Finite Group Databasebeta

← #26116789

Subgroups — #26116789

Order 12032.

568 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 109 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#20254160C211yesmaximal, characteristic
2C2×#15689458C211yesmaximal, characteristic
3#20244373C211yesmaximal, characteristic
4#20243478C211yesmaximal, characteristic
5#20241960C211yesmaximal, characteristic
6#20241236C211yesmaximal, characteristic
7C47×#2502C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689458—22no—
10#15689458—22no—
11#15689458—21no—
12#15689456—21no—
13#15689362—22no—
14#15689279—21no—
15#15689278—21no—
16C2×#13093957—21no—
17C2×#13093947—21no—
18C2×C22⋊(C47⋊C8)C2211yescharacteristic
19C2×C22⋊(C47⋊C8)C2211yescharacteristic
20#15688983C2211yescharacteristic
21#15688889—22no—
22#15688885—22no—
23#15688860—22no—
24C2×C22⋊Q8×C47C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#398—22no—
27C47×#391—21no—
28C47×#385—22no—
29C47×#384—21no—
30C4×C22⋊C4×C47C2211yescharacteristic
31C22×Dic94D411yescharacteristic
32C2×#10625337—22no—
33C22×C47⋊C8—21no—
34C22×C47⋊C8D412yes—
35#13094003—21no—
36#13094003—21no—
37#13094003—21no—
38#13094003—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47D411yescharacteristic
41C2×Dic47⋊C4—21no—
42C2×C22⋊Dic47—21no—
43C2×C22⋊Dic47—21no—
44#13093957—22no—
45#13093957—24no—
46#13093948—22no—
47#13093947—22no—
48#13093947—24no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52C22⋊(C47⋊C8)D412yes—
53C47⋊#95—21no—
54C22×Q8×C47D411yescharacteristic
55C23×C4×C47—21no—
56C23×C4×C47C2311yescharacteristic
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×C47—21no—
64C2×C4⋊C4×C47—21no—
65C2×C4⋊C4×C47—21no—
66C2×C4⋊C4×C47—21no—
67C2×C4⋊C4×C47—24no—
68C2×C4⋊C4×C47D412yes—
69C2×C4⋊C4×C47D411yescharacteristic
70C2×C22⋊C4×C47—21no—
71C2×C22⋊C4×C47D412yes—
72C2×C42×C47—21no—
73#95×C47—24no—
74#95×C47—21no—
75C24×C47C2×D411yescharacteristic
76C2×Q8×C47—21no—
77C2×Q8×C47—21no—
78C2×Q8×C47—42no—
79C22×C4×C47—22no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—22no—
83C22×C4×C47—42no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47—82no—
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C22×Dic47—21no—
94C22×Dic47—21no—
95C22×Dic47—41no—
96C2×Dic94—21no—
97C2×Dic94—21no—
98C2×Dic94—42no—
99C4⋊C4×C47—22no—
100C4⋊C4×C47—21no—
101C4⋊C4×C47—22no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—41no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—84no—
108C22⋊C4×C47—22no—
109C22⋊C4×C47—41no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47Q1614yescomplemented
112C42×C47—42no—
113C22⋊Dic47—41no—
114C22⋊Dic47—41no—
115C22⋊Dic47—42no—
116#10625337—82no—
117C4⋊Dic47—21no—
118C4⋊Dic47—21no—
119C4⋊Dic47—42no—
120Dic47⋊C4—41no—
121Dic47⋊C4—41no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—42no—
125C2×C47⋊C8—42no—
126C23×C47—41no—
127C23×C47—41no—
128C23×C47#12011yescharacteristic
129C23×C47#12111yescharacteristic
130C23×C47#12711yescharacteristic
131C23×C47C2×Q1612yes—
132C23×C47#13712yes—
133Q8×C47—42no—
134Q8×C47—81no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—24no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—44no—
146C2×C4×C47—42no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—42no—
150C2×C4×C47—82no—
151C2×C4×C47—82no—
152C2×C4×C47#12011yescharacteristic
153C2×C4×C47#12011yescharacteristic
154C2×C4×C47#12111yescharacteristic
155C2×C4×C47#12111yescharacteristic
156C2×Dic47—41no—
157C2×Dic47—41no—
158C2×Dic47—81no—
159C2×Dic47—81no—
160C2×Dic47—81no—
161C2×Dic47—81no—
162C2×Dic47—81no—
163Dic94—42no—
164Dic94—42no—
165C47⋊C8—42no—
166C47⋊C8—42no—
167#9959—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—42no—
175C22×C47—42no—
176C22×C47#39211yescharacteristic
177C22×C47#45112yes—
178C22×C47#46212yes—
179C22×C47#49411yescharacteristic
180C22×C47#49711yescharacteristic
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—44no—
185C4×C47—81no—
186C4×C47—82no—
187C4×C47—81no—
