Finite Group Databasebeta

← #26113572

Subgroups — #26113572

Order 12032.

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

normal subgroups only

This group has 424 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#20254594C211yesmaximal, characteristic
2#20244331C211yesmaximal, characteristic
3#20241990C211yesmaximal, characteristic
4#20241938C211yesmaximal, characteristic
5#20241252C211yesmaximal, characteristic
6#20241244C211yesmaximal, characteristic
7C47×#2336C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689533C412yescomplemented
10#15689493C412yescomplemented
11#15689458—21no—
12#15689453—21no—
13#15689427—22no—
14#15689180C2211yescharacteristic
15#15689180C2211yescharacteristic
16C4×#10625329C2211yescharacteristic
17#15688883—21no—
18#15688883—21no—
19#15688865—22no—
20#15688859—21no—
21#15688859—21no—
22C2×#125×C47C2211yescharacteristic
23C47×#389—21no—
24C47×#387—21no—
25C47×#387—21no—
26C47×#387—21no—
27C47×#387—21no—
28C47×#383C2211yescharacteristic
29C4×C4⋊C4×C47—21no—
30C4×C4⋊C4×C47C2211yescharacteristic
31C2×#10625329—21no—
32#13094024C2×C411yescharacteristic
33#13094024C2×C411yescharacteristic
34#13094016—21no—
35#13094003—21no—
36#13094003—21no—
37C4×Dic94—21no—
38C4×Dic94—21no—
39#13093983—21no—
40#13093957—22no—
41#13093955—22no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44#13093947—22no—
45#13093947—22no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48#13093935—21no—
49#13093935—21no—
50#13093935C2×C412yes—
51C4×C47⋊C8—21no—
52#125×C47—21no—
53#125×C47—21no—
54#125×C47C2×C412yes—
55#125×C47D412yes—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—21no—
61C2×C4⋊C4×C47D411yescharacteristic
62C2×C4⋊C4×C47D411yescharacteristic
63C2×C42×C47—21no—
64C2×C42×C47—21no—
65C2×C42×C47D411yescharacteristic
66C2×C42×C47D411yescharacteristic
67C2×C42×C47C2311yescharacteristic
68#95×C47—21no—
69#95×C47—21no—
70#95×C47—21no—
71#95×C47—21no—
72#95×C47—21no—
73#95×C47—21no—
74#95×C47—21no—
75#95×C47D411yescharacteristic
76#95×C47D411yescharacteristic
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—21no—
82C22×C4×C47—21no—
83C22×C4×C47C2×D411yescharacteristic
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D411yescharacteristic
86C2×Dic94—21no—
87C2×Dic94—21no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—22no—
90C4⋊C4×C47—22no—
91C4⋊C4×C47—22no—
92C4⋊C4×C47—21no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—42no—
95C4⋊C4×C47—42no—
96C4⋊C4×C47—81no—
97C4⋊C4×C47C22⋊C412yes—
98C4⋊C4×C47C22⋊C412yes—
99C42×C47—42no—
100C42×C47—41no—
101C42×C47—41no—
102C42×C47—81no—
103C42×C47C22×C411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C42×C47#4111yescharacteristic
107C22⋊Dic47—41no—
108C4⋊Dic47—21no—
109C4⋊Dic47—21no—
110C4⋊Dic47—21no—
111C4⋊Dic47—21no—
112Dic47⋊C4—41no—
113Dic47⋊C4—41no—
114C4×Dic47—41no—
115#10625329—22no—
116#10625329—41no—
117C2×C47⋊C8—22no—
118C2×C47⋊C8—41no—
119C2×C47⋊C8—41no—
120C23×C47#12011yescharacteristic
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—22no—
124C2×C4×C47—22no—
125C2×C4×C47—22no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47C2×C22⋊C411yescharacteristic
144C2×C4×C47C4×D411yescharacteristic
145C2×C4×C47C4×D411yescharacteristic
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#12311yescharacteristic
149C2×Dic47—41no—
150C2×Dic47—41no—
151Dic94—42no—
152Dic94—81no—
153C47⋊C8—42no—
154C47⋊C8—42no—
155C47⋊C8—81no—
156#9380—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47#38611yescharacteristic
160C22×C47#45511yescharacteristic
161C22×C47#45611yescharacteristic
162C4×C47—21no—
163C4×C47—21no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—42no—
