Finite Group Databasebeta

← #26133148

Subgroups — #26133148

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252289C211yesmaximal, characteristic
2#20245017C211yesmaximal, characteristic, complemented
3C2×#15688907C211yesmaximal, characteristic, complemented
4#20245003C211yesmaximal, characteristic, complemented
5#20241478C211yesmaximal, characteristic, complemented
6#20241473C211yesmaximal, characteristic, complemented
7C47×#3407C211yesmaximal, characteristic, complemented
8#15689921C411yescharacteristic
9C2×#13094030C2211yescharacteristic
10C2×#13093964C2211yescharacteristic
11C22×C47⋊C16C2211yescharacteristic
12#15689243—22no—
13#15689238—22no—
14#15689171C411yescharacteristic
15#15689140C411yescharacteristic
16#15689139C411yescharacteristic
17#15689078C2211yescharacteristic
18#15688908—22no—
19#15688907—22no—
20(C2×D47)⋊C16—22no—
21C2×#131×C47C2211yescharacteristic
22C47×#504—22no—
23C47×#504—22no—
24C2×M32×C47—22no—
25C2×M32×C47C2211yescharacteristic
26C2×M32×C47C2211yescharacteristic
27C2×#10625350C2×C411yescharacteristic
28#13094055—21no—
29#13094054C812yes—
30#13094050—21no—
31#13094049C812yes—
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—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46C2×C47⋊C16—21no—
47C2×C47⋊C16—21no—
48#13093959—22no—
49#13093951—22no—
50#13093950—22no—
51#13093934—21no—
52C8×Dic47—21no—
53C2×#41×C47C2×C411yescharacteristic
54#131×C47—21no—
55#131×C47—21no—
56#131×C47D412yes—
57#131×C47D412yes—
58C2×M16×C47—21no—
59C2×M16×C47C2×C411yescharacteristic
60C22×C8×C47—21no—
61C22×C8×C47C2311yescharacteristic
62M32×C47—24no—
63M32×C47—21no—
64M32×C47—21no—
65M32×C47—21no—
66M32×C47D412yes—
67C2×C16×C47—24no—
68C2×C16×C47—21no—
69C2×C16×C47D412yes—
70#41×C47—21no—
71#41×C47—21no—
72#41×C47C22⋊C412yes—
73#41×C47C2×C812yes—
74C2×Q8×C47M1611yescharacteristic
75C2×D4×C47—21no—
76C2×D4×C47M1611yescharacteristic
77C22×C4×C47—21no—
78C22×C4×C47C22×C411yescharacteristic
79C2×C22⋊D47—21no—
80#10625350—21no—
81#10625350—21no—
82#10625350—21no—
83#10625350—21no—
84C2×D188C2×C811yescharacteristic
85C2×C4×D47—21no—
86C2×Dic94C2×C811yescharacteristic
87M16×C47—21no—
88M16×C47—22no—
89M16×C47—22no—
90M16×C47C22⋊C412yes—
91C2×C8×C47—21no—
92C2×C8×C47—22no—
93C2×C8×C47—22no—
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47#4111yescharacteristic
99C2×C8×C47#4111yescharacteristic
100C22⋊Dic47—21no—
101#10625337—22no—
102#10625336—22no—
103#10625335—22no—
104#10625334—22no—
105C4⋊Dic47—21no—
106C4×Dic47—21no—
107#10625329—21no—
108#10625329—21no—
109#10625329—21no—
110#10625329C2×C812yes—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—21no—
113#10625324—41no—
114#10625324—41no—
115C8×D47—41no—
116C8×D47—41no—
117C16×C47—22no—
118C16×C47—22no—
119C47⋊C16—41no—
120C47⋊C16—41no—
121C23×C47—21no—
122C23×C47#13111yescharacteristic
123C22×D47—21no—
124Q8×C47—21no—
125Q8×C47C22⋊C812yes—
126D4×C47—22no—
127D4×C47—22no—
128D4×C47—21no—
129D4×C47C22⋊C812yes—
130C2×C4×C47—21no—
131C2×C4×C47—22no—
132C2×C4×C47—22no—
133C2×C4×C47C2×C22⋊C411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47C22×C811yescharacteristic
137C2×C4×C47C2×M1611yescharacteristic
138C2×C4×C47#13111yescharacteristic
139C22⋊D47—41no—
140C22⋊D47—41no—
141C2×Dic47—21no—
142C2×Dic47—41no—
143D188—21no—
144D188—21no—
145C4×D47—41no—
146C4×D47—41no—
147Dic94—21no—
148Dic94—21no—
149C8×C47—22no—
150C8×C47#12011yescharacteristic
151C8×C47#12111yescharacteristic
152C8×C47#12111yescharacteristic
153C8×C47#12311yescharacteristic
154C47⋊C8—22no—
155C47⋊C8—41no—
156#12979—471nomaximal, Sylow
157C22×C47—22no—
158C22×C47—41no—
159C22×C47—81no—
160C22×C47#40811yescharacteristic
161C22×C47C8×D411yescharacteristic
162C22×C47#43511yescharacteristic
163C2×D47—42no—
164C2×D47—41no—
165C4×C47—22no—
166C4×C47#38611yescharacteristic
167C4×C47C2×C22⋊C811yescharacteristic
168C4×C47C8×D411yescharacteristic
169C4×C47#43511yescharacteristic
170Dic47—41no—
171Dic47—81no—
172#3407D4711yescharacteristic, complemented
173#2955—471no—
174#2117—471no—
175C2×#350—471no—
176#2113—471no—
177#1379—471no—
178#1378—471no—
179C2×C47—21no—
180C2×C47—21no—
