Finite Group Databasebeta

← C23×#10206 #19157832

Subgroups — C23×#10206 #19157832

Order 5888.

560 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3246C211yesmaximal, characteristic
2C23×#3229C211yesmaximal, characteristic, complemented
3C23×#3152C211yesmaximal, characteristic
4C23×#1988C211yesmaximal, characteristic, complemented
5C23×#1986C211yesmaximal, characteristic, complemented
6C23×#1502C211yesmaximal, characteristic
7C23×#1283C211yesmaximal, characteristic, complemented
8C2×C23×#137—22no—
9C23×#533C2211yescharacteristic
10C23×#532C2211yescharacteristic
11C2×C23×C4⋊Q8C2211yescharacteristic
12C23×#498—21no—
13C23×#497—22no—
14C23×#494—21no—
15C23×#492—21no—
16C23×#487—21no—
17C23×#483—22no—
18C23×#476—21no—
19C23×#475—22no—
20C23×#475—21no—
21C23×#465—22no—
22C23×#461—21no—
23C23×#460—21no—
24C23×#431C2211yescharacteristic
25C2×C23×C4⋊C8C2211yescharacteristic
26C4×M16×C23C2211yescharacteristic
27C4×C4⋊C4×C23C2211yescharacteristic
28C23×#340—22no—
29C23×#339—22no—
30C23×#327—22no—
31C2×#41×C23—21no—
32C22×Q8×C23—21no—
33C23×#137—44no—
34C23×#137—44no—
35C2×Q16×C23—24no—
36C2×SD16×C23—24no—
37C2×SD16×C23—22no—
38C2×M16×C23D412yes—
39C22×C8×C23D412yes—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8—21no—
42C23×C4⋊Q8—21no—
43C23×C4⋊Q8D411yescharacteristic
44C23×C4⋊Q8D411yescharacteristic
45C23×C4⋊Q8D412yes—
46C23×C4⋊Q8D411yescharacteristic
47C23×#127D411yescharacteristic
48C23×#125—21no—
49C23×#124—21no—
50C23×C22⋊Q8—21no—
51C23×C22⋊Q8—21no—
52C23×#121—21no—
53C23×#121—21no—
54C4×Q8×C23—21no—
55C4×Q8×C23—21no—
56C4×D4×C23—21no—
57C4×D4×C23—21no—
58C23×#117—21no—
59C2×C4⋊C4×C23—21no—
60C2×C4⋊C4×C23—21no—
61C2×C4⋊C4×C23—21no—
62C2×C42×C23—21no—
63C2×C42×C23C2311yescharacteristic
64C23×#106—22no—
65C23×#106—24no—
66C23×C4⋊C8—21no—
67C23×C4⋊C8—21no—
68C23×C4⋊C8—21no—
69C23×#103—22no—
70C23×#103—24no—
71C23×#102—22no—
72C23×C22⋊C8D412yes—
73C23×#97—21no—
74C4×C8×C23—21no—
75C4×C8×C23—21no—
76C23×#95—21no—
77#41×C23—42no—
78#41×C23—42no—
79C2×Q8×C23—21no—
80C2×Q8×C23—21no—
81C2×Q8×C23—21no—
82C2×Q8×C23—22no—
83C2×Q8×C23—41no—
84C2×Q8×C23—42no—
85C2×Q8×C23—42no—
86C2×D4×C23—21no—
87C2×D4×C23—21no—
88C2×D4×C23—21no—
89C2×D4×C23—41no—
90C22×C4×C23—41no—
91C22×C4×C23—41no—
92C22×C4×C23—41no—
93C22×C4×C23C2×D411yescharacteristic
94C22×C4×C23C2×D411yescharacteristic
95C22×C4×C23C2×D411yescharacteristic
96Q16×C23—48no—
97SD16×C23—48no—
98SD16×C23—82no—
99M16×C23—22no—
100M16×C23SD1614yescomplemented
101C2×C8×C23—22no—
102C2×C8×C23—22no—
103C2×C8×C23—22no—
104C2×C8×C23—42no—
105C4⋊C4×C23—22no—
106C4⋊C4×C23—21no—
107C4⋊C4×C23—22no—
108C4⋊C4×C23—21no—
109C4⋊C4×C23—21no—
110C4⋊C4×C23—22no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—21no—
114C4⋊C4×C23—41no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C4⋊C4×C23—41no—
119C22⋊C4×C23—41no—
120C22⋊C4×C23—41no—
121C22⋊C4×C23—41no—
122C42×C23—41no—
123C42×C23—42no—
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126C42×C23C2×D411yescharacteristic
127C42×C23#4111yescharacteristic
128#10206C2311yesmaximal, characteristic, direct product factor, Sylow
129C23×C23—41no—
130C23×C23#12711yescharacteristic
131Q8×C23—42no—
132Q8×C23—44no—
133Q8×C23—42no—
134Q8×C23—41no—
135Q8×C23—41no—
136Q8×C23—82no—
137D4×C23—42no—
138D4×C23—41no—
139D4×C23—41no—
140D4×C23—81no—
141D4×C23—81no—
142D4×C23—81no—
143C2×C4×C23—21no—
144C2×C4×C23—21no—
145C2×C4×C23—41no—
146C2×C4×C23—42no—
147C2×C4×C23—41no—
148C2×C4×C23—41no—
149C2×C4×C23—41no—
150C2×C4×C23—42no—
151C2×C4×C23—81no—
152C2×C4×C23—81no—
153C2×C4×C23—81no—
154C2×C4×C23—81no—
155C2×C4×C23—82no—
156C2×C4×C23#12011yescharacteristic
157C2×C4×C23#12011yescharacteristic
158C2×C4×C23#12011yescharacteristic
159C2×C4×C23#12111yescharacteristic
160C2×C4×C23#12111yescharacteristic
161C2×C4×C23#12111yescharacteristic
162C2×C4×C23C2×SD1612yes—
163C2×C4×C23AGL(1,ℤ/8)12yes—
164C8×C23—24no—
165C8×C23—42no—
166C8×C23—42no—
167#3246C2×C2311yescharacteristic
168#3229C2×C2311yescharacteristic, complemented
169#3152C2×C2311yescharacteristic
170#1988C2×C2311yescharacteristic, complemented
171#1986C2×C2311yescharacteristic, complemented
172#1502C2×C2311yescharacteristic
