Finite Group Databasebeta

← #26134470

Subgroups — #26134470

Order 12032.

410 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252289C211yesmaximal, characteristic
2#20245146C211yesmaximal, characteristic, complemented
3#20245016C211yesmaximal, characteristic, complemented
4#20245015C211yesmaximal, characteristic, complemented
5#20245001C211yesmaximal, characteristic, complemented
6#20244992C211yesmaximal, characteristic, complemented
7C47×#2113C211yesmaximal, characteristic, complemented
8#15689921C411yescharacteristic
9C2×#13094030C2211yescharacteristic
10C2×#13093964C2211yescharacteristic
11C2×#13093964C2211yescharacteristic
12#15689243—22no—
13#15689238—22no—
14#15689171C411yescharacteristic
15#15689140C411yescharacteristic
16#15689139C411yescharacteristic
17#15689078C2211yescharacteristic
18Dic47⋊C16—22no—
19#15688907—22no—
20#15688906—22no—
21(C2×D47)⋊C16—22no—
22#15688900—22no—
23C2×#131×C47C2211yescharacteristic
24C2×M32×C47C2211yescharacteristic
25C22×C16×C47C2211yescharacteristic
26C47×C22⋊C16—22no—
27C2×#10625350C2×C411yescharacteristic
28#13094055—21no—
29#13094054—21no—
30#13094050—21no—
31#13094049—21no—
32#13094044—22no—
33#13094030—21no—
34#13094030—21no—
35#13094030—21no—
36#13094030—21no—
37C2×#10625329C2×C411yescharacteristic
38C2×#10625329C2×C411yescharacteristic
39C2×#10625324—21no—
40C2×C8×D47—21no—
41#13093983C2×C411yescharacteristic
42#13093964—21no—
43#13093964—21no—
44#13093964—21no—
45#13093964—21no—
46C2×C47⋊C16—21no—
47C2×C47⋊C16D412yes—
48#13093959—22no—
49#13093951—22no—
50#13093950—22no—
51#13093934D412yes—
52C8×Dic47D412yes—
53C2×#41×C47C2×C411yescharacteristic
54#131×C47—21no—
55#131×C47—21no—
56#131×C47—21no—
57#131×C47—21no—
58C2×M16×C47—21no—
59C2×M16×C47C2×C411yescharacteristic
60C22×C8×C47—21no—
61C22×C8×C47C2311yescharacteristic
62M32×C47—21no—
63M32×C47—21no—
64C2×C16×C47—21no—
65C2×C16×C47—21no—
66C2×C16×C47—21no—
67C2×C16×C47D412yes—
68#41×C47—21no—
69#41×C47—21no—
70#41×C47—21no—
71#41×C47—21no—
72C2×Q8×C47M1611yescharacteristic
73C2×D4×C47—21no—
74C2×D4×C47M1611yescharacteristic
75C22×C4×C47—21no—
76C22×C4×C47C22×C411yescharacteristic
77C2×C22⋊D47—21no—
78#10625350—21no—
79#10625350—21no—
80#10625350—21no—
81#10625350—21no—
82C2×D188M1611yescharacteristic
83C2×C4×D47—21no—
84C2×Dic94M1611yescharacteristic
85M16×C47—21no—
86M16×C47—21no—
87M16×C47—41no—
88M16×C47—41no—
89C2×C8×C47—21no—
90C2×C8×C47—41no—
91C2×C8×C47—41no—
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47#4111yescharacteristic
97C2×C8×C47#4111yescharacteristic
98C22⋊Dic47—21no—
99#10625337—22no—
100#10625336—22no—
101#10625335—22no—
102#10625334—22no—
103C4⋊Dic47—21no—
104C4×Dic47C22⋊C412yes—
105#10625329—21no—
106#10625329—21no—
107#10625329—21no—
108#10625329—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8C22⋊C412yes—
111#10625324—41no—
112#10625324—41no—
113C8×D47—41no—
114C8×D47—41no—
115C16×C47—22no—
116C16×C47—41no—
117C47⋊C16—22no—
118C47⋊C16—41no—
119C23×C47—21no—
120C23×C47#13111yescharacteristic
121C22×D47—21no—
122Q8×C47—21no—
123Q8×C47—21no—
124D4×C47—21no—
125D4×C47—21no—
126D4×C47—41no—
127D4×C47—41no—
128C2×C4×C47—21no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47C2×C22⋊C411yescharacteristic
132C2×C4×C47C4×D411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47C2×M1611yescharacteristic
135C2×C4×C47C2×M1611yescharacteristic
136C2×C4×C47#13111yescharacteristic
137C22⋊D47—41no—
138C22⋊D47—41no—
139C2×Dic47—22no—
140C2×Dic47—21no—
141D188—21no—
142D188—21no—
143C4×D47—41no—
144C4×D47—41no—
145Dic94—21no—
146Dic94—21no—
147C8×C47—41no—
148C8×C47#12111yescharacteristic
149C8×C47#12111yescharacteristic
150C8×C47#12411yescharacteristic
151C8×C47#12711yescharacteristic
152C47⋊C8—22no—
153C47⋊C8—41no—
154#13198—471nomaximal, Sylow
155C22×C47—41no—
156C22×C47—42no—
157C22×C47—41no—
158C22×C47#40811yescharacteristic
159C22×C47#43511yescharacteristic
160C22×C47#43511yescharacteristic
161C2×D47—42no—
162C2×D47—41no—
163C4×C47—41no—
164C4×C47#39011yescharacteristic
165C4×C47#40711yescharacteristic
166C4×C47#43511yescharacteristic
167C4×C47#43511yescharacteristic
168Dic47—42no—
169Dic47—41no—
170#2955—471no—
171#2151—471no—
172#2117—471no—
