Finite Group Databasebeta

← C23×#10196 #19157822

Subgroups — C23×#10196 #19157822

Order 5888.

546 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#3245C211yesmaximal, characteristic
2C23×#3206C211yesmaximal, characteristic, complemented
3C23×#3141C211yesmaximal, characteristic, complemented
4C23×#1987C211yesmaximal, characteristic, complemented
5C23×#1980C211yesmaximal, characteristic, complemented
6C23×#1502C211yesmaximal, characteristic, complemented
7C23×#1283C211yesmaximal, characteristic, complemented
8C23×#532C2211yescharacteristic
9C23×#532C2211yescharacteristic
10C2×C23×C4⋊Q8C2211yescharacteristic
11C23×#492—21no—
12C23×#489—22no—
13C23×#487—21no—
14C23×#487—21no—
15C23×#486—21no—
16C23×#474—21no—
17C23×#474—21no—
18C23×#466—22no—
19C23×#461—22no—
20C23×#459—21no—
21C23×#459—21no—
22C23×#448—22no—
23C23×#431C2211yescharacteristic
24C2×C23×C4⋊C8C2211yescharacteristic
25C2×C23×#104—22no—
26C23×#417—22no—
27C4×M16×C23C2211yescharacteristic
28C4×C4⋊C4×C23C2211yescharacteristic
29C23×#340—22no—
30C23×#327—22no—
31C2×#41×C23—21no—
32C2×#41×C23—21no—
33C22×Q8×C23—21no—
34C2×SD16×C23—24no—
35C2×D8×C23—22no—
36C2×M16×C23D412yes—
37C22×C8×C23D412yes—
38C23×C4⋊Q8—21no—
39C23×C4⋊Q8—21no—
40C23×C4⋊Q8—21no—
41C23×C4⋊Q8D412yes—
42C23×C4⋊Q8D411yescharacteristic
43C23×C4⋊Q8D411yescharacteristic
44C23×#127D411yescharacteristic
45C23×#127D411yescharacteristic
46C23×#124—21no—
47C23×#124—21no—
48C23×#121—21no—
49C23×#121—21no—
50C23×#121—21no—
51C23×#121—21no—
52C4×D4×C23—21no—
53C4×D4×C23—21no—
54C4×D4×C23—21no—
55C4×D4×C23—21no—
56C2×C4⋊C4×C23—21no—
57C2×C4⋊C4×C23—21no—
58C2×C4⋊C4×C23—21no—
59C2×C42×C23—21no—
60C2×C42×C23C2311yescharacteristic
61C23×#107—22no—
62C23×C4⋊C8—21no—
63C23×C4⋊C8—21no—
64C23×C4⋊C8—21no—
65C23×#104—44no—
66C23×#103—24no—
67C23×#103—24no—
68C23×#102—24no—
69C23×#102—22no—
70C23×#102—22no—
71C23×C22⋊C8D412yes—
72C23×#97—21no—
73C4×C8×C23—21no—
74C4×C8×C23—21no—
75C23×#95—21no—
76#41×C23—42no—
77#41×C23—42no—
78#41×C23—42no—
79#41×C23—42no—
80C2×Q8×C23—21no—
81C2×Q8×C23—22no—
82C2×Q8×C23—21no—
83C2×Q8×C23—41no—
84C2×Q8×C23—82no—
85C2×D4×C23—21no—
86C2×D4×C23—21no—
87C2×D4×C23—21no—
88C2×D4×C23—21no—
89C2×D4×C23—21no—
90C2×D4×C23—21no—
91C2×D4×C23—41no—
92C2×D4×C23—41no—
93C22×C4×C23—41no—
94C22×C4×C23—41no—
95C22×C4×C23—41no—
96C22×C4×C23—41no—
97C22×C4×C23C2×D411yescharacteristic
98C22×C4×C23C2×D411yescharacteristic
99C22×C4×C23C2×D411yescharacteristic
100SD16×C23—84no—
101D8×C23—82no—
102M16×C23—22no—
103M16×C23—22no—
104C2×C8×C23—22no—
105C2×C8×C23—22no—
106C2×C8×C23—22no—
107C2×C8×C23—42no—
108C4⋊C4×C23—22no—
109C4⋊C4×C23—22no—
110C4⋊C4×C23—21no—
111C4⋊C4×C23—21no—
112C4⋊C4×C23—21no—
113C4⋊C4×C23—22no—
114C4⋊C4×C23—21no—
115C4⋊C4×C23—41no—
116C4⋊C4×C23—41no—
117C4⋊C4×C23—41no—
118C22⋊C4×C23—41no—
119C22⋊C4×C23—41no—
120C22⋊C4×C23—41no—
