Finite Group Databasebeta

← #26116803

Subgroups — #26116803

Order 12032.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15689453C211yesmaximal, characteristic
3#20244373C211yesmaximal, characteristic
4#20243474C211yesmaximal, characteristic
5#20241960C211yesmaximal, characteristic
6#20241236C211yesmaximal, characteristic
7C47×#2503C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689458—21no—
10#15689456—21no—
11#15689453—22no—
12#15689453—22no—
13#15689362—22no—
14#15689279—21no—
15#15689278—21no—
16C2×#13093955—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×#400—22no—
27C47×#391—21no—
28C47×#382—21no—
29C47×#381—22no—
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#13093955—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×C47D412yes—
68C2×C4⋊C4×C47D411yescharacteristic
69C2×C22⋊C4×C47—21no—
70C2×C22⋊C4×C47D412yes—
71C2×C42×C47—21no—
72#95×C47—24no—
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—42no—
87C22×C4×C47—42no—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47#4111yescharacteristic
94C22×Dic47—21no—
95C22×Dic47—21no—
96C22×Dic47—41no—
97C2×Dic94—21no—
98C2×Dic94—21no—
99C2×Dic94—42no—
100C4⋊C4×C47—22no—
101C4⋊C4×C47—21no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47—41no—
105C4⋊C4×C47—41no—
106C4⋊C4×C47—41no—
107C4⋊C4×C47—41no—
108C22⋊C4×C47—22no—
109C22⋊C4×C47—22no—
110C22⋊C4×C47—41no—
111C22⋊C4×C47—41no—
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×C47#13512yes—
132C23×C47#13712yes—
133Q8×C47—42no—
134Q8×C47—81no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—42no—
149C2×C4×C47—44no—
150C2×C4×C47—82no—
151C2×C4×C47—81no—
152C2×C4×C47—81no—
153C2×C4×C47#12011yescharacteristic
154C2×C4×C47#12011yescharacteristic
155C2×C4×C47#12111yescharacteristic
156C2×C4×C47#12111yescharacteristic
157C2×Dic47—41no—
158C2×Dic47—41no—
159C2×Dic47—81no—
160C2×Dic47—81no—
161C2×Dic47—81no—
162C2×Dic47—81no—
163C2×Dic47—81no—
164Dic94—42no—
165Dic94—42no—
166C47⋊C8—42no—
167C47⋊C8—42no—
168#9961—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—42no—
176C22×C47—42no—
177C22×C47#39211yescharacteristic
178C22×C47#45212yes—
179C22×C47#46412yes—
180C22×C47#49511yescharacteristic
181C22×C47#49711yescharacteristic
182C4×C47—21no—
183C4×C47—21no—
184C4×C47—21no—
185C4×C47—82no—
186C4×C47—81no—
187C4×C47—82no—
188C4×C47—81no—
189C4×C47#45812yes—
190Dic47—81no—
191Dic47—81no—
192Dic47—161no—
193#3189—471no—
194C2×#483—471no—
195#2503D4711yescharacteristic
196#1996—471no—
197#1823—471no—
198#1520—471no—
199#1282—471no—
200C2×C47—21no—
201C2×C47—22no—
202C2×C47—22no—
203C2×C47—21no—
204C2×C47#200911yescharacteristic
205C2×C47#201511yescharacteristic
206C2×C47#202211yescharacteristic
207C2×C22⋊Q8—471no—
208C2×C22⋊Q8C2×D4711yescharacteristic
209C22×C4⋊C4C2×D4711yescharacteristic
210#484—941no—
211#484—941no—
212#483—942no—
213#483—942no—
214#467—942no—
215#451—941no—
216#451—941no—
217C2×#106—941no—
218C2×#103—941no—
219C2×C22⋊C8—471no—
220C2×C22⋊C8—471no—
221#400—22no—
222#391—21no—
223#382—21no—
224#381—22no—
225#380—471no—
226C4×C22⋊C4C2×D4711yescharacteristic
227#342—942no—
228#340—942no—
229#328—942no—
230C47#996111yescharacteristic, complemented, Sylow
231C22×Q8—471no—
232C22×Q8C22⋊D4711yescharacteristic
233C23×C4—21no—
234C23×C4C22×D4711yescharacteristic
235C2×Q16—942no—
236C22×C8—472no—
237C22×C8—941no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—941no—
243C22⋊Q8—941no—
244C22⋊Q8—941no—
245C22⋊Q8—941no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
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—942no—
263#106—944no—
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—24no—
273#95—21no—
274#95—941no—
275C24C2×C22⋊D4711yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—941no—
280C2×Q8—941no—
281C2×Q8—1882no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—941no—
292C22×C4—941no—
293C22×C4—1881no—
294C22×C4D4×D4711yescharacteristic
295C22×C4D4×D4711yescharacteristic
296C22×C4D4×D4711yescharacteristic
297C22×C4#1062535211yescharacteristic
298C22×C4C2×C22⋊D4711yescharacteristic
299C22×C4C2×C22⋊D4711yescharacteristic
300Q16—3762no—
301C2×C8—942no—
302C2×C8—942no—
303C2×C8—1882no—
304C2×C8—1882no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—941no—
314C4⋊C4—941no—
315C4⋊C4—1882no—
316C4⋊C4—1881no—
317C4⋊C4—1881no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—1882no—
323C22⋊C4—1881no—
324C22⋊C4—1881no—
325C42—42no—
326C23—41no—
327C23—41no—
328C23#1309399311yescharacteristic
329C23#1309399511yescharacteristic
330C23#1309402011yescharacteristic
331C23#1309404412yes—
332C23#1309405412yes—
333Q8—42no—
334Q8—81no—
335Q8—1882no—
336Q8—1882no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—3761no—
356C2×C4—3761no—
357C2×C4—3761no—
358C2×C4—3761no—
359C2×C4—3761no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4#1309399211yescharacteristic
363C2×C4#1309399211yescharacteristic
364C2×C4#1309399611yescharacteristic
365C2×C4#1309399611yescharacteristic
366C8—1882no—
367C8—1882no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—42no—
375C22—42no—
376C22#1568903111yescharacteristic
377C22#1568928512yes—
378C22#1568934212yes—
379C22#1568950511yescharacteristic
380C22#1568951811yescharacteristic
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—81no—
387C4—81no—
388C4—3761no—
389C4—3761no—
390C4—7521no—
391C4#1568931212yes—
392C2—22no—
393C2—21no—
394C2—21no—
395C2—22no—
396C2#2024444811yescharacteristic
397C2#2024448211yescharacteristic
398C2#2024451611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2503C2
socleC22×C47#392
prime radicalC24×C47C2×D4
hypercenter#2503D47
hypocenterC47#9961
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2503

Complements

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

NH ≅ G/Ntype
230. C47#9961semidirect

Maximal subgroups

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

subgroupindex
#202541602
C2×#156894532
#202443732
#202434742
#202419602
#202412362
C47×#25032
#996147