Finite Group Databasebeta

← #26127307

Subgroups — #26127307

Order 12032.

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

normal subgroups only

This group has 522 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#20243376C211yesmaximal, characteristic, complemented
2#20241709C211yesmaximal, characteristic, complemented
3#20241388C212yesmaximal, complemented
4#20241385C212yesmaximal, complemented
5C47×#2543C211yesmaximal, characteristic
6#15689176—21no—
7C2×#13093939—21no—
8#15688983—21no—
9#15688890C2212yes—
10#15688847—21no—
11#15688847C2212yes—
12C2×#121×C47—21no—
13C2×#121×C47C411yescharacteristic, complemented
14C2×C4×D4×C47C2211yescharacteristic
15C47×#398C411yescharacteristic, complemented
16C47×#397C2211yescharacteristic
17C47×#394C411yescharacteristic, complemented
18C47×#392C2211yescharacteristic
19C47×#388—21no—
20C47×#386—21no—
21C47×#386—21no—
22C2×#95×C47C411yescharacteristic, complemented
23C2×C4⋊Dic47—41no—
24C2×C22⋊Dic47—22no—
25C4⋊(C47⋊C8)—41no—
26#13093939—42no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—41no—
29C47⋊#95—41no—
30C22×D4×C47—21no—
31C22×D4×C47—21no—
32C22×D4×C47C2×C411yescharacteristic
33C22×D4×C47D411yescharacteristic
34C23×C4×C47—21no—
35C23×C4×C47C2×C411yescharacteristic
36C23×C4×C47D411yescharacteristic
37#121×C47—42no—
38#121×C47—42no—
39#121×C47—41no—
40#121×C47—41no—
41C4×D4×C47—41no—
42C4×D4×C47—41no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47—21no—
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47C2311yescharacteristic
53C2×C42×C47D411yescharacteristic
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47C2×C411yescharacteristic
58#95×C47C2×C411yescharacteristic
59C24×C47—21no—
60C24×C47C22⋊C411yescharacteristic
61C24×C47C2×D411yescharacteristic
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—42no—
65C2×D4×C47—41no—
66C2×D4×C47—42no—
67C2×D4×C47—42no—
68C2×D4×C47—42no—
69C2×D4×C47—41no—
70C2×D4×C47—41no—
71C2×D4×C47—81no—
72C2×D4×C47—81no—
73C2×D4×C47D812yes—
74C2×D4×C47SD1612yes—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—42no—
83C22×C4×C47—42no—
84C22×C4×C47—81no—
85C22×C4×C47—81no—
86C22×C4×C47—81no—
87C22×C4×C47C22⋊C411yescharacteristic
88C22×C4×C47C22⋊C411yescharacteristic
89C22×C4×C47C22⋊C411yescharacteristic
90C22×C4×C47C22×C411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×Dic47—42no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—42no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—42no—
99C22⋊C4×C47—41no—
100C22⋊C4×C47—41no—
101C22⋊C4×C47—41no—
102C22⋊C4×C47—41no—
103C22⋊C4×C47—42no—
104C22⋊C4×C47—81no—
105C42×C47—21no—
106C42×C47—21no—
107C22⋊Dic47—42no—
108C22⋊Dic47—42no—
109C4⋊Dic47—82no—
110C2×C47⋊C8—42no—
111C23×C47—21no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47—41no—
123C23×C47—41no—
124C23×C47—41no—
125C23×C47—41no—
126C23×C47#10212yes—
127C23×C47C2×C22⋊C411yescharacteristic
128C23×C47C2×D811yescharacteristic
129C23×C47C2×SD1611yescharacteristic
130D4×C47—42no—
131D4×C47—42no—
132D4×C47—82no—
133D4×C47—82no—
134D4×C47—82no—
135D4×C47—84no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—22no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47—41no—
152C2×C4×C47—41no—
153C2×C4×C47—41no—
154C2×C4×C47—41no—
155C2×C4×C47—81no—
156C2×C4×C47—81no—
157C2×C4×C47—81no—
158C2×C4×C47—84no—
159C2×C4×C47#9911yescharacteristic
160C2×C4×C47#9911yescharacteristic
161C2×C4×C47#10212yes—
162C2×Dic47—82no—
163C2×Dic47—82no—
164C2×Dic47—82no—
165C47⋊C8—82no—
166#11876—471nomaximal, Sylow
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—22no—
171C22×C47—21no—
172C22×C47—42no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—42no—
176C22×C47—41no—
177C22×C47—41no—
178C22×C47—41no—
179C22×C47—41no—
180C22×C47—81no—
181C22×C47—81no—
182C22×C47—81no—
183C22×C47—81no—
184C22×C47—81no—
185C22×C47—81no—
186C22×C47—81no—
187C22×C47—81no—
188C22×C47C2×#9911yescharacteristic
