Finite Group Databasebeta

← #26112323

Subgroups — #26112323

Order 12032.

560 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 113 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#20249203C211yesmaximal, characteristic
2#20246005C211yesmaximal, characteristic
3#20241960C211yesmaximal, characteristic
4C2×#15688848C211yesmaximal, characteristic
5#20241256C211yesmaximal, characteristic
6#20241255C211yesmaximal, characteristic
7C47×#1520C211yesmaximal, characteristic
8C2×#13093980—21no—
9C22×Dic47⋊C4C2211yescharacteristic
10C2×C22⋊(C47⋊C8)C2211yescharacteristic
11#15689061C412yescomplemented
12#15689056—22no—
13#15689042C412yescomplemented
14#15689020—22no—
15#15689018—22no—
16#15688995—21no—
17#15688984—21no—
18#15688983C2211yescharacteristic
19#15688983C2211yescharacteristic
20#15688981—21no—
21#15688860—22no—
22#15688848—22no—
23C2×C22⋊Q8×C47C2211yescharacteristic
24C2×C22⋊C8×C47C2211yescharacteristic
25C47×#380C2211yescharacteristic
26C47×#328—22no—
27C23×Dic47D411yescharacteristic
28C23×Dic47D411yescharacteristic
29C22×C47⋊C8—21no—
30#13093980—42no—
31#13093980—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⋊C4—22no—
40C2×Dic47⋊C4—22no—
41C2×Dic47⋊C4C2×C412yes—
42C2×Dic47⋊C4D412yes—
43C2×C22⋊Dic47—21no—
44C2×C22⋊Dic47—21no—
45C2×C4×Dic47—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C47⋊#95—22no—
49C47⋊#95—22no—
50C47⋊#95—22no—
51C47⋊#95—22no—
52C47⋊#95C2×C412yes—
53C47⋊#95D412yes—
54C22×Q8×C47C2×C411yescharacteristic
55C23×C4×C47C2311yescharacteristic
56C22×C8×C47—21no—
57C22⋊Q8×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C22⋊Q8×C47—21no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47C2×C411yescharacteristic
64C2×C22⋊C4×C47—21no—
65C2×C22⋊C4×C47—21no—
66C22⋊C8×C47—21no—
67C22⋊C8×C47—21no—
68#95×C47D412yes—
69C24×C47C2×D411yescharacteristic
70C2×Q8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—81no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—22no—
76C22×C4×C47—81no—
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47#4111yescharacteristic
82C22×C4×C47#4111yescharacteristic
83C22×Dic47—21no—
84C22×Dic47—21no—
85C22×Dic47—21no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—21no—
89C22×Dic47—22no—
90C22×Dic47—22no—
91C22×Dic47—81no—
92C22×Dic47C22⋊C412yes—
93C22×Dic47C22⋊C412yes—
94C22×Dic47SD1612yes—
95C22×Dic47Q1612yes—
96C2×C8×C47—41no—
97C2×C8×C47—81no—
98C4⋊C4×C47—21no—
99C4⋊C4×C47—21no—
100C4⋊C4×C47—41no—
101C4⋊C4×C47—41no—
102C4⋊C4×C47—81no—
103C22⋊C4×C47—41no—
104C22⋊C4×C47—41no—
105C22⋊C4×C47—41no—
106C22⋊C4×C47—41no—
107C22⋊Dic47—41no—
108C22⋊Dic47—41no—
109C22⋊Dic47—41no—
110C22⋊Dic47—41no—
111C4⋊Dic47—81no—
112C4⋊Dic47—81no—
113Dic47⋊C4—22no—
114Dic47⋊C4—22no—
115Dic47⋊C4—22no—
116Dic47⋊C4—22no—
117Dic47⋊C4—44no—
118Dic47⋊C4—82no—
119Dic47⋊C4—82no—
120C4×Dic47—44no—
121C2×C47⋊C8—41no—
122C2×C47⋊C8—81no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47C2×C22⋊C411yescharacteristic
126C23×C47#12011yescharacteristic
127C23×C47#12311yescharacteristic
128C23×C47C2×SD1611yescharacteristic
129C23×C47C2×Q1611yescharacteristic
130C23×C47AGL(1,ℤ/8)11yescharacteristic
131C23×C47#13711yescharacteristic
132Q8×C47—81no—
133Q8×C47—81no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—42no—
147C2×C4×C47C4×D411yescharacteristic
148C2×C4×C47C4×D411yescharacteristic
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×Dic47—22no—
152C2×Dic47—22no—
153C2×Dic47—22no—
154C2×Dic47—22no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—22no—
158C2×Dic47—42no—
159C2×Dic47—42no—
160C2×Dic47—42no—
161C2×Dic47—42no—
162C2×Dic47—42no—
163C2×Dic47—42no—
164C2×Dic47—42no—
165C2×Dic47—42no—
166C2×Dic47—81no—
167C2×Dic47#10314yes—
168C8×C47—81no—
169C47⋊C8—81no—
170#9200—471nomaximal, Sylow
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47—21no—
177C22×C47—41no—
178C22×C47—41no—
179C22×C47—41no—
180C22×C47—41no—
181C22×C47#38611yescharacteristic
182C22×C47C2×#10311yescharacteristic
183C22×C47#41711yescharacteristic
184C22×C47#45011yescharacteristic
185C22×C47#45111yescharacteristic
186C22×C47#48311yescharacteristic
187C22×C47#48411yescharacteristic
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—82no—
193C4×C47—81no—
194C4×C47—81no—
195Dic47—24no—
196Dic47—42no—
197Dic47—82no—
198Dic47—82no—
199#2627—471no—
200#2316—471no—
201#1520—471no—
202#1520D4711yescharacteristic
