Finite Group Databasebeta

← C23×#9160 #19156786

Subgroups — C23×#9160 #19156786

Order 5888.

450 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3116C211yesmaximal, characteristic
2C23×#2971C211yesmaximal, characteristic
3C23×#1759C211yesmaximal, characteristic
4C23×#1526C211yesmaximal, characteristic, complemented
5C23×#1506C211yesmaximal, characteristic, complemented
6C23×#1300C211yesmaximal, characteristic, complemented
7C23×#1291C211yesmaximal, characteristic, complemented
8C23×#533C2211yescharacteristic
9C4×#41×C23C2211yescharacteristic
10C23×#485—21no—
11C23×#477C411yescharacteristic, complemented
12C23×#476C411yescharacteristic, complemented
13C23×#462C411yescharacteristic, complemented
14C23×#461C411yescharacteristic, complemented
15C23×#452—21no—
16C8×Q8×C23—21no—
17C23×#441—21no—
18C23×#440—21no—
19C23×#436—22no—
20C23×#436—21no—
21C23×#432—21no—
22C23×#432C2211yescharacteristic
23C23×#432C2211yescharacteristic
24C2×C23×#104—21no—
25C4×M16×C23—21no—
26C4×M16×C23C2211yescharacteristic
27C4×M16×C23C2211yescharacteristic
28C23×#376C2211yescharacteristic
29C23×#339—21no—
30C23×#330—21no—
31C23×#330—21no—
32C23×#328—22no—
33C2×#41×C23D411yescharacteristic
34C2×Q16×C23—21no—
35C2×SD16×C23—21no—
36C2×M16×C23—22no—
37C2×M16×C23—21no—
38C2×M16×C23—21no—
39C23×C4⋊Q8C2×C411yescharacteristic
40C23×C4⋊Q8C2×C411yescharacteristic
41C23×#125—21no—
42C23×C22⋊Q8—21no—
43C23×C22⋊Q8—21no—
44C4×Q8×C23—21no—
45C4×Q8×C23—21no—
46C4×Q8×C23C2×C411yescharacteristic
47C4×Q8×C23D411yescharacteristic
48C4×D4×C23—21no—
49C4×D4×C23—21no—
50C4×D4×C23C2×C411yescharacteristic
51C4×D4×C23D411yescharacteristic
52C23×#117—21no—
53C23×#117—21no—
54C23×#117D411yescharacteristic
55C2×C42×C23—21no—
56C2×C42×C23—21no—
57C2×C42×C23C2311yescharacteristic
58C23×#107—21no—
59C23×#106—21no—
60C23×C4⋊C8—22no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8C2×C411yescharacteristic
63C23×C4⋊C8C2×C411yescharacteristic
64C23×#104—24no—
65C23×#103—21no—
66C23×#103—21no—
67C23×#103—21no—
68C23×#102—21no—
69C23×C22⋊C8—22no—
70C23×C22⋊C8—21no—
71C23×C22⋊C8D412yes—
72C23×#97—21no—
73C23×#97—21no—
74C23×#97—21no—
75C23×#97D411yescharacteristic
76C23×#97D411yescharacteristic
77C4×C8×C23—22no—
78C4×C8×C23—21no—
79C4×C8×C23—21no—
80C23×#95—21no—
81C23×#95—21no—
82#41×C23—22no—
83#41×C23—41no—
84C2×Q8×C23—21no—
85C2×Q8×C23—21no—
86C2×Q8×C23C22⋊C411yescharacteristic
87C2×D4×C23—21no—
88C2×D4×C23C22⋊C411yescharacteristic
89C22×C4×C23—21no—
90C22×C4×C23—21no—
91C22×C4×C23—21no—
92C22×C4×C23—41no—
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23C2×D411yescharacteristic
95C22×C4×C23#4111yescharacteristic
96Q16×C23—42no—
97SD16×C23—42no—
98M16×C23—22no—
99M16×C23—42no—
100M16×C23—44no—
101M16×C23—41no—
102C2×C8×C23—21no—
103C2×C8×C23—21no—
104C2×C8×C23—21no—
105C2×C8×C23—22no—
106C2×C8×C23—21no—
107C2×C8×C23—41no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23C22⋊C411yescharacteristic
117C4⋊C4×C23C22⋊C411yescharacteristic
118C22⋊C4×C23—21no—
119C22⋊C4×C23—21no—
