Finite Group Databasebeta

← C23×#9116 #19156742

Subgroups — C23×#9116 #19156742

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C23×#432C211yesmaximal, characteristic
2C23×#2855C211yesmaximal, characteristic
3C2×C23×#328C212yesmaximal
4C23×#1434C211yesmaximal, characteristic
5C23×#1294C212yesmaximal
6C2×C23×C22⋊Q8C2211yescharacteristic
7C22×C42×C23C2211yescharacteristic
8C23×#432—24no—
9C2×C23×C4⋊C8—21no—
10C2×C23×C22⋊C8C2212yes—
11C2×C23×#97—21no—
12C23×#391C411yescharacteristic, complemented
13C23×#391C411yescharacteristic, complemented
14C23×#389C411yescharacteristic, complemented
15C23×#389C411yescharacteristic, complemented
16C23×#387—21no—
17C23×#387—21no—
18C23×#385—21no—
19C23×#381—21no—
20C23×#380C2211yescharacteristic
21C2×C23×#95—21no—
22C2×C23×#95—21no—
23C2×C23×#95C2212yes—
24C23×#328—24no—
25C22×Q8×C23C2×C411yescharacteristic
26C23×C4×C23—22no—
27C23×C4×C23D411yescharacteristic
28C23×C4×C23D411yescharacteristic
29C23×C4×C23C2311yescharacteristic
30C22×C8×C23—22no—
31C23×C22⋊Q8—22no—
32C23×C22⋊Q8—22no—
33C2×C4⋊C4×C23—21no—
34C2×C4⋊C4×C23—21no—
35C2×C4⋊C4×C23—21no—
36C2×C4⋊C4×C23C2×C411yescharacteristic
37C2×C4⋊C4×C23C2×C411yescharacteristic
38C2×C4⋊C4×C23C2×C411yescharacteristic
39C2×C22⋊C4×C23—21no—
40C2×C22⋊C4×C23—21no—
41C2×C42×C23—22no—
42C2×C42×C23—22no—
43C2×C42×C23C2×C411yescharacteristic
44C2×C42×C23C2×C411yescharacteristic
45C2×C42×C23D411yescharacteristic
46C2×C42×C23D411yescharacteristic
47C23×C4⋊C8—24no—
48C23×C22⋊C8—22no—
49C23×C22⋊C8D414yes—
50C23×#97—24no—
51C23×#95—24no—
52C23×#95—21no—
53C23×#95—21no—
54C23×#95—21no—
55C23×#95—21no—
56C23×#95—21no—
57C23×#95—21no—
58C23×#95—21no—
59C23×#95—22no—
60C23×#95—22no—
61C23×#95—42no—
62C24×C23C2×D411yescharacteristic
63C2×Q8×C23—22no—
64C2×Q8×C23—41no—
65C2×Q8×C23—41no—
66C22×C4×C23—22no—
67C22×C4×C23—22no—
68C22×C4×C23—21no—
69C22×C4×C23—21no—
70C22×C4×C23—22no—
71C22×C4×C23—22no—
72C22×C4×C23—24no—
73C22×C4×C23—21no—
74C22×C4×C23—21no—
75C22×C4×C23—21no—
76C22×C4×C23—21no—
77C22×C4×C23—41no—
78C22×C4×C23—41no—
79C22×C4×C23—41no—
80C22×C4×C23—42no—
81C22×C4×C23—42no—
82C22×C4×C23—42no—
83C22×C4×C23—42no—
84C22×C4×C23—42no—
85C22×C4×C23—42no—
86C22×C4×C23C22⋊C411yescharacteristic
87C22×C4×C23C22⋊C411yescharacteristic
88C22×C4×C23C22⋊C411yescharacteristic
89C22×C4×C23C22⋊C411yescharacteristic
90C22×C4×C23C22×C411yescharacteristic
91C22×C4×C23C2×D411yescharacteristic
92C22×C4×C23C2×D412yes—
93C22×C4×C23#4112yes—
94C2×C8×C23—24no—
95C2×C8×C23—42no—
96C2×C8×C23—42no—
97C4⋊C4×C23—22no—
98C4⋊C4×C23—41no—
99C4⋊C4×C23—41no—
100C4⋊C4×C23—42no—
101C4⋊C4×C23—42no—
102C4⋊C4×C23—41no—
103C4⋊C4×C23—41no—
104C22⋊C4×C23—42no—
105C22⋊C4×C23—41no—
106C22⋊C4×C23—41no—
107C42×C23—22no—
108C42×C23—22no—
109C42×C23—21no—
110C42×C23—21no—
111C42×C23—21no—
112C42×C23—21no—
