Finite Group Databasebeta

← #26316792

Subgroups — #26316792

Order 12032.

770 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20249363C211yesmaximal, characteristic, complemented
2#20244776C212yesmaximal, complemented
3#20244762C212yesmaximal, complemented
4#20244213C211yesmaximal, characteristic
5#20244195C211yesmaximal, characteristic
6#20243649C212yesmaximal, complemented
7#20243635C212yesmaximal, complemented
8#20243550C211yesmaximal, characteristic, complemented
9#20243521C211yesmaximal, characteristic, complemented
10#20243245C211yesmaximal, characteristic, complemented
11C47×#2328C211yesmaximal, characteristic
12C2×#13093957C2212yes—
13C2×#13093955C2212yes—
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×C4⋊(C47⋊C8)C2211yescharacteristic
16C2×#13093945C2212yes—
17C2×#13093945C2212yes—
18C2×#13093945C2212yes—
19C2×#13093945C2212yes—
20C2×#13093935C2211yescharacteristic
21C2×C4×C47⋊C8C2211yescharacteristic
22#15689065—22no—
23#15689057—22no—
24#15689021C2212yes—
25#15689021C2212yes—
26#15688993C2211yescharacteristic
27#15688989C2211yescharacteristic
28#15688885C2212yes—
29#15688885C2212yes—
30#15688878C2211yescharacteristic
31#15688878C2211yescharacteristic
32C2×#127×C47C2211yescharacteristic
33C2×C4×D4×C47C2212yes—
34C47×#390C2212yes—
35C47×#386—22no—
36C47×#386—22no—
37C47×#382C2211yescharacteristic
38C47×#376C2211yescharacteristic
39C22×C47⋊C8C2311yescharacteristic
40C22×C47⋊C8C2311yescharacteristic
41C22×C47⋊C8C2311yescharacteristic
42C22×C47⋊C8C2311yescharacteristic
43C2×C4⋊Dic47—22no—
44C2×C4⋊Dic47C2312yes—
45C2×C4⋊Dic47C2312yes—
46C2×Dic47⋊C4—22no—
47C2×Dic47⋊C4—22no—
48#13093957—42no—
49#13093955—42no—
50C4⋊(C47⋊C8)—22no—
51C4⋊(C47⋊C8)—22no—
52#13093945—24no—
53#13093945—24no—
54#13093945—42no—
55#13093945—42no—
56#13093935—22no—
57C4×C47⋊C8—22no—
58C47⋊#95—22no—
59C47⋊#95—22no—
60C47⋊#95—22no—
61C22×D4×C47—22no—
62C22×D4×C47C2312yes—
63C23×C4×C47—22no—
64#127×C47—22no—
65#127×C47Q814yes—
66C4×D4×C47—42no—
67C4×D4×C47—42no—
68C2×C4⋊C4×C47C2312yes—
69C2×C22⋊C4×C47—22no—
70C2×C22⋊C4×C47—22no—
71C2×C22⋊C4×C47—22no—
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C2×C42×C47C2311yescharacteristic
75#95×C47—22no—
76#95×C47D412yes—
77#95×C47D412yes—
78C24×C47—22no—
79C2×D4×C47—28no—
80C2×D4×C47—22no—
81C2×D4×C47—22no—
82C2×D4×C47—22no—
83C2×D4×C47—22no—
84C2×D4×C47—22no—
85C2×D4×C47—82no—
86C2×D4×C47—82no—
87C2×D4×C47C2×Q814yes—
88C22×C4×C47—22no—
89C22×C4×C47—44no—
90C22×C4×C47—82no—
91C22×C4×C47—82no—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D411yescharacteristic
94C22×C4×C47C2×D412yes—
95C22×C4×C47C2×D412yes—
96C22×C4×C47C2411yescharacteristic
97C22×Dic47—22no—
98C22×Dic47—22no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—22no—
101C22⋊C4×C47—44no—
102C22⋊C4×C47—82no—
103C22⋊C4×C47—82no—
104C42×C47—21no—
105C42×C47—21no—
106C42×C47—21no—
107C42×C47—21no—
108C42×C47C2×Q812yes—
109C42×C47#4112yes—
110C4⋊Dic47—22no—
111C4⋊Dic47—22no—
112C4⋊Dic47—22no—
113C4⋊Dic47—22no—
114C4⋊Dic47—82no—
115Dic47⋊C4—82no—
116Dic47⋊C4—82no—
117C2×C47⋊C8—21no—
