Finite Group Databasebeta

← C23×#12768 #19160394

Subgroups — C23×#12768 #19160394

Order 5888.

778 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×#3545C211yesmaximal, characteristic
2C2×C23×#354C211yesmaximal, characteristic, complemented
3C23×#2121C212yesmaximal, complemented
4C23×#1913C212yesmaximal, complemented
5C23×#1885C211yesmaximal, characteristic, complemented
6C23×#563C411yescharacteristic, complemented
7C23×#562—21no—
8C23×H(3,𝔽4)C411yescharacteristic, complemented
9C23×#555—21no—
10C23×#551—22no—
11C23×#551C412yescomplemented
12C23×#546—22no—
13C23×#542—22no—
14C23×#538—22no—
15C23×#536—21no—
16C23×#534C2212yes—
17C23×#529—21no—
18C2×C23×#124C2211yescharacteristic
19C23×#518C2211yescharacteristic
20C23×#419—21no—
21C2×C23×#100C2211yescharacteristic
22C2×C23×#99—22no—
23C2×C23×#99C2212yes—
24C23×#354—21no—
25C23×#354—21no—
26C23×#352—22no—
27C23×AΣL(1,𝔽16)—22no—
28C23×#342—22no—
29C23×#339—21no—
30C2×#41×C23—21no—
31C22×Q8×C23D411yescharacteristic
32C22×D4×C23D412yes—
33C22×D4×C23C2311yescharacteristic
34C2×M16×C23—21no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8—21no—
37C23×#126—22no—
38C23×#126—22no—
39C23×#126—22no—
40C23×#125—21no—
41C23×#124—22no—
42C23×#124—21no—
43C23×#124—21no—
44C23×#124—22no—
45C23×#124—22no—
46C23×#124—21no—
47C23×#124C2×C411yescharacteristic
48C23×#124C2×C411yescharacteristic
49C23×#124D411yescharacteristic
50C23×#124D411yescharacteristic
51C23×#123—22no—
52C23×#123—22no—
53C23×#123—22no—
54C23×#123—22no—
55C23×#123—22no—
56C23×#123C2×C412yes—
57C23×#123D412yes—
58C23×C22⋊Q8—21no—
59C23×C22⋊Q8—22no—
60C23×C22⋊Q8—22no—
61C23×C22⋊Q8—22no—
62C23×C22⋊Q8—22no—
63C23×#121—22no—
64C23×#121—22no—
65C23×#121—21no—
66C23×#121—22no—
67C23×#121—22no—
68C23×#120—22no—
69C23×#120—22no—
70C23×#120C2×C412yes—
71C23×#120D412yes—
72C4×Q8×C23—21no—
73C4×D4×C23—22no—
74C4×D4×C23—22no—
75C4×D4×C23—21no—
76C4×D4×C23—22no—
77C4×D4×C23—42no—
78C23×#117—21no—
79C23×#117—22no—
80C23×#117—22no—
81C23×#117—21no—
82C2×C22⋊C4×C23—22no—
83C2×C22⋊C4×C23—22no—
84C2×C22⋊C4×C23—22no—
85C2×C22⋊C4×C23—22no—
86C2×C22⋊C4×C23D412yes—
87C2×C42×C23D411yescharacteristic
88C23×#103—42no—
89C23×#100—22no—
90C23×#99—22no—
91C23×#99—22no—
92C23×#99—44no—
93C24×C23C22⋊C412yes—
94C24×C23C2×D412yes—
95#41×C23—81no—
96#41×C23—81no—
97C2×Q8×C23—21no—
98C2×Q8×C23—21no—
99C2×Q8×C23—21no—
100C2×Q8×C23—41no—
101C2×Q8×C23—41no—
102C2×Q8×C23C22⋊C411yescharacteristic
103C2×Q8×C23C22⋊C411yescharacteristic
104C2×D4×C23—22no—
105C2×D4×C23—21no—
106C2×D4×C23—21no—
107C2×D4×C23—22no—
108C2×D4×C23—22no—
109C2×D4×C23—22no—
110C2×D4×C23—41no—
111C2×D4×C23—41no—
112C2×D4×C23—42no—
113C2×D4×C23—42no—
114C2×D4×C23C22×C411yescharacteristic
115C2×D4×C23C2×D411yescharacteristic
116C2×D4×C23C2×D412yes—
117C22×C4×C23—21no—
118C22×C4×C23—22no—
119C22×C4×C23—22no—
120C22×C4×C23—22no—
121C22×C4×C23—21no—
122C22×C4×C23—42no—
123C22×C4×C23—42no—
124C22×C4×C23C22⋊C412yes—
125C22×C4×C23C2×D411yescharacteristic
126M16×C23—41no—
127M16×C23—41no—
128C2×C8×C23—41no—
129C4⋊C4×C23—22no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—21no—
135C4⋊C4×C23—21no—
136C4⋊C4×C23—22no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—22no—
139C22⋊C4×C23—22no—
