Finite Group Databasebeta

← #26119163

Subgroups — #26119163

Order 12032.

602 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 105 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#20254160C211yesmaximal, characteristic
2#20251727C211yesmaximal, characteristic
3#20243456C211yesmaximal, characteristic
4#20241960C211yesmaximal, characteristic
5#20241959C211yesmaximal, characteristic
6#20241255C211yesmaximal, characteristic
7C47×#2855C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689458—21no—
10#15689456—21no—
11#15689362—22no—
12#15689279—21no—
13#15689278—21no—
14C2×C22⋊(C47⋊C8)C2211yescharacteristic
15C2×C22⋊(C47⋊C8)C2211yescharacteristic
16#15689029—21no—
17#15689021—21no—
18#15689007—22no—
19#15688996—21no—
20#15688986—21no—
21#15688983C2211yescharacteristic
22#15688878—22no—
23#15688860—22no—
24#15688859—22no—
25C2×C22⋊Q8×C47C2211yescharacteristic
26C22×C42×C47C2211yescharacteristic
27C47×#391—21no—
28C47×#389—21no—
29C47×#387—22no—
30C47×#385—21no—
31C47×#381—21no—
32C47×#380C2211yescharacteristic
33C22×Dic94D411yescharacteristic
34C2×#10625337—22no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8D412yes—
37#13094003—21no—
38#13094003—21no—
39#13094003—21no—
40#13094003—21no—
41C2×C4⋊Dic47—21no—
42C2×C4⋊Dic47D411yescharacteristic
43C2×Dic47⋊C4—22no—
44C2×Dic47⋊C4—21no—
45C2×C22⋊Dic47—21no—
46C2×C22⋊Dic47—21no—
47#13093957—22no—
48#13093948—22no—
49#13093947—22no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)—21no—
53C22⋊(C47⋊C8)—21no—
54C47⋊#95—22no—
55C47⋊#95—22no—
56C47⋊#95—22no—
57C47⋊#95D412yes—
58C22×Q8×C47D411yescharacteristic
59C23×C4×C47D411yescharacteristic
60C23×C4×C47D411yescharacteristic
61C23×C4×C47C2311yescharacteristic
62C22⋊Q8×C47—21no—
63C22⋊Q8×C47—21no—
64C22⋊Q8×C47—21no—
65C22⋊Q8×C47—21no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C22⋊C4×C47—21no—
71C2×C22⋊C4×C47—21no—
72C2×C42×C47—21no—
73C2×C42×C47—21no—
74C2×C42×C47—21no—
75C2×C42×C47—21no—
76C2×C42×C47—21no—
77C2×C42×C47—21no—
78#95×C47—22no—
79#95×C47—22no—
80#95×C47—22no—
81#95×C47D412yes—
82C24×C47C2×D411yescharacteristic
83C2×Q8×C47—21no—
84C2×Q8×C47—21no—
85C2×Q8×C47—42no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—21no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47—22no—
97C22×C4×C47—21no—
98C22×C4×C47—42no—
99C22×C4×C47—42no—
100C22×C4×C47Q1612yescomplemented
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47#4111yescharacteristic
107C22×Dic47—21no—
108C22×Dic47—22no—
109C22×Dic47—21no—
110C2×Dic94—21no—
111C2×Dic94—21no—
112C2×Dic94—42no—
113C4⋊C4×C47—21no—
114C4⋊C4×C47—21no—
115C4⋊C4×C47—42no—
116C4⋊C4×C47—44no—
117C4⋊C4×C47—41no—
118C4⋊C4×C47—41no—
119C22⋊C4×C47—41no—
120C22⋊C4×C47—41no—
121C22⋊C4×C47—41no—
122C22⋊C4×C47—41no—
123C42×C47—41no—
124C42×C47—41no—
125C42×C47—41no—
126C42×C47—41no—
127C22⋊Dic47—41no—
128C22⋊Dic47—41no—
129C22⋊Dic47—41no—
130C22⋊Dic47—41no—
131#10625337—82no—
132C4⋊Dic47—21no—
133C4⋊Dic47—21no—
134C4⋊Dic47—42no—
135Dic47⋊C4—44no—
136Dic47⋊C4—41no—
137Dic47⋊C4—41no—
138C2×C47⋊C8—22no—
139C2×C47⋊C8—42no—
140C2×C47⋊C8—42no—
