Finite Group Databasebeta

← #26107175

Subgroups — #26107175

Order 12032.

804 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13093998C211yesmaximal, characteristic
2C2×#15688858C212yesmaximal
3C4×C47⋊#95C211yesmaximal, characteristic, complemented
4#20241268C212yesmaximal, complemented
5C47×#1749C211yesmaximal, characteristic, complemented
6C2×#13093998C2211yescharacteristic
7C2×C4×D188C411yescharacteristic, complemented
8C2×C4×C22⋊D47—21no—
9C22×C4×Dic47—21no—
10#15689025C411yescharacteristic, complemented
11#15689021C411yescharacteristic, complemented
12#15689014—21no—
13#15689002—21no—
14#15688983C2211yescharacteristic
15C4×C4⋊Dic47C411yescharacteristic, complemented
16C4×(C2×D47)⋊C4—21no—
17C2×C47⋊#95—21no—
18C2×C47⋊#95C2212yes—
19#15688858C418yescomplemented
20C22×C42×C47C2211yescharacteristic
21C2×C22⋊C8×C47C2212yes—
22C2×#97×C47—21no—
23C47×#336—21no—
24C22×D188C2×C411yescharacteristic
25C22×C22⋊D47—21no—
26C23×Dic47—22no—
27C22×C4×D47—21no—
28#13093998—22no—
29#13093998C2×C414yes—
30C4×D188—42no—
31C4×C22⋊D47—44no—
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×C4⋊Dic47C2×C411yescharacteristic
34C2×C4⋊Dic47C2×C411yescharacteristic
35C2×Dic47⋊C4—21no—
36C2×C22⋊Dic47—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C2×(C2×D47)⋊C4—21no—
40C2×(C2×D47)⋊C4—21no—
41C2×C4×Dic47—24no—
42C2×C4×Dic47—21no—
43C2×C4×Dic47—42no—
44C2×C4×Dic47—42no—
45C47⋊#95—24no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C47⋊#95—22no—
50C47⋊#95C2×C414yescomplemented
51C23×C4×C47D411yescharacteristic
52C23×C4×C47D411yescharacteristic
53C23×C4×C47C2311yescharacteristic
54C22×C8×C47—22no—
55C2×C42×C47—42no—
56C2×C42×C47C2×C411yescharacteristic
57C2×C42×C47C2×C411yescharacteristic
58C2×C42×C47D411yescharacteristic
59C2×C42×C47D411yescharacteristic
60C22⋊C8×C47—22no—
61C22⋊C8×C47C2×C414yescomplemented
62#97×C47—42no—
63C24×C47C2×D411yescharacteristic
64C23×D47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C22×C412yes—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4112yes—
82C2×C22⋊D47—24no—
83C2×C22⋊D47—42no—
84C2×C22⋊D47—41no—
85C2×C22⋊D47—41no—
86C22×Dic47—24no—
87C22×Dic47—24no—
88C22×Dic47—21no—
89C22×Dic47—24no—
90C22×Dic47—21no—
91C22×Dic47—21no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—42no—
95C22×Dic47—42no—
96C2×D188—21no—
97C2×D188—41no—
98C2×D188—41no—
99C2×D188C4212yescomplemented
100C2×C4×D47—42no—
101C2×C4×D47—42no—
102C2×C4×D47—42no—
103C2×C8×C47—24no—
104C2×C8×C47—42no—
105C2×C8×C47—42no—
106C42×C47—22no—
107C42×C47—22no—
108C42×C47—41no—
109C42×C47—41no—
110C22⋊Dic47—42no—
111(C2×D47)⋊C4—24no—
112(C2×D47)⋊C4—42no—
113(C2×D47)⋊C4—42no—
114(C2×D47)⋊C4—42no—
115C4⋊Dic47—21no—
116C4⋊Dic47—41no—
117C4⋊Dic47—41no—
118C4⋊Dic47C4212yescomplemented
119Dic47⋊C4—42no—
120C4×Dic47—48no—
121C4×Dic47—42no—
122C23×C47—21no—
123C23×C47—21no—
124C23×C47—21no—
125C23×C47—21no—
126C23×C47C2×C22⋊C411yescharacteristic
127C23×C47C4×D412yes—
128C23×C47AGL(1,ℤ/8)11yescharacteristic
129C23×C47AGL(1,ℤ/8)12yes—
130C23×C47#13711yescharacteristic
131C22×D47—22no—
132C22×D47—41no—
133C22×D47—41no—
134C22×D47—41no—
135C22×D47—41no—
136C22×D47—41no—
137C22×D47—41no—
138C2×C4×C47—24no—
139C2×C4×C47—24no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—21no—
