Finite Group Databasebeta

← C23×#9681 #19157307

Subgroups — C23×#9681 #19157307

Order 5888.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#2850C211yesmaximal, characteristic
2C4×C23×C22⋊C8C211yesmaximal, characteristic, complemented
3C2×C23×#328C212yesmaximal, complemented
4C2×C23×#327C212yesmaximal
5C23×#1434C211yesmaximal, characteristic, complemented
6C2×C23×#124C411yescharacteristic, complemented
7C2×C23×C22⋊Q8C2211yescharacteristic
8C2×C23×#121C2211yescharacteristic
9C22×C42×C23C2211yescharacteristic
10C2×C23×C22⋊C8C2212yes—
11C2×C4×C8×C23—21no—
12C23×#391C411yescharacteristic, complemented
13C23×#390C411yescharacteristic, complemented
14C23×#388—21no—
15C23×#388—21no—
16C23×#386—21no—
17C23×#386—21no—
18C23×#382C411yescharacteristic, complemented
19C2×C23×#95—21no—
20C2×C23×#95—21no—
21C2×C23×#95C2212yes—
22C23×#336—21no—
23C23×#328—24no—
24C23×#327—24no—
25C22×Q8×C23C2×C411yescharacteristic
26C22×D4×C23—21no—
27C22×D4×C23C2×C411yescharacteristic
28C23×C4×C23—22no—
29C23×C4×C23D411yescharacteristic
30C23×C4×C23D411yescharacteristic
31C23×C4×C23C2311yescharacteristic
32C22×C8×C23—22no—
33C23×#124—42no—
34C23×C22⋊Q8—22no—
35C23×C22⋊Q8—22no—
36C23×#121—22no—
37C23×#121—22no—
38C2×C4⋊C4×C23—21no—
39C2×C4⋊C4×C23C2×C411yescharacteristic
40C2×C4⋊C4×C23C2×C411yescharacteristic
41C2×C22⋊C4×C23—21no—
42C2×C22⋊C4×C23—21no—
43C2×C22⋊C4×C23—21no—
44C2×C22⋊C4×C23—21no—
45C2×C22⋊C4×C23—21no—
46C2×C22⋊C4×C23—21no—
47C2×C42×C23—42no—
48C2×C42×C23C2×C411yescharacteristic
49C2×C42×C23C2×C411yescharacteristic
50C2×C42×C23D411yescharacteristic
51C2×C42×C23D411yescharacteristic
52C23×C22⋊C8—22no—
53C23×C22⋊C8D414yes—
54C4×C8×C23—42no—
55C23×#95—24no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—21no—
60C23×#95—22no—
61C23×#95—22no—
62C23×#95—42no—
63C24×C23—21no—
64C24×C23C2×D411yescharacteristic
65C2×Q8×C23—22no—
66C2×Q8×C23—41no—
67C2×Q8×C23—41no—
68C2×D4×C23—22no—
69C2×D4×C23—41no—
70C2×D4×C23—41no—
71C2×D4×C23—41no—
72C2×D4×C23—41no—
73C2×D4×C23—42no—
74C2×D4×C23—42no—
75C22×C4×C23—22no—
76C22×C4×C23—22no—
77C22×C4×C23—22no—
78C22×C4×C23—22no—
79C22×C4×C23—24no—
80C22×C4×C23—21no—
81C22×C4×C23—21no—
82C22×C4×C23—21no—
83C22×C4×C23—41no—
84C22×C4×C23—41no—
85C22×C4×C23—41no—
86C22×C4×C23—41no—
87C22×C4×C23—42no—
88C22×C4×C23—42no—
89C22×C4×C23—42no—
90C22×C4×C23—42no—
91C22×C4×C23—42no—
92C22×C4×C23—42no—
93C22×C4×C23C22⋊C411yescharacteristic
94C22×C4×C23C22⋊C411yescharacteristic
95C22×C4×C23C22⋊C411yescharacteristic
96C22×C4×C23C22⋊C411yescharacteristic
97C22×C4×C23C22×C411yescharacteristic
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D412yes—
100C22×C4×C23#4112yes—
101C2×C8×C23—24no—
102C2×C8×C23—42no—
103C2×C8×C23—42no—
104C4⋊C4×C23—22no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23—42no—
107C22⋊C4×C23—42no—
108C22⋊C4×C23—42no—
109C22⋊C4×C23—41no—
110C22⋊C4×C23—41no—
111C22⋊C4×C23—42no—
112C22⋊C4×C23—42no—
113C22⋊C4×C23—42no—
114C42×C23—22no—
