Finite Group Databasebeta

← #26133706

Subgroups — #26133706

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253108C211yesmaximal, characteristic
2#20245125C211yesmaximal, characteristic, complemented
3C2×#15688931C211yesmaximal, characteristic, complemented
4#20243442C211yesmaximal, characteristic, complemented
5#20241500C211yesmaximal, characteristic, complemented
6#20241484C211yesmaximal, characteristic, complemented
7C2×C47×#504C211yesmaximal, characteristic, complemented
8#15689921—21no—
9C2×#13094046C2211yescharacteristic
10#15689398C412yescomplemented
11#15689393C412yescomplemented
12#15689386—22no—
13C2×#13093960C2211yescharacteristic
14C2×#13093960C2211yescharacteristic
15#15689161C2211yescharacteristic
16#15689140—21no—
17#15689105—21no—
18#15688934—22no—
19#15688931—21no—
20#15688931—21no—
21#15688895—21no—
22C2×#131×C47C2211yescharacteristic
23C47×#504—22no—
24C47×#504—22no—
25C2×M32×C47—21no—
26C2×M32×C47C2211yescharacteristic
27C22×C16×C47—21no—
28C22×C16×C47C2211yescharacteristic
29C2×#10625350—21no—
30#13094055—41no—
31#13094054—22no—
32#13094050—41no—
33#13094049—22no—
34#13094046—21no—
35#13094046—21no—
36#13094046—21no—
37#13094046—21no—
38#13094044—42no—
39C2×Dic188C2×C411yescharacteristic
40C2×#10625325—21no—
41C2×D376C2×C411yescharacteristic
42C2×#10625329—21no—
43C2×#10625329—21no—
44#13093983—21no—
45#13093960—21no—
46#13093960—21no—
47#13093960—21no—
48#13093960C2×C412yes—
49#13093958—21no—
50#13093956—21no—
51#13093951—42no—
52#13093942—22no—
53#13093940—22no—
54C2×#41×C47D411yescharacteristic
55#131×C47—21no—
56#131×C47—21no—
57#131×C47C2×C412yes—
58#131×C47D412yes—
59C2×M16×C47—21no—
60C2×M16×C47D411yescharacteristic
61C22×C8×C47—21no—
62C22×C8×C47C2311yescharacteristic
63M32×C47—24no—
64M32×C47—21no—
65M32×C47D412yes—
66C2×C16×C47—24no—
67C2×C16×C47—21no—
68C2×C16×C47—21no—
69C2×C16×C47—21no—
70C2×C16×C47D411yescharacteristic
71C2×C16×C47D411yescharacteristic
72#41×C47—21no—
73#41×C47—21no—
74#41×C47C22⋊C412yes—
75#41×C47D812yes—
76C2×Q8×C47SD1611yescharacteristic
77C2×D4×C47—21no—
78C2×D4×C47SD1611yescharacteristic
79C22×C4×C47—21no—
80C22×C4×C47C2×D411yescharacteristic
81C2×C22⋊D47—41no—
82#10625350—41no—
83#10625350—41no—
84#10625350—41no—
85#10625350—41no—
86C2×D188—21no—
87C2×C4×D47—41no—
88C2×Dic94—21no—
89M16×C47—21no—
90M16×C47—22no—
91M16×C47—22no—
92M16×C47C22⋊C412yes—
93C2×C8×C47—21no—
94C2×C8×C47—22no—
95C2×C8×C47—22no—
96C2×C8×C47C22×C411yescharacteristic
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47#4111yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C22⋊Dic47—41no—
103#10625337—42no—
104#10625336—42no—
105#10625335—42no—
106#10625334—42no—
107C4⋊Dic47—41no—
108C4×Dic47—41no—
109#10625329—22no—
110#10625329—41no—
111#10625329—41no—
112#10625329—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115Dic188—21no—
116Dic188—21no—
117D376—21no—
118D376—21no—
119#10625325—41no—
120#10625325—41no—
121C16×C47—21no—
122C16×C47—22no—
123C16×C47—21no—
124C23×C47—21no—
125C23×C47#13511yescharacteristic
126C22×D47—41no—
127Q8×C47—21no—
128Q8×C47#10212yes—
129D4×C47—22no—
130D4×C47—22no—
131D4×C47—21no—
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
142C22⋊D47—81no—
143C22⋊D47—81no—
144C2×Dic47—41no—
145C2×Dic47—81no—
146D188—41no—
147D188—41no—
148C4×D47—81no—
149C4×D47—81no—
150Dic94—41no—
151Dic94—41no—
152C8×C47—22no—
153C8×C47C4×D411yescharacteristic
154C8×C47C4×D411yescharacteristic
155C8×C47#12011yescharacteristic
156C8×C47#12311yescharacteristic
157C47⋊C8—42no—
158C47⋊C8—81no—
159#13071—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—41no—
162C22×C47—42no—
163C22×C47#41611yescharacteristic
164C22×C47#46511yescharacteristic
165C22×C47#46611yescharacteristic
166C2×D47—81no—
167C2×D47—161no—
168C4×C47—22no—
169C4×C47#38611yescharacteristic
170C4×C47C2×#10211yescharacteristic
171C4×C47#46511yescharacteristic
172C4×C47#46611yescharacteristic
173Dic47—81no—
174Dic47—161no—
175C2×#504D4711yescharacteristic, complemented
176#3067—471no—
177#2145—471no—
178C2×#359—471no—
179#1814—471no—
180#1388—471no—
181#1382—471no—
182C2×C47—21no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—42no—