120C22⋊C4×C23—21no—
121C22⋊C4×C23—41no—
122C42×C23—21no—
123C42×C23—21no—
124C42×C23—21no—
125C42×C23—22no—
126C42×C23—41no—
127C42×C23—41no—
128C42×C23C22×C411yescharacteristic
129C42×C23C2×D411yescharacteristic
130C42×C23C2×D411yescharacteristic
131C42×C23#4111yescharacteristic
132#9160C2311yesmaximal, characteristic, direct product factor, Sylow
133C23×C23—21no—
134C23×C23#12111yescharacteristic
135Q8×C23—21no—
136Q8×C23—41no—
137Q8×C23—41no—
138Q8×C23—81no—
139Q8×C23#10412yes—
140D4×C23—21no—
141D4×C23—21no—
142D4×C23—42no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—21no—
146C2×C4×C23—21no—
147C2×C4×C23—21no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—42no—
151C2×C4×C23—42no—
152C2×C4×C23—42no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—81no—
156C2×C4×C23—81no—
157C2×C4×C23—81no—
158C2×C4×C23C2×C22⋊C411yescharacteristic
159C2×C4×C23C4×D411yescharacteristic
160C2×C4×C23C4×D411yescharacteristic
161C2×C4×C23#12011yescharacteristic
162C2×C4×C23#12111yescharacteristic
163C2×C4×C23#12311yescharacteristic
164C8×C23—22no—
165C8×C23—42no—
166C8×C23—41no—
167C8×C23—42no—
168C8×C23—41no—
169C8×C23—41no—
170#3116C2×C2311yescharacteristic
171#2971C2×C2311yescharacteristic
172#1759C2×C2311yescharacteristic
173#1526C2×C2311yescharacteristic, complemented
174#1506C2×C2311yescharacteristic, complemented
175#1300C2×C2311yescharacteristic, complemented
176#1291C2×C2311yescharacteristic, complemented
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23—41no—
183C22×C23#38611yescharacteristic
184C4×C23—21no—
185C4×C23—22no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—21no—
189C4×C23—21no—
190C4×C23—41no—
191C4×C23—41no—
192C4×C23—82no—
193C4×C23—81no—
194C4×C23—81no—
195C4×C23C2×#10411yescharacteristic
196C4×C23#42111yescharacteristic
197#533C22×C2311yescharacteristic
198C4×#41C22×C2311yescharacteristic
199#485—21no—
200#477C4×C2311yescharacteristic, complemented
201#476C4×C2311yescharacteristic, complemented
202#462C4×C2311yescharacteristic, complemented
203#461C4×C2311yescharacteristic, complemented
204#452—21no—
205C8×Q8—21no—
206#441—21no—
207#440—21no—
208#436—21no—
209#436—22no—
210#432—21no—
211#432C22×C2311yescharacteristic
212#432C22×C2311yescharacteristic
213C2×#104—21no—
214C4×M16—21no—
215C4×M16C22×C2311yescharacteristic
216C4×M16C22×C2311yescharacteristic
217#376C22×C2311yescharacteristic
218#339—21no—
219#330—21no—
220#330—21no—
221#328—22no—
222C2×C23—41no—
223C2×C23—81no—
224C2×C23#187111yescharacteristic
225C2×C23#190211yescharacteristic
226C2×C23#190611yescharacteristic
227C2×#41D4×C2311yescharacteristic
228C2×Q16—21no—
229C2×SD16—21no—
230C2×M16—22no—
231C2×M16—21no—
232C2×M16—21no—
233C4⋊Q8C2×C4×C2311yescharacteristic
234C4⋊Q8C2×C4×C2311yescharacteristic
235#125—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C2×C4×C2311yescharacteristic
241C4×Q8D4×C2311yescharacteristic
242C4×D4—21no—
243C4×D4—21no—
244C4×D4C2×C4×C2311yescharacteristic
245C4×D4D4×C2311yescharacteristic
