Finite Group Databasebeta

← #26316214

Subgroups — #26316214

Order 12032.

798 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246113C211yesmaximal, characteristic
2#20244531C212yesmaximal, complemented
3#20244521C212yesmaximal, complemented
4#20244211C211yesmaximal, characteristic
5#20244193C211yesmaximal, characteristic
6#20243523C211yesmaximal, characteristic, complemented
7#20243518C211yesmaximal, characteristic, complemented
8#20243395C212yesmaximal, complemented
9#20243385C212yesmaximal, complemented
10#20243245C211yesmaximal, characteristic, complemented
11C47×#2660C211yesmaximal, characteristic, complemented
12C2×#13094021C2211yescharacteristic
13C2×C4×D188C2212yes—
14C2×C4⋊(C47⋊C8)C2212yes—
15C2×#13093944C2212yes—
16C2×#13093944C2212yes—
17C2×#13093944C2212yes—
18C2×#13093944C2212yes—
19C2×#13093935C2211yescharacteristic
20C2×C4×C47⋊C8C2211yescharacteristic
21#15689025C2212yes—
22#15689001—22no—
23#15689001—22no—
24#15688989C2211yescharacteristic
25#15688885C2212yes—
26#15688885C2212yes—
27#15688885C2212yes—
28#15688885C2212yes—
29#15688878C2212yes—
30C2×C4⋊Q8×C47C2211yescharacteristic
31C2×#125×C47C2211yescharacteristic
32C47×#398—22no—
33C47×#395—22no—
34C47×#384C2211yescharacteristic
35C47×#384C2211yescharacteristic
36C47×#382C2211yescharacteristic
37C47×#382C2211yescharacteristic
38C47×#376C2211yescharacteristic
39C22×D188—22no—
40C22×D188C2312yes—
41C22×C4×D47—22no—
42C22×C47⋊C8C2312yes—
43C22×C47⋊C8C2312yes—
44#13094021—22no—
45#13094021Q814yescomplemented
46C4×D188—24no—
47C4×D188—24no—
48C2×C4⋊Dic47C2312yes—
49C2×(C2×D47)⋊C4—22no—
50C2×(C2×D47)⋊C4—22no—
51C2×(C2×D47)⋊C4—22no—
52C4⋊(C47⋊C8)—24no—
53#13093944—42no—
54#13093944—42no—
55#13093944—42no—
56#13093944—42no—
57#13093935—22no—
58C4×C47⋊C8—22no—
59C47⋊#95—22no—
60C22×Q8×C47—21no—
61C4⋊Q8×C47—42no—
62#125×C47—42no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47C2311yescharacteristic
69C2×C4⋊C4×C47C2311yescharacteristic
70C2×C4⋊C4×C47C2311yescharacteristic
71C2×C4⋊C4×C47C2311yescharacteristic
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C2×C42×C47C2311yescharacteristic
75#95×C47—22no—
76#95×C47—22no—
77#95×C47D412yes—
78#95×C47D412yes—
79C23×D47—22no—
80C2×Q8×C47—42no—
81C2×Q8×C47—82no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D412yes—
89C22×C4×C47C2×D412yes—
90C22×C4×C47C2411yescharacteristic
91C22×Dic47—22no—
92C2×D188—22no—
93C2×D188—22no—
94C2×D188—22no—
95C2×D188—22no—
96C2×D188—22no—
97C2×D188—28no—
98C2×D188—82no—
99C2×D188—82no—
100C2×D188C2×Q814yes—
101C2×C4×D47—44no—
102C2×C4×D47—82no—
103C2×C4×D47—82no—
104C4⋊C4×C47—22no—
105C4⋊C4×C47—22no—
106C4⋊C4×C47—22no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—42no—
109C4⋊C4×C47—42no—
110C4⋊C4×C47—42no—
111C4⋊C4×C47—82no—
112C4⋊C4×C47—82no—
113C42×C47—22no—
114C42×C47C2×Q812yes—
115C42×C47#4112yes—
116C42×C47#4114yes—
117(C2×D47)⋊C4—44no—
