Finite Group Databasebeta

← #26133197

Subgroups — #26133197

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689539C211yesmaximal, characteristic, complemented
2#20252289C211yesmaximal, characteristic
3#20245017C211yesmaximal, characteristic, complemented
4#20245009C211yesmaximal, characteristic, complemented
5#20241476C211yesmaximal, characteristic, complemented
6#20241473C211yesmaximal, characteristic, complemented
7C2×C47×#350C211yesmaximal, characteristic, complemented
8#15689921C411yescharacteristic
9C2×#13094030C2211yescharacteristic
10#15689539—22no—
11#15689539—22no—
12C2×#13093964C2211yescharacteristic
13C2×#13093964C2211yescharacteristic
14C2×#13093963—21no—
15C2×C16×D47—21no—
16#15689243—22no—
17#15689238—22no—
18#15689171C411yescharacteristic
19#15689140C411yescharacteristic
20#15689139C411yescharacteristic
21#15689078C2211yescharacteristic
22#15688908—22no—
23#15688905—22no—
24C2×#131×C47C2211yescharacteristic
25C2×M32×C47C2211yescharacteristic
26C22×C16×C47C2211yescharacteristic
27C47×#350—21no—
28C47×#350—21no—
29C2×#10625350C2×C411yescharacteristic
30#13094055C812yes—
31#13094054—21no—
32#13094050C812yes—
33#13094049—21no—
34#13094044—22no—
35#13094030—21no—
36#13094030—21no—
37#13094030D412yes—
38#13094030D412yes—
39C2×#10625329C2×C411yescharacteristic
40C2×#10625329C2×C411yescharacteristic
41C2×#10625324—21no—
42C2×C8×D47—21no—
43#13093983C2×C411yescharacteristic
44#13093964—21no—
45#13093964—21no—
46#13093964—21no—
47#13093964D412yes—
48#13093963—24no—
49C2×C47⋊C16—21no—
50C2×C47⋊C16—21no—
51C16×D47—24no—
52#13093959—22no—
53#13093951—22no—
54#13093950—22no—
55#13093934—21no—
56C8×Dic47—21no—
57C2×#41×C47C2×C411yescharacteristic
58#131×C47—21no—
59#131×C47—21no—
60#131×C47—21no—
61#131×C47—21no—
62C2×M16×C47—21no—
63C2×M16×C47C2×C411yescharacteristic
64C22×C8×C47—21no—
65C22×C8×C47C2311yescharacteristic
66M32×C47—21no—
67M32×C47—21no—
68C2×C16×C47—21no—
69C2×C16×C47—21no—
70C2×C16×C47—21no—
71C2×C16×C47D411yescharacteristic
72C2×C16×C47D411yescharacteristic
73#41×C47—21no—
74#41×C47—21no—
75#41×C47—21no—
76#41×C47—21no—
77C2×Q8×C47C2×C811yescharacteristic
78C2×D4×C47—21no—
79C2×D4×C47C2×C811yescharacteristic
80C22×C4×C47—21no—
81C22×C4×C47C22×C411yescharacteristic
82C2×C22⋊D47—21no—
83#10625350—21no—
84#10625350—21no—
85#10625350C22⋊C412yes—
86#10625350C2×C812yes—
87C2×D188M1611yescharacteristic
88C2×C4×D47—21no—
89C2×Dic94M1611yescharacteristic
90M16×C47—21no—
91M16×C47—21no—
92M16×C47—41no—
93M16×C47—41no—
94C2×C8×C47—21no—
95C2×C8×C47—41no—
96C2×C8×C47—41no—
97C2×C8×C47C2×D411yescharacteristic
98C2×C8×C47C2×D411yescharacteristic
99C2×C8×C47C2×D411yescharacteristic
100C2×C8×C47C2×D411yescharacteristic
101C2×C8×C47#4111yescharacteristic
102C2×C8×C47#4111yescharacteristic
103C22⋊Dic47—21no—
104#10625337—22no—
105#10625336—22no—
106#10625335—22no—
107#10625334—22no—
108C4⋊Dic47—21no—
