Finite Group Databasebeta

← C23×#9321 #19156947

Subgroups — C23×#9321 #19156947

Order 5888.

474 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3079C211yesmaximal, characteristic
2C23×#2989C211yesmaximal, characteristic
3C23×#1981C211yesmaximal, characteristic, complemented
4C23×#1806C211yesmaximal, characteristic
5C23×#1549C211yesmaximal, characteristic, complemented
6C23×#1303C211yesmaximal, characteristic, complemented
7C23×#1292C211yesmaximal, characteristic, complemented
8C2×C23×C4⋊Q8C2211yescharacteristic
9C4×#41×C23C2211yescharacteristic
10C23×#477C412yescomplemented
11C23×#476C412yescomplemented
12C23×#475—21no—
13C23×#474—21no—
14C23×#446—21no—
15C23×#436—21no—
16C23×#435—22no—
17C23×#432—21no—
18C23×#431C2211yescharacteristic
19C23×#428—21no—
20C23×#423—21no—
21C23×#423C2211yescharacteristic
22C23×#423C2211yescharacteristic
23C2×C23×C4⋊C8C2211yescharacteristic
24C2×C23×#104—21no—
25C23×#417—21no—
26C2×C23×#103—22no—
27C4×C4⋊C4×C23C2211yescharacteristic
28C23×#339—21no—
29C23×#332—21no—
30C23×#332—21no—
31C2×#41×C23D411yescharacteristic
32C22×Q8×C23—21no—
33C2×M16×C23—21no—
34C2×M16×C23—22no—
35C2×M16×C23—21no—
36C22×C8×C23D412yes—
37C23×C4⋊Q8—21no—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8C2×C412yes—
41C4×Q8×C23C2×C411yescharacteristic
42C4×Q8×C23D411yescharacteristic
43C4×D4×C23—21no—
44C4×D4×C23—21no—
45C4×D4×C23C2×C411yescharacteristic
46C4×D4×C23D411yescharacteristic
47C23×#117—21no—
48C23×#117D411yescharacteristic
49C2×C4⋊C4×C23—21no—
50C2×C4⋊C4×C23—21no—
51C2×C4⋊C4×C23—21no—
52C2×C42×C23—21no—
53C2×C42×C23—21no—
54C2×C42×C23C2311yescharacteristic
55C23×#107—22no—
56C23×#106—22no—
57C23×C4⋊C8—22no—
58C23×C4⋊C8—21no—
59C23×C4⋊C8—21no—
60C23×C4⋊C8—21no—
61C23×C4⋊C8—21no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8C2×C412yes—
64C23×C4⋊C8D411yescharacteristic
65C23×C4⋊C8D411yescharacteristic
66C23×#104—42no—
67C23×#103—22no—
68C23×#103—44no—
69C23×#102—22no—
70C23×C22⋊C8—22no—
71C23×C22⋊C8—21no—
72C23×#97—22no—
73C4×C8×C23—21no—
74C23×#95—21no—
75#41×C23—21no—
76#41×C23—21no—
77#41×C23—21no—
78#41×C23—21no—
79C2×Q8×C23—22no—
80C2×Q8×C23—42no—
81C2×Q8×C23—42no—
82C2×Q8×C23—41no—
83C2×Q8×C23C22⋊C411yescharacteristic
84C2×D4×C23—21no—
85C2×D4×C23C22⋊C411yescharacteristic
86C22×C4×C23—21no—
87C22×C4×C23—21no—
88C22×C4×C23—21no—
89C22×C4×C23—41no—
90C22×C4×C23—41no—
91C22×C4×C23C2×D411yescharacteristic
92C22×C4×C23C2×D411yescharacteristic
93C22×C4×C23#4111yescharacteristic
94M16×C23—41no—
95M16×C23—41no—
96M16×C23—41no—
97M16×C23—41no—
98M16×C23—82no—
99C2×C8×C23—22no—
100C2×C8×C23—22no—
101C2×C8×C23—21no—
102C2×C8×C23—21no—
103C2×C8×C23—42no—
104C2×C8×C23—41no—
105C4⋊C4×C23—21no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—41no—
111C4⋊C4×C23—41no—
112C4⋊C4×C23—41no—
113C4⋊C4×C23C22⋊C411yescharacteristic
114C4⋊C4×C23C22⋊C411yescharacteristic
115C22⋊C4×C23—21no—
116C22⋊C4×C23—21no—
117C22⋊C4×C23—21no—
118C42×C23—21no—
