Finite Group Databasebeta

← C23×#12757 #19160383

Subgroups — C23×#12757 #19160383

Order 5888.

736 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3538C211yesmaximal, characteristic
2C23×#2135C211yesmaximal, characteristic, complemented
3C23×#2123C211yesmaximal, characteristic, complemented
4C23×#2120C211yesmaximal, characteristic
5C23×#1915C211yesmaximal, characteristic, complemented
6C23×#1909C211yesmaximal, characteristic, complemented
7C23×#1891C211yesmaximal, characteristic
8C23×#568C2211yescharacteristic
9C23×#563—21no—
10C23×#563C412yescomplemented
11C23×#552—21no—
12C23×#552—21no—
13C23×#551—21no—
14C23×#551C412yescomplemented
15D4×Q8×C23—21no—
16D4×Q8×C23—21no—
17C23×#544—22no—
18C23×#541—21no—
19C23×#539—22no—
20C23×#536C2211yescharacteristic
21C23×#536C2211yescharacteristic
22C23×#525C2211yescharacteristic
23C2×C23×C22⋊Q8—21no—
24C23×#419—21no—
25C23×#413C2211yescharacteristic
26C23×#413C2211yescharacteristic
27C2×C23×#101—21no—
28C23×#410C2211yescharacteristic
29C23×#407—21no—
30C23×#356—22no—
31C23×#352—21no—
32C23×#352—22no—
33C23×AΣL(1,𝔽16)—21no—
34C23×#344—21no—
35C23×#344—21no—
36C23×H(3,ℤ/4)—21no—
37C2×#41×C23C2311yescharacteristic
38C22×Q8×C23—21no—
39C22×Q8×C23D411yescharacteristic
40C22×Q8×C23D411yescharacteristic
41C23×C4×C23D411yescharacteristic
42C23×#131—42no—
43C2×M16×C23—21no—
44C2×M16×C23—21no—
45C2×M16×C23—21no—
46C2×M16×C23—21no—
47C2×M16×C23—21no—
48C23×C4⋊Q8—21no—
49C23×C4⋊Q8—22no—
50C23×C4⋊Q8—22no—
51C23×C4⋊Q8—21no—
52C23×C4⋊Q8D412yes—
53C23×#126—22no—
54C23×#126—22no—
55C23×#126—22no—
56C23×#126—22no—
57C23×#125—21no—
58C23×#125—22no—
59C23×#125—22no—
60C23×#125—22no—
61C23×#125—22no—
62C23×#124—21no—
63C23×#124C2×C412yes—
64C23×#123—22no—
65C23×#123—21no—
66C23×#123—21no—
67C23×#123—22no—
68C23×#123C2×C412yes—
69C23×#123C2×C411yescharacteristic
70C23×#123D411yescharacteristic
71C23×#123D412yes—
72C23×C22⋊Q8—22no—
73C23×C22⋊Q8—21no—
74C23×C22⋊Q8—21no—
75C23×C22⋊Q8—22no—
76C23×C22⋊Q8—22no—
77C23×C22⋊Q8—22no—
78C23×C22⋊Q8—21no—
79C23×C22⋊Q8—22no—
80C23×C22⋊Q8—22no—
81C23×C22⋊Q8—21no—
82C23×C22⋊Q8—21no—
83C23×C22⋊Q8—21no—
84C23×C22⋊Q8—21no—
85C23×#121—21no—
86C23×#120C2×C411yescharacteristic
87C23×#120D411yescharacteristic
88C4×Q8×C23—21no—
89C4×Q8×C23—21no—
90C4×D4×C23—22no—
91C4×D4×C23—22no—
92C4×D4×C23—21no—
93C4×D4×C23—21no—
94C23×#117—22no—
95C23×#117—22no—
96C23×#117—21no—
97C23×#117D411yescharacteristic
98C23×#117D411yescharacteristic
99C23×#117D411yescharacteristic
100C2×C4⋊C4×C23—21no—
101C2×C4⋊C4×C23—22no—
102C2×C22⋊C4×C23—22no—
103C23×#103—42no—
104C23×#101—21no—
105C23×#101—21no—
106C23×#101—42no—
107C23×#100—21no—
108C23×#100—21no—
109C23×#99—21no—
110C23×#99—21no—
111C23×C22⋊C8—42no—
112C24×C23C22⋊C411yescharacteristic
113#41×C23—41no—
114#41×C23—41no—
115C2×Q8×C23—22no—
116C2×Q8×C23—21no—
