Finite Group Databasebeta

← #26112755

Subgroups — #26112755

Order 12032.

478 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252444C211yesmaximal, characteristic
2#20241892C211yesmaximal, characteristic
3C2×#15688860C211yesmaximal, characteristic, complemented
4#20241752C211yesmaximal, characteristic, complemented
5#20241276C211yesmaximal, characteristic, complemented
6#20241230C211yesmaximal, characteristic, complemented
7C47×#3190C211yesmaximal, characteristic
8#15689195—21no—
9#15689195—21no—
10D4×C47⋊C8—21no—
11D4×C47⋊C8—21no—
12#15689184—21no—
13#15689090—21no—
14#15689087C2211yescharacteristic
15C2×C22⋊(C47⋊C8)C2211yescharacteristic
16C2×C22⋊(C47⋊C8)C2211yescharacteristic
17C2×C47⋊#95C2211yescharacteristic
18#15688860—21no—
19#15688860—21no—
20#15688860—21no—
21#15688857—21no—
22#15688849—21no—
23#15688849—21no—
24C47×#525C2211yescharacteristic
25C2×C22⋊Q8×C47C2211yescharacteristic
26C47×#485—21no—
27C47×#483C411yescharacteristic, complemented
28C47×#480C411yescharacteristic, complemented
29C47×#467—21no—
30C47×#465—21no—
31C47×#452—21no—
32C47×#451C411yescharacteristic, complemented
33C47×#448C411yescharacteristic, complemented
34C2×C22⋊C8×C47C2211yescharacteristic
35C23×Dic47—21no—
36C2×#10625329—21no—
37C2×#10625329—21no—
38C22×C47⋊C8—21no—
39C22×C47⋊C8—21no—
40C22×C47⋊C8—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C22⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)D411yescharacteristic
50C22⋊(C47⋊C8)D411yescharacteristic
51C22⋊(C47⋊C8)D411yescharacteristic
52C22⋊(C47⋊C8)D411yescharacteristic
53#13093935—21no—
54C4×C47⋊C8—21no—
55C47⋊#95—21no—
56C47⋊#95—21no—
57C2×#41×C47D411yescharacteristic
58C22×Q8×C47C2×C411yescharacteristic
59C23×C4×C47C2311yescharacteristic
60C2×Q16×C47—21no—
61C2×SD16×C47—21no—
62C22×C8×C47—21no—
63#123×C47—21no—
64C22⋊Q8×C47—21no—
65C22⋊Q8×C47—21no—
66C22⋊Q8×C47—21no—
67C22⋊Q8×C47—21no—
68C22⋊Q8×C47C2×C411yescharacteristic
69C22⋊Q8×C47D411yescharacteristic
70#121×C47C2×C411yescharacteristic
71#121×C47D411yescharacteristic
72#120×C47—21no—
73C4×D4×C47—21no—
74C4×D4×C47—21no—
75#117×C47D411yescharacteristic
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47C2×C411yescharacteristic
78C2×C22⋊C4×C47—21no—
79#107×C47—21no—
80#106×C47—21no—
81#103×C47—21no—
82#103×C47—21no—
83#103×C47—21no—
84#102×C47—21no—
85C22⋊C8×C47—21no—
86C22⋊C8×C47C2×C411yescharacteristic
87C22⋊C8×C47C2×C411yescharacteristic
88C24×C47#4111yescharacteristic
89#41×C47—81no—
90C2×Q8×C47—21no—
91C2×Q8×C47—21no—
92C2×Q8×C47—81no—
93C2×Q8×C47C22⋊C411yescharacteristic
94C2×D4×C47—21no—
95C2×D4×C47—21no—
96C2×D4×C47—21no—
97C2×D4×C47C22⋊C411yescharacteristic
98C22×C4×C47—21no—
99C22×C4×C47—21no—
100C22×C4×C47—21no—
101C22×C4×C47—81no—
102C22×C4×C47C22×C411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×Dic47—22no—
109C22×Dic47—41no—
110C22×Dic47—41no—
111C22×Dic47—81no—
112C22×Dic47—81no—
113Q16×C47—81no—
114SD16×C47—81no—
115C2×C8×C47—21no—
116C2×C8×C47—21no—
117C2×C8×C47—81no—
118C4⋊C4×C47—21no—
119C4⋊C4×C47—21no—
120C4⋊C4×C47—21no—
121C4⋊C4×C47—21no—
122C4⋊C4×C47—41no—
123C4⋊C4×C47—41no—
124C4⋊C4×C47C22⋊C411yescharacteristic
125C4⋊C4×C47C22⋊C411yescharacteristic
126C22⋊C4×C47—21no—
127C22⋊C4×C47—21no—
128C22⋊C4×C47—21no—
129C22⋊C4×C47—21no—
130C22⋊C4×C47—21no—
131C22⋊C4×C47—41no—
132C22⋊C4×C47—41no—
133C42×C47—21no—
134#10625329—81no—
135#10625329—81no—
136C2×C47⋊C8—21no—
