Finite Group Databasebeta

← #26109809

Subgroups — #26109809

Order 12032.

546 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246002C211yesmaximal, characteristic
2#20241715C211yesmaximal, characteristic
3#20241295C212yesmaximal, complemented
4#20241293C212yesmaximal, complemented
5C47×#1462C211yesmaximal, characteristic
6C22×C22⋊Dic47C411yescharacteristic
7C22×C22⋊Dic47C411yescharacteristic
8#15689087C2212yes—
9#15688998—21no—
10#15688998—21no—
11#15688998—21no—
12#15688998—21no—
13#15688983C411yescharacteristic
14#15688983C411yescharacteristic
15#15688983C2211yescharacteristic
16#15688983C2211yescharacteristic
17#15688847—21no—
18#15688847—21no—
19C22×C22⋊C4×C47C2211yescharacteristic
20C47×#407C2212yes—
21C47×#323—21no—
22C47×#323—21no—
23C23×Dic47—21no—
24C23×Dic47—21no—
25C2×#10625329—22no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C22⋊Dic47—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—42no—
33C2×C22⋊Dic47—42no—
34C2×C22⋊Dic47C2×C411yescharacteristic
35C2×C22⋊Dic47C2×C411yescharacteristic
36C2×C22⋊Dic47C2×C411yescharacteristic
37C2×C22⋊Dic47C2×C411yescharacteristic
38C22⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)—22no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C25×C47C2×C411yescharacteristic
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47C2311yescharacteristic
47C2×M16×C47—22no—
48C2×C22⋊C4×C47—22no—
49C2×C22⋊C4×C47—22no—
50C2×C22⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C22⋊C4×C47D411yescharacteristic
54C22⋊C8×C47—22no—
55C22⋊C8×C47—22no—
56C24×C47—22no—
57C24×C47—22no—
58C24×C47—21no—
59C24×C47—21no—
60C24×C47—81no—
61C24×C47—81no—
62C24×C47C22⋊C411yescharacteristic
63C24×C47C22⋊C411yescharacteristic
64C24×C47C22×C411yescharacteristic
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—81no—
68C22×C4×C47—81no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4112yes—
74C22×C4×C47#4112yes—
75C22×Dic47—21no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C22×Dic47—42no—
80C22×Dic47—42no—
81C22×Dic47—81no—
82C22×Dic47—81no—
83C22×Dic47—81no—
84C22×Dic47—81no—
85M16×C47—82no—
86C2×C8×C47—42no—
87C22⋊C4×C47—22no—
88C22⋊C4×C47—22no—
89C22⋊C4×C47—22no—
90C22⋊C4×C47—22no—
91C22⋊Dic47—42no—
92C22⋊Dic47—42no—
93C22⋊Dic47—41no—
94C22⋊Dic47—41no—
95C22⋊Dic47—41no—
96C22⋊Dic47—41no—
97C4⋊Dic47—81no—
98C4⋊Dic47—81no—
99C4⋊Dic47—81no—
100C4⋊Dic47—81no—
101#10625329—82no—
102C2×C47⋊C8—42no—
103C23×C47—24no—
104C23×C47—24no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—81no—
118C23×C47—81no—
119C23×C47—81no—
120C23×C47—81no—
121C23×C47—84no—
122C23×C47—84no—
123C23×C47#10012yes—
124C23×C47#10112yes—
125C23×C47C2×C22⋊C411yescharacteristic
126C23×C47#11711yescharacteristic
127C23×C47#11711yescharacteristic
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—42no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—81no—
137C2×C4×C47—81no—
138C2×C4×C47#12312yes—
139C2×C4×C47#12312yes—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—84no—
153C2×Dic47—84no—
154C8×C47—82no—
155C47⋊C8—82no—
156#8756—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—42no—
168C22×C47—42no—
169C22×C47—42no—
170C22×C47—42no—
171C22×C47—42no—
172C22×C47—42no—
173C22×C47—41no—
174C22×C47—41no—
175C22×C47—41no—
176C22×C47—41no—
177C22×C47—84no—
178C22×C47—84no—
179C22×C47#38111yescharacteristic
180C22×C47C2×#10011yescharacteristic
181C22×C47C2×#10111yescharacteristic
182C4×C47—22no—
183C4×C47—22no—
184C4×C47—41no—
185C4×C47—41no—
186Dic47—81no—
187Dic47—81no—
188Dic47—81no—
189Dic47—81no—
190#2315—471no—
191#1462—471no—
192#1462D4711yescharacteristic
193#1305—472no—
194#1305—472no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—41no—
