Finite Group Databasebeta

← #26134485

Subgroups — #26134485

Order 12032.

420 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#20252089C211yesmaximal, characteristic
2#20245153C211yesmaximal, characteristic, complemented
3#20245016C211yesmaximal, characteristic, complemented
4C2×#15688904C211yesmaximal, characteristic, complemented
5C2×#15688904C211yesmaximal, characteristic, complemented
6#20244992C211yesmaximal, characteristic, complemented
7C47×#2117C211yesmaximal, characteristic, complemented
8C22×#10625324C2211yescharacteristic
9#15689630C411yescharacteristic
10C2×#13093964C2211yescharacteristic
11C2×#13093964C2211yescharacteristic
12#15689199—22no—
13#15689194—22no—
14#15689151C411yescharacteristic
15#15689090C411yescharacteristic
16#15689086C411yescharacteristic
17#15689078C2211yescharacteristic
18#15688943—22no—
19#15688907—22no—
20#15688904—22no—
21#15688904—22no—
22#15688900—22no—
23C2×#131×C47C2211yescharacteristic
24C2×M32×C47C2211yescharacteristic
25C2×M32×C47C2211yescharacteristic
26C47×#350—22no—
27C22×C4×D47C2×C411yescharacteristic
28C2×#10625329C2×C411yescharacteristic
29C2×#10625324—21no—
30C2×#10625324—21no—
31C2×#10625324—21no—
32C2×#10625324—21no—
33C2×#10625324—21no—
34C2×#10625324—21no—
35C22×C47⋊C8C2×C411yescharacteristic
36#13094005—21no—
37#13094000—21no—
38#13093995—21no—
39#13093992—21no—
40C4×C22⋊D47—22no—
41#13093983C2×C411yescharacteristic
42#13093964—21no—
43#13093964—21no—
44#13093964—21no—
45#13093964D412yescomplemented
46C2×C47⋊C16—21no—
47C2×C47⋊C16—21no—
48Dic47⋊C8—22no—
49C22⋊(C47⋊C8)—22no—
50(C2×D47)⋊C8—22no—
51#13093934D412yes—
52C8×Dic47D412yes—
53C2×#41×C47C2×C411yescharacteristic
54#131×C47—21no—
55#131×C47—21no—
56#131×C47—21no—
57#131×C47—21no—
58C2×M16×C47—21no—
59C2×M16×C47C2×C411yescharacteristic
60C22×C8×C47—21no—
61C22×C8×C47C2311yescharacteristic
62M32×C47—21no—
63M32×C47—21no—
64M32×C47—21no—
65M32×C47D412yescomplemented
66C2×C16×C47—21no—
67C2×C16×C47—21no—
68C23×D47M1611yescharacteristic
69#41×C47—21no—
70#41×C47—21no—
71#41×C47—21no—
72#41×C47—21no—
73C2×Q8×C47M1611yescharacteristic
74C2×D4×C47—21no—
75C2×D4×C47M1611yescharacteristic
76C22×C4×C47—21no—
77C22×C4×C47C22×C411yescharacteristic
78C2×C22⋊D47—22no—
79C22×Dic47M1611yescharacteristic
80C2×C4×D47—21no—
81C2×C4×D47—21no—
82C2×C4×D47—21no—
83C2×C4×D47—21no—
84C2×C4×D47—21no—
85C2×C4×D47—21no—
86M16×C47—21no—
87M16×C47—21no—
88M16×C47—41no—
89M16×C47—41no—
90C2×C8×C47—21no—
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—
107C2×C47⋊C8—21no—
108C2×C47⋊C8—21no—
109C2×C47⋊C8—21no—
110C2×C47⋊C8C22⋊C412yes—
111#10625324—41no—
112#10625324—41no—
113#10625324—41no—
114#10625324—41no—
115C16×C47—22no—
116C16×C47—41no—
117C47⋊C16—22no—
118C47⋊C16—41no—
119C23×C47—21no—
120C23×C47C2×M1611yescharacteristic
121C22×D47—21no—
122C22×D47—21no—
123C22×D47—21no—
124C22×D47—81no—
125Q8×C47—21no—
126Q8×C47—21no—
127D4×C47—21no—
128D4×C47—21no—
129D4×C47—41no—
130D4×C47—41no—
131C2×C4×C47—21no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
134C2×C4×C47C2×C22⋊C411yescharacteristic
135C2×C4×C47C4×D411yescharacteristic
136C2×C4×C47C4×D411yescharacteristic
137C2×C4×C47C2×M1611yescharacteristic
138C2×C4×C47#13111yescharacteristic
139C2×C4×C47#13111yescharacteristic
140C22⋊D47—82no—
141C2×Dic47—22no—
142C2×Dic47—21no—
143C2×Dic47—21no—
144C2×Dic47—21no—
145C4×D47—41no—
146C4×D47—41no—
147C4×D47—41no—
148C4×D47—41no—
149C8×C47—41no—
150C8×C47#12111yescharacteristic
151C8×C47#12111yescharacteristic
152C8×C47#12411yescharacteristic
153C8×C47#12711yescharacteristic
154C47⋊C8—22no—
155C47⋊C8—41no—
156#13201—471nomaximal, Sylow
157C22×C47—41no—
158C22×C47—42no—
159C22×C47—41no—
160C22×C47#40711yescharacteristic
161C22×C47#43511yescharacteristic
162C22×C47#43511yescharacteristic
163C2×D47—41no—
164C2×D47—41no—
165C2×D47—41no—
166C2×D47—81no—
167C2×D47—81no—
168C4×C47—41no—
169C4×C47#39011yescharacteristic
170C4×C47#40811yescharacteristic
171C4×C47#43511yescharacteristic
172C4×C47#43511yescharacteristic
173Dic47—42no—
174Dic47—41no—
175#2930—471no—
176#2153—471no—
177#2117—471no—
178#2117D4711yescharacteristic, complemented
179C2×#349—471no—
180C2×#349—471no—
181#2110—471no—
182C2×C47—21no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—81no—