119C42×C23—21no—
120C42×C23—21no—
121C42×C23—41no—
122C42×C23—41no—
123C42×C23C22×C411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126C42×C23#4111yescharacteristic
127#9321C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—21no—
129C23×C23#12111yescharacteristic
130Q8×C23—21no—
131Q8×C23—21no—
132Q8×C23—44no—
133Q8×C23—81no—
134Q8×C23—81no—
135D4×C23—21no—
136D4×C23—21no—
137D4×C23—41no—
138D4×C23—41no—
139C2×C4×C23—21no—
140C2×C4×C23—21no—
141C2×C4×C23—21no—
142C2×C4×C23—21no—
143C2×C4×C23—21no—
144C2×C4×C23—41no—
145C2×C4×C23—41no—
146C2×C4×C23—41no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—42no—
150C2×C4×C23—42no—
151C2×C4×C23—81no—
152C2×C4×C23—81no—
153C2×C4×C23—82no—
154C2×C4×C23—81no—
155C2×C4×C23C2×C22⋊C411yescharacteristic
156C2×C4×C23C4×D411yescharacteristic
157C2×C4×C23C4×D411yescharacteristic
158C2×C4×C23#12011yescharacteristic
159C2×C4×C23#12111yescharacteristic
160C2×C4×C23#12311yescharacteristic
161C2×C4×C23AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C23#13711yescharacteristic
163C2×C4×C23#13711yescharacteristic
164C2×C4×C23#13711yescharacteristic
165C8×C23—42no—
166C8×C23—42no—
167C8×C23—41no—
168C8×C23—41no—
169C8×C23—81no—
170#3079C2×C2311yescharacteristic
171#2989C2×C2311yescharacteristic
172#1981C2×C2311yescharacteristic, complemented
173#1806C2×C2311yescharacteristic
174#1549C2×C2311yescharacteristic, complemented
175#1303C2×C2311yescharacteristic, complemented
176#1292C2×C2311yescharacteristic, complemented
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—41no—
180C22×C23—41no—
181C22×C23—41no—
182C22×C23#38611yescharacteristic
183C22×C23#46811yescharacteristic
184C22×C23#47011yescharacteristic
185C4×C23—21no—
186C4×C23—21no—
187C4×C23—21no—
188C4×C23—41no—
189C4×C23—41no—
190C4×C23—41no—
191C4×C23—82no—
192C4×C23—82no—
193C4×C23#41711yescharacteristic
194C4×C23#41911yescharacteristic
195C4×C23#45311yescharacteristic
196C4×C23#45411yescharacteristic
197C4×C23#46811yescharacteristic
198C4×C23#47011yescharacteristic
199C2×C4⋊Q8C22×C2311yescharacteristic
200C4×#41C22×C2311yescharacteristic
201#477C4×C2312yescomplemented
202#476C4×C2312yescomplemented
203#475—21no—
204#474—21no—
205#446—21no—
206#436—21no—
207#435—22no—
208#432—21no—
209#431C22×C2311yescharacteristic
210#428—21no—
211#423—21no—
212#423C22×C2311yescharacteristic
213#423C22×C2311yescharacteristic
214C2×C4⋊C8C22×C2311yescharacteristic
215C2×#104—21no—
216#417—21no—
217C2×#103—22no—
218C4×C4⋊C4C22×C2311yescharacteristic
219#339—21no—
220#332—21no—
221#332—21no—
222C2×C23—21no—
223C2×C23—21no—
224C2×C23—81no—
225C2×C23#188311yescharacteristic
226C2×C23#189411yescharacteristic
227C2×C23#190211yescharacteristic
228C2×#41D4×C2311yescharacteristic
229C22×Q8—21no—
230C2×M16—21no—
231C2×M16—22no—
232C2×M16—21no—
233C22×C8D4×C2312yes—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8C2×C4×C2312yes—
238C4×Q8C2×C4×C2311yescharacteristic
