Finite Group Databasebeta

← #26108957

Subgroups — #26108957

Order 12032.

570 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689007C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic
3C2×#15688848C212yesmaximal
4#20241230C212yesmaximal
5C47×#1797C211yesmaximal, characteristic
6C22×Dic47⋊C4C411yescharacteristic
7C22×Dic47⋊C4C2211yescharacteristic
8C2×C4⋊(C47⋊C8)—21no—
9C2×C22⋊(C47⋊C8)C2212yes—
10#15689007—21no—
11#15689007—21no—
12#15689007—21no—
13#15689007—21no—
14C2×C47⋊#95C411yescharacteristic
15C2×C47⋊#95C411yescharacteristic
16C2×C47⋊#95C411yescharacteristic
17C2×C47⋊#95C2211yescharacteristic
18#15688878—21no—
19#15688848—24no—
20C22×C42×C47C2211yescharacteristic
21C2×C4⋊C8×C47—21no—
22C2×C22⋊C8×C47C2212yes—
23C47×#336—21no—
24C23×Dic47C2×C411yescharacteristic
25C23×Dic47C2×C411yescharacteristic
26C23×Dic47C2×C411yescharacteristic
27C23×Dic47C2×C411yescharacteristic
28C22×C47⋊C8—22no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×Dic47⋊C4—42no—
36C2×Dic47⋊C4D412yes—
37C2×Dic47⋊C4Q812yes—
38C4⋊(C47⋊C8)—42no—
39C22⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—22no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C23×C4×C47C2×C411yescharacteristic
50C23×C4×C47C2×C411yescharacteristic
51C23×C4×C47C2311yescharacteristic
52C22×C8×C47—22no—
53C2×C42×C47—42no—
54C2×C42×C47D411yescharacteristic
55C2×C42×C47D411yescharacteristic
56C2×C42×C47D411yescharacteristic
57C2×C42×C47D411yescharacteristic
58C4⋊C8×C47—42no—
59C22⋊C8×C47—22no—
60C22⋊C8×C47—22no—
61C24×C47C22×C411yescharacteristic
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47C22⋊C411yescharacteristic
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×Q811yescharacteristic
78C22×C4×C47#4111yescharacteristic
79C22×C4×C47#4111yescharacteristic
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83C22×Dic47—21no—
84C22×Dic47—22no—
85C22×Dic47—22no—
86C22×Dic47—21no—
87C22×Dic47—21no—
88C22×Dic47—22no—
89C22×Dic47—22no—
90C22×Dic47—41no—
91C22×Dic47—41no—
92C22×Dic47—41no—
93C22×Dic47—41no—
94C22×Dic47—41no—
95C22×Dic47—41no—
96C22×Dic47C4⋊C412yes—
97C22×Dic47C4⋊C412yes—
98C2×C8×C47—42no—
99C2×C8×C47—42no—
100C2×C8×C47—42no—
101C42×C47—22no—
102C42×C47—22no—
103C42×C47—41no—
104C42×C47—41no—
105Dic47⋊C4—22no—
106Dic47⋊C4—22no—
107Dic47⋊C4—22no—
108Dic47⋊C4—22no—
109Dic47⋊C4—42no—
110Dic47⋊C4—42no—
111C2×C47⋊C8—42no—
112C2×C47⋊C8—42no—
113C2×C47⋊C8—42no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—21no—
117C23×C47—21no—
118C23×C47C2×C22⋊C411yescharacteristic
119C23×C47C2×C4⋊C411yescharacteristic
120C23×C47#11711yescharacteristic
121C23×C47#13112yes—
122C23×C47#13112yes—
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—21no—
136C2×C4×C47—21no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—44no—
142C2×C4×C47—44no—
143C2×C4×C47#9912yes—
144C2×C4×C47#10112yes—
145C2×C4×C47#12111yescharacteristic
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47C22⋊Q811yescharacteristic
148C2×C4×C47C22⋊Q811yescharacteristic
149C2×Dic47—22no—
150C2×Dic47—22no—
151C2×Dic47—24no—
152C2×Dic47—42no—
153C2×Dic47—41no—
154C2×Dic47—41no—
155C2×Dic47—41no—
156C2×Dic47—41no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159C2×Dic47—41no—
160C2×Dic47—41no—
161C2×Dic47—41no—
162C2×Dic47—41no—
163C2×Dic47—41no—
164C2×Dic47—41no—
165C2×Dic47—82no—
166C2×Dic47—82no—
167C8×C47—82no—
168C47⋊C8—82no—
169#8562—471nomaximal, Sylow
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47—22no—
175C22×C47—22no—
176C22×C47—22no—
177C22×C47—22no—
178C22×C47#38011yescharacteristic
179C22×C47#40811yescharacteristic
180C22×C47#40811yescharacteristic
181C22×C47C2×#9911yescharacteristic
182C22×C47#41011yescharacteristic
183C22×C47C2×#10111yescharacteristic
184C22×C47#41311yescharacteristic
185C22×C47#42412yes—
186C22×C47#43312yes—
187C4×C47—21no—
188C4×C47—21no—
189C4×C47—21no—
190C4×C47—21no—
191C4×C47—22no—
192C4×C47—22no—
193C4×C47—41no—
194C4×C47—41no—
195Dic47—42no—
196Dic47—42no—
197Dic47—81no—
198Dic47—81no—
199C2×#387—471no—
200#1797—471no—
201#1797D4711yescharacteristic
202C2×#324—472no—
203#1280—472no—
204C2×C47—22no—
205C2×C47—22no—
206C2×C47#146811yescharacteristic
207C2×C47#147011yescharacteristic
208C2×C47#147211yescharacteristic
209C2×C47#147311yescharacteristic
