Finite Group Databasebeta

← #26126564

Subgroups — #26126564

Order 12032.

592 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 107 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#20249193C211yesmaximal, characteristic
2C2×#15688890C211yesmaximal, characteristic
3#20241413C212yesmaximal, complemented
4#20241410C212yesmaximal, complemented
5C2×C47×#323C211yesmaximal, characteristic
6C2×#13094003C411yescharacteristic, complemented
7C22×C22⋊Dic47—21no—
8#15689059C411yescharacteristic, complemented
9#15689059C2211yescharacteristic
10#15689046—21no—
11#15689046—21no—
12#15689035C411yescharacteristic, complemented
13#15689035C2211yescharacteristic
14#15688996—21no—
15#15688983C411yescharacteristic, complemented
16#15688979—21no—
17#15688890—21no—
18#15688890C2212yes—
19C22×C22⋊C4×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#323—21no—
22C47×#323C2212yes—
23C22×Dic94C2×C411yescharacteristic, complemented
24C23×Dic47—41no—
25#13094003—41no—
26#13094003—41no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic, complemented
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—24no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C2×C22⋊Dic47—21no—
39C2×C22⋊Dic47—21no—
40C2×C22⋊Dic47—41no—
41C2×C22⋊Dic47—42no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95C2×C411yescharacteristic, complemented
45C47⋊#95C2×C411yescharacteristic, complemented
46C25×C47D411yescharacteristic
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C22×C8×C47—41no—
50C2×C22⋊C4×C47—41no—
51C2×C22⋊C4×C47—41no—
52C2×C22⋊C4×C47C2×C411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D411yescharacteristic
55C2×C22⋊C4×C47C2311yescharacteristic
56C22⋊C8×C47—22no—
57C22⋊C8×C47—41no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—22no—
61C24×C47—81no—
62C24×C47—81no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C22⋊C411yescharacteristic
65C24×C47SD1612yes—
66C24×C47Q1612yes—
67C24×C47C2×D411yescharacteristic
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—81no—
73C22×C4×C47—81no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×Dic47—21no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—44no—
83C22×Dic47—42no—
84C22×Dic47—82no—
85C22×Dic47—82no—
86C2×Dic94—41no—
87C2×Dic94—81no—
88C2×C8×C47—42no—
89C2×C8×C47—82no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C22⋊Dic47—24no—
95C22⋊Dic47—44no—
96C22⋊Dic47—41no—
97C22⋊Dic47—41no—
98C22⋊Dic47—42no—
99C22⋊Dic47—42no—
100C22⋊Dic47—81no—
101C22⋊Dic47—81no—
102C22⋊Dic47—81no—
103C22⋊Dic47—81no—
104C4⋊Dic47—41no—
105C4⋊Dic47—81no—
106Dic47⋊C4—42no—
107Dic47⋊C4—81no—
108Dic47⋊C4—81no—
109C23×C47—22no—
110C23×C47—22no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—22no—
114C23×C47—22no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—42no—
120C23×C47—42no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—81no—
126C23×C47—81no—
127C23×C47—82no—
128C23×C47—82no—
129C23×C47—84no—
130C23×C47#9911yescharacteristic
131C23×C47#9911yescharacteristic
132C23×C47#10312yes—
133C23×C47#10312yes—
134C23×C47#10412yes—
135C23×C47C2×C22⋊C411yescharacteristic
136C23×C47C2×SD1611yescharacteristic
137C23×C47C2×Q1611yescharacteristic
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—42no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—81no—
148C2×C4×C47—81no—
149C2×C4×C47—81no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—44no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—41no—
162C2×Dic47—41no—
163C2×Dic47—88no—
164C2×Dic47—82no—
165C2×Dic47—82no—
166Dic94—82no—
167C8×C47—82no—
168#11716—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—22no—
177C22×C47—21no—
178C22×C47—42no—
179C22×C47—42no—
180C22×C47—42no—
181C22×C47—41no—
182C22×C47—41no—
183C22×C47—44no—
184C22×C47—44no—
185C22×C47—41no—
186C22×C47—41no—
187C22×C47—41no—
188C22×C47—41no—
189C22×C47—81no—
190C22×C47—81no—
191C22×C47—82no—
192C22×C47—82no—
193C22×C47#32812yes—
194C22×C47#32812yes—
195C22×C47C2×#9911yescharacteristic
196C22×C47C2×#10311yescharacteristic
197C22×C47C2×#10411yescharacteristic
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—41no—
201C4×C47—41no—
202Dic47—81no—
