Finite Group Databasebeta

← #26109261

Subgroups — #26109261

Order 12032.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252082C211yesmaximal, characteristic
2#20243417C211yesmaximal, characteristic, complemented
3#20241932C212yesmaximal
4#20241295C212yesmaximal, complemented
5C47×#1810C211yesmaximal, characteristic, complemented
6C2×#13094030C411yescharacteristic, complemented
7#15689634C2211yescharacteristic
8#15689194—21no—
9#15689194—21no—
10C8×C22⋊D47—21no—
11C8×C22⋊D47—21no—
12C2×#13093958—21no—
13#15689149C411yescharacteristic, complemented
14#15689089C411yescharacteristic, complemented
15#15689087C2211yescharacteristic
16#15689078C411yescharacteristic, complemented
17#15688983C2212yes—
18#15688886—21no—
19#15688861—22no—
20#15688858—22no—
21C23×C8×C47C2211yescharacteristic
22C2×#108×C47—21no—
23C47×#407C2212yes—
24C47×#341—21no—
25C2×#10625350C2×C411yescharacteristic
26#13094030—42no—
27C2×#10625329C2×C411yescharacteristic
28C2×#10625329C2×C411yescharacteristic
29C2×#10625324—21no—
30C2×C8×D47—21no—
31#13094007—21no—
32#13094003D411yescharacteristic
33#13094003Q811yescharacteristic
34#13093998D411yescharacteristic
35#13093998Q811yescharacteristic
36#13093993—21no—
37C4×C22⋊D47—21no—
38C4×C22⋊D47—21no—
39#13093983C2×C411yescharacteristic
40C2×C4⋊Dic47—22no—
41C2×C22⋊Dic47—22no—
42#13093958—42no—
43Dic47⋊C8—21no—
44Dic47⋊C8—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47(C2×D47)⋊C8—21no—
48(C2×D47)⋊C8—21no—
49#13093934—21no—
50C8×Dic47—21no—
51C47⋊#95—22no—
52C23×C4×C47C2311yescharacteristic
53C2×M16×C47—22no—
54C22×C8×C47—42no—
55C22×C8×C47C2×C411yescharacteristic
56C22×C8×C47C2×C411yescharacteristic
57C22×C8×C47D411yescharacteristic
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C22×C8×C47D411yescharacteristic
61#108×C47—42no—
62C22⋊C8×C47—22no—
63C22⋊C8×C47—22no—
64C24×C47#4111yescharacteristic
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47C22×C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×Q811yescharacteristic
72C22×C4×C47#4111yescharacteristic
73C2×C22⋊D47—21no—
74C2×C22⋊D47—21no—
75C2×C22⋊D47—21no—
76C22×Dic47—42no—
77#10625350—41no—
78#10625350—41no—
79C2×D188C4⋊C411yescharacteristic
80C2×C4×D47—21no—
81C2×Dic94C4⋊C411yescharacteristic
82M16×C47—42no—
83M16×C47—42no—
84C2×C8×C47—22no—
85C2×C8×C47—22no—
86C2×C8×C47—22no—
87C2×C8×C47—41no—
88C2×C8×C47—41no—
89C2×C8×C47—41no—
90C2×C8×C47—42no—
91C2×C8×C47C22⋊C411yescharacteristic
92C2×C8×C47C22⋊C411yescharacteristic
93C2×C8×C47C22⋊C411yescharacteristic
94C2×C8×C47C22⋊C411yescharacteristic
95C22⋊Dic47—21no—
96C22⋊Dic47—21no—
97C22⋊Dic47—21no—
98C22⋊Dic47—82no—
99(C2×D47)⋊C4—21no—
100(C2×D47)⋊C4—21no—
101C4⋊Dic47—42no—
102C4⋊Dic47—42no—
103C4⋊Dic47C4⋊C411yescharacteristic
104C4⋊Dic47C4⋊C411yescharacteristic
105Dic47⋊C4—21no—
106Dic47⋊C4—21no—
107C4×Dic47—21no—
108#10625329—41no—
109#10625329—41no—
110C2×C47⋊C8—21no—
111C2×C47⋊C8—21no—
112#10625324—42no—
113C8×D47—42no—
114C23×C47—21no—
115C23×C47—21no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47#11711yescharacteristic
119C23×C47#12111yescharacteristic
120C23×C47C22⋊Q811yescharacteristic
121C22×D47—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—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47C2×C22⋊C411yescharacteristic
132C2×C4×C47C2×C4⋊C411yescharacteristic
133C2×C4×C47#12111yescharacteristic
134C2×C4×C47C22⋊Q811yescharacteristic
135C22⋊D47—42no—
136C22⋊D47—42no—
137C22⋊D47—41no—
138C22⋊D47—41no—
139C2×Dic47—21no—
140C2×Dic47—21no—
141C2×Dic47—21no—
142C2×Dic47—82no—
143C2×Dic47—82no—
144C2×Dic47—82no—
145D188—41no—
146C4×D47—41no—
147C4×D47—41no—
148Dic94—41no—
149C8×C47—21no—
150C8×C47—21no—
151C8×C47—21no—
152C8×C47—82no—
153C8×C47#9912yes—
154C47⋊C8—41no—
155C47⋊C8—41no—
156#8628—471nomaximal, Sylow
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—42no—
166C22×C47—42no—
167C22×C47#38011yescharacteristic
168C22×C47#47211yescharacteristic
169C22×C47#47311yescharacteristic
170C2×D47—41no—
171C2×D47—41no—
172C2×D47—41no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47C2×#9911yescharacteristic
177C4×C47#41011yescharacteristic
178Dic47—41no—
179Dic47—41no—
180Dic47—41no—
181Dic47—162no—
182#2929—471no—
183#1810D4711yescharacteristic, complemented