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—42no—
154C2×C4×C47—42no—
155C2×C4×C47—44no—
156C2×C4×C47#9912yes—
157C2×C4×C47#10412yes—
158C2×C4×C47#10412yes—
159C2×C4×C47C2×C4211yescharacteristic
160C2×C4×C47#11711yescharacteristic
161C2×C4×C47C4×D412yes—
162C22⋊D47—44no—
163C22⋊D47—44no—
164C2×Dic47—28no—
165C2×Dic47—22no—
166C2×Dic47—44no—
167C2×Dic47—44no—
168C2×Dic47—44no—
169C2×Dic47—44no—
170C2×Dic47—44no—
171C2×Dic47—41no—
172C2×Dic47—41no—
173C2×Dic47—44no—
174C2×Dic47—41no—
175C2×Dic47—41no—
176C2×Dic47—41no—
177C2×Dic47—41no—
178C2×Dic47—41no—
179C2×Dic47—41no—
180D188—42no—
181D188—42no—
182C4×D47—84no—
183C8×C47—44no—
184#8218—471nomaximal, Sylow
185C22×C47—22no—
186C22×C47—22no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—22no—
190C22×C47—22no—
191C22×C47—21no—
192C22×C47—21no—
193C22×C47C4×C22⋊C411yescharacteristic
194C22×C47C2×#9911yescharacteristic
195C22×C47#41011yescharacteristic
196C22×C47#41711yescharacteristic
197C22×C47#41811yescharacteristic
198C22×C47C2×#10411yescharacteristic
199C22×C47C2×#10411yescharacteristic
200C22×C47#44012yes—
201C22×C47#44212yes—
202C2×D47—42no—
203C2×D47—42no—
204C2×D47—42no—
205C2×D47—81no—
206C2×D47—81no—
207C2×D47—81no—
208C2×D47—81no—
209C4×C47—22no—
210C4×C47—21no—
211C4×C47—21no—
212C4×C47—22no—
213C4×C47—22no—
214C4×C47—22no—
215C4×C47—41no—
216Dic47—48no—
217Dic47—42no—
218Dic47—81no—
219Dic47—81no—
220C4×#121—471no—
221#1749D4711yescharacteristic, complemented
222C2×#327—472no—
223C4×#95—471no—
224#1294—472no—
225C2×C47—22no—
226C2×C47—22no—
227C2×C47#150411yescharacteristic
228C2×C47#150511yescharacteristic
229C2×C47#150711yescharacteristic
230C2×C47#150911yescharacteristic
231C2×C47C4×#9911yescharacteristic
232C2×C47C4×#10411yescharacteristic
233C2×C47#176211yescharacteristic
234D47—82no—
235C2×#121—471no—
236C2×C4×D4—471no—
237C2×C4×D4—941no—
238C22×C42—941no—
239C22×C42C2×D4711yescharacteristic
240C2×C22⋊C8C2×D4712yes—
241C2×#97—21no—
242#390—471no—
243#389—471no—
244#388—941no—
245#386—941no—
246#380—471no—
247C4×C4⋊C4—471no—
248C4×C22⋊C4—941no—
249C2×#95—472no—
250C2×#95—941no—
251#336—21no—
252#327—478no—
253C47#821811yescharacteristic, complemented, Sylow
254C22×D4—471no—
255C22×D4—941no—
256C23×C4—942no—
257C23×C4—941no—
258C23×C4D18811yescharacteristic
259C23×C4D18811yescharacteristic
260C23×C4C22×D4711yescharacteristic
261C22×C8—22no—
262#121—474no—
263#121—942no—
264C4×D4—1882no—
265C4×D4—1884no—
266C2×C4⋊C4—471no—
267C2×C4⋊C4—471no—
268C2×C4⋊C4—471no—
269C2×C4⋊C4—941no—
270C2×C22⋊C4—941no—
271C2×C22⋊C4—941no—
272C2×C22⋊C4—941no—
273C2×C22⋊C4—941no—
274C2×C22⋊C4—941no—
275C2×C42—42no—
276C2×C42—944no—
277C2×C42—941no—
278C2×C42—1882no—
279C2×C42—1882no—
280C2×C42C4×D4711yescharacteristic
281C2×C42C4×D4711yescharacteristic
282C2×C42C22⋊D4711yescharacteristic
283C2×C42C22⋊D4711yescharacteristic
284C22⋊C8—22no—
285C22⋊C8C4×D4714yescomplemented
286#97—42no—
287#95—474no—
288#95—944no—
289#95—941no—
290#95—941no—
291#95—942no—
292#95—941no—
293C24—941no—
294C24C2×D18811yescharacteristic
295C2×D4—472no—
296C2×D4—944no—
297C2×D4—941no—
298C2×D4—1881no—
299C2×D4—1881no—
300C2×D4—1882no—
301C2×D4—1881no—
302C2×D4—1881no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—944no—
313C22×C4—944no—
314C22×C4—941no—
315C22×C4—944no—
316C22×C4—941no—
