Finite Group Databasebeta

← C23×#9325 #19156951

Subgroups — C23×#9325 #19156951

Order 5888.

650 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3109C211yesmaximal, characteristic
2C23×#2275C211yesmaximal, characteristic, complemented
3C23×#1980C211yesmaximal, characteristic, complemented
4C23×#1505C211yesmaximal, characteristic, complemented
5C23×#1490C211yesmaximal, characteristic, complemented
6C23×#1300C211yesmaximal, characteristic, complemented
7C23×#1285C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C2×C4×D4×C23—21no—
10C2×C4×D4×C23C2211yescharacteristic
11C23×#481—21no—
12C23×#476C412yescomplemented
13C23×#459C412yescomplemented
14C23×#447—21no—
15C23×#442—22no—
16C23×#432C2211yescharacteristic
17C23×#432C2211yescharacteristic
18C2×C23×#104—22no—
19C4×M16×C23C2211yescharacteristic
20C4×C4⋊C4×C23—21no—
21C4×C4⋊C4×C23C2211yescharacteristic
22C4×C22⋊C4×C23—21no—
23C4×C22⋊C4×C23—22no—
24C23×#376C2211yescharacteristic
25C2×C23×#95—21no—
26C23×#327—21no—
27C23×#327—21no—
28C23×D4⋊C8—22no—
29C2×#41×C23—21no—
30C22×D4×C23D411yescharacteristic
31C23×C4×C23—22no—
32C23×C4×C23—21no—
33C2×D8×C23—22no—
34C2×M16×C23—21no—
35C23×C4⋊Q8C2×C411yescharacteristic, complemented
36C23×#127C2×C411yescharacteristic, complemented
37C23×#124—21no—
38C23×#121—21no—
39C23×#121—21no—
40C4×D4×C23—28no—
41C4×D4×C23—21no—
42C4×D4×C23—21no—
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C4×D4×C23C2×C412yes—
46C4×D4×C23D412yes—
47C2×C4⋊C4×C23—21no—
48C2×C4⋊C4×C23D411yescharacteristic
49C2×C22⋊C4×C23—22no—
50C2×C22⋊C4×C23—21no—
51C2×C42×C23—21no—
52C2×C42×C23—22no—
53C2×C42×C23D412yes—
54C2×C42×C23C2311yescharacteristic
55C23×#106—22no—
56C23×C4⋊C8—21no—
57C23×C4⋊C8C2×C412yescomplemented
58C23×#104—44no—
59C23×#102—22no—
60C23×#102—22no—
61C23×C22⋊C8—21no—
62C23×C22⋊C8—21no—
63C23×#97—21no—
64C23×#97—21no—
65C23×#97—21no—
66C4×C8×C23—21no—
67C23×#95—22no—
68C23×#95—21no—
69C23×#95—21no—
70C23×#95D411yescharacteristic
71C23×#95D411yescharacteristic
72C24×C23—21no—
73#41×C23—42no—
74#41×C23—42no—
75C2×Q8×C23—21no—
76C2×D4×C23—24no—
77C2×D4×C23—21no—
78C2×D4×C23—21no—
79C2×D4×C23—21no—
80C2×D4×C23—21no—
81C2×D4×C23—41no—
82C2×D4×C23C22⋊C412yes—
83C22×C4×C23—21no—
84C22×C4×C23—21no—
85C22×C4×C23—22no—
86C22×C4×C23—21no—
87C22×C4×C23—28no—
88C22×C4×C23—22no—
89C22×C4×C23—22no—
90C22×C4×C23—41no—
91C22×C4×C23—82no—
92C22×C4×C23—82no—
93C22×C4×C23—82no—
94C22×C4×C23C2×D411yescharacteristic
95C22×C4×C23C2×D411yescharacteristic
96C22×C4×C23C2×D411yescharacteristic
97D8×C23—44no—
98M16×C23—22no—
99M16×C23—41no—
100C2×C8×C23—22no—
101C2×C8×C23—41no—
102C2×C8×C23—41no—
103C4⋊C4×C23—24no—
104C4⋊C4×C23—21no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23C22⋊C412yes—
108C22⋊C4×C23—28no—
109C22⋊C4×C23—22no—
110C22⋊C4×C23—22no—
111C22⋊C4×C23—41no—
112C22⋊C4×C23—41no—
113C42×C23—24no—
114C42×C23—42no—
115C42×C23—82no—
116C42×C23C22×C411yescharacteristic
117C42×C23C2×D411yescharacteristic
118C42×C23#4111yescharacteristic
119C42×C23#4111yescharacteristic
120#9325C2311yesmaximal, characteristic, direct product factor, Sylow
