Finite Group Databasebeta

← #26113731

Subgroups — #26113731

Order 12032.

606 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 99 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#20248546C211yesmaximal, characteristic
2#20245986C211yesmaximal, characteristic
3#20241909C212yesmaximal, complemented
4#20241273C212yesmaximal, complemented
5C47×#1462C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7#15689048C412yescomplemented
8#15689035C412yescomplemented
9#15689017—22no—
10#15689016—24no—
11#15688986—21no—
12#15688983—21no—
13#15688979—21no—
14#15688979—21no—
15C4×C22⋊Dic47C2211yescharacteristic
16C4×C22⋊Dic47C2212yes—
17#15688861—24no—
18C22×C22⋊C4×C47C2211yescharacteristic
19C47×#407C2212yes—
20C47×#323—22no—
21C22×Dic94C2×C411yescharacteristic
22#13094003—22no—
23#13094003—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×Dic47⋊C4—21no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C4×Dic47—22no—
35C2×C4×Dic47—21no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95C2×C412yes—
40C47⋊#95D414yes—
41C25×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C2×M16×C47—22no—
45C2×C22⋊C4×C47—22no—
46C2×C22⋊C4×C47—22no—
47C2×C22⋊C4×C47C2×C412yes—
48C2×C22⋊C4×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—42no—
54C24×C47—42no—
55C24×C47—81no—
56C24×C47C22⋊C412yes—
57C24×C47C2×D411yescharacteristic
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—81no—
62C22×C4×C47C22⋊C412yes—
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D412yes—
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47#4112yes—
67C22×Dic47—22no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—24no—
71C2×Dic94—22no—
72C2×Dic94—81no—
73M16×C47—82no—
74C2×C8×C47—42no—
75C22⋊C4×C47—24no—
76C22⋊C4×C47—24no—
77C22⋊Dic47—44no—
78C22⋊Dic47—42no—
79C22⋊Dic47—44no—
80C22⋊Dic47—41no—
81C22⋊Dic47—41no—
82C22⋊Dic47—82no—
83C22⋊Dic47—82no—
84C22⋊Dic47—82no—
85C4⋊Dic47—22no—
86C4⋊Dic47—82no—
87Dic47⋊C4—42no—
88C4×Dic47—82no—
89C4×Dic47—81no—
90C23×C47—22no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—41no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47—42no—
100C23×C47—42no—
101C23×C47—42no—
102C23×C47—81no—
103C23×C47—81no—
104C23×C47—88no—
105C23×C47C2×C22⋊C411yescharacteristic
106C23×C47#12011yescharacteristic
107C23×C47#12311yescharacteristic
108C2×C4×C47—22no—
109C2×C4×C47—42no—
110C2×C4×C47—41no—
111C2×C4×C47—42no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47—81no—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12112yes—
118C2×Dic47—42no—
119C2×Dic47—44no—
120C2×Dic47—42no—
121C2×Dic47—44no—
122C2×Dic47—42no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—44no—
128Dic94—82no—
129C8×C47—82no—
130#9411—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—21no—
138C22×C47—44no—
139C22×C47—44no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—42no—
143C22×C47—82no—
144C22×C47—84no—
145C22×C47—84no—
146C22×C47#38611yescharacteristic
147C22×C47#42112yes—
148C22×C47#45512yes—
149C4×C47—22no—
150C4×C47—22no—
151C4×C47—42no—
152Dic47—84no—
153Dic47—82no—
154Dic47—81no—
155Dic47—81no—
156#2561—471no—
157#2312—471no—
158#1508—472no—
159#1462D4711yescharacteristic
160#1297—472no—
161C2×C47—22no—
162C2×C47—22no—
163C2×C47—41no—
164C2×C47—82no—
165C2×C47#188511yescharacteristic
166C2×C47#189711yescharacteristic
167C2×C47#191311yescharacteristic
168C2×C22⋊Q8—471no—
169C22×C22⋊C4C2×D4711yescharacteristic
170#407C2×D4712yes—
171#397—472no—
172#393—472no—
173#388—944no—
174#388—942no—
175#381—941no—
176#380—941no—
177#379—941no—
178#379—941no—
179C4×C22⋊C4—471no—
180C4×C22⋊C4—472no—
181#328—944no—
182#323—22no—
183C47#941111yescharacteristic, complemented, Sylow
184C25D18811yescharacteristic
185C22×Q8—471no—
186C23×C4D18811yescharacteristic
187C23×C4C22×D4711yescharacteristic
188C2×M16—22no—
189C22⋊Q8—942no—
190C22⋊Q8—942no—
191C2×C4⋊C4—471no—
192C2×C4⋊C4—942no—
193C2×C4⋊C4—941no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—942no—
197C2×C22⋊C4—942no—
198C2×C22⋊C4—942no—
199C2×C22⋊C4—942no—
200C2×C22⋊C4—941no—
201C2×C22⋊C4—942no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4C4×D4712yes—
204C2×C22⋊C4C22⋊D4712yes—
205C2×C42—942no—
206C2×C42—941no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209#95—472no—
210#95—474no—
211#95—942no—
212#95—942no—
213#95—942no—
214C24—21no—
215C24—21no—
216C24—42no—
217C24—42no—
218C24—81no—
219C24(C2×D47)⋊C412yes—
220C24C2×D18811yescharacteristic
221C2×Q8—942no—
222C2×Q8—3761no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4—942no—
228C22×C4—941no—
229C22×C4—941no—
230C22×C4—944no—
231C22×C4(C2×D47)⋊C412yes—
232C22×C4C2×C4×D4711yescharacteristic
233C22×C4D4×D4712yes—
234C22×C4#1062535212yes—
235C22×C4C2×C22⋊D4711yescharacteristic
236M16—82no—
237C2×C8—42no—
238C4⋊C4—942no—
239C4⋊C4—3762no—
240C4⋊C4—1882no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—3762no—
244C22⋊C4—3762no—
245C22⋊C4—3762no—
246C22⋊C4—1882no—
247C22⋊C4—1884no—
248C22⋊C4—1884no—
249C22⋊C4—1881no—
250C22⋊C4—1881no—
251C42—3762no—
252C42—3761no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—42no—
265C23—81no—
266C23—81no—
267C23—88no—
268C23C2×(C2×D47)⋊C411yescharacteristic
269C23#1309399411yescharacteristic
270C23#1309400811yescharacteristic
271Q8—3762no—
272C2×C4—22no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—81no—
280C2×C4—1882no—
281C2×C4—1884no—
282C2×C4—1882no—
283C2×C4—1882no—
284C2×C4—1884no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1884no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4(C22⋊D47)⋊C412yes—
291C2×C4#1309399612yes—
292C8—82no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—44no—
301C22—44no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—84no—
307C22—84no—
308C22#1568900011yescharacteristic
309C22#1568913912yes—
310C22#1568930312yes—
311C4—22no—
312C4—22no—
313C4—42no—
314C4—3764no—
315C4—3762no—
316C4—3761no—
317C4—3761no—
318C2—22no—
319C2—22no—
320C2—41no—
321C2—82no—
322C2#2024377411yescharacteristic
323C2#2024383411yescharacteristic
324C2#2024392811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1462C2
socleC22×C47#386
prime radicalC25×C47D4
cosocleC23×C4×C47C23
hypercenter#1462D47
hypocenterC47#9411
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20241909C2semidirect
4. #20241273C2semidirect
7. #15689048C4semidirect
8. #15689035C4semidirect
183. C47#9411semidirect

Maximal subgroups

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

subgroupindex
#202485462
#202459862
#202419092
#202412732
C47×#14622
#941147