169C4×C47—42no—
170C4×C47—42no—
171C4×C47—81no—
172C4×C47—81no—
173C4×C47—81no—
174Dic47—81no—
175Dic47—81no—
176#3236—471no—
177#2336D4711yescharacteristic
178#1986—471no—
179#1528—471no—
180#1515—471no—
181#1288—471no—
182#1285—471no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47#188211yescharacteristic
186C2×C47#190611yescharacteristic
187C2×C47#191211yescharacteristic
188#533—471no—
189C2×#125C2×D4711yescharacteristic
190#501—472no—
191#491—472no—
192#484—941no—
193#483—941no—
194#479—942no—
195#432—471no—
196#432—471no—
197C4×M16—471no—
198#389—21no—
199#387—21no—
200#387—21no—
201#387—21no—
202#387—21no—
203#383C2×D4711yescharacteristic
204C4×C4⋊C4—21no—
205C4×C4⋊C4C2×D4711yescharacteristic
206#339—941no—
207#339—941no—
208#330—942no—
209#328—941no—
210#328—941no—
211C47#938011yescharacteristic, complemented, Sylow
212C2×M16—941no—
213C4⋊Q8—471no—
214C4⋊Q8—471no—
215#125—21no—
216#125—21no—
217#125—941no—
218#125C4×D4712yes—
219#125D18812yes—
220C22⋊Q8—941no—
221C22⋊Q8—941no—
222C4×Q8—941no—
223C4×Q8—941no—
224#117—941no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C22⋊D4711yescharacteristic
231C2×C4⋊C4C22⋊D4711yescharacteristic
232C2×C42—21no—
233C2×C42—21no—
234C2×C42C22⋊D4711yescharacteristic
235C2×C42C22⋊D4711yescharacteristic
236C2×C42C22×D4711yescharacteristic
237#107—942no—
238#106—942no—
239C4⋊C8—941no—
240C4⋊C8—941no—
241#103—942no—
242#103—942no—
243C22⋊C8—941no—
244C22⋊C8—941no—
245#97—472no—
246#97—941no—
247#97—941no—
248C4×C8—941no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95C22⋊D4711yescharacteristic
257#95C22⋊D4711yescharacteristic
258C2×Q8—941no—
259C2×Q8—941no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4C2×C22⋊D4711yescharacteristic
267C22×C4C2×C22⋊D4711yescharacteristic
268C22×C4C2×C22⋊D4711yescharacteristic
269M16—942no—
270M16—1881no—
271C2×C8—942no—
272C2×C8—1881no—
273C2×C8—1881no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—81no—
283C4⋊C4—941no—
284C4⋊C4—941no—
285C4⋊C4—941no—
286C4⋊C4—941no—
287C4⋊C4—1881no—
288C4⋊C4—1881no—
289C4⋊C4(C2×D47)⋊C412yes—
290C4⋊C4(C2×D47)⋊C412yes—
291C22⋊C4—1881no—
292C42—41no—
293C42—41no—
294C42—42no—
295C42—81no—
296C42—1881no—
297C42C2×C4×D4711yescharacteristic
298C42C2×D18811yescharacteristic
299C42#1062535011yescharacteristic
300C42#1062535011yescharacteristic
301C23#1309399311yescharacteristic
302Q8—3761no—
303Q8—1882no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—1881no—
327C2×C4—1881no—
328C2×C4C2×(C2×D47)⋊C411yescharacteristic
329C2×C4C4×C22⋊D4711yescharacteristic
330C2×C4C4×C22⋊D4711yescharacteristic
331C2×C4#1309399811yescharacteristic
332C2×C4#1309399811yescharacteristic
333C2×C4#1309400711yescharacteristic
334C8—3761no—
335C8—1882no—
336C8—1882no—
337C22—21no—
338C22—21no—
339C22#1568900211yescharacteristic
340C22#1568930111yescharacteristic
341C22#1568930611yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—3761no—
355C4—3761no—
356C2—21no—
357C2—21no—
358C2#2024375711yescharacteristic
359C2#2024388911yescharacteristic
360C2#2024392511yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2336C2
socleC22×C47#386
prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#2336D47
hypocenterC47#9380
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2336

Complements

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

NH ≅ G/Ntype
9. #15689533C4semidirect
10. #15689493C4semidirect
211. C47#9380semidirect

Maximal subgroups

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

subgroupindex
#202545942
#202443312
#202419902
#202419382
#202412522
#202412442
C47×#23362
#938047