Finite Group Databasebeta

← #26216579

Subgroups — #26216579

Order 12032.

758 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#117×Dic47C211yesmaximal, characteristic
2#20243532C212yesmaximal, complemented
3#20243503C212yesmaximal, complemented
4#20243454C212yesmaximal
5#20243370C212yesmaximal
6#20243248C211yesmaximal, characteristic, complemented
7#20243246C211yesmaximal, characteristic, complemented
8#20243154C211yesmaximal, characteristic
9#20243136C211yesmaximal, characteristic, complemented
10#20243133C211yesmaximal, characteristic, complemented
11C47×#1750C211yesmaximal, characteristic
12C2×#13093983C2211yescharacteristic
13C22×C4×Dic47C2211yescharacteristic
14C2×Dic47⋊C8C2212yes—
15C2×Dic47⋊C8C2212yes—
16C2×C22⋊(C47⋊C8)C2212yes—
17C2×#13093935C2211yescharacteristic
18C2×C4×C47⋊C8C2211yescharacteristic
19C4⋊C4×Dic47C412yes—
20C4⋊C4×Dic47C412yes—
21C22⋊C4×Dic47C412yes—
22C22⋊C4×Dic47C412yes—
23#15688969C2211yescharacteristic
24#15688969C2211yescharacteristic
25C42×Dic47C2211yescharacteristic
26C42×Dic47C2211yescharacteristic
27#15688878C2212yes—
28#15688877C2212yes—
29#15688877C2212yes—
30#15688877C2212yes—
31#15688877C2212yes—
32#15688877C2212yes—
33#15688877C2212yes—
34C2×#117×C47C2211yescharacteristic
35C2×C22⋊C8×C47C2212yes—
36C2×#97×C47C2211yescharacteristic
37C2×C4×C8×C47C2211yescharacteristic
38C47×#336C2212yes—
39C23×Dic47C2×C412yes—
40C22×C47⋊C8C2312yes—
41C22×C47⋊C8C2312yes—
42#13093983—24no—
43C2×C4⋊Dic47C2×C412yes—
44C2×C22⋊Dic47C2×C412yes—
45C2×C4×Dic47—24no—
46C2×C4×Dic47C2×C412yes—
47C2×C4×Dic47C2×C412yes—
48C2×C4×Dic47C2×C412yes—
49C2×C4×Dic47C2×C412yes—
50C2×C4×Dic47C2×C412yes—
51C2×C4×Dic47C2311yescharacteristic
52C2×C4×Dic47C2311yescharacteristic
53C2×C4×Dic47C2311yescharacteristic
54C2×C4×Dic47C2311yescharacteristic
55Dic47⋊C8—24no—
56Dic47⋊C8—24no—
57C22⋊(C47⋊C8)—24no—
58#13093935—22no—
59C4×C47⋊C8—22no—
60C47⋊#95C2×C412yes—
61C47⋊#95C2×C412yes—
62C47⋊#95C2×C412yes—
63C47⋊#95C2×C412yes—
64C23×C4×C47C2311yescharacteristic
65C22×C8×C47C2312yes—
66C22×C8×C47C2312yes—
67#117×C47—22no—
68#117×C47—22no—
69C2×C4⋊C4×C47C2×C411yescharacteristic
70C2×C4⋊C4×C47C2×C411yescharacteristic
71C2×C22⋊C4×C47C2×C411yescharacteristic
72C2×C22⋊C4×C47C2×C411yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C2×C42×C47C2311yescharacteristic
75C22⋊C8×C47—24no—
76#97×C47—22no—
77C4×C8×C47—22no—
78C24×C47C22×C411yescharacteristic
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—22no—
82C22×C4×C47C22×C411yescharacteristic
83C22×C4×C47C22×C411yescharacteristic
84C22×C4×C47C22×C411yescharacteristic
85C22×C4×C47C22×C411yescharacteristic
86C22×C4×C47C22×C411yescharacteristic
87C22×C4×C47#4114yes—
88C22×C4×C47C2411yescharacteristic
89C22×Dic47—24no—
90C22×Dic47—24no—
91C22×Dic47—42no—
92C22×Dic47C22×C412yes—
93C22×Dic47C22×C412yes—
94C22×Dic47C22×C412yes—
95C22×Dic47C22×C412yes—
96C2×C8×C47—22no—
97C2×C8×C47—22no—
98C2×C8×C47—22no—
99C2×C8×C47—22no—
100C2×C8×C47—22no—
101C2×C8×C47—22no—
102C4⋊C4×C47—22no—
103C4⋊C4×C47—22no—
104C22⋊C4×C47—22no—
105C22⋊C4×C47—22no—
106C42×C47—21no—
107C42×C47—21no—