186C2×C47#195511yescharacteristic
187D47—81no—
188C2×#131C2×D4711yescharacteristic
189C22×M16—471no—
190#525—471no—
191C2×M32—471no—
192C2×M32—471no—
193C2×M32C2×D4711yescharacteristic
194C2×M32C2×D4711yescharacteristic
195#436—942no—
196#435—942no—
197#424—471no—
198#408—471no—
199#407—471no—
200#405—471no—
201#364—942no—
202#350—22no—
203#350—942no—
204#349—942no—
205#349—942no—
206#347—942no—
207C47#1320111yescharacteristic, complemented, Sylow
208C2×#41C4×D4711yescharacteristic
209C23×C4—471no—
210#131—21no—
211#131—21no—
212#131—21no—
213#131—21no—
214C2×M16—21no—
215C2×M16—471no—
216C2×M16—941no—
217C2×M16—941no—
218C2×M16—941no—
219C2×M16—941no—
220C2×M16—941no—
221C2×M16—941no—
222C2×M16C4×D4711yescharacteristic
223C22×C8—21no—
224C22×C8—471no—
225C22×C8C22×D4711yescharacteristic
226#123—941no—
227C22⋊Q8—941no—
228#121—941no—
229#120—941no—
230C4×D4—942no—
231#117—471no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—472no—
236M32—941no—
237M32—941no—
238M32—941no—
239M32C22⋊D4712yescomplemented
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—941no—
243C2×C16—941no—
244C4⋊C8—942no—
245C22⋊C8—942no—
246C22⋊C8—942no—
247#97—472no—
248C4×C8—472no—
249C24—471no—
250#41—21no—
251#41—21no—
252#41—21no—
253#41—21no—
254C2×Q8#1062532411yescharacteristic
255C2×D4—21no—
256C2×D4—942no—
257C2×D4#1062532411yescharacteristic
258C22×C4—21no—
259C22×C4—471no—
260C22×C4—941no—
261C22×C4—941no—
262C22×C4—941no—
263C22×C4—941no—
264C22×C4—941no—
265C22×C4—941no—
266C22×C4C2×C4×D4711yescharacteristic
267M16—21no—
268M16—21no—
269M16—41no—
270M16—41no—
271M16—941no—
272M16—941no—
273M16—1881no—
274M16—1881no—
275M16—1881no—
276M16—1881no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—472no—
281C2×C8—941no—
282C2×C8—941no—
283C2×C8—941no—
284C2×C8#1062535011yescharacteristic
285C2×C8D4×D4711yescharacteristic
286C2×C8D4×D4711yescharacteristic
287C2×C8D4×D4711yescharacteristic
288C2×C8#1062535211yescharacteristic
289C2×C8C2×C22⋊D4711yescharacteristic
290C4⋊C4—942no—
291C4⋊C4—941no—
292C22⋊C4—942no—
293C22⋊C4—942no—
294C22⋊C4—941no—
295C42—472no—
296C16—22no—
297C16—41no—
298C16—942no—
299C16—1881no—
300C23—21no—
301C23—941no—
302C23—941no—
303C23—941no—
304C23—3761no—
305C23M16×D4711yescharacteristic
306Q8—21no—
307Q8—21no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—3762no—
313C2×C4—21no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—942no—
317C2×C4—941no—
318C2×C4—941no—
319C2×C4—941no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4C22⋊C4×D4711yescharacteristic
325C2×C4(C22⋊D47)⋊C411yescharacteristic
326C2×C4C4×C22⋊D4711yescharacteristic
327C2×C4C2×#1062532411yescharacteristic
328C2×C4#1309403011yescharacteristic
329C2×C4#1309403211yescharacteristic
330C8—41no—
331C8—942no—
332C8—1881no—
333C8#1309399611yescharacteristic
334C8#1309399711yescharacteristic
335C8#1309401011yescharacteristic
336C8#1309402011yescharacteristic
337C22—41no—
338C22—42no—
339C22—41no—
340C22—3761no—
341C22—3761no—
342C22—1881no—
343C22—1881no—
344C22—1881no—
345C22#1568908511yescharacteristic
346C22#1568919211yescharacteristic
347C22#1568919711yescharacteristic
348C4—41no—
349C4—1882no—
350C4—1881no—
351C4#1568902311yescharacteristic
352C4#1568908811yescharacteristic
353C4#1568919111yescharacteristic
354C4#1568919411yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—3761no—
360C2#2024418411yescharacteristic
subgroupG/H
centerC4#15689023
commutatorC22×C4×C47C22×C4
FrattiniC22×C8C22×D47
FittingC47×#2117C2
socleC2×C47#1955
prime radical#15689630C4
cosocleC22×C8×C47C23
hypercenter#2117D47
hypocenterC47#13201
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2117

Complements

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

NH ≅ G/Ntype
2. #20245153C2semidirect
3. #20245016C2semidirect
4. C2×#15688904C2semidirect
5. C2×#15688904C2semidirect
6. #20244992C2semidirect
7. C47×#2117C2semidirect
45. #13093964D4semidirect
65. M32×C47D4semidirect
178. #2117D47semidirect
207. C47#13201semidirect
239. M32C22⋊D47semidirect

Maximal subgroups

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

subgroupindex
#202520892
#202451532
#202450162
C2×#156889042
#202449922
C47×#21172
#1320147