Finite Group Databasebeta

← C23×#12720 #19160346

Subgroups — C23×#12720 #19160346

Order 5888.

694 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3455C211yesmaximal, characteristic
2C23×#2132C211yesmaximal, characteristic, complemented
3C23×#2126C211yesmaximal, characteristic, complemented
4C23×#2120C212yesmaximal, complemented
5C23×#1827C212yesmaximal, complemented
6C23×#542—21no—
7C23×#542—21no—
8C23×#541—21no—
9C23×#540—21no—
10C23×#539—22no—
11C23×#538—22no—
12C23×#533C411yescharacteristic, complemented
13C23×#532C2211yescharacteristic
14C23×#529—22no—
15C23×#529—22no—
16C23×#525C411yescharacteristic, complemented
17C23×#525C2211yescharacteristic
18C4×#41×C23C411yescharacteristic, complemented
19C4×#41×C23C2211yescharacteristic
20C2×C23×#117C411yescharacteristic, complemented
21C23×#413C2212yes—
22C23×#410C2212yes—
23C23×#356—21no—
24C23×#355—21no—
25C23×#354—21no—
26C23×#353—21no—
27C23×#352—22no—
28C23×AΣL(1,𝔽16)—22no—
29C23×#339—22no—
30C23×H(3,ℤ/4)—22no—
31C2×#41×C23C2×C411yescharacteristic
32C2×#41×C23C2311yescharacteristic
33C23×C4×C23C2×C411yescharacteristic
34C2×M16×C23—22no—
35C23×C4⋊Q8—21no—
36C23×C4⋊Q8D411yescharacteristic
37C23×#127D411yescharacteristic
38C23×#126—22no—
39C23×#126—22no—
40C23×#126—22no—
41C23×#126—22no—
42C23×#125—22no—
43C23×#125—22no—
44C23×#125—21no—
45C23×#124—22no—
46C23×#124—22no—
47C23×#124D411yescharacteristic
48C23×#124D411yescharacteristic
49C23×#123—22no—
50C23×#123—22no—
51C23×#123—22no—
52C23×#123—22no—
53C23×#123—21no—
54C23×#123D412yes—
55C23×C22⋊Q8—22no—
56C23×C22⋊Q8—22no—
57C23×C22⋊Q8—21no—
58C23×C22⋊Q8—21no—
59C23×C22⋊Q8—22no—
60C23×#121—22no—
61C23×#121—22no—
62C23×#121—21no—
63C23×#121—21no—
64C23×#121—22no—
65C23×#120—21no—
66C23×#120D412yes—
67C4×Q8×C23—21no—
68C4×Q8×C23—21no—
69C4×Q8×C23—22no—
70C4×D4×C23—21no—
71C4×D4×C23—22no—
72C4×D4×C23—21no—
73C4×D4×C23—22no—
74C4×D4×C23—22no—
75C4×D4×C23—22no—
76C4×D4×C23—22no—
77C23×#117—21no—
78C23×#117—21no—
79C23×#117—21no—
80C23×#117—21no—
81C23×#117—21no—
82C23×#117—22no—
83C23×#117C2×C411yescharacteristic
84C23×#117C2×C411yescharacteristic
85C23×#117D412yes—
86C2×C4⋊C4×C23—21no—
87C2×C22⋊C4×C23—21no—
88C2×C42×C23—21no—
89C2×C42×C23C2×C411yescharacteristic
90C2×C42×C23C2×C411yescharacteristic
91C2×C42×C23D412yes—
92C23×#101—22no—
93C23×#100—22no—
94C23×#99—22no—
95C23×#99—22no—
96C24×C23C22⋊C411yescharacteristic
97#41×C23—41no—
98#41×C23—41no—
99#41×C23—41no—
100#41×C23—41no—
101C2×Q8×C23—22no—
102C2×Q8×C23C22⋊C411yescharacteristic
103C2×Q8×C23C2×D411yescharacteristic
104C2×D4×C23—21no—
105C2×D4×C23—22no—
106C2×D4×C23—22no—
107C2×D4×C23—22no—
108C2×D4×C23C22⋊C411yescharacteristic
109C2×D4×C23C2×D411yescharacteristic
110C2×D4×C23C2×D412yes—
111C22×C4×C23—22no—
112C22×C4×C23—21no—
113C22×C4×C23—21no—
114C22×C4×C23—22no—
115C22×C4×C23—21no—
116C22×C4×C23—21no—
117C22×C4×C23—21no—
118C22×C4×C23—41no—
119C22×C4×C23—41no—
120C22×C4×C23C22⋊C411yescharacteristic
121C22×C4×C23C22×C411yescharacteristic
122C22×C4×C23C2×D412yes—
123M16×C23—42no—
124M16×C23—42no—
125C2×C8×C23—42no—
126C4⋊C4×C23—21no—
127C4⋊C4×C23—21no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—22no—
131C4⋊C4×C23—22no—
132C4⋊C4×C23—22no—
133C4⋊C4×C23—22no—
134C4⋊C4×C23—22no—
135C4⋊C4×C23—21no—
136C4⋊C4×C23—21no—
137C4⋊C4×C23—22no—
138C4⋊C4×C23—22no—
139C4⋊C4×C23—22no—
140C4⋊C4×C23—42no—
141C22⋊C4×C23—21no—
142C22⋊C4×C23—21no—
