Finite Group Databasebeta

← #26119024

Subgroups — #26119024

Order 12032.

576 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 109 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#20254529C211yesmaximal, characteristic
2#20251784C211yesmaximal, characteristic
3#20244366C211yesmaximal, characteristic
4#20242060C211yesmaximal, characteristic
5#20242059C211yesmaximal, characteristic
6#20241220C211yesmaximal, characteristic
7C47×#3229C211yesmaximal, characteristic
8C2×#13094024C2211yescharacteristic
9C2×#13094022—22no—
10#15689519—21no—
11#15689473—21no—
12#15689451—22no—
13#15689408—22no—
14#15689176C2211yescharacteristic
15#15689147C2211yescharacteristic
16#15688992—22no—
17#15688992—22no—
18C42×Dic47C2211yescharacteristic
19#15688882—22no—
20#15688869—22no—
21#15688868—22no—
22C47×#533C2211yescharacteristic
23C2×C4⋊Q8×C47C2211yescharacteristic
24C47×#498—21no—
25C47×#487—21no—
26C47×#483—22no—
27C47×#475—22no—
28C47×#431C2211yescharacteristic
29C22×Dic94—21no—
30C2×#10625329—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024—21no—
35#13094022—44no—
36#13094022—44no—
37C2×C4⋊Dic47—21no—
38C2×C4⋊Dic47D412yes—
39C2×Dic47⋊C4—28no—
40C2×C4×Dic47—21no—
41C2×C4×Dic47—21no—
42C2×C4×Dic47D412yes—
43#13093955—24no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)D412yes—
47#13093948—24no—
48C22⋊(C47⋊C8)—21no—
49C4×C47⋊C8—21no—
50C22×Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47—21no—
53C4⋊Q8×C47—21no—
54C4⋊Q8×C47—21no—
55C4⋊Q8×C47D411yescharacteristic
56C4⋊Q8×C47D411yescharacteristic
57#125×C47—21no—
58C22⋊Q8×C47—21no—
59C22⋊Q8×C47—21no—
60C4×Q8×C47—21no—
61C4×Q8×C47—21no—
62#117×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47D412yes—
65C2×C42×C47C2311yescharacteristic
66#106×C47—24no—
67C4⋊C8×C47D412yes—
68#103×C47—24no—
69C22⋊C8×C47—21no—
70C4×C8×C47—21no—
71C2×Q8×C47—21no—
72C2×Q8×C47—21no—
73C2×Q8×C47—41no—
74C2×Q8×C47—41no—
75C2×Q8×C47—42no—
76C2×Q8×C47—42no—
77C22×C4×C47—22no—
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×C4×C47C2×D411yescharacteristic
81C22×Dic47—22no—
82C22×Dic47—22no—
83C2×Dic94—41no—
84C2×Dic94—41no—
85C2×Dic94—42no—
86C2×Dic94—42no—
87C2×C8×C47—22no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C42×C47—41no—
100C42×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47#4111yescharacteristic
104C4⋊Dic47—22no—
105C4⋊Dic47—22no—
106C4⋊Dic47—41no—
107C4⋊Dic47—41no—
108Dic47⋊C4—416no—
109C4×Dic47—24no—
110C4×Dic47—24no—
111C4×Dic47—22no—
112C4×Dic47SD1614yes—
113#10625329—42no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—41no—
116C23×C47#12011yescharacteristic
117Q8×C47—42no—
118Q8×C47—42no—
119Q8×C47—82no—
120Q8×C47—81no—
121Q8×C47—81no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—44no—
128C2×C4×C47#12011yescharacteristic
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12111yescharacteristic
131C2×C4×C47#12111yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47#12711yescharacteristic
134C2×C4×C47C2×SD1612yes—
135C2×C4×C47AGL(1,ℤ/8)12yes—
136C2×Dic47—24no—
137C2×Dic47—24no—
138C2×Dic47—24no—
139C2×Dic47—42no—
140C2×Dic47—44no—
141Dic94—82no—
142Dic94—81no—
143Dic94—81no—
144C8×C47—42no—
145C47⋊C8—42no—
146C47⋊C8—81no—
147#10299—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47#39211yescharacteristic
151C22×C47#45012yes—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47—81no—
155C4×C47—81no—
156C4×C47—82no—
157C4×C47#45612yes—
158C4×C47#45612yes—
159C4×C47#46012yes—
160C4×C47#49211yescharacteristic
161C4×C47#49611yescharacteristic
162Dic47—28no—
163Dic47—82no—
164#3229—471no—
