Finite Group Databasebeta

← #26116871

Subgroups — #26116871

Order 12032.

632 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#15689451C212yesmaximal
2C2×#15688860C212yesmaximal
3#20241655C211yesmaximal, characteristic
4#20241231C211yesmaximal, characteristic
5C47×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7#15689451—24no—
8#15689451—24no—
9C2×#13093955—22no—
10C2×#13093948—22no—
11C2×C22⋊(C47⋊C8)C2212yes—
12C2×C47⋊#95—22no—
13C2×C47⋊#95C2211yescharacteristic
14#15688860—24no—
15C2×C22⋊Q8×C47C2212yes—
16C47×#398—22no—
17C47×#395—22no—
18C47×#380—22no—
19C2×#95×C47C2211yescharacteristic
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47D412yes—
27#13093955—28no—
28#13093948—28no—
29C22⋊(C47⋊C8)—22no—
30C22⋊(C47⋊C8)D414yes—
31C47⋊#95—22no—
32C47⋊#95—44no—
33C22×Q8×C47D412yes—
34C23×C4×C47D412yes—
35C23×C4×C47C2311yescharacteristic
36C22⋊Q8×C47—22no—
37C22⋊Q8×C47—22no—
38C22⋊Q8×C47—22no—
39C22⋊Q8×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—22no—
42C2×C4⋊C4×C47—22no—
43C2×C4⋊C4×C47D412yes—
44C2×C22⋊C4×C47—22no—
45#95×C47—24no—
46#95×C47—22no—
47C24×C47C2×D411yescharacteristic
48C2×Q8×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—44no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—22no—
55C22×C4×C47—42no—
56C22×C4×C47—42no—
57C22×C4×C47—42no—
58C22×C4×C47SD1614yes—
59C22×C4×C47C2×D412yes—
60C22×C4×C47C2×D411yescharacteristic
61C22×C4×C47C2×D412yes—
62C22×C4×C47#4111yescharacteristic
63C22×Dic47—22no—
64C22×Dic47—22no—
65C22×Dic47—41no—
66C22×Dic47—42no—
67C22×Dic47—41no—
68C22×Dic47—41no—
69C22×Dic47—41no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73C4⋊C4×C47—22no—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—82no—
78C4⋊C4×C47—82no—
79C22⋊C4×C47—42no—
80C22⋊C4×C47—42no—
81C4⋊Dic47—24no—
82C4⋊Dic47—42no—
83C2×C47⋊C8—24no—
84C2×C47⋊C8—44no—
85C23×C47—21no—
86C23×C47—21no—
87C23×C47—22no—
88C23×C47#12011yescharacteristic
89C23×C47#12111yescharacteristic
90C23×C47#12711yescharacteristic
91C23×C47C2×SD1612yes—
92C23×C47#13712yes—
93Q8×C47—44no—
94Q8×C47—82no—
95C2×C4×C47—24no—
96C2×C4×C47—24no—
97C2×C4×C47—41no—
98C2×C4×C47—42no—
99C2×C4×C47—41no—
100C2×C4×C47—42no—
101C2×C4×C47—42no—
102C2×C4×C47—44no—
103C2×C4×C47—42no—
104C2×C4×C47—42no—
105C2×C4×C47—42no—
106C2×C4×C47—42no—
107C2×C4×C47—44no—
108C2×C4×C47—82no—
109C2×C4×C47#12012yes—
110C2×C4×C47#12112yes—
111C2×Dic47—44no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—44no—
115C2×Dic47—82no—
116C2×Dic47—81no—
117C2×Dic47—82no—
118C2×Dic47—81no—
119C2×Dic47—81no—
120C2×Dic47—81no—
121C47⋊C8—44no—
122#9973—471nomaximal, Sylow
123C22×C47—22no—
124C22×C47—22no—
125C22×C47—22no—
126C22×C47—22no—
127C22×C47—21no—
128C22×C47—21no—
129C22×C47—21no—
130C22×C47—21no—
131C22×C47#39211yescharacteristic
132C22×C47#44812yes—
133C22×C47#45511yescharacteristic
134C22×C47#45611yescharacteristic
135C22×C47#45812yes—
136C22×C47#46112yes—
137C22×C47#49211yescharacteristic
138C22×C47#49711yescharacteristic
139C4×C47—22no—
140C4×C47—22no—
141C4×C47—44no—
142C4×C47—82no—
143C4×C47—82no—
144Dic47—82no—
145Dic47—82no—
146C2×#483—472no—
