Finite Group Databasebeta

← #26104533

Subgroups — #26104533

Order 12032.

556 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 103 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#20245926C211yesmaximal, characteristic
2#20243149C211yesmaximal, characteristic
3#20243080C211yesmaximal, characteristic
4#20241214C211yesmaximal, characteristic, complemented
5#20241212C211yesmaximal, characteristic, complemented
6#20241210C211yesmaximal, characteristic, complemented
7C47×#1272C211yesmaximal, characteristic, complemented
8C22×#10625324—21no—
9C2×#13094000C411yescharacteristic
10C2×Q8×Dic47C411yescharacteristic
11C2×C42×D47C2211yescharacteristic
12C2×(C2×D47)⋊C8—21no—
13C2×#13093934—21no—
14#15689021C2211yescharacteristic
15#15689010—21no—
16#15688997—21no—
17#15688980C411yescharacteristic
18C4⋊C4×Dic47C411yescharacteristic
19#15688974—21no—
20#15688970—21no—
21#15688878C2211yescharacteristic
22#15688878C2211yescharacteristic
23#15688877—21no—
24C47×#391C2211yescharacteristic
25C47×#336C2211yescharacteristic
26C47×#336C2211yescharacteristic
27C22×C4×D47C2×C411yescharacteristic
28C22×C4×D47D411yescharacteristic
29C22×C4×D47D411yescharacteristic
30C2×#10625324—24no—
31C2×#10625324—42no—
32C2×#10625324—42no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8—21no—
35#13094000—22no—
36#13094000—22no—
37Q8×Dic47—42no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×C4⋊Dic47C2×C411yescharacteristic
41C2×Dic47⋊C4—21no—
42C2×Dic47⋊C4—21no—
43C2×(C2×D47)⋊C4—21no—
44C2×(C2×D47)⋊C4—21no—
45C2×C4×Dic47—21no—
46C2×C4×Dic47—21no—
47C2×C4×Dic47C2×C411yescharacteristic
48C2×C4×Dic47D411yescharacteristic
49C2×C4×Dic47D411yescharacteristic
50C42×D47—42no—
51(C2×D47)⋊C8—24no—
52#13093934—42no—
53C47⋊#95—21no—
54C47⋊#95—21no—
55C47⋊#95—21no—
56C22×Q8×C47C2×C411yescharacteristic
57C22×C8×C47—21no—
58C22×C8×C47—21no—
59C2×C4⋊C4×C47C2×C411yescharacteristic
60C2×C42×C47C2311yescharacteristic
61#95×C47—21no—
62#95×C47—21no—
63C23×D47C22⋊C411yescharacteristic
64C2×Q8×C47—21no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47M1612yescomplemented
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47C22⋊C411yescharacteristic
76C22×Dic47C22⋊C411yescharacteristic
77C22×Dic47C22⋊C411yescharacteristic
78C2×C4×D47—22no—
79C2×C4×D47—21no—
80C2×C4×D47—21no—
81C2×C4×D47—22no—
82C2×C4×D47—21no—
83C2×C4×D47—21no—
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86C2×C4×D47—22no—
87C2×C4×D47—41no—
88C2×C4×D47—41no—
89C2×C4×D47—41no—
90C2×C8×C47—22no—
91C2×C8×C47—22no—
92C2×C8×C47—41no—
93C2×C8×C47—41no—
94C2×C8×C47—41no—
95C2×C8×C47—41no—
96C4⋊C4×C47—41no—
97C42×C47—41no—
98(C2×D47)⋊C4—24no—
99(C2×D47)⋊C4—42no—
100C4⋊Dic47—21no—
101C4⋊Dic47—42no—
102C4⋊Dic47—41no—
103C4⋊Dic47M1612yescomplemented
104Dic47⋊C4—24no—
105Dic47⋊C4—41no—
106Dic47⋊C4—41no—
107C4×Dic47—22no—
108C4×Dic47—22no—
109C4×Dic47—41no—
110C4×Dic47—42no—
111C4×Dic47—42no—
112C2×C47⋊C8—22no—
113C2×C47⋊C8—22no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C2×C47⋊C8—41no—
118#10625324—48no—