109C4×Dic47—21no—
110#10625329—21no—
111#10625329—21no—
112#10625329C22⋊C412yes—
113#10625329C2×C812yes—
114C2×C47⋊C8—21no—
115C2×C47⋊C8—21no—
116#10625324—22no—
117#10625324—22no—
118C8×D47—22no—
119C8×D47—22no—
120C16×C47—21no—
121C16×C47—21no—
122C16×C47—41no—
123C47⋊C16—22no—
124C47⋊C16—41no—
125C23×C47—21no—
126C23×C47#13111yescharacteristic
127C22×D47—21no—
128Q8×C47—21no—
129Q8×C47—21no—
130D4×C47—21no—
131D4×C47—21no—
132D4×C47—41no—
133D4×C47—41no—
134C2×C4×C47—21no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47C2×C22⋊C411yescharacteristic
138C2×C4×C47C4×D411yescharacteristic
139C2×C4×C47C4×D411yescharacteristic
140C2×C4×C47C22×C811yescharacteristic
141C2×C4×C47C2×M1611yescharacteristic
142C2×C4×C47#13111yescharacteristic
143C22⋊D47—22no—
144C22⋊D47—22no—
145C2×Dic47—21no—
146C2×Dic47—41no—
147D188—21no—
148D188C22⋊C812yes—
149C4×D47—22no—
150C4×D47—22no—
151Dic94—21no—
152Dic94C22⋊C812yes—
153C8×C47—41no—
154C8×C47#12011yescharacteristic
155C8×C47#12111yescharacteristic
156C8×C47#12111yescharacteristic
157C8×C47#12311yescharacteristic
158C47⋊C8—22no—
159C47⋊C8—22no—
160#12987—471nomaximal, Sylow
161C22×C47—41no—
162C22×C47—41no—
163C22×C47—81no—
164C22×C47#40811yescharacteristic
165C22×C47C8×D411yescharacteristic
166C22×C47#43511yescharacteristic
167C2×D47—22no—
168C2×D47—42no—
169C4×C47—41no—
170C4×C47#38611yescharacteristic
171C4×C47C2×C22⋊C811yescharacteristic
172C4×C47C8×D411yescharacteristic
173C4×C47#43511yescharacteristic
174Dic47—22no—
175Dic47—81no—
176C2×#504—471no—
177#2955—471no—
178#2117—471no—
179C2×#350D4711yescharacteristic, complemented
180#2115—471no—
181#1379—471no—
182#1378—471no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—81no—
187C2×C47#186911yescharacteristic
188D47—42no—
189#575—471no—
190C2×#131—471no—
191C2×#131C2×D4711yescharacteristic
192#504—942no—
193#504—942no—
194C2×M32—471no—
195C2×M32—471no—
196C2×M32—941no—
197C2×M32C2×D4711yescharacteristic
198C22×C16—941no—
199C22×C16C2×D4711yescharacteristic
200#444—942no—
201#443—942no—
202#430—471no—
203#421—471no—
204#421—471no—
205#405—471no—
206#350—21no—
207#350—21no—
208#350—942no—
209#349—942no—
210C47#1298711yescharacteristic, complemented, Sylow
211C2×#41—471no—
212C2×#41C4×D4711yescharacteristic
213#137—472no—
214#137—941no—
215AGL(1,ℤ/8)—472no—
216AGL(1,ℤ/8)—941no—
217#135—942no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222#131—472no—
223#131—472no—
224#131—941no—
225#131—941no—
226C2×M16—21no—
227C2×M16—471no—
228C2×M16—471no—
229C2×M16—941no—
230C2×M16C4×D4711yescharacteristic
231C22×C8—21no—
232C22×C8—941no—
233C22×C8C22×D4711yescharacteristic
234#117—471no—
235M32—21no—
236M32—21no—
237M32—472no—
238M32—944no—
239M32—941no—
240M32—941no—
241M32—941no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—944no—
246C2×C16—941no—
247C2×C16—941no—
248C2×C16C22⋊D4711yescharacteristic
