Finite Group Databasebeta

← C23×#12717 #19160343

Subgroups — C23×#12717 #19160343

Order 5888.

944 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3499C211yesmaximal, characteristic
2C23×#2132C211yesmaximal, characteristic, complemented
3C23×#2126C211yesmaximal, characteristic, complemented
4C23×#2123C212yesmaximal, complemented
5C23×#1796C212yesmaximal, complemented
6C2×C23×2−1+4C411yescharacteristic, complemented
7C2×C23×2−1+4C2211yescharacteristic
8D4×Q8×C23—22no—
9C23×#547—22no—
10C23×#547—24no—
11C23×#536—21no—
12C23×#536—21no—
13C23×#536C411yescharacteristic, complemented
14C23×#536C2211yescharacteristic
15C23×#532—21no—
16C23×#532C2211yescharacteristic
17C2×C23×C4⋊Q8—21no—
18C2×C23×C4⋊Q8C411yescharacteristic, complemented
19C4×#41×C23C411yescharacteristic, complemented
20C23×#421—22no—
21C23×#419—22no—
22C23×#413C2212yes—
23C23×#410C2212yes—
24C23×#356—21no—
25C23×#355—21no—
26C23×#354—21no—
27C23×#353—21no—
28C23×#352—24no—
29C23×2−1+4—42no—
30C23×2−1+4—42no—
31C23×2−1+4—42no—
32C23×2−1+4—42no—
33C2×#41×C23—21no—
34C2×#41×C23—21no—
35C2×#41×C23—22no—
36C2×#41×C23—21no—
37C2×#41×C23—22no—
38C2×#41×C23C2×C411yescharacteristic
39C2×#41×C23C2×C411yescharacteristic
40C2×#41×C23D412yes—
41C2×#41×C23C2311yescharacteristic
42C22×Q8×C23—21no—
43C22×Q8×C23—21no—
44C22×Q8×C23—21no—
45C22×Q8×C23C2×C411yescharacteristic
46C22×Q8×C23C2×C411yescharacteristic
47C22×Q8×C23D412yes—
48C2×M16×C23—22no—
49C23×C4⋊Q8—24no—
50C23×C4⋊Q8—22no—
51C23×C4⋊Q8—21no—
52C23×C4⋊Q8—21no—
53C23×C4⋊Q8—21no—
54C23×C4⋊Q8—21no—
55C23×C4⋊Q8D411yescharacteristic
56C23×#127D411yescharacteristic
57C23×#124—21no—
58C23×#124—21no—
59C23×#124D411yescharacteristic
60C23×#124D411yescharacteristic
61C23×#123—24no—
62C23×#123—24no—
63C23×#123—22no—
64C23×#123D414yes—
65C23×C22⋊Q8—24no—
66C23×C22⋊Q8—24no—
67C23×C22⋊Q8—22no—
68C23×C22⋊Q8—22no—
69C23×#121—24no—
70C23×#121—22no—
71C4×Q8×C23—21no—
72C4×D4×C23—24no—
73C4×D4×C23—21no—
74C4×D4×C23—22no—
75C4×D4×C23—22no—
76C23×#117—21no—
77C23×#117—22no—
78C23×#117C2×C411yescharacteristic
79C2×C4⋊C4×C23—24no—
80C2×C4⋊C4×C23—22no—
81C2×C42×C23—21no—
82C2×C42×C23C2×C411yescharacteristic
83C23×#104—44no—
84C23×#103—44no—
85C23×#101—22no—
86C23×#100—22no—
87C23×#99—22no—
88C23×#99—22no—
89#41×C23—22no—
90#41×C23—22no—
91#41×C23—24no—
92#41×C23—41no—
93#41×C23—41no—
94#41×C23—42no—
95#41×C23—44no—
96#41×C23—44no—
97#41×C23—41no—
98#41×C23—41no—
99#41×C23—48no—
100#41×C23—48no—
101#41×C23—42no—
102#41×C23—42no—
103C2×Q8×C23—22no—
104C2×Q8×C23—21no—
105C2×Q8×C23—21no—
106C2×Q8×C23—21no—
107C2×Q8×C23—22no—
108C2×Q8×C23—24no—