119C23×C47C2×C22⋊C411yescharacteristic
120C22×D47—21no—
121C22×D47—21no—
122C22×D47—21no—
123C22×D47—21no—
124C22×D47—21no—
125C22×D47—21no—
126C22×D47#10412yes—
127Q8×C47—42no—
128Q8×C47—42no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—22no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47C2×M1611yescharacteristic
142C2×C4×C47C2×M1611yescharacteristic
143C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C47#13711yescharacteristic
145C2×C4×C47#13711yescharacteristic
146C2×C4×C47#13711yescharacteristic
147C2×Dic47—21no—
148C2×Dic47—21no—
149C2×Dic47—21no—
150C2×Dic47—21no—
151C2×Dic47—21no—
152C2×Dic47—22no—
153C2×Dic47—21no—
154C2×Dic47—21no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—41no—
160C2×Dic47#9912yes—
161C2×Dic47#10412yes—
162C4×D47—24no—
163C4×D47—24no—
164C4×D47—42no—
165C4×D47—42no—
166C4×D47—42no—
167C4×D47—42no—
168C8×C47—42no—
169C8×C47—42no—
170C47⋊C8—42no—
171C47⋊C8—42no—
172#7634—471nomaximal, Sylow
173C22×C47#40711yescharacteristic
174C22×C47C2×#9911yescharacteristic
175C22×C47#41311yescharacteristic
176C22×C47#41711yescharacteristic
177C22×C47#41911yescharacteristic
178C22×C47C2×#10411yescharacteristic
179C22×C47C2×#10411yescharacteristic
180C2×D47—22no—
181C2×D47—22no—
182C2×D47—22no—
183C2×D47—41no—
184C2×D47—41no—
185C2×D47—41no—
186C2×D47—41no—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—42no—
194C4×C47—81no—
195Dic47—22no—
196Dic47—22no—
197Dic47—41no—
198Dic47—42no—
199Dic47—81no—
200C4×C22⋊Q8—471no—
201#1749—471no—
202#1728—471no—
203#1272—471no—
204#1272D4711yescharacteristic, complemented
205#1271—471no—
206#1271—471no—
207C2×C47#146511yescharacteristic
208C2×C47#148911yescharacteristic
209C2×C47#149111yescharacteristic
210C2×C47#150411yescharacteristic
211C2×C47#150811yescharacteristic
212C2×C47#152411yescharacteristic
213C2×C47#155911yescharacteristic
214D47—42no—
215C22×M16—941no—
216C2×C22⋊Q8—471no—
217C2×C4×Q8—471no—
218C22×C42—471no—
219C2×C22⋊C8—941no—
220C2×#97—941no—
221#391C2×D4711yescharacteristic
222#389—471no—
223#387—941no—
224#385—941no—
225#380—471no—
226C4×C4⋊C4—471no—
227C4×C4⋊C4—941no—
228C4×C22⋊C4—941no—
229#336—471no—
230#336—471no—
231#336—941no—
232#336C2×D4711yescharacteristic
233#336C2×D4711yescharacteristic
234C47#763411yescharacteristic, complemented, Sylow
235C22×Q8C4×D4711yescharacteristic
236C23×C4—471no—
237C23×C4—471no—
238C23×C4—471no—
239C2×M16—944no—
240C2×M16—1882no—
241C2×M16—1882no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—941no—
245C22×C8—941no—
246C22⋊Q8—942no—
247C22⋊Q8—942no—
248C4×Q8—1882no—
249C2×C4⋊C4—471no—
250C2×C4⋊C4—471no—
251C2×C4⋊C4—941no—
252C2×C4⋊C4—941no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4C4×D4711yescharacteristic
255C2×C22⋊C4—941no—
256C2×C22⋊C4—941no—
257C2×C42—471no—
258C2×C42—471no—
259C2×C42—471no—
260C2×C42—941no—
261C2×C42—941no—
262C2×C42—1882no—
263C2×C42C22×D4711yescharacteristic
264C22⋊C8—944no—
265#97—1882no—
266#95—21no—
267#95—21no—
268#95—941no—
269#95—941no—
270#95—941no—