108C42×C47#4112yes—
109C42×C47#4112yes—
110C22⋊Dic47—24no—
111C4⋊Dic47—24no—
112C4×Dic47—24no—
113C4×Dic47—22no—
114C4×Dic47—22no—
115C4×Dic47—22no—
116C4×Dic47—24no—
117C4×Dic47—24no—
118C4×Dic47—42no—
119C4×Dic47—42no—
120C4×Dic47#4114yes—
121C2×C47⋊C8—22no—
122C2×C47⋊C8—22no—
123C2×C47⋊C8—22no—
124C2×C47⋊C8—22no—
125C2×C47⋊C8—22no—
126C2×C47⋊C8—22no—
127C23×C47—21no—
128C23×C47—21no—
129C23×C47—22no—
130C23×C47—22no—
131C23×C47#11712yes—
132C23×C47C23×C411yescharacteristic
133C2×C4×C47—21no—
134C2×C4×C47—21no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—24no—
138C2×C4×C47—24no—
139C2×C4×C47—21no—
140C2×C4×C47—21no—
141C2×C4×C47—22no—
142C2×C4×C47—22no—
143C2×C4×C47#11712yes—
144C2×C4×C47#11712yes—
145C2×C4×C47#11712yes—
146C2×C4×C47C2×#4112yes—
147C2×C4×C47C2×#4111yescharacteristic
148C2×C4×C47C2×#4111yescharacteristic
149C2×C4×C47C2×#4112yes—
150C2×Dic47—22no—
151C2×Dic47—22no—
152C2×Dic47—22no—
153C2×Dic47—22no—
154C2×Dic47—22no—
155C2×Dic47—22no—
156C2×Dic47—22no—
157C2×Dic47—22no—
158C2×Dic47—22no—
159C2×Dic47—22no—
160C2×Dic47—48no—
161C2×Dic47#11714yes—
162C2×Dic47#13114yes—
163C8×C47—42no—
164C8×C47—42no—
165C47⋊C8—42no—
166C47⋊C8—42no—
167#20780—471nomaximal, Sylow
168C22×C47—24no—
169C22×C47—22no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47—42no—
173C22×C47C2×#11711yescharacteristic
174C22×C47C2×#11711yescharacteristic
175C22×C47C2×#11711yescharacteristic
176C22×C47C2×#13112yes—
177C22×C47#56812yes—
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—41no—
182C4×C47—41no—
183C4×C47#52912yes—
184C4×C47#52912yes—
185Dic47—24no—
186Dic47—24no—
187Dic47—42no—
188Dic47—42no—
189C4×#117—471no—
190#1840—472no—
191#1831—472no—
192#1817—472no—
193#1798—472no—
194#1768—471no—
195#1768—471no—
196#1750—471no—
197#1750D4711yescharacteristic
198#1745—471no—
199#1744—471no—
200C2×C47—42no—
201C2×C47#228311yescharacteristic
202C2×C47#292211yescharacteristic
203C2×C47#292311yescharacteristic
204C2×C47#292412yes—
205C2×C47#292512yes—
206C2×#117—471no—
207C2×#117C2×D4711yescharacteristic
208C22×C42—471no—
209C2×C4⋊C8—472no—
210C2×C4⋊C8—472no—
211C2×C22⋊C8—472no—
212C2×C22⋊C8C2×D4712yes—
213C2×#97—471no—
214C2×#97C2×D4711yescharacteristic
215C2×C4×C8—471no—
216C2×C4×C8C2×D4711yescharacteristic
217C4×C4⋊C4—472no—
218C4×C4⋊C4—472no—
219C4×C22⋊C4—472no—
220C4×C22⋊C4—472no—
221#376—471no—
222#376—471no—
223C43—471no—
224C43—471no—
225#336—472no—
226#336—472no—
227#336—472no—
228#336—472no—
229#336—472no—
230#336—472no—
231#336—472no—
232#336C2×D4712yes—
233C47#2078011yescharacteristic, complemented, Sylow
234C23×C4—472no—
235C23×C4C22×D4711yescharacteristic
236C22×C8—472no—
237C22×C8—472no—
238C22×C8C22×D4712yes—
239C22×C8C22×D4712yes—
240#117—22no—
241#117—22no—
242#117—944no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4C4×D4711yescharacteristic
245C2×C4⋊C4C4×D4711yescharacteristic
246C2×C22⋊C4—472no—
247C2×C22⋊C4C4×D4711yescharacteristic
248C2×C22⋊C4C4×D4711yescharacteristic
249C2×C42—471no—
250C2×C42—472no—
251C2×C42—472no—
252C2×C42—472no—
253C2×C42—471no—
254C2×C42—471no—