239C4×Q8D4×C2311yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C2×C4×C2311yescharacteristic
243C4×D4D4×C2311yescharacteristic
244#117—21no—
245#117D4×C2311yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C23×C2311yescharacteristic
252#107—22no—
253#106—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8C2×C4×C2312yes—
261C4⋊C8D4×C2311yescharacteristic
262C4⋊C8D4×C2311yescharacteristic
263#104—42no—
264#103—22no—
265#103—44no—
266#102—22no—
267C22⋊C8—22no—
268C22⋊C8—21no—
269#97—22no—
270C4×C8—21no—
271#95—21no—
272C23#932111yescharacteristic, direct product factor, Sylow
273#41—21no—
274#41—21no—
275#41—21no—
276#41—21no—
277C2×Q8—22no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8C22⋊C4×C2311yescharacteristic
282C2×D4—21no—
283C2×D4C22⋊C4×C2311yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×D4×C2311yescharacteristic
290C22×C4C2×D4×C2311yescharacteristic
291C22×C4#41×C2311yescharacteristic
292M16—41no—
293M16—41no—
294M16—41no—
295M16—41no—
296M16—82no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4C22⋊C4×C2311yescharacteristic
312C4⋊C4C22⋊C4×C2311yescharacteristic
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—41no—
320C42—41no—
321C42C22×C4×C2311yescharacteristic
322C42C2×D4×C2311yescharacteristic
323C42C2×D4×C2311yescharacteristic
324C42#41×C2311yescharacteristic
325C23—21no—
326C23C23×#12111yescharacteristic
327Q8—21no—
328Q8—21no—
329Q8—44no—
330Q8—81no—
331Q8—81no—
332D4—21no—
333D4—21no—
334D4—41no—
335D4—41no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—81no—
349C2×C4—82no—
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
358C2×C4C23×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C23×#13711yescharacteristic
360C2×C4C23×#13711yescharacteristic
361C2×C4C23×#13711yescharacteristic
362C8—42no—
363C8—42no—
364C8—41no—
365C8—41no—
366C8—81no—
367C22—21no—
368C22—21no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22C23×#38611yescharacteristic
373C22C23×#46811yescharacteristic
374C22C23×#47011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4—82no—
383C4C23×#41711yescharacteristic
384C4C23×#41911yescharacteristic
385C4C23×#45311yescharacteristic
386C4C23×#45411yescharacteristic
387C4C23×#46811yescharacteristic
388C4C23×#47011yescharacteristic
389C2—21no—
390C2—21no—
391C2—81no—
392C2C23×#188311yescharacteristic
393C2C23×#189411yescharacteristic
394C2C23×#190211yescharacteristic
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
3. C23×#1981C2semidirect
5. C23×#1549C2semidirect
6. C23×#1303C2semidirect
7. C23×#1292C2semidirect
10. C23×#477C4semidirect
11. C23×#476C4semidirect
127. #9321C23direct (N × H)
172. #1981C2×C23semidirect
174. #1549C2×C23semidirect
175. #1303C2×C23semidirect
176. #1292C2×C23semidirect
201. #477C4×C23semidirect
202. #476C4×C23semidirect
272. C23#9321direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#30792
C23×#29892
C23×#19812
C23×#18062
C23×#15492
C23×#13032
C23×#12922
#932123