Finite Group Databasebeta

← #26113194

Subgroups — #26113194

Order 12032.

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

normal subgroups only

This group has 574 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#20253510C211yesmaximal, characteristic
2#20244306C211yesmaximal, characteristic
3#20241895C211yesmaximal, characteristic
4#20241835C211yesmaximal, characteristic, complemented
5#20241281C211yesmaximal, characteristic, complemented
6#20241244C211yesmaximal, characteristic, complemented
7C47×#2276C211yesmaximal, characteristic, complemented
8#15689673C2211yescharacteristic
9#15689436—21no—
10#15689418C412yescomplemented
11#15689323C412yescomplemented
12#15689262—21no—
13#15689221—22no—
14#15689180C2211yescharacteristic
15#15689180C2211yescharacteristic
16C2×#13093952—22no—
17C4×#10625329C2211yescharacteristic
18#15688856—21no—
19#15688856—21no—
20Q8⋊(C47⋊C8)—22no—
21C2×C4×Q8×C47—21no—
22C2×C4×Q8×C47C2211yescharacteristic
23C4×C4⋊C4×C47—21no—
24C4×C4⋊C4×C47—21no—
25C4×C4⋊C4×C47—22no—
26C4×C4⋊C4×C47C2211yescharacteristic
27C47×#376—21no—
28C47×#376C2211yescharacteristic
29C2×#10625350—21no—
30C2×#10625336—22no—
31C2×#10625329—21no—
32#13094024C2×C411yescharacteristic
33#13094021C2×C411yescharacteristic
34#13094012—21no—
35#13093998—21no—
36#13093998—21no—
37C4×D188—21no—
38C4×D188—21no—
39#13093957—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)C2×C412yes—
42#13093952—44no—
43#13093948—22no—
44#13093944—22no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47#13093935—21no—
48#13093935—21no—
49#13093935—21no—
50C4×C47⋊C8—21no—
51C22×Q8×C47D411yescharacteristic
52C4×Q8×C47—28no—
53C4×Q8×C47—21no—
54C4×Q8×C47—21no—
55C4×Q8×C47C2×C412yes—
56C4×Q8×C47D412yes—
57C2×C4⋊C4×C47—22no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47D411yescharacteristic
61C2×C42×C47—22no—
62C2×C42×C47—21no—
63C2×C42×C47—21no—
64C2×C42×C47—22no—
65C2×C42×C47D412yes—
66C2×C42×C47C2311yescharacteristic
67#95×C47—22no—
68#95×C47—21no—
69#95×C47—21no—
70#95×C47D411yescharacteristic
71#95×C47D411yescharacteristic
72C2×Q8×C47—24no—
73C2×Q8×C47—21no—
74C2×Q8×C47C22⋊C412yes—
75C22×C4×C47—21no—
76C22×C4×C47—21no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—22no—
80C22×C4×C47C2×D411yescharacteristic
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C2×C22⋊D47—41no—
84#10625350—42no—
85#10625350—42no—
86C2×D188—21no—
87C2×D188—21no—
88C2×D188—21no—
89C2×C4×D47—41no—
90C2×Dic94—21no—
91C4⋊C4×C47—28no—
92C4⋊C4×C47—24no—
93C4⋊C4×C47—22no—
94C4⋊C4×C47—22no—
95C4⋊C4×C47—21no—
96C4⋊C4×C47C22⋊C412yes—
97C42×C47—28no—
98C42×C47—24no—
99C42×C47—22no—
100C42×C47—22no—
101C42×C47—42no—
102C42×C47—82no—
103C42×C47C22×C411yescharacteristic
104C42×C47C2×D411yescharacteristic
105C42×C47#4111yescharacteristic
106C42×C47#4111yescharacteristic
107#10625336—44no—
108(C2×D47)⋊C4—41no—
109(C2×D47)⋊C4—41no—
110C4⋊Dic47—21no—
111C4⋊Dic47—21no—
112#10625329—22no—
113#10625329—41no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—41no—
116C2×C47⋊C8—41no—
117C23×C47#12011yescharacteristic
118C22×D47—41no—
119Q8×C47—24no—
120Q8×C47—22no—
121Q8×C47#10414yescomplemented
122C2×C4×C47—24no—
123C2×C4×C47—22no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—22no—
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—24no—
130C2×C4×C47—22no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—42no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47C2×C22⋊C411yescharacteristic
141C2×C4×C47C4×D411yescharacteristic
142C2×C4×C47C4×D411yescharacteristic
143C2×C4×C47#12111yescharacteristic
144C2×C4×C47#12111yescharacteristic
145C2×C4×C47#12311yescharacteristic
146C22⋊D47—81no—
147C22⋊D47—81no—
148C2×Dic47—41no—
149D188—42no—
150D188—42no—
151D188—81no—
152C4×D47—82no—
153Dic94—42no—
154C47⋊C8—42no—
155C47⋊C8—42no—
156C47⋊C8—81no—
157#9326—471nomaximal, Sylow
158C22×C47—21no—
159C22×C47—21no—
160C22×C47#38611yescharacteristic
161C22×C47#45311yescharacteristic
162C22×C47#45411yescharacteristic
163C2×D47—81no—
164C2×D47—81no—
165C2×D47—81no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—24no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—44no—
173C4×C47—42no—
174C4×C47—42no—
