Finite Group Databasebeta

← C23×#12761 #19160387

Subgroups — C23×#12761 #19160387

Order 5888.

936 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#551C211yesmaximal, characteristic
2C2×C23×#354C211yesmaximal, characteristic, complemented
3C2×C23×#352C211yesmaximal, characteristic, complemented
4C2×C23×AΣL(1,𝔽16)C211yesmaximal, characteristic
5C23×#1890C211yesmaximal, characteristic, complemented
6C23×#1889C211yesmaximal, characteristic
7C23×#1885C211yesmaximal, characteristic, complemented
8C23×#551—22no—
9C23×#551—22no—
10C23×#551—22no—
11C23×#551C414yescomplemented
12C2×C23×#126—21no—
13C2×C23×#124C2211yescharacteristic
14C2×C23×#123—21no—
15C2×C23×#123C2211yescharacteristic
16C2×C23×C22⋊Q8—21no—
17C2×C23×#120C2211yescharacteristic
18C23×#518C2211yescharacteristic
19C2×C4×D4×C23—21no—
20C2×C23×#117—21no—
21C22×C22⋊C4×C23—21no—
22C23×#419—21no—
23C2×C23×#100C2211yescharacteristic
24C2×C23×#99C2211yescharacteristic
25C2×C23×#99C2211yescharacteristic
26C23×#381—21no—
27C23×#379—21no—
28C23×#354—22no—
29C23×#352—22no—
30C23×AΣL(1,𝔽16)—22no—
31C23×#342—21no—
32C23×#342—21no—
33C23×#339—21no—
34C25×C23D411yescharacteristic
35C22×Q8×C23D411yescharacteristic
36C22×D4×C23—21no—
37C22×D4×C23C2311yescharacteristic
38C23×C4×C23—21no—
39C23×C4×C23—21no—
40C23×C4×C23D411yescharacteristic
41C2×M16×C23—21no—
42C23×#126—24no—
43C23×#126—24no—
44C23×#124—21no—
45C23×#124—21no—
46C23×#124C2×C412yes—
47C23×#124D412yes—
48C23×#123—24no—
49C23×#123—21no—
50C23×#123—21no—
51C23×#123—24no—
52C23×#123C2×C412yes—
53C23×#123D412yes—
54C23×C22⋊Q8—24no—
55C23×C22⋊Q8—24no—
56C23×#120—21no—
57C23×#120—21no—
58C23×#120C2×C412yes—
59C23×#120D412yes—
60C4×D4×C23—24no—
61C4×D4×C23—24no—
62C4×D4×C23—42no—
63C23×#117—24no—
64C23×#117—24no—
65C23×#117—21no—
66C2×C4⋊C4×C23—21no—
67C2×C4⋊C4×C23—21no—
68C2×C4⋊C4×C23—21no—
69C2×C4⋊C4×C23—21no—
70C2×C22⋊C4×C23—24no—
71C2×C22⋊C4×C23—21no—
72C2×C22⋊C4×C23—21no—
73C2×C22⋊C4×C23—24no—
74C2×C22⋊C4×C23—21no—
75C2×C22⋊C4×C23—21no—
76C2×C22⋊C4×C23—21no—
77C2×C22⋊C4×C23—21no—
78C2×C22⋊C4×C23—21no—
79C2×C22⋊C4×C23—21no—
80C2×C22⋊C4×C23—21no—
81C2×C22⋊C4×C23—21no—
82C2×C22⋊C4×C23—42no—
83C2×C22⋊C4×C23D411yescharacteristic
84C2×C22⋊C4×C23D411yescharacteristic
85C2×C42×C23—21no—
86C2×C42×C23D411yescharacteristic
87C23×#103—42no—
88C23×#100—21no—
89C23×#100—21no—
90C23×#99—21no—
91C23×#99—21no—
92C23×#99—21no—
93C23×#99—21no—
94C23×#95—42no—
95C24×C23—21no—
96C24×C23—21no—
97C24×C23—41no—
98C24×C23—44no—
99C24×C23—41no—
100C24×C23C22⋊C412yes—
101C24×C23C2×D411yescharacteristic
102C24×C23C2×D411yescharacteristic
103C2×Q8×C23—21no—
104C2×Q8×C23—41no—
105C2×Q8×C23—41no—
106C2×Q8×C23C22⋊C412yescomplemented
107C2×D4×C23—22no—
108C2×D4×C23—22no—
109C2×D4×C23—41no—
110C2×D4×C23—41no—
111C2×D4×C23—44no—
