Finite Group Databasebeta

← #26112246

Subgroups — #26112246

Order 12032.

530 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 103 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#20249305C211yesmaximal, characteristic
2C4×#13093939C211yesmaximal, characteristic
3#20241909C212yesmaximal, complemented
4#20241242C212yesmaximal, complemented
5C47×#2975C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C2×#13093939—21no—
8#15689053C411yescharacteristic, complemented
9#15689051—21no—
10#15689048C411yescharacteristic, complemented
11#15689043C411yescharacteristic, complemented
12#15689037—21no—
13#15689035C411yescharacteristic, complemented
14#15689007—21no—
15#15688996—21no—
16#15688986C2211yescharacteristic
17C4×C22⋊Dic47C2212yes—
18#15688890—21no—
19#15688861—24no—
20C2×C4×D4×C47C2211yescharacteristic
21C47×#436—24no—
22C47×#431—21no—
23C47×#407—21no—
24C47×#407C2212yes—
25C22×Dic94C2×C411yescharacteristic
26#13094003—22no—
27#13094003—22no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C4×Dic47—22no—
38#13093939—24no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—22no—
45C47⋊#95C2×C411yescharacteristic
46C47⋊#95C2×C411yescharacteristic
47C22×D4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C2×M16×C47—22no—
51C2×M16×C47—22no—
52C4×D4×C47—22no—
53C4×D4×C47—22no—
54C2×C4⋊C4×C47C2×C411yescharacteristic
55C2×C22⋊C4×C47C2×C411yescharacteristic
56C2×C22⋊C4×C47D411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C4⋊C8×C47—22no—
59C22⋊C8×C47—22no—
60C22⋊C8×C47—22no—
61C22⋊C8×C47D414yes—
62C4×C8×C47—22no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C2×D411yescharacteristic
65C2×D4×C47—22no—
66C2×D4×C47—22no—
67C2×D4×C47—41no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—81no—
71C22×C4×C47—81no—
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D412yes—
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47#4112yes—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—42no—
84C2×Dic94—22no—
85C2×Dic94—81no—
86M16×C47—82no—
87M16×C47—82no—
88C2×C8×C47—24no—
89C4⋊C4×C47—22no—
90C22⋊C4×C47—22no—
91C22⋊C4×C47—22no—
92C42×C47—21no—
93C42×C47—21no—
94C22⋊Dic47—24no—
95C22⋊Dic47—42no—
96C22⋊Dic47—42no—
97C22⋊Dic47—42no—
98C22⋊Dic47—81no—
99C22⋊Dic47—81no—
100C4⋊Dic47—22no—
101Dic47⋊C4—42no—
102Dic47⋊C4—41no—
103Dic47⋊C4—41no—
104Dic47⋊C4—81no—
105C4×Dic47—82no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47C2×C22⋊C411yescharacteristic
116C23×C47#12112yes—
117D4×C47—44no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—81no—
130C2×C4×C47#10412yes—
131C2×C4×C47C4×D412yes—
132C2×C4×C47#12011yescharacteristic
133C2×C4×C47#12311yescharacteristic
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—44no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—82no—
148C2×Dic47—82no—
149Dic94—82no—
150C8×C47—44no—
151#9184—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—22no—
155C22×C47—22no—
156C22×C47—22no—
157C22×C47—42no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—81no—
162C22×C47—81no—
163C22×C47—81no—
164C22×C47#38611yescharacteristic
165C22×C47C2×#10411yescharacteristic
166C22×C47#42111yescharacteristic
167C4×C47—22no—
168C4×C47—21no—
169C4×C47—22no—
170C4×C47—21no—
171C4×C47—41no—
172C4×C47—41no—
173Dic47—81no—
174Dic47—81no—
175Dic47—84no—
176Dic47—81no—
177Dic47—81no—
178#2975D4711yescharacteristic
179#2638—471no—
180C4×#99—471no—
181#1508—472no—
182#1284—472no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—42no—
186C2×C47—81no—
187C2×C47#187111yescharacteristic
