Finite Group Databasebeta

← #26126627

Subgroups — #26126627

Order 12032.

500 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#13093939C211yesmaximal, characteristic, complemented
2#20241706C211yesmaximal, characteristic, complemented
3#20241409C212yesmaximal, complemented
4#20241385C212yesmaximal, complemented
5C47×#2631C211yesmaximal, characteristic
6C2×#13093939—21no—
7C2×C22⋊(C47⋊C8)—21no—
8C4×C22⋊Dic47—21no—
9#15688890C2212yes—
10#15688849—21no—
11#15688847C2212yes—
12C2×#124×C47C411yescharacteristic, complemented
13C2×C4×D4×C47C2211yescharacteristic
14C47×#397—21no—
15C47×#393C411yescharacteristic, complemented
16C47×#393C2211yescharacteristic
17C47×#392—21no—
18C47×#392C411yescharacteristic, complemented
19C47×#392C2211yescharacteristic
20C47×#386—21no—
21C47×#386—21no—
22C47×#383C411yescharacteristic, complemented
23C22×C47⋊C8—41no—
24C2×C22⋊Dic47—22no—
25C2×C4×Dic47—41no—
26#13093939—24no—
27C22⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—41no—
29C47⋊#95—41no—
30C22×Q8×C47C2×C411yescharacteristic
31C22×D4×C47—21no—
32C22×D4×C47—21no—
33C22×D4×C47C2×C411yescharacteristic
34C22×D4×C47D411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47D411yescharacteristic
37#124×C47—41no—
38#124×C47—41no—
39C4×D4×C47—41no—
40C4×D4×C47—41no—
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C22⋊C4×C47—21no—
43C2×C22⋊C4×C47—21no—
44C2×C22⋊C4×C47—21no—
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C22⋊C4×C47—21no—
48C2×C22⋊C4×C47C2×C411yescharacteristic
49C2×C22⋊C4×C47C2311yescharacteristic
50C2×C42×C47C2×C411yescharacteristic
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47C2×C411yescharacteristic
56#95×C47C2×C411yescharacteristic
57C24×C47—21no—
58C24×C47C22⋊C411yescharacteristic
59C24×C47C2×D411yescharacteristic
60C2×Q8×C47—41no—
61C2×Q8×C47—81no—
62C2×D4×C47—22no—
63C2×D4×C47—22no—
64C2×D4×C47—41no—
65C2×D4×C47—41no—
66C2×D4×C47—42no—
67C2×D4×C47—42no—
68C2×D4×C47—81no—
69C2×D4×C47—81no—
70C2×D4×C47—81no—
71C2×D4×C47—81no—
72C2×D4×C47—81no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—81no—
81C22×C4×C47—81no—
82C22×C4×C47C22⋊C411yescharacteristic
83C22×C4×C47C22⋊C411yescharacteristic
84C22×C4×C47C22⋊C411yescharacteristic
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×Dic47—42no—
88C4⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—41no—
94C22⋊C4×C47—81no—
95C22⋊C4×C47—81no—
96C22⋊C4×C47—81no—
97C22⋊C4×C47—81no—
98C42×C47—21no—
99C42×C47—21no—
100C22⋊Dic47—24no—
101C22⋊Dic47—42no—
102C4×Dic47—82no—
103C2×C47⋊C8—42no—
104C2×C47⋊C8—82no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47#10412yes—
121C23×C47C2×C22⋊C411yescharacteristic
122C23×C47#13512yes—
123Q8×C47—82no—
124D4×C47—44no—
125D4×C47—82no—
126D4×C47—84no—
127D4×C47—84no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—41no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—81no—
150C2×C4×C47—81no—
151C2×C4×C47#9911yescharacteristic
152C2×C4×C47#9911yescharacteristic
153C2×Dic47—44no—
154C2×Dic47—82no—
155C2×Dic47—82no—
156C47⋊C8—82no—
157#11730—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—22no—
161C22×C47—22no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—81no—
169C22×C47—81no—
170C22×C47—81no—
171C22×C47—81no—
172C22×C47—81no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47—81no—
176C22×C47—81no—
177C22×C47C2×#9911yescharacteristic
178C22×C47#41611yescharacteristic
179C22×C47C2×#10411yescharacteristic
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—41no—
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—41no—
186C4×C47—81no—
187C4×C47—81no—
188C4×C47—81no—
189Dic47—84no—
190#2631D4711yescharacteristic
191C4×#99—471no—
192#1459—471no—