271C24—471no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8#1062532412yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—471no—
279C22×C4—471no—
280C22×C4—471no—
281C22×C4—942no—
282C22×C4—942no—
283C22×C4—942no—
284C22×C4—941no—
285C22×C4—941no—
286C22×C4—941no—
287C22×C4—941no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—941no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4—1881no—
295C22×C4C2×C4×D4711yescharacteristic
296C22×C4D4×D4711yescharacteristic
297C22×C4D4×D4711yescharacteristic
298M16—1888no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—942no—
306C2×C8—942no—
307C2×C8—1881no—
308C2×C8—1881no—
309C2×C8—1881no—
310C2×C8—1881no—
311C4⋊C4—41no—
312C4⋊C4—472no—
313C4⋊C4—944no—
314C4⋊C4—941no—
315C4⋊C4—1881no—
316C4⋊C4—1882no—
317C4⋊C4—1881no—
318C4⋊C4—1881no—
319C22⋊C4—944no—
320C22⋊C4—1882no—
321C42—41no—
322C42—942no—
323C42—942no—
324C42—1881no—
325C42—1882no—
326C42—1882no—
327C23—472no—
328C23—941no—
329C23—941no—
330C23—941no—
331C23—941no—
332C23—941no—
333C23—941no—
334C23C22⋊C4×D4711yescharacteristic
335Q8—42no—
336Q8—42no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—472no—
350C2×C4—472no—
351C2×C4—944no—
352C2×C4—944no—
353C2×C4—941no—
354C2×C4—941no—
355C2×C4—941no—
356C2×C4—941no—
357C2×C4—942no—
358C2×C4—941no—
359C2×C4—941no—
360C2×C4—941no—
361C2×C4—1882no—
362C2×C4—1882no—
363C2×C4—1882no—
364C2×C4—1882no—
365C2×C4—1881no—
366C2×C4—1881no—
367C2×C4—1881no—
368C2×C4—1881no—
369C2×C4—1881no—
370C2×C4M16×D4711yescharacteristic
371C2×C4C2×#1062532411yescharacteristic
372C2×C4#1309405111yescharacteristic
373C2×C4#1309405311yescharacteristic
374C2×C4#1309405611yescharacteristic
375C2×C4#1309405611yescharacteristic
376C8—42no—
377C8—42no—
378C8—1882no—
379C8—1882no—
380C22—942no—
381C22—942no—
382C22—942no—
383C22—1881no—
384C22—1881no—
385C22—1881no—
386C22—1881no—
387C22#1568908511yescharacteristic
388C22#99×D4711yescharacteristic
389C22#1568910411yescharacteristic
390C22#1568912311yescharacteristic
391C22#1568913011yescharacteristic
392C22#104×D4711yescharacteristic
393C22#104×D4711yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—42no—
401C4—81no—
402C4—942no—
403C4—942no—
404C4—3761no—
405C4—1881no—
406C4—1882no—
407C2—1882no—
408C2#2024172311yescharacteristic
409C2#2024182611yescharacteristic
410C2#2024183311yescharacteristic
411C2#2024189111yescharacteristic
412C2#2024190611yescharacteristic
413C2#2024197311yescharacteristic
414C2#2024211511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1272C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1272D47
hypocenterC47#7634
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #20241214C2semidirect
5. #20241212C2semidirect
6. #20241210C2semidirect
7. C47×#1272C2semidirect
67. C2×Q8×C47M16semidirect
103. C4⋊Dic47M16semidirect
204. #1272D47semidirect
234. C47#7634semidirect
275. C2×Q8#10625324semidirect

Maximal subgroups

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

subgroupindex
#202459262
#202431492
#202430802
#202412142
#202412122
#202412102
C47×#12722
#763447