112C2×D4×C23C22×C411yescharacteristic
113C2×D4×C23C2×D411yescharacteristic
114C2×D4×C23C2×D411yescharacteristic
115C2×D4×C23C2×D411yescharacteristic
116C22×C4×C23—24no—
117C22×C4×C23—21no—
118C22×C4×C23—21no—
119C22×C4×C23—21no—
120C22×C4×C23—21no—
121C22×C4×C23—21no—
122C22×C4×C23—24no—
123C22×C4×C23—21no—
124C22×C4×C23—22no—
125C22×C4×C23—22no—
126C22×C4×C23—44no—
127C22×C4×C23—41no—
128C22×C4×C23—41no—
129C22×C4×C23—41no—
130C22×C4×C23—41no—
131C22×C4×C23—81no—
132C22×C4×C23—81no—
133C22×C4×C23C22⋊C412yescomplemented
134C22×C4×C23C2×D411yescharacteristic
135M16×C23—41no—
136M16×C23—41no—
137C2×C8×C23—41no—
138C4⋊C4×C23—24no—
139C4⋊C4×C23—24no—
140C4⋊C4×C23—24no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C22⋊C4×C23—24no—
145C22⋊C4×C23—24no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—22no—
149C22⋊C4×C23—22no—
150C22⋊C4×C23—22no—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—21no—
153C22⋊C4×C23—21no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—22no—
156C22⋊C4×C23—41no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23—41no—
159C22⋊C4×C23—41no—
160C22⋊C4×C23—41no—
161C22⋊C4×C23—41no—
162C22⋊C4×C23—41no—
163C22⋊C4×C23—84no—
164C22⋊C4×C23C22⋊C412yes—
165C22⋊C4×C23C22⋊C412yescomplemented
166C42×C23—24no—
167C42×C23—21no—
168C42×C23—41no—
169C42×C23C22⋊C412yescomplemented
170#12761C2311yesmaximal, characteristic, direct product factor, Sylow
171C23×C23—24no—
172C23×C23—22no—
173C23×C23—21no—
174C23×C23—21no—
175C23×C23—21no—
176C23×C23—21no—
177C23×C23—42no—
178C23×C23—42no—
179C23×C23—44no—
180C23×C23—42no—
181C23×C23—42no—
182C23×C23—44no—
183C23×C23—41no—
184C23×C23—41no—
185C23×C23—44no—
186C23×C23—81no—
187C23×C23—84no—
188C23×C23—82no—
189C23×C23C2×C22⋊C411yescharacteristic
190C23×C23C2×C22⋊C411yescharacteristic
191C23×C23#12011yescharacteristic
192C23×C23#12011yescharacteristic
193C23×C23#12011yescharacteristic
194Q8×C23—42no—
195Q8×C23—81no—
196D4×C23—41no—
197D4×C23—41no—
198D4×C23—41no—
199D4×C23—41no—
200D4×C23—44no—
201D4×C23—44no—
202C2×C4×C23—21no—
203C2×C4×C23—21no—
204C2×C4×C23—22no—
205C2×C4×C23—22no—
206C2×C4×C23—22no—
207C2×C4×C23—22no—
208C2×C4×C23—22no—
209C2×C4×C23—42no—
210C2×C4×C23—42no—
211C2×C4×C23—48no—
212C2×C4×C23—42no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—41no—
216C2×C4×C23—41no—
217C2×C4×C23—42no—
218C2×C4×C23—42no—
219C2×C4×C23—81no—
220C2×C4×C23—84no—
221C2×C4×C23—82no—
222C2×C4×C23—82no—
223C2×C4×C23—81no—
224C2×C4×C23—81no—
225C2×C4×C23—82no—
226C2×C4×C23—82no—
227C2×C4×C23C2×C22⋊C411yescharacteristic
228C2×C4×C23#12011yescharacteristic
229C8×C23—81no—
230C2×#551C2×C2311yescharacteristic
231C2×#354C2×C2311yescharacteristic, complemented
232C2×#352C2×C2311yescharacteristic, complemented
