Finite Group Databasebeta

← #26127302

Subgroups — #26127302

Order 12032.

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

normal subgroups only

This group has 536 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#20241706C211yesmaximal, characteristic, complemented
3#20241388C212yesmaximal, complemented
4#20241385C212yesmaximal, complemented
5C47×#2535C211yesmaximal, characteristic
6C2×#13093939—21no—
7C2×C22⋊(C47⋊C8)—21no—
8#15688983—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×#121×C47—21no—
13C2×C4×D4×C47C411yescharacteristic, complemented
14C2×C4×D4×C47C2211yescharacteristic
15C47×#397C411yescharacteristic, complemented
16C47×#397C2211yescharacteristic
17C47×#392C411yescharacteristic, complemented
18C47×#392C2211yescharacteristic
19C47×#386—21no—
20C47×#386—21no—
21C47×#382C411yescharacteristic, complemented
22C47×#380—21no—
23C22×C47⋊C8—41no—
24C2×C4⋊Dic47—41no—
25C2×C22⋊Dic47—22no—
26#13093939—42no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(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×C47D411yescharacteristic
36C23×C4×C47D411yescharacteristic
37#121×C47—42no—
38#121×C47—42no—
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C4×D4×C47—41no—
42C4×D4×C47—41no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
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×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47C2311yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
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—41no—
65C2×D4×C47—42no—
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—42no—
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—81no—
104C22⋊C4×C47—81no—
105C42×C47—21no—
106C42×C47—21no—
107C22⋊Dic47—42no—
108C22⋊Dic47—42no—
109C4⋊Dic47—82no—
110C2×C47⋊C8—42no—
111C2×C47⋊C8—82no—
112C23×C47—21no—
113C23×C47—21no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—22no—
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—41no—
127C23×C47#10212yes—
128C23×C47C2×C22⋊C411yescharacteristic
129C23×C47C2×D811yescharacteristic
130C23×C47C2×SD1611yescharacteristic
131D4×C47—42no—
132D4×C47—42no—
133D4×C47—42no—
134D4×C47—42no—
135D4×C47—84no—
136D4×C47—82no—
137D4×C47—82no—
138C2×C4×C47—21no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—21no—
142C2×C4×C47—21no—
143C2×C4×C47—41no—
144C2×C4×C47—42no—
145C2×C4×C47—42no—
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—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—81no—
159C2×C4×C47—81no—
160C2×C4×C47—84no—
161C2×C4×C47#9911yescharacteristic
162C2×C4×C47#9911yescharacteristic
163C2×C4×C47#10212yes—
164C2×Dic47—82no—
165C2×Dic47—82no—
166C2×Dic47—82no—
167C47⋊C8—82no—
168#11875—471nomaximal, Sylow
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—22no—
173C22×C47—21no—
174C22×C47—42no—
175C22×C47—41no—
176C22×C47—41no—
177C22×C47—42no—
178C22×C47—42no—
179C22×C47—41no—
180C22×C47—41no—
181C22×C47—41no—
182C22×C47—41no—
183C22×C47—42no—
184C22×C47—81no—
185C22×C47—81no—
186C22×C47—81no—
187C22×C47—81no—
188C22×C47—81no—
189C22×C47—81no—
190C22×C47C2×#9911yescharacteristic
191C22×C47C2×#10211yescharacteristic
192C22×C47#42111yescharacteristic
193C4×C47—21no—
194C4×C47—21no—
195C4×C47—22no—
196C4×C47—41no—
197C4×C47—41no—
198C4×C47—41no—
199C4×C47—81no—
200C4×C47—81no—
201C4×C47—81no—
202Dic47—162no—
203#2535D4711yescharacteristic
204#1799—471no—
205#1459—471no—
206#1344—472no—
207#1343—472no—
208C2×C47—21no—
209C2×C47—21no—
210C2×C47—41no—
211C2×C47—42no—
212C2×C47—41no—
213C2×C47—82no—