175C4×C47—42no—
176C4×C47C2×#10412yes—
177Dic47—81no—
178#3112—471no—
179#2276D4711yescharacteristic, complemented
180#1980—471no—
181#1505—471no—
182#1491—471no—
183#1300—471no—
184#1285—471no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#188311yescharacteristic
188C2×C47#189611yescharacteristic
189C2×C47#190511yescharacteristic
190D47—161no—
191#532—471no—
192C2×C4×Q8—21no—
193C2×C4×Q8C2×D4711yescharacteristic
194#480—941no—
195#477—472no—
196#460—472no—
197#448—941no—
198#440—942no—
199#432—471no—
200#432—471no—
201C2×#104—942no—
202C4×M16—471no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2×D4711yescharacteristic
207#376—21no—
208#376C2×D4711yescharacteristic
209#327—941no—
210#327—941no—
211Q8⋊C8—942no—
212C47#932611yescharacteristic, complemented, Sylow
213C2×#41—941no—
214C22×Q8C22⋊D4711yescharacteristic
215C2×SD16—942no—
216C2×M16—941no—
217C4⋊Q8—471no—
218#127—471no—
219#124—941no—
220#121—941no—
221#121—941no—
222C4×Q8—28no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8C4×D4712yes—
226C4×Q8D18812yes—
227C4×D4—941no—
228C4×D4—941no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C22⋊D4711yescharacteristic
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42C22⋊D4712yes—
238C2×C42C22×D4711yescharacteristic
239#107—942no—
240C4⋊C8—472no—
241C4⋊C8—941no—
242#104—1884no—
243#103—942no—
244#102—942no—
245C22⋊C8—941no—
246C22⋊C8—941no—
247#97—941no—
248#97—941no—
249#97—941no—
250C4×C8—941no—
251#95—22no—
252#95—21no—
253#95—21no—
254#95C22⋊D4711yescharacteristic
255#95C22⋊D4711yescharacteristic
256#41—1882no—
257#41—1882no—
258C2×Q8—24no—
259C2×Q8—21no—
260C2×Q8—941no—
261C2×Q8(C2×D47)⋊C412yes—
262C2×D4—941no—
263C2×D4—941no—
264C2×D4—941no—
265C2×D4—1881no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—1881no—
272C22×C4C2×C22⋊D4711yescharacteristic
273C22×C4C2×C22⋊D4711yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275SD16—1884no—
276M16—942no—
277M16—1881no—
278C2×C8—942no—
279C2×C8—1881no—
280C2×C8—1881no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—941no—
287C4⋊C4—941no—
288C4⋊C4(C2×D47)⋊C412yes—
289C22⋊C4—1881no—
290C22⋊C4—1881no—
291C42—28no—
292C42—24no—
293C42—22no—
294C42—22no—
295C42—42no—
296C42—82no—
297C42C2×C4×D4711yescharacteristic
298C42C2×D18811yescharacteristic
299C42#1062535011yescharacteristic
300C42#1062535011yescharacteristic
301C23—1881no—
302C23#1309399311yescharacteristic
303Q8—24no—
304Q8—22no—
305Q8—1882no—
306Q8#1309395214yescomplemented
307D4—3761no—
308D4—3761no—
309D4—3761no—
310D4—1882no—
311D4—1882no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—3762no—
331C2×C4—1881no—
332C2×C4C2×(C2×D47)⋊C411yescharacteristic
333C2×C4C4×C22⋊D4711yescharacteristic
334C2×C4C4×C22⋊D4711yescharacteristic
335C2×C4#1309399811yescharacteristic
336C2×C4#1309399811yescharacteristic
337C2×C4#1309400711yescharacteristic
338C8—3761no—
339C8—1882no—
340C8—1882no—
341C22—21no—
342C22—21no—
343C22—3761no—
344C22—3761no—
345C22—3761no—
346C22#1568900211yescharacteristic
347C22#1568929111yescharacteristic
348C22#1568929611yescharacteristic
349C4—21no—
350C4—21no—
351C4—24no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—44no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—3761no—
360C4C2×#1309395212yes—
361C2—21no—
362C2—21no—
363C2—7521no—
364C2#2024376411yescharacteristic
365C2#2024383111yescharacteristic
366C2#2024388211yescharacteristic
subgroupG/H
centerC22#15689002
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#2276C2
socleC22×C47#386
prime radical#15689673C22
cosocleC2×C42×C47C23
hypercenter#2276D47
hypocenterC47#9326
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2276

Complements

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

NH ≅ G/Ntype
4. #20241835C2semidirect
5. #20241281C2semidirect
6. #20241244C2semidirect
7. C47×#2276C2semidirect
10. #15689418C4semidirect
11. #15689323C4semidirect
121. Q8×C47#104semidirect
179. #2276D47semidirect
212. C47#9326semidirect
306. Q8#13093952semidirect

Maximal subgroups

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

subgroupindex
#202535102
#202443062
#202418952
#202418352
#202412812
#202412442
C47×#22762
#932647