Finite Group Databasebeta

← #26112319

Subgroups — #26112319

Order 12032.

684 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 127 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#20249193C211yesmaximal, characteristic
2#20246002C211yesmaximal, characteristic
3#20241961C212yesmaximal, complemented
4#20241256C212yesmaximal, complemented
5C2×C47×#323C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C22×C22⋊Dic47—21no—
8#15689059C412yescomplemented
9#15689046—22no—
10#15689035C412yescomplemented
11#15688998—24no—
12#15688996—21no—
13#15688983—21no—
14#15688983C2211yescharacteristic
15#15688983C2212yes—
16#15688979—21no—
17#15688861—24no—
18C22×C22⋊C4×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#323—22no—
21C22×Dic94C2×C411yescharacteristic, complemented
22C23×Dic47—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47C2×C411yescharacteristic, complemented
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—22no—
35C2×C22⋊Dic47—22no—
36C2×C22⋊Dic47—22no—
37C2×C22⋊Dic47—42no—
38C2×C22⋊Dic47—42no—
39C47⋊#95—22no—
40C47⋊#95—22no—
41C47⋊#95C2×C412yescomplemented
42C47⋊#95D414yes—
43C25×C47D411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47C2311yescharacteristic
46C22×C8×C47—22no—
47C2×C22⋊C4×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47C2×C412yes—
50C2×C22⋊C4×C47D412yes—
51C22⋊C8×C47—22no—
52C22⋊C8×C47—22no—
53C24×C47—21no—
54C24×C47—21no—
55C24×C47—21no—
56C24×C47—21no—
57C24×C47—81no—
58C24×C47—81no—
59C24×C47C22⋊C412yes—
60C24×C47SD1612yes—
61C24×C47Q1612yes—
62C24×C47C2×D411yescharacteristic
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—81no—
66C22×C4×C47—81no—
67C22×C4×C47C22⋊C412yes—
68C22×C4×C47C22×C411yescharacteristic
69C22×C4×C47C2×D412yes—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4112yes—
72C22×Dic47—21no—
73C22×Dic47—22no—
74C22×Dic47—21no—
75C22×Dic47—24no—
76C22×Dic47—44no—
77C22×Dic47—82no—
78C22×Dic47—82no—
79C2×Dic94—22no—
80C2×Dic94—81no—
81C2×C8×C47—42no—
82C2×C8×C47—82no—
83C22⋊C4×C47—24no—
84C22⋊C4×C47—24no—
85C22⋊Dic47—44no—
86C22⋊Dic47—41no—
87C22⋊Dic47—41no—
88C22⋊Dic47—42no—
89C22⋊Dic47—41no—
90C22⋊Dic47—42no—
91C22⋊Dic47—41no—
92C22⋊Dic47—82no—
93C22⋊Dic47—82no—
94C4⋊Dic47—22no—
95C4⋊Dic47—81no—
96C4⋊Dic47—82no—
97C4⋊Dic47—82no—
98Dic47⋊C4—42no—
99Dic47⋊C4—82no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—42no—
109C23×C47—42no—
110C23×C47—42no—
111C23×C47—42no—
112C23×C47—81no—
113C23×C47—81no—
114C23×C47—82no—
115C23×C47—82no—
116C23×C47—82no—
117C23×C47—82no—
118C23×C47#10314yes—
119C23×C47C2×C22⋊C411yescharacteristic
120C23×C47#12011yescharacteristic
121C23×C47#12311yescharacteristic
122C23×C47C2×SD1611yescharacteristic
123C23×C47C2×Q1611yescharacteristic
124C23×C47AGL(1,ℤ/8)11yescharacteristic
125C23×C47#13711yescharacteristic
126C2×C4×C47—22no—
127C2×C4×C47—42no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—81no—
134C2×C4×C47C4×D412yes—
135C2×C4×C47#12112yes—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—44no—
140C2×Dic47—44no—
141C2×Dic47—42no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—42no—
145C2×Dic47—44no—
146C2×Dic47—88no—
147Dic94—82no—
148C8×C47—82no—
149#9199—471nomaximal, Sylow
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—24no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—44no—
162C22×C47—44no—
163C22×C47—44no—
164C22×C47—42no—
165C22×C47—42no—
166C22×C47—82no—
167C22×C47—82no—
168C22×C47—82no—
169C22×C47#38611yescharacteristic
170C22×C47C2×#10311yescharacteristic
171C22×C47#41711yescharacteristic
172C22×C47#45011yescharacteristic
173C22×C47#45111yescharacteristic
174C22×C47#48311yescharacteristic
175C22×C47#48411yescharacteristic
176C4×C47—22no—
177C4×C47—22no—
178C4×C47—42no—
179Dic47—82no—
180Dic47—84no—
181Dic47—81no—