109C2×Q8×C23—41no—
110C2×Q8×C23—41no—
111C2×Q8×C23—42no—
112C2×Q8×C23—41no—
113C2×Q8×C23—41no—
114C2×Q8×C23—44no—
115C2×Q8×C23—44no—
116C2×Q8×C23—42no—
117C2×Q8×C23—42no—
118C2×Q8×C23C22⋊C411yescharacteristic
119C2×Q8×C23C22⋊C412yes—
120C2×Q8×C23C22⋊C411yescharacteristic
121C2×Q8×C23C2×D411yescharacteristic
122C2×D4×C23—21no—
123C2×D4×C23—22no—
124C2×D4×C23—21no—
125C2×D4×C23—22no—
126C2×D4×C23—22no—
127C2×D4×C23C22⋊C411yescharacteristic
128C2×D4×C23C22⋊C411yescharacteristic
129C2×D4×C23C2×D411yescharacteristic
130C2×D4×C23C2×D412yes—
131C22×C4×C23—21no—
132C22×C4×C23—22no—
133C22×C4×C23—22no—
134C22×C4×C23—22no—
135C22×C4×C23—21no—
136C22×C4×C23—21no—
137C22×C4×C23—22no—
138C22×C4×C23—21no—
139C22×C4×C23—22no—
140C22×C4×C23C22⋊C412yes—
141C22×C4×C23C22×C411yescharacteristic
142C22×C4×C23C2×D412yes—
143M16×C23—42no—
144M16×C23—42no—
145C2×C8×C23—42no—
146C4⋊C4×C23—24no—
147C4⋊C4×C23—24no—
148C4⋊C4×C23—24no—
149C4⋊C4×C23—22no—
150C4⋊C4×C23—21no—
151C4⋊C4×C23—22no—
152C4⋊C4×C23—24no—
153C4⋊C4×C23—42no—
154C22⋊C4×C23—22no—
155C22⋊C4×C23—24no—
156C22⋊C4×C23—22no—
157C22⋊C4×C23—24no—
158C22⋊C4×C23—22no—
159C22⋊C4×C23—42no—
160C22⋊C4×C23C22⋊C412yes—
161C42×C23—22no—
162C42×C23—21no—
163C42×C23—21no—
164C42×C23—42no—
165C42×C23C22⋊C411yescharacteristic
166C42×C23C22⋊C411yescharacteristic
167#12717C2311yesmaximal, characteristic, direct product factor, Sylow
168C23×C23—21no—
169C23×C23—21no—
170C23×C23C2×C22⋊C411yescharacteristic
171C23×C23#12012yes—
172Q8×C23—21no—
173Q8×C23—22no—
174Q8×C23—22no—
175Q8×C23—41no—
176Q8×C23—44no—
177Q8×C23—41no—
178Q8×C23—42no—
179Q8×C23—42no—
180Q8×C23—44no—
181Q8×C23—41no—
182Q8×C23—41no—
183Q8×C23#9912yes—
184D4×C23—24no—
185D4×C23—21no—
186D4×C23—42no—
187D4×C23—41no—
188D4×C23—41no—
189D4×C23—44no—
190D4×C23—44no—
191D4×C23—42no—
192D4×C23—44no—
193D4×C23#9912yes—
194C2×C4×C23—24no—
195C2×C4×C23—21no—
196C2×C4×C23—21no—
197C2×C4×C23—22no—
198C2×C4×C23—22no—
199C2×C4×C23—21no—
200C2×C4×C23—21no—
201C2×C4×C23—22no—
202C2×C4×C23—21no—
203C2×C4×C23—22no—
204C2×C4×C23—21no—
205C2×C4×C23—22no—
206C2×C4×C23—44no—
207C2×C4×C23—44no—
208C2×C4×C23—42no—
209C2×C4×C23—44no—
210C2×C4×C23—44no—
211C2×C4×C23—42no—
212C2×C4×C23—44no—
213C2×C4×C23—41no—
214C2×C4×C23—41no—
215C2×C4×C23—42no—
216C2×C4×C23—82no—
217C2×C4×C23C2×C22⋊C411yescharacteristic
218C2×C4×C23C2×C22⋊C411yescharacteristic
219C2×C4×C23#12012yes—
220C8×C23—82no—
221#3499C2×C2311yescharacteristic
222#2132C2×C2311yescharacteristic, complemented
223#2126C2×C2311yescharacteristic, complemented
224#2123C2×C2312yescomplemented
225#1796C2×C2312yescomplemented