117C2×Q8×C23—21no—
118C2×Q8×C23—21no—
119C2×Q8×C23—41no—
120C2×Q8×C23—41no—
121C2×Q8×C23—41no—
122C2×Q8×C23—41no—
123C2×Q8×C23—81no—
124C2×Q8×C23—81no—
125C2×Q8×C23C22⋊C412yes—
126C2×Q8×C23C2×D411yescharacteristic
127C2×D4×C23—21no—
128C2×D4×C23—21no—
129C2×D4×C23C22×C411yescharacteristic
130C2×D4×C23C2×D411yescharacteristic
131C2×D4×C23C2×D411yescharacteristic
132C22×C4×C23—22no—
133C22×C4×C23—22no—
134C22×C4×C23—21no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—41no—
138C22×C4×C23—41no—
139C22×C4×C23C22⋊C412yes—
140C22×C4×C23C22⋊C411yescharacteristic
141C22×C4×C23C2×D411yescharacteristic
142C22×C4×C23C2×D411yescharacteristic
143C22×C4×C23C2×D411yescharacteristic
144M16×C23—22no—
145M16×C23—41no—
146M16×C23—41no—
147M16×C23—41no—
148M16×C23—41no—
149M16×C23—41no—
150M16×C23—41no—
151M16×C23—41no—
152C2×C8×C23—22no—
153C2×C8×C23—22no—
154C2×C8×C23—41no—
155C2×C8×C23—41no—
156C2×C8×C23—41no—
157C4⋊C4×C23—22no—
158C4⋊C4×C23—22no—
159C4⋊C4×C23—22no—
160C4⋊C4×C23—22no—
161C4⋊C4×C23—22no—
162C4⋊C4×C23—22no—
163C4⋊C4×C23—22no—
164C4⋊C4×C23—22no—
165C4⋊C4×C23—22no—
166C4⋊C4×C23—22no—
167C4⋊C4×C23—21no—
168C4⋊C4×C23—22no—
169C4⋊C4×C23—22no—
170C4⋊C4×C23—22no—
171C4⋊C4×C23—21no—
172C4⋊C4×C23—21no—
173C4⋊C4×C23—21no—
174C4⋊C4×C23—21no—
175C4⋊C4×C23—41no—
176C22⋊C4×C23—22no—
177C22⋊C4×C23—22no—
178C22⋊C4×C23—22no—
179C22⋊C4×C23—22no—
180C22⋊C4×C23—21no—
181C22⋊C4×C23—22no—
182C22⋊C4×C23—21no—
183C22⋊C4×C23—21no—
184C22⋊C4×C23—22no—
185C22⋊C4×C23—21no—
186C22⋊C4×C23—21no—
187C22⋊C4×C23—41no—
188C22⋊C4×C23C22⋊C411yescharacteristic
189C22⋊C4×C23C22⋊C411yescharacteristic
190C22⋊C4×C23C22⋊C412yes—
191C42×C23—22no—
192C42×C23—22no—
193C42×C23—21no—
194C42×C23—21no—
195C42×C23—41no—
196C42×C23C22⋊C412yes—
197#12757C2311yesmaximal, characteristic, direct product factor, Sylow
198C23×C23—21no—
199C23×C23—41no—
200C23×C23—42no—
201C23×C23C2×C22⋊C411yescharacteristic
202C23×C23C2×C22⋊C411yescharacteristic
203C23×C23#12011yescharacteristic
204Q8×C23—41no—
205Q8×C23—42no—
206Q8×C23—41no—
207Q8×C23—41no—
208Q8×C23—41no—
209Q8×C23—41no—
210Q8×C23—81no—
211Q8×C23—82no—
212D4×C23—41no—
213D4×C23—41no—
214D4×C23—41no—
215D4×C23—42no—
216D4×C23—42no—
217C2×C4×C23—21no—
218C2×C4×C23—21no—
219C2×C4×C23—21no—
220C2×C4×C23—21no—
221C2×C4×C23—21no—
222C2×C4×C23—21no—
223C2×C4×C23—21no—
224C2×C4×C23—21no—
225C2×C4×C23—22no—
226C2×C4×C23—22no—
227C2×C4×C23—22no—
228C2×C4×C23—22no—
229C2×C4×C23—21no—
230C2×C4×C23—44no—
231C2×C4×C23—41no—
232C2×C4×C23—41no—
233C2×C4×C23—41no—
234C2×C4×C23—42no—
235C2×C4×C23—81no—
236C2×C4×C23—81no—
237C2×C4×C23—82no—
238C2×C4×C23—81no—
239C2×C4×C23C2×C22⋊C411yescharacteristic
240C2×C4×C23#12011yescharacteristic
241C2×C4×C23#12011yescharacteristic
242C2×C4×C23#12011yescharacteristic
243C8×C23—42no—
244C8×C23—81no—
245C8×C23—81no—
