Finite Group Databasebeta

← #26116358

Subgroups — #26116358

Order 12032.

598 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243150C211yesmaximal, characteristic
2#20241618C211yesmaximal, characteristic
3#20241266C212yesmaximal
4#20241266C212yesmaximal
5C47×#2851C211yesmaximal, characteristic
6C22×C4⋊Dic47—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×#13093935—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2212yes—
11#15688878—21no—
12C2×#125×C47C411yescharacteristic, complemented
13C22×C42×C47C2211yescharacteristic
14C47×#389C411yescharacteristic, complemented
15C47×#389C411yescharacteristic, complemented
16C47×#387—21no—
17C47×#387—21no—
18C47×#385—21no—
19C47×#385—21no—
20C47×#382C411yescharacteristic, complemented
21C47×#380C2211yescharacteristic
22C47×#380C2211yescharacteristic
23C23×Dic47—22no—
24C22×C47⋊C8—22no—
25C2×C4⋊Dic47—24no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47—42no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)Q814yes—
30#13093935—42no—
31C47⋊#95—24no—
32C47⋊#95—42no—
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47C2311yescharacteristic
36#125×C47—42no—
37C2×C4⋊C4×C47—21no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47C2×C411yescharacteristic
42C2×C4⋊C4×C47C2×C411yescharacteristic
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47C2×C411yescharacteristic
45C2×C22⋊C4×C47—21no—
46C2×C22⋊C4×C47—21no—
47C2×C42×C47—42no—
48C2×C42×C47C2×C411yescharacteristic
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58C24×C47C2×D411yescharacteristic
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22×C411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×C4×C47C2×Q812yes—
80C22×C4×C47#4112yes—
81C22×Dic47—24no—
82C22×Dic47—42no—
83C22×Dic47—42no—
84C22×Dic47—42no—
85C22×Dic47—42no—
86C22×Dic47—42no—
87C22×Dic47—42no—
88C4⋊C4×C47—42no—
89C4⋊C4×C47—42no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47—41no—
92C4⋊C4×C47—41no—
93C4⋊C4×C47—41no—
94C4⋊C4×C47—42no—
95C4⋊C4×C47—42no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—41no—
100C42×C47—22no—
101C42×C47—22no—
102C42×C47—41no—
103C42×C47—41no—
104C4⋊Dic47—48no—
105C2×C47⋊C8—24no—
106C2×C47⋊C8—42no—
107C2×C47⋊C8—42no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—21no—
112C23×C47C2×C22⋊C411yescharacteristic
113C23×C47#12411yescharacteristic
114C23×C47C4⋊Q811yescharacteristic
115C23×C47AGL(1,ℤ/8)11yescharacteristic
116C23×C47AGL(1,ℤ/8)11yescharacteristic
117C23×C47#13711yescharacteristic
118C23×C47#13711yescharacteristic
119C2×C4×C47—24no—
120C2×C4×C47—24no—
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—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—42no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—41no—
148C2×C4×C47—41no—
149C2×C4×C47—41no—
150C2×C4×C47—41no—
151C2×C4×C47#9912yes—
152C2×C4×C47#10412yes—
153C2×C4×C47C4×Q812yes—
154C2×C4×C47C22⋊Q812yes—
155C2×Dic47—44no—
156C2×Dic47—44no—
157C2×Dic47—44no—
158C2×Dic47—82no—
159C2×Dic47—82no—
160C2×Dic47—82no—
161C2×Dic47—82no—
162C47⋊C8—44no—
163#9887—471nomaximal, Sylow
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—21no—
175C22×C47—21no—
176C22×C47#39111yescharacteristic
177C22×C47C2×#9911yescharacteristic
178C22×C47#41011yescharacteristic
179C22×C47#41711yescharacteristic
180C22×C47#41711yescharacteristic
181C22×C47C2×#10411yescharacteristic
182C22×C47#42111yescharacteristic
183C22×C47#49011yescharacteristic
184C22×C47#49111yescharacteristic
185C22×C47#50111yescharacteristic
186C22×C47#50111yescharacteristic
187C4×C47—22no—
188C4×C47—22no—
189C4×C47—22no—
190C4×C47—22no—
191C4×C47—41no—
192C4×C47—41no—
193C4×C47—81no—
194C4×C47—81no—
195C4×C47—81no—
196C4×C47—81no—
197Dic47—84no—
198#2851D4711yescharacteristic
199#1749—471no—
200C4⋊#95—471no—
201#1294—472no—
202#1294—472no—
203C2×C47—21no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47—21no—
207C2×C47#150411yescharacteristic
208C2×C47#150911yescharacteristic