188C2×C47#190911yescharacteristic
189C2×C47#191311yescharacteristic
190C2×C22⋊Q8—471no—
191C2×C4×D4C2×D4711yescharacteristic
192#436—24no—
193#431—21no—
194C2×#99—941no—
195#407—21no—
196#407C2×D4712yes—
197#398—471no—
198#397—471no—
199#397—941no—
200#395—471no—
201#394—941no—
202#393—471no—
203#387—941no—
204#385—941no—
205#381—471no—
206C4×C22⋊C4—472no—
207#342—941no—
208#328—944no—
209C47#918411yescharacteristic, complemented, Sylow
210C22×Q8—471no—
211C22×D4D18811yescharacteristic
212C23×C4C22⋊D4711yescharacteristic
213C23×C4C22×D4711yescharacteristic
214C2×M16—22no—
215C2×M16—22no—
216C22⋊Q8—942no—
217C22⋊Q8—942no—
218C4×D4—22no—
219C4×D4—22no—
220C2×C4⋊C4—471no—
221C2×C4⋊C4—941no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4C4×D4711yescharacteristic
225C2×C22⋊C4—942no—
226C2×C22⋊C4—941no—
227C2×C22⋊C4—941no—
228C2×C22⋊C4—941no—
229C2×C22⋊C4—941no—
230C2×C22⋊C4C4×D4711yescharacteristic
231C2×C22⋊C4D18811yescharacteristic
232C2×C42—942no—
233C2×C42C22⋊D4711yescharacteristic
234C4⋊C8—22no—
235#99—944no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8C22⋊D4714yes—
239C4×C8—22no—
240#95—471no—
241#95—471no—
242#95—941no—
243#95—941no—
244#95—941no—
245#95—941no—
246#95—941no—
247#95—942no—
248C24(C2×D47)⋊C411yescharacteristic
249C24C2×D18811yescharacteristic
250C2×Q8—942no—
251C2×Q8—3761no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—941no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—1882no—
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4(C2×D47)⋊C411yescharacteristic
266C22×C4(C2×D47)⋊C411yescharacteristic
267C22×C4C2×C4×D4711yescharacteristic
268C22×C4#1062535012yes—
269C22×C4C2×C22⋊D4712yes—
270C22×C4C2×C22⋊D4711yescharacteristic
271M16—82no—
272M16—82no—
273C2×C8—24no—
274C4⋊C4—22no—
275C4⋊C4—942no—
276C4⋊C4—3761no—
277C4⋊C4—1882no—
278C4⋊C4—1881no—
279C4⋊C4—1881no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—944no—
283C22⋊C4—3761no—
284C22⋊C4—3761no—
285C22⋊C4—1882no—
286C22⋊C4—1882no—
287C22⋊C4—1882no—
288C42—21no—
289C42—21no—
290C42—3762no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C2×(C2×D47)⋊C411yescharacteristic
301C23#1309399812yes—
302Q8—3762no—
303D4—44no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—3762no—
317C2×C4—3762no—
318C2×C4—1884no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4—1881no—
329C2×C4—1881no—
330C2×C4—1881no—
331C2×C4#1309395012yes—
332C2×C4C4×C22⋊D4712yes—
333C2×C4#1309399311yescharacteristic
334C2×C4#1309400711yescharacteristic
335C8—44no—
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#1568900211yescharacteristic
349C22C2×#1309395011yescharacteristic
350C22#1568913911yescharacteristic
351C4—21no—
352C4—21no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—3761no—
358C4—3761no—
359C4—3764no—
360C4—3761no—
361C4—3761no—
362C2—21no—
363C2—21no—
364C2—42no—
365C2—81no—
366C2#2024369011yescharacteristic
367C2#2024391011yescharacteristic
368C2#2024393011yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2975C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#2975D47
hypocenterC47#9184
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2975

Complements

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

NH ≅ G/Ntype
3. #20241909C2semidirect
4. #20241242C2semidirect
8. #15689053C4semidirect
10. #15689048C4semidirect
11. #15689043C4semidirect
13. #15689035C4semidirect
209. C47#9184semidirect

Maximal subgroups

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

subgroupindex
#202493052
C4×#130939392
#202419092
#202412422
C47×#29752
#918447