋊C4C2×D4×C2312yes—
448C4⋊C4C2×D4×C2312yes—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C42—24no—
456C42—21no—
457C42—21no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—22no—
463C23—21no—
464C23C23×#12012yes—
465C23C2×SD16×C2312yes—
466C23C2×SD16×C2312yes—
467C23C23×#13712yes—
468C23C22×D4×C2311yescharacteristic
469Q8—24no—
470Q8—41no—
471Q8—41no—
472Q8—84no—
473D4—28no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4C23×#12012yes—
505C2×C4C2×SD16×C2312yes—
506C2×C4C22×D4×C2311yescharacteristic
507C2×C4C22×D4×C2311yescharacteristic
508C2×C4C2×#41×C2312yes—
509C2×C4C23×2−1+412yes—
510C8—24no—
511C8—42no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—24no—
516C22—22no—
517C22—24no—
518C22—41no—
519C22—41no—
520C22—42no—
521C22C23×#44814yes—
522C22C2×C23×#12011yescharacteristic
523C22C23×#54712yes—
524C22C22×SD16×C2311yescharacteristic
525C22C2×C23×#13711yescharacteristic
526C22C23×#57712yes—
527C4—22no—
528C4—22no—
529C4—41no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—82no—
536C4—82no—
537C4C23×#52511yescharacteristic
538C4C23×#53611yescharacteristic
539C4C23×#54712yes—
540C2—24no—
541C2—42no—
542C2C23×#240611yescharacteristic
543C2C2×C23×#44811yescharacteristic
544C2C23×#300911yescharacteristic
545C2C23×#333512yes—
546C2C23×#334412yes—
subgroupG/H
center, socleC23×C23C22×D4
commutator, Frattini, cosocleC22×C4C24×C23
prime radicalC22×D4×C23D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C23 ◁ C2×C4×D4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C23×#483C2semidirect
3. C22×C23×#106C2semidirect
5. C23×#2058C2semidirect
6. C23×#2035C2semidirect
7. C23×#1892C2semidirect
9. C23×#1865C2semidirect
10. C23×#1853C2semidirect
11. C23×#1824C2semidirect
184. #24097C23direct (N × H)
240. C2×#483C2×C23semidirect
241. C22×#106C2×C23semidirect
243. #2058C2×C23semidirect
244. #2035C2×C23semidirect
245. #1892C2×C23semidirect
247. #1865C2×C23semidirect
248. #1853C2×C23semidirect
249. #1824C2×C23semidirect
385. C23#24097direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#5542
C2×C23×#4832
C22×C23×#1062
C23×#24862
C23×#20582
C23×#20352
C23×#18922
C23×#18692
C23×#18652
C23×#18532
C23×#18242
#2409723