137C2×C47⋊C8—21no—
138C2×C47⋊C8—21no—
139C2×C47⋊C8—21no—
140C2×C47⋊C8—41no—
141C2×C47⋊C8—81no—
142C2×C47⋊C8—81no—
143C2×C47⋊C8—81no—
144C23×C47—21no—
145C23×C47—21no—
146C23×C47—21no—
147C23×C47—21no—
148C23×C47—21no—
149C23×C47—41no—
150C23×C47C4×D411yescharacteristic
151C23×C47#12111yescharacteristic
152C23×C47#12311yescharacteristic
153Q8×C47—41no—
154Q8×C47—41no—
155Q8×C47—41no—
156Q8×C47—81no—
157D4×C47—41no—
158D4×C47—41no—
159D4×C47—41no—
160D4×C47—41no—
161D4×C47—41no—
162D4×C47—81no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—41no—
169C2×C4×C47—41no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47—41no—
173C2×C4×C47—41no—
174C2×C4×C47—41no—
175C2×C4×C47—41no—
176C2×C4×C47—41no—
177C2×C4×C47—41no—
178C2×C4×C47—81no—
179C2×C4×C47C2×C22⋊C411yescharacteristic
180C2×C4×C47C4×D411yescharacteristic
181C2×C4×C47#12011yescharacteristic
182C2×C4×C47#12111yescharacteristic
183C2×Dic47—42no—
184C2×Dic47—81no—
185C2×Dic47—82no—
186C2×Dic47—81no—
187C2×Dic47—82no—
188C8×C47—41no—
189C8×C47—41no—
190C47⋊C8—41no—
191C47⋊C8—41no—
192C47⋊C8—41no—
193C47⋊C8—41no—
194C47⋊C8—81no—
195#9263—471nomaximal, Sylow
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—21no—
203C22×C47—41no—
204C22×C47—41no—
205C22×C47—41no—
206C22×C47—41no—
207C22×C47—41no—
208C22×C47—41no—
209C22×C47—41no—
210C22×C47#38611yescharacteristic
211C22×C47#44311yescharacteristic
212C22×C47#44411yescharacteristic
213C22×C47#47111yescharacteristic
214C22×C47#47311yescharacteristic
215C4×C47—21no—
216C4×C47—21no—
217C4×C47—21no—
218C4×C47—21no—
219C4×C47—41no—
220C4×C47—41no—
221C4×C47—41no—
222C4×C47—81no—
223C4×C47—81no—
224Dic47—82no—
225#3190D4711yescharacteristic
226#2977—471no—
227#1504—471no—
228C2×#328—471no—
229#1472—471no—
230#1298—471no—
231#1280—471no—
232C2×C47—21no—
233C2×C47—21no—
234C2×C47—21no—
235C2×C47—21no—
236C2×C47—41no—
237C2×C47—81no—
238C2×C47#187611yescharacteristic
239C2×C47#190811yescharacteristic
240C2×C47#191311yescharacteristic
241#525C2×D4711yescharacteristic
242C2×C22⋊Q8C2×D4711yescharacteristic
243#485—21no—
244#483Dic4711yescharacteristic, complemented
245#480Dic4711yescharacteristic, complemented
246#467—21no—
247#465—21no—
248#452—21no—
249#451Dic4711yescharacteristic, complemented
250#448Dic4711yescharacteristic, complemented
251#435—941no—
252#435—941no—
253C8×D4—941no—
254C8×D4—941no—
255#433—941no—
256#408—941no—
257#407—471no—
258C2×C22⋊C8—471no—
259C2×C22⋊C8—471no—
260C2×C22⋊C8C2×D4711yescharacteristic
261C2×#95—471no—
262#328—941no—
263#328—941no—
264#328—941no—
265#327—941no—
266#324—941no—
267#324—941no—
268C47#926311yescharacteristic, complemented, Sylow
269C2×#41C22⋊D4711yescharacteristic
270C22×Q8C2×Dic4711yescharacteristic
271C23×C4—941no—
272C23×C4C22×D4711yescharacteristic
273C2×Q16—21no—
274C2×SD16—21no—
275C2×M16—941no—
276C2×M16—941no—
277C22×C8—21no—
278C22×C8—941no—
279C22×C8—941no—
280C22×C8—941no—
281#123—21no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286C22⋊Q8C2×Dic4711yescharacteristic
287C22⋊Q8C22⋊D4711yescharacteristic
288#121C2×Dic4711yescharacteristic
289#121C22⋊D4711yescharacteristic
290#120—21no—
291C4×D4—21no—
292C4×D4—21no—
293#117C22⋊D4711yescharacteristic
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×Dic4711yescharacteristic
296C2×C22⋊C4—21no—
297#107—21no—
298#106—21no—
299C4⋊C8—941no—
300C4⋊C8—941no—
301#103—21no—
302#103—21no—