121C23×C23—24no—
122C23×C23—22no—
123C23×C23—42no—
124C23×C23—41no—
125C23×C23—41no—
126C23×C23—41no—
127C23×C23#12011yescharacteristic
128Q8×C23—42no—
129D4×C23—24no—
130D4×C23—22no—
131D4×C23—42no—
132D4×C23—42no—
133D4×C23—81no—
134D4×C23—81no—
135D4×C23—81no—
136D4×C23#10414yescomplemented
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—22no—
140C2×C4×C23—22no—
141C2×C4×C23—22no—
142C2×C4×C23—24no—
143C2×C4×C23—22no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—42no—
147C2×C4×C23—42no—
148C2×C4×C23—41no—
149C2×C4×C23—42no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—82no—
154C2×C4×C23—82no—
155C2×C4×C23—82no—
156C2×C4×C23—88no—
157C2×C4×C23C2×C22⋊C411yescharacteristic
158C2×C4×C23C4×D411yescharacteristic
159C2×C4×C23C4×D411yescharacteristic
160C2×C4×C23#12111yescharacteristic
161C2×C4×C23#12111yescharacteristic
162C2×C4×C23#12311yescharacteristic
163C8×C23—42no—
164C8×C23—42no—
165C8×C23—81no—
166#3109C2×C2311yescharacteristic
167#2275C2×C2311yescharacteristic, complemented
168#1980C2×C2311yescharacteristic, complemented
169#1505C2×C2311yescharacteristic, complemented
170#1490C2×C2311yescharacteristic, complemented
171#1300C2×C2311yescharacteristic, complemented
172#1285C2×C2311yescharacteristic, complemented
173C22×C23—21no—
174C22×C23—21no—
175C22×C23—24no—
176C22×C23—44no—
177C22×C23—44no—
178C22×C23—42no—
179C22×C23—42no—
180C22×C23—81no—
181C22×C23—81no—
182C22×C23—81no—
183C22×C23#38611yescharacteristic
184C22×C23#45311yescharacteristic
185C22×C23#45411yescharacteristic
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—21no—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—42no—
192C4×C23—44no—
193C4×C23—42no—
194C4×C23—42no—
195C4×C23—81no—
196C4×C23C2×#10412yes—
197#532C22×C2311yescharacteristic
198C2×C4×D4—21no—
199C2×C4×D4C22×C2311yescharacteristic
200#481—21no—
201#476C4×C2312yescomplemented
202#459C4×C2312yescomplemented
203#447—21no—
204#442—22no—
205#432C22×C2311yescharacteristic
206#432C22×C2311yescharacteristic
207C2×#104—22no—
208C4×M16C22×C2311yescharacteristic
209C4×C4⋊C4—21no—
210C4×C4⋊C4C22×C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4—21no—
213#376C22×C2311yescharacteristic
214C2×#95—21no—
215#327—21no—
216#327—21no—
217D4⋊C8—22no—
218C2×C23—21no—
219C2×C23—21no—
220C2×C23—44no—
221C2×C23—161no—
222C2×C23#188311yescharacteristic
223C2×C23#189611yescharacteristic
224C2×C23#190411yescharacteristic
225C2×#41—21no—
226C22×D4D4×C2311yescharacteristic
227C23×C4—22no—
228C23×C4—21no—
229C2×D8—22no—
230C2×M16—21no—
231C4⋊Q8C2×C4×C2311yescharacteristic, complemented
232#127C2×C4×C2311yescharacteristic, complemented
233#124—21no—
234#121—21no—
235#121—21no—
236C4×D4—28no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C2×C4×C2312yes—
242C4×D4D4×C2312yes—
243C2×C4⋊C4—21no—
244C2×C4⋊C4D4×C2311yescharacteristic
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42—22no—
249C2×C42D4×C2312yes—
250C2×C42C23×C2311yescharacteristic
251#106—22no—
252C4⋊C8—21no—
253C4⋊C8C2×C4×C2312yescomplemented
