Finite Group Databasebeta

← #26134480

Subgroups — #26134480

Order 12032.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252082C211yesmaximal, characteristic
2#20245153C211yesmaximal, characteristic, complemented
3#20245017C211yesmaximal, characteristic, complemented
4#20245015C211yesmaximal, characteristic, complemented
5#20244999C211yesmaximal, characteristic, complemented
6#20244992C211yesmaximal, characteristic, complemented
7C47×#2112C211yesmaximal, characteristic, complemented
8C2×#13094030C2211yescharacteristic
9#15689634C411yescharacteristic
10C2×#13093964C2211yescharacteristic
11C2×#13093964C2211yescharacteristic
12#15689194—22no—
13C8×C22⋊D47—22no—
14#15689149C411yescharacteristic
15#15689089C411yescharacteristic
16#15689087C411yescharacteristic
17#15689078C2211yescharacteristic
18#15688943—22no—
19#15688908—22no—
20#15688906—22no—
21C22⋊(C47⋊C16)—22no—
22#15688900—22no—
23C23×C8×C47C2211yescharacteristic
24C2×M32×C47C2211yescharacteristic
25C2×M32×C47C2211yescharacteristic
26C47×C22⋊C16—22no—
27C2×#10625350C2×C411yescharacteristic
28#13094030—21no—
29#13094030—21no—
30#13094030—21no—
31#13094030—21no—
32C2×#10625329C2×C411yescharacteristic
33C2×#10625329C2×C411yescharacteristic
34C2×#10625324—21no—
35C2×C8×D47—21no—
36#13094007—21no—
37#13094003—21no—
38#13093998—21no—
39#13093993—21no—
40C4×C22⋊D47—22no—
41#13093983C2×C411yescharacteristic
42#13093964—21no—
43#13093964—21no—
44#13093964—21no—
45#13093964—21no—
46C2×C47⋊C16—21no—
47C2×C47⋊C16D412yescomplemented
48Dic47⋊C8—22no—
49C22⋊(C47⋊C8)—22no—
50(C2×D47)⋊C8—22no—
51#13093934D412yes—
52C8×Dic47D412yes—
53C23×C4×C47C2×C411yescharacteristic
54C22×C8×C47—21no—
55C22×C8×C47—21no—
56C22×C8×C47—21no—
57C22×C8×C47—21no—
58C22×C8×C47—21no—
59C22×C8×C47—21no—
60C22×C8×C47C2×C411yescharacteristic
61C22×C8×C47C2311yescharacteristic
62M32×C47—21no—
63M32×C47—21no—
64M32×C47—21no—
65M32×C47—21no—
66C2×C16×C47—21no—
67C2×C16×C47D412yescomplemented
68C24×C47M1611yescharacteristic
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47—21no—
75C22×C4×C47M1611yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C2×C22⋊D47—22no—
78C2×C22⋊D47—21no—
79#10625350—21no—
80#10625350—21no—
81#10625350—21no—
82#10625350—21no—
83C2×D188M1611yescharacteristic
84C2×C4×D47—21no—
85C2×Dic94M1611yescharacteristic
86C2×C8×C47—21no—
87C2×C8×C47—21no—
88C2×C8×C47—21no—
89C2×C8×C47—41no—
90C2×C8×C47—41no—
91C2×C8×C47—41no—
92C2×C8×C47—41no—
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2×D411yescharacteristic
96C2×C8×C47C2×D411yescharacteristic
97C2×C8×C47#4111yescharacteristic
98C2×C8×C47#4111yescharacteristic
99C22⋊Dic47—22no—
100C22⋊Dic47—21no—
101(C2×D47)⋊C4—22no—
102C4⋊Dic47—21no—
103Dic47⋊C4—22no—
104C4×Dic47C22⋊C412yes—
105#10625329—21no—
106#10625329—21no—
107#10625329—21no—
108#10625329—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8C22⋊C412yes—
111#10625324—41no—
112#10625324—41no—
113C8×D47—41no—
114C8×D47—41no—
115C16×C47—22no—
116C16×C47—41no—
117C47⋊C16—22no—
118C47⋊C16—41no—
119C23×C47—21no—
120C23×C47—21no—
121C23×C47—21no—
122C23×C47—42no—
123C23×C47C2×M1611yescharacteristic
124C22×D47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47C2×C22⋊C411yescharacteristic
133C2×C4×C47C4×D411yescharacteristic
134C2×C4×C47C4×D411yescharacteristic
135C2×C4×C47C2×M1611yescharacteristic
136C2×C4×C47#13111yescharacteristic
137C2×C4×C47#13111yescharacteristic
138C22⋊D47—44no—
139C22⋊D47—41no—
140C22⋊D47—41no—
141C2×Dic47—22no—
142C2×Dic47—21no—
143D188—21no—
144D188—21no—
145C4×D47—41no—
146C4×D47—41no—
147Dic94—21no—
148Dic94—21no—
149C8×C47—41no—
150C8×C47#12111yescharacteristic
151C8×C47#12111yescharacteristic
152C8×C47#12411yescharacteristic
153C8×C47#12711yescharacteristic
154C47⋊C8—22no—
155C47⋊C8—41no—
156#13200—471nomaximal, Sylow
157C22×C47—22no—
158C22×C47—22no—
159C22×C47—22no—
160C22×C47—41no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47#40711yescharacteristic
166C22×C47#43511yescharacteristic
167C22×C47#43511yescharacteristic
168C2×D47—42no—
169C2×D47—41no—
170C4×C47—41no—
171C4×C47#39011yescharacteristic