121C22⋊C4×C23—41no—
122C42×C23—42no—
123C42×C23C2×D411yescharacteristic
124C42×C23C2×D411yescharacteristic
125C42×C23C2×D411yescharacteristic
126C42×C23#4111yescharacteristic
127#10196C2311yesmaximal, characteristic, direct product factor, Sylow
128C23×C23—41no—
129C23×C23—41no—
130C23×C23#12711yescharacteristic
131Q8×C23—44no—
132Q8×C23—42no—
133Q8×C23—42no—
134Q8×C23—82no—
135D4×C23—42no—
136D4×C23—42no—
137D4×C23—41no—
138D4×C23—41no—
139D4×C23—41no—
140D4×C23—41no—
141D4×C23—42no—
142D4×C23—81no—
143D4×C23—81no—
144D4×C23—81no—
145D4×C23—81no—
146D4×C23—81no—
147C2×C4×C23—21no—
148C2×C4×C23—21no—
149C2×C4×C23—41no—
150C2×C4×C23—42no—
151C2×C4×C23—41no—
152C2×C4×C23—41no—
153C2×C4×C23—42no—
154C2×C4×C23—81no—
155C2×C4×C23—82no—
156C2×C4×C23—81no—
157C2×C4×C23—82no—
158C2×C4×C23—82no—
159C2×C4×C23#12011yescharacteristic
160C2×C4×C23#12011yescharacteristic
161C2×C4×C23#12011yescharacteristic
162C2×C4×C23#12111yescharacteristic
163C2×C4×C23#12111yescharacteristic
164C2×C4×C23#12111yescharacteristic
165C2×C4×C23#13512yes—
166C2×C4×C23AGL(1,ℤ/8)12yes—
167C8×C23—42no—
168C8×C23—42no—
169C8×C23—42no—
170#3245C2×C2311yescharacteristic
171#3206C2×C2311yescharacteristic, complemented
172#3141C2×C2311yescharacteristic, complemented
173#1987C2×C2311yescharacteristic, complemented
174#1980C2×C2311yescharacteristic, complemented
175#1502C2×C2311yescharacteristic, complemented
176#1283C2×C2311yescharacteristic, complemented
177C22×C23—21no—
178C22×C23—21no—
179C22×C23—81no—
180C22×C23—81no—
181C22×C23—81no—
182C22×C23—81no—
183C22×C23—81no—
184C22×C23—81no—
185C22×C23#39211yescharacteristic
186C22×C23#49511yescharacteristic
187C22×C23#49611yescharacteristic
188C4×C23—21no—
189C4×C23—21no—
190C4×C23—21no—
191C4×C23—82no—
192C4×C23—82no—
193C4×C23—81no—
194C4×C23—81no—
195C4×C23#44912yes—
196C4×C23#45511yescharacteristic
197C4×C23#45611yescharacteristic
198C4×C23#46312yes—
199#532C22×C2311yescharacteristic
200#532C22×C2311yescharacteristic
201C2×C4⋊Q8C22×C2311yescharacteristic
202#492—21no—
203#489—22no—
204#487—21no—
205#487—21no—
206#486—21no—
207#474—21no—
208#474—21no—
209#466—22no—
210#461—22no—
211#459—21no—
212#459—21no—
213#448—22no—
214#431C22×C2311yescharacteristic
215C2×C4⋊C8C22×C2311yescharacteristic
216C2×#104—22no—
217#417—22no—
218C4×M16C22×C2311yescharacteristic
219C4×C4⋊C4C22×C2311yescharacteristic
220#340—22no—
221#327—22no—
222C2×C23—21no—
223C2×C23—21no—
224C2×C23—161no—
225C2×C23—161no—
226C2×C23#200711yescharacteristic
227C2×C23#200811yescharacteristic
228C2×C23#201211yescharacteristic
229C2×#41—21no—
230C2×#41—21no—
231C22×Q8—21no—
232C2×SD16—24no—
233C2×D8—22no—
234C2×M16D4×C2312yes—
235C22×C8D4×C2312yes—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8D4×C2312yes—
240C4⋊Q8D4×C2311yescharacteristic
241C4⋊Q8D4×C2311yescharacteristic