255C2×C42—471no—
256C2×C42—472no—
257C2×C42—472no—
258C2×C42—944no—
259C2×C42C22×D4711yescharacteristic
260C2×C42C22×D4711yescharacteristic
261C4⋊C8—944no—
262C4⋊C8—944no—
263C22⋊C8—24no—
264C22⋊C8—944no—
265#97—22no—
266#97—942no—
267C4×C8—22no—
268C4×C8—942no—
269#95—472no—
270#95—472no—
271#95—472no—
272#95—472no—
273C24C2×C4×D4711yescharacteristic
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—472no—
278C22×C4—472no—
279C22×C4—472no—
280C22×C4—472no—
281C22×C4—944no—
282C22×C4—944no—
283C22×C4—1882no—
284C22×C4C2×C4×D4711yescharacteristic
285C22×C4C2×C4×D4711yescharacteristic
286C22×C4C2×C4×D4711yescharacteristic
287C22×C4C2×C4×D4711yescharacteristic
288C22×C4C2×C4×D4711yescharacteristic
289C22×C4#1062535014yes—
290C22×C4C23×D4711yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—942no—
298C2×C8—942no—
299C2×C8—942no—
300C2×C8—942no—
301C2×C8—942no—
302C2×C8—942no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—944no—
306C22⋊C4—22no—
307C22⋊C4—22no—
308C22⋊C4—944no—
309C42—21no—
310C42—21no—
311C42—474no—
312C42—944no—
313C42—944no—
314C42—944no—
315C42—942no—
316C42—942no—
317C42—942no—
318C42—1882no—
319C42—1882no—
320C42#1062535212yes—
321C42#1062535212yes—
322C23—21no—
323C23—21no—
324C23—22no—
325C23—22no—
326C23#1309398112yes—
327C23C22×C4×D4711yescharacteristic
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—474no—
339C2×C4—474no—
340C2×C4—942no—
341C2×C4—942no—
342C2×C4—942no—
343C2×C4—942no—
344C2×C4—942no—
345C2×C4—942no—
346C2×C4—942no—
347C2×C4—942no—
348C2×C4—942no—
349C2×C4—942no—
350C2×C4—1888no—
351C2×C4#1309398012yes—
352C2×C4#1309398012yes—
353C2×C4#1309398112yes—
354C2×C4#41×D4712yes—
355C2×C4C2×#1062535211yescharacteristic
356C2×C4C2×#1062535211yescharacteristic
357C2×C4C2×#1062535012yes—
358C8—42no—
359C8—42no—
360C8—1882no—
361C8—1882no—
362C22—22no—
363C22—24no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22#117×D4711yescharacteristic
368C22C2×#1309398011yescharacteristic
369C22C2×#1309398111yescharacteristic
370C22#131×D4712yes—
371C22#1568988812yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—944no—
378C4—944no—
379C4—1882no—
380C4—1882no—
381C4#1568965312yes—
382C4#1568965312yes—
383C2—42no—
384C2#2024578211yescharacteristic
385C2#2025202811yescharacteristic
386C2#2025203511yescharacteristic
387C2#2025204312yes—
388C2#2025205112yes—
subgroupG/H
center, FrattiniC22×C4C23×D47
commutator, socleC23×C47C23×C4
FittingC47×#1750C2
prime radicalC24×C47C22×C4
cosocleC22×C4×C47C24
hypercenter#1750D47
hypocenterC47#20780
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ #1750

Complements

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

NH ≅ G/Ntype
2. #20243532C2semidirect
3. #20243503C2semidirect
6. #20243248C2semidirect
7. #20243246C2semidirect
9. #20243136C2semidirect
10. #20243133C2semidirect
233. C47#20780semidirect

Maximal subgroups

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

subgroupindex
#117×Dic472
#202435322
#202435032
#202434542
#202433702
#202432482
#202432462
#202431542
#202431362
#202431332
C47×#17502
#2078047