182Dic47—81no—
183#2626—471no—
184#2315—471no—
185#1520—472no—
186C2×#323D4711yescharacteristic
187#1289—472no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47—44no—
195C2×C47#187211yescharacteristic
196C2×C47#191111yescharacteristic
197C2×C47#191311yescharacteristic
198C2×C22⋊Q8—471no—
199C22×C22⋊C4—941no—
200C22×C22⋊C4C2×D4711yescharacteristic
201C2×C22⋊C8C2×D4712yes—
202#399—472no—
203#396—942no—
204#393—472no—
205#385—944no—
206#385—941no—
207#380—471no—
208#380—472no—
209#380—941no—
210#379—941no—
211#328—944no—
212#323—22no—
213C47#919911yescharacteristic, complemented, Sylow
214C25D18811yescharacteristic
215C22×Q8—471no—
216C23×C4—942no—
217C23×C4D18811yescharacteristic
218C23×C4C22×D4711yescharacteristic
219C22×C8—22no—
220C22⋊Q8—942no—
221C22⋊Q8—942no—
222C2×C4⋊C4—471no—
223C2×C4⋊C4—942no—
224C2×C4⋊C4—942no—
225C2×C4⋊C4—942no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—941no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—942no—
231C2×C22⋊C4—942no—
232C2×C22⋊C4—942no—
233C2×C22⋊C4—942no—
234C2×C22⋊C4—941no—
235C2×C22⋊C4—941no—
236C2×C22⋊C4—1882no—
237C2×C22⋊C4—1882no—
238C2×C22⋊C4C4×D4712yes—
239C2×C22⋊C4C22⋊D4712yes—
240C22⋊C8—22no—
241C22⋊C8—22no—
242#95—472no—
243#95—474no—
244#95—942no—
245#95—942no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24#1062532512yes—
253C24Dic18812yes—
254C24(C2×D47)⋊C412yes—
255C24C2×D18811yescharacteristic
256C2×Q8—942no—
257C2×Q8—3761no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—942no—
263C22×C4—941no—
264C22×C4—941no—
265C22×C4—944no—
266C22×C4—3762no—
267C22×C4—3762no—
268C22×C4—1884no—
269C22×C4(C2×D47)⋊C412yes—
270C22×C4C2×C4×D4711yescharacteristic
271C22×C4D4×D4712yes—
272C22×C4#1062535212yes—
273C22×C4C2×C22⋊D4711yescharacteristic
274C2×C8—42no—
275C2×C8—82no—
276C4⋊C4—942no—
277C4⋊C4—3762no—
278C4⋊C4—3762no—
279C4⋊C4—3761no—
280C4⋊C4—3762no—
281C4⋊C4—1882no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—3762no—
285C22⋊C4—3762no—
286C22⋊C4—1884no—
287C22⋊C4—1881no—
288C22⋊C4—1881no—
289C22⋊C4—1882no—
290C22⋊C4—1881no—
291C22⋊C4—1882no—
292C22⋊C4—1881no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1309394914yes—
312C23C2×(C2×D47)⋊C411yescharacteristic
313C23#1309399411yescharacteristic
314C23#1309400811yescharacteristic
315C23C2×#1062532511yescharacteristic
316C23C2×Dic18811yescharacteristic
317C23#1309405211yescharacteristic
318C23#1309405711yescharacteristic
319Q8—3762no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—3768no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1884no—
334C2×C4—1884no—
335C2×C4—1881no—
336C2×C4—1881no—
337C2×C4—1884no—
338C2×C4—1882no—
339C2×C4(C22⋊D47)⋊C412yes—
340C2×C4#1309399612yes—
341C8—82no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1568900011yescharacteristic
362C22C2×#1309394911yescharacteristic
363C22#1568912511yescharacteristic
364C22#1568927611yescharacteristic
365C22#1568928111yescharacteristic
366C22#1568945411yescharacteristic
367C22#1568945911yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—3764no—
372C4—3762no—
373C4—3761no—
374C4—3761no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2024369311yescharacteristic
383C2#2024391811yescharacteristic
384C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#323C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#323D47
hypocenterC47#9199
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20241961C2semidirect
4. #20241256C2semidirect
8. #15689059C4semidirect
10. #15689035C4semidirect
21. C22×Dic94C2×C4semidirect
28. C2×C4⋊Dic47C2×C4semidirect
41. C47⋊#95C2×C4semidirect
213. C47#9199semidirect

Maximal subgroups

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

subgroupindex
#202491932
#202460022
#202419612
#202412562
C2×C47×#3232
#919947