209C2×C47#151211yescharacteristic
210C2×C47#151311yescharacteristic
211C2×C47#198611yescharacteristic
212C2×C47#199211yescharacteristic
213C2×C47#199611yescharacteristic
214C2×#125Dic4711yescharacteristic, complemented
215C22×C4⋊C4—941no—
216C22×C42C2×D4711yescharacteristic
217C2×C22⋊C8—472no—
218C2×#97—941no—
219#389Dic4711yescharacteristic, complemented
220#389Dic4711yescharacteristic, complemented
221#387—21no—
222#387—21no—
223#385—21no—
224#385—21no—
225#382Dic4711yescharacteristic, complemented
226#380C2×D4711yescharacteristic
227#380C2×D4711yescharacteristic
228C2×#95—472no—
229C2×#95—941no—
230#336—941no—
231C47#988711yescharacteristic, complemented, Sylow
232C23×C4—942no—
233C23×C4C22⋊D4711yescharacteristic
234C23×C4C22⋊D4711yescharacteristic
235C23×C4C22×D4711yescharacteristic
236C22×C8—942no—
237#125—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—944no—
243C2×C4⋊C4—1882no—
244C2×C4⋊C4—1882no—
245C2×C4⋊C4C2×Dic4711yescharacteristic
246C2×C4⋊C4C2×Dic4711yescharacteristic
247C2×C4⋊C4C2×Dic4711yescharacteristic
248C2×C4⋊C4C2×Dic4711yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—42no—
252C2×C42C2×Dic4711yescharacteristic
253C2×C42C2×Dic4711yescharacteristic
254C2×C42C22⋊D4711yescharacteristic
255C2×C42C22⋊D4711yescharacteristic
256C22⋊C8—474no—
257C22⋊C8—942no—
258#97—1882no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—944no—
266#95—1882no—
267C24C2×C22⋊D4711yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—944no—
283C22×C4—1882no—
284C22×C4—1882no—
285C22×C4—1882no—
286C22×C4—1882no—
287C22×C4—1882no—
288C22×C4—1882no—
289C22×C4C22⋊Dic4711yescharacteristic
290C22×C4C22⋊Dic4711yescharacteristic
291C22×C4C22⋊Dic4711yescharacteristic
292C22×C4C22⋊Dic4711yescharacteristic
293C22×C4Q8×D4712yes—
294C22×C4#1062535412yes—
295C22×C4C22×Dic4711yescharacteristic
296C22×C4C2×C22⋊D4711yescharacteristic
297C2×C8—944no—
298C2×C8—1882no—
299C2×C8—1882no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—1888no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—41no—
316C42—41no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊Dic4711yescharacteristic
322C23#1309401311yescharacteristic
323C23#1309402211yescharacteristic
324C23#1309404911yescharacteristic
325C23#1309404911yescharacteristic
326C23#1309405411yescharacteristic
327C23#1309405411yescharacteristic
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—3762no—
361C2×C4—3762no—
362C2×C4—3762no—
363C2×C4—3762no—
364C2×C4—1884no—
365C2×C4—1884no—
366C2×C4—1884no—
367C2×C4#1309393912yes—
368C2×C4#1309395112yes—
369C2×C4Q8×Dic4712yes—
370C2×C4#1309400012yes—
371C8—1884no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#1568903011yescharacteristic
385C22C2×#1309393911yescharacteristic
386C22#1568909711yescharacteristic
387C22#1568912411yescharacteristic
388C22#1568912411yescharacteristic
389C22C2×#1309395111yescharacteristic
390C22#1568914011yescharacteristic
391C22#1568948911yescharacteristic
392C22#1568949311yescharacteristic
393C22#1568953311yescharacteristic
394C22#1568953311yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—3764no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#2024189211yescharacteristic
411C2#2024191311yescharacteristic
412C2#2024192611yescharacteristic
413C2#2024193111yescharacteristic
414C2#2024433311yescharacteristic
415C2#2024435811yescharacteristic
416C2#2024437411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2851C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#2851D47
hypocenterC47#9887
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C47C4semidirect
14. C47×#389C4semidirect
15. C47×#389C4semidirect
20. C47×#382C4semidirect
214. C2×#125Dic47semidirect
219. #389Dic47semidirect
220. #389Dic47semidirect
225. #382Dic47semidirect
231. C47#9887semidirect

Maximal subgroups

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

subgroupindex
#202431502
#202416182
#202412662
C47×#28512
#988747