Finite Group Databasebeta

← C23×#9263 #19156889

Subgroups — C23×#9263 #19156889

Order 5888.

478 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3190C211yesmaximal, characteristic
2C23×#2977C211yesmaximal, characteristic
3C23×#1504C211yesmaximal, characteristic
4C2×C23×#328C211yesmaximal, characteristic, complemented
5C23×#1472C211yesmaximal, characteristic, complemented
6C23×#1298C211yesmaximal, characteristic, complemented
7C23×#1280C211yesmaximal, characteristic, complemented
8C23×#525C2211yescharacteristic
9C2×C23×C22⋊Q8C2211yescharacteristic
10C23×#485—21no—
11C23×#483C411yescharacteristic, complemented
12C23×#480C411yescharacteristic, complemented
13C23×#467—21no—
14C23×#465—21no—
15C23×#452—21no—
16C23×#451C411yescharacteristic, complemented
17C23×#448C411yescharacteristic, complemented
18C23×#435—21no—
19C23×#435—21no—
20C8×D4×C23—21no—
21C8×D4×C23—21no—
22C23×#433—21no—
23C23×#408—21no—
24C23×#407C2211yescharacteristic
25C2×C23×C22⋊C8C2211yescharacteristic
26C2×C23×C22⋊C8C2211yescharacteristic
27C2×C23×C22⋊C8C2211yescharacteristic
28C2×C23×#95C2211yescharacteristic
29C23×#328—21no—
30C23×#328—21no—
31C23×#328—21no—
32C23×#327—21no—
33C23×#324—21no—
34C23×#324—21no—
35C2×#41×C23D411yescharacteristic
36C22×Q8×C23C2×C411yescharacteristic
37C23×C4×C23—21no—
38C23×C4×C23C2311yescharacteristic
39C2×Q16×C23—21no—
40C2×SD16×C23—21no—
41C2×M16×C23—21no—
42C2×M16×C23—21no—
43C22×C8×C23—21no—
44C22×C8×C23—21no—
45C22×C8×C23—21no—
46C22×C8×C23—21no—
47C23×#123—21no—
48C23×C22⋊Q8—21no—
49C23×C22⋊Q8—21no—
50C23×C22⋊Q8—21no—
51C23×C22⋊Q8—21no—
52C23×C22⋊Q8C2×C411yescharacteristic
53C23×C22⋊Q8D411yescharacteristic
54C23×#121C2×C411yescharacteristic
55C23×#121D411yescharacteristic
56C23×#120—21no—
57C4×D4×C23—21no—
58C4×D4×C23—21no—
59C23×#117D411yescharacteristic
60C2×C4⋊C4×C23—21no—
61C2×C4⋊C4×C23C2×C411yescharacteristic
62C2×C22⋊C4×C23—21no—
63C23×#107—21no—
64C23×#106—21no—
65C23×C4⋊C8—21no—
66C23×C4⋊C8—21no—
67C23×#103—21no—
68C23×#103—21no—
69C23×#103—21no—
70C23×#102—21no—
71C23×C22⋊C8—21no—
72C23×C22⋊C8—21no—
73C23×C22⋊C8—21no—
74C23×C22⋊C8—21no—
75C23×C22⋊C8—21no—
76C23×C22⋊C8—21no—
77C23×C22⋊C8—21no—
78C23×C22⋊C8C2×C411yescharacteristic
79C23×C22⋊C8C2×C411yescharacteristic
80C23×C22⋊C8D411yescharacteristic
81C23×C22⋊C8D411yescharacteristic
82C23×C22⋊C8D411yescharacteristic
83C23×C22⋊C8D411yescharacteristic
84C23×#97—21no—
85C4×C8×C23—21no—
86C23×#95—21no—
87C23×#95—21no—
88C24×C23#4111yescharacteristic
89#41×C23—81no—
90C2×Q8×C23—21no—
91C2×Q8×C23—21no—
92C2×Q8×C23—81no—
93C2×Q8×C23C22⋊C411yescharacteristic
94C2×D4×C23—21no—
95C2×D4×C23—21no—
96C2×D4×C23—21no—
97C2×D4×C23C22⋊C411yescharacteristic
98C22×C4×C23—21no—
99C22×C4×C23—21no—
100C22×C4×C23—22no—
101C22×C4×C23—21no—
102C22×C4×C23—41no—
103C22×C4×C23—41no—
104C22×C4×C23—81no—
105C22×C4×C23—81no—
106C22×C4×C23—81no—
107C22×C4×C23C22×C411yescharacteristic
108C22×C4×C23C2×D411yescharacteristic
109C22×C4×C23C2×D411yescharacteristic
110C22×C4×C23C2×D411yescharacteristic
111C22×C4×C23C2×D411yescharacteristic
112C22×C4×C23#4111yescharacteristic
113Q16×C23—81no—
114SD16×C23—81no—
115M16×C23—81no—
116M16×C23—81no—
117C2×C8×C23—21no—
118C2×C8×C23—21no—
119C2×C8×C23—21no—
120C2×C8×C23—21no—
121C2×C8×C23—21no—
122C2×C8×C23—21no—
123C2×C8×C23—41no—
124C2×C8×C23—81no—
125C2×C8×C23—81no—
126C2×C8×C23—81no—
127C2×C8×C23—81no—
128C4⋊C4×C23—21no—
129C4⋊C4×C23—21no—
130C4⋊C4×C23—21no—
131C4⋊C4×C23—21no—
