Finite Group Databasebeta

← #26133449

Subgroups — #26133449

Order 12032.

476 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×#15689240C211yesmaximal, characteristic
2C2×#15688939C211yesmaximal, characteristic, complemented
3#20245100C211yesmaximal, characteristic
4#20244250C211yesmaximal, characteristic, complemented
5#20241497C211yesmaximal, characteristic, complemented
6#20241469C211yesmaximal, characteristic, complemented
7C2×C47×#510C211yesmaximal, characteristic
8C2×#13094031—21no—
9C2×#13093964C2211yescharacteristic
10#15689240—22no—
11#15689240—22no—
12C2×#13093960C2211yescharacteristic
13C2×#13093958C2211yescharacteristic
14C2×#13093951—21no—
15C2×#13093948—21no—
16C2×C8×Dic47C2211yescharacteristic
17#15689030—21no—
18#15688939—22no—
19#15688927—21no—
20#15688924—21no—
21C2×#135×C47C2211yescharacteristic
22C22×Q16×C47C2211yescharacteristic
23C47×#510—22no—
24C47×#510C414yescomplemented
25C2×Q32×C47—21no—
26C2×SD32×C47—21no—
27C2×M32×C47C2211yescharacteristic
28#13094031—24no—
29#13094031—42no—
30C2×#10625329—21no—
31C2×#10625329—41no—
32C22×C47⋊C8—41no—
33C22×C47⋊C8D411yescharacteristic
34C2×C4⋊Dic47—21no—
35C2×C4×Dic47D411yescharacteristic
36#13093964—21no—
37#13093964—21no—
38C2×C47⋊C16—21no—
39#13093960—21no—
40#13093960D412yes—
41#13093958—21no—
42#13093958—21no—
43#13093951—24no—
44#13093948—41no—
45#13093948—41no—
46C8×Dic47—21no—
47C8×Dic47—21no—
48C47⋊#95—42no—
49C2×#41×C47—21no—
50C22×Q8×C47—21no—
51#135×C47—21no—
52#135×C47—21no—
53#135×C47C2×C412yes—
54#135×C47D412yes—
55C2×Q16×C47—21no—
56C2×Q16×C47—21no—
57C2×Q16×C47—21no—
58C2×Q16×C47—21no—
59C2×Q16×C47—21no—
60C2×Q16×C47C2×C412yes—
61C2×Q16×C47D411yescharacteristic
62C2×SD16×C47—21no—
63C2×D8×C47D411yescharacteristic
64C22×C8×C47C2311yescharacteristic
65Q32×C47—24no—
66SD32×C47—24no—
67M32×C47—21no—
68M32×C47C2×C412yes—
69C2×C16×C47—21no—
70#41×C47—22no—
71#41×C47—22no—
72#41×C47—41no—
73#41×C47—41no—
74C2×Q8×C47—22no—
75C2×Q8×C47—21no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C2×Q8×C47—41no—
80C2×Q8×C47—41no—
81C2×D4×C47—21no—
82C2×D4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47C2×D411yescharacteristic
86C22×Dic47—21no—
87C22×Dic47—41no—
88Q16×C47—22no—
89Q16×C47—22no—
90Q16×C47—21no—
91Q16×C47—41no—
92Q16×C47—41no—
93Q16×C47C22⋊C412yes—
94SD16×C47—22no—
95SD16×C47—22no—
96D8×C47—21no—
97D8×C47C22⋊C412yes—
98C2×C8×C47C22×C411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47C2×D411yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C2×C8×C47#4111yescharacteristic
104C4⋊Dic47—41no—
105C4⋊Dic47—41no—
106C4×Dic47—21no—
107C4×Dic47—21no—
108#10625329—22no—
109#10625329—44no—
110#10625329—41no—
111C2×C47⋊C8—21no—
112C2×C47⋊C8—21no—
113C2×C47⋊C8—44no—
114C2×C47⋊C8—41no—
115C2×C47⋊C8D811yescharacteristic
116C2×C47⋊C8D811yescharacteristic
117C16×C47—22no—
118C47⋊C16—41no—
119C23×C47—41no—
120C23×C47AGL(1,ℤ/8)11yescharacteristic
121Q8×C47—22no—
122Q8×C47—41no—
123Q8×C47—42no—
124Q8×C47—42no—
125Q8×C47—81no—
126Q8×C47—81no—
127D4×C47—22no—
128D4×C47—42no—
129D4×C47—42no—
130D4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—81no—
136C2×C4×C47—81no—
137C2×C4×C47C4×D411yescharacteristic
138C2×C4×C47#12011yescharacteristic
139C2×C4×C47#12111yescharacteristic
140C2×C4×C47C2×D811yescharacteristic
141C2×C4×C47#13511yescharacteristic
142C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
143C2×Dic47—22no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—82no—
147C2×Dic47—81no—
148C8×C47C2×C22⋊C411yescharacteristic
149C8×C47C4×D411yescharacteristic
150C8×C47#12111yescharacteristic
151C8×C47#12311yescharacteristic
152C47⋊C8—21no—
153C47⋊C8—21no—
154C47⋊C8—42no—
155#13028—471nomaximal, Sylow
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—42no—
159C22×C47—81no—
160C22×C47—81no—
161C22×C47#44211yescharacteristic
162C22×C47#44911yescharacteristic
163C22×C47#45911yescharacteristic
164C4×C47—42no—
165C4×C47—82no—
166C4×C47#38611yescharacteristic
167C4×C47C4×D811yescharacteristic
168C4×C47#44711yescharacteristic
169C4×C47#46311yescharacteristic
170Dic47—42no—
171Dic47—161no—
172C2×#510D4711yescharacteristic
173C2×#443—471no—
