Finite Group Databasebeta

← #26112331

Subgroups — #26112331

Order 12032.

612 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 115 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#20249193C211yesmaximal, characteristic
2#20241959C212yesmaximal, complemented
3C2×#15688847C211yesmaximal, characteristic
4#20241255C212yesmaximal, complemented
5C47×#2312C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689059C412yescomplemented
9#15689046—22no—
10#15689035C412yescomplemented
11#15688996—21no—
12#15688983C2211yescharacteristic
13#15688979—21no—
14#15688859—24no—
15#15688847—22no—
16C22×C22⋊C4×C47C2211yescharacteristic
17C47×#388—24no—
18C47×#380C2212yes—
19C47×#379—21no—
20C4×C22⋊C4×C47—21no—
21C22×Dic94C2×C411yescharacteristic
22C22×C47⋊C8—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×Dic47⋊C4—21no—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C22⋊(C47⋊C8)—22no—
34C22⋊(C47⋊C8)—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C25×C47D411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47C2311yescharacteristic
40C2×C4⋊C4×C47—22no—
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47—22no—
43C2×C22⋊C4×C47—22no—
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47C2×C412yes—
47C2×C22⋊C4×C47D412yes—
48C2×C42×C47—22no—
49#95×C47—22no—
50#95×C47D414yes—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—22no—
54C24×C47—22no—
55C24×C47—81no—
56C24×C47—81no—
57C24×C47C22⋊C412yes—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—24no—
62C22×C4×C47—81no—
63C22×C4×C47—81no—
64C22×C4×C47C22⋊C412yes—
65C22×C4×C47C22×C411yescharacteristic
66C22×C4×C47C2×D412yes—
67C22×C4×C47C2×D411yescharacteristic
68C22×C4×C47#4112yes—
69C22×Dic47—21no—
70C22×Dic47—22no—
71C22×Dic47—21no—
72C2×Dic94—22no—
73C2×Dic94—81no—
74C4⋊C4×C47—82no—
75C22⋊C4×C47—24no—
76C22⋊C4×C47—24no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C22⋊C4×C47—82no—
80C22⋊C4×C47—82no—
81C42×C47—82no—
82C22⋊Dic47—41no—
83C22⋊Dic47—41no—
84C22⋊Dic47—42no—
85C22⋊Dic47—82no—
86C22⋊Dic47—82no—
87C4⋊Dic47—22no—
88C4⋊Dic47—81no—
89Dic47⋊C4—42no—
90Dic47⋊C4—82no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—82no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—42no—
103C23×C47—42no—
104C23×C47—81no—
105C23×C47—81no—
106C23×C47—84no—
107C23×C47—84no—
108C23×C47C2×C22⋊C411yescharacteristic
109C23×C47#12011yescharacteristic
110C23×C47#12311yescharacteristic
111C23×C47#13512yes—
112C23×C47#13712yes—
113C2×C4×C47—22no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—44no—
121C2×C4×C47—44no—
122C2×C4×C47—44no—
123C2×C4×C47—81no—
124C2×C4×C47C4×D412yes—
125C2×C4×C47#12112yes—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—41no—
131C2×Dic47—41no—
132C2×Dic47—42no—
133Dic94—82no—
134C47⋊C8—82no—
135#9201—471nomaximal, Sylow
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—44no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—82no—
149C22×C47—84no—
150C22×C47—84no—
151C22×C47#38611yescharacteristic
152C22×C47#41611yescharacteristic
153C22×C47#41911yescharacteristic
154C22×C47#45212yes—
155C22×C47#48512yes—
156C4×C47—22no—
157C4×C47—22no—
158C4×C47—42no—
159C4×C47—84no—
160Dic47—82no—
161Dic47—81no—
162Dic47—81no—
163#2626—471no—
164#2312D4711yescharacteristic
165#1520—472no—
166C2×#323—471no—
167#1289—472no—
168C2×C47—21no—
169C2×C47—22no—
170C2×C47—21no—
171C2×C47—22no—
172C2×C47—82no—
173C2×C47#187311yescharacteristic
174C2×C47#191111yescharacteristic
175C2×C47#191311yescharacteristic
176C2×C22⋊Q8—471no—
177C22×C22⋊C4C2×D4711yescharacteristic
178C2×C22⋊C8—472no—
179#399—472no—
180#396—942no—
181#393—472no—
182#388—24no—
183#385—941no—
184#380—471no—
185#380C2×D4712yes—
186#379—21no—
187#379—941no—
188C4×C22⋊C4—21no—
189#328—944no—
190#323—942no—
191C47#920111yescharacteristic, complemented, Sylow
192C25C22⋊D4711yescharacteristic
193C22×Q8—471no—
194C23×C4C22⋊D4711yescharacteristic
195C23×C4C22×D4711yescharacteristic
196C22×C8—942no—
197C22⋊Q8—942no—
198C22⋊Q8—942no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—471no—
201C2×C4⋊C4—942no—
202C2×C4⋊C4—941no—
203C2×C4⋊C4—941no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—942no—
210C2×C22⋊C4—942no—
211C2×C22⋊C4—941no—
212C2×C22⋊C4—941no—
213C2×C22⋊C4C4×D4712yes—
214C2×C22⋊C4D18812yes—
215C2×C42—22no—
216C22⋊C8—942no—
217C22⋊C8—942no—
218#95—22no—
219#95—472no—
220#95—942no—
221#95C22⋊D4714yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24(C2×D47)⋊C412yes—
229C24C2×C22⋊D4711yescharacteristic
230C2×Q8—942no—
231C2×Q8—3761no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—942no—
238C22×C4—941no—
239C22×C4—941no—
240C22×C4(C2×D47)⋊C412yes—
241C22×C4C2×C4×D4711yescharacteristic
242C22×C4C2×D18811yescharacteristic
243C22×C4#1062535012yes—
244C22×C4C2×C22⋊D4712yes—
245C2×C8—3762no—
246C2×C8—1882no—
247C4⋊C4—82no—
248C4⋊C4—942no—
249C4⋊C4—3762no—
250C4⋊C4—3761no—
251C4⋊C4—1882no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—3762no—
259C22⋊C4—3762no—
260C22⋊C4—1881no—
261C22⋊C4—1881no—
262C22⋊C4—1882no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×(C2×D47)⋊C411yescharacteristic
280C23#1309399311yescharacteristic
281C23#1309400711yescharacteristic
282C23#1309404412yes—
283C23#1309405412yes—
284Q8—3762no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—1882no—
297C2×C4—1882no—
298C2×C4—1882no—
299C2×C4—1882no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1882no—
303C2×C4C4×C22⋊D4712yes—
304C2×C4#1309399812yes—
305C8—3762no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1568900211yescharacteristic
322C22#1568911711yescharacteristic
323C22#1568913111yescharacteristic
324C22#1568928512yes—
325C22#1568946412yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—3762no—
331C4—3761no—
332C4—3761no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2024370011yescharacteristic
339C2#2024392011yescharacteristic
340C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2312C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#2312D47
hypocenterC47#9201
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #20241959C2semidirect
4. #20241255C2semidirect
8. #15689059C4semidirect
10. #15689035C4semidirect
191. C47#9201semidirect

Maximal subgroups

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

subgroupindex
#202491932
#202419592
C2×#156888472
#202412552
C47×#23122
#920147