132C4⋊C4×C23—41no—
133C4⋊C4×C23—41no—
134C4⋊C4×C23C22⋊C411yescharacteristic
135C4⋊C4×C23C22⋊C411yescharacteristic
136C22⋊C4×C23—21no—
137C22⋊C4×C23—21no—
138C22⋊C4×C23—21no—
139C22⋊C4×C23—21no—
140C22⋊C4×C23—21no—
141C22⋊C4×C23—41no—
142C22⋊C4×C23—41no—
143C42×C23—21no—
144#9263C2311yesmaximal, characteristic, direct product factor, Sylow
145C23×C23—21no—
146C23×C23—21no—
147C23×C23—21no—
148C23×C23—21no—
149C23×C23—21no—
150C23×C23—41no—
151C23×C23C4×D411yescharacteristic
152C23×C23#12111yescharacteristic
153C23×C23#12311yescharacteristic
154Q8×C23—41no—
155Q8×C23—41no—
156Q8×C23—41no—
157Q8×C23—81no—
158D4×C23—41no—
159D4×C23—41no—
160D4×C23—41no—
161D4×C23—41no—
162D4×C23—41no—
163D4×C23—81no—
164C2×C4×C23—21no—
165C2×C4×C23—21no—
166C2×C4×C23—21no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—41no—
170C2×C4×C23—41no—
171C2×C4×C23—41no—
172C2×C4×C23—41no—
173C2×C4×C23—41no—
174C2×C4×C23—41no—
175C2×C4×C23—42no—
176C2×C4×C23—41no—
177C2×C4×C23—41no—
178C2×C4×C23—41no—
179C2×C4×C23—41no—
180C2×C4×C23—81no—
181C2×C4×C23—82no—
182C2×C4×C23—81no—
183C2×C4×C23—81no—
184C2×C4×C23—82no—
185C2×C4×C23C2×C22⋊C411yescharacteristic
186C2×C4×C23C4×D411yescharacteristic
187C2×C4×C23#12011yescharacteristic
188C2×C4×C23#12111yescharacteristic
189C8×C23—41no—
190C8×C23—41no—
191C8×C23—41no—
192C8×C23—41no—
193C8×C23—41no—
194C8×C23—41no—
195C8×C23—81no—
196#3190C2×C2311yescharacteristic
197#2977C2×C2311yescharacteristic
198#1504C2×C2311yescharacteristic
199C2×#328C2×C2311yescharacteristic, complemented
200#1472C2×C2311yescharacteristic, complemented
201#1298C2×C2311yescharacteristic, complemented
202#1280C2×C2311yescharacteristic, complemented
203C22×C23—21no—
204C22×C23—21no—
205C22×C23—21no—
206C22×C23—21no—
207C22×C23—21no—
208C22×C23—21no—
209C22×C23—21no—
210C22×C23—41no—
211C22×C23—41no—
212C22×C23—41no—
213C22×C23—41no—
214C22×C23—41no—
215C22×C23—41no—
216C22×C23—41no—
217C22×C23#38611yescharacteristic
218C22×C23#44311yescharacteristic
219C22×C23#44411yescharacteristic
220C22×C23#47111yescharacteristic
221C22×C23#47311yescharacteristic
222C4×C23—21no—
223C4×C23—21no—
224C4×C23—21no—
225C4×C23—21no—
226C4×C23—41no—
227C4×C23—41no—
228C4×C23—41no—
229C4×C23—82no—
230C4×C23—81no—
231C4×C23—81no—
232#525C22×C2311yescharacteristic
233C2×C22⋊Q8C22×C2311yescharacteristic
234#485—21no—
235#483C4×C2311yescharacteristic, complemented
236#480C4×C2311yescharacteristic, complemented
237#467—21no—
238#465—21no—
239#452—21no—
240#451C4×C2311yescharacteristic, complemented
241#448C4×C2311yescharacteristic, complemented
242#435—21no—
243#435—21no—
244C8×D4—21no—
245C8×D4—21no—
246#433—21no—
247#408—21no—
248#407C22×C2311yescharacteristic
249C2×C22⋊C8C22×C2311yescharacteristic
250C2×C22⋊C8C22×C2311yescharacteristic
251C2×C22⋊C8C22×C2311yescharacteristic
252C2×#95C22×C2311yescharacteristic
253#328—21no—
254#328—21no—
255#328—21no—
256#327—21no—
257#324—21no—
258#324—21no—
259C2×C23—21no—
260C2×C23—21no—
261C2×C23—21no—
262C2×C23—21no—
263C2×C23—41no—
264C2×C23—81no—
265C2×C23#187611yescharacteristic
266C2×C23#190811yescharacteristic
267C2×C23#191311yescharacteristic
268C2×#41D4×C2311yescharacteristic
269C22×Q8C2×C4×C2311yescharacteristic
270C23×C4—21no—
271C23×C4C23×C2311yescharacteristic
272C2×Q16—21no—
273C2×SD16—21no—
274C2×M16—21no—
275C2×M16—21no—
276C22×C8—21no—
277C22×C8—21no—
278C22×C8—21no—
279C22×C8—21no—
280#123—21no—
