Finite Group Databasebeta

← #26112353

Subgroups — #26112353

Order 12032.

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

normal subgroups only

This group has 578 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#20249132C211yesmaximal, characteristic
2#20245990C211yesmaximal, characteristic, complemented
3#20241956C211yesmaximal, characteristic
4C2×#15688848C211yesmaximal, characteristic, complemented
5#20241256C211yesmaximal, characteristic, complemented
6#20241255C211yesmaximal, characteristic, complemented
7C47×#1519C211yesmaximal, characteristic
8C2×#13094005—21no—
9C22×Dic47⋊C4C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11#15689054C412yescomplemented
12#15689046—22no—
13#15689037C412yescomplemented
14#15689010—22no—
15#15689004—22no—
16#15689003—21no—
17#15688984—21no—
18#15688983C2211yescharacteristic
19#15688983C2211yescharacteristic
20C22⋊C4×Dic47—21no—
21#15688857—22no—
22#15688848—22no—
23C2×#121×C47C2211yescharacteristic
24C2×C22⋊C8×C47C2211yescharacteristic
25C47×#380C2211yescharacteristic
26C47×#327—22no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30#13094005—42no—
31#13094005—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×Dic47⋊C4—21no—
37C2×Dic47⋊C4—21no—
38C2×Dic47⋊C4—22no—
39C2×Dic47⋊C4C2×C412yes—
40C2×Dic47⋊C4D412yes—
41C2×C22⋊Dic47—21no—
42C2×C22⋊Dic47—21no—
43C2×C22⋊Dic47—22no—
44C2×C22⋊Dic47—22no—
45C2×C4×Dic47—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C47⋊#95—22no—
49C47⋊#95—24no—
50C47⋊#95—22no—
51C47⋊#95C2×C412yes—
52C47⋊#95D412yes—
53C22×D4×C47—21no—
54C22×D4×C47C2×C411yescharacteristic
55C23×C4×C47C2311yescharacteristic
56C22×C8×C47—21no—
57#121×C47—21no—
58#121×C47—21no—
59#121×C47—21no—
60#121×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47C2×C411yescharacteristic
63C2×C22⋊C4×C47—21no—
64C2×C22⋊C4×C47—21no—
65C22⋊C8×C47—21no—
66C22⋊C8×C47—21no—
67#95×C47D412yes—
68C24×C47—21no—
69C24×C47C2×D411yescharacteristic
70C2×D4×C47—21no—
71C2×D4×C47—21no—
72C2×D4×C47—41no—
73C2×D4×C47—41no—
74C2×D4×C47—41no—
75C2×D4×C47—41no—
76C2×D4×C47—41no—
77C2×D4×C47—41no—
78C2×D4×C47—81no—
79C22×C4×C47—21no—
80C22×C4×C47—22no—
81C22×C4×C47—81no—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47#4111yescharacteristic
87C22×C4×C47#4111yescharacteristic
88C22×Dic47—21no—
89C22×Dic47—21no—
90C22×Dic47—21no—
91C22×Dic47—21no—
92C22×Dic47—21no—
93C22×Dic47—21no—
94C22×Dic47—21no—
95C22×Dic47—21no—
96C22×Dic47—22no—
97C22×Dic47—22no—
98C22×Dic47—81no—
99C22×Dic47C22⋊C412yes—
100C22×Dic47C22⋊C412yes—
101C2×C8×C47—41no—
102C2×C8×C47—81no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—21no—
105C4⋊C4×C47—81no—
106C22⋊C4×C47—41no—
107C22⋊C4×C47—41no—
108C22⋊C4×C47—41no—
109C22⋊C4×C47—41no—
110C22⋊Dic47—44no—
111C22⋊Dic47—41no—
112C22⋊Dic47—41no—
113C22⋊Dic47—41no—
114C22⋊Dic47—41no—
115C22⋊Dic47—82no—
116C4⋊Dic47—81no—
117C4⋊Dic47—81no—
118Dic47⋊C4—22no—
119Dic47⋊C4—22no—
120Dic47⋊C4—22no—
121Dic47⋊C4—22no—
122Dic47⋊C4—44no—
123C4×Dic47—82no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—81no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47—42no—