198C2×C47—41no—
199C2×C47—42no—
200C2×C47—42no—
201C2×C47#182111yescharacteristic
202C2×C47#182811yescharacteristic
203C2×C47#182911yescharacteristic
204C22×C22⋊C4—471no—
205C22×C22⋊C4—471no—
206C22×C22⋊C4C2×D4711yescharacteristic
207#407—472no—
208#407C2×D4712yes—
209#385—941no—
210#385—941no—
211#385—941no—
212#385—941no—
213#380—471no—
214#380—471no—
215#380—471no—
216#380—471no—
217#323—21no—
218#323—21no—
219#323—941no—
220#323—941no—
221C47#875611yescharacteristic, complemented, Sylow
222C25C4×D4711yescharacteristic
223C23×C4—941no—
224C23×C4—941no—
225C23×C4C4×D4711yescharacteristic
226C23×C4C22×D4711yescharacteristic
227C2×M16—22no—
228C2×M16—942no—
229C2×C4⋊C4—941no—
230C2×C4⋊C4—941no—
231C2×C4⋊C4—941no—
232C2×C4⋊C4—941no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—471no—
236C2×C22⋊C4—471no—
237C2×C22⋊C4—471no—
238C2×C22⋊C4—471no—
239C2×C22⋊C4—941no—
240C2×C22⋊C4—941no—
241C2×C22⋊C4—1882no—
242C2×C22⋊C4—1882no—
243C2×C22⋊C4D18811yescharacteristic
244C2×C22⋊C4D18811yescharacteristic
245C2×C22⋊C4C22⋊D4711yescharacteristic
246C2×C22⋊C4C22⋊D4711yescharacteristic
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—942no—
250C22⋊C8—942no—
251#95—941no—
252#95—941no—
253#95—941no—
254#95—941no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—81no—
260C24—81no—
261C24(C2×D47)⋊C411yescharacteristic
262C24(C2×D47)⋊C411yescharacteristic
263C24C2×C4×D4711yescharacteristic
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—941no—
269C22×C4—941no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4—3761no—
273C22×C4—3761no—
274C22×C4—3761no—
275C22×C4—3761no—
276C22×C4—1882no—
277C22×C4—1882no—
278C22×C4(C2×D47)⋊C411yescharacteristic
279C22×C4(C2×D47)⋊C411yescharacteristic
280C22×C4C2×D18811yescharacteristic
281C22×C4#1062535212yes—
282C22×C4#1062535212yes—
283C22×C4C2×C22⋊D4711yescharacteristic
284M16—82no—
285M16—3762no—
286C2×C8—42no—
287C2×C8—1882no—
288C4⋊C4—3761no—
289C4⋊C4—3761no—
290C4⋊C4—3761no—
291C4⋊C4—3761no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—1882no—
297C22⋊C4—1882no—
298C22⋊C4—1881no—
299C22⋊C4—1881no—
300C22⋊C4—1881no—
301C22⋊C4—1881no—
302C23—24no—
303C23—24no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—84no—
321C23—84no—
322C23#1309394012yes—
323C23#1309394212yes—
324C23C2×(C2×D47)⋊C411yescharacteristic
325C23#1309398011yescharacteristic
326C23#1309398011yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—3764no—
338C2×C4—3764no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1881no—
345C2×C4—1881no—
346C2×C4—1881no—
347C2×C4—1881no—
348C2×C4—1881no—
349C2×C4—1881no—
350C2×C4—1881no—
351C2×C4#1309400512yes—
352C2×C4#1309400812yes—
353C8—82no—
354C8—3762no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—84no—
376C22—84no—
377C22#1568898411yescharacteristic
378C22C2×#1309394011yescharacteristic
379C22C2×#1309394211yescharacteristic
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—3761no—
385C4—3761no—
386C4—3761no—
387C4—3761no—
388C2—21no—
389C2—21no—
390C2—41no—
391C2—41no—
392C2—42no—
393C2—42no—
394C2#2024346611yescharacteristic
395C2#2024349411yescharacteristic
396C2#2024349711yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC24×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1462C2
socleC22×C47#381
prime radicalC25×C47C2×C4
cosocleC23×C4×C47C23
hypercenter#1462D47
hypocenterC47#8756
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20241295C2semidirect
4. #20241293C2semidirect
221. C47#8756semidirect

Maximal subgroups

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

subgroupindex
#202460022
#202417152
#202412952
#202412932
C47×#14622
#875647