193#1351—472no—
194#1343—472no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—42no—
198C2×C47—42no—
199C2×C47—161no—
200C2×C47#150111yescharacteristic
201C2×C47#212211yescharacteristic
202C2×C47#212711yescharacteristic
203C2×#124Dic4711yescharacteristic, complemented
204C2×C4×D4C2×D4711yescharacteristic
205C2×#99—941no—
206C2×C22⋊C8—941no—
207#397—21no—
208#393Dic4711yescharacteristic, complemented
209#393C2×D4711yescharacteristic
210#392—21no—
211#392Dic4711yescharacteristic, complemented
212#392C2×D4711yescharacteristic
213#386—21no—
214#386—21no—
215#383Dic4711yescharacteristic, complemented
216C4×C22⋊C4—941no—
217#342—472no—
218#324—941no—
219#323—472no—
220C47#1173011yescharacteristic, complemented, Sylow
221C22×Q8C2×Dic4711yescharacteristic
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C2×Dic4711yescharacteristic
225C22×D4C22⋊D4711yescharacteristic
226C23×C4C22⋊D4711yescharacteristic
227C23×C4C22⋊D4711yescharacteristic
228C22×C8—1881no—
229#124—41no—
230#124—41no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4C22⋊D4711yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—942no—
241C2×C22⋊C4C2×Dic4711yescharacteristic
242C2×C22⋊C4C22×D4711yescharacteristic
243C2×C42—1881no—
244C2×C42C2×Dic4711yescharacteristic
245#99—944no—
246C22⋊C8—942no—
247C22⋊C8—1881no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—1881no—
253#95C2×Dic4711yescharacteristic
254#95C2×Dic4711yescharacteristic
255C24—21no—
256C24C22⋊Dic4711yescharacteristic
257C24C2×C22⋊D4711yescharacteristic
258C2×Q8—41no—
259C2×Q8—81no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—81no—
267C2×D4—81no—
268C2×D4—81no—
269C2×D4—81no—
270C2×D4—81no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—1882no—
281C22×C4C22⋊Dic4711yescharacteristic
282C22×C4C22⋊Dic4711yescharacteristic
283C22×C4C22⋊Dic4711yescharacteristic
284C22×C4C22×Dic4711yescharacteristic
285C22×C4C2×C22⋊D4711yescharacteristic
286C2×C8—3762no—
287C2×C8—1882no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C22⋊C4—81no—
298C22⋊C4—944no—
299C22⋊C4—1882no—
300C42—21no—
301C42—21no—
302C42—3762no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23#1309395112yes—
319C23C2×C22⋊Dic4711yescharacteristic
320C23#1309404412yes—
321Q8—82no—
322D4—44no—
323D4—82no—
324D4—84no—
325D4—84no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—3762no—
350C2×C4—3762no—
351C2×C4—1884no—
352C2×C4#1309393911yescharacteristic
353C2×C4#1309393911yescharacteristic
354C8—3762no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—42no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22C2×#1309393911yescharacteristic
375C22#1568911911yescharacteristic
376C22C2×#1309395111yescharacteristic
377C4—21no—
378C4—21no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—3764no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—161no—
392C2#2024187911yescharacteristic
393C2#2024503811yescharacteristic
394C2#2024505711yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2631C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2631D47
hypocenterC47#11730
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2631

Complements

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

NH ≅ G/Ntype
1. C4×#13093939C2semidirect
2. #20241706C2semidirect
3. #20241409C2semidirect
4. #20241385C2semidirect
12. C2×#124×C47C4semidirect
15. C47×#393C4semidirect
18. C47×#392C4semidirect
22. C47×#383C4semidirect
203. C2×#124Dic47semidirect
208. #393Dic47semidirect
211. #392Dic47semidirect
215. #383Dic47semidirect
220. C47#11730semidirect

Maximal subgroups

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

subgroupindex
C4×#130939392
#202417062
#202414092
#202413852
C47×#26312
#1173047