129C23×C47—42no—
130C23×C47—41no—
131C23×C47—41no—
132C23×C47—41no—
133C23×C47C2×C22⋊C411yescharacteristic
134C23×C47#12011yescharacteristic
135C23×C47#12311yescharacteristic
136C23×C47#13511yescharacteristic
137C23×C47#13511yescharacteristic
138C23×C47AGL(1,ℤ/8)11yescharacteristic
139C23×C47AGL(1,ℤ/8)11yescharacteristic
140D4×C47—81no—
141D4×C47—81no—
142D4×C47—81no—
143D4×C47—81no—
144D4×C47—81no—
145D4×C47—81no—
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—42no—
153C2×C4×C47—42no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—42no—
158C2×C4×C47C4×D411yescharacteristic
159C2×C4×C47C4×D411yescharacteristic
160C2×C4×C47#12111yescharacteristic
161C2×C4×C47#12111yescharacteristic
162C2×Dic47—22no—
163C2×Dic47—22no—
164C2×Dic47—22no—
165C2×Dic47—22no—
166C2×Dic47—42no—
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—41no—
171C2×Dic47—42no—
172C2×Dic47—42no—
173C2×Dic47—42no—
174C2×Dic47—42no—
175C2×Dic47—42no—
176C2×Dic47—42no—
177C2×Dic47—42no—
178C2×Dic47—81no—
179C8×C47—81no—
180C47⋊C8—81no—
181#9206—471nomaximal, Sylow
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47—41no—
191C22×C47—41no—
192C22×C47—82no—
193C22×C47—82no—
194C22×C47—81no—
195C22×C47—81no—
196C22×C47—81no—
197C22×C47#38611yescharacteristic
198C22×C47#41611yescharacteristic
199C22×C47#41811yescharacteristic
200C22×C47#44911yescharacteristic
201C22×C47#44911yescharacteristic
202C22×C47#48211yescharacteristic
203C22×C47#48211yescharacteristic
204C4×C47—21no—
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—82no—
209C4×C47—81no—
210Dic47—42no—
211Dic47—42no—
212Dic47—82no—
213Dic47—82no—
214#2618—471no—
215#2313—471no—
216#1519—471no—
217#1519D4711yescharacteristic
218C2×#324—471no—
219#1289—471no—
220#1289—471no—
221C2×C47—21no—
222C2×C47—21no—
223C2×C47—21no—
224C2×C47—21no—
225C2×C47—21no—
226C2×C47—21no—
227C2×C47—161no—
228C2×C47#187511yescharacteristic
229C2×C47#191211yescharacteristic
230C2×C47#191311yescharacteristic
231C2×#123—941no—
232C2×#121C2×D4711yescharacteristic
233C22×C4⋊C4—471no—
234C2×C22⋊C8—471no—
235C2×C22⋊C8C2×D4711yescharacteristic
236#398—472no—
237#396—942no—
238#394—472no—
239#387—942no—
240#387—942no—
241#386—941no—
242#381—941no—
243#380—471no—
244#380—471no—
245#380C2×D4711yescharacteristic
246C4×C22⋊C4—941no—
247#327—22no—
248#327—942no—
249#324—942no—
250C47#920611yescharacteristic, complemented, Sylow
251C22×D4—21no—
252C22×D4C4×D4711yescharacteristic
253C23×C4—471no—
254C23×C4—471no—
255C23×C4C22×D4711yescharacteristic
256C22×C8—21no—
257C22×C8—941no—
258#123—1882no—
259#123—1882no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—472no—
266C2×C4⋊C4—472no—
267C2×C4⋊C4—941no—
268C2×C4⋊C4—941no—
269C2×C4⋊C4—941no—
270C2×C4⋊C4—941no—
271C2×C4⋊C4—941no—
272C2×C4⋊C4—941no—
273C2×C4⋊C4—942no—
274C2×C4⋊C4C4×D4711yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—942no—
278C2×C22⋊C4—942no—
279C2×C22⋊C4—941no—
280C2×C22⋊C4—941no—
281C2×C42—942no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—941no—
285C22⋊C8—941no—
286#95—472no—
287#95—472no—
288#95—942no—
289#95—942no—
290#95—944no—