118C2×C47⋊C8—21no—
119C2×C47⋊C8—21no—
120C2×C47⋊C8—21no—
121C2×C47⋊C8—21no—
122C2×C47⋊C8—21no—
123C2×C47⋊C8—21no—
124C2×C47⋊C8—21no—
125C2×C47⋊C8—21no—
126C2×C47⋊C8—21no—
127C2×C47⋊C8#4112yes—
128C2×C47⋊C8#4112yes—
129C23×C47—24no—
130C23×C47—42no—
131C23×C47—42no—
132C23×C47—42no—
133C23×C47—42no—
134C23×C47—42no—
135C23×C47—42no—
136C23×C47C22×D411yescharacteristic
137D4×C47—24no—
138D4×C47—24no—
139D4×C47—42no—
140D4×C47—42no—
141D4×C47—88no—
142C2×C4×C47—22no—
143C2×C4×C47—22no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—22no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47—22no—
151C2×C4×C47—42no—
152C2×C4×C47—42no—
153C2×C4×C47—42no—
154C2×C4×C47—88no—
155C2×C4×C47C22⋊Q814yes—
156C2×C4×C47C22×Q811yescharacteristic
157C2×C4×C47C2×#4111yescharacteristic
158C2×C4×C47C2×#4111yescharacteristic
159C2×C4×C47C2×#4111yescharacteristic
160C2×C4×C47H(5,𝔽2)11yescharacteristic
161C2×C4×C47H(5,𝔽2)11yescharacteristic
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—42no—
165C2×Dic47—42no—
166C2×Dic47—42no—
167C2×Dic47—42no—
168C47⋊C8—22no—
169C47⋊C8—22no—
170C47⋊C8—41no—
171C47⋊C8—41no—
172#29201—471nomaximal, Sylow
173C22×C47—44no—
174C22×C47—44no—
175C22×C47—44no—
176C22×C47—82no—
177C22×C47—82no—
178C22×C47—82no—
179C22×C47—82no—
180C22×C47C2×C22⋊Q811yescharacteristic
181C22×C47#53211yescharacteristic
182C22×C47#53511yescharacteristic
183C22×C47#57612yes—
184C22×C47#57712yes—
185C4×C47—24no—
186C4×C47—41no—
187C4×C47—41no—
188C4×C47—42no—
189C4×C47—82no—
190C4×C47#55311yescharacteristic
191C4×C47#55311yescharacteristic
192C4×C47#55411yescharacteristic
193C4×C47#55411yescharacteristic
194Dic47—82no—
195Dic47—82no—
196#2647—471no—
197#2328D4711yescharacteristic
198#2067—472no—
199#2065—472no—
200#1961—471no—
201#1957—471no—
202#1864—472no—
203#1862—472no—
204#1846—471no—
205#1836—471no—
206#1768—471no—
207C2×C47—84no—
208C2×C47#256611yescharacteristic
209C2×C47#308112yes—
210C2×C47#317411yescharacteristic
211C2×C47#317611yescharacteristic
212C2×C47#322412yes—
213C2×#127C2×D4711yescharacteristic
214C2×C4×D4C2×D4712yes—
215C2×#107—472no—
216C2×#106—472no—
217C2×C4⋊C8—471no—
218C2×C4⋊C8—471no—
219C2×#102—472no—
220C2×#102—472no—
221C2×#102—472no—
222C2×#102—472no—
223C2×#97—471no—
224C2×C4×C8—471no—
225#400—942no—
226#398—942no—
227#390C2×D4712yes—
228#389—472no—
229#389—472no—
230#386—22no—
231#386—22no—
232#384—471no—
233#382—471no—
234#382C2×D4711yescharacteristic
235#376C2×D4711yescharacteristic
236#340—472no—
237#340—472no—
238#336—471no—
239#336—471no—
240C47#2920111yescharacteristic, complemented, Sylow
241C22×D4—22no—
242C22×D4C22×D4712yes—
243C23×C4—22no—
244C22×C8—471no—
245C22×C8—471no—
246C22×C8—471no—
247C22×C8—471no—
248#127—22no—
249#127Dic9414yes—
250C4×D4—42no—
251C4×D4—42no—
252C2×C4⋊C4—472no—
253C2×C4⋊C4—472no—
254C2×C4⋊C4—942no—
255C2×C4⋊C4—942no—
256C2×C4⋊C4—942no—
257C2×C4⋊C4C22×D4712yes—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C42C22×D4711yescharacteristic
262C2×C42C22×D4711yescharacteristic
263C2×C42C22×D4711yescharacteristic
264#107—1882no—