186C2×C47#189311yescharacteristic
187D47—161no—
188#575—941no—
189C2×#135—471no—
190C2×#131C2×D4711yescharacteristic
191#504—22no—
192#504—22no—
193C2×M32—21no—
194C2×M32C2×D4711yescharacteristic
195C22×C16—21no—
196C22×C16C2×D4711yescharacteristic
197#473—472no—
198#472—472no—
199#471—942no—
200C2×#108—471no—
201C2×#108—471no—
202#427—471no—
203#421—941no—
204#413—941no—
205#360—942no—
206#359—941no—
207#359—941no—
208#344—941no—
209C47#1307111yescharacteristic, complemented, Sylow
210C2×#41—941no—
211C2×#41D18811yescharacteristic
212#137—942no—
213#137—1881no—
214AGL(1,ℤ/8)—942no—
215AGL(1,ℤ/8)—1881no—
216#135—941no—
217#135—941no—
218#135—941no—
219#135—941no—
220#135—1882no—
221C2×Q16—471no—
222C2×SD16—941no—
223C2×D8—471no—
224#131—21no—
225#131—21no—
226#131C4×D4712yes—
227#131C22⋊D4712yes—
228C2×M16—21no—
229C2×M16—941no—
230C2×M16—941no—
231C2×M16D18811yescharacteristic
232C22×C8—21no—
233C22×C8C22×D4711yescharacteristic
234#117—941no—
235M32—24no—
236M32—21no—
237M32C22⋊D4712yes—
238C2×C16—24no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16C22⋊D4711yescharacteristic
243C2×C16C22⋊D4711yescharacteristic
244#108—472no—
245#108—941no—
246#108—941no—
247#108—941no—
248#107—941no—
249#106—941no—
250#104—1882no—
251#101—942no—
252#100—942no—
253#41—21no—
254#41—21no—
255#41—1881no—
256#41—1881no—
257#41—1881no—
258#41—1881no—
259#41D37612yes—
260#41(C2×D47)⋊C412yes—
261C2×Q8—941no—
262C2×Q8#1062532511yescharacteristic
263C2×D4—21no—
264C2×D4—941no—
265C2×D4—1881no—
266C2×D4#1062532511yescharacteristic
267C22×C4—21no—
268C22×C4—1881no—
269C22×C4C2×D18811yescharacteristic
270Q16—941no—
271Q16—941no—
272Q16—1882no—
273SD16—1882no—
274SD16—1882no—
275SD16—1881no—
276SD16—1881no—
277D8—941no—
278D8—941no—
279D8—1882no—
280M16—21no—
281M16—22no—
282M16—22no—
283M16—942no—
284M16—1881no—
285M16—1881no—
286M16—1881no—
287M16(C2×D47)⋊C412yes—
288C2×C8—21no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—1881no—
292C2×C8—1881no—
293C2×C8C2×C4×D4711yescharacteristic
294C2×C8#1062535011yescharacteristic
295C2×C8#1062535011yescharacteristic
296C2×C8C2×C22⋊D4711yescharacteristic
297C2×C8C2×C22⋊D4711yescharacteristic
298C2×C8C2×C22⋊D4711yescharacteristic
299C4⋊C4—1881no—
300C22⋊C4—1881no—
301C42—1881no—
302C16—21no—
303C16—22no—
304C16—21no—
305C23—21no—
306C23—1881no—
307C23#1309404611yescharacteristic
308Q8—21no—
309Q8—1881no—
310Q8—1881no—
311Q8#1309394612yes—
312D4—22no—
313D4—22no—
314D4—21no—
315D4—3761no—
316D4—3761no—
317D4—1881no—
318D4—1881no—
319D4#1309394612yes—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—3761no—
324C2×C4—3761no—
325C2×C4—3761no—
326C2×C4—1881no—
327C2×C4C2×(C2×D47)⋊C411yescharacteristic
328C2×C4#1309399811yescharacteristic
329C2×C4#1309399811yescharacteristic
330C2×C4C2×D37611yescharacteristic
331C2×C4C2×#1062532511yescharacteristic
332C2×C4#1309404611yescharacteristic
333C8—22no—
334C8—3761no—
335C8—1882no—
336C8C4×C22⋊D4711yescharacteristic
337C8C4×C22⋊D4711yescharacteristic
338C8#1309399311yescharacteristic
339C8#1309400711yescharacteristic
340C22—22no—
341C22—41no—
342C22—42no—
343C22—3761no—
344C22—7521no—
345C22#1568912011yescharacteristic
346C22#1568935211yescharacteristic
347C22#1568935911yescharacteristic
348C4—22no—
349C4—3761no—
350C4—7521no—
351C4#1568900211yescharacteristic
352C4C2×#1309394611yescharacteristic
353C4#1568935211yescharacteristic
354C4#1568935911yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—7521no—
360C2#2024381611yescharacteristic
subgroupG/H
centerC4#15689359
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#504C2
socleC2×C47#1893
prime radical#20253108C2
cosocleC22×C8×C47C23
hypercenterC2×#504D47
hypocenterC47#13071
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. #20245125C2semidirect
3. C2×#15688931C2semidirect
4. #20243442C2semidirect
5. #20241500C2semidirect
6. #20241484C2semidirect
7. C2×C47×#504C2semidirect
10. #15689398C4semidirect
11. #15689393C4semidirect
175. C2×#504D47semidirect
209. C47#13071semidirect

Maximal subgroups

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

subgroupindex
#202531082
#202451252
C2×#156889312
#202434422
#202415002
#202414842
C2×C47×#5042
#1307147