226C22×C23—21no—
227C22×C23—22no—
228C22×C23—22no—
229C22×C23—41no—
230C22×C23—42no—
231C22×C23—42no—
232C22×C23—42no—
233C22×C23—41no—
234C22×C23#37911yescharacteristic
235C4×C23—22no—
236C4×C23—22no—
237C4×C23—21no—
238C4×C23—22no—
239C4×C23—42no—
240C4×C23—41no—
241C4×C23—42no—
242C4×C23—41no—
243C4×C23—42no—
244C4×C23—162no—
245C4×C23C2×#9911yescharacteristic
246C4×C23C2×#9911yescharacteristic
247C2×2−1+4C4×C2311yescharacteristic, complemented
248C2×2−1+4C22×C2311yescharacteristic
249D4×Q8—22no—
250#547—22no—
251#547—24no—
252#536—21no—
253#536—21no—
254#536C4×C2311yescharacteristic, complemented
255#536C22×C2311yescharacteristic
256#532—21no—
257#532C22×C2311yescharacteristic
258C2×C4⋊Q8—21no—
259C2×C4⋊Q8C4×C2311yescharacteristic, complemented
260C4×#41C4×C2311yescharacteristic, complemented
261#421—22no—
262#419—22no—
263#413C22×C2312yes—
264#410C22×C2312yes—
265#356—21no—
266#355—21no—
267#354—21no—
268#353—21no—
269#352—24no—
270C2×C23—21no—
271C2×C23—42no—
272C2×C23—41no—
273C2×C23—42no—
274C2×C23—81no—
275C2×C23#178111yescharacteristic
2762−1+4—42no—
2772−1+4—42no—
2782−1+4—42no—
2792−1+4—42no—
280C2×#41—21no—
281C2×#41—21no—
282C2×#41—21no—
283C2×#41—22no—
284C2×#41—22no—
285C2×#41C2×C4×C2311yescharacteristic
286C2×#41C2×C4×C2311yescharacteristic
287C2×#41D4×C2312yes—
288C2×#41C23×C2311yescharacteristic
289C22×Q8—21no—
290C22×Q8—21no—
291C22×Q8—21no—
292C22×Q8C2×C4×C2311yescharacteristic
293C22×Q8C2×C4×C2311yescharacteristic
294C22×Q8D4×C2312yes—
295C2×M16—22no—
296C4⋊Q8—24no—
297C4⋊Q8—21no—
298C4⋊Q8—22no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8D4×C2311yescharacteristic
303#127D4×C2311yescharacteristic
304#124—21no—
305#124—21no—
306#124D4×C2311yescharacteristic
307#124D4×C2311yescharacteristic
308#123—24no—
309#123—24no—
310#123—22no—
311#123D4×C2314yes—
312C22⋊Q8—24no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316#121—24no—
317#121—22no—
318C4×Q8—21no—
319C4×D4—24no—
320C4×D4—21no—
321C4×D4—22no—
322C4×D4—22no—
323#117—21no—
324#117—22no—
325#117C2×C4×C2311yescharacteristic
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C42—21no—
329C2×C42C2×C4×C2311yescharacteristic
330#104—44no—
331#103—44no—
332#101—22no—
333#100—22no—
334#99—22no—
335#99—22no—
336C23#1271711yescharacteristic, direct product factor, Sylow
337#41—22no—
338#41—22no—
339#41—24no—
340#41—41no—
341#41—41no—
342#41—42no—
343#41—44no—
344#41—44no—
345#41—41no—
346#41—41no—
347#41—48no—
348#41—48no—
349#41—42no—
350#41—42no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—42no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8C22⋊C4×C2312yes—