184#1808—471no—
185#1513—472no—
186#1305—472no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—41no—
190C2×C47—41no—
191C2×C47#179911yescharacteristic
192C2×C47#181411yescharacteristic
193C2×C47#181611yescharacteristic
194D47—81no—
195C2×#131—471no—
196C23×C8C2×D4711yescharacteristic
197#525—471no—
198#435—941no—
199#435—941no—
200C8×D4—941no—
201C8×D4—941no—
202C2×#108—21no—
203C2×#107—941no—
204#424—471no—
205#408—471no—
206#407—471no—
207#407C2×D4712yes—
208#405—471no—
209#380—472no—
210#341—21no—
211#340—941no—
212#328—942no—
213#327—942no—
214C47#862811yescharacteristic, complemented, Sylow
215C2×#41—471no—
216C23×C4C22×D4711yescharacteristic
217#131—1882no—
218C2×M16—22no—
219C2×M16—471no—
220C2×M16—471no—
221C2×M16—941no—
222C22×C8—42no—
223C22×C8—941no—
224C22×C8C4×D4711yescharacteristic
225C22×C8C4×D4711yescharacteristic
226C22×C8D18811yescharacteristic
227C22×C8D18811yescharacteristic
228C22×C8C22⋊D4711yescharacteristic
229C22×C8C22⋊D4711yescharacteristic
230#123—941no—
231C22⋊Q8—471no—
232C22⋊Q8—471no—
233#121—471no—
234#121—471no—
235#120—941no—
236C4×D4—941no—
237C4×D4—941no—
238#117—471no—
239C2×C4⋊C4—942no—
240C2×C22⋊C4—942no—
241#108—42no—
242#107—1882no—
243C4⋊C8—941no—
244C4⋊C8—941no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—941no—
248C22⋊C8—941no—
249C22⋊C8—941no—
250C22⋊C8—941no—
251#97—941no—
252C4×C8—941no—
253#95—942no—
254C24#1062535411yescharacteristic
255#41—1881no—
256#41—1881no—
257C2×Q8—471no—
258C2×D4—471no—
259C2×D4—941no—
260C2×D4—941no—
261C2×D4—941no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—941no—
265C22×C4—1882no—
266C22×C4C2×C4×D4711yescharacteristic
267C22×C4C2×D18811yescharacteristic
268C22×C4D4×D4711yescharacteristic
269C22×C4#1062535211yescharacteristic
270C22×C4Q8×D4711yescharacteristic
271C22×C4C2×C22⋊D4711yescharacteristic
272M16—42no—
273M16—42no—
274M16—1882no—
275M16—1881no—
276M16—1881no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—42no—
284C2×C8—941no—
285C2×C8—941no—
286C2×C8—1882no—
287C2×C8(C2×D47)⋊C411yescharacteristic
288C2×C8(C2×D47)⋊C411yescharacteristic
289C2×C8(C2×D47)⋊C411yescharacteristic
290C2×C8(C2×D47)⋊C411yescharacteristic
291C4⋊C4—471no—
292C4⋊C4—471no—
293C4⋊C4—941no—
294C4⋊C4—941no—
295C4⋊C4—1882no—
296C4⋊C4—1882no—
297C22⋊C4—941no—
298C22⋊C4—941no—
299C22⋊C4—941no—
300C22⋊C4—941no—
301C22⋊C4—941no—
302C22⋊C4—3762no—
303C42—941no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—941no—
309C23#1309398211yescharacteristic
310C23#1309399911yescharacteristic
311C23#1309400011yescharacteristic
312Q8—1881no—
313D4—1881no—
314D4—1882no—
315D4—1882no—
316D4—1881no—
317D4—1881no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—941no—
328C2×C4—941no—
329C2×C4—941no—
330C2×C4—3762no—
331C2×C4—3762no—
332C2×C4—3762no—
333C2×C4—1881no—
334C2×C4—1881no—
335C2×C4C2×(C2×D47)⋊C411yescharacteristic
336C2×C4C4⋊C4×D4711yescharacteristic
337C2×C4#1309399511yescharacteristic
338C2×C4#1309400411yescharacteristic
339C8—21no—
340C8—21no—
341C8—21no—
342C8—82no—
343C8—1881no—
344C8—1881no—
345C8#1309393812yes—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—1881no—
357C22—1881no—
358C22—1881no—
359C22#1568898011yescharacteristic
360C22#1568939411yescharacteristic
361C22#1568939911yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—1881no—
366C4—1881no—
367C4—1881no—
368C4—7522no—
369C4C2×#1309393811yescharacteristic
370C4#1568909611yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—3761no—
376C2#2024337311yescharacteristic
377C2#2024344311yescharacteristic
378C2#2024344911yescharacteristic
subgroupG/H
centerC22#15688980
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1810C2
socleC22×C47#380
prime radical#15689634C22
cosocleC23×C4×C47C23
hypercenter#1810D47
hypocenterC47#8628
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. #20243417C2semidirect
4. #20241295C2semidirect
5. C47×#1810C2semidirect
6. C2×#13094030C4semidirect
13. #15689149C4semidirect
14. #15689089C4semidirect
16. #15689078C4semidirect
183. #1810D47semidirect
214. C47#8628semidirect

Maximal subgroups

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

subgroupindex
#202520822
#202434172
#202419322
#202412952
C47×#18102
#862847