233C2×AΣL(1,𝔽16)C2×C2311yescharacteristic
234#1890C2×C2311yescharacteristic, complemented
235#1889C2×C2311yescharacteristic
236#1885C2×C2311yescharacteristic, complemented
237C22×C23—22no—
238C22×C23—24no—
239C22×C23—21no—
240C22×C23—21no—
241C22×C23—21no—
242C22×C23—21no—
243C22×C23—41no—
244C22×C23—41no—
245C22×C23—41no—
246C22×C23—41no—
247C22×C23—42no—
248C22×C23—44no—
249C22×C23—44no—
250C22×C23—42no—
251C22×C23—42no—
252C22×C23—44no—
253C22×C23—42no—
254C22×C23—81no—
255C22×C23—81no—
256C22×C23—84no—
257C22×C23—82no—
258C22×C23#37911yescharacteristic
259C22×C23B(4,𝔽2)11yescharacteristic
260C22×C23#45811yescharacteristic
261C4×C23—21no—
262C4×C23—21no—
263C4×C23—42no—
264C4×C23—42no—
265C4×C23—42no—
266C4×C23—42no—
267C4×C23—42no—
268C4×C23—82no—
269C4×C23—161no—
270C4×C23—161no—
271#551—22no—
272#551—22no—
273#551—22no—
274#551C4×C2314yescomplemented
275C2×#126—21no—
276C2×#124C22×C2311yescharacteristic
277C2×#123—21no—
278C2×#123C22×C2311yescharacteristic
279C2×C22⋊Q8—21no—
280C2×#120C22×C2311yescharacteristic
281#518C22×C2311yescharacteristic
282C2×C4×D4—21no—
283C2×#117—21no—
284C22×C22⋊C4—21no—
285#419—21no—
286C2×#100C22×C2311yescharacteristic
287C2×#99C22×C2311yescharacteristic
288C2×#99C22×C2311yescharacteristic
289#381—21no—
290#379—21no—
291#354—22no—
292#352—22no—
293AΣL(1,𝔽16)—22no—
294#342—21no—
295#342—21no—
296#339—21no—
297C2×C23—21no—
298C2×C23—21no—
299C2×C23—41no—
300C2×C23—41no—
301C2×C23—44no—
302C2×C23—41no—
303C2×C23—41no—
304C2×C23#179011yescharacteristic
305C2×C23#219712yes—
306C25D4×C2311yescharacteristic
307C22×Q8D4×C2311yescharacteristic
308C22×D4—21no—
309C22×D4C23×C2311yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4D4×C2311yescharacteristic
313C2×M16—21no—
314#126—24no—
315#126—24no—
316#124—21no—
317#124—21no—
318#124C2×C4×C2312yes—
319#124D4×C2312yes—
320#123—24no—
321#123—21no—
322#123—21no—
323#123—24no—
324#123C2×C4×C2312yes—
325#123D4×C2312yes—
326C22⋊Q8—24no—
327C22⋊Q8—24no—
328#120—21no—
329#120—21no—
330#120C2×C4×C2312yes—
331#120D4×C2312yes—
332C4×D4—24no—
333C4×D4—24no—
334C4×D4—42no—
335#117—24no—
336#117—24no—
337#117—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4D4×C2311yescharacteristic
356C2×C22⋊C4D4×C2311yescharacteristic
357C2×C42—21no—
358C2×C42D4×C2311yescharacteristic
359#103—42no—
360#100—21no—
361#100—21no—
362#99—21no—
363#99—21no—
364#99—21no—
365#99—21no—
366#95—42no—
367C23#1276111yescharacteristic, direct product factor, Sylow
368C24—21no—
369C24—21no—
370C24—41no—
371C24—44no—
372C24—41no—
373C24C22⋊C4×C2312yes—
374C24C2×D4×C2311yescharacteristic
375C24C2×D4×C2311yescharacteristic
376C2×Q8—21no—
377C2×Q8—41no—
378C2×Q8—41no—
379C2×Q8C22⋊C4×C2312yescomplemented
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—44no—