203C2×#324—471no—
204#1289—471no—
205#1289—471no—
206C2×C47—21no—
207C2×C47—21no—
208C2×C47—21no—
209C2×C47—21no—
210C2×C47—21no—
211C2×C47—21no—
212C2×C47#187211yescharacteristic
213C2×C47#191211yescharacteristic
214C2×C47#191311yescharacteristic
215C2×C22⋊Q8C2×D4711yescharacteristic
216C2×#117—941no—
217C22×C4⋊C4—471no—
218C2×C22⋊C8—471no—
219C2×C22⋊C8C2×D4711yescharacteristic
220#400—472no—
221#398—942no—
222#395—472no—
223#389—942no—
224#389—942no—
225#385—941no—
226#381—941no—
227#380—471no—
228#380—471no—
229#380—941no—
230#380C2×D4711yescharacteristic
231#328—22no—
232#328—942no—
233#324—942no—
234C47#920011yescharacteristic, complemented, Sylow
235C22×Q8C4×D4711yescharacteristic
236C23×C4—471no—
237C23×C4—471no—
238C23×C4C22×D4711yescharacteristic
239C22×C8—21no—
240C22×C8—941no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245#117—1882no—
246#117—1882no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—472no—
250C2×C4⋊C4—472no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4—941no—
255C2×C4⋊C4—942no—
256C2×C4⋊C4—942no—
257C2×C4⋊C4—941no—
258C2×C4⋊C4—941no—
259C2×C4⋊C4—942no—
260C2×C4⋊C4C4×D4711yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—941no—
264C2×C22⋊C4—941no—
265C2×C42—942no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—941no—
269C22⋊C8—941no—
270#95—472no—
271#95—472no—
272#95—942no—
273#95—942no—
274#95—942no—
275#95—942no—
276#95C22⋊D4712yes—
277C24D4×D4711yescharacteristic
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—81no—
285C22×C4—472no—
286C22×C4—472no—
287C22×C4—472no—
288C22×C4—472no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—941no—
292C22×C4—941no—
293C22×C4—941no—
294C22×C4—941no—
295C22×C4—942no—
296C22×C4—942no—
297C22×C4—3761no—
298C22×C4C2×C4×D4711yescharacteristic
299C22×C4#1062535011yescharacteristic
300C22×C4D4×D4711yescharacteristic
301C22×C4D4×D4711yescharacteristic
302C22×C4#1062535211yescharacteristic
303C22×C4C2×C22⋊D4711yescharacteristic
304C2×C8—41no—
305C2×C8—81no—
306C2×C8—3761no—
307C2×C8—1881no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—81no—
313C4⋊C4—942no—
314C4⋊C4—942no—
315C4⋊C4—942no—
316C4⋊C4—942no—
317C4⋊C4—3762no—
318C4⋊C4—3762no—
319C4⋊C4—3761no—
320C4⋊C4—3761no—
321C4⋊C4—1884no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—1881no—
327C22⋊C4—1881no—
328C22⋊C4—1881no—
329C22⋊C4—1881no—
330C42—1884no—
331C23—41no—
332C23—41no—
333C23C22⋊C4×D4711yescharacteristic
334C23#1309399211yescharacteristic
335C23#1309400611yescharacteristic
336C23SD16×D4711yescharacteristic
337C23Q16×D4711yescharacteristic
338C23#1309405111yescharacteristic
339C23#1309405611yescharacteristic
340Q8—81no—
341Q8—81no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—474no—
356C2×C4—942no—
357C2×C4—942no—
358C2×C4—942no—
359C2×C4—942no—
360C2×C4—942no—
361C2×C4—942no—
362C2×C4—942no—
363C2×C4—3761no—
364C2×C4—1882no—
365C2×C4—1882no—
366C2×C4—1882no—
367C2×C4—1882no—
368C2×C4—1882no—
369C2×C4—1882no—
370C2×C4—1882no—
371C2×C4—1882no—
372C2×C4(C22⋊D47)⋊C411yescharacteristic
373C2×C4C4×C22⋊D4711yescharacteristic
374C2×C4#1309399511yescharacteristic
375C2×C4#1309399711yescharacteristic
376C8—81no—
377C8—3761no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22#1568899911yescharacteristic
389C22#103×D4711yescharacteristic
390C22#1568912311yescharacteristic
391C22#1568927211yescharacteristic
392C22#1568927711yescharacteristic
393C22#1568945211yescharacteristic
394C22#1568945711yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—82no—
400C4—81no—
401C4—81no—
402C4—944no—
403C4—3762no—
404C4—3762no—
405C4—1882no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2—21no—
411C2—21no—
412C2#2024369211yescharacteristic
413C2#2024392211yescharacteristic
414C2#2024392711yescharacteristic
subgroupG/H
centerC22#15688999
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1520C2
socleC22×C47#386
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1520D47
hypocenterC47#9200
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1520

Complements

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

NH ≅ G/Ntype
11. #15689061C4semidirect
13. #15689042C4semidirect
234. C47#9200semidirect

Maximal subgroups

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

subgroupindex
#202492032
#202460052
#202419602
C2×#156888482
#202412562
#202412552
C47×#15202
#920047