254#104—44no—
255#102—22no—
256#102—22no—
257C22⋊C8—21no—
258C22⋊C8—21no—
259#97—21no—
260#97—21no—
261#97—21no—
262C4×C8—21no—
263#95—22no—
264#95—21no—
265#95—21no—
266#95D4×C2311yescharacteristic
267#95D4×C2311yescharacteristic
268C23#932511yescharacteristic, direct product factor, Sylow
269C24—21no—
270#41—42no—
271#41—42no—
272C2×Q8—21no—
273C2×D4—24no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4C22⋊C4×C2312yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×D4×C2311yescharacteristic
292C22×C4C2×D4×C2311yescharacteristic
293C22×C4C2×D4×C2311yescharacteristic
294D8—44no—
295M16—22no—
296M16—41no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C4⋊C4—24no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4C22⋊C4×C2312yes—
305C22⋊C4—28no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C42—24no—
311C42—42no—
312C42—82no—
313C42C22×C4×C2311yescharacteristic
314C42C2×D4×C2311yescharacteristic
315C42#41×C2311yescharacteristic
316C42#41×C2311yescharacteristic
317C23—24no—
318C23—22no—
319C23—42no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23C23×#12011yescharacteristic
324Q8—42no—
325D4—24no—
326D4—22no—
327D4—42no—
328D4—42no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4C23×#10414yescomplemented
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—88no—
353C2×C4C2×C22⋊C4×C2311yescharacteristic
354C2×C4C4×D4×C2311yescharacteristic
355C2×C4C4×D4×C2311yescharacteristic
356C2×C4C23×#12111yescharacteristic
357C2×C4C23×#12111yescharacteristic
358C2×C4C23×#12311yescharacteristic
359C8—42no—
360C8—42no—
361C8—81no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C23×#38611yescharacteristic
373C22C23×#45311yescharacteristic
374C22C23×#45411yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—44no—
382C4—42no—
383C4—42no—
384C4—81no—
385C4C2×C23×#10412yes—
386C2—21no—
387C2—21no—
388C2—44no—
389C2—161no—
390C2C23×#188311yescharacteristic
391C2C23×#189611yescharacteristic
392C2C23×#190411yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3109C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C23 ◁ C2×C42×C23 ◁ G

Complements

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

NH ≅ G/Ntype
2. C23×#2275C2semidirect
3. C23×#1980C2semidirect
4. C23×#1505C2semidirect
5. C23×#1490C2semidirect
6. C23×#1300C2semidirect
7. C23×#1285C2semidirect
12. C23×#476C4semidirect
13. C23×#459C4semidirect
35. C23×C4⋊Q8C2×C4semidirect
36. C23×#127C2×C4semidirect
57. C23×C4⋊C8C2×C4semidirect
120. #9325C23direct (N × H)
136. D4×C23#104semidirect
167. #2275C2×C23semidirect
168. #1980C2×C23semidirect
169. #1505C2×C23semidirect
170. #1490C2×C23semidirect
171. #1300C2×C23semidirect
172. #1285C2×C23semidirect
201. #476C4×C23semidirect
202. #459C4×C23semidirect
231. C4⋊Q8C2×C4×C23semidirect
232. #127C2×C4×C23semidirect
253. C4⋊C8C2×C4×C23semidirect
268. C23#9325direct (N × H)
332. D4C23×#104semidirect

Maximal subgroups

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

subgroupindex
C23×#31092
C23×#22752
C23×#19802
C23×#15052
C23×#14902
C23×#13002
C23×#12852
#932523