143C22⋊C4×C23—21no—
144C22⋊C4×C23—21no—
145C22⋊C4×C23—22no—
146C22⋊C4×C23—22no—
147C22⋊C4×C23—22no—
148C22⋊C4×C23—21no—
149C22⋊C4×C23—22no—
150C22⋊C4×C23—22no—
151C22⋊C4×C23—22no—
152C22⋊C4×C23—22no—
153C22⋊C4×C23—22no—
154C22⋊C4×C23—42no—
155C22⋊C4×C23C22⋊C412yes—
156C42×C23—22no—
157C42×C23—21no—
158C42×C23—21no—
159C42×C23—21no—
160C42×C23—22no—
161C42×C23—42no—
162C42×C23C22⋊C411yescharacteristic
163C42×C23C22⋊C411yescharacteristic
164C42×C23C22⋊C411yescharacteristic
165C42×C23C22⋊C412yes—
166C42×C23C22⋊C411yescharacteristic
167#12720C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23—21no—
169C23×C23—21no—
170C23×C23—41no—
171C23×C23—81no—
172C23×C23C2×C22⋊C411yescharacteristic
173C23×C23#12012yes—
174Q8×C23—41no—
175Q8×C23—41no—
176Q8×C23—82no—
177D4×C23—42no—
178D4×C23—41no—
179D4×C23—41no—
180D4×C23—42no—
181D4×C23—82no—
182D4×C23—82no—
183D4×C23—82no—
184C2×C4×C23—22no—
185C2×C4×C23—21no—
186C2×C4×C23—22no—
187C2×C4×C23—21no—
188C2×C4×C23—21no—
189C2×C4×C23—21no—
190C2×C4×C23—21no—
191C2×C4×C23—22no—
192C2×C4×C23—21no—
193C2×C4×C23—22no—
194C2×C4×C23—21no—
195C2×C4×C23—21no—
196C2×C4×C23—42no—
197C2×C4×C23—41no—
198C2×C4×C23—41no—
199C2×C4×C23—41no—
200C2×C4×C23—41no—
201C2×C4×C23—41no—
202C2×C4×C23—81no—
203C2×C4×C23—82no—
204C2×C4×C23—82no—
205C2×C4×C23—81no—
206C2×C4×C23—82no—
207C2×C4×C23C2×C22⋊C411yescharacteristic
208C2×C4×C23C2×C22⋊C411yescharacteristic
209C2×C4×C23#12012yes—
210C8×C23—82no—
211#3455C2×C2311yescharacteristic
212#2132C2×C2311yescharacteristic, complemented
213#2126C2×C2311yescharacteristic, complemented
214#2120C2×C2312yescomplemented
215#1827C2×C2312yescomplemented
216C22×C23—21no—
217C22×C23—22no—
218C22×C23—41no—
219C22×C23—42no—
220C22×C23—42no—
221C22×C23—41no—
222C22×C23—41no—
223C22×C23—41no—
224C22×C23—41no—
225C22×C23—82no—
226C22×C23#37911yescharacteristic
227C4×C23—22no—
228C4×C23—21no—
229C4×C23—41no—
230C4×C23—41no—
231C4×C23—41no—
232C4×C23—42no—
233C4×C23—42no—
234C4×C23—41no—
235C4×C23—41no—
236C4×C23—41no—
237C4×C23—162no—
238C4×C23#41011yescharacteristic
239C4×C23#41011yescharacteristic
240#542—21no—
241#542—21no—
242#541—21no—
243#540—21no—
244#539—22no—
245#538—22no—
246#533C4×C2311yescharacteristic, complemented
247#532C22×C2311yescharacteristic
248#529—22no—
249#529—22no—
250#525C4×C2311yescharacteristic, complemented
251#525C22×C2311yescharacteristic
252C4×#41C4×C2311yescharacteristic, complemented
253C4×#41C22×C2311yescharacteristic
254C2×#117C4×C2311yescharacteristic, complemented
255#413C22×C2312yes—
256#410C22×C2312yes—
257#356—21no—
258#355—21no—
259#354—21no—
260#353—21no—
261#352—22no—
262AΣL(1,𝔽16)—22no—
263#339—22no—
264H(3,ℤ/4)—22no—
265C2×C23—21no—
266C2×C23—42no—
267C2×C23—41no—
268C2×C23—81no—
269C2×C23—81no—
270C2×C23#178211yescharacteristic
271C2×#41C2×C4×C2311yescharacteristic
272C2×#41C23×C2311yescharacteristic
273C23×C4C2×C4×C2311yescharacteristic
274C2×M16—22no—
275C4⋊Q8—21no—
276C4⋊Q8D4×C2311yescharacteristic
277#127D4×C2311yescharacteristic
278#126—22no—
279#126—22no—
280#126—22no—
281#126—22no—
282#125—22no—
283#125—22no—
284#125—21no—
285#124—22no—
286#124—22no—
287#124D4×C2311yescharacteristic
288#124D4×C2311yescharacteristic
289#123—22no—
290#123—22no—
291#123—22no—
292#123—22no—
293#123—21no—
294#123D4×C2312yes—
295C22⋊Q8—22no—
296C22⋊Q8—22no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—22no—