115C42×C23—22no—
116C42×C23—41no—
117C42×C23—41no—
118#9681C2311yesmaximal, characteristic, direct product factor, Sylow
119C23×C23—21no—
120C23×C23—21no—
121C23×C23—21no—
122C23×C23—21no—
123C23×C23—41no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23—41no—
128C23×C23—41no—
129C23×C23—41no—
130C23×C23C2×C22⋊C411yescharacteristic
131C23×C23#12411yescharacteristic
132C23×C23#12711yescharacteristic
133C23×C23#13512yes—
134C23×C23#13512yes—
135Q8×C23—82no—
136D4×C23—82no—
137D4×C23—82no—
138D4×C23—82no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—21no—
150C2×C4×C23—21no—
151C2×C4×C23—21no—
152C2×C4×C23—21no—
153C2×C4×C23—42no—
154C2×C4×C23—42no—
155C2×C4×C23—44no—
156C2×C4×C23—44no—
157C2×C4×C23—44no—
158C2×C4×C23—44no—
159C2×C4×C23—44no—
160C2×C4×C23—41no—
161C2×C4×C23—41no—
162C2×C4×C23—41no—
163C2×C4×C23—41no—
164C2×C4×C23—41no—
165C2×C4×C23—41no—
166C2×C4×C23—41no—
167C2×C4×C23—41no—
168C2×C4×C23—41no—
169C2×C4×C23—82no—
170C2×C4×C23—82no—
171C2×C4×C23—82no—
172C2×C4×C23—82no—
173C2×C4×C23#9912yes—
174C2×C4×C23#10412yes—
175C2×C4×C23C4×D412yes—
176C2×C4×C23#12112yes—
177C8×C23—44no—
178#2850C2×C2311yescharacteristic
179C4×C22⋊C8C2×C2311yescharacteristic, complemented
180C2×#328C2×C2312yescomplemented
181C2×#327C2×C2312yes—
182#1434C2×C2311yescharacteristic, complemented
183C22×C23—21no—
184C22×C23—21no—
185C22×C23—21no—
186C22×C23—21no—
187C22×C23—22no—
188C22×C23—22no—
189C22×C23—22no—
190C22×C23—22no—
191C22×C23—81no—
192C22×C23—81no—
193C22×C23—81no—
194C22×C23—81no—
195C22×C23—81no—
196C22×C23—81no—
197C22×C23—81no—
198C22×C23#39011yescharacteristic
199C22×C23C2×#9911yescharacteristic
200C22×C23#41011yescharacteristic
201C22×C23#41611yescharacteristic
202C22×C23#41611yescharacteristic
203C22×C23C2×#10411yescharacteristic
204C22×C23#42111yescharacteristic
205C22×C23#48812yes—
206C22×C23#49512yes—
207C4×C23—22no—
208C4×C23—22no—
209C4×C23—22no—
210C4×C23—22no—
211C4×C23—41no—
212C4×C23—41no—
213C4×C23—84no—
214C4×C23—81no—
215C4×C23—81no—
216C4×C23—81no—
217C2×#124C4×C2311yescharacteristic, complemented
218C2×C22⋊Q8C22×C2311yescharacteristic
219C2×#121C22×C2311yescharacteristic
220C22×C42C22×C2311yescharacteristic
221C2×C22⋊C8C22×C2312yes—
222C2×C4×C8—21no—
223#391C4×C2311yescharacteristic, complemented
224#390C4×C2311yescharacteristic, complemented
225#388—21no—
226#388—21no—
227#386—21no—
228#386—21no—
229#382C4×C2311yescharacteristic, complemented
230C2×#95—21no—
231C2×#95—21no—
232C2×#95C22×C2312yes—
233#336—21no—
234#328—24no—
235#327—24no—
236C2×C23—22no—
237C2×C23—22no—
238C2×C23—161no—
239C2×C23#150011yescharacteristic
240C2×C23#150111yescharacteristic
241C2×C23#151611yescharacteristic
242C2×C23#152111yescharacteristic
243C2×C23#195811yescharacteristic
244C2×C23#196411yescharacteristic
245C2×C23#198011yescharacteristic
246C22×Q8C2×C4×C2311yescharacteristic
247C22×D4—21no—
248C22×D4C2×C4×C2311yescharacteristic
249C23×C4—22no—