281C22⋊Q8—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8C2×C4×C2311yescharacteristic
286C22⋊Q8D4×C2311yescharacteristic
287#121C2×C4×C2311yescharacteristic
288#121D4×C2311yescharacteristic
289#120—21no—
290C4×D4—21no—
291C4×D4—21no—
292#117D4×C2311yescharacteristic
293C2×C4⋊C4—21no—
294C2×C4⋊C4C2×C4×C2311yescharacteristic
295C2×C22⋊C4—21no—
296#107—21no—
297#106—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300#103—21no—
301#103—21no—
302#103—21no—
303#102—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8C2×C4×C2311yescharacteristic
312C22⋊C8C2×C4×C2311yescharacteristic
313C22⋊C8D4×C2311yescharacteristic
314C22⋊C8D4×C2311yescharacteristic
315C22⋊C8D4×C2311yescharacteristic
316C22⋊C8D4×C2311yescharacteristic
317#97—21no—
318C4×C8—21no—
319#95—21no—
320#95—21no—
321C23#926311yescharacteristic, direct product factor, Sylow
322C24#41×C2311yescharacteristic
323#41—81no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8C22⋊C4×C2311yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C22⋊C4×C2311yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4C22×C4×C2311yescharacteristic
342C22×C4C2×D4×C2311yescharacteristic
343C22×C4C2×D4×C2311yescharacteristic
344C22×C4C2×D4×C2311yescharacteristic
345C22×C4C2×D4×C2311yescharacteristic
346C22×C4#41×C2311yescharacteristic
347Q16—81no—
348SD16—81no—
349M16—81no—
350M16—81no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—41no—
358C2×C8—81no—
359C2×C8—81no—
360C2×C8—81no—
361C2×C8—81no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C22⋊C4×C2311yescharacteristic
369C4⋊C4C22⋊C4×C2311yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C42—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—41no—
384C23C4×D4×C2311yescharacteristic
385C23C23×#12111yescharacteristic
386C23C23×#12311yescharacteristic
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—81no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—81no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—82no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4C2×C22⋊C4×C2311yescharacteristic
419C2×C4C4×D4×C2311yescharacteristic
420C2×C4C23×#12011yescharacteristic
421C2×C4C23×#12111yescharacteristic
422C8—41no—
423C8—41no—
424C8—41no—
425C8—41no—
426C8—41no—
427C8—41no—
428C8—81no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22C23×#38611yescharacteristic
444C22C23×#44311yescharacteristic
445C22C23×#44411yescharacteristic
446C22C23×#47111yescharacteristic
447C22C23×#47311yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—82no—
456C4—81no—
457C4—81no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—81no—
464C2C23×#187611yescharacteristic
465C2C23×#190811yescharacteristic
466C2C23×#191311yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC23×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C23×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
4. C2×C23×#328C2semidirect
5. C23×#1472C2semidirect
6. C23×#1298C2semidirect
7. C23×#1280C2semidirect
11. C23×#483C4semidirect
12. C23×#480C4semidirect
16. C23×#451C4semidirect
17. C23×#448C4semidirect
144. #9263C23direct (N × H)
199. C2×#328C2×C23semidirect
200. #1472C2×C23semidirect
201. #1298C2×C23semidirect
202. #1280C2×C23semidirect
235. #483C4×C23semidirect
236. #480C4×C23semidirect
240. #451C4×C23semidirect
241. #448C4×C23semidirect
321. C23#9263direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31902
C23×#29772
C23×#15042
C2×C23×#3282
C23×#14722
C23×#12982
C23×#12802
#926323