214C2×C47#151911yescharacteristic
215C2×C47#212011yescharacteristic
216C2×C47#212711yescharacteristic
217C2×#121—21no—
218C2×C4×D4Dic4711yescharacteristic, complemented
219C2×C4×D4C2×D4711yescharacteristic
220C2×#99—941no—
221C2×C22⋊C8—941no—
222#397Dic4711yescharacteristic, complemented
223#397C2×D4711yescharacteristic
224#392Dic4711yescharacteristic, complemented
225#392C2×D4711yescharacteristic
226#386—21no—
227#386—21no—
228#382Dic4711yescharacteristic, complemented
229#380—21no—
230#380—941no—
231#342—472no—
232#324—941no—
233#323—472no—
234C47#1187511yescharacteristic, complemented, Sylow
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2×Dic4711yescharacteristic
238C22×D4C22⋊D4711yescharacteristic
239C23×C4—21no—
240C23×C4C22⋊D4711yescharacteristic
241C23×C4C22⋊D4711yescharacteristic
242C22×C8—1881no—
243#121—42no—
244#121—42no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—1881no—
251C2×C4⋊C4C2×Dic4711yescharacteristic
252C2×C4⋊C4C22⋊D4711yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—942no—
259C2×C22⋊C4C2×Dic4711yescharacteristic
260C2×C22⋊C4C22×D4711yescharacteristic
261C2×C42C2×Dic4711yescharacteristic
262#99—1882no—
263C22⋊C8—942no—
264C22⋊C8—1881no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—1881no—
269#95C2×Dic4711yescharacteristic
270#95C2×Dic4711yescharacteristic
271C24—21no—
272C24C22⋊Dic4711yescharacteristic
273C24C2×C22⋊D4711yescharacteristic
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C2×D4—81no—
285C2×D4#1062533412yes—
286C2×D4#1062533512yes—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—1882no—
300C22×C4C22⋊Dic4711yescharacteristic
301C22×C4C22⋊Dic4711yescharacteristic
302C22×C4C22⋊Dic4711yescharacteristic
303C22×C4C22×Dic4711yescharacteristic
304C22×C4C2×C22⋊D4711yescharacteristic
305C2×C8—3762no—
306C2×C8—1882no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—3762no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C22⋊C4—1882no—
321C22⋊C4—1882no—
322C42—21no—
323C42—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23#1309394512yes—
340C23C2×C22⋊Dic4711yescharacteristic
341C23C2×#1062533411yescharacteristic
342C23C2×#1062533511yescharacteristic
343D4—42no—
344D4—42no—
345D4—42no—
346D4—42no—
347D4—84no—
348D4—82no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—3762no—
374C2×C4—3762no—
375C2×C4—3762no—
376C2×C4#1309393911yescharacteristic
377C2×C4#1309393911yescharacteristic
378C2×C4#1309394512yes—
379C8—3762no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22C2×#1309393911yescharacteristic
402C22C2×#1309394511yescharacteristic
403C22#1568914011yescharacteristic
404C4—22no—
405C4—21no—
406C4—21no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—7522no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—42no—
418C2—41no—
419C2—82no—
420C2#2024195611yescharacteristic
421C2#2024502911yescharacteristic
422C2#2024505711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2535C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2535D47
hypocenterC47#11875
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #20243376C2semidirect
2. #20241706C2semidirect
3. #20241388C2semidirect
4. #20241385C2semidirect
13. C2×C4×D4×C47C4semidirect
15. C47×#397C4semidirect
17. C47×#392C4semidirect
21. C47×#382C4semidirect
218. C2×C4×D4Dic47semidirect
222. #397Dic47semidirect
224. #392Dic47semidirect
228. #382Dic47semidirect
234. C47#11875semidirect

Maximal subgroups

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

subgroupindex
#202433762
#202417062
#202413882
#202413852
C47×#25352
#1187547