Finite Group Databasebeta

← #26133718

Subgroups — #26133718

Order 12032.

478 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×#15689540C211yesmaximal, characteristic, complemented
2C2×#15689393C211yesmaximal, characteristic
3C2×#15688935C211yesmaximal, characteristic, complemented
4C2×#15688923C211yesmaximal, characteristic, complemented
5#20241496C211yesmaximal, characteristic, complemented
6#20241481C211yesmaximal, characteristic, complemented
7C47×#1816C211yesmaximal, characteristic, complemented
8C2×#13094049—21no—
9C22×D376C2211yescharacteristic
10#15689540—22no—
11#15689540—22no—
12C2×#13093964—21no—
13C2×#13093964C2211yescharacteristic
14C22×C47⋊C16—21no—
15C22×C47⋊C16C2211yescharacteristic
16#15689393—22no—
17#15689393C414yescomplemented
18C2×#13093960C2211yescharacteristic
19C2×#13093940—21no—
20#15688935—22no—
21#15688923—21no—
22#15688923—21no—
23C2×#131×C47C2211yescharacteristic
24C2×#108×C47C2211yescharacteristic
25C2×#107×C47C2211yescharacteristic
26C47×#417—21no—
27C47×#343—21no—
28C22×D188—21no—
29#13094049—24no—
30#13094049—42no—
31C2×#10625336—41no—
32C2×D376—21no—
33C2×D376—21no—
34C2×D376—21no—
35C2×D376—21no—
36C2×D376—21no—
37C2×D376C2×C412yes—
38C2×#10625334—41no—
39C2×#10625329—21no—
40#13093964—24no—
41#13093964—21no—
42#13093964D412yes—
43C2×C47⋊C16—24no—
44C2×C47⋊C16—21no—
45C2×C47⋊C16—21no—
46C2×C47⋊C16—21no—
47C2×C47⋊C16D411yescharacteristic
48C2×C47⋊C16D411yescharacteristic
49#13093960—21no—
50#13093960C2×C412yes—
51#13093940—24no—
52C2×#41×C47D411yescharacteristic
53#131×C47—21no—
54#131×C47—21no—
55#131×C47C2×C412yes—
56#131×C47D412yes—
57C2×M16×C47—21no—
58C2×M16×C47—21no—
59C2×M16×C47D411yescharacteristic
60C22×C8×C47—21no—
61C22×C8×C47C2311yescharacteristic
62C2×C4⋊C4×C47—21no—
63#108×C47—21no—
64#108×C47—21no—
65#107×C47—21no—
66#107×C47—21no—
67#103×C47—41no—
68#102×C47—41no—
69C23×D47—41no—
70#41×C47—21no—
71#41×C47—21no—
72#41×C47C22⋊C412yes—
73#41×C47SD1612yes—
74C2×Q8×C47D811yescharacteristic
75C2×D4×C47—21no—
76C2×D4×C47D811yescharacteristic
77C22×C4×C47—21no—
78C22×C4×C47—41no—
79C22×C4×C47C2×D411yescharacteristic
80C2×D188—22no—
81C2×D188—41no—
82C2×D188—41no—
83C2×D188—41no—
84C2×D188—41no—
85C2×D188—41no—
86M16×C47—21no—
87M16×C47—22no—
88M16×C47—22no—
89M16×C47—41no—
90M16×C47—41no—
91M16×C47C22⋊C412yes—
92C2×C8×C47—21no—
93C2×C8×C47—22no—
94C2×C8×C47—22no—
95C2×C8×C47—41no—
96C2×C8×C47C22×C411yescharacteristic
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47#4111yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C4⋊C4×C47—41no—
103C4⋊C4×C47—41no—
104#10625336—44no—
105#10625334—44no—
106#10625329—22no—
107#10625329—41no—
108C2×C47⋊C8—41no—
109D376—22no—
110D376—22no—
111D376—41no—
112D376—41no—
113C47⋊C16—21no—
114C47⋊C16—22no—
115C47⋊C16—21no—
116C23×C47—21no—
117C23×C47#13511yescharacteristic
118C22×D47—42no—
119C22×D47—81no—
120C22×D47—81no—
121C22×D47—161no—
122C22×D47—161no—
123Q8×C47—21no—
124Q8×C47#10212yes—
125D4×C47—22no—
126D4×C47—22no—
127D4×C47—21no—
128D4×C47#10212yes—
129C2×C4×C47—21no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—81no—
133C2×C4×C47—161no—
134C2×C4×C47C2×C22⋊C411yescharacteristic
135C2×C4×C47#12111yescharacteristic
136C2×C4×C47#12111yescharacteristic
137C2×C4×C47C2×D811yescharacteristic
138C2×C4×C47C2×SD1611yescharacteristic
139C2×C4×C47#13511yescharacteristic
140D188—42no—
141D188—42no—
142D188—81no—
143D188—81no—
144C8×C47—22no—
145C8×C47—81no—
146C8×C47C4×D411yescharacteristic
147C8×C47C4×D411yescharacteristic
148C8×C47#12011yescharacteristic
149C8×C47#12311yescharacteristic
150C47⋊C8—42no—
151#13073—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—22no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47#41611yescharacteristic
158C22×C47#46511yescharacteristic
159C22×C47#46611yescharacteristic
160C2×D47—82no—
161C2×D47—162no—
162C2×D47—161no—
163C2×D47—161no—
164C2×D47—162no—
165C4×C47—22no—
166C4×C47—161no—
167C4×C47#38611yescharacteristic
168C4×C47C2×#10211yescharacteristic
169C4×C47#46511yescharacteristic
170C4×C47#46611yescharacteristic
171C2×#504—471no—
172C2×#472—471no—
173C2×#361—471no—
174C2×#357—471no—
175#1816D4711yescharacteristic, complemented
176#1387—471no—