250C23×C4D4×C2311yescharacteristic
251C23×C4D4×C2311yescharacteristic
252C23×C4C23×C2311yescharacteristic
253C22×C8—22no—
254#124—42no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257#121—22no—
258#121—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C2×C4×C2311yescharacteristic
261C2×C4⋊C4C2×C4×C2311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—42no—
269C2×C42C2×C4×C2311yescharacteristic
270C2×C42C2×C4×C2311yescharacteristic
271C2×C42D4×C2311yescharacteristic
272C2×C42D4×C2311yescharacteristic
273C22⋊C8—22no—
274C22⋊C8D4×C2314yes—
275C4×C8—42no—
276#95—24no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95—42no—
284C23#968111yescharacteristic, direct product factor, Sylow
285C24—21no—
286C24C2×D4×C2311yescharacteristic
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×D4—22no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4C22⋊C4×C2311yescharacteristic
316C22×C4C22⋊C4×C2311yescharacteristic
317C22×C4C22⋊C4×C2311yescharacteristic
318C22×C4C22⋊C4×C2311yescharacteristic
319C22×C4C22×C4×C2311yescharacteristic
320C22×C4C2×D4×C2311yescharacteristic
321C22×C4C2×D4×C2312yes—
322C22×C4#41×C2312yes—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23C2×C22⋊C4×C2311yescharacteristic
352C23C23×#12411yescharacteristic
353C23C23×#12711yescharacteristic
354C23C23×#13512yes—
355C23C23×#13512yes—
356Q8—82no—
357D4—82no—
358D4—82no—
359D4—82no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4C23×#9912yes—
395C2×C4C23×#10412yes—
396C2×C4C4×D4×C2312yes—
397C2×C4C23×#12112yes—
398C8—44no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22C23×#39011yescharacteristic
415C22C2×C23×#9911yescharacteristic
416C22C23×#41011yescharacteristic
417C22C23×#41611yescharacteristic
418C22C23×#41611yescharacteristic
419C22C2×C23×#10411yescharacteristic
420C22C23×#42111yescharacteristic
421C22C23×#48812yes—
422C22C23×#49512yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—84no—
430C4—81no—
431C4—81no—
432C4—81no—
433C2—22no—
434C2—22no—
435C2—161no—
436C2C23×#150011yescharacteristic
437C2C23×#150111yescharacteristic
438C2C23×#151611yescharacteristic
439C2C23×#152111yescharacteristic
440C2C23×#195811yescharacteristic
441C2C23×#196411yescharacteristic
442C2C23×#198011yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC2×C23×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C23 ◁ C22×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C23×C22⋊C8C2semidirect
3. C2×C23×#328C2semidirect
5. C23×#1434C2semidirect
6. C2×C23×#124C4semidirect
12. C23×#391C4semidirect
13. C23×#390C4semidirect
18. C23×#382C4semidirect
118. #9681C23direct (N × H)
179. C4×C22⋊C8C2×C23semidirect
180. C2×#328C2×C23semidirect
182. #1434C2×C23semidirect
217. C2×#124C4×C23semidirect
223. #391C4×C23semidirect
224. #390C4×C23semidirect
229. #382C4×C23semidirect
284. C23#9681direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#28502
C4×C23×C22⋊C82
C2×C23×#3282
C2×C23×#3272
C23×#14342
#968123