173#2117—471no—
174#2113—471no—
175#2113D4711yescharacteristic, complemented
176#2110—471no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47—81no—
181C2×C47#195511yescharacteristic
182D47—81no—
183#575—471no—
184C2×#131—471no—
185C2×#131C2×D4711yescharacteristic
186C2×M32—471no—
187C2×M32—471no—
188C2×M32C2×D4711yescharacteristic
189C22×C16C2×D4711yescharacteristic
190#444—942no—
191#443—942no—
192#430—471no—
193#421—471no—
194#421—471no—
195#405—471no—
196C4⋊C16—942no—
197#350—942no—
198#350—942no—
199C22⋊C16—22no—
200C22⋊C16—942no—
201#347—942no—
202C47#1319811yescharacteristic, complemented, Sylow
203C2×#41—471no—
204C2×#41C4×D4711yescharacteristic
205#137—941no—
206#137—941no—
207AGL(1,ℤ/8)—941no—
208AGL(1,ℤ/8)—941no—
209#135—942no—
210#131—21no—
211#131—21no—
212#131—21no—
213#131—21no—
214#131—941no—
215#131—941no—
216#131—941no—
217#131—941no—
218C2×M16—21no—
219C2×M16—471no—
220C2×M16—471no—
221C2×M16—941no—
222C2×M16C4×D4711yescharacteristic
223C22×C8—21no—
224C22×C8—941no—
225C22×C8C22×D4711yescharacteristic
226#117—471no—
227M32—21no—
228M32—21no—
229M32—941no—
230M32—941no—
231M32—941no—
232M32—941no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—472no—
237C2×C16—941no—
238C2×C16C22⋊D4712yes—
239#108—942no—
240#104—942no—
241#104—942no—
242#97—472no—
243C4×C8—472no—
244#41—21no—
245#41—21no—
246#41—21no—
247#41—21no—
248#41—941no—
249#41—941no—
250#41—941no—
251#41—941no—
252C2×Q8—471no—
253C2×Q8#1062532411yescharacteristic
254C2×D4—21no—
255C2×D4—471no—
256C2×D4—941no—
257C2×D4#1062532411yescharacteristic
258C22×C4—21no—
259C22×C4—941no—
260C22×C4C2×C4×D4711yescharacteristic
261Q16—942no—
262SD16—942no—
263SD16—942no—
264D8—942no—
265M16—21no—
266M16—21no—
267M16—41no—
268M16—41no—
269M16—941no—
270M16—941no—
271M16—941no—
272M16—941no—
273M16—1881no—
274M16—1881no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—472no—
279C2×C8—941no—
280C2×C8—1881no—
281C2×C8—1881no—
282C2×C8#1062535011yescharacteristic
283C2×C8D4×D4711yescharacteristic
284C2×C8D4×D4711yescharacteristic
285C2×C8D4×D4711yescharacteristic
286C2×C8#1062535211yescharacteristic
287C2×C8C2×C22⋊D4711yescharacteristic
288C4⋊C4—941no—
289C22⋊C4—941no—
290C42—472no—
291C16—22no—
292C16—41no—
293C16—942no—
294C16—1881no—
295C23—21no—
296C23—941no—
297C23#1309403011yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—941no—
301Q8—941no—
302D4—21no—
303D4—21no—
304D4—41no—
305D4—41no—
306D4—941no—
307D4—941no—
308D4—1881no—
309D4—1881no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—942no—
314C2×C4—941no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4C22⋊C4×D4711yescharacteristic
318C2×C4(C22⋊D47)⋊C411yescharacteristic
319C2×C4C4×C22⋊D4711yescharacteristic
320C2×C4M16×D4711yescharacteristic
321C2×C4C2×#1062532411yescharacteristic
322C2×C4#1309403211yescharacteristic
323C8—41no—
324C8—942no—
325C8—1881no—
326C8#1309399611yescharacteristic
327C8#1309399711yescharacteristic
328C8#1309401011yescharacteristic
329C8#1309402011yescharacteristic
330C22—41no—
331C22—42no—
332C22—41no—
333C22—1882no—
334C22—1881no—
335C22#1568908811yescharacteristic
336C22#1568919411yescharacteristic
337C22#1568919711yescharacteristic
338C4—41no—
339C4—1882no—
340C4—1881no—
341C4#1568902311yescharacteristic
342C4#1568908511yescharacteristic
343C4#1568919111yescharacteristic
344C4#1568919211yescharacteristic
345C2—21no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—3761no—
350C2#2024418411yescharacteristic
subgroupG/H
centerC8#13094020
commutatorC22×C4×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2113C2
socleC2×C47#1955
prime radical#15689921C4
cosocleC22×C8×C47C23
hypercenter#2113D47
hypocenterC47#13198
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2113

Complements

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

NH ≅ G/Ntype
2. #20245146C2semidirect
3. #20245016C2semidirect
4. #20245015C2semidirect
5. #20245001C2semidirect
6. #20244992C2semidirect
7. C47×#2113C2semidirect
175. #2113D47semidirect
202. C47#13198semidirect

Maximal subgroups

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

subgroupindex
#202522892
#202451462
#202450162
#202450152
#202450012
#202449922
C47×#21132
#1319847