246#3538C2×C2311yescharacteristic
247#2135C2×C2311yescharacteristic, complemented
248#2123C2×C2311yescharacteristic, complemented
249#2120C2×C2311yescharacteristic
250#1915C2×C2311yescharacteristic, complemented
251#1909C2×C2311yescharacteristic, complemented
252#1891C2×C2311yescharacteristic
253C22×C23—22no—
254C22×C23—21no—
255C22×C23—21no—
256C22×C23—21no—
257C22×C23—41no—
258C22×C23—41no—
259C22×C23—41no—
260C22×C23—41no—
261C22×C23—42no—
262C22×C23—41no—
263C22×C23—81no—
264C22×C23#37911yescharacteristic
265C4×C23—21no—
266C4×C23—21no—
267C4×C23—21no—
268C4×C23—42no—
269C4×C23—42no—
270C4×C23—41no—
271C4×C23—41no—
272C4×C23—41no—
273C4×C23—42no—
274C4×C23—42no—
275C4×C23—161no—
276C4×C23B(4,𝔽2)11yescharacteristic
277C4×C23#45811yescharacteristic
278#568C22×C2311yescharacteristic
279#563—21no—
280#563C4×C2312yescomplemented
281#552—21no—
282#552—21no—
283#551—21no—
284#551C4×C2312yescomplemented
285D4×Q8—21no—
286D4×Q8—21no—
287#544—22no—
288#541—21no—
289#539—22no—
290#536C22×C2311yescharacteristic
291#536C22×C2311yescharacteristic
292#525C22×C2311yescharacteristic
293C2×C22⋊Q8—21no—
294#419—21no—
295#413C22×C2311yescharacteristic
296#413C22×C2311yescharacteristic
297C2×#101—21no—
298#410C22×C2311yescharacteristic
299#407—21no—
300#356—22no—
301#352—21no—
302#352—22no—
303AΣL(1,𝔽16)—21no—
304#344—21no—
305#344—21no—
306H(3,ℤ/4)—21no—
307C2×C23—21no—
308C2×C23—41no—
309C2×C23—41no—
310C2×C23—42no—
311C2×C23—41no—
312C2×C23#179011yescharacteristic
313C2×#41C23×C2311yescharacteristic
314C22×Q8—21no—
315C22×Q8D4×C2311yescharacteristic
316C22×Q8D4×C2311yescharacteristic
317C23×C4D4×C2311yescharacteristic
318#131—42no—
319C2×M16—21no—
320C2×M16—21no—
321C2×M16—21no—
322C2×M16—21no—
323C2×M16—21no—
324C4⋊Q8—21no—
325C4⋊Q8—22no—
326C4⋊Q8—22no—
327C4⋊Q8—21no—
328C4⋊Q8D4×C2312yes—
329#126—22no—
330#126—22no—
331#126—22no—
332#126—22no—
333#125—22no—
334#125—22no—
335#125—21no—
336#125—22no—
337#125—22no—
338#124—21no—
339#124C2×C4×C2312yes—
340#123—22no—
341#123—21no—
342#123—21no—
343#123—22no—
344#123C2×C4×C2311yescharacteristic
345#123C2×C4×C2312yes—
346#123D4×C2311yescharacteristic
347#123D4×C2312yes—
348C22⋊Q8—22no—
349C22⋊Q8—21no—
350C22⋊Q8—21no—
351C22⋊Q8—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—21no—
355C22⋊Q8—22no—
356C22⋊Q8—22no—
357C22⋊Q8—21no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—21no—
361#121—21no—
362#120C2×C4×C2311yescharacteristic
363#120D4×C2311yescharacteristic
364C4×Q8—21no—
365C4×Q8—21no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—21no—
369C4×D4—21no—
370#117—22no—
371#117—22no—
372#117—21no—
373#117D4×C2311yescharacteristic
374#117D4×C2311yescharacteristic
375#117D4×C2311yescharacteristic
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C22⋊C4—22no—
379#103—42no—
380#101—21no—
381#101—21no—
382#101—42no—
