Finite Group Databasebeta

← #26107976

Subgroups — #26107976

Order 12032.

780 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243474C212yesmaximal, complemented
2#20243376C211yesmaximal, characteristic
3#20241960C212yesmaximal, complemented
4#20241706C211yesmaximal, characteristic
5C2×C47×#552C211yesmaximal, characteristic
6C2×#13093939—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15688983C2212yes—
9#15688890—21no—
10#15688885—24no—
11#15688860—24no—
12#15688849—21no—
13#15688847—21no—
14C47×#552—24no—
15C47×#552—24no—
16C2×#126×C47—21no—
17C2×#125×C47—21no—
18C2×#125×C47C411yescharacteristic, complemented
19C2×#123×C47—21no—
20C2×C22⋊Q8×C47C411yescharacteristic, complemented
21C2×C22⋊Q8×C47C2211yescharacteristic
22C2×C4×Q8×C47C411yescharacteristic, complemented
23C2×C4×D4×C47C2211yescharacteristic
24C2×#117×C47—21no—
25C2×#117×C47C411yescharacteristic, complemented
26C22×C4⋊C4×C47C2211yescharacteristic
27C22×C47⋊C8—22no—
28C2×C4⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30#13093939—42no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—22no—
33C47⋊#95—22no—
34C22×Q8×C47C2×C411yescharacteristic
35C22×D4×C47D411yescharacteristic
36C23×C4×C47—21no—
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39#126×C47—28no—
40#125×C47—28no—
41#125×C47—22no—
42#125×C47—22no—
43#123×C47—28no—
44C22⋊Q8×C47—22no—
45C22⋊Q8×C47—22no—
46C22⋊Q8×C47—22no—
47C22⋊Q8×C47D414yes—
48C4×Q8×C47—22no—
49C4×Q8×C47—22no—
50C4×D4×C47—22no—
51C4×D4×C47—22no—
52#117×C47—28no—
53#117×C47—22no—
54#117×C47—22no—
55C2×C4⋊C4×C47—21no—
56C2×C4⋊C4×C47—21no—
57C2×C4⋊C4×C47—21no—
58C2×C4⋊C4×C47—21no—
59C2×C4⋊C4×C47—21no—
60C2×C4⋊C4×C47—22no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47C2×C411yescharacteristic
63C2×C4⋊C4×C47C2×C411yescharacteristic
64C2×C4⋊C4×C47C2×C411yescharacteristic
65C2×C4⋊C4×C47D411yescharacteristic
66C2×C4⋊C4×C47D414yes—
67C2×C22⋊C4×C47—21no—
68C2×C22⋊C4×C47—21no—
69C2×C22⋊C4×C47—21no—
70C2×C22⋊C4×C47D411yescharacteristic
71C2×C22⋊C4×C47D411yescharacteristic
72C2×C42×C47—21no—
73C2×C42×C47—21no—
74C2×C42×C47C2×C411yescharacteristic
75C24×C47C22⋊C411yescharacteristic
76C24×C47C2×D411yescharacteristic
77C2×Q8×C47—21no—
78C2×Q8×C47—42no—
79C2×Q8×C47C22⋊C412yes—
80C2×D4×C47—22no—
81C2×D4×C47—22no—
82C2×D4×C47—21no—
83C2×D4×C47—21no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—21no—
87C22×C4×C47—21no—
88C22×C4×C47—24no—
89C22×C4×C47—21no—
90C22×C4×C47—21no—
91C22×C4×C47—42no—
92C22×C4×C47—42no—
93C22×C4×C47—42no—
94C22×C4×C47—41no—
95C22×C4×C47—41no—
96C22×C4×C47C22⋊C411yescharacteristic
97C22×C4×C47C22⋊C411yescharacteristic
98C22×C4×C47C22⋊C411yescharacteristic
99C22×C4×C47C22×C411yescharacteristic
100C22×C4×C47C2×D412yes—
101C22×C4×C47C2×D412yes—
102C22×C4×C47C2×D411yescharacteristic
103C22×Dic47—42no—
104C4⋊C4×C47—24no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—24no—
107C4⋊C4×C47—24no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—21no—
110C4⋊C4×C47—21no—
111C4⋊C4×C47—24no—
112C4⋊C4×C47—24no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47C22⋊C412yes—
115C4⋊C4×C47C22⋊C412yes—