141C2×C47⋊C8—41no—
142C23×C47—41no—
143C23×C47—41no—
144C23×C47#12011yescharacteristic
145C23×C47#12011yescharacteristic
146C23×C47#12111yescharacteristic
147C23×C47C2×Q1611yescharacteristic
148C23×C47#13511yescharacteristic
149C23×C47#13711yescharacteristic
150C23×C47#13711yescharacteristic
151Q8×C47—42no—
152Q8×C47—81no—
153C2×C4×C47—22no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—22no—
161C2×C4×C47—22no—
162C2×C4×C47—41no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
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—42no—
174C2×C4×C47—42no—
175C2×C4×C47—42no—
176C2×C4×C47—42no—
177C2×C4×C47—41no—
178C2×C4×C47—41no—
179C2×C4×C47—42no—
180C2×C4×C47#12011yescharacteristic
181C2×C4×C47#12111yescharacteristic
182C2×C4×C47#12111yescharacteristic
183C2×C4×C47#12711yescharacteristic
184C2×Dic47—42no—
185C2×Dic47—42no—
186C2×Dic47—42no—
187C2×Dic47—42no—
188C2×Dic47—41no—
189C2×Dic47—41no—
190C2×Dic47—42no—
191Dic94—42no—
192Dic94—81no—
193C47⋊C8—42no—
194C47⋊C8—81no—
195#10325—471nomaximal, Sylow
196C22×C47—21no—
197C22×C47—21no—
198C22×C47—21no—
199C22×C47—21no—
200C22×C47—21no—
201C22×C47—21no—
202C22×C47—41no—
203C22×C47—41no—
204C22×C47—41no—
205C22×C47—41no—
206C22×C47#39211yescharacteristic
207C22×C47#45111yescharacteristic
208C22×C47#45111yescharacteristic
209C22×C47#45211yescharacteristic
210C22×C47#45211yescharacteristic
211C22×C47#46711yescharacteristic
212C22×C47#47011yescharacteristic
213C4×C47—21no—
214C4×C47—21no—
215C4×C47—21no—
216C4×C47—42no—
217C4×C47—42no—
218C4×C47—82no—
219C4×C47—81no—
220C4×C47—81no—
221C4×C47#45812yes—
222Dic47—82no—
223Dic47—81no—
224Dic47—81no—
225#3189—471no—
226#2855—471no—
227#2855D4711yescharacteristic
228#1817—471no—
229#1520—471no—
230#1520—471no—
231#1289—471no—
232C2×C47—21no—
233C2×C47—21no—
234C2×C47—21no—
235C2×C47—21no—
236C2×C47—21no—
237C2×C47—21no—
238C2×C47#200111yescharacteristic
239C2×C47#201011yescharacteristic
240C2×C47#202211yescharacteristic
241C2×C22⋊Q8—471no—
242C2×C22⋊Q8C2×D4711yescharacteristic
243C22×C42C2×D4711yescharacteristic
244#484—941no—
245#484—941no—
246#467—942no—
247#451—941no—
248#451—941no—
249C2×C22⋊C8—471no—
250C2×C22⋊C8—471no—
251#391—21no—
252#391—941no—
253#389—21no—
254#389—941no—
255#387—22no—
256#387—942no—
257#385—21no—
258#385—941no—
259#381—21no—
260#381—941no—
261#380—471no—
262#380C2×D4711yescharacteristic
263#336—942no—
264#328—942no—
265#328—942no—
266C47#1032511yescharacteristic, complemented, Sylow
267C22×Q8—471no—
268C22×Q8C22⋊D4711yescharacteristic
269C23×C4C22⋊D4711yescharacteristic
270C23×C4C22⋊D4711yescharacteristic
271C23×C4C22×D4711yescharacteristic
272C2×Q16—942no—
273C22×C8—472no—
274C22×C8—941no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8—941no—
280C22⋊Q8—941no—
281C22⋊Q8—941no—
282C22⋊Q8—941no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—471no—
287C2×C4⋊C4—942no—
288C2×C4⋊C4—941no—
289C2×C4⋊C4—941no—
290C2×C4⋊C4C22⋊D4711yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—941no—
294C2×C22⋊C4—941no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301#107—942no—