242#127D4×C2311yescharacteristic
243#127D4×C2311yescharacteristic
244#124—21no—
245#124—21no—
246#121—21no—
247#121—21no—
248#121—21no—
249#121—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C42—21no—
258C2×C42C23×C2311yescharacteristic
259#107—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263#104—44no—
264#103—24no—
265#103—24no—
266#102—24no—
267#102—22no—
268#102—22no—
269C22⋊C8D4×C2312yes—
270#97—21no—
271C4×C8—21no—
272C4×C8—21no—
273#95—21no—
274C23#1019611yescharacteristic, direct product factor, Sylow
275#41—42no—
276#41—42no—
277#41—42no—
278#41—42no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—41no—
283C2×Q8—82no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C2×D4×C2311yescharacteristic
297C22×C4C2×D4×C2311yescharacteristic
298C22×C4C2×D4×C2311yescharacteristic
299SD16—84no—
300D8—82no—
301M16—22no—
302M16—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—42no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C42—42no—
322C42C2×D4×C2311yescharacteristic
323C42C2×D4×C2311yescharacteristic
324C42C2×D4×C2311yescharacteristic
325C42#41×C2311yescharacteristic
326C23—41no—
327C23—41no—
328C23C23×#12711yescharacteristic
329Q8—44no—
330Q8—42no—
331Q8—42no—
332Q8—82no—
333D4—42no—
334D4—42no—
335D4—41no—
336D4—41no—
337D4—41no—
338D4—41no—
339D4—42no—
340D4—81no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4C23×#12011yescharacteristic
358C2×C4C23×#12011yescharacteristic
359C2×C4C23×#12011yescharacteristic
360C2×C4C23×#12111yescharacteristic
361C2×C4C23×#12111yescharacteristic
362C2×C4C23×#12111yescharacteristic
363C2×C4C23×#13512yes—
364C2×C4C23×AGL(1,ℤ/8)12yes—
365C8—42no—
366C8—42no—
367C8—42no—
368C22—21no—
369C22—21no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22C23×#39211yescharacteristic
377C22C23×#49511yescharacteristic
378C22C23×#49611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—82no—
383C4—82no—
384C4—81no—
385C4—81no—
386C4C23×#44912yes—
387C4C23×#45511yescharacteristic
388C4C23×#45611yescharacteristic
389C4C23×#46312yes—
390C2—21no—
391C2—21no—
392C2—161no—
393C2—161no—
394C2C23×#200711yescharacteristic
395C2C23×#200811yescharacteristic
396C2C23×#201211yescharacteristic
subgroupG/H
center, socleC22×C23#392
commutator, Frattini, cosocleC2×C42C23×C23
prime radicalC23×#3245C2
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×#3206C2semidirect
3. C23×#3141C2semidirect
4. C23×#1987C2semidirect
5. C23×#1980C2semidirect
6. C23×#1502C2semidirect
7. C23×#1283C2semidirect
127. #10196C23direct (N × H)
171. #3206C2×C23semidirect
172. #3141C2×C23semidirect
173. #1987C2×C23semidirect
174. #1980C2×C23semidirect
175. #1502C2×C23semidirect
176. #1283C2×C23semidirect
274. C23#10196direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#32452
C23×#32062
C23×#31412
C23×#19872
C23×#19802
C23×#15022
C23×#12832
#1019623