Finite Group Databasebeta

← #26133722

Subgroups — #26133722

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×#15688936C211yesmaximal, characteristic, complemented
3C2×#15688924C211yesmaximal, characteristic, complemented
4#20243452C211yesmaximal, characteristic, complemented
5#20241498C211yesmaximal, characteristic, complemented
6#20241481C211yesmaximal, characteristic, complemented
7C2×C47×#472C211yesmaximal, characteristic
8#15689540—22no—
9#15689540—22no—
10C2×#13093964—21no—
11C2×#13093964C2211yescharacteristic
12C22×C47⋊C16—21no—
13C22×C47⋊C16C2211yescharacteristic
14C2×#13093960C2211yescharacteristic
15C2×#13093958C2211yescharacteristic
16#15689124—21no—
17#15688936—22no—
18#15688924—21no—
19#15688924—21no—
20#15688891—21no—
21C2×AGL(1,ℤ/8)×C47—21no—
22C22×D8×C47C2211yescharacteristic
23C2×#131×C47C2211yescharacteristic
24C47×#472—22no—
25C47×#472C414yescomplemented
26C2×#108×C47C2211yescharacteristic
27C2×#100×C47—21no—
28C2×#10625329—21no—
29C2×C4⋊Dic47—21no—
30#13093964—24no—
31#13093964—21no—
32#13093964D412yes—
33C2×C47⋊C16—24no—
34C2×C47⋊C16—21no—
35C2×C47⋊C16—21no—
36C2×C47⋊C16—21no—
37C2×C47⋊C16D411yescharacteristic
38C2×C47⋊C16D411yescharacteristic
39#13093960—21no—
40#13093960—21no—
41#13093958—21no—
42#13093958—21no—
43#13093948—41no—
44#13093945—41no—
45C2×#41×C47D411yescharacteristic
46C22×D4×C47—21no—
47AGL(1,ℤ/8)×C47—24no—
48AGL(1,ℤ/8)×C47—42no—
49C2×SD16×C47—41no—
50C2×D8×C47—21no—
51C2×D8×C47—21no—
52C2×D8×C47—21no—
53C2×D8×C47—21no—
54C2×D8×C47—21no—
55C2×D8×C47—41no—
56C2×D8×C47C2×C412yes—
57#131×C47—21no—
58#131×C47—21no—
59#131×C47C2×C412yes—
60#131×C47D412yes—
61C2×M16×C47—21no—
62C2×M16×C47—21no—
63C2×M16×C47D411yescharacteristic
64C22×C8×C47—21no—
65C22×C8×C47C2311yescharacteristic
66#108×C47—21no—
67#108×C47C2×C412yes—
68#100×C47—24no—
69C24×C47—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×C47—22no—
77C2×D4×C47—41no—
78C2×D4×C47—41no—
79C2×D4×C47—41no—
80C2×D4×C47—41no—
81C2×D4×C47—41no—
82C2×D4×C47D811yescharacteristic
83C22×C4×C47—21no—
84C22×C4×C47C2×D411yescharacteristic
85C22×Dic47—41no—
86SD16×C47—44no—
87D8×C47—22no—
88D8×C47—22no—
89D8×C47—44no—
90D8×C47—41no—
91D8×C47—41no—
92M16×C47—21no—
93M16×C47—22no—
94M16×C47—22no—
95M16×C47—22no—
96M16×C47—41no—
97M16×C47C22⋊C412yes—
98C2×C8×C47—21no—
99C2×C8×C47—22no—
100C2×C8×C47—22no—
101C2×C8×C47—41no—
102C2×C8×C47C22×C411yescharacteristic
103C2×C8×C47C2×D411yescharacteristic
104C2×C8×C47C2×D411yescharacteristic
105C2×C8×C47C2×D411yescharacteristic
106C2×C8×C47#4111yescharacteristic
107C2×C8×C47#4111yescharacteristic
108C4⋊Dic47—41no—
109C4⋊Dic47—41no—
110#10625329—41no—
111#10625329—41no—
112C2×C47⋊C8—41no—
113C47⋊C16—21no—
114C47⋊C16—22no—
115C47⋊C16—21no—
116C23×C47—21no—
117C23×C47—42no—
118C23×C47—81no—
119C23×C47—81no—
120C23×C47—161no—
121C23×C47—161no—
122C23×C47#13511yescharacteristic
123Q8×C47—21no—
124Q8×C47#10212yes—
125D4×C47—22no—
126D4×C47—22no—
127D4×C47—21no—
128D4×C47—42no—
129D4×C47—42no—
130D4×C47—81no—
131D4×C47—81no—
132D4×C47#10212yes—
133C2×C4×C47—21no—
134C2×C4×C47—22no—
135C2×C4×C47—22no—
136C2×C4×C47C2×C22⋊C411yescharacteristic
137C2×C4×C47#12111yescharacteristic
138C2×C4×C47#12111yescharacteristic
139C2×C4×C47C2×D811yescharacteristic
140C2×C4×C47C2×SD1611yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×Dic47—81no—
143C2×Dic47—161no—
144C8×C47—22no—
145C8×C47—42no—
146C8×C47C4×D411yescharacteristic
147C8×C47C4×D411yescharacteristic
148C8×C47#12011yescharacteristic
149C8×C47#12311yescharacteristic
150C47⋊C8—81no—
151#13073—471nomaximal, Sylow
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—22no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—82no—
158C22×C47—162no—
159C22×C47—162no—
160C22×C47—161no—
161C22×C47—161no—
162C22×C47#41611yescharacteristic
163C22×C47#46511yescharacteristic
164C22×C47#46611yescharacteristic
165C4×C47—22no—
166C4×C47#38611yescharacteristic
167C4×C47C2×#10211yescharacteristic
168C4×C47#46511yescharacteristic