189C22×C47C2×#10211yescharacteristic
190C22×C47#42111yescharacteristic
191C4×C47—21no—
192C4×C47—21no—
193C4×C47—22no—
194C4×C47—42no—
195C4×C47—41no—
196C4×C47—41no—
197C4×C47—41no—
198C4×C47—81no—
199C4×C47—81no—
200Dic47—162no—
201#2543D4711yescharacteristic
202#1799—471no—
203#1460—471no—
204#1344—472no—
205#1343—472no—
206C2×C47—21no—
207C2×C47—21no—
208C2×C47—41no—
209C2×C47—42no—
210C2×C47—41no—
211C2×C47—161no—
212C2×C47#151911yescharacteristic
213C2×C47#212111yescharacteristic
214C2×C47#212611yescharacteristic
215C2×#121—21no—
216C2×#121Dic4711yescharacteristic, complemented
217C2×C4×D4C2×D4711yescharacteristic
218#431—941no—
219C2×#99—941no—
220#398Dic4711yescharacteristic, complemented
221#397C2×D4711yescharacteristic
222#394Dic4711yescharacteristic, complemented
223#392C2×D4711yescharacteristic
224#388—21no—
225#386—21no—
226#386—21no—
227#380—941no—
228C2×#95Dic4711yescharacteristic, complemented
229#342—472no—
230#323—472no—
231#323—941no—
232C47#1187611yescharacteristic, complemented, Sylow
233C22×D4—21no—
234C22×D4—21no—
235C22×D4C2×Dic4711yescharacteristic
236C22×D4C22⋊D4711yescharacteristic
237C23×C4—21no—
238C23×C4C2×Dic4711yescharacteristic
239C23×C4C22⋊D4711yescharacteristic
240#121—42no—
241#121—42no—
242#121—41no—
243#121—41no—
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1881no—
248C2×C4⋊C4C2×Dic4711yescharacteristic
249C2×C4⋊C4C2×Dic4711yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—942no—
256C2×C22⋊C4C22⋊D4711yescharacteristic
257C2×C22⋊C4C22×D4711yescharacteristic
258C2×C42C22⋊D4711yescharacteristic
259C4⋊C8—1881no—
260#99—1882no—
261C22⋊C8—942no—
262C4×C8—1881no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—1881no—
267#95C2×Dic4711yescharacteristic
268#95C2×Dic4711yescharacteristic
269C24—21no—
270C24C22⋊Dic4711yescharacteristic
271C24C2×C22⋊D4711yescharacteristic
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4#1062533412yes—
284C2×D4#1062533512yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—1882no—
298C22×C4C22⋊Dic4711yescharacteristic
299C22×C4C22⋊Dic4711yescharacteristic
300C22×C4C22⋊Dic4711yescharacteristic
301C22×C4C22×Dic4711yescharacteristic
302C22×C4C2×C22⋊D4711yescharacteristic
303C2×C8—1882no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—3762no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—81no—
317C22⋊C4—1882no—
318C22⋊C4—1882no—
319C42—21no—
320C42—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1309394512yes—
337C23C2×C22⋊Dic4711yescharacteristic
338C23C2×#1062533411yescharacteristic
339C23C2×#1062533511yescharacteristic
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344D4—82no—
345D4—84no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—84no—
369C2×C4—3762no—
370C2×C4—3762no—
371C2×C4—3762no—
372C2×C4#1309393911yescharacteristic
373C2×C4#1309393911yescharacteristic
374C2×C4#1309394512yes—
375C8—3762no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—21no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22C2×#1309393911yescharacteristic
398C22C2×#1309394511yescharacteristic
399C22#1568914011yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4—81no—
409C4—7522no—
410C2—21no—
411C2—21no—
412C2—41no—
413C2—42no—
414C2—41no—
415C2—161no—
416C2#2024195611yescharacteristic
417C2#2024503411yescharacteristic
418C2#2024505311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2543C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2543D47
hypocenterC47#11876
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #20243376C2semidirect
2. #20241709C2semidirect
3. #20241388C2semidirect
4. #20241385C2semidirect
13. C2×#121×C47C4semidirect
15. C47×#398C4semidirect
17. C47×#394C4semidirect
22. C2×#95×C47C4semidirect
216. C2×#121Dic47semidirect
220. #398Dic47semidirect
222. #394Dic47semidirect
228. C2×#95Dic47semidirect
232. C47#11876semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417092
#202413882
#202413852
C47×#25432
#1187647