303#103—21no—
304#102—21no—
305C22⋊C8—21no—
306C22⋊C8—471no—
307C22⋊C8—471no—
308C22⋊C8—471no—
309C22⋊C8—471no—
310C22⋊C8—941no—
311C22⋊C8—941no—
312C22⋊C8—941no—
313C22⋊C8—941no—
314C22⋊C8—941no—
315C22⋊C8—941no—
316C22⋊C8C2×Dic4711yescharacteristic
317C22⋊C8C2×Dic4711yescharacteristic
318#97—941no—
319C4×C8—941no—
320#95—941no—
321#95—941no—
322C24#1062535211yescharacteristic
323#41—81no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—81no—
327C2×Q8C22⋊Dic4711yescharacteristic
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4C22⋊Dic4711yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—81no—
336C22×C4—942no—
337C22×C4—3761no—
338C22×C4—3761no—
339C22×C4—1881no—
340C22×C4—1881no—
341C22×C4D4×D4711yescharacteristic
342C22×C4D4×D4711yescharacteristic
343C22×C4#1062535211yescharacteristic
344C22×C4C22×Dic4711yescharacteristic
345C22×C4C2×C22⋊D4711yescharacteristic
346C22×C4C2×C22⋊D4711yescharacteristic
347Q16—81no—
348SD16—81no—
349M16—3761no—
350M16—3761no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—81no—
354C2×C8—941no—
355C2×C8—941no—
356C2×C8—941no—
357C2×C8—941no—
358C2×C8—3761no—
359C2×C8—3761no—
360C2×C8—3761no—
361C2×C8—1881no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C22⋊Dic4711yescharacteristic
369C4⋊C4C22⋊Dic4711yescharacteristic
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—21no—
373C22⋊C4—21no—
374C22⋊C4—21no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C42—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—41no—
384C23D4×Dic4711yescharacteristic
385C23#1309399611yescharacteristic
386C23#1309400511yescharacteristic
387Q8—41no—
388Q8—41no—
389Q8—41no—
390Q8—81no—
391D4—41no—
392D4—41no—
393D4—41no—
394D4—41no—
395D4—41no—
396D4—81no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—81no—
413C2×C4—3761no—
414C2×C4—3762no—
415C2×C4—3761no—
416C2×C4—3762no—
417C2×C4—1882no—
418C2×C4C2×C22⋊Dic4711yescharacteristic
419C2×C4D4×Dic4711yescharacteristic
420C2×C4#1309399211yescharacteristic
421C2×C4#1309399611yescharacteristic
422C8—41no—
423C8—41no—
424C8—3761no—
425C8—1881no—
426C8—1881no—
427C8—1881no—
428C8—1881no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22#1568900311yescharacteristic
444C22#1568924011yescharacteristic
445C22#1568924511yescharacteristic
446C22#1568938811yescharacteristic
447C22#1568939611yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—81no—
456C4—81no—
457C4—3762no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—81no—
464C2#2024371711yescharacteristic
465C2#2024390511yescharacteristic
466C2#2024393111yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#3190C2
socleC22×C47#386
prime radicalC47×#525C22
cosocleC23×C4×C47C23
hypercenter#3190D47
hypocenterC47#9263
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3190

Complements

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

NH ≅ G/Ntype
3. C2×#15688860C2semidirect
4. #20241752C2semidirect
5. #20241276C2semidirect
6. #20241230C2semidirect
27. C47×#483C4semidirect
28. C47×#480C4semidirect
32. C47×#451C4semidirect
33. C47×#448C4semidirect
244. #483Dic47semidirect
245. #480Dic47semidirect
249. #451Dic47semidirect
250. #448Dic47semidirect
268. C47#9263semidirect

Maximal subgroups

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

subgroupindex
#202524442
#202418922
C2×#156888602
#202417522
#202412762
#202412302
C47×#31902
#926347