113#9116C2311yesmaximal, characteristic, direct product factor, Sylow
114C23×C23—21no—
115C23×C23—21no—
116C23×C23—21no—
117C23×C23—21no—
118C23×C23C2×C22⋊C411yescharacteristic
119C23×C23#12112yes—
120C23×C23AGL(1,ℤ/8)11yescharacteristic
121C23×C23#13711yescharacteristic
122C23×C23#13712yes—
123Q8×C23—82no—
124C2×C4×C23—21no—
125C2×C4×C23—21no—
126C2×C4×C23—21no—
127C2×C4×C23—21no—
128C2×C4×C23—21no—
129C2×C4×C23—21no—
130C2×C4×C23—21no—
131C2×C4×C23—21no—
132C2×C4×C23—21no—
133C2×C4×C23—21no—
134C2×C4×C23—21no—
135C2×C4×C23—21no—
136C2×C4×C23—21no—
137C2×C4×C23—21no—
138C2×C4×C23—21no—
139C2×C4×C23—42no—
140C2×C4×C23—42no—
141C2×C4×C23—44no—
142C2×C4×C23—42no—
143C2×C4×C23—42no—
144C2×C4×C23—44no—
145C2×C4×C23—44no—
146C2×C4×C23—44no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—41no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—41no—
154C2×C4×C23—41no—
155C2×C4×C23—41no—
156C2×C4×C23—41no—
157C2×C4×C23—41no—
158C2×C4×C23—41no—
159C2×C4×C23—82no—
160C2×C4×C23—82no—
161C2×C4×C23—82no—
162C2×C4×C23—82no—
163C2×C4×C23#10412yes—
164C2×C4×C23C4×D412yes—
165C2×C4×C23#12011yescharacteristic
166C2×C4×C23#12311yescharacteristic
167C8×C23—44no—
168C2×#432C2×C2311yescharacteristic
169#2855C2×C2311yescharacteristic
170C2×#328C2×C2312yes—
171#1434C2×C2311yescharacteristic
172#1294C2×C2312yes—
173C22×C23—22no—
174C22×C23—22no—
175C22×C23—22no—
176C22×C23—21no—
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—21no—
180C22×C23—22no—
181C22×C23#38611yescharacteristic
182C22×C23#41011yescharacteristic
183C22×C23#41011yescharacteristic
184C22×C23#41711yescharacteristic
185C22×C23#41911yescharacteristic
186C22×C23C2×#10411yescharacteristic
187C22×C23#42111yescharacteristic
188C22×C23#46812yes—
189C22×C23#47012yes—
190C4×C23—22no—
191C4×C23—21no—
192C4×C23—22no—
193C4×C23—21no—
194C4×C23—41no—
195C4×C23—41no—
196C4×C23—41no—
197C4×C23—84no—
198C4×C23—81no—
199C4×C23—81no—
200C4×C23—81no—
201C4×C23—81no—
202C2×C22⋊Q8C22×C2311yescharacteristic
203C22×C42C22×C2311yescharacteristic
204#432—24no—
205C2×C4⋊C8—21no—
206C2×C22⋊C8C22×C2312yes—
207C2×#97—21no—
208#391C4×C2311yescharacteristic, complemented
209#391C4×C2311yescharacteristic, complemented
210#389C4×C2311yescharacteristic, complemented
211#389C4×C2311yescharacteristic, complemented
212#387—21no—
213#387—21no—
214#385—21no—
215#381—21no—
216#380C22×C2311yescharacteristic
217C2×#95—21no—
218C2×#95—21no—
219C2×#95C22×C2312yes—
220#328—24no—
221C2×C23—22no—
222C2×C23—22no—
223C2×C23#150611yescharacteristic
224C2×C23#150911yescharacteristic
225C2×C23#151211yescharacteristic
226C2×C23#151511yescharacteristic
227C2×C23#187111yescharacteristic
228C2×C23#189411yescharacteristic
229C2×C23#189911yescharacteristic
230C22×Q8C2×C4×C2311yescharacteristic
231C23×C4—22no—
