Finite Group Databasebeta

← C23×#12765 #19160391

Subgroups — C23×#12765 #19160391

Order 5888.

852 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3481C211yesmaximal, characteristic
2C23×#2132C211yesmaximal, characteristic, complemented
3C23×#2120C212yesmaximal, complemented
4C23×#1891C212yesmaximal, complemented
5C23×#1884C211yesmaximal, characteristic, complemented
6C23×#568C2211yescharacteristic, complemented
7C23×#559C411yescharacteristic, complemented
8C23×#554—21no—
9C23×#553—22no—
10C23×#551—21no—
11C23×#551C412yescomplemented
12C23×#546—22no—
13D42×C23C411yescharacteristic, complemented
14C23×#532C2211yescharacteristic
15C23×#529—22no—
16C23×#525—21no—
17C23×#525C2212yes—
18C4×#41×C23—21no—
19C2×C4×D4×C23—22no—
20C23×#421—21no—
21C23×#413C2212yescomplemented
22C23×#410C2211yescharacteristic, complemented
23C23×#407—22no—
24C23×#356—21no—
25C23×#355—21no—
26C23×#352—22no—
27C23×AΣL(1,𝔽16)—22no—
28C23×#344—22no—
29C23×#339—21no—
30C2×#41×C23—21no—
31C2×#41×C23D411yescharacteristic
32C2×#41×C23C2311yescharacteristic
33C22×D4×C23—22no—
34C23×C4×C23—21no—
35C23×C4×C23D412yes—
36C23×#131—42no—
37C2×M16×C23—22no—
38C2×M16×C23—21no—
39C2×M16×C23—22no—
40C23×C4⋊Q8D411yescharacteristic
41C23×#127C2×C411yescharacteristic
42C23×#126—22no—
43C23×#126—22no—
44C23×#125—22no—
45C23×#124—22no—
46C23×#124C2×C411yescharacteristic
47C23×#124D411yescharacteristic
48C23×#123—22no—
49C23×#123—21no—
50C23×#123—21no—
51C23×#123—22no—
52C23×#123C2×C412yes—
53C23×#123D412yes—
54C23×C22⋊Q8—22no—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—22no—
57C23×#121—22no—
58C23×#121—22no—
59C23×#121—22no—
60C23×#121—22no—
61C23×#120—21no—
62C23×#120—21no—
63C23×#120C2×C412yes—
64C23×#120D412yes—
65C4×Q8×C23—21no—
66C4×Q8×C23—21no—
67C4×D4×C23—22no—
68C4×D4×C23—22no—
69C4×D4×C23—22no—
70C4×D4×C23—21no—
71C4×D4×C23—22no—
72C4×D4×C23—22no—
73C4×D4×C23—22no—
74C4×D4×C23—22no—
75C4×D4×C23—22no—
76C4×D4×C23—22no—
77C4×D4×C23—21no—
78C23×#117—21no—
79C23×#117—21no—
80C23×#117—22no—
81C23×#117—21no—
82C23×#117D412yes—
83C2×C4⋊C4×C23—22no—
84C2×C22⋊C4×C23—22no—
85C2×C22⋊C4×C23—22no—
86C2×C42×C23—22no—
87C2×C42×C23D411yescharacteristic
88C23×#104—42no—
89C23×#101—22no—
90C23×#100—22no—
91C23×#99—22no—
92C23×C22⋊C8—44no—
93C24×C23—21no—
94C24×C23C22⋊C412yes—
95#41×C23—41no—
96#41×C23—41no—
97#41×C23—41no—
98#41×C23—41no—
99#41×C23—44no—
100C2×Q8×C23—21no—
101C2×Q8×C23C22⋊C411yescharacteristic
102C2×Q8×C23C2×D411yescharacteristic
103C2×D4×C23—21no—
104C2×D4×C23—22no—
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—22no—
108C2×D4×C23—21no—
109C2×D4×C23—48no—
110C2×D4×C23C22⋊C411yescharacteristic
111C2×D4×C23C22×C411yescharacteristic
112C2×D4×C23C2×D412yes—
113C22×C4×C23—22no—
114C22×C4×C23—22no—
115C22×C4×C23—21no—
116C22×C4×C23—22no—
117C22×C4×C23—21no—
118C22×C4×C23—22no—
119C22×C4×C23—21no—
120C22×C4×C23—22no—
121C22×C4×C23—42no—
122C22×C4×C23—42no—
123C22×C4×C23—81no—
124C22×C4×C23—81no—
125C22×C4×C23C22⋊C412yes—
126C22×C4×C23C2×D411yescharacteristic
127C22×C4×C23C2×D412yes—
128M16×C23—22no—
129M16×C23—42no—
130M16×C23—41no—
131M16×C23—42no—
132M16×C23—42no—
133C2×C8×C23—24no—
134C2×C8×C23—41no—
135C2×C8×C23—42no—
136C4⋊C4×C23—21no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—21no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—22no—
141C4⋊C4×C23—22no—
142C4⋊C4×C23—22no—