246#117—21no—
247#117—21no—
248#117D4×C2311yescharacteristic
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C23×C2311yescharacteristic
252#107—21no—
253#106—21no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8C2×C4×C2311yescharacteristic
257C4⋊C8C2×C4×C2311yescharacteristic
258#104—24no—
259#103—21no—
260#103—21no—
261#103—21no—
262#102—21no—
263C22⋊C8—22no—
264C22⋊C8—21no—
265C22⋊C8D4×C2312yes—
266#97—21no—
267#97—21no—
268#97—21no—
269#97D4×C2311yescharacteristic
270#97D4×C2311yescharacteristic
271C4×C8—22no—
272C4×C8—21no—
273C4×C8—21no—
274#95—21no—
275#95—21no—
276C23#916011yescharacteristic, direct product factor, Sylow
277#41—22no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C22⋊C4×C2311yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊C4×C2311yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4C2×D4×C2311yescharacteristic
289C22×C4C2×D4×C2311yescharacteristic
290C22×C4#41×C2311yescharacteristic
291Q16—42no—
292SD16—42no—
293M16—22no—
294M16—42no—
295M16—44no—
296M16—41no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊C4×C2311yescharacteristic
312C4⋊C4C22⋊C4×C2311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—41no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—41no—
322C42—41no—
323C42C22×C4×C2311yescharacteristic
324C42C2×D4×C2311yescharacteristic
325C42C2×D4×C2311yescharacteristic
326C42#41×C2311yescharacteristic
327C23—21no—
328C23C23×#12111yescharacteristic
329Q8—21no—
330Q8—41no—
331Q8—41no—
332Q8—81no—
333Q8C23×#10412yes—
334D4—21no—
335D4—21no—
336D4—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4C2×C22⋊C4×C2311yescharacteristic
353C2×C4C4×D4×C2311yescharacteristic
354C2×C4C4×D4×C2311yescharacteristic
355C2×C4C23×#12011yescharacteristic
356C2×C4C23×#12111yescharacteristic
357C2×C4C23×#12311yescharacteristic
358C8—22no—
359C8—42no—
360C8—41no—
361C8—42no—
362C8—41no—
363C8—41no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C23×#38611yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—82no—
380C4—81no—
381C4—81no—
382C4C2×C23×#10411yescharacteristic
383C4C23×#42111yescharacteristic
384C2—41no—
385C2—81no—
386C2C23×#187111yescharacteristic
387C2C23×#190211yescharacteristic
388C2C23×#190611yescharacteristic
subgroupG/H
center, socleC22×C23#386
commutatorC42C22×C4×C23
Frattini, cosocleC2×C42C23×C23
prime radicalC2×#41×C23D4
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
4. C23×#1526C2semidirect
5. C23×#1506C2semidirect
6. C23×#1300C2semidirect
7. C23×#1291C2semidirect
11. C23×#477C4semidirect
12. C23×#476C4semidirect
13. C23×#462C4semidirect
14. C23×#461C4semidirect
132. #9160C23direct (N × H)
173. #1526C2×C23semidirect
174. #1506C2×C23semidirect
175. #1300C2×C23semidirect
176. #1291C2×C23semidirect
200. #477C4×C23semidirect
201. #476C4×C23semidirect
202. #462C4×C23semidirect
203. #461C4×C23semidirect
276. C23#9160direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#31162
C23×#29712
C23×#17592
C23×#15262
C23×#15062
C23×#13002
C23×#12912
#916023