317C22×C4—941no—
318C22×C4—1882no—
319C22×C4—1882no—
320C22×C4—1882no—
321C22×C4—1882no—
322C22×C4—1882no—
323C22×C4—1882no—
324C22×C4—1882no—
325C22×C4(C2×D47)⋊C411yescharacteristic
326C22×C4(C2×D47)⋊C411yescharacteristic
327C22×C4(C2×D47)⋊C411yescharacteristic
328C22×C4(C2×D47)⋊C411yescharacteristic
329C22×C4C2×C4×D4711yescharacteristic
330C22×C4C2×C4×D4712yes—
331C22×C4#1062535012yes—
332C22×C4C2×C22⋊D4711yescharacteristic
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C4⋊C4—472no—
337C4⋊C4—941no—
338C4⋊C4—1881no—
339C4⋊C4—1881no—
340C4⋊C4—1882no—
341C22⋊C4—944no—
342C22⋊C4—1882no—
343C22⋊C4—1882no—
344C22⋊C4—1882no—
345C22⋊C4—1882no—
346C42—22no—
347C42—22no—
348C42—41no—
349C42—41no—
350C42—1888no—
351C42—1882no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—942no—
357C23—1881no—
358C23—1881no—
359C23—1881no—
360C23—1881no—
361C23—1881no—
362C23—1881no—
363C23C2×(C2×D47)⋊C411yescharacteristic
364C23C4×D18812yes—
365C23#1309405212yes—
366C23#1309405211yescharacteristic
367C23#1309405711yescharacteristic
368D4—1882no—
369D4—1882no—
370D4—1884no—
371D4—1884no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—948no—
391C2×C4—942no—
392C2×C4—3764no—
393C2×C4—1881no—
394C2×C4—1881no—
395C2×C4—1881no—
396C2×C4—1881no—
397C2×C4—1881no—
398C2×C4—1881no—
399C2×C4—1884no—
400C2×C4—1884no—
401C2×C4—1884no—
402C2×C4—1884no—
403C2×C4—1884no—
404C2×C4—1884no—
405C2×C4—1881no—
406C2×C4—1881no—
407C2×C4#1309393812yes—
408C2×C4#1309395012yes—
409C2×C4#1309395012yes—
410C2×C4C42×D4711yescharacteristic
411C2×C4#1309398111yescharacteristic
412C2×C4C4×C22⋊D4712yes—
413C8—44no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—3761no—
423C22—3761no—
424C22—3761no—
425C22—3761no—
426C22—1882no—
427C22—1882no—
428C22—1882no—
429C22C4×(C2×D47)⋊C411yescharacteristic
430C22C2×#1309393811yescharacteristic
431C22#1568909611yescharacteristic
432C22#1568912511yescharacteristic
433C22#1568912911yescharacteristic
434C22C2×#1309395011yescharacteristic
435C22C2×#1309395011yescharacteristic
436C22#1568922512yes—
437C22#1568923612yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—22no—
444C4—41no—
445C4—3761no—
446C4—3761no—
447C4—1888no—
448C4—1882no—
449C2—22no—
450C2—22no—
451C2—3762no—
452C2#2024189311yescharacteristic
453C2#2024189711yescharacteristic
454C2#2024190511yescharacteristic
455C2#2024191411yescharacteristic
456C2C4×#1309393811yescharacteristic
457C2C4×#1309395011yescharacteristic
458C2#2024321411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC2×C4×C47C2×C42
FrattiniC23×C4C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C22
cosocleC23×C4×C47C23
hypercenter#1749D47
hypocenterC47#8218
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. C4×C47⋊#95C2semidirect
4. #20241268C2semidirect
5. C47×#1749C2semidirect
7. C2×C4×D188C4semidirect
10. #15689025C4semidirect
11. #15689021C4semidirect
15. C4×C4⋊Dic47C4semidirect
19. #15688858C4semidirect
50. C47⋊#95C2×C4semidirect
61. C22⋊C8×C47C2×C4semidirect
99. C2×D188C42semidirect
118. C4⋊Dic47C42semidirect
221. #1749D47semidirect
253. C47#8218semidirect
285. C22⋊C8C4×D47semidirect

Maximal subgroups

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

subgroupindex
C4×#130939982
C2×#156888582
C4×C47⋊#952
#202412682
C47×#17492
#821847