265#106—1882no—
266C4⋊C8—942no—
267C4⋊C8—942no—
268#102—944no—
269#102—944no—
270#102—1882no—
271#102—1882no—
272#97—942no—
273C4×C8—942no—
274#95—22no—
275#95—942no—
276#95—942no—
277#95—942no—
278#95C22⋊D4712yes—
279#95C22⋊D4712yes—
280C24—22no—
281C2×D4—28no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4C2×Dic9414yes—
290C22×C4—22no—
291C22×C4—44no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—942no—
295C22×C4—942no—
296C22×C4C2×C22⋊D4711yescharacteristic
297C22×C4C2×C22⋊D4712yes—
298C22×C4C2×C22⋊D4711yescharacteristic
299C22×C4C2×C22⋊D4712yes—
300C22×C4C23×D4711yescharacteristic
301C2×C8—472no—
302C2×C8—472no—
303C2×C8—941no—
304C2×C8—941no—
305C2×C8—941no—
306C2×C8—941no—
307C2×C8—941no—
308C2×C8—941no—
309C2×C8—941no—
310C2×C8—941no—
311C2×C8—941no—
312C2×C8—941no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—942no—
316C4⋊C4—942no—
317C4⋊C4—942no—
318C4⋊C4—942no—
319C4⋊C4—3762no—
320C4⋊C4—3762no—
321C4⋊C4—3762no—
322C22⋊C4—44no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42C2×Dic9412yes—
330C42#1062535012yes—
331C23—24no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23C22×C22⋊D4711yescharacteristic
339D4—24no—
340D4—24no—
341D4—42no—
342D4—42no—
343D4—88no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—88no—
357C2×C4—1882no—
358C2×C4—1882no—
359C2×C4—1882no—
360C2×C4—1882no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4#1309400314yes—
364C2×C4C22×Dic9411yescharacteristic
365C2×C4#41×D4711yescharacteristic
366C2×C4#41×D4711yescharacteristic
367C2×C4C2×#1062535011yescharacteristic
368C2×C4#1309406911yescharacteristic
369C2×C4#1309406911yescharacteristic
370C8—942no—
371C8—942no—
372C8—1881no—
373C8—1881no—
374C22—44no—
375C22—44no—
376C22—44no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22C2×#1309400311yescharacteristic
382C22#1568966911yescharacteristic
383C22#1568969011yescharacteristic
384C22#1568992612yes—
385C22#1568993412yes—
386C4—24no—
387C4—42no—
388C4—41no—
389C4—41no—
390C4—82no—
391C4—3762no—
392C4—3762no—
393C4#1568982311yescharacteristic
394C4#1568982311yescharacteristic
395C4#1568983211yescharacteristic
396C4#1568983211yescharacteristic
397C2—84no—
398C2#2024858411yescharacteristic
399C2#2025321912yes—
400C2#2025401011yescharacteristic
401C2#2025403211yescharacteristic
402C2#2025448212yes—
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2328C2
socleC23×C47C22×D4
prime radicalC2×#127×C47C22
hypercenter#2328D47
hypocenterC47#29201
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2328

Complements

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

NH ≅ G/Ntype
1. #20249363C2semidirect
2. #20244776C2semidirect
3. #20244762C2semidirect
6. #20243649C2semidirect
7. #20243635C2semidirect
8. #20243550C2semidirect
9. #20243521C2semidirect
10. #20243245C2semidirect
240. C47#29201semidirect

Maximal subgroups

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

subgroupindex
#202493632
#202447762
#202447622
#202442132
#202441952
#202436492
#202436352
#202435502
#202435212
#202432452
C47×#23282
#2920147