Finite Group Databasebeta

← #26113306

Subgroups — #26113306

Order 12032.

478 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 99 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#20254209C211yesmaximal, characteristic
2#20244372C211yesmaximal, characteristic
3#20241920C211yesmaximal, characteristic
4#20241908C211yesmaximal, characteristic
5#20241294C211yesmaximal, characteristic
6#20241240C211yesmaximal, characteristic
7C47×#2378C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689453—21no—
10#15689451C412yescomplemented
11#15689379—22no—
12#15689279—21no—
13#15689278C412yescomplemented
14#15689087C2211yescharacteristic
15#15689087C2211yescharacteristic
16C4×C22⋊Dic47C2211yescharacteristic
17#15688889—21no—
18#15688889—21no—
19#15688860—22no—
20#15688859—21no—
21#15688859—21no—
22C2×C22⋊Q8×C47C2211yescharacteristic
23C47×#391—21no—
24C47×#389—21no—
25C47×#388—21no—
26C47×#388—21no—
27C47×#387—21no—
28C47×#387—21no—
29C47×#381C2211yescharacteristic
30C47×#380C2211yescharacteristic
31C22×Dic94C2×C411yescharacteristic
32C2×#10625337—22no—
33C2×#10625329—21no—
34C2×#10625329—21no—
35#13094003—21no—
36#13094003—21no—
37#13094003—21no—
38#13094003—21no—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×Dic47⋊C4—21no—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47—21no—
43C2×C4×Dic47—21no—
44#13093955—22no—
45#13093948—22no—
46#13093947—22no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)C2×C412yes—
51C47⋊#95—21no—
52C22×Q8×C47D411yescharacteristic
53C23×C4×C47C2311yescharacteristic
54C22⋊Q8×C47—21no—
55C22⋊Q8×C47—21no—
56C22⋊Q8×C47C2×C412yes—
57C22⋊Q8×C47D412yes—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47D411yescharacteristic
66C2×C22⋊C4×C47D411yescharacteristic
67C2×C42×C47—21no—
68C2×C42×C47—21no—
69#95×C47—21no—
70#95×C47—21no—
71#95×C47—21no—
72#95×C47—21no—
73#95×C47—21no—
74#95×C47—21no—
75#95×C47D411yescharacteristic
76#95×C47D411yescharacteristic
77C24×C47C2×D411yescharacteristic
78C2×Q8×C47—21no—
79C2×Q8×C47—81no—
80C2×Q8×C47C22⋊C412yes—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—81no—
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—21no—
95C22×Dic47—21no—
96C22×Dic47—41no—
97C2×Dic94—21no—
98C2×Dic94—21no—
99C2×Dic94—81no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—81no—
104C4⋊C4×C47—81no—
105C4⋊C4×C47—81no—
106C4⋊C4×C47C22⋊C412yes—
107C22⋊C4×C47—22no—
108C22⋊C4×C47—22no—
109C22⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—81no—
112C42×C47—81no—
113C42×C47—81no—
114C22⋊Dic47—22no—
115C22⋊Dic47—41no—
116C22⋊Dic47—41no—
117C22⋊Dic47—41no—
118#10625337—82no—
119C4⋊Dic47—21no—
120C4⋊Dic47—21no—
121Dic47⋊C4—41no—
122Dic47⋊C4—41no—
123C4×Dic47—81no—
124#10625329—81no—
125#10625329—81no—
126C2×C47⋊C8—22no—
127C2×C47⋊C8—41no—
128C23×C47—21no—
129C23×C47—21no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47C4×D411yescharacteristic
133C23×C47#12011yescharacteristic
134C23×C47#12111yescharacteristic
135Q8×C47—42no—
136Q8×C47—81no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47C2×C22⋊C411yescharacteristic
162C2×C4×C47C4×D411yescharacteristic
163C2×C4×C47#12111yescharacteristic
164C2×C4×C47#12311yescharacteristic
165C2×Dic47—42no—
166C2×Dic47—41no—
167C2×Dic47—41no—
168C2×Dic47—81no—
169C2×Dic47—81no—
170C2×Dic47—81no—
171C2×Dic47—81no—
172Dic94—42no—
173Dic94—81no—
174C47⋊C8—42no—
175C47⋊C8—81no—
176#9342—471nomaximal, Sylow
177C22×C47—21no—
178C22×C47—21no—
179C22×C47—21no—
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—81no—
186C22×C47—81no—
187C22×C47#38611yescharacteristic
188C22×C47#45511yescharacteristic
189C22×C47#45611yescharacteristic
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—21no—
194C4×C47—42no—
195C4×C47—42no—
196C4×C47—81no—
197C4×C47—81no—
198C4×C47—81no—
199C4×C47—81no—
200Dic47—82no—
201Dic47—81no—
202Dic47—81no—
203#3195—471no—
204#2378D4711yescharacteristic
205#1996—471no—
206#1511—471no—
207#1508—471no—