181C2×C47—21no—
182C2×C47—42no—
183C2×C47#186911yescharacteristic
184D47—81no—
185#575—471no—
186C2×#131—471no—
187C2×#131C2×D4711yescharacteristic
188#504—22no—
189#504—22no—
190C2×M32—22no—
191C2×M32—471no—
192C2×M32C2×D4711yescharacteristic
193C2×M32C2×D4711yescharacteristic
194C22×C16—471no—
195#444—942no—
196#443—942no—
197#430—471no—
198#421—471no—
199#421—471no—
200#405—471no—
201#350—942no—
202#350—942no—
203C22⋊C16—942no—
204C47#1297911yescharacteristic, complemented, Sylow
205C2×#41—471no—
206C2×#41C4×D4711yescharacteristic
207#137—472no—
208#137—941no—
209AGL(1,ℤ/8)—472no—
210AGL(1,ℤ/8)—941no—
211#135—942no—
212#131—21no—
213#131—21no—
214#131—941no—
215#131—941no—
216#131—941no—
217#131—941no—
218#131D18812yes—
219#131C22⋊D4712yes—
220C2×M16—21no—
221C2×M16—471no—
222C2×M16—471no—
223C2×M16—941no—
224C2×M16C4×D4711yescharacteristic
225C22×C8—21no—
226C22×C8—941no—
227C22×C8C22×D4711yescharacteristic
228#117—471no—
229M32—24no—
230M32—21no—
231M32—21no—
232M32—21no—
233M32—941no—
234M32—941no—
235M32C22⋊D4712yes—
236C2×C16—24no—
237C2×C16—21no—
238C2×C16—941no—
239C2×C16—941no—
240C2×C16—941no—
241C2×C16—941no—
242C2×C16C22⋊D4712yes—
243#108—942no—
244#104—942no—
245#104—942no—
246#97—941no—
247C4×C8—941no—
248#41—21no—
249#41—21no—
250#41—941no—
251#41—941no—
252#41—941no—
253#41—941no—
254#41C8×D4712yes—
255#41(C2×D47)⋊C412yes—
256C2×Q8—471no—
257C2×Q8#1062532411yescharacteristic
258C2×D4—21no—
259C2×D4—471no—
260C2×D4—941no—
261C2×D4#1062532411yescharacteristic
262C22×C4—21no—
263C22×C4—941no—
264C22×C4C2×C4×D4711yescharacteristic
265Q16—942no—
266SD16—942no—
267SD16—942no—
268D8—942no—
269M16—21no—
270M16—22no—
271M16—22no—
272M16—472no—
273M16—941no—
274M16—941no—
275M16—941no—
276M16—1881no—
277M16—1881no—
278M16(C2×D47)⋊C412yes—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—941no—
283C2×C8—941no—
284C2×C8—1881no—
285C2×C8—1881no—
286C2×C8C2×D18811yescharacteristic
287C2×C8#1062535011yescharacteristic
288C2×C8#1062535011yescharacteristic
289C2×C8C2×C22⋊D4711yescharacteristic
290C2×C8C2×C22⋊D4711yescharacteristic
291C2×C8C2×C22⋊D4711yescharacteristic
292C4⋊C4—941no—
293C22⋊C4—941no—
294C42—941no—
295C16—22no—
296C16—22no—
297C16—1881no—
298C16—1881no—
299C23—21no—
300C23—941no—
301C23#1309403011yescharacteristic
302Q8—21no—
303Q8—941no—
304Q8—941no—
305Q8(C2×D47)⋊C812yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—941no—
310D4—941no—
311D4—1881no—
312D4—1881no—
313D4(C2×D47)⋊C812yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—941no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4C2×(C2×D47)⋊C411yescharacteristic
322C2×C4C4×C22⋊D4711yescharacteristic
323C2×C4C4×C22⋊D4711yescharacteristic
324C2×C4C2×C8×D4711yescharacteristic
325C2×C4C2×#1062532411yescharacteristic
326C2×C4#1309403011yescharacteristic
327C8—22no—
328C8—942no—
329C8—1881no—
330C8#1309399311yescharacteristic
331C8#1309399811yescharacteristic
332C8#1309399811yescharacteristic
333C8#1309400711yescharacteristic
334C22—22no—
335C22—41no—
336C22—81no—
337C22—1882no—
338C22—1881no—
339C22#1568908911yescharacteristic
340C22C8×C22⋊D4711yescharacteristic
341C22#1568919411yescharacteristic
342C4—22no—
343C4—3761no—
344C4—1881no—
345C4#1568900211yescharacteristic
346C4C2×(C2×D47)⋊C811yescharacteristic
347C4C8×C22⋊D4711yescharacteristic
348C4#1568919411yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—3761no—
354C2#2024368011yescharacteristic
subgroupG/H
centerC8#13093993
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#3407C2
socleC2×C47#1869
prime radical#15689921C4
cosocleC22×C8×C47C23
hypercenter#3407D47
hypocenterC47#12979
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C22×C8 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. #20245017C2semidirect
3. C2×#15688907C2semidirect
4. #20245003C2semidirect
5. #20241478C2semidirect
6. #20241473C2semidirect
7. C47×#3407C2semidirect
172. #3407D47semidirect
204. C47#12979semidirect

Maximal subgroups

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

subgroupindex
#202522892
#202450172
C2×#156889072
#202450032
#202414782
#202414732
C47×#34072
#1297947