300#121—22no—
301#121—22no—
302#121—21no—
303#121—21no—
304#121—22no—
305#120—21no—
306#120D4×C2312yes—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—22no—
310C4×D4—21no—
311C4×D4—22no—
312C4×D4—21no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317#117—21no—
318#117—21no—
319#117—21no—
320#117—21no—
321#117—21no—
322#117—22no—
323#117C2×C4×C2311yescharacteristic
324#117C2×C4×C2311yescharacteristic
325#117D4×C2312yes—
326C2×C4⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329C2×C42C2×C4×C2311yescharacteristic
330C2×C42C2×C4×C2311yescharacteristic
331C2×C42D4×C2312yes—
332#101—22no—
333#100—22no—
334#99—22no—
335#99—22no—
336C23#1272011yescharacteristic, direct product factor, Sylow
337C24C22⋊C4×C2311yescharacteristic
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342C2×Q8—22no—
343C2×Q8C22⋊C4×C2311yescharacteristic
344C2×Q8C2×D4×C2311yescharacteristic
345C2×D4—21no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4C22⋊C4×C2311yescharacteristic
350C2×D4C2×D4×C2311yescharacteristic
351C2×D4C2×D4×C2312yes—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4C22⋊C4×C2311yescharacteristic
362C22×C4C22×C4×C2311yescharacteristic
363C22×C4C2×D4×C2312yes—
364M16—42no—
365M16—42no—
366C2×C8—42no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C22⋊C4—21no—
383C22⋊C4—21no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—21no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—42no—
396C22⋊C4C22⋊C4×C2312yes—
397C42—22no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—22no—
402C42—42no—
403C42C22⋊C4×C2311yescharacteristic
404C42C22⋊C4×C2311yescharacteristic
405C42C22⋊C4×C2311yescharacteristic
406C42C22⋊C4×C2311yescharacteristic
407C42C22⋊C4×C2312yes—
408C23—21no—
409C23—21no—
410C23—41no—
411C23—81no—
412C23C2×C22⋊C4×C2311yescharacteristic
413C23C23×#12012yes—
414Q8—41no—
415Q8—41no—
416Q8—82no—
417D4—42no—
418D4—41no—
419D4—41no—
420D4—42no—
421D4—82no—
422D4—82no—
423D4—82no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4C2×C22⋊C4×C2311yescharacteristic
448C2×C4C2×C22⋊C4×C2311yescharacteristic
449C2×C4C23×#12012yes—
450C8—82no—
451C22—21no—
452C22—22no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—82no—
461C22C23×#37911yescharacteristic
462C4—22no—
463C4—21no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—42no—
468C4—42no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—162no—
473C4C23×#41011yescharacteristic
474C4C23×#41011yescharacteristic
475C2—21no—
476C2—42no—
477C2—41no—
478C2—81no—
479C2—81no—
480C2C23×#178211yescharacteristic
subgroupG/H
centerC4×C23#410
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
socleC2×C23#1782
prime radicalC23×#3455C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C42×C23 ◁ C4×#41×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2132C2semidirect
3. C23×#2126C2semidirect
4. C23×#2120C2semidirect
5. C23×#1827C2semidirect
12. C23×#533C4semidirect
16. C23×#525C4semidirect
18. C4×#41×C23C4semidirect
20. C2×C23×#117C4semidirect
167. #12720C23direct (N × H)
212. #2132C2×C23semidirect
213. #2126C2×C23semidirect
214. #2120C2×C23semidirect
215. #1827C2×C23semidirect
246. #533C4×C23semidirect
250. #525C4×C23semidirect
252. C4×#41C4×C23semidirect
254. C2×#117C4×C23semidirect
336. C23#12720direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34552
C23×#21322
C23×#21262
C23×#21202
C23×#18272
#1272023