Finite Group Databasebeta

← #26104869

Subgroups — #26104869

Order 12032.

500 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 111 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#20251698C211yesmaximal, characteristic
2#20243082C211yesmaximal, characteristic
3#20241212C212yesmaximal, complemented
4#20241210C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic
6C22×#10625329—21no—
7C2×#13094016C411yescharacteristic
8#15689021C2211yescharacteristic
9#15689021C2211yescharacteristic
10#15689007—21no—
11#15689007—21no—
12#15688996—21no—
13#15688996—21no—
14#15688989C411yescharacteristic
15#15688983C411yescharacteristic
16#15688983C411yescharacteristic
17#15688878—21no—
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C4⋊C8×C47—21no—
21C2×C22⋊C8×C47—21no—
22C47×#336C2212yes—
23C2×#10625329—24no—
24C2×#10625329—42no—
25C2×#10625329—42no—
26C22×C47⋊C8—22no—
27#13094016—42no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C22×C8×C47—22no—
48C2×C42×C47—42no—
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—42no—
54C22⋊C8×C47—42no—
55C24×C47C22⋊C411yescharacteristic
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—21no—
74C22×Dic47—21no—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C2×C8×C47—42no—
78C2×C8×C47—42no—
79C2×C8×C47—42no—
80C42×C47—22no—
81C42×C47—22no—
82C42×C47—41no—
83C42×C47—41no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—22no—
87C4⋊Dic47—22no—
88C4⋊Dic47—41no—
89C4⋊Dic47—41no—
90Dic47⋊C4—42no—
91Dic47⋊C4—42no—
92Dic47⋊C4—41no—
93Dic47⋊C4—41no—
94Dic47⋊C4—41no—
95Dic47⋊C4—41no—
96#10625329—48no—
97C2×C47⋊C8—24no—
98C2×C47⋊C8—42no—
99C2×C47⋊C8—42no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47#10412yes—
107C23×C47C2×C22⋊C411yescharacteristic
108C2×C4×C47—24no—
109C2×C4×C47—24no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—44no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47#9912yes—
127C2×C4×C47#10112yes—
128C2×C4×C47#13112yes—
129C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C47#13711yescharacteristic
132C2×C4×C47#13711yescharacteristic
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C8×C47—82no—
146C47⋊C8—44no—
147#7711—471nomaximal, Sylow
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—22no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47#40811yescharacteristic
156C22×C47C2×#9911yescharacteristic
157C22×C47C2×#10111yescharacteristic
158C22×C47#41711yescharacteristic
159C22×C47#41711yescharacteristic
160C22×C47C2×#10411yescharacteristic
161C22×C47#42111yescharacteristic
162C4×C47—22no—
163C4×C47—22no—
164C4×C47—21no—
165C4×C47—22no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—21no—
169C4×C47—41no—
170Dic47—81no—
171Dic47—81no—
172Dic47—81no—
173Dic47—81no—
174#2851—471no—
175#1797D4711yescharacteristic
176#1728—471no—
177#1271—472no—
178#1271—472no—
179C2×C47—42no—
180C2×C47#147211yescharacteristic
181C2×C47#148311yescharacteristic
182C2×C47#149511yescharacteristic
183C2×C47#150411yescharacteristic
184C2×C47#151311yescharacteristic
185C2×C47#152811yescharacteristic
186C2×C47#156111yescharacteristic
187C22×M16—941no—
188C2×#125—471no—
189C22×C42C2×D4711yescharacteristic
190C2×C4⋊C8—21no—
191C2×C22⋊C8—21no—
192#389—471no—
193#389—471no—
194#387—941no—
195#387—941no—
196#385—941no—
197#385—941no—
198#382—471no—
199#380—471no—
200#380—471no—
201#336—472no—
202#336—941no—
203#336C2×D4712yes—
204C47#771111yescharacteristic, complemented, Sylow
205C23×C4C4×D4711yescharacteristic
206C23×C4D18811yescharacteristic
207C23×C4C22⋊D4711yescharacteristic
208C2×M16—944no—
209C2×M16—1882no—
210C2×M16—1882no—
211C22×C8—22no—
212C22×C8—942no—
213#125—1882no—
214C2×C4⋊C4—471no—
215C2×C4⋊C4—471no—
216C2×C4⋊C4—471no—
217C2×C4⋊C4—471no—
218C2×C4⋊C4—941no—
219C2×C4⋊C4—941no—
220C2×C4⋊C4—941no—
221C2×C4⋊C4—941no—
222C2×C22⋊C4—941no—
223C2×C22⋊C4—941no—
224C2×C42—42no—
225C2×C42C4×D4711yescharacteristic
226C2×C42D18811yescharacteristic
227C2×C42C22⋊D4711yescharacteristic
228C2×C42C22×D4711yescharacteristic
229C4⋊C8—42no—
230C22⋊C8—42no—
231#95—941no—
232#95—941no—
233#95—941no—
234#95—941no—
235#95—941no—
236#95—941no—
237C24(C2×D47)⋊C411yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—941no—
250C22×C4—941no—
251C22×C4—941no—
252C22×C4—941no—
253C22×C4(C2×D47)⋊C411yescharacteristic
254C22×C4(C2×D47)⋊C411yescharacteristic
255C22×C4(C2×D47)⋊C411yescharacteristic
256C22×C4C2×C4×D4711yescharacteristic
257C22×C4C2×D18811yescharacteristic
258C22×C4C2×C22⋊D4711yescharacteristic
259M16—1888no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—944no—
264C2×C8—1882no—
265C2×C8—1882no—
266C4⋊C4—942no—
267C4⋊C4—942no—
268C4⋊C4—1881no—
269C4⋊C4—1881no—
270C4⋊C4—1882no—
271C4⋊C4—1882no—
272C4⋊C4—1881no—
273C4⋊C4—1881no—
274C4⋊C4—1881no—
275C4⋊C4—1881no—
276C22⋊C4—1882no—
277C22⋊C4—1882no—
278C42—22no—
279C42—22no—
280C42—41no—
281C42—41no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23#1309395012yes—
289C23C2×(C2×D47)⋊C411yescharacteristic
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—1881no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4#1309393812yes—
321C2×C4#1309394212yes—
322C2×C4#1309403012yes—
323C2×C4#1309405011yescharacteristic
324C2×C4#1309405211yescharacteristic
325C2×C4#1309405511yescharacteristic
326C2×C4#1309405711yescharacteristic
327C8—82no—
328C8—1884no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1568908911yescharacteristic
337C22C2×#1309393811yescharacteristic
338C22C2×#1309394211yescharacteristic
339C22#1568912211yescharacteristic
340C22#1568912511yescharacteristic
341C22C2×#1309395011yescharacteristic
342C22#1568914111yescharacteristic
343C4—21no—
344C4—21no—
345C4—22no—
346C4—22no—
347C4—21no—
348C4—22no—
349C4—21no—
350C4—41no—
351C4—3761no—
352C4—3761no—
353C4—3761no—
354C4—3761no—
355C2—42no—
356C2#2024175111yescharacteristic
357C2#2024179911yescharacteristic
358C2#2024185411yescharacteristic
359C2#2024189311yescharacteristic
360C2#2024192911yescharacteristic
361C2#2024199111yescharacteristic
362C2#2024212511yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1797D47
hypocenterC47#7711
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
3. #20241212C2semidirect
4. #20241210C2semidirect
204. C47#7711semidirect

Maximal subgroups

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

subgroupindex
#202516982
#202430822
#202412122
#202412102
C47×#17972
#771147