210C2×C47#179811yescharacteristic
211C2×C47#179911yescharacteristic
212C2×C47#180111yescharacteristic
213C22×C4⋊C4—471no—
214C22×C4⋊C4—471no—
215C22×C42C2×D4711yescharacteristic
216C2×C4⋊C8—21no—
217C2×C4⋊C8—941no—
218C2×C22⋊C8—472no—
219C2×C22⋊C8C2×D4712yes—
220#387—941no—
221#387—941no—
222#387—941no—
223#387—941no—
224C2×#95—471no—
225C2×#95—471no—
226C2×#95—471no—
227C2×#95—471no—
228#336—21no—
229#336—941no—
230#324—944no—
231C47#856211yescharacteristic, complemented, Sylow
232C23×C4—471no—
233C23×C4—471no—
234C23×C4—471no—
235C23×C4—471no—
236C23×C4C4×D4711yescharacteristic
237C23×C4C4×D4711yescharacteristic
238C23×C4C22×D4711yescharacteristic
239C22×C8—22no—
240C22×C8—942no—
241C2×C4⋊C4—472no—
242C2×C4⋊C4—472no—
243C2×C4⋊C4—941no—
244C2×C4⋊C4—941no—
245C2×C4⋊C4—941no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4—941no—
249C2×C4⋊C4—1882no—
250C2×C42—42no—
251C2×C42D18811yescharacteristic
252C2×C42D18811yescharacteristic
253C2×C42C22⋊D4711yescharacteristic
254C2×C42C22⋊D4711yescharacteristic
255C4⋊C8—42no—
256C4⋊C8—1882no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—942no—
260C22⋊C8—942no—
261#95—941no—
262#95—941no—
263#95—941no—
264#95—941no—
265#95—941no—
266#95—941no—
267#95—941no—
268#95—941no—
269C24C2×C4×D4711yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—472no—
279C22×C4—472no—
280C22×C4—941no—
281C22×C4—941no—
282C22×C4—941no—
283C22×C4—941no—
284C22×C4—942no—
285C22×C4—942no—
286C22×C4—942no—
287C22×C4—942no—
288C22×C4—941no—
289C22×C4—941no—
290C22×C4—1881no—
291C22×C4—1881no—
292C22×C4—1881no—
293C22×C4—1881no—
294C22×C4—1881no—
295C22×C4—1881no—
296C22×C4(C2×D47)⋊C411yescharacteristic
297C22×C4(C2×D47)⋊C411yescharacteristic
298C22×C4(C2×D47)⋊C411yescharacteristic
299C22×C4(C2×D47)⋊C411yescharacteristic
300C22×C4C2×D18811yescharacteristic
301C22×C4D4×D4711yescharacteristic
302C22×C4#1062535211yescharacteristic
303C22×C4Q8×D4711yescharacteristic
304C22×C4#1062535411yescharacteristic
305C22×C4C2×C22⋊D4711yescharacteristic
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—1882no—
310C2×C8—1882no—
311C2×C8—1882no—
312C4⋊C4—942no—
313C4⋊C4—942no—
314C4⋊C4—942no—
315C4⋊C4—942no—
316C4⋊C4—1882no—
317C4⋊C4—1882no—
318C42—22no—
319C42—22no—
320C42—41no—
321C42—41no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23C2×(C2×D47)⋊C411yescharacteristic
327C23C4⋊C4×D4711yescharacteristic
328C23#1309398211yescharacteristic
329C23#1309403012yes—
330C23#1309403212yes—
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—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—942no—
352C2×C4—942no—
353C2×C4—944no—
354C2×C4—3762no—
355C2×C4—3762no—
356C2×C4—1881no—
357C2×C4—1881no—
358C2×C4—1881no—
359C2×C4—1881no—
360C2×C4—1881no—
361C2×C4—1881no—
362C2×C4—1882no—
363C2×C4—1881no—
364C2×C4—1881no—
365C2×C4—1881no—
366C2×C4—1881no—
367C2×C4—1882no—
368C2×C4—1882no—
369C2×C4#1309393812yes—
370C2×C4#1309394212yes—
371C2×C4#1309399511yescharacteristic
372C2×C4#1309399911yescharacteristic
373C2×C4#1309400011yescharacteristic
374C2×C4#1309400411yescharacteristic
375C8—82no—
376C8—3762no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22#1568898011yescharacteristic
386C22#1568908911yescharacteristic
387C22#1568909111yescharacteristic
388C22C2×#1309393811yescharacteristic
389C22#1568909611yescharacteristic
390C22C2×#1309394211yescharacteristic
391C22#1568910511yescharacteristic
392C22#1568914812yes—
393C22#1568918312yes—
394C4—22no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—41no—
401C4—41no—
402C4—3761no—
403C4—3761no—
404C4—1882no—
405C4—1882no—
406C2—22no—
407C2—22no—
408C2#2024173711yescharacteristic
409C2#2024174511yescharacteristic
410C2#2024175111yescharacteristic
411C2#2024175511yescharacteristic
412C2#2024336911yescharacteristic
413C2#2024337311yescharacteristic
414C2#2024338111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutator, prime radicalC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
cosocleC23×C4×C47C23
hypercenter#1797D47
hypocenterC47#8562
LCSG ▷ C24×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
231. C47#8562semidirect

Maximal subgroups

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

subgroupindex
C2×#156890072
#202433682
C2×#156888482
#202412302
C47×#17972
#856247