Finite Group Databasebeta

← #26109263

Subgroups — #26109263

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252083C211yesmaximal, characteristic
2#20243424C211yesmaximal, characteristic, complemented
3#20243411C211yesmaximal, characteristic, complemented
4#20241931C211yesmaximal, characteristic
5#20241295C211yesmaximal, characteristic, complemented
6#20241294C211yesmaximal, characteristic, complemented
7C47×#1513C211yesmaximal, characteristic
8C2×#13094031C411yescharacteristic, complemented
9C23×C47⋊C8C2211yescharacteristic
10#15689195—21no—
11#15689195—21no—
12D4×C47⋊C8—21no—
13D4×C47⋊C8—21no—
14C2×#13093959—21no—
15C2×#13093955—21no—
16#15689150C411yescharacteristic, complemented
17#15689090C411yescharacteristic, complemented
18#15689087C2211yescharacteristic
19#15689087C2211yescharacteristic
20#15689079C411yescharacteristic, complemented
21#15688983C2211yescharacteristic
22#15688887—21no—
23#15688885—21no—
24#15688860—21no—
25#15688857—21no—
26C47×#525C2211yescharacteristic
27C47×#407C2211yescharacteristic
28C47×#380C2211yescharacteristic
29C47×#328—21no—
30C47×#327—21no—
31#13094031—42no—
32C2×#10625329—21no—
33C2×#10625329—21no—
34C2×#10625329C2×C411yescharacteristic
35C2×#10625329C2×C411yescharacteristic
36C22×C47⋊C8—21no—
37C22×C47⋊C8—42no—
38C22×C47⋊C8C2×C411yescharacteristic
39C22×C47⋊C8C2×C411yescharacteristic
40C22×C47⋊C8D411yescharacteristic
41C22×C47⋊C8D411yescharacteristic
42C22×C47⋊C8D411yescharacteristic
43C22×C47⋊C8D411yescharacteristic
44C2×C4⋊Dic47—21no—
45C2×C22⋊Dic47—21no—
46#13093959—42no—
47#13093955—42no—
48C4⋊(C47⋊C8)—21no—
49C4⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)—21no—
53C22⋊(C47⋊C8)—21no—
54C22⋊(C47⋊C8)—21no—
55C22⋊(C47⋊C8)—21no—
56#13093935—21no—
57C4×C47⋊C8—21no—
58C47⋊#95—21no—
59C2×#41×C47C2×C411yescharacteristic
60C23×C4×C47C2311yescharacteristic
61C2×M16×C47—21no—
62#123×C47—21no—
63C22⋊Q8×C47D411yescharacteristic
64C22⋊Q8×C47Q811yescharacteristic
65#121×C47D411yescharacteristic
66#121×C47Q811yescharacteristic
67#120×C47—21no—
68C4×D4×C47—21no—
69C4×D4×C47—21no—
70#117×C47C2×C411yescharacteristic
71C2×C4⋊C4×C47—21no—
72C2×C22⋊C4×C47—21no—
73C22⋊C8×C47—21no—
74C22⋊C8×C47—21no—
75#95×C47—21no—
76C24×C47#4111yescharacteristic
77#41×C47—41no—
78#41×C47—41no—
79C2×Q8×C47C4⋊C411yescharacteristic
80C2×D4×C47—21no—
81C2×D4×C47—21no—
82C2×D4×C47—21no—
83C2×D4×C47C4⋊C411yescharacteristic
84C22×C4×C47—21no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47—41no—
88C22×C4×C47C22×C411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×Q811yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—41no—
95M16×C47—41no—
96M16×C47—41no—
97C2×C8×C47—41no—
98C4⋊C4×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47C4⋊C411yescharacteristic
103C4⋊C4×C47C4⋊C411yescharacteristic
104C22⋊C4×C47—21no—
105C22⋊C4×C47—21no—
106C22⋊C4×C47—21no—
107C22⋊C4×C47—21no—
108C22⋊C4×C47—21no—
109C22⋊C4×C47—81no—
110C42×C47—21no—
111C22⋊Dic47—81no—
112C4⋊Dic47—41no—
113C4⋊Dic47—41no—
114#10625329—42no—
115#10625329—41no—
116#10625329—41no—
117#10625329—41no—