173#1283C2×C2311yescharacteristic, complemented
174C22×C23—21no—
175C22×C23—21no—
176C22×C23—81no—
177C22×C23—81no—
178C22×C23—81no—
179C22×C23#39211yescharacteristic
180C22×C23#49211yescharacteristic
181C22×C23#49611yescharacteristic
182C4×C23—21no—
183C4×C23—21no—
184C4×C23—21no—
185C4×C23—82no—
186C4×C23—81no—
187C4×C23—81no—
188C4×C23—81no—
189C4×C23—82no—
190C4×C23#45012yes—
191C4×C23#45511yescharacteristic
192C4×C23#45611yescharacteristic
193C4×C23#46012yes—
194C2×#137—22no—
195#533C22×C2311yescharacteristic
196#532C22×C2311yescharacteristic
197C2×C4⋊Q8C22×C2311yescharacteristic
198#498—21no—
199#497—22no—
200#494—21no—
201#492—21no—
202#487—21no—
203#483—22no—
204#476—21no—
205#475—22no—
206#475—21no—
207#465—22no—
208#461—21no—
209#460—21no—
210#431C22×C2311yescharacteristic
211C2×C4⋊C8C22×C2311yescharacteristic
212C4×M16C22×C2311yescharacteristic
213C4×C4⋊C4C22×C2311yescharacteristic
214#340—22no—
215#339—22no—
216#327—22no—
217C2×C23—21no—
218C2×C23—21no—
219C2×C23—161no—
220C2×C23#200711yescharacteristic
221C2×C23#200911yescharacteristic
222C2×C23#201411yescharacteristic
223C2×#41—21no—
224C22×Q8—21no—
225#137—44no—
226#137—44no—
227C2×Q16—24no—
228C2×SD16—24no—
229C2×SD16—22no—
230C2×M16D4×C2312yes—
231C22×C8D4×C2312yes—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8D4×C2312yes—
236C4⋊Q8D4×C2311yescharacteristic
237C4⋊Q8D4×C2311yescharacteristic
238C4⋊Q8D4×C2311yescharacteristic
239#127D4×C2311yescharacteristic
240#125—21no—
241#124—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244#121—21no—
245#121—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×D4—21no—
249C4×D4—21no—
250#117—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C42—21no—
255C2×C42C23×C2311yescharacteristic
256#106—22no—
257#106—24no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261#103—22no—
262#103—24no—
263#102—22no—
264C22⋊C8D4×C2312yes—
265#97—21no—
266C4×C8—21no—
267C4×C8—21no—
268#95—21no—
269C23#1020611yescharacteristic, direct product factor, Sylow
270#41—42no—
271#41—42no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C2×D4×C2311yescharacteristic
287C22×C4C2×D4×C2311yescharacteristic
288C22×C4C2×D4×C2311yescharacteristic
289Q16—48no—
290SD16—48no—
291SD16—82no—
292M16—22no—
293M16SD16×C2314yescomplemented
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—42no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—42no—
317C42C2×D4×C2311yescharacteristic
318C42C2×D4×C2311yescharacteristic
319C42C2×D4×C2311yescharacteristic
320C42#41×C2311yescharacteristic
321C23—41no—
322C23C23×#12711yescharacteristic
323Q8—42no—
324Q8—44no—
325Q8—42no—
326Q8—41no—
327Q8—41no—
328Q8—82no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334D4—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4C23×#12011yescharacteristic
349C2×C4C23×#12011yescharacteristic
350C2×C4C23×#12011yescharacteristic
351C2×C4C23×#12111yescharacteristic
352C2×C4C23×#12111yescharacteristic
353C2×C4C23×#12111yescharacteristic
354C2×C4C2×SD16×C2312yes—
355C2×C4C23×AGL(1,ℤ/8)12yes—
356C8—24no—
357C8—42no—
358C8—42no—
359C22—21no—
360C22—21no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22C23×#39211yescharacteristic
365C22C23×#49211yescharacteristic
366C22C23×#49611yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—82no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—82no—
375C4C23×#45012yes—
376C4C23×#45511yescharacteristic
377C4C23×#45611yescharacteristic
378C4C23×#46012yes—
379C2—21no—
380C2—21no—
381C2—161no—
382C2C23×#200711yescharacteristic
383C2C23×#200911yescharacteristic
384C2C23×#201411yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#532C22
LCSG ▷ C2×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×#3229C2semidirect
4. C23×#1988C2semidirect
5. C23×#1986C2semidirect
7. C23×#1283C2semidirect
100. M16×C23SD16semidirect
128. #10206C23direct (N × H)
168. #3229C2×C23semidirect
170. #1988C2×C23semidirect
171. #1986C2×C23semidirect
173. #1283C2×C23semidirect
269. C23#10206direct (N × H)
293. M16SD16×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#32462
C23×#32292
C23×#31522
C23×#19882
C23×#19862
C23×#15022
C23×#12832
#1020623