118(C2×D47)⋊C4—82no—
119(C2×D47)⋊C4—82no—
120C4⋊Dic47—22no—
121C4⋊Dic47—22no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—22no—
128C23×C47C22×D411yescharacteristic
129C22×D47—24no—
130C22×D47—42no—
131C22×D47—42no—
132C22×D47—42no—
133C22×D47—42no—
134C22×D47—42no—
135C22×D47—42no—
136Q8×C47—84no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—22no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—22no—
144C2×C4×C47—22no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—42no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47C22⋊Q814yes—
154C2×C4×C47#12414yes—
155C2×C4×C47C22×Q811yescharacteristic
156C2×C4×C47C2×#4112yes—
157C2×C4×C47C2×#4111yescharacteristic
158C2×C4×C47H(5,𝔽2)12yes—
159C2×Dic47—42no—
160C2×Dic47—42no—
161C2×Dic47—42no—
162D188—24no—
163D188—24no—
164D188—42no—
165D188—42no—
166D188—88no—
167C4×D47—88no—
168C47⋊C8—42no—
169C47⋊C8—42no—
170#29152—471nomaximal, Sylow
171C22×C47C2×C22⋊Q811yescharacteristic
172C22×C47C2×#12411yescharacteristic
173C22×C47#53511yescharacteristic
174C22×C47#57612yes—
175C22×C47#57712yes—
176C2×D47—44no—
177C2×D47—44no—
178C2×D47—44no—
179C2×D47—82no—
180C2×D47—82no—
181C2×D47—82no—
182C2×D47—82no—
183C4×C47—24no—
184C4×C47—24no—
185C4×C47—41no—
186C4×C47—41no—
187C4×C47—81no—
188C4×C47—81no—
189C4×C47—81no—
190C4×C47—81no—
191C4×C47#55412yes—
192C4×C47#55512yes—
193Dic47—82no—
194#2660D4711yescharacteristic, complemented
195#2328—471no—
196#2025—472no—
197#2023—472no—
198#1961—471no—
199#1957—471no—
200#1836—471no—
201#1835—471no—
202#1804—472no—
203#1802—472no—
204#1768—471no—
205C2×C47#256311yescharacteristic
206C2×C47#308112yes—
207C2×C47#313612yes—
208C2×C47#322011yescharacteristic
209C2×C47#322111yescharacteristic
210D47—84no—
211C2×C4⋊Q8C2×D4711yescharacteristic
212C2×#127—471no—
213C2×#125C2×D4711yescharacteristic
214C2×C4×D4—472no—
215C2×C4⋊C8—472no—
216C2×#102—472no—
217C2×#102—472no—
218C2×#102—472no—
219C2×#102—472no—
220C2×#97—471no—
221C2×C4×C8—471no—
222#398—22no—
223#395—22no—
224#390—472no—
225#386—942no—
226#386—942no—
227#384C2×D4711yescharacteristic
228#384C2×D4711yescharacteristic
229#382—471no—
230#382C2×D4711yescharacteristic
231#382C2×D4711yescharacteristic
232#376C2×D4711yescharacteristic
233#340—472no—
234#340—472no—
235#340—472no—
236#340—472no—
237#336—472no—
238C47#2915211yescharacteristic, complemented, Sylow
239C22×Q8—21no—
240C22×D4—472no—
241C22×D4—942no—
242C23×C4—942no—
243C22×C8—472no—
244C22×C8—472no—
245C4⋊Q8—42no—
246#127—474no—
247#127—942no—
248#125—42no—
249C4×D4—944no—
250C4×D4—944no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—472no—
257C2×C4⋊C4C22×D4711yescharacteristic
258C2×C4⋊C4C22×D4711yescharacteristic
259C2×C4⋊C4C22×D4711yescharacteristic
260C2×C4⋊C4C22×D4711yescharacteristic
261C2×C22⋊C4—942no—
262C2×C22⋊C4—942no—
263C2×C22⋊C4—942no—
264C2×C42C22×D4711yescharacteristic
265C2×C42C22×D4711yescharacteristic