291#95C22⋊D4712yes—
292C24—21no—
293C24D4×D4711yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—81no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—81no—
306C22×C4—472no—
307C22×C4—472no—
308C22×C4—941no—
309C22×C4—941no—
310C22×C4—941no—
311C22×C4—941no—
312C22×C4—941no—
313C22×C4—941no—
314C22×C4—941no—
315C22×C4—941no—
316C22×C4—942no—
317C22×C4—942no—
318C22×C4—3761no—
319C22×C4C2×C4×D4711yescharacteristic
320C22×C4#1062535011yescharacteristic
321C22×C4D4×D4711yescharacteristic
322C22×C4D4×D4711yescharacteristic
323C22×C4#1062535211yescharacteristic
324C22×C4C2×C22⋊D4711yescharacteristic
325C2×C8—41no—
326C2×C8—81no—
327C2×C8—3761no—
328C2×C8—1881no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—81no—
332C4⋊C4—942no—
333C4⋊C4—942no—
334C4⋊C4—942no—
335C4⋊C4—942no—
336C4⋊C4—3761no—
337C4⋊C4—3761no—
338C4⋊C4—1884no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—3762no—
344C22⋊C4—1884no—
345C22⋊C4—1881no—
346C22⋊C4—1881no—
347C22⋊C4—1881no—
348C22⋊C4—1881no—
349C42—3762no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23C22⋊C4×D4711yescharacteristic
358C23#1309399211yescharacteristic
359C23#1309400611yescharacteristic
360C23#1309404311yescharacteristic
361C23#1309404511yescharacteristic
362C23#1309404811yescharacteristic
363C23#1309405111yescharacteristic
364D4—81no—
365D4—81no—
366D4—81no—
367D4—81no—
368D4—81no—
369D4—81no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—942no—
383C2×C4—942no—
384C2×C4—942no—
385C2×C4—942no—
386C2×C4—3761no—
387C2×C4—1882no—
388C2×C4—1882no—
389C2×C4—1882no—
390C2×C4—1882no—
391C2×C4—1882no—
392C2×C4—1882no—
393C2×C4—1882no—
394C2×C4—1881no—
395C2×C4—1881no—
396C2×C4—1881no—
397C2×C4—1881no—
398C2×C4—1882no—
399C2×C4(C22⋊D47)⋊C411yescharacteristic
400C2×C4C4×C22⋊D4711yescharacteristic
401C2×C4#1309399511yescharacteristic
402C2×C4#1309399711yescharacteristic
403C8—81no—
404C8—3761no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—82no—
416C22—82no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22#1568899911yescharacteristic
421C22#1568911611yescharacteristic
422C22#1568912611yescharacteristic
423C22#1568926511yescharacteristic
424C22#1568926711yescharacteristic
425C22#1568944511yescharacteristic
426C22#1568944611yescharacteristic
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—82no—
432C4—81no—
433C4—3762no—
434C4—3762no—
435C4—1882no—
436C4—1882no—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—161no—
444C2#2024370711yescharacteristic
445C2#2024392211yescharacteristic
446C2#2024392711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1519C2
socleC22×C47#386
prime radicalC2×#121×C47C22
cosocleC23×C4×C47C23
hypercenter#1519D47
hypocenterC47#9206
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #20245990C2semidirect
4. C2×#15688848C2semidirect
5. #20241256C2semidirect
6. #20241255C2semidirect
11. #15689054C4semidirect
13. #15689037C4semidirect
250. C47#9206semidirect

Maximal subgroups

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

subgroupindex
#202491322
#202459902
#202419562
C2×#156888482
#202412562
#202412552
C47×#15192
#920647