367C2×Q8C22⋊C4×C2311yescharacteristic
368C2×Q8C22⋊C4×C2311yescharacteristic
369C2×Q8C2×D4×C2311yescharacteristic
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4C22⋊C4×C2311yescharacteristic
376C2×D4C22⋊C4×C2311yescharacteristic
377C2×D4C2×D4×C2311yescharacteristic
378C2×D4C2×D4×C2312yes—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4C22⋊C4×C2312yes—
389C22×C4C22×C4×C2311yescharacteristic
390C22×C4C2×D4×C2312yes—
391M16—42no—
392M16—42no—
393C2×C8—42no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—42no—
402C22⋊C4—22no—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—22no—
407C22⋊C4—42no—
408C22⋊C4C22⋊C4×C2312yes—
409C42—22no—
410C42—21no—
411C42—21no—
412C42—42no—
413C42C22⋊C4×C2311yescharacteristic
414C42C22⋊C4×C2311yescharacteristic
415C23—21no—
416C23—21no—
417C23C2×C22⋊C4×C2311yescharacteristic
418C23C23×#12012yes—
419Q8—21no—
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—44no—
424Q8—41no—
425Q8—42no—
426Q8—42no—
427Q8—44no—
428Q8—41no—
429Q8—41no—
430Q8C23×#9912yes—
431D4—24no—
432D4—21no—
433D4—42no—
434D4—41no—
435D4—41no—
436D4—44no—
437D4—44no—
438D4—42no—
439D4—44no—
440D4C23×#9912yes—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—82no—
464C2×C4C2×C22⋊C4×C2311yescharacteristic
465C2×C4C2×C22⋊C4×C2311yescharacteristic
466C2×C4C23×#12012yes—
467C8—82no—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—41no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22C23×#37911yescharacteristic
477C4—22no—
478C4—22no—
479C4—21no—
480C4—22no—
481C4—42no—
482C4—41no—
483C4—42no—
484C4—41no—
485C4—42no—
486C4—162no—
487C4C2×C23×#9911yescharacteristic
488C4C2×C23×#9911yescharacteristic
489C2—21no—
490C2—42no—
491C2—42no—
492C2—41no—
493C2—81no—
494C2C23×#178111yescharacteristic
subgroupG/H
center, socleC2×C23#1781
commutatorC22×C4C22×C4×C23
Frattini, cosocleC2×#41C23×C23
prime radicalC23×#3499C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ C2×C23×2−1+4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2132C2semidirect
3. C23×#2126C2semidirect
4. C23×#2123C2semidirect
5. C23×#1796C2semidirect
6. C2×C23×2−1+4C4semidirect
13. C23×#536C4semidirect
18. C2×C23×C4⋊Q8C4semidirect
19. C4×#41×C23C4semidirect
167. #12717C23direct (N × H)
222. #2132C2×C23semidirect
223. #2126C2×C23semidirect
224. #2123C2×C23semidirect
225. #1796C2×C23semidirect
247. C2×2−1+4C4×C23semidirect
254. #536C4×C23semidirect
259. C2×C4⋊Q8C4×C23semidirect
260. C4×#41C4×C23semidirect
336. C23#12717direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#34992
C23×#21322
C23×#21262
C23×#21232
C23×#17962
#1271723