Finite Group Databasebeta

← #26116454

Subgroups — #26116454

Order 12032.

640 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251698C211yesmaximal, characteristic
2C4×C47⋊#95C211yesmaximal, characteristic
3#20241268C212yesmaximal
4#20241268C212yesmaximal
5C47×#1797C211yesmaximal, characteristic
6C2×#13094016C411yescharacteristic, complemented
7C22×C4×Dic47—21no—
8#15689021C411yescharacteristic, complemented
9#15689021C411yescharacteristic, complemented
10#15689007—21no—
11#15689007—21no—
12#15688996—21no—
13#15688996—21no—
14#15688989C411yescharacteristic, complemented
15#15688983C2211yescharacteristic
16#15688983C2211yescharacteristic
17C2×C47⋊#95—21no—
18C2×C47⋊#95C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C4⋊C8×C47—21no—
21C2×C22⋊C8×C47C2212yes—
22C47×#336—21no—
23C23×Dic47—22no—
24#13094016—42no—
25C2×C4⋊Dic47C2×C411yescharacteristic
26C2×C4⋊Dic47C2×C411yescharacteristic
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×C22⋊Dic47—21no—
35C2×C4×Dic47—24no—
36C2×C4×Dic47—42no—
37C2×C4×Dic47—42no—
38C47⋊#95—24no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—22no—
46C47⋊#95Q814yes—
47C23×C4×C47D411yescharacteristic
48C23×C4×C47D411yescharacteristic
49C23×C4×C47C2311yescharacteristic
50C22×C8×C47—22no—
51C2×C42×C47—42no—
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47C2×C411yescharacteristic
54C2×C42×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C4⋊C8×C47—42no—
57C22⋊C8×C47—24no—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22×C411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47C2×Q812yes—
75C22×C4×C47#4112yes—
76C22×Dic47—24no—
77C22×Dic47—24no—
78C22×Dic47—24no—
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—42no—
84C22×Dic47—42no—
85C22×Dic47—42no—
86C22×Dic47—42no—
87C2×C8×C47—42no—
88C2×C8×C47—42no—
89C2×C8×C47—42no—
90C42×C47—22no—
91C42×C47—22no—
92C42×C47—41no—
93C42×C47—41no—
94C22⋊Dic47—42no—
95C22⋊Dic47—41no—
96C22⋊Dic47—41no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101Dic47⋊C4—42no—
102Dic47⋊C4—42no—
103Dic47⋊C4—42no—
104Dic47⋊C4—42no—
105C4×Dic47—48no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47C2×C22⋊C411yescharacteristic
111C23×C47C22⋊Q812yes—
112C23×C47#13512yes—
113C23×C47AGL(1,ℤ/8)11yescharacteristic
114C23×C47#13711yescharacteristic
115C2×C4×C47—24no—
116C2×C4×C47—24no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—44no—
133C2×C4×C47#9912yes—
134C2×C4×C47#10412yes—
135C2×C4×C47#10412yes—
136C2×C4×C47C4×Q812yes—
137C2×C4×C47#12411yescharacteristic
138C2×C4×C47C4⋊Q811yescharacteristic
139C2×Dic47—28no—
140C2×Dic47—44no—
141C2×Dic47—44no—
142C2×Dic47—44no—
143C2×Dic47—44no—
144C2×Dic47—44no—
145C2×Dic47—44no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—41no—
158C8×C47—82no—
159#9910—471nomaximal, Sylow
160C22×C47—22no—
161C22×C47—22no—
162C22×C47—22no—
163C22×C47—22no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47#39111yescharacteristic
169C22×C47C2×#9911yescharacteristic
170C22×C47#41011yescharacteristic
171C22×C47#41611yescharacteristic
172C22×C47#41711yescharacteristic
173C22×C47C2×#10411yescharacteristic
174C22×C47C2×#10411yescharacteristic
175C22×C47#47812yes—
176C22×C47#47912yes—
177C4×C47—21no—
178C4×C47—22no—
179C4×C47—21no—
180C4×C47—22no—
181C4×C47—22no—
182C4×C47—22no—
183C4×C47—41no—
184Dic47—48no—
185Dic47—81no—
186Dic47—81no—
187Dic47—81no—
188Dic47—81no—
189#2851—471no—
190#1797D4711yescharacteristic
191C4×#95—471no—
192#1294—472no—
193#1294—472no—
194C2×C47—22no—
195C2×C47—22no—
196C2×C47#150011yescharacteristic
197C2×C47#150111yescharacteristic
198C2×C47#150411yescharacteristic
199C2×C47#150911yescharacteristic
200C2×C47#198911yescharacteristic
201C2×C47#199411yescharacteristic
202C2×C47#199611yescharacteristic