203Dic47—81no—
204Dic47—84no—
205Dic47—81no—
206Dic47—81no—
207#2626—471no—
208C2×#342—471no—
209C2×#323D4711yescharacteristic
210#1352—472no—
211#1351—472no—
212C2×C47—21no—
213C2×C47—22no—
214C2×C47—21no—
215C2×C47—22no—
216C2×C47—41no—
217C2×C47—42no—
218C2×C47—41no—
219C2×C47C2×#32811yescharacteristic
220C2×C47#212111yescharacteristic
221C2×C47#212711yescharacteristic
222C2×C22⋊Q8—471no—
223C22×C22⋊C4—941no—
224C22×C22⋊C4C2×D4711yescharacteristic
225C2×C22⋊C8—21no—
226#399—471no—
227#399—471no—
228#396—941no—
229#396—941no—
230#393—471no—
231#393—471no—
232#385—941no—
233#380—471no—
234#379—941no—
235#342—472no—
236#342—941no—
237#323—21no—
238#323C2×D4712yes—
239C47#1171611yescharacteristic, complemented, Sylow
240C25D18811yescharacteristic
241C22×Q8—471no—
242C23×C4—1881no—
243C23×C4C4×D4711yescharacteristic
244C23×C4C22⋊D4711yescharacteristic
245C22×C8—41no—
246C22⋊Q8—1881no—
247C22⋊Q8—1881no—
248C2×C4⋊C4—471no—
249C2×C4⋊C4—941no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—944no—
256C2×C22⋊C4—942no—
257C2×C22⋊C4—941no—
258C2×C22⋊C4—941no—
259C2×C22⋊C4—941no—
260C2×C22⋊C4—941no—
261C2×C22⋊C4—941no—
262C2×C22⋊C4—941no—
263C2×C22⋊C4—1881no—
264C2×C22⋊C4—1882no—
265C2×C22⋊C4C4×D4711yescharacteristic
266C2×C22⋊C4D18811yescharacteristic
267C2×C22⋊C4C22⋊D4711yescharacteristic
268C2×C22⋊C4C22×D4711yescharacteristic
269C22⋊C8—22no—
270C22⋊C8—41no—
271#95—471no—
272#95—471no—
273#95—941no—
274#95—941no—
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24#1062532512yes—
281C24Dic18812yes—
282C24(C2×D47)⋊C411yescharacteristic
283C24(C2×D47)⋊C411yescharacteristic
284C24C2×D18811yescharacteristic
285C2×Q8—3761no—
286C2×Q8—1881no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—941no—
294C22×C4—941no—
295C22×C4—941no—
296C22×C4—941no—
297C22×C4—3762no—
298C22×C4—3762no—
299C22×C4—1882no—
300C22×C4—1884no—
301C22×C4(C2×D47)⋊C411yescharacteristic
302C22×C4(C2×D47)⋊C411yescharacteristic
303C22×C4C2×C4×D4711yescharacteristic
304C22×C4C2×C22⋊D4711yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—3761no—
308C4⋊C4—3761no—
309C4⋊C4—3761no—
310C4⋊C4—1881no—
311C4⋊C4—1882no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—944no—
317C22⋊C4—3761no—
318C22⋊C4—3761no—
319C22⋊C4—3761no—
320C22⋊C4—3761no—
321C22⋊C4—1884no—
322C22⋊C4—1881no—
323C22⋊C4—1881no—
324C22⋊C4—1882no—
325C22⋊C4—1882no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#1309393811yescharacteristic
348C23#1309393811yescharacteristic
349C23#1309394912yes—
350C23#1309394912yes—
351C23#1309395012yes—
352C23C2×(C2×D47)⋊C411yescharacteristic
353C23C2×#1062532511yescharacteristic
354C23C2×Dic18811yescharacteristic
355Q8—3762no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—3768no—
369C2×C4—3762no—
370C2×C4—3762no—
371C2×C4—1884no—
372C2×C4—1881no—
373C2×C4—1881no—
374C2×C4—1881no—
375C2×C4—1881no—
376C2×C4—1881no—
377C2×C4—1881no—
378C2×C4—1881no—
379C2×C4—1881no—
380C2×C4—1881no—
381C2×C4—1881no—
382C2×C4—1881no—
383C2×C4—1881no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22#1568886112yes—
410C22#1568886112yes—
411C22C2×#1309393811yescharacteristic
412C22C2×#1309394911yescharacteristic
413C22C2×#1309395011yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—3761no—
419C4—3761no—
420C4—3764no—
421C4—3761no—
422C4—3761no—
423C2—21no—
424C2—21no—
425C2—22no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×#1568886111yescharacteristic
431C2#2024503511yescharacteristic
432C2#2024505511yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#323C2
socleC22×C47C2×#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#323D47
hypocenterC47#11716
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20241413C2semidirect
4. #20241410C2semidirect
6. C2×#13094003C4semidirect
8. #15689059C4semidirect
12. #15689035C4semidirect
15. #15688983C4semidirect
23. C22×Dic94C2×C4semidirect
28. C2×C4⋊Dic47C2×C4semidirect
44. C47⋊#95C2×C4semidirect
45. C47⋊#95C2×C4semidirect
239. C47#11716semidirect

Maximal subgroups

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

subgroupindex
#202491932
C2×#156888902
#202414132
#202414102
C2×C47×#3232
#1171647