140C22⋊C4×C23—22no—
141C22⋊C4×C23—22no—
142C22⋊C4×C23—22no—
143C22⋊C4×C23—22no—
144C22⋊C4×C23—22no—
145C22⋊C4×C23—22no—
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—24no—
153C22⋊C4×C23—41no—
154C22⋊C4×C23—42no—
155C22⋊C4×C23—42no—
156C22⋊C4×C23C22⋊C412yes—
157C22⋊C4×C23C22⋊C412yes—
158C42×C23—21no—
159C42×C23—22no—
160C42×C23—21no—
161C42×C23—21no—
162C42×C23—41no—
163C42×C23C22⋊C411yescharacteristic
164C42×C23C22⋊C411yescharacteristic
165#12768C2311yesmaximal, characteristic, direct product factor, Sylow
166C23×C23—22no—
167C23×C23—22no—
168C23×C23—22no—
169C23×C23—42no—
170C23×C23—82no—
171C23×C23—82no—
172C23×C23C2×C22⋊C412yes—
173C23×C23#12012yes—
174C23×C23#12011yescharacteristic
175Q8×C23—41no—
176Q8×C23—41no—
177Q8×C23—81no—
178Q8×C23—81no—
179Q8×C23—81no—
180D4×C23—41no—
181D4×C23—42no—
182D4×C23—41no—
183D4×C23—42no—
184D4×C23—42no—
185D4×C23—82no—
186D4×C23—82no—
187D4×C23—82no—
188D4×C23—81no—
189D4×C23—81no—
190C2×C4×C23—21no—
191C2×C4×C23—21no—
192C2×C4×C23—21no—
193C2×C4×C23—21no—
194C2×C4×C23—22no—
195C2×C4×C23—21no—
196C2×C4×C23—22no—
197C2×C4×C23—22no—
198C2×C4×C23—21no—
199C2×C4×C23—41no—
200C2×C4×C23—41no—
201C2×C4×C23—42no—
202C2×C4×C23—42no—
203C2×C4×C23—82no—
204C2×C4×C23—81no—
205C2×C4×C23—82no—
206C2×C4×C23—82no—
207C2×C4×C23—82no—
208C2×C4×C23—162no—
209C2×C4×C23—162no—
210C2×C4×C23C2×C22⋊C411yescharacteristic
211C2×C4×C23#12011yescharacteristic
212C8×C23—81no—
213#3545C2×C2311yescharacteristic
214C2×#354C2×C2311yescharacteristic, complemented
215#2121C2×C2312yescomplemented
216#1913C2×C2312yescomplemented
217#1885C2×C2311yescharacteristic, complemented
218C22×C23—22no—
219C22×C23—22no—
220C22×C23—41no—
221C22×C23—42no—
222C22×C23—42no—
223C22×C23—42no—
224C22×C23—42no—
225C22×C23—42no—
226C22×C23—82no—
227C22×C23—82no—
228C22×C23—162no—
229C22×C23#37911yescharacteristic
230C22×C23B(4,𝔽2)11yescharacteristic
231C22×C23#45811yescharacteristic
232C4×C23—21no—
233C4×C23—21no—
234C4×C23—41no—
235C4×C23—42no—
236C4×C23—42no—
237C4×C23—41no—
238C4×C23—41no—
239C4×C23—41no—
240C4×C23—42no—
241C4×C23—82no—
242C4×C23—162no—
243#563C4×C2311yescharacteristic, complemented
244#562—21no—
245H(3,𝔽4)C4×C2311yescharacteristic, complemented
246#555—21no—
247#551—22no—
248#551C4×C2312yescomplemented
249#546—22no—
250#542—22no—
251#538—22no—
252#536—21no—
253#534C22×C2312yes—
254#529—21no—
255C2×#124C22×C2311yescharacteristic
256#518C22×C2311yescharacteristic
257#419—21no—
258C2×#100C22×C2311yescharacteristic
259C2×#99—22no—
260C2×#99C22×C2312yes—
261#354—21no—
262#354—21no—
263#352—22no—
264AΣL(1,𝔽16)—22no—
265#342—22no—
266#339—21no—
267C2×C23—21no—
268C2×C23—21no—
269C2×C23—21no—
270C2×C23—42no—
271C2×C23—42no—
272C2×C23—82no—
273C2×C23#179011yescharacteristic
274C2×#41—21no—
275C22×Q8D4×C2311yescharacteristic
276C22×D4D4×C2312yes—
277C22×D4C23×C2311yescharacteristic
278C2×M16—21no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281#126—22no—
282#126—22no—
283#126—22no—
284#125—21no—
285#124—21no—
286#124—22no—
287#124—21no—
288#124—22no—
289#124—22no—
290#124—21no—
291#124C2×C4×C2311yescharacteristic
292#124C2×C4×C2311yescharacteristic
293#124D4×C2311yescharacteristic
294#124D4×C2311yescharacteristic
295#123—22no—
296#123—22no—