174C2×#362—471no—
175#2139—471no—
176#1969—471no—
177#1387—471no—
178#1376—471no—
179C2×C47—21no—
180C2×C47—21no—
181C2×C47—82no—
182C2×C47#187911yescharacteristic
183C2×C47#221211yescharacteristic
184C2×C47#221411yescharacteristic
185C2×#135C2×D4711yescharacteristic
186C22×Q16C2×D4711yescharacteristic
187C2×#131—941no—
188#510—22no—
189#510Dic4714yescomplemented
190C2×Q32—21no—
191C2×SD32—21no—
192C2×M32—471no—
193C2×M32C2×D4711yescharacteristic
194#443—942no—
195#443—942no—
196C2×#108—471no—
197C2×#107—471no—
198C2×#104—941no—
199C2×#103—941no—
200C2×C4×C8—471no—
201#391—941no—
202#362—942no—
203#358—941no—
204#357—941no—
205C47#1302811yescharacteristic, complemented, Sylow
206C2×#41—21no—
207C22×Q8—21no—
208#135—21no—
209#135—21no—
210#135C2×Dic4712yes—
211#135C22⋊D4712yes—
212C2×Q16—21no—
213C2×Q16—21no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16C2×Dic4712yes—
218C2×Q16C22⋊D4711yescharacteristic
219C2×SD16—21no—
220C2×D8C22⋊D4711yescharacteristic
221#131—944no—
222#131—1882no—
223C2×M16—941no—
224C2×M16—1881no—
225C22×C8—471no—
226C22×C8—1881no—
227C22×C8C22×D4711yescharacteristic
228C2×C4⋊C4—941no—
229C2×C42—471no—
230Q32—24no—
231SD32—24no—
232M32—21no—
233M32—941no—
234M32—941no—
235M32C2×Dic4712yes—
236C2×C16—21no—
237C2×C16—941no—
238#108—472no—
239#108—941no—
240#107—941no—
241#107—941no—
242#104—944no—
243#103—1881no—
244#103—1881no—
245C4×C8—941no—
246C4×C8—941no—
247#95—1882no—
248#41—22no—
249#41—22no—
250#41—41no—
251#41—41no—
252C2×Q8—22no—
253C2×Q8—21no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—21no—
260C2×D4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—941no—
264C22×C4—1881no—
265C22×C4D4×D4711yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—21no—
269Q16—41no—
270Q16—41no—
271Q16C22⋊Dic4712yes—
272SD16—22no—
273SD16—22no—
274D8—21no—
275D8C22⋊Dic4712yes—
276M16—942no—
277M16—1884no—
278M16—1881no—
279C2×C8—471no—
280C2×C8—471no—
281C2×C8—941no—
282C2×C8—941no—
283C2×C8—1884no—
284C2×C8—1881no—
285C2×C8D4×D4711yescharacteristic
286C2×C8#1062535211yescharacteristic
287C2×C8#1062535211yescharacteristic
288C2×C8C22×Dic4711yescharacteristic
289C2×C8C2×C22⋊D4711yescharacteristic
290C2×C8C2×C22⋊D4711yescharacteristic
291C4⋊C4—1881no—
292C4⋊C4—1881no—
293C42—941no—
294C42—941no—
295C16—22no—
296C16—1881no—
297C23—41no—
298C23#1309404811yescharacteristic
299Q8—22no—
300Q8—41no—
301Q8—42no—
302Q8—42no—
303Q8—81no—
304Q8—81no—
305D4—22no—
306D4—42no—
307D4—42no—
308D4—41no—
309C2×C4—41no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—942no—
316C2×C4—3762no—
317C2×C4—3761no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4D4×Dic4711yescharacteristic
321C2×C4#1309399211yescharacteristic
322C2×C4#1309399611yescharacteristic
323C2×C4D8×D4711yescharacteristic
324C2×C4#1309404311yescharacteristic
325C2×C4#1309404811yescharacteristic
326C8—941no—
327C8—941no—
328C8—1882no—
329C8C2×C22⋊Dic4711yescharacteristic
330C8D4×Dic4711yescharacteristic
331C8#1309399611yescharacteristic
332C8#1309400511yescharacteristic
333C22—21no—
334C22—21no—
335C22—42no—
336C22—81no—
337C22—81no—
338C22#1568923511yescharacteristic
339C22#1568926511yescharacteristic
340C22#1568931611yescharacteristic
341C4—42no—
342C4—82no—
343C4—1882no—
344C4—7521no—
345C4#1568900311yescharacteristic
346C4D8×Dic4711yescharacteristic
347C4#1568925511yescharacteristic
348C4#1568933611yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2#2024373811yescharacteristic
353C2#2024545911yescharacteristic
354C2#2024546711yescharacteristic
subgroupG/H
centerC22#15689316
commutatorC2×C8×C47C22×C4
FrattiniC22×C8C22×D47
FittingC2×C47×#510C2
socleC22×C47#459
prime radicalC2×#135×C47C22
cosocleC22×C8×C47C23
hypercenterC2×#510D47
hypocenterC47#13028
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
2. C2×#15688939C2semidirect
4. #20244250C2semidirect
5. #20241497C2semidirect
6. #20241469C2semidirect
24. C47×#510C4semidirect
189. #510Dic47semidirect
205. C47#13028semidirect

Maximal subgroups

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

subgroupindex
C2×#156892402
C2×#156889392
#202451002
#202442502
#202414972
#202414692
C2×C47×#5102
#1302847