208#1305—471no—
209#1284—471no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47—41no—
213C2×C47—41no—
214C2×C47#188211yescharacteristic
215C2×C47#190511yescharacteristic
216C2×C47#191311yescharacteristic
217C2×C22⋊Q8—471no—
218C2×C22⋊Q8C2×D4711yescharacteristic
219#483—472no—
220#483—941no—
221#470—942no—
222#451—472no—
223#451—941no—
224#407—471no—
225#407—471no—
226#391—21no—
227#389—21no—
228#388—21no—
229#388—21no—
230#387—21no—
231#387—21no—
232#381C2×D4711yescharacteristic
233#380C2×D4711yescharacteristic
234C4×C22⋊C4—471no—
235#342—941no—
236#342—941no—
237#328—942no—
238#328—941no—
239#328—941no—
240C47#934211yescharacteristic, complemented, Sylow
241C22×Q8—471no—
242C22×Q8C22⋊D4711yescharacteristic
243C23×C4C22×D4711yescharacteristic
244C2×Q16—942no—
245C2×M16—941no—
246C2×M16—941no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—941no—
250C22⋊Q8—941no—
251C22⋊Q8—941no—
252C22⋊Q8—941no—
253C22⋊Q8C4×D4712yes—
254C22⋊Q8D18812yes—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—471no—
260C2×C4⋊C4—941no—
261C2×C4⋊C4C22⋊D4711yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—941no—
265C2×C22⋊C4—941no—
266C2×C22⋊C4C22⋊D4711yescharacteristic
267C2×C22⋊C4C22⋊D4711yescharacteristic
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—941no—
271#106—942no—
272#103—942no—
273#103—942no—
274C22⋊C8—472no—
275C22⋊C8—941no—
276C22⋊C8—941no—
277C22⋊C8—941no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—941no—
285#95C22⋊D4711yescharacteristic
286#95C22⋊D4711yescharacteristic
287C24C2×C22⋊D4711yescharacteristic
288C2×Q8—21no—
289C2×Q8—81no—
290C2×Q8—941no—
291C2×Q8—941no—
292C2×Q8—3761no—
293C2×Q8(C2×D47)⋊C412yes—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—81no—
301C22×C4—941no—
302C22×C4—941no—
303C22×C4—1881no—
304C22×C4C2×C4×D4711yescharacteristic
305C22×C4C2×D18811yescharacteristic
306C22×C4#1062535011yescharacteristic
307C22×C4#1062535011yescharacteristic
308C22×C4C2×C22⋊D4711yescharacteristic
309C22×C4C2×C22⋊D4711yescharacteristic
310Q16—3762no—
311M16—3761no—
312M16—3761no—
313C2×C8—942no—
314C2×C8—1881no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—81no—
319C4⋊C4—81no—
320C4⋊C4—81no—
321C4⋊C4—941no—
322C4⋊C4—941no—
323C4⋊C4—1881no—
324C4⋊C4—1881no—
325C4⋊C4(C2×D47)⋊C412yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—81no—
331C22⋊C4—942no—
332C22⋊C4—1881no—
333C22⋊C4—1881no—
334C22⋊C4—1881no—
335C42—81no—
336C42—81no—
337C42—3761no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23C4×C22⋊D4711yescharacteristic
343C23#1309399311yescharacteristic
344C23#1309399811yescharacteristic
345Q8—42no—
346Q8—81no—
347Q8—3761no—
348Q8—1882no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—3761no—
374C2×C4—3761no—
375C2×C4—3761no—
376C2×C4—3761no—
377C2×C4—1882no—
378C2×C4—1881no—
379C2×C4—1881no—
380C2×C4C2×(C2×D47)⋊C411yescharacteristic
381C2×C4C4×C22⋊D4711yescharacteristic
382C2×C4#1309399811yescharacteristic
383C2×C4#1309400711yescharacteristic
384C8—3761no—
385C8—1882no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—81no—
395C22—81no—
396C22#1568900211yescharacteristic
397C22#1568930111yescharacteristic
398C22#1568930611yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4—81no—
409C4—3762no—
410C4—3761no—
411C4—3761no—
412C2—21no—
413C2—21no—
414C2—41no—
415C2—41no—
416C2#2024375711yescharacteristic
417C2#2024388211yescharacteristic
418C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2378C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2378D47
hypocenterC47#9342
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2378

Complements

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

NH ≅ G/Ntype
10. #15689451C4semidirect
13. #15689278C4semidirect
240. C47#9342semidirect

Maximal subgroups

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

subgroupindex
#202542092
#202443722
#202419202
#202419082
#202412942
#202412402
C47×#23782
#934247