143C4⋊C4×C23—22no—
144C4⋊C4×C23—22no—
145C4⋊C4×C23—41no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—22no—
149C22⋊C4×C23—21no—
150C22⋊C4×C23—21no—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—22no—
156C22⋊C4×C23—22no—
157C22⋊C4×C23—41no—
158C22⋊C4×C23C22⋊C412yes—
159C22⋊C4×C23C22⋊C412yes—
160C42×C23—22no—
161C42×C23—22no—
162C42×C23—21no—
163C42×C23—22no—
164C42×C23—41no—
165C42×C23C22⋊C411yescharacteristic
166C42×C23C22⋊C411yescharacteristic
167#12765C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23—22no—
169C23×C23—21no—
170C23×C23—42no—
171C23×C23—44no—
172C23×C23—44no—
173C23×C23—81no—
174C23×C23C2×C22⋊C412yes—
175C23×C23#12011yescharacteristic
176Q8×C23—41no—
177Q8×C23—41no—
178Q8×C23—42no—
179D4×C23—24no—
180D4×C23—41no—
181D4×C23—42no—
182D4×C23—41no—
183D4×C23—44no—
184D4×C23—44no—
185D4×C23—44no—
186D4×C23—44no—
187D4×C23—81no—
188C2×C4×C23—21no—
189C2×C4×C23—21no—
190C2×C4×C23—22no—
191C2×C4×C23—21no—
192C2×C4×C23—22no—
193C2×C4×C23—21no—
194C2×C4×C23—22no—
195C2×C4×C23—22no—
196C2×C4×C23—22no—
197C2×C4×C23—21no—
198C2×C4×C23—44no—
199C2×C4×C23—42no—
200C2×C4×C23—44no—
201C2×C4×C23—42no—
202C2×C4×C23—41no—
203C2×C4×C23—42no—
204C2×C4×C23—41no—
205C2×C4×C23—42no—
206C2×C4×C23—82no—
207C2×C4×C23—81no—
208C2×C4×C23—82no—
209C2×C4×C23—81no—
210C2×C4×C23C2×C22⋊C411yescharacteristic
211C2×C4×C23#12012yes—
212C2×C4×C23#12011yescharacteristic
213C8×C23—42no—
214C8×C23—82no—
215#3481C2×C2311yescharacteristic
216#2132C2×C2311yescharacteristic, complemented
217#2120C2×C2312yescomplemented
218#1891C2×C2312yescomplemented
219#1884C2×C2311yescharacteristic, complemented
220C22×C23—24no—
221C22×C23—22no—
222C22×C23—21no—
223C22×C23—41no—
224C22×C23—42no—
225C22×C23—42no—
226C22×C23—42no—
227C22×C23—44no—
228C22×C23—42no—
229C22×C23—41no—
230C22×C23—82no—
231C22×C23—84no—
232C22×C23#37911yescharacteristic
233C4×C23—22no—
234C4×C23—22no—
235C4×C23—21no—
236C4×C23—41no—
237C4×C23—42no—
238C4×C23—42no—
239C4×C23—42no—
240C4×C23—41no—
241C4×C23—161no—
242C4×C23B(4,𝔽2)11yescharacteristic
243C4×C23#45811yescharacteristic
244#568C22×C2311yescharacteristic, complemented
245#559C4×C2311yescharacteristic, complemented
246#554—21no—
247#553—22no—
248#551—21no—
249#551C4×C2312yescomplemented
250#546—22no—
251D42C4×C2311yescharacteristic, complemented
252#532C22×C2311yescharacteristic
253#529—22no—
254#525—21no—
255#525C22×C2312yes—
256C4×#41—21no—
257C2×C4×D4—22no—
258#421—21no—
259#413C22×C2312yescomplemented
260#410C22×C2311yescharacteristic, complemented
261#407—22no—
262#356—21no—
263#355—21no—
264#352—22no—
265AΣL(1,𝔽16)—22no—
266#344—22no—
267#339—21no—
268C2×C23—21no—
269C2×C23—41no—
270C2×C23—42no—
271C2×C23—44no—
272C2×C23—81no—
273C2×C23#179011yescharacteristic
274C2×#41—21no—
275C2×#41D4×C2311yescharacteristic
276C2×#41C23×C2311yescharacteristic
277C22×D4—22no—
278C23×C4—21no—
279C23×C4D4×C2312yes—
280#131—42no—
281C2×M16—22no—
282C2×M16—21no—
283C2×M16—22no—
284C4⋊Q8D4×C2311yescharacteristic
285#127C2×C4×C2311yescharacteristic
286#126—22no—
287#126—22no—
288#125—22no—
289#124—22no—
290#124C2×C4×C2311yescharacteristic
291#124D4×C2311yescharacteristic
292#123—21no—
293#123—22no—
294#123—21no—
295#123—22no—
296#123C2×C4×C2312yes—
297#123D4×C2312yes—
298C22⋊Q8—22no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301#121—22no—
302#121—22no—