177#1381—471no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47—42no—
181C2×C47#189311yescharacteristic
182C2×C47#222111yescharacteristic
183C2×C47#222311yescharacteristic
184D47—162no—
185C2×AGL(1,ℤ/8)—941no—
186C22×D8—471no—
187C2×#131C2×D4711yescharacteristic
188#504—942no—
189#504—942no—
190C2×M32—471no—
191C2×M32—941no—
192C22×C16—471no—
193C22×C16—941no—
194#472—474no—
195#472—942no—
196C2×#108—471no—
197C2×#108C2×D4711yescharacteristic
198C2×#107C2×D4711yescharacteristic
199#417—21no—
200C2×#100—941no—
201#361—942no—
202#357—941no—
203#357—941no—
204#343—21no—
205C47#1307311yescharacteristic, complemented, Sylow
206C2×#41C22⋊D4711yescharacteristic
207C22×D4—941no—
208AGL(1,ℤ/8)—944no—
209AGL(1,ℤ/8)—1882no—
210C2×SD16—1881no—
211C2×D8—472no—
212C2×D8—941no—
213C2×D8—941no—
214C2×D8—941no—
215C2×D8—941no—
216C2×D8—941no—
217C2×D8—1881no—
218#131—21no—
219#131—21no—
220#131C4×D4712yes—
221#131D18812yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—941no—
225C2×M16C22⋊D4711yescharacteristic
226C22×C8—21no—
227C22×C8C22×D4711yescharacteristic
228C2×C4⋊C4—21no—
229M32—472no—
230M32—944no—
231M32—941no—
232C2×C16—471no—
233C2×C16—471no—
234C2×C16—944no—
235C2×C16—941no—
236C2×C16—941no—
237C2×C16—941no—
238#108—21no—
239#108—21no—
240#108—472no—
241#108—941no—
242#107—21no—
243#107—21no—
244#103—41no—
245#102—41no—
246#100—944no—
247C24—1881no—
248#41—21no—
249#41—21no—
250#41(C2×D47)⋊C412yes—
251#41#1062533612yes—
252C2×Q8#1062533411yescharacteristic
253C2×D4—21no—
254C2×D4—942no—
255C2×D4—1881no—
256C2×D4—1881no—
257C2×D4—1881no—
258C2×D4—1881no—
259C2×D4—1881no—
260C2×D4#1062533411yescharacteristic
261C22×C4—21no—
262C22×C4—41no—
263C22×C4C2×C22⋊D4711yescharacteristic
264SD16—1884no—
265D8—942no—
266D8—942no—
267D8—1884no—
268D8—1881no—
269D8—1881no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—41no—
274M16—41no—
275M16—942no—
276M16—1881no—
277M16(C2×D47)⋊C412yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—1881no—
283C2×C8C2×C4×D4711yescharacteristic
284C2×C8C2×D18811yescharacteristic
285C2×C8D4×D4711yescharacteristic
286C2×C8D4×D4711yescharacteristic
287C2×C8#1062535211yescharacteristic
288C2×C8#1062535211yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—41no—
291C16—941no—
292C16—941no—
293C16—942no—
294C23—21no—
295C23—3761no—
296C23—3761no—
297C23—1882no—
298C23—7521no—
299C23—7521no—
300C23#1309404411yescharacteristic
301Q8—21no—
302Q8#1309394412yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—3761no—
307D4—3761no—
308D4—1882no—
309D4—1882no—
310D4#1309394412yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4C2×(C2×D47)⋊C411yescharacteristic
317C2×C4#1309399611yescharacteristic
318C2×C4#1309399611yescharacteristic
319C2×C4C2×#1062533411yescharacteristic
320C2×C4C2×#1062533611yescharacteristic
321C2×C4#1309404411yescharacteristic
322C8—22no—
323C8—81no—
324C8—1882no—
325C8(C22⋊D47)⋊C411yescharacteristic
326C8(C22⋊D47)⋊C411yescharacteristic
327C8#1309399411yescharacteristic
328C8#1309400811yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—3762no—
335C22—7522no—
336C22—7522no—
337C22—7521no—
338C22—7521no—
339C22#1568911711yescharacteristic
340C22#1568935411yescharacteristic
341C22#1568935711yescharacteristic
342C4—22no—
343C4—161no—
344C4#1568900011yescharacteristic
345C4C2×#1309394411yescharacteristic
346C4#1568935411yescharacteristic
347C4#1568935711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—7522no—
352C2#2024381411yescharacteristic
353C2#2024550611yescharacteristic
354C2#2024552011yescharacteristic
subgroupG/H
centerC22#15689357
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#1816C2
socleC22×C47#466
prime radicalC2×#15689393C2
cosocleC22×C8×C47C23
hypercenter#1816D47
hypocenterC47#13073
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ #1816

Complements

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

NH ≅ G/Ntype
1. C2×#15689540C2semidirect
3. C2×#15688935C2semidirect
4. C2×#15688923C2semidirect
5. #20241496C2semidirect
6. #20241481C2semidirect
7. C47×#1816C2semidirect
17. #15689393C4semidirect
175. #1816D47semidirect
205. C47#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#156895402
C2×#156893932
C2×#156889352
C2×#156889232
#202414962
#202414812
C47×#18162
#1307347