302#103—942no—
303#103—942no—
304C22⋊C8—941no—
305C22⋊C8—941no—
306C22⋊C8—941no—
307C22⋊C8—941no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—472no—
312#95—942no—
313#95—942no—
314#95—942no—
315#95C22⋊D4712yes—
316C24C2×C22⋊D4711yescharacteristic
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—42no—
320C2×Q8—941no—
321C2×Q8—941no—
322C2×Q8—1882no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—942no—
338C22×C4—941no—
339C22×C4—941no—
340C22×C4#1062533712yescomplemented
341C22×C4D4×D4711yescharacteristic
342C22×C4D4×D4711yescharacteristic
343C22×C4D4×D4711yescharacteristic
344C22×C4#1062535211yescharacteristic
345C22×C4C2×C22⋊D4711yescharacteristic
346C22×C4C2×C22⋊D4711yescharacteristic
347Q16—3762no—
348C2×C8—942no—
349C2×C8—1882no—
350C2×C8—1881no—
351C2×C8—1882no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—941no—
359C4⋊C4—941no—
360C4⋊C4—1884no—
361C4⋊C4—1882no—
362C4⋊C4—1881no—
363C4⋊C4—1881no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—1881no—
369C22⋊C4—1881no—
370C22⋊C4—1881no—
371C22⋊C4—1881no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C23—41no—
377C23—41no—
378C23#1309399211yescharacteristic
379C23#1309399311yescharacteristic
380C23#1309399611yescharacteristic
381C23C2×#1062533711yescharacteristic
382C23#1309404411yescharacteristic
383C23#1309405411yescharacteristic
384C23#1309405511yescharacteristic
385Q8—42no—
386Q8—81no—
387Q8—3761no—
388Q8—1882no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
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—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—1882no—
417C2×C4—1882no—
418C2×C4—1882no—
419C2×C4—1882no—
420C2×C4—1882no—
421C2×C4—1881no—
422C2×C4—1881no—
423C2×C4#1309399211yescharacteristic
424C2×C4#1309399511yescharacteristic
425C2×C4#1309399611yescharacteristic
426C2×C4#1309402011yescharacteristic
427C8—3761no—
428C8—1882no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—21no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22#1568903111yescharacteristic
440C22#1568927811yescharacteristic
441C22#1568927911yescharacteristic
442C22#1568928311yescharacteristic
443C22#1568928511yescharacteristic
444C22#1568936211yescharacteristic
445C22#1568937911yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—42no—
450C4—42no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—3762no—
455C4—3761no—
456C4—3761no—
457C4#1568931212yes—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—21no—
464C2#2024439611yescharacteristic
465C2#2024445311yescharacteristic
466C2#2024451611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2855C2
socleC22×C47#392
prime radicalC24×C47C2×D4
hypercenter#2855D47
hypocenterC47#10325
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2855

Complements

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

NH ≅ G/Ntype
100. C22×C4×C47Q16semidirect
266. C47#10325semidirect
340. C22×C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202541602
#202517272
#202434562
#202419602
#202419592
#202412552
C47×#28552
#1032547