172C4×C47#40811yescharacteristic
173C4×C47#43511yescharacteristic
174C4×C47#43511yescharacteristic
175Dic47—42no—
176Dic47—41no—
177#2929—471no—
178#2153—471no—
179#2117—471no—
180#2117—471no—
181#2112—471no—
182#2112D4711yescharacteristic, complemented
183#2110—471no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47—42no—
188C2×C47#195511yescharacteristic
189D47—81no—
190C2×#131—471no—
191C23×C8C2×D4711yescharacteristic
192#525—471no—
193C2×M32—471no—
194C2×M32—471no—
195C2×M32C2×D4711yescharacteristic
196C2×M32C2×D4711yescharacteristic
197#435—942no—
198C8×D4—942no—
199#424—471no—
200#408—471no—
201#407—471no—
202#405—471no—
203#364—942no—
204#350—942no—
205#350—942no—
206C22⋊C16—22no—
207C22⋊C16—942no—
208#347—942no—
209C47#1320011yescharacteristic, complemented, Sylow
210C2×#41—471no—
211C23×C4C4×D4711yescharacteristic
212#131—941no—
213#131—941no—
214#131—941no—
215#131—941no—
216C2×M16—471no—
217C2×M16—471no—
218C2×M16—941no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—941no—
226C22×C8C4×D4711yescharacteristic
227C22×C8C22×D4711yescharacteristic
228#123—941no—
229C22⋊Q8—941no—
230#121—941no—
231#120—941no—
232C4×D4—942no—
233#117—471no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—941no—
239M32—941no—
240M32—941no—
241M32—941no—
242C2×C16—21no—
243C2×C16—472no—
244C2×C16—941no—
245C2×C16C22⋊D4712yescomplemented
246C4⋊C8—942no—
247C22⋊C8—942no—
248C22⋊C8—942no—
249#97—472no—
250C4×C8—472no—
251C24#1062532411yescharacteristic
252#41—941no—
253#41—941no—
254#41—941no—
255#41—941no—
256C2×Q8—471no—
257C2×D4—471no—
258C2×D4—942no—
259C2×D4—941no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—941no—
267C22×C4#1062532411yescharacteristic
268C22×C4C2×C4×D4711yescharacteristic
269M16—941no—
270M16—941no—
271M16—941no—
272M16—941no—
273M16—1881no—
274M16—1881no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—472no—
283C2×C8—941no—
284C2×C8—1881no—
285C2×C8—1881no—
286C2×C8#1062535011yescharacteristic
287C2×C8D4×D4711yescharacteristic
288C2×C8D4×D4711yescharacteristic
289C2×C8D4×D4711yescharacteristic
290C2×C8#1062535211yescharacteristic
291C2×C8C2×C22⋊D4711yescharacteristic
292C4⋊C4—942no—
293C4⋊C4—941no—
294C22⋊C4—942no—
295C22⋊C4—942no—
296C22⋊C4—941no—
297C42—472no—
298C16—22no—
299C16—41no—
300C16—942no—
301C16—1881no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—941no—
307C23C2×#1062532411yescharacteristic
308Q8—941no—
309Q8—941no—
310D4—941no—
311D4—941no—
312D4—1884no—
313D4—1881no—
314D4—1881no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—942no—
323C2×C4—941no—
324C2×C4—1881no—
325C2×C4—1881no—
326C2×C4C22⋊C4×D4711yescharacteristic
327C2×C4(C22⋊D47)⋊C411yescharacteristic
328C2×C4C4×C22⋊D4711yescharacteristic
329C2×C4M16×D4711yescharacteristic
330C2×C4#1309403011yescharacteristic
331C2×C4#1309403211yescharacteristic
332C8—41no—
333C8—942no—
334C8—1881no—
335C8#1309399611yescharacteristic
336C8#1309399711yescharacteristic
337C8#1309401011yescharacteristic
338C8#1309402011yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—1882no—
348C22—1881no—
349C22#1568908511yescharacteristic
350C22#1568919111yescharacteristic
351C22#1568919411yescharacteristic
352C4—41no—
353C4—1882no—
354C4—1881no—
355C4#1568902311yescharacteristic
356C4#1568908811yescharacteristic
357C4#1568919211yescharacteristic
358C4#1568919711yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—3761no—
364C2#2024418411yescharacteristic
subgroupG/H
centerC8#13094020
commutatorC22×C4×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2112C2
socleC2×C47#1955
prime radical#15689634C4
cosocleC22×C8×C47C23
hypercenter#2112D47
hypocenterC47#13200
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #20245153C2semidirect
3. #20245017C2semidirect
4. #20245015C2semidirect
5. #20244999C2semidirect
6. #20244992C2semidirect
7. C47×#2112C2semidirect
47. C2×C47⋊C16D4semidirect
67. C2×C16×C47D4semidirect
182. #2112D47semidirect
209. C47#13200semidirect
245. C2×C16C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202520822
#202451532
#202450172
#202450152
#202449992
#202449922
C47×#21122
#1320047