Finite Group Databasebeta

← #26190473

Subgroups — #26190473

Order 12032.

950 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 183 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254209C212yesmaximal
2#20248768C211yesmaximal, characteristic
3#20242396C214yesmaximal, complemented
4#20242362C214yesmaximal, complemented
5#20241908C212yesmaximal
6#20241718C211yesmaximal, characteristic
7C2×C47×#544C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689453—22no—
10#15689451C2214yes—
11#15689379—24no—
12#15689279—22no—
13#15689278C2214yes—
14#15689087C2212yes—
15#15689053C2212yes—
16#15689043C2212yes—
17#15689018—22no—
18#15688996—21no—
19#15688996—21no—
20C4×C22⋊Dic47C2211yescharacteristic
21#15688860C2218yes—
22#15688849C2214yes—
23C47×#544—24no—
24C47×#544—22no—
25C47×#544C2214yes—
26C2×C4⋊Q8×C47—22no—
27C2×#125×C47—22no—
28C2×C22⋊Q8×C47—21no—
29C2×C22⋊Q8×C47C2212yes—
30C2×#117×C47—21no—
31C22×C4⋊C4×C47C2211yescharacteristic
32C22×Dic94C2311yescharacteristic
33C2×#10625337—24no—
34C2×#10625329—22no—
35#13094003—22no—
36#13094003—22no—
37C2×C4⋊Dic47C2311yescharacteristic
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4—21no—
40C2×Dic47⋊C4—22no—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47—21no—
43C2×C4×Dic47—21no—
44#13093955—24no—
45#13093948—24no—
46#13093947—24no—
47C22⋊(C47⋊C8)—22no—
48C22⋊(C47⋊C8)C2314yes—
49C47⋊#95—22no—
50C47⋊#95—22no—
51C47⋊#95C2312yes—
52C22×Q8×C47—21no—
53C22×Q8×C47D412yes—
54C23×C4×C47D411yescharacteristic
55C23×C4×C47D411yescharacteristic
56C23×C4×C47C2311yescharacteristic
57C4⋊Q8×C47—28no—
58C4⋊Q8×C47—28no—
59#125×C47—28no—
60#125×C47—28no—
61C22⋊Q8×C47—24no—
62C22⋊Q8×C47—24no—
63C22⋊Q8×C47—22no—
64C22⋊Q8×C47—22no—
65C22⋊Q8×C47D414yes—
66C22⋊Q8×C47C2314yes—
67#117×C47—24no—
68#117×C47—24no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—22no—
73C2×C4⋊C4×C47—22no—
74C2×C4⋊C4×C47—22no—
75C2×C4⋊C4×C47—22no—
76C2×C4⋊C4×C47D412yes—
77C2×C4⋊C4×C47D412yes—
78C2×C4⋊C4×C47C2312yes—
79C2×C22⋊C4×C47—22no—
80C2×C42×C47—22no—
81C24×C47C2×D411yescharacteristic
82C2×Q8×C47—24no—
83C2×Q8×C47—22no—
84C2×Q8×C47—82no—
85C2×Q8×C47—82no—
86C2×Q8×C47C2×D414yes—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—24no—
90C22×C4×C47—22no—
91C22×C4×C47—22no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—81no—
95C22×C4×C47—81no—
96C22×C4×C47C2×D412yes—
97C22×C4×C47C2×D412yes—
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D412yes—
100C22×C4×C47C2×D412yes—
101C22×C4×C47C2411yescharacteristic
102C22×Dic47—22no—
103C22×Dic47—21no—
104C22×Dic47—21no—
105C2×Dic94—22no—
106C2×Dic94—81no—
107C4⋊C4×C47—24no—
108C4⋊C4×C47—24no—
109C4⋊C4×C47—24no—
110C4⋊C4×C47—28no—
111C4⋊C4×C47—24no—
112C4⋊C4×C47—24no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—24no—
115C4⋊C4×C47—24no—
116C4⋊C4×C47C2×D414yes—
117C22⋊C4×C47—24no—
118C22⋊C4×C47—24no—
119C42×C47—28no—
120C22⋊Dic47—24no—
121C22⋊Dic47—41no—
122C22⋊Dic47—41no—
123#10625337—84no—
124C4⋊Dic47—22no—
125Dic47⋊C4—42no—
126Dic47⋊C4—44no—
127Dic47⋊C4—82no—
128C4×Dic47—81no—
129#10625329—82no—
130C2×C47⋊C8—24no—
131C23×C47—22no—
132C23×C47—22no—
133C23×C47—41no—
134C23×C47#12011yescharacteristic
135C23×C47#12011yescharacteristic
136C23×C47C22×D411yescharacteristic
137Q8×C47—44no—
138Q8×C47—82no—
139Q8×C47—84no—
140C2×C4×C47—24no—
141C2×C4×C47—24no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—24no—
145C2×C4×C47—24no—
146C2×C4×C47—22no—
147C2×C4×C47—42no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—44no—
153C2×C4×C47—44no—
154C2×C4×C47—42no—
155C2×C4×C47—81no—
156C2×C4×C47#12012yes—
157C2×C4×C47#13714yes—
158C2×C4×C47C22×D412yes—
159C2×Dic47—24no—
160C2×Dic47—42no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163C2×Dic47—41no—
164C2×Dic47—41no—
165C2×Dic47—42no—
166Dic94—44no—
167C47⋊C8—44no—
168#18327—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—44no—
177C22×C47#45612yes—
178C22×C47C2×#12011yescharacteristic