266C2×C42C22×D4711yescharacteristic
267C4⋊C8—944no—
268#102—1882no—
269#102—1882no—
270#102—1882no—
271#102—1882no—
272#97—942no—
273C4×C8—942no—
274#95—22no—
275#95—22no—
276#95—942no—
277#95C22⋊D4712yes—
278#95C22⋊D4712yes—
279C24—942no—
280C2×Q8—42no—
281C2×Q8—82no—
282C2×D4—474no—
283C2×D4—948no—
284C2×D4—942no—
285C2×D4—942no—
286C2×D4—942no—
287C2×D4—942no—
288C2×D4—942no—
289C2×D4—3762no—
290C2×D4—3762no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—942no—
296C22×C4—3762no—
297C22×C4—3762no—
298C22×C4—1884no—
299C22×C4C2×C22⋊D4711yescharacteristic
300C22×C4C2×C22⋊D4711yescharacteristic
301C22×C4C2×C22⋊D4712yes—
302C22×C4C2×C22⋊D4712yes—
303C22×C4C23×D4711yescharacteristic
304C2×C8—942no—
305C2×C8—942no—
306C2×C8—942no—
307C2×C8—942no—
308C2×C8—942no—
309C2×C8—942no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—942no—
320C4⋊C4—942no—
321C22⋊C4—3762no—
322C22⋊C4—3762no—
323C22⋊C4—1884no—
324C42—22no—
325C42Q8×D4712yes—
326C42#1062535412yes—
327C42#1062535414yes—
328C23—944no—
329C23—1882no—
330C23—1882no—
331C23—1882no—
332C23—1882no—
333C23—1882no—
334C23—1882no—
335C23C22×C22⋊D4711yescharacteristic
336Q8—84no—
337D4—944no—
338D4—944no—
339D4—3768no—
340D4—1882no—
341D4—1882no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—3768no—
359C2×C4—1882no—
360C2×C4—1882no—
361C2×C4—1882no—
362C2×C4#1309400014yes—
363C2×C4#1309401314yes—
364C2×C4C2×Q8×D4711yescharacteristic
365C2×C4C2×#1062535412yes—
366C2×C4C2×#1062535411yescharacteristic
367C2×C4#1309406912yes—
368C8—1882no—
369C8—1882no—
370C22—3762no—
371C22—3762no—
372C22—3762no—
373C22—3762no—
374C22—1884no—
375C22—1884no—
376C22—1884no—
377C22C2×#1309400011yescharacteristic
378C22C2×#1309401311yescharacteristic
379C22#1568969011yescharacteristic
380C22#1568992612yes—
381C22#1568993112yes—
382C4—24no—
383C4—24no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—3762no—
391C4#1568983412yes—
392C4#1568983912yes—
393C2—3764no—
394C2#2024856211yescharacteristic
395C2#2025321312yes—
396C2#2025374712yes—
397C2#2025443811yescharacteristic
398C2#2025444911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2660C2
socleC23×C47C22×D4
prime radicalC2×#13094021C22
hypercenter#2660D47
hypocenterC47#29152
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2660

Complements

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

NH ≅ G/Ntype
2. #20244531C2semidirect
3. #20244521C2semidirect
6. #20243523C2semidirect
7. #20243518C2semidirect
8. #20243395C2semidirect
9. #20243385C2semidirect
10. #20243245C2semidirect
11. C47×#2660C2semidirect
45. #13094021Q8semidirect
194. #2660D47semidirect
238. C47#29152semidirect

Maximal subgroups

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

subgroupindex
#202461132
#202445312
#202445212
#202442112
#202441932
#202435232
#202435182
#202433952
#202433852
#202432452
C47×#26602
#2915247