203C2×#125—471no—
204C22×C42—941no—
205C22×C42C2×D4711yescharacteristic
206C2×C4⋊C8—21no—
207C2×C22⋊C8C2×D4712yes—
208#389—471no—
209#389—471no—
210#387—941no—
211#387—941no—
212#385—941no—
213#385—941no—
214#382—471no—
215#380—471no—
216#380—471no—
217C2×#95—472no—
218C2×#95—941no—
219#336—21no—
220C47#991011yescharacteristic, complemented, Sylow
221C23×C4—942no—
222C23×C4D18811yescharacteristic
223C23×C4D18811yescharacteristic
224C23×C4C22×D4711yescharacteristic
225C22×C8—22no—
226#125—1882no—
227C2×C4⋊C4—471no—
228C2×C4⋊C4—471no—
229C2×C4⋊C4—471no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4—941no—
232C2×C4⋊C4—941no—
233C2×C4⋊C4—941no—
234C2×C4⋊C4—941no—
235C2×C22⋊C4—941no—
236C2×C22⋊C4—941no—
237C2×C42—42no—
238C2×C42—944no—
239C2×C42—1882no—
240C2×C42—1882no—
241C2×C42C4×D4711yescharacteristic
242C2×C42C4×D4711yescharacteristic
243C2×C42C22⋊D4711yescharacteristic
244C2×C42C22⋊D4711yescharacteristic
245C4⋊C8—42no—
246C22⋊C8—24no—
247#95—474no—
248#95—944no—
249#95—941no—
250#95—941no—
251#95—941no—
252#95—941no—
253#95—942no—
254#95—941no—
255#95—941no—
256C24C2×D18811yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—944no—
267C22×C4—944no—
268C22×C4—944no—
269C22×C4—941no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—941no—
273C22×C4—1882no—
274C22×C4—1882no—
275C22×C4—1882no—
276C22×C4—1882no—
277C22×C4(C2×D47)⋊C411yescharacteristic
278C22×C4(C2×D47)⋊C411yescharacteristic
279C22×C4(C2×D47)⋊C411yescharacteristic
280C22×C4(C2×D47)⋊C411yescharacteristic
281C22×C4C2×C4×D4711yescharacteristic
282C22×C4#1062535212yes—
283C22×C4Q8×D4712yes—
284C22×C4C2×C22⋊D4711yescharacteristic
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C4⋊C4—1881no—
291C4⋊C4—1881no—
292C4⋊C4—1882no—
293C4⋊C4—1882no—
294C4⋊C4—1882no—
295C4⋊C4—1882no—
296C22⋊C4—1882no—
297C22⋊C4—1881no—
298C22⋊C4—1881no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C42—1888no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×(C2×D47)⋊C411yescharacteristic
309C23#1309400412yes—
310C23#1309404612yes—
311C23#1309405211yescharacteristic
312C23#1309405711yescharacteristic
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—948no—
332C2×C4—1881no—
333C2×C4—1881no—
334C2×C4—1881no—
335C2×C4—1881no—
336C2×C4—1881no—
337C2×C4—1881no—
338C2×C4—1881no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1884no—
345C2×C4—1884no—
346C2×C4—1884no—
347C2×C4—1884no—
348C2×C4—1884no—
349C2×C4—1884no—
350C2×C4#1309393812yes—
351C2×C4#1309395012yes—
352C2×C4#1309395012yes—
353C2×C4#1309398912yes—
354C2×C4#1309401111yescharacteristic
355C2×C4#1309402211yescharacteristic
356C8—82no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#1568902811yescharacteristic
366C22C2×#1309393811yescharacteristic
367C22#1568909611yescharacteristic
368C22#1568912011yescharacteristic
369C22#1568912511yescharacteristic
370C22C2×#1309395011yescharacteristic
371C22C2×#1309395011yescharacteristic
372C22#1568942412yes—
373C22#1568943112yes—
374C4—21no—
375C4—22no—
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—3761no—
382C4—3761no—
383C4—3761no—
384C4—3761no—
385C4—1888no—
386C2—22no—
387C2—22no—
388C2#2024187511yescharacteristic
389C2#2024188011yescharacteristic
390C2#2024189311yescharacteristic
391C2#2024191411yescharacteristic
392C2#2024434311yescharacteristic
393C2#2024436411yescharacteristic
394C2#2024437211yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1797D47
hypocenterC47#9910
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
6. C2×#13094016C4semidirect
8. #15689021C4semidirect
9. #15689021C4semidirect
14. #15688989C4semidirect
220. C47#9910semidirect

Maximal subgroups

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

subgroupindex
#202516982
C4×C47⋊#952
#202412682
C47×#17972
#991047