147#2659D4711yescharacteristic
148C2×#328—472no—
149#1445—471no—
150#1280—471no—
151C2×C47—21no—
152C2×C47—22no—
153C2×C47—21no—
154C2×C47#160812yes—
155C2×C47#161112yes—
156C2×C47#200711yescharacteristic
157C2×C47#201311yescharacteristic
158C2×C47#202211yescharacteristic
159C2×C22⋊Q8C2×D4712yes—
160C22×C4⋊C4—471no—
161#483—944no—
162#483—944no—
163C2×#106—942no—
164C2×#103—942no—
165C2×C22⋊C8—472no—
166#398—22no—
167#395—22no—
168#380—22no—
169C2×#95—471no—
170C2×#95—942no—
171C2×#95C2×D4711yescharacteristic
172#328—944no—
173C47#997311yescharacteristic, complemented, Sylow
174C22×Q8C22⋊D4712yes—
175C23×C4—941no—
176C23×C4—941no—
177C23×C4C22⋊D4712yes—
178C23×C4C22×D4711yescharacteristic
179C22×C8—942no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—472no—
188C2×C4⋊C4—942no—
189C2×C4⋊C4—942no—
190C2×C4⋊C4—941no—
191C2×C4⋊C4C22⋊D4712yes—
192C2×C22⋊C4—22no—
193#106—948no—
194#103—948no—
195C22⋊C8—474no—
196C22⋊C8—942no—
197#95—24no—
198#95—22no—
199#95—942no—
200#95—1884no—
201C24C2×C22⋊D4711yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—44no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—942no—
213C22×C4—942no—
214C22×C4—1881no—
215C22×C4—1882no—
216C22×C4—1881no—
217C22×C4—1881no—
218C22×C4—1881no—
219C22×C4—1882no—
220C22×C4—1882no—
221C22×C4—1882no—
222C22×C4#1062533614yes—
223C22×C4D4×D4711yescharacteristic
224C22×C4D4×D4712yes—
225C22×C4#1062535211yescharacteristic
226C22×C4C2×C22⋊D4712yes—
227C2×C8—944no—
228C2×C8—1884no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—944no—
236C4⋊C4—1882no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C23—21no—
240C23—21no—
241C23—22no—
242C23#1309399311yescharacteristic
243C23#1309399511yescharacteristic
244C23#1309402011yescharacteristic
245C23C2×#1062533612yes—
246C23#1309405512yes—
247Q8—44no—
248Q8—82no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—82no—
263C2×C4—3761no—
264C2×C4—3762no—
265C2×C4—3761no—
266C2×C4—3761no—
267C2×C4—3761no—
268C2×C4—3762no—
269C2×C4—1882no—
270C2×C4—1884no—
271C2×C4—1882no—
272C2×C4—1884no—
273C2×C4#1309399212yes—
274C2×C4#1309399612yes—
275C8—1884no—
276C22—21no—
277C22—21no—
278C22—22no—
279C22—22no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—22no—
284C22#1568903111yescharacteristic
285C22#1568926212yes—
286C22#1568930111yescharacteristic
287C22#1568930611yescharacteristic
288C22#1568931312yes—
289C22#1568932512yes—
290C22#1568949711yescharacteristic
291C22#1568951511yescharacteristic
292C4—22no—
293C4—22no—
294C4—44no—
295C4—82no—
296C4—82no—
297C4—3762no—
298C4—3762no—
299C2—21no—
300C2—21no—
301C2—22no—
302C2#2024237912yes—
303C2#2024239612yes—
304C2#2024443511yescharacteristic
305C2#2024447111yescharacteristic
306C2#2024451711yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2659C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#2659D47
hypocenterC47#9973
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
173. C47#9973semidirect

Maximal subgroups

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

subgroupindex
C2×#156894512
C2×#156888602
#202416552
#202412312
C47×#26592
#997347