385C2×D4C22×C4×C2311yescharacteristic
386C2×D4C2×D4×C2311yescharacteristic
387C2×D4C2×D4×C2311yescharacteristic
388C2×D4C2×D4×C2311yescharacteristic
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4C22⋊C4×C2312yescomplemented
407C22×C4C2×D4×C2311yescharacteristic
408M16—41no—
409M16—41no—
410C2×C8—41no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—21no—
426C22⋊C4—21no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—84no—
437C22⋊C4C22⋊C4×C2312yes—
438C22⋊C4C22⋊C4×C2312yescomplemented
439C42—24no—
440C42—21no—
441C42—41no—
442C42C22⋊C4×C2312yescomplemented
443C23—24no—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—44no—
455C23—41no—
456C23—41no—
457C23—44no—
458C23—81no—
459C23—84no—
460C23—82no—
461C23C2×C22⋊C4×C2311yescharacteristic
462C23C2×C22⋊C4×C2311yescharacteristic
463C23C23×#12011yescharacteristic
464C23C23×#12011yescharacteristic
465C23C23×#12011yescharacteristic
466Q8—42no—
467Q8—81no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—41no—
472D4—44no—
473D4—44no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—48no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—81no—
492C2×C4—84no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4C2×C22⋊C4×C2311yescharacteristic
500C2×C4C23×#12011yescharacteristic
501C8—81no—
502C22—22no—
503C22—24no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—44no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—42no—
519C22—81no—
520C22—81no—
521C22—84no—
522C22—82no—
523C22C23×#37911yescharacteristic
524C22C23×B(4,𝔽2)11yescharacteristic
525C22C23×#45811yescharacteristic
526C4—21no—
527C4—21no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—42no—
532C4—42no—
533C4—82no—
534C4—161no—
535C4—161no—
536C2—21no—
537C2—21no—
538C2—41no—
539C2—41no—
540C2—44no—
541C2—41no—
542C2—41no—
543C2C23×#179011yescharacteristic
544C2C23×#219712yes—
subgroupG/H
center, socleC22×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC2×C23×#120C22
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. C2×C23×#354C2semidirect
3. C2×C23×#352C2semidirect
5. C23×#1890C2semidirect
7. C23×#1885C2semidirect
11. C23×#551C4semidirect
106. C2×Q8×C23C22⋊C4semidirect
133. C22×C4×C23C22⋊C4semidirect
165. C22⋊C4×C23C22⋊C4semidirect
169. C42×C23C22⋊C4semidirect
170. #12761C23direct (N × H)
231. C2×#354C2×C23semidirect
232. C2×#352C2×C23semidirect
234. #1890C2×C23semidirect
236. #1885C2×C23semidirect
274. #551C4×C23semidirect
367. C23#12761direct (N × H)
379. C2×Q8C22⋊C4×C23semidirect
406. C22×C4C22⋊C4×C23semidirect
438. C22⋊C4C22⋊C4×C23semidirect
442. C42C22⋊C4×C23semidirect

Maximal subgroups

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

subgroupindex
C2×C23×#5512
C2×C23×#3542
C2×C23×#3522
C2×C23×AΣL(1,𝔽16)2
C23×#18902
C23×#18892
C23×#18852
#1276123