188C4×C47#45812yes—
189Dic47—81no—
190Dic47—81no—
191Dic47—161no—
192#3189—471no—
193C2×#484—471no—
194#2502D4711yescharacteristic
195#1996—471no—
196#1824—471no—
197#1520—471no—
198#1282—471no—
199C2×C47—21no—
200C2×C47—22no—
201C2×C47—22no—
202C2×C47—21no—
203C2×C47#200811yescharacteristic
204C2×C47#201611yescharacteristic
205C2×C47#202211yescharacteristic
206C2×C22⋊Q8—471no—
207C2×C22⋊Q8C2×D4711yescharacteristic
208C22×C4⋊C4C2×D4711yescharacteristic
209#484—942no—
210#484—941no—
211#484—942no—
212#484—941no—
213#467—942no—
214#451—941no—
215#451—941no—
216C2×#107—941no—
217C2×#103—941no—
218C2×C22⋊C8—471no—
219C2×C22⋊C8—471no—
220#398—22no—
221#391—21no—
222#385—22no—
223#384—21no—
224#380—471no—
225C4×C22⋊C4C2×D4711yescharacteristic
226#342—942no—
227#340—942no—
228#328—942no—
229C47#995911yescharacteristic, complemented, Sylow
230C22×Q8—471no—
231C22×Q8C22⋊D4711yescharacteristic
232C23×C4—21no—
233C23×C4C22×D4711yescharacteristic
234C2×Q16—942no—
235C22×C8—472no—
236C22×C8—941no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—941no—
242C22⋊Q8—941no—
243C22⋊Q8—941no—
244C22⋊Q8—941no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—471no—
253C2×C4⋊C4—941no—
254C2×C4⋊C4—941no—
255C2×C4⋊C4C22⋊D4712yes—
256C2×C4⋊C4C22⋊D4711yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—941no—
259C2×C22⋊C4—941no—
260C2×C22⋊C4C22⋊D4712yes—
261C2×C42—21no—
262#107—944no—
263#107—942no—
264#103—944no—
265#103—942no—
266#103—942no—
267C22⋊C8—472no—
268C22⋊C8—941no—
269C22⋊C8—941no—
270C22⋊C8—941no—
271#95—24no—
272#95—21no—
273#95—941no—
274C24C2×C22⋊D4711yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—42no—
278C2×Q8—941no—
279C2×Q8—941no—
280C2×Q8—1882no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—82no—
289C22×C4—941no—
290C22×C4—941no—
291C22×C4—1881no—
292C22×C4D4×D4711yescharacteristic
293C22×C4D4×D4711yescharacteristic
294C22×C4D4×D4711yescharacteristic
295C22×C4#1062535211yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297C22×C4C2×C22⋊D4711yescharacteristic
298Q16—3762no—
299C2×C8—942no—
300C2×C8—942no—
301C2×C8—1882no—
302C2×C8—1882no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—84no—
312C4⋊C4—941no—
313C4⋊C4—941no—
314C4⋊C4—1882no—
315C4⋊C4—1881no—
316C4⋊C4—1881no—
317C22⋊C4—22no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—1882no—
321C22⋊C4—1881no—
322C22⋊C4—1881no—
323C22⋊C4#1062533714yescomplemented
324C42—42no—
325C23—41no—
326C23—41no—
327C23#1309399311yescharacteristic
328C23#1309399511yescharacteristic
329C23#1309402011yescharacteristic
330C23C2×#1062533712yes—
331C23#1309405512yes—
332Q8—42no—
333Q8—81no—
334Q8—1882no—
335Q8—1882no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—3761no—
354C2×C4—3761no—
355C2×C4—3761no—
356C2×C4—3761no—
357C2×C4—3761no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4#1309399211yescharacteristic
361C2×C4#1309399211yescharacteristic
362C2×C4#1309399611yescharacteristic
363C2×C4#1309399611yescharacteristic
364C8—1882no—
365C8—1882no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1568903111yescharacteristic
375C22#1568927912yes—
376C22#1568933012yes—
377C22#1568950211yescharacteristic
378C22#1568951511yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—81no—
384C4—82no—
385C4—81no—
386C4—3761no—
387C4—3761no—
388C4—7521no—
389C4#1568931212yes—
390C2—22no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2#2024444111yescharacteristic
395C2#2024448811yescharacteristic
396C2#2024451611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2502C2
socleC22×C47#392
prime radicalC24×C47C2×D4
hypercenter#2502D47
hypocenterC47#9959
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2502

Complements

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

NH ≅ G/Ntype
111. C22⋊C4×C47Q16semidirect
229. C47#9959semidirect
323. C22⋊C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202541602
C2×#156894582
#202443732
#202434782
#202419602
#202412362
C47×#25022
#995947