Finite Group Databasebeta

← #26133531

Subgroups — #26133531

Order 12032.

508 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
1C2×#15689558C211yesmaximal, characteristic
2C2×#15688935C211yesmaximal, characteristic, complemented
3C2×#15688933C211yesmaximal, characteristic, complemented
4#20244264C211yesmaximal, characteristic, complemented
5#20241488C211yesmaximal, characteristic, complemented
6#20241468C211yesmaximal, characteristic, complemented
7C2×C47×#444C211yesmaximal, characteristic, complemented
8C22×D376C2211yescharacteristic
9#15689558—22no—
10#15689558—22no—
11C2×#13093967C412yescomplemented
12C2×#13093965C412yescomplemented
13C2×#13093964C2211yescharacteristic
14C2×#13093964C2211yescharacteristic
15C2×#13093956C2211yescharacteristic
16C2×#13093946—21no—
17#15689025—21no—
18#15688935—22no—
19#15688933—22no—
20C2×#135×C47C2211yescharacteristic
21C2×#131×C47—21no—
22C47×#444—22no—
23C47×#444—22no—
24C2×#108×C47C2211yescharacteristic
25C2×#104×C47—21no—
26C2×#97×C47C2211yescharacteristic
27C22×D188—21no—
28C2×D376—21no—
29C2×D376—21no—
30C2×D376—21no—
31C2×D376—21no—
32C2×D376—21no—
33C2×D376C2×C412yes—
34C2×C4⋊Dic47—21no—
35C2×(C2×D47)⋊C4—42no—
36#13093967—24no—
37#13093965—24no—
38#13093964—21no—
39#13093964—21no—
40#13093964—21no—
41#13093964—21no—
42C2×C47⋊C16—21no—
43C2×C47⋊C16C2×C412yes—
44#13093956—21no—
45#13093956—21no—
46#13093946—42no—
47C2×#41×C47—21no—
48#135×C47—21no—
49#135×C47—21no—
50#135×C47D412yes—
51#135×C47D412yes—
52C2×Q16×C47C2×C411yescharacteristic
53C2×SD16×C47—21no—
54C2×D8×C47C2×C411yescharacteristic
55#131×C47—24no—
56#131×C47—42no—
57C2×M16×C47—21no—
58C2×M16×C47—41no—
59C22×C8×C47—41no—
60C22×C8×C47D411yescharacteristic
61C22×C8×C47C2311yescharacteristic
62C2×C42×C47D411yescharacteristic
63#108×C47—21no—
64#108×C47D412yes—
65#104×C47—24no—
66#97×C47—21no—
67#97×C47—21no—
68C23×D47—41no—
69#41×C47—22no—
70#41×C47—22no—
71#41×C47—41no—
72#41×C47—41no—
73C2×Q8×C47—21no—
74C2×D4×C47—21no—
75C2×D4×C47—41no—
76C22×C4×C47—21no—
77C22×C4×C47—41no—
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—41no—
80C2×D188—22no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—41no—
84C2×D188—41no—
85C2×D188—41no—
86Q16×C47—21no—
87Q16×C47C22⋊C412yescomplemented
88SD16×C47—22no—
89SD16×C47—22no—
90D8×C47—21no—
91D8×C47C22⋊C412yescomplemented
92M16×C47—22no—
93M16×C47—44no—
94M16×C47—41no—
95C2×C8×C47—21no—
96C2×C8×C47—21no—
97C2×C8×C47—44no—
98C2×C8×C47—41no—
99C2×C8×C47D812yes—
100C2×C8×C47C22×C411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47C2×D411yescharacteristic
103C2×C8×C47C2×D411yescharacteristic
104C2×C8×C47#4111yescharacteristic
105C2×C8×C47#4111yescharacteristic
106C42×C47—21no—
107C42×C47—21no—
108(C2×D47)⋊C4—84no—
109C4⋊Dic47—41no—
110C4⋊Dic47—41no—
111D376—22no—
112D376—22no—
113D376—41no—
114D376—41no—
115C47⋊C16—22no—
116C47⋊C16—41no—
117C23×C47—41no—
118C23×C47AGL(1,ℤ/8)11yescharacteristic
119C22×D47—42no—
120C22×D47—82no—
121C22×D47—82no—
122C22×D47—81no—
123C22×D47—81no—
124Q8×C47—22no—
125Q8×C47—41no—
126D4×C47—22no—
127D4×C47—42no—
128D4×C47—42no—
129D4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47#12011yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47C2×D811yescharacteristic
139C2×C4×C47#13511yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
141C2×Dic47—82no—
142C2×Dic47—81no—
143D188—42no—
144D188—42no—
145D188—81no—
146D188—81no—
147C8×C47—22no—
148C8×C47—42no—
149C8×C47C2×C22⋊C411yescharacteristic
150C8×C47C4×D411yescharacteristic
151C8×C47#12111yescharacteristic
152C8×C47#12311yescharacteristic
153#13040—471nomaximal, Sylow
154C22×C47—22no—
155C22×C47—42no—
156C22×C47—82no—
157C22×C47#44211yescharacteristic
158C22×C47#44911yescharacteristic
159C22×C47#45911yescharacteristic
160C2×D47—82no—
161C2×D47—84no—
162C2×D47—161no—
163C2×D47—161no—
164C2×D47—162no—
165C4×C47—42no—
166C4×C47—81no—
167C4×C47#38611yescharacteristic
168C4×C47C4×D811yescharacteristic
169C4×C47#44711yescharacteristic
170C4×C47#46311yescharacteristic
171Dic47—161no—
172C2×#509—471no—
173C2×#444D4711yescharacteristic, complemented
174C2×#361—471no—
175C2×#360—471no—
176#1972—471no—
177#1383—471no—
178#1375—471no—
179C2×C47—21no—
180C2×C47—21no—