179C22×C47C2×#13712yes—
180C4×C47—24no—
181C4×C47—22no—
182C4×C47—22no—
183C4×C47—44no—
184C4×C47—42no—
185C4×C47—44no—
186Dic47—44no—
187Dic47—81no—
188Dic47—81no—
189C2×#544D4711yescharacteristic
190#3195—472no—
191#2583—471no—
192#1611—474no—
193#1605—474no—
194#1508—472no—
195#1463—471no—
196C2×C47—22no—
197C2×C47—22no—
198C2×C47—41no—
199C2×C47#300411yescharacteristic
200C2×C47C2×#45611yescharacteristic
201C2×C47#302611yescharacteristic
202#544—24no—
203#544—22no—
204#544C2×D4714yes—
205C2×C4⋊Q8—22no—
206C2×#125—22no—
207C2×C22⋊Q8—21no—
208C2×C22⋊Q8—471no—
209C2×C22⋊Q8C2×D4712yes—
210C2×#117—21no—
211C22×C4⋊C4C2×D4711yescharacteristic
212#483—474no—
213#483—942no—
214#470—944no—
215#451—474no—
216#451—942no—
217#407—472no—
218#398—472no—
219#395—472no—
220#389—942no—
221#385—941no—
222#385—941no—
223C4×C22⋊C4—471no—
224#328—478no—
225#324—474no—
226C47#1832711yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—471no—
229C22×Q8C22⋊D4712yes—
230C23×C4C22⋊D4711yescharacteristic
231C23×C4C22⋊D4711yescharacteristic
232C23×C4C22×D4711yescharacteristic
233C2×Q16—944no—
234C2×M16—942no—
235C4⋊Q8—28no—
236C4⋊Q8—28no—
237#125—28no—
238#125—28no—
239C22⋊Q8—24no—
240C22⋊Q8—24no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—942no—
244C22⋊Q8—942no—
245C22⋊Q8C22⋊D4714yes—
246C22⋊Q8C22×D4714yes—
247#117—24no—
248#117—24no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—471no—
257C2×C4⋊C4—942no—
258C2×C4⋊C4—941no—
259C2×C4⋊C4—942no—
260C2×C4⋊C4C22⋊D4712yes—
261C2×C4⋊C4C22⋊D4712yes—
262C2×C4⋊C4C22×D4712yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—941no—
265C2×C22⋊C4—941no—
266C2×C42—22no—
267C2×C42—941no—
268#106—944no—
269#103—944no—
270#103—944no—
271C22⋊C8—474no—
272C22⋊C8—942no—
273#95—472no—
274#95—942no—
275#95—942no—
276C24C2×C22⋊D4711yescharacteristic
277C2×Q8—24no—
278C2×Q8—22no—
279C2×Q8—82no—
280C2×Q8—82no—
281C2×Q8—942no—
282C2×Q8—3761no—
283C2×Q8C2×C22⋊D4714yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—942no—
294C22×C4—941no—
295C22×C4—941no—
296C22×C4C2×C22⋊D4712yes—
297C22×C4C2×C22⋊D4712yes—
298C22×C4C2×C22⋊D4711yescharacteristic
299C22×C4C2×C22⋊D4712yes—
300C22×C4C2×C22⋊D4712yes—
301C22×C4C23×D4711yescharacteristic
302Q16—3764no—
303M16—3762no—
304C2×C8—944no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—942no—
315C4⋊C4—3762no—
316C4⋊C4—1882no—
317C4⋊C4—1884no—
318C4⋊C4C2×C22⋊D4714yes—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—944no—
322C22⋊C4—1881no—
323C22⋊C4—1881no—
324C42—28no—
325C42—3761no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23#1309399311yescharacteristic
330C23#1309399311yescharacteristic
331C23C22×C22⋊D4711yescharacteristic
332Q8—44no—
333Q8—82no—
334Q8—84no—
335Q8—1884no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—944no—
353C2×C4—1882no—
354C2×C4—1881no—
355C2×C4—1881no—
356C2×C4—1882no—
357C2×C4—1882no—
358C2×C4—1882no—
359C2×C4#1309399312yes—
360C2×C4#1309405514yes—
361C2×C4C22×C22⋊D4712yes—
362C8—1884no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22#1568930612yes—
372C22C2×#1309399311yescharacteristic
373C22C2×#1309405512yes—
374C4—22no—
375C4—22no—
376C4—24no—
377C4—44no—
378C4—42no—
379C4—44no—
380C4—3761no—
381C4—3761no—
382C4—1884no—
383C2—22no—
384C2—22no—
385C2—41no—
386C2#2025262211yescharacteristic
387C2C2×#1568930611yescharacteristic
388C2#2025278511yescharacteristic
subgroupG/H
centerC22C2×#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#544C2
socleC22×C47C2×#120
prime radicalC24×C47C2×D4
hypercenterC2×#544D47
hypocenterC47#18327
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#544

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #20242396C2semidirect
4. #20242362C2semidirect
226. C47#18327semidirect

Maximal subgroups

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

subgroupindex
#202542092
#202487682
#202423962
#202423622
#202419082
#202417182
C2×C47×#5442
#1832747