232C23×C4D4×C2311yescharacteristic
233C23×C4D4×C2311yescharacteristic
234C23×C4C23×C2311yescharacteristic
235C22×C8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×C4×C2311yescharacteristic
242C2×C4⋊C4C2×C4×C2311yescharacteristic
243C2×C4⋊C4C2×C4×C2311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42C2×C4×C2311yescharacteristic
249C2×C42C2×C4×C2311yescharacteristic
250C2×C42D4×C2311yescharacteristic
251C2×C42D4×C2311yescharacteristic
252C4⋊C8—24no—
253C22⋊C8—22no—
254C22⋊C8D4×C2314yes—
255#97—24no—
256#95—24no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—22no—
266#95—42no—
267C23#911611yescharacteristic, direct product factor, Sylow
268C24C2×D4×C2311yescharacteristic
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—41no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C22⋊C4×C2311yescharacteristic
293C22×C4C22⋊C4×C2311yescharacteristic
294C22×C4C22⋊C4×C2311yescharacteristic
295C22×C4C22⋊C4×C2311yescharacteristic
296C22×C4C22×C4×C2311yescharacteristic
297C22×C4C2×D4×C2311yescharacteristic
298C22×C4C2×D4×C2312yes—
299C22×C4#41×C2312yes—
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C2×C22⋊C4×C2311yescharacteristic
324C23C23×#12112yes—
325C23C23×AGL(1,ℤ/8)11yescharacteristic
326C23C23×#13711yescharacteristic
327C23C23×#13712yes—
328Q8—82no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4C23×#10412yes—
369C2×C4C4×D4×C2312yes—
370C2×C4C23×#12011yescharacteristic
371C2×C4C23×#12311yescharacteristic
372C8—44no—
373C22—22no—
374C22—22no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—22no—
381C22C23×#38611yescharacteristic
382C22C23×#41011yescharacteristic
383C22C23×#41011yescharacteristic
384C22C23×#41711yescharacteristic
385C22C23×#41911yescharacteristic
386C22C2×C23×#10411yescharacteristic
387C22C23×#42111yescharacteristic
388C22C23×#46812yes—
389C22C23×#47012yes—
390C4—21no—
391C4—22no—
392C4—21no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—84no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C2—22no—
403C2—22no—
404C2C23×#150611yescharacteristic
405C2C23×#150911yescharacteristic
406C2C23×#151211yescharacteristic
407C2C23×#151511yescharacteristic
408C2C23×#187111yescharacteristic
409C2C23×#189411yescharacteristic
410C2C23×#189911yescharacteristic
subgroupG/H
center, socleC23×C23C2×C22⋊C4
commutatorC22×C4C22×C4×C23
Frattini, cosocleC23×C4C23×C23
prime radicalC24×C23C2×D4
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
12. C23×#391C4semidirect
13. C23×#391C4semidirect
14. C23×#389C4semidirect
15. C23×#389C4semidirect
113. #9116C23direct (N × H)
208. #391C4×C23semidirect
209. #391C4×C23semidirect
210. #389C4×C23semidirect
211. #389C4×C23semidirect
267. C23#9116direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C23×#4322
C23×#28552
C2×C23×#3282
C23×#14342
C23×#12942
#911623