181C2×C47—82no—
182C2×C47#187911yescharacteristic
183C2×C47AGL(1,ℤ/16)12yes—
184D47—162no—
185C2×#135C2×D4711yescharacteristic
186C22×D8—471no—
187C2×#131—21no—
188#509—942no—
189#509—942no—
190C2×SD32—472no—
191C2×D16—472no—
192C2×M32—471no—
193C2×M32—471no—
194#444—22no—
195#444—22no—
196C2×#108C2×D4711yescharacteristic
197C2×#106—471no—
198C2×#104—21no—
199C2×#102—941no—
200C2×#97C2×D4711yescharacteristic
201#390—941no—
202#361—942no—
203#360—942no—
204C47#1304011yescharacteristic, complemented, Sylow
205C2×#41—21no—
206C22×D4—941no—
207#135—21no—
208#135—21no—
209#135D18812yes—
210#135C22⋊D4712yes—
211C2×Q16C4×D4711yescharacteristic
212C2×SD16—21no—
213C2×D8—472no—
214C2×D8—941no—
215C2×D8—941no—
216C2×D8—941no—
217C2×D8—941no—
218C2×D8—941no—
219C2×D8C4×D4711yescharacteristic
220#131—24no—
221#131—42no—
222C2×M16—21no—
223C2×M16—41no—
224C22×C8—41no—
225C22×C8C22⋊D4711yescharacteristic
226C22×C8C22×D4711yescharacteristic
227C2×C4⋊C4—941no—
228C2×C22⋊C4—1882no—
229C2×C42C22⋊D4711yescharacteristic
230SD32—944no—
231D16—944no—
232M32—941no—
233M32—941no—
234M32—941no—
235M32—941no—
236C2×C16—472no—
237C2×C16—941no—
238#108—21no—
239#108C22⋊D4712yes—
240#106—941no—
241#106—941no—
242#104—24no—
243#102—1882no—
244#97—21no—
245#97—21no—
246C24—1881no—
247#41—22no—
248#41—22no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—942no—
255C2×D4—1881no—
256C2×D4—1881no—
257C2×D4—1881no—
258C2×D4—1881no—
259C2×D4—1881no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—1881no—
263C22×C4C2×C22⋊D4711yescharacteristic
264Q16—21no—
265Q16(C2×D47)⋊C412yescomplemented
266SD16—22no—
267SD16—22no—
268D8—21no—
269D8—942no—
270D8—942no—
271D8—1881no—
272D8—1881no—
273D8(C2×D47)⋊C412yescomplemented
274M16—22no—
275M16—44no—
276M16—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—44no—
280C2×C8—41no—
281C2×C8#1062533412yes—
282C2×C8C2×C4×D4711yescharacteristic
283C2×C8C2×D18811yescharacteristic
284C2×C8#1062535011yescharacteristic
285C2×C8#1062535011yescharacteristic
286C2×C8C2×C22⋊D4711yescharacteristic
287C2×C8C2×C22⋊D4711yescharacteristic
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C22⋊C4—3764no—
291C42—21no—
292C42—21no—
293C16—942no—
294C16—1881no—
295C23—41no—
296C23—3762no—
297C23—3762no—
298C23—3761no—
299C23—3761no—
300C23—1882no—
301C23#1309404911yescharacteristic
302Q8—22no—
303Q8—41no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—41no—
308D4—3761no—
309D4—3761no—
310D4—1882no—
311D4—1882no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—3762no—
318C2×C4—3761no—
319C2×C4C4×C22⋊D4711yescharacteristic
320C2×C4#1309399311yescharacteristic
321C2×C4#1309399811yescharacteristic
322C2×C4C2×#1062533411yescharacteristic
323C2×C4#1309404411yescharacteristic
324C2×C4#1309405011yescharacteristic
325C8—22no—
326C8—42no—
327C8C2×(C2×D47)⋊C411yescharacteristic
328C8C4×C22⋊D4711yescharacteristic
329C8#1309399811yescharacteristic
330C8#1309400711yescharacteristic
331C22—22no—
332C22—42no—
333C22—82no—
334C22—3762no—
335C22—3764no—
336C22—7522no—
337C22—7521no—
338C22—7521no—
339C22#1568923311yescharacteristic
340C22#1568926811yescharacteristic
341C22#1568931811yescharacteristic
342C4—42no—
343C4—81no—
344C4—7521no—
345C4#1568900211yescharacteristic
346C4C4×#1062533411yescharacteristic
347C4#1568925711yescharacteristic
348C4#1568933811yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2—7522no—
353C2#2024373611yescharacteristic
354C2#2024528612yes—
subgroupG/H
centerC22#15689233
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#444C2
socleC22×C47#442
prime radicalC2×#15689558C2
cosocleC22×C8×C47C23
hypercenterC2×#444D47
hypocenterC47#13040
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. C2×#15688935C2semidirect
3. C2×#15688933C2semidirect
4. #20244264C2semidirect
5. #20241488C2semidirect
6. #20241468C2semidirect
7. C2×C47×#444C2semidirect
11. C2×#13093967C4semidirect
12. C2×#13093965C4semidirect
87. Q16×C47C22⋊C4semidirect
91. D8×C47C22⋊C4semidirect
173. C2×#444D47semidirect
204. C47#13040semidirect
265. Q16(C2×D47)⋊C4semidirect
273. D8(C2×D47)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#156895582
C2×#156889352
C2×#156889332
#202442642
#202414882
#202414682
C2×C47×#4442
#1304047