118#10625329—41no—
119C2×C47⋊C8—22no—
120C2×C47⋊C8—22no—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—21no—
123C2×C47⋊C8—21no—
124C2×C47⋊C8—41no—
125C2×C47⋊C8—41no—
126C2×C47⋊C8—41no—
127C2×C47⋊C8—42no—
128C2×C47⋊C8—41no—
129C2×C47⋊C8C22⋊C411yescharacteristic
130C2×C47⋊C8C22⋊C411yescharacteristic
131C2×C47⋊C8C22⋊C411yescharacteristic
132C2×C47⋊C8C22⋊C411yescharacteristic
133C23×C47—21no—
134C23×C47—21no—
135C23×C47—21no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47#11711yescharacteristic
139C23×C47#12111yescharacteristic
140C23×C47C22⋊Q811yescharacteristic
141Q8×C47—41no—
142D4×C47—41no—
143D4×C47—42no—
144D4×C47—42no—
145D4×C47—41no—
146D4×C47—41no—
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—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—81no—
162C2×C4×C47—81no—
163C2×C4×C47—81no—
164C2×C4×C47C2×C22⋊C411yescharacteristic
165C2×C4×C47C2×C4⋊C411yescharacteristic
166C2×C4×C47#12111yescharacteristic
167C2×C4×C47C22⋊Q811yescharacteristic
168C2×Dic47—81no—
169C2×Dic47—81no—
170C2×Dic47—81no—
171C8×C47—81no—
172C47⋊C8—21no—
173C47⋊C8—21no—
174C47⋊C8—21no—
175C47⋊C8—41no—
176C47⋊C8—41no—
177C47⋊C8—81no—
178C47⋊C8#9912yes—
179#8629—471nomaximal, Sylow
180C22×C47—21no—
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—42no—
189C22×C47—42no—
190C22×C47—41no—
191C22×C47—41no—
192C22×C47—41no—
193C22×C47#38011yescharacteristic
194C22×C47#47111yescharacteristic
195C22×C47#47111yescharacteristic
196C4×C47—21no—
197C4×C47—21no—
198C4×C47—21no—
199C4×C47—41no—
200C4×C47—41no—
201C4×C47—41no—
202C4×C47—161no—
203C4×C47C2×#9911yescharacteristic
204C4×C47#41011yescharacteristic
205Dic47—161no—
206#2929—471no—
207#1810—471no—
208#1807—471no—
209#1513—471no—
210#1513D4711yescharacteristic
211#1305—471no—
212#1305—471no—
213C2×C47—21no—
214C2×C47—21no—
215C2×C47—41no—
216C2×C47—41no—
217C2×C47—81no—
218C2×C47#179911yescharacteristic
219C2×C47#181411yescharacteristic
220C2×C47#181511yescharacteristic
221C2×#131—471no—
222C23×C8—471no—
223#525C2×D4711yescharacteristic
224#435—941no—
225#435—941no—
226C8×D4—941no—
227C8×D4—941no—
228C2×#108—941no—
229C2×#106—941no—
230#424—471no—
231#408—471no—
232#407—471no—
233#407—471no—
234#407C2×D4711yescharacteristic
235#405—471no—
236#380—471no—
237#380C2×D4711yescharacteristic
238#341—941no—
239#340—941no—
240#328—21no—
241#328—941no—
242#327—21no—
243#327—941no—
244C47#862911yescharacteristic, complemented, Sylow
245C2×#41C4×D4711yescharacteristic
246C23×C4C22×D4711yescharacteristic
247#131—1882no—
248C2×M16—21no—
249C2×M16—471no—
250C2×M16—471no—
251C2×M16—941no—
252C2×M16—941no—
253C22×C8—471no—
254C22×C8—471no—
255C22×C8—471no—
256C22×C8—471no—
257C22×C8—471no—
258C22×C8—471no—
259C22×C8—941no—
260C22×C8—1882no—
261#123—21no—
262C22⋊Q8Dic9411yescharacteristic
263C22⋊Q8C22⋊D4711yescharacteristic
264#121Dic9411yescharacteristic
265#121C22⋊D4711yescharacteristic
266#120—21no—
267C4×D4—21no—
268C4×D4—21no—
269#117C4×D4711yescharacteristic
270C2×C4⋊C4—21no—
271C2×C4⋊C4—941no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—941no—