116C4⋊C4×C47C22⋊C412yes—
117C22⋊C4×C47—24no—
118C22⋊C4×C47—24no—
119C22⋊C4×C47—24no—
120C22⋊C4×C47—22no—
121C22⋊C4×C47—22no—
122C42×C47—24no—
123C42×C47—24no—
124C42×C47—21no—
125C42×C47—21no—
126C22⋊Dic47—42no—
127C22⋊Dic47—42no—
128C4⋊Dic47—44no—
129C2×C47⋊C8—44no—
130C2×C47⋊C8—42no—
131C23×C47—21no—
132C23×C47—21no—
133C23×C47—21no—
134C23×C47—41no—
135C23×C47—41no—
136C23×C47—41no—
137C23×C47—41no—
138C23×C47C2×C22⋊C411yescharacteristic
139C23×C47#12012yes—
140C23×C47#13512yes—
141C23×C47AGL(1,ℤ/8)11yescharacteristic
142C23×C47#13711yescharacteristic
143Q8×C47—42no—
144Q8×C47—42no—
145D4×C47—42no—
146D4×C47—42no—
147C2×C4×C47—24no—
148C2×C4×C47—22no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—22no—
154C2×C4×C47—21no—
155C2×C4×C47—22no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—41no—
159C2×C4×C47—42no—
160C2×C4×C47—42no—
161C2×C4×C47—42no—
162C2×C4×C47—42no—
163C2×C4×C47—41no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—44no—
168C2×C4×C47—44no—
169C2×C4×C47—42no—
170C2×C4×C47—41no—
171C2×C4×C47—41no—
172C2×C4×C47C2×C22⋊C411yescharacteristic
173C2×C4×C47C2×C22⋊C411yescharacteristic
174C2×C4×C47#12012yes—
175C2×Dic47—82no—
176C2×Dic47—82no—
177C2×Dic47—82no—
178C47⋊C8—82no—
179#8382—471nomaximal, Sylow
180C22×C47—22no—
181C22×C47—21no—
182C22×C47—22no—
183C22×C47—21no—
184C22×C47—42no—
185C22×C47—41no—
186C22×C47—41no—
187C22×C47—41no—
188C22×C47—41no—
189C22×C47—41no—
190C22×C47—81no—
191C22×C47—81no—
192C22×C47#37911yescharacteristic
193C22×C47#41611yescharacteristic
194C22×C47#41711yescharacteristic
195C22×C47#44912yes—
196C22×C47#45212yes—
197C4×C47—22no—
198C4×C47—21no—
199C4×C47—41no—
200C4×C47—42no—
201C4×C47—41no—
202C4×C47—42no—
203C4×C47—41no—
204C4×C47—42no—
205C4×C47—42no—
206C4×C47#41012yes—
207Dic47—162no—
208C2×#552D4711yescharacteristic
209#1823—472no—
210#1799—471no—
211#1520—472no—
212#1459—471no—
213C2×C47—22no—
214C2×C47—21no—
215C2×C47—21no—
216C2×C47—22no—
217C2×C47—81no—
218C2×C47#178211yescharacteristic
219C2×C47#178811yescharacteristic
220C2×C47#179611yescharacteristic
221#552—24no—
222#552—24no—
223C2×#126—21no—
224C2×#125—21no—
225C2×#125Dic4711yescharacteristic, complemented
226C2×#123—21no—
227C2×C22⋊Q8Dic4711yescharacteristic, complemented
228C2×C22⋊Q8C2×D4711yescharacteristic
229C2×C4×Q8Dic4711yescharacteristic, complemented
230C2×C4×D4C2×D4711yescharacteristic
231C2×#117—21no—
232C2×#117Dic4711yescharacteristic, complemented
233C22×C4⋊C4C2×D4711yescharacteristic
234C2×#99—941no—
235C2×C22⋊C8—472no—
236#380—472no—
237#342—941no—
238#340—944no—
239#328—944no—
240#324—941no—
241#323—941no—
242C47#838211yescharacteristic, complemented, Sylow
243C22×Q8C2×Dic4711yescharacteristic
244C22×D4C22⋊D4711yescharacteristic
245C23×C4—21no—
246C23×C4C2×Dic4711yescharacteristic
247C23×C4C22×D4711yescharacteristic
248C22×C8—942no—
249#126—28no—
250#125—28no—
251#125—22no—
252#125—22no—
253#123—28no—
254C22⋊Q8—22no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8C22⋊D4714yes—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×D4—22no—
261C4×D4—22no—
262#117—28no—