303#121—22no—
304#121—22no—
305#120—21no—
306#120—21no—
307#120C2×C4×C2312yes—
308#120D4×C2312yes—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×D4—21no—
312C4×D4—22no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—21no—
322#117—21no—
323#117—21no—
324#117—22no—
325#117—21no—
326#117D4×C2312yes—
327C2×C4⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C42—22no—
331C2×C42D4×C2311yescharacteristic
332#104—42no—
333#101—22no—
334#100—22no—
335#99—22no—
336C22⋊C8—44no—
337C23#1276511yescharacteristic, direct product factor, Sylow
338C24—21no—
339C24C22⋊C4×C2312yes—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—44no—
345C2×Q8—21no—
346C2×Q8C22⋊C4×C2311yescharacteristic
347C2×Q8C2×D4×C2311yescharacteristic
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—48no—
355C2×D4C22⋊C4×C2311yescharacteristic
356C2×D4C22×C4×C2311yescharacteristic
357C2×D4C2×D4×C2312yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C22⋊C4×C2312yes—
371C22×C4C2×D4×C2311yescharacteristic
372C22×C4C2×D4×C2312yes—
373M16—22no—
374M16—42no—
375M16—41no—
376M16—42no—
377M16—42no—
378C2×C8—24no—
379C2×C8—41no—
380C2×C8—42no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—21no—
395C22⋊C4—21no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—41no—
403C22⋊C4C22⋊C4×C2312yes—
404C22⋊C4C22⋊C4×C2312yes—
405C42—22no—
406C42—22no—
407C42—21no—
408C42—22no—
409C42—41no—
410C42C22⋊C4×C2311yescharacteristic
411C42C22⋊C4×C2311yescharacteristic
412C23—22no—
413C23—21no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—81no—
418C23C2×C22⋊C4×C2312yes—
419C23C23×#12011yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—42no—
423D4—24no—
424D4—41no—
425D4—42no—
426D4—41no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—44no—
431D4—81no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4C2×C22⋊C4×C2311yescharacteristic
455C2×C4C23×#12012yes—
456C2×C4C23×#12011yescharacteristic
457C8—42no—
458C8—82no—
459C22—24no—
460C22—22no—
461C22—21no—
462C22—41no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—42no—
468C22—41no—
469C22—82no—
470C22—84no—
471C22C23×#37911yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—41no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—41no—
480C4—161no—
481C4C23×B(4,𝔽2)11yescharacteristic
482C4C23×#45811yescharacteristic
483C2—21no—
484C2—41no—
485C2—44no—
486C2—42no—
487C2—81no—
488C2C23×#179011yescharacteristic
subgroupG/H
centerC4×C23B(4,𝔽2)
commutatorC2×D4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
socleC2×C23#1790
prime radicalC23×#3481C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×C4×C23 ◁ C2×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2132C2semidirect
3. C23×#2120C2semidirect
4. C23×#1891C2semidirect
5. C23×#1884C2semidirect
6. C23×#568C22semidirect
7. C23×#559C4semidirect
11. C23×#551C4semidirect
13. D42×C23C4semidirect
21. C23×#413C22semidirect
22. C23×#410C22semidirect
167. #12765C23direct (N × H)
216. #2132C2×C23semidirect
217. #2120C2×C23semidirect
218. #1891C2×C23semidirect
219. #1884C2×C23semidirect
244. #568C22×C23semidirect
245. #559C4×C23semidirect
249. #551C4×C23semidirect
251. D42C4×C23semidirect
259. #413C22×C23semidirect
260. #410C22×C23semidirect
337. C23#12765direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34812
C23×#21322
C23×#21202
C23×#18912
C23×#18842
#1276523