274#108—1882no—
275#106—1882no—
276C4⋊C8—941no—
277C4⋊C8—941no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—941no—
281C22⋊C8—941no—
282C22⋊C8—941no—
283C22⋊C8—941no—
284C22⋊C8—941no—
285C22⋊C8—941no—
286#97—941no—
287C4×C8—941no—
288#95—21no—
289#95—941no—
290C24#1062535211yescharacteristic
291#41—41no—
292#41—41no—
293C2×Q8Dic47⋊C411yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4Dic47⋊C411yescharacteristic
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1881no—
303C22×C4C2×Dic9411yescharacteristic
304C22×C4C2×C4×D4711yescharacteristic
305C22×C4D4×D4711yescharacteristic
306C22×C4D4×D4711yescharacteristic
307C22×C4#1062535211yescharacteristic
308C22×C4C2×C22⋊D4711yescharacteristic
309M16—41no—
310M16—41no—
311M16—1882no—
312M16—1881no—
313M16—1881no—
314M16—1881no—
315M16—1881no—
316C2×C8—41no—
317C2×C8—471no—
318C2×C8—471no—
319C2×C8—471no—
320C2×C8—471no—
321C2×C8—942no—
322C2×C8—942no—
323C2×C8—942no—
324C2×C8—941no—
325C2×C8—941no—
326C2×C8—1881no—
327C2×C8—1881no—
328C2×C8—1881no—
329C2×C8—1882no—
330C2×C8—1881no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1881no—
336C4⋊C4—1881no—
337C4⋊C4Dic47⋊C411yescharacteristic
338C4⋊C4Dic47⋊C411yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—81no—
345C22⋊C4—3761no—
346C42—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23#1309398011yescharacteristic
353C23#1309399611yescharacteristic
354C23#1309400111yescharacteristic
355Q8—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359D4—41no—
360D4—41no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—3761no—
379C2×C4—3761no—
380C2×C4—3761no—
381C2×C4C22⋊C4×D4711yescharacteristic
382C2×C4C2×Dic47⋊C411yescharacteristic
383C2×C4#1309399611yescharacteristic
384C2×C4#1309400111yescharacteristic
385C8—81no—
386C8—472no—
387C8—941no—
388C8—941no—
389C8—941no—
390C8—3761no—
391C8—1881no—
392C8—1881no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22#1568898111yescharacteristic
407C22#1568938411yescharacteristic
408C22#1568938811yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—161no—
416C4—7521no—
417C4#99×D4711yescharacteristic
418C4#1568909511yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—81no—
424C2#2024337411yescharacteristic
425C2#2024344011yescharacteristic
426C2#2024344511yescharacteristic
subgroupG/H
centerC22#15688981
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1513C2
socleC22×C47#380
prime radicalC47×#525C22
cosocleC23×C4×C47C23
hypercenter#1513D47
hypocenterC47#8629
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1513

Complements

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

NH ≅ G/Ntype
2. #20243424C2semidirect
3. #20243411C2semidirect
5. #20241295C2semidirect
6. #20241294C2semidirect
8. C2×#13094031C4semidirect
16. #15689150C4semidirect
17. #15689090C4semidirect
20. #15689079C4semidirect
244. C47#8629semidirect

Maximal subgroups

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

subgroupindex
#202520832
#202434242
#202434112
#202419312
#202412952
#202412942
C47×#15132
#862947