263#117—22no—
264#117—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—942no—
273C2×C4⋊C4C2×Dic4711yescharacteristic
274C2×C4⋊C4C2×Dic4711yescharacteristic
275C2×C4⋊C4C2×Dic4711yescharacteristic
276C2×C4⋊C4C22⋊D4711yescharacteristic
277C2×C4⋊C4C22⋊D4714yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—942no—
282C2×C22⋊C4C22⋊D4711yescharacteristic
283C2×C22⋊C4C22⋊D4711yescharacteristic
284C2×C42—21no—
285C2×C42—21no—
286C2×C42C2×Dic4711yescharacteristic
287#99—1882no—
288C22⋊C8—942no—
289C22⋊C8—942no—
290#95—942no—
291C24C22⋊Dic4711yescharacteristic
292C24C2×C22⋊D4711yescharacteristic
293C2×Q8—21no—
294C2×Q8—42no—
295C2×Q8C22⋊Dic4712yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1882no—
313C22×C4C22⋊Dic4711yescharacteristic
314C22×C4C22⋊Dic4711yescharacteristic
315C22×C4C22⋊Dic4711yescharacteristic
316C22×C4C22×Dic4711yescharacteristic
317C22×C4C2×C22⋊D4712yes—
318C22×C4C2×C22⋊D4712yes—
319C22×C4C2×C22⋊D4711yescharacteristic
320C2×C8—1884no—
321C2×C8—1882no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—1884no—
333C4⋊C4C22⋊Dic4712yes—
334C4⋊C4C22⋊Dic4712yes—
335C4⋊C4C22⋊Dic4712yes—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—1882no—
342C22⋊C4—1882no—
343C42—24no—
344C42—24no—
345C42—21no—
346C42—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23C2×C22⋊Dic4711yescharacteristic
355C23#1309399312yes—
356C23#1309404412yes—
357C23#1309404911yescharacteristic
358C23#1309405411yescharacteristic
359Q8—42no—
360Q8—42no—
361D4—42no—
362D4—42no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—3762no—
389C2×C4—3762no—
390C2×C4—3762no—
391C2×C4C2×C22⋊Dic4711yescharacteristic
392C2×C4C2×C22⋊Dic4711yescharacteristic
393C2×C4#1309399312yes—
394C8—3762no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—22no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22#1568897911yescharacteristic
408C22#1568911911yescharacteristic
409C22#1568912411yescharacteristic
410C22#1568926812yes—
411C22#1568928512yes—
412C4—22no—
413C4—21no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—42no—
418C4—42no—
419C4—42no—
420C4—41no—
421C4—7522no—
422C4#1568909712yes—
423C2—22no—
424C2—21no—
425C2—21no—
426C2—22no—
427C2—81no—
428C2#2024330611yescharacteristic
429C2#2024332811yescharacteristic
430C2#2024336311yescharacteristic
subgroupG/H
centerC22#15688979
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC2×C47×#552C2
socleC22×C47#379
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenterC2×#552D47
hypocenterC47#8382
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#552

Complements

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

NH ≅ G/Ntype
1. #20243474C2semidirect
3. #20241960C2semidirect
18. C2×#125×C47C4semidirect
20. C2×C22⋊Q8×C47C4semidirect
22. C2×C4×Q8×C47C4semidirect
25. C2×#117×C47C4semidirect
225. C2×#125Dic47semidirect
227. C2×C22⋊Q8Dic47semidirect
229. C2×C4×Q8Dic47semidirect
232. C2×#117Dic47semidirect
242. C47#8382semidirect

Maximal subgroups

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

subgroupindex
#202434742
#202433762
#202419602
#202417062
C2×C47×#5522
#838247