169C4×C47#46611yescharacteristic
170Dic47—161no—
171C2×#504—471no—
172C2×#472D4711yescharacteristic
173C2×#361—471no—
174C2×#357—471no—
175#1816—471no—
176#1387—471no—
177#1381—471no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47—42no—
181C2×C47—162no—
182C2×C47#189311yescharacteristic
183C2×C47#222111yescharacteristic
184C2×C47#222311yescharacteristic
185C2×AGL(1,ℤ/8)—21no—
186C22×D8C2×D4711yescharacteristic
187C2×#131C2×D4711yescharacteristic
188#504—942no—
189#504—942no—
190C2×M32—471no—
191C2×M32—941no—
192C22×C16—471no—
193C22×C16—941no—
194#472—22no—
195#472Dic4714yescomplemented
196C2×#108—471no—
197C2×#108C2×D4711yescharacteristic
198C2×#107—471no—
199#417—941no—
200C2×#100—21no—
201#361—942no—
202#357—941no—
203#357—941no—
204#343—941no—
205C47#1307311yescharacteristic, complemented, Sylow
206C2×#41C22⋊D4711yescharacteristic
207C22×D4—21no—
208AGL(1,ℤ/8)—24no—
209AGL(1,ℤ/8)—42no—
210C2×SD16—41no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—41no—
217C2×D8C2×Dic4712yes—
218#131—21no—
219#131—21no—
220#131C2×Dic4712yes—
221#131C22⋊D4712yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—941no—
225C2×M16C22⋊D4711yescharacteristic
226C22×C8—21no—
227C22×C8C22×D4711yescharacteristic
228C2×C4⋊C4—941no—
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—941no—
240#108—941no—
241#108C2×Dic4712yes—
242#107—941no—
243#107—941no—
244#103—1881no—
245#102—1881no—
246#100—24no—
247C24—41no—
248#41—21no—
249#41—21no—
250#41#1062533512yes—
251#41C22⋊Dic4712yes—
252C2×Q8#1062533411yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4#1062533411yescharacteristic
261C22×C4—21no—
262C22×C4—1881no—
263C22×C4C2×C22⋊D4711yescharacteristic
264SD16—44no—
265D8—22no—
266D8—22no—
267D8—44no—
268D8—41no—
269D8—41no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—1881no—
276M16—1881no—
277M16C22⋊Dic4712yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—1881no—
283C2×C8D4×D4711yescharacteristic
284C2×C8D4×D4711yescharacteristic
285C2×C8#1062535211yescharacteristic
286C2×C8#1062535211yescharacteristic
287C2×C8C22×Dic4711yescharacteristic
288C2×C8C2×C22⋊D4711yescharacteristic
289C4⋊C4—1881no—
290C4⋊C4—1881no—
291C16—941no—
292C16—941no—
293C16—942no—
294C23—21no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—161no—
299C23—161no—
300C23#1309404411yescharacteristic
301Q8—21no—
302Q8#1309394512yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4#1309394512yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—3761no—
315C2×C4—7521no—
316C2×C4C2×C22⋊Dic4711yescharacteristic
317C2×C4#1309399611yescharacteristic
318C2×C4#1309399611yescharacteristic
319C2×C4C2×#1062533411yescharacteristic
320C2×C4C2×#1062533511yescharacteristic
321C2×C4#1309404411yescharacteristic
322C8—22no—
323C8—42no—
324C8—3761no—
325C8D4×Dic4711yescharacteristic
326C8D4×Dic4711yescharacteristic
327C8#1309399211yescharacteristic
328C8#1309400511yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#1568911911yescharacteristic
340C22#1568935011yescharacteristic
341C22#1568935711yescharacteristic
342C4—22no—
343C4—7521no—
344C4#1568900311yescharacteristic
345C4C2×#1309394511yescharacteristic
346C4#1568935011yescharacteristic
347C4#1568935711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—162no—
352C2#2024381711yescharacteristic
353C2#2024550611yescharacteristic
354C2#2024551611yescharacteristic
subgroupG/H
centerC22#15689357
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
Fitting, prime radicalC2×C47×#472C2
socleC22×C47#466
cosocleC22×C8×C47C23
hypercenterC2×#472D47
hypocenterC47#13073
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
1. C2×#15689540C2semidirect
2. C2×#15688936C2semidirect
3. C2×#15688924C2semidirect
4. #20243452C2semidirect
5. #20241498C2semidirect
6. #20241481C2semidirect
25. C47×#472C4semidirect
195. #472Dic47semidirect
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×#156889362
C2×#156889242
#202434522
#202414982
#202414812
C2×C47×#4722
#1307347