249C2×C16C22⋊D4711yescharacteristic
250#108—942no—
251#104—942no—
252#104—942no—
253#97—941no—
254C4×C8—941no—
255#41—21no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—472no—
260#41—472no—
261#41—941no—
262#41—941no—
263C2×Q8—471no—
264C2×Q8C8×D4711yescharacteristic
265C2×D4—21no—
266C2×D4—471no—
267C2×D4—941no—
268C2×D4C8×D4711yescharacteristic
269C22×C4—21no—
270C22×C4—941no—
271C22×C4C2×C4×D4711yescharacteristic
272Q16—942no—
273SD16—942no—
274SD16—942no—
275D8—942no—
276M16—21no—
277M16—21no—
278M16—41no—
279M16—41no—
280M16—472no—
281M16—472no—
282M16—941no—
283M16—941no—
284M16—942no—
285M16—942no—
286C2×C8—21no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—941no—
290C2×C8—941no—
291C2×C8—942no—
292C2×C8—942no—
293C2×C8#1062535011yescharacteristic
294C2×C8D4×D4711yescharacteristic
295C2×C8D4×D4711yescharacteristic
296C2×C8D4×D4711yescharacteristic
297C2×C8#1062535211yescharacteristic
298C2×C8C2×C22⋊D4711yescharacteristic
299C4⋊C4—941no—
300C22⋊C4—941no—
301C42—941no—
302C16—21no—
303C16—21no—
304C16—41no—
305C16—942no—
306C16—1881no—
307C23—21no—
308C23—941no—
309C23#1309403211yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—472no—
313Q8—941no—
314D4—21no—
315D4—21no—
316D4—41no—
317D4—41no—
318D4—472no—
319D4—942no—
320D4—942no—
321D4—941no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—941no—
326C2×C4—942no—
327C2×C4—942no—
328C2×C4—1881no—
329C2×C4C22⋊C4×D4711yescharacteristic
330C2×C4(C22⋊D47)⋊C411yescharacteristic
331C2×C4C4×C22⋊D4711yescharacteristic
332C2×C4C2×C8×D4711yescharacteristic
333C2×C4M16×D4711yescharacteristic
334C2×C4#1309403011yescharacteristic
335C8—41no—
336C8—942no—
337C8—942no—
338C8#1309399211yescharacteristic
339C8#1309399511yescharacteristic
340C8#1309399711yescharacteristic
341C8#1309400611yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—942no—
346C22—1882no—
347C22#1568908811yescharacteristic
348C22(C22⋊D47)⋊C811yescharacteristic
349C22#1568919211yescharacteristic
350C4—41no—
351C4—942no—
352C4—3761no—
353C4#1568899911yescharacteristic
354C4C22⋊C8×D4711yescharacteristic
355C4C8×C22⋊D4711yescharacteristic
356C4#1568919711yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—1882no—
362C2#2024367711yescharacteristic
subgroupG/H
centerC8#13093995
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC2×C47×#350C2
socleC2×C47#1869
prime radical#15689921C4
cosocleC22×C8×C47C23
hypercenterC2×#350D47
hypocenterC47#12987
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. C2×#15689539C2semidirect
3. #20245017C2semidirect
4. #20245009C2semidirect
5. #20241476C2semidirect
6. #20241473C2semidirect
7. C2×C47×#350C2semidirect
179. C2×#350D47semidirect
210. C47#12987semidirect

Maximal subgroups

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

subgroupindex
C2×#156895392
#202522892
#202450172
#202450092
#202414762
#202414732
C2×C47×#3502
#1298747