383#100—21no—
384#100—21no—
385#99—21no—
386#99—21no—
387C22⋊C8—42no—
388C23#1275711yescharacteristic, direct product factor, Sylow
389C24C22⋊C4×C2311yescharacteristic
390#41—41no—
391#41—41no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8C22⋊C4×C2312yes—
403C2×Q8C2×D4×C2311yescharacteristic
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22×C4×C2311yescharacteristic
407C2×D4C2×D4×C2311yescharacteristic
408C2×D4C2×D4×C2311yescharacteristic
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4C22⋊C4×C2312yes—
417C22×C4C22⋊C4×C2311yescharacteristic
418C22×C4C2×D4×C2311yescharacteristic
419C22×C4C2×D4×C2311yescharacteristic
420C22×C4C2×D4×C2311yescharacteristic
421M16—22no—
422M16—41no—
423M16—41no—
424M16—41no—
425M16—41no—
426M16—41no—
427M16—41no—
428M16—41no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—41no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—21no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—22no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—41no—
465C22⋊C4C22⋊C4×C2311yescharacteristic
466C22⋊C4C22⋊C4×C2311yescharacteristic
467C22⋊C4C22⋊C4×C2312yes—
468C42—22no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—41no—
473C42C22⋊C4×C2312yes—
474C23—21no—
475C23—41no—
476C23—42no—
477C23C2×C22⋊C4×C2311yescharacteristic
478C23C2×C22⋊C4×C2311yescharacteristic
479C23C23×#12011yescharacteristic
480Q8—41no—
481Q8—42no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—81no—
487Q8—82no—
488D4—41no—
489D4—41no—
490D4—41no—
491D4—42no—
492D4—42no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4C2×C22⋊C4×C2311yescharacteristic
516C2×C4C23×#12011yescharacteristic
517C2×C4C23×#12011yescharacteristic
518C2×C4C23×#12011yescharacteristic
519C8—42no—
520C8—81no—
521C8—81no—
522C22—22no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—42no—
531C22—41no—
532C22—81no—
533C22C23×#37911yescharacteristic
534C4—21no—
535C4—21no—
536C4—21no—
537C4—42no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—42no—
543C4—42no—
544C4—161no—
545C4C23×B(4,𝔽2)11yescharacteristic
546C4C23×#45811yescharacteristic
547C2—21no—
548C2—41no—
549C2—42no—
550C2—41no—
551C2—41no—
552C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×D4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#525C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2135C2semidirect
3. C23×#2123C2semidirect
5. C23×#1915C2semidirect
6. C23×#1909C2semidirect
10. C23×#563C4semidirect
14. C23×#551C4semidirect
197. #12757C23direct (N × H)
247. #2135C2×C23semidirect
248. #2123C2×C23semidirect
250. #1915C2×C23semidirect
251. #1909C2×C23semidirect
280. #563C4×C23semidirect
284. #551C4×C23semidirect
388. C23#12757direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35382
C23×#21352
C23×#21232
C23×#21202
C23×#19152
C23×#19092
C23×#18912
#1275723