297#123—22no—
298#123—22no—
299#123—22no—
300#123C2×C4×C2312yes—
301#123D4×C2312yes—
302C22⋊Q8—21no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307#121—21no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121—22no—
312#120—22no—
313#120—22no—
314#120C2×C4×C2312yes—
315#120D4×C2312yes—
316C4×Q8—21no—
317C4×D4—21no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—42no—
322#117—21no—
323#117—22no—
324#117—22no—
325#117—21no—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4D4×C2312yes—
331C2×C42D4×C2311yescharacteristic
332#103—42no—
333#100—22no—
334#99—22no—
335#99—22no—
336#99—44no—
337C23#1276811yescharacteristic, direct product factor, Sylow
338C24C22⋊C4×C2312yes—
339C24C2×D4×C2312yes—
340#41—81no—
341#41—81no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8C22⋊C4×C2311yescharacteristic
348C2×Q8C22⋊C4×C2311yescharacteristic
349C2×D4—22no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C22×C4×C2311yescharacteristic
360C2×D4C2×D4×C2311yescharacteristic
361C2×D4C2×D4×C2312yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4C22⋊C4×C2312yes—
370C22×C4C2×D4×C2311yescharacteristic
371M16—41no—
372M16—41no—
373C2×C8—41no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—24no—
398C22⋊C4—41no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4C22⋊C4×C2312yes—
402C22⋊C4C22⋊C4×C2312yes—
403C42—21no—
404C42—22no—
405C42—21no—
406C42—21no—
407C42—41no—
408C42C22⋊C4×C2311yescharacteristic
409C42C22⋊C4×C2311yescharacteristic
410C23—22no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—82no—
415C23—82no—
416C23C2×C22⋊C4×C2312yes—
417C23C23×#12012yes—
418C23C23×#12011yescharacteristic
419Q8—41no—
420Q8—41no—
421Q8—81no—
422Q8—81no—
423Q8—81no—
424D4—41no—
425D4—42no—
426D4—41no—
427D4—42no—
428D4—42no—
429D4—82no—
430D4—81no—
431D4—82no—
432D4—82no—
433D4—81no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—162no—
453C2×C4—162no—
454C2×C4C2×C22⋊C4×C2311yescharacteristic
455C2×C4C23×#12011yescharacteristic
456C8—81no—
457C22—22no—
458C22—22no—
459C22—41no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—82no—
466C22—82no—
467C22—162no—
468C22C23×#37911yescharacteristic
469C22C23×B(4,𝔽2)11yescharacteristic
470C22C23×#45811yescharacteristic
471C4—21no—
472C4—21no—
473C4—41no—
474C4—42no—
475C4—42no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—42no—
480C4—82no—
481C4—162no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—42no—
486C2—42no—
487C2—82no—
488C2C23×#179011yescharacteristic
subgroupG/H
center, socleC2×C23#1790
commutatorC2×D4C22×C4×C23
Frattini, cosocleC22×D4C23×C23
prime radicalC23×#3545C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×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. C23×#2121C2semidirect
4. C23×#1913C2semidirect
5. C23×#1885C2semidirect
6. C23×#563C4semidirect
8. C23×H(3,𝔽4)C4semidirect
11. C23×#551C4semidirect
165. #12768C23direct (N × H)
214. C2×#354C2×C23semidirect
215. #2121C2×C23semidirect
216. #1913C2×C23semidirect
217. #1885C2×C23semidirect
243. #563C4×C23semidirect
245. H(3,𝔽4)C4×C23semidirect
248. #551C4×C23semidirect
337. C23#12768direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#35452
C2×C23×#3542
C23×#21212
C23×#19132
C23×#18852
#1276823