165#3229D4711yescharacteristic
166#2868—471no—
167#1994—471no—
168#1546—471no—
169#1546—471no—
170#1276—471no—
171C2×C47—22no—
172C2×C47#200511yescharacteristic
173C2×C47#200511yescharacteristic
174C2×C47#201411yescharacteristic
175#533C2×D4711yescharacteristic
176C2×C4⋊Q8—471no—
177C2×C4⋊Q8—942no—
178C2×C4⋊Q8C2×D4711yescharacteristic
179#498—21no—
180#498—941no—
181#487—21no—
182#487—941no—
183#483—22no—
184#483—942no—
185#475—22no—
186#475—942no—
187#431—471no—
188#431C2×D4711yescharacteristic
189#423—471no—
190#384—942no—
191#384—942no—
192C43—471no—
193#339—942no—
194#332—942no—
195#332—942no—
196C47#1029911yescharacteristic, complemented, Sylow
197C22×Q8—21no—
198C22×Q8—941no—
199C2×M16—941no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—941no—
205C4⋊Q8—941no—
206C4⋊Q8—941no—
207C4⋊Q8—941no—
208C4⋊Q8—1884no—
209C4⋊Q8—1884no—
210C4⋊Q8C22⋊D4711yescharacteristic
211C4⋊Q8C22⋊D4711yescharacteristic
212#125—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217#117—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—472no—
220C2×C4⋊C4—948no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4C22⋊D4712yes—
223C2×C42—472no—
224C2×C42—941no—
225C2×C42—941no—
226C2×C42C22×D4711yescharacteristic
227#106—24no—
228#106—944no—
229C4⋊C8—472no—
230C4⋊C8—941no—
231C4⋊C8—941no—
232C4⋊C8C22⋊D4712yes—
233#103—24no—
234#103—944no—
235C22⋊C8—21no—
236C22⋊C8—941no—
237C4×C8—21no—
238C4×C8—941no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—1881no—
246C2×Q8—1881no—
247C2×Q8—1882no—
248C2×Q8—1882no—
249C22×C4—22no—
250C22×C4—942no—
251C22×C4—942no—
252C22×C4D4×D4711yescharacteristic
253C22×C4D4×D4711yescharacteristic
254C22×C4C2×C22⋊D4711yescharacteristic
255M16—1882no—
256C2×C8—22no—
257C2×C8—942no—
258C2×C8—1881no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—21no—
262C4⋊C4—22no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—942no—
270C4⋊C4—942no—
271C4⋊C4—18816no—
272C4⋊C4—1881no—
273C4⋊C4—1881no—
274C22⋊C4—41no—
275C42—41no—
276C42—474no—
277C42—944no—
278C42—944no—
279C42—942no—
280C42D4×D4711yescharacteristic
281C42#1062535211yescharacteristic
282C42C2×C22⋊D4711yescharacteristic
283C42C2×C22⋊D4711yescharacteristic
284C23#1309399211yescharacteristic
285Q8—42no—
286Q8—42no—
287Q8—81no—
288Q8—81no—
289Q8—82no—
290Q8—3761no—
291Q8—3761no—
292Q8—3762no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—944no—
300C2×C4—944no—
301C2×C4—944no—
302C2×C4—1884no—
303C2×C4—1882no—
304C2×C4#1309399211yescharacteristic
305C2×C4#1309399311yescharacteristic
306C2×C4#1309399511yescharacteristic
307C2×C4#1309399611yescharacteristic
308C2×C4#1309399611yescharacteristic
309C2×C4#1309402011yescharacteristic
310C2×C4SD16×D4712yes—
311C2×C4#1309405112yes—
312C8—42no—
313C8—3761no—
314C8—1882no—
315C22—21no—
316C22—21no—
317C22#1568903111yescharacteristic
318C22#1568927212yes—
319C4—21no—
320C4—21no—
321C4—81no—
322C4—81no—
323C4—82no—
324C4—948no—
325C4—3762no—
326C4#1568930412yes—
327C4#1568930612yes—
328C4#1568932112yes—
329C4#1568949411yescharacteristic
330C4#1568950911yescharacteristic
331C2—22no—
332C2#2024442111yescharacteristic
333C2#2024442211yescharacteristic
334C2#2024447411yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3229C2
socleC22×C47#392
prime radicalC23×C47#120
hypercenter#3229D47
hypocenterC47#10299
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
196. C47#10299semidirect

Maximal subgroups

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

subgroupindex